A Survey on Oscillation of Impulsive Ordinary Differential Equations Academic research paper on "Mathematics"

Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 354841,52 pages doi:10.1155/2010/354841

Review Article

A Survey on Oscillation of Impulsive Ordinary Differential Equations

Ravi P. Agarwal,1,2 Fatma Karakoc,3 and Agacik Zafer4

1 Department of Mathematical Sciences, FL Institute of Technology, Melbourne, FL 32901, USA

2 Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

3 Department of Mathematics, Faculty of Sciences, Ankara University, 06100 Ankara, Turkey

4 Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

Correspondence should be addressed to Ravi P. Agarwal, agarwal@fit.edu Received 1 December 2009; Accepted 3 March 2010 Academic Editor: Leonid Berezansky

Copyright © 2010 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper summarizes a series of results on the oscillation of impulsive ordinary differential equations. We consider linear, half-linear, super-half-linear, and nonlinear equations. Several oscillation criteria are given. The Sturmian comparison theory for linear and half linear equations is also included.

1. Introduction

Impulsive differential equations, that is, differential equations involving impulse effect, appear as a natural description of observed evolution phenomena of several real world problems. There are many good monographs on the impulsive differential equations [16]. It is known that many biological phenomena, involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulates systems, do exhibit impulse effects. Let us describe the Kruger-Thiemer model [7] for drug distribution to show how impulses occur naturally. It is assumed that the drug, which is administered orally, is first dissolved into the gastrointestinal tract. The drug is then absorbed into the so-called apparent volume of distribution and finally eliminated from the system by the kidneys. Let x(t) and y(t) denote the amounts of drug at time t in the gastrointestinal tract and apparent volume of distribution, respectively, and let k1 and k2 be the relevant rate constants. For simplicity, assume that ki / k2. The dynamic description of this model is then given by

X = -k1x,

y = -k2y + k1x.

In [8], the authors postulate the following control problem. At discrete instants of time 0 = t0 < t1 < t2 < ■■■ < tN, the drug is ingested in amounts 60,61,62,...,6N. This imposes the following boundary conditions:

x(t+) = x(t-) + 6, i = 0,1,2,...,N,

y(t+) = y(t-), (1.2)

x(0-) = y(0-) = 0.

To achieve the desired therapeutic effect, it is required that the amount of drug in the apparent volume of distribution never goes below a constant level or plateau m, say, during the time interval [t1,tN+1], where tN+1 > tN. Thus, we have the constraint

y(t) > m, t e [ti,tN+i]. (1.3)

It is also assumed that only nonnegative amounts of the drug can be given. Then, a control vector is a point u = (61,62,...,6N) in the nonnegative orthant of Euclidean space of dimension N + 1. Hence, ^ = {u e RN+1 : u > 0}. Finally, the biological cost function f (6) = (1/2) ^N0 6t2 minimizes both the side effects and the cost of the drug. The problem is to find infu>0f (u) subject to (1.1)—(1.3).

The first investigation on the oscillation theory of impulsive differential equations was published in 1989 [9]. In that paper Gopalsamy and Zhang consider impulsive delay differential equations of the form

x'(t) + ax(t - t) = ^6(t - tj), t /tj, (1.4)

x'(t) + p(t)x(t - t)= 0, t /ti,

-^t-) = bix(t-), i e N = {1,2,...},

where 0 < ti < t2 < ■■■ < tj ^ to as j ^ to, and t is a positive real number. Sufficient conditions are obtained for the asymptotic stability of the zero solution of (1.4) and existence of oscillatory solutions of (1.5). However, it seems that the problem of oscillation of ordinary differential equations with impulses has received attention much later [10]. Although, the theory of impulsive differential equations has been well established, the oscillation theory of such equations has developed rather slowly. To the best of our knowledge, except one paper [11], all of the investigations have been on differential equations subject to fixed moments of impulse effect. In [11], second-order differential equations with random impulses were dealt with, and there are no papers on the oscillation of differential equations with impulses at variable times.

In this survey paper, our aim is to present the results (within our reach) obtained so far on the oscillation theory of impulsive ordinary differential equations. The paper is organized as follows. Section 2 includes notations, definitions, and some well-known oscillation theorems needed in later sections. In Section 3, we are concerned with linear impulsive differential equations. In Section 4, we deal with nonlinear impulsive differential equations.

2. Preliminaries

In this section, we introduce notations, definitions, and some well-known results which will be used in this survey paper.

Let J = [t0, to) for some fixed t0 > 0 and {tk}TO=1 be a sequence in J such that t0 < t1 < t2 < ■■■ <tk< tkk1 < ■■■ and limk^TOtk = to.

By PLC (X,Y), we denote the set of all functions f : X ^ Y which are continuous for t = tk and continuous from the left with discontinuities of the first kind at t = tk. Similarly, PLCr(X,Y) is the set of functions f : X ^ Y having derivative f(r e PLC(X,Y).One has X c J, Y = R", or Y = R. In case Y = R, we simply write PLC(x) for PLC(X, R). As usual, C(X,Y) denotes the set of continuous functions from X to Y.

Consider the system of first-order impulsive ordinary differential equations having impulses at fixed moments of the form

X = f (t,x), t /tk,

Ax\t=tk = fk(x), k e N, where f (t,x) e C(J x R",R"), fk e C(R",R"), and

Ax^ = x(t+k) - x(tk) (2.2)

with x(t±) = limt^t±x(t). The notation Ax(tk) in place of Ax\t=tk is also used. For simplicity, it is usually assumed that x(tk) = x(t-).

The qualitative theory of impulsive ordinary differential equations of the form (2.1) can be found in [1-6,12].

Definition 2.1. A function y e PLC1(J, R") is said to be a solution of (2.1) in an interval (a, ¡5) if y(t) satisfies (2.1) for t e (a,p).

For t0 e (a,5), we may impose the initial condition

x(tk) = xo. (2.3)

Each solution y(t) of (2.1) which is defined in the interval (a,fi) and satisfying the condition y(tk) = x0 is said to be a solution of the initial value problem (2.1)-(2.3).

Note that if t0 e (tk,tkk1), then the solution of the initial value problem (2.1)-(2.3) coincides with the solution of

x' = f (t,x), x(t0) = x0 (2.4)

on (tk, tkk1).

Definition 2.2. A real-valued function x(t), not necessarily a solution, is said to be oscillatory, if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. A differential equation is called oscillatory if all its solutions are oscillatory.

For our purpose we now state some well-known results on oscillation of second-order ordinary differential equations without impulses.

Theorem 2.3 (see [13]). Let p(t) > 0. Then, the equation

y" + P(t)y = 0 (2.5)

is oscillatory if

lirninf tj p(s)ds> 4 (2.6)

t —> oo

and nonoscillatory if

tj p(s)ds < 4 for large t. (2.7)

Theorem 2.4 (see [14]). Let r and p be continuous functions and r (t),p(t) > 0.If

lim f = i, lim f p(s)ds = <x>, (2.8)

-■nij i r(s) tni J i

then the equation

ir(t)y')' + p(t)y = 0 (2.9)

is oscillatory.

Theorem 2.5 (see [15]). Let f be a positive and continuously differentiable function for t > 0, and let f'(t) < 0. If

t2n-1f (t)dt < i, (2.10)

then the equation

y" + f (t)y2n-1 = 0 (2.11)

has nonoscillatory solutions, where n> 1 is an integer.

Theorem 2.6 (see [15]). Let f (t) be a positive and continuous function for t > 0, and n > 1 an integer. Then every solution of (2.11) is oscillatory if and only if

tf (t)dt = i. (2.12)

3. Linear Equations

In this section, we consider the oscillation problem for first-, second-, and higher-order linear impulsive differential equations. Moreover, the Sturm type comparison theorems for second-order linear impulsive differential equations are included.

3.1. Oscillation of First-Order Linear Equations

Let us consider the linear impulsive differential equation

X + a(t)x = 0, t /tk,

Ax|t=tk + akx = 0, k e N, together with the corresponding inequalities:

x' + a(t)x < 0, t / tk,

Ax|t=tk + akx < 0, k e N,

x' + a(t)x > 0, t / tk,

k (3.3)

Ax|t=tk + akx > 0, k e N.

The following theorems are proved in [1].

Theorem 3.1. Let a e PLC(/) and 1 - ak = 0, k e N. Then the following assertions are equivalent.

(1) The sequence {1 - ak} has infinitely many negative terms.

(2) The inequality (3.2) has no eventually positive solution.

(3) The inequality (3.3) has no eventually negative solution.

(4) Each nonzero solution of (3.1) is oscillatory.

Proof. (1)^(2). Let the sequence {1 - ak} have infinitely many negative terms. Let us suppose that the assertion (2) is not true; that is, the inequality (3.2) has an eventually positive solution x(t), t > T0. Let tk > T0 be such that 1 - ak < 0. Then, it follows from (3.2) that

x(t+) < (1 - ak)x(tk) < 0, (3.4)

which is a contradiction.

(2)^(3). The validity of this relation follows from the fact that if x(t) is a solution of the inequality (3.2), then -x(t) is a solution of the inequality (3.3) and vice versa.

(2) and (3)^(4). In fact, if (3.1) has neither an eventually positive nor an eventually negative solution, then each nonzero solution of (3.1) is oscillatory.

(4)^(1). If x(t) is an oscillatory solution of (3.3), then it follows from the equality

x(t) = x(ta) expi - a(s)ds 1 (1 - ak) (3.5)

\ 'i0 / t0<tk<t

that the sequence {1 - ak} has infinitely many negative terms. □

The following theorem can be proved similarly.

Theorem 3.2. Let a e PLC(J), and 1 - ak f 0, k e N. Then the following assertions are equivalent.

(1) The sequence {1 - ak} has finitely many negative terms.

(2) The inequality (3.2) has an eventually positive solution.

(3) The inequality (3.3) has an eventually negative solution.

(4) Each nonzero solution of (3.1) is nonoscillatory.

It is known that (3.1) without impulses has no oscillatory solutions. But (3.1) (with impulses) can have oscillatory solutions. So, impulse actions determine the oscillatory properties of first-order linear differential equations.

3.2. Sturmian Theory for Second-Order Linear Equations

It is well-known that Sturm comparison theory plays an important role in the study of qualitative properties of the solutions of both linear and nonlinear equations. The first paper on the Sturm theory of impulsive differential equations was published in 1996. In [10], Bainov et al. derived a Sturmian type comparison theorem, a zeros-separation theorem, and a dichotomy theorem for second-order linear impulsive differential equations. Recently, the theory has been extended in various directions in [16-18], with emphasis on Picone's formulas, Wirtinger type inequalities, and Leighton type comparison theorems.

We begin with a series of results contained in [1, 10]. The second-order linear impulsive differential equations considered are

where p and q are continuous for t e J, tf tk, and they have a discontinuity of the first kind at the points tk e J, where they are continuous from the left.

The main result is the following theorem, which is also valid for differential inequalities.

Theorem 3.3 (see [1,10]). Suppose the following. (1) Equation (3.7) has a solution y(t) such that

x" + p(t)x f 0, t/tk, Ax(tk) f 0, Ax'(tk) + pkx(tk) f 0, k e N,

y" + q(t)y f 0, t/tk, Ay(tk) f 0, Ay'(tk) + qky(tk) f 0, k e N,

y(t) > 0, t e (a, b), y(a+) f y(b~) f 0.

(2) The following inequalities are valid:

p(t) > q(t), pk > qk, t e (a,b), tk e (a,b).

(3) p(t) > q(t) in a subinterval of (a,b) or pk > qk for some tk e (a,b). Then (3.6) has no positive solution x(t) defined on (a, b).

Proof. Assume that (3.6) has a solution x(t) such that x(t) > 0, t e (a,b). Then from the relation

(x'(t)y(t) - x(t)y'(t))' = x"(t)y(t) - x(t)y"(t), t e (a, b), t f tk, (3.10)

an integration yields

x'(b")y(b") - x(b-)y'(b-) - x'(a+)y(a+) + x(a+)y'(a+)

(* (3.11)

= [x"(t)y(t) - x(t)y"(t)] dt + 2 [Ax'(tk)y(tk) - x(tk)Ay'(tk)].

'a a<tk<b

From (3.6), (3.7), condition (1), and the above inequality, we conclude that C b

0 < (q(t) - p(t))x(t)y(t)dt + ^ (qk - pk)x(tk)y(tk). (3.12)

J a a<h.<h

But, from conditions (2) and (3), it follows that the right side of the above inequality is negative, which leads to a contradiction. This completes the proof. □

The following corollaries follow easily from Theorem 3.3.

Corollary 3.4 (Comparison Theorem). Suppose the following.

(1) Equation (3.7) has a solution y(t) such that

y(t) f 0, t e (a,b), y(a+) = y(b~) = 0. (3.13)

(2) The following inequalities are valid:

p(t) > q(t), pk > qk, t e (a,b), tk e (a,b). (3.14)

(3) p(t) > q(t) in some subinterval of (a,b) or pk > qk for some tk e (a,b). Then, each solution x(t) of (3.6) has at least one zero in (a, b).

Corollary 3.5. If conditions (1) and (2) of Corollary 3.4 are satisfied, then one has the following.

(1) Each solution x(t) of (3.6) for which |x(a+)| + |x(b-) | > 0 has at least one zero in (a, b).

(2) Each solution x(t) of (3.6) has at least one zero in [a, b].

Corollary 3.6 (Oscillation Theorem). Suppose the following.

(1) There exists a solution y(t) of (3.7) and a sequence of disjoint intervals (an,bn) c J such that

lim an = go, y(an) = y (b-) = 0, and y(t) f 0 (3.15)

for t e (an,bn), n e N.

(2) The following inequalities are valid for t e (a„, b„), tk e (a„, b„), and n e N;

p(t) > q(t), pk > qk-

(3.16)

Then all solutions of (3.6) are oscillatory, and moreover, they change sign in each interval [a„, b„].

Corollary 3.7 (Comparison Theorem). Let the inequalities p(t) > q(t), pk > qk hold for t>T > a and tk > T. Then, all solutions of (3.7) are nonoscillatory if (3.6) has a nonoscillatory solution.

Corollary 3.8 (Separation Theorem). The zeros of two linearly independent solutions of (3.6) separate one another; that is, the two solutions have no common zeros, and if a,b (a < b) are two consecutive zeros of one of the solutions, then the interval (a, b) contains exactly one zero of the other solution.

Corollary 3.9 (Dichotomy Theorem). All solutions of (3.6) are oscillatory or nonoscillatory.

We can use Corollary 3.7, to deduce the following oscillation result for the equation:

Theorem 3.10 (see [1,10]). Suppose the following.

(1) The function f (t,u,v) is such that if z(t) is a continuous function for t > a having a piecewise continuous derivative z'(t) for t > a, then the function f (t,z(t),Z(t)) is piecewise continuous for t > a.

(2) The following inequalities are valid:

Then every solution z(t) of (3.17) defined for t > T > a is oscillatory if (3.7) has an oscillatory solution.

Recently, by establishing a Picone's formula and a Wirtinger type inequality, Ozbekler and Zafer [17] have obtained similar results for second-order linear impulsive differential equations of the form

z" + f(t,z,z') = 0, t /tk, Az\t=h = 0, Az'|t=tk + fk (z,z') = 0.

(3.17)

uf (t,u,v) > q(t)u2, t > a, (u,v) e R2, ufk (u,v) > qku2, tk > a, (u,v) e R2.

(3.18)

l[x] = (k(t)x')' + r (t)x' + p(t)x = 0, t /6i,

lo [x] = A (k(t)x') + pix = 0, t = 0i, L[y] = (m(t)y')' + s(t)y' + q(t)y = 0, t /dt, L0 [y] = A(m(t)y') + qiy = 0, t = di,

(3.19)

where [pi] and [qi] are real sequences, k,m,r,s,p,q e PLC(I) with k(t) > 0 and m(t) > 0 for all t e I c J.

Let I0 be a nondegenerate subinterval of I. In what follows we shall make use of the following condition:

k(t) / m(t) whenever r(t) / s(t) Vt e I0. (H)

It is well-known that condition (H) is crucial in obtaining a Picone's formula in the case when impulses are absent. If (H) fails to hold, then Wirtinger, Leighton, and Sturm-Picone type results require employing a so-called "device of Picard." We will show how this is possible for impulsive differential equations as well.

Let (H) be satisfied. Suppose that x,y e C(I0) such that x', y' e PLC(Io) and kx', my' e PLC!(Io). These conditions simply mean that x and y are in the domain of l,l0 and L,L0, respectively. If y(t) / 0 for any t e I0, then we may define

w(t) = -(-) [y(t)k(t)x'(t) - x(t)m(t)y'(t)] for t e I0. (3.21)

For clarity, we suppress the variable t. Clearly,

2 / x \2 sy' w = (k - m)( x') + (q - p)x2 + m\x'--yj + x2

+ ~{yl[x] -xL[y]}, t = Gi, (3.22)

Aw = ^lo [x] - px)--{Lo [y] - qyj, t = Gi.

In view of (3.19) and (3.20) it is not difficult to see from (3.22) that

w' = (k - m)(x')2 + (q - p)x2 + ^^x' - - sx(x' - ^^^

+ (s - r)xx' + — {yl[x] - xL[y]} y

-={q- p- w-m - imm)x2+(k - Ax'+2 (323)

+ x'y- xy'- 2mxy) + y tyl[x] - xL[y]}, t

Aw = (qi - pi)x2 + x {yl0[x] - xL0[y]}, t = Gi. y

Employing the identity

w(ß) - w(a)=\ w'(t)dt + Aw{di)

a< et<ß

(3.24)

the following Picone's formula is easily obtained.

Theorem 3.11 (Picone's formula [17]). Let (H) be satisfied. Suppose that x,y e C(I0) such that x',y' e PLC(I0) and kx',my' e PLC!(I0). If y(t) f 0 for any t e I0 and [a,p] c I0, then

(ykx' - xmy')

q - p -

(s - r)

4( k - m) 4m

x2 + (k - m)\ x' +

+ J2{xy - xy~ 2m

(s - r) 2(k - m)

{yZ[x] - xL[y]} >dt

(3.25)

a< 0i<ß

{fli - Pi)x2 + -{yZü[x] - xLü [y]} y

In a similar manner one may derive a Wirtinger type inequality.

Theorem 3.12 (Wirtinger type inequality [17]). If there exists a solution x of (3.19) such that x / 0 on (a,b), then

W[n] = J jpn2 - k(i{- 2kUj Idt + Pin2 < 0, n e ®-rk,

(3.26)

Qrk = {n e C[a,b] : rn' e PLC [a, b], kr[ e PLC1[a,b], q(a) = n(b) = 0}. (3.27)

Corollary 3.13. If there exists an q e Q,rk such that W [q] > 0, then every solution x of (3.19) has a zero in (a,b).

Corollary 3.14. Suppose that for a given t\ > t0 there exists an interval (a, b) c [t\, <x>) and a function n e Q.rk for which W[q] > 0. Then (3.19) is oscillatory.

Next, we give a Leighton type comparison theorem.

Theorem 3.15 (Leighton type comparison [17]). Suppose that there exists a solution x e Qrk of (3.19). If (H) is satisfied with (a,b) c I0 and

L[x] :=

(s - r)__

4(k - m) 4m

x2 + (k - m)

s-r 2(k - m)

E (qi- PÔx2 > 0

a< ei<b

then every solution y of (3.20) must have at least one zero in (a, b).

Proof. Let a = a + e and ¡5 = b - e e I0. Since x and y are solutions of (3.19) and (3.20), respectively, we have Z[x] = Z0[x] = L[y] = L0[y] = 0. Employing Picone's formula (3.25), we see that

— {ykx' - xmy')

\q - p -

J a+e ^

(s - r)2

4(k - m) 4m J

x2 + (k - m)\ x' +

+ yj|xy - xy- —xy

E (q- p^ x2-

a+e< d¡<b-e

(s - r) . 2(k - m)'

(3.29)

The functions under integral sign are all integrable, and regardless of the values of y(a) or y(b), the left-hand side of (3.29) tends to zero as e ^ 0+. Clearly, (3.29) results in

L[x] < 0, (3.30)

which contradicts (3.28). □

Corollary 3.16 (Sturm-Picone type comparison). Let x be a solution of (3.19) having two consecutive zeros a,b e I0. Suppose that (H) holds, and

q > p +

k > m,

(S - r)2

4(k - m) 4m

(3.31)

(3.32)

for all t e [a,b], and

qi > pi

(3.33)

for all i e N for which di e [a, b].

If either (3.31) or (3.32) is strict in a subinterval of [a,b] or (3.33) is strict for some i e N, then every solution y of (3.20) must have at least one zero on (a, b).

Corollary 3.17. Suppose that conditions (3.31)-(3.32) are satisfied for all t e [t„, to) for some integer U > t0, and that (3.33) is satisfied for all i e N for which di > t„. If one of the inequalities (3.31)-(3.33) is strict, then (3.20) is oscillatory whenever any solution x of (3.19) is oscillatory.

As a consequence of Theorem 3.15 and Corollary 3.16, we have the following oscillation result.

Corollary 3.18. Suppose for a given t\ > t0 there exists an interval (a, b) c [t\, to) for which that condition of either Theorem 3.15 or Corollary 3.16 are satisfied. Then (3.20) is oscillatory.

If (H) does not hold, we introduce a setting, which is based on a device of Picard, leading to different versions of Corollary 3.16. Indeed, for any h e PLC1 (I), we have

4:(x2h) = 2xx'h + x2 h', t = d. (3.34)

v := — (ykx' - xmy') + x2h, t e I. (3.35)

It follows that

, [ (s - r + 2h)2 s2\ 2 n J , s - r + 2h

v = < q - p + h--—-----— \x +(k - m)\ x + ----

4(k - m) 4m 2(k - m)

+ ynxy-xy-2mxn' tf6u

Av = (qi - p^x2 + x2Ah, t = Qi. Assuming that r,s e PLC1 (I), the choice of h = (r - s)/2 yields

(3.36)

; I s' - r' s2 I 2 . , „2 ml, , s I

v = \q-p- - 4m \x +(k-m)(x) + yi\xy-xy- , t=0i'

2lx'y - xy'- sxy L2

(3.37)

Av =\ qi - Pi - 1 (As - Ar) [x2, t = di.

Then, we have the following result.

Theorem 3.19 (Device of Picard [17]). Let r,s e PLC1(I) and let x be a solution of (3.19) having two consecutive zeros a and b in I. Suppose that

k > m, (3.38)

q > p + 1 (s'-r') + ^ (3.39)

are satisfied for all t e [a,b], and that

qi > pt + 2 (As - Ar), t = di, (3.40)

for all i e N for which di e [a, b].

If either (3.38) or (3.39) is strict in a subinterval of [a,b] or (3.40) is strict for some i, then any solution y of (3.20) must have at least one zero in (a, b).

Corollary 3.20. Suppose that (3.38)-(3.39) are satisfied for all t e [t,, to) for some integer t, > t0, and that (3.40) is satisfied for all i e N for which Qi > t, .If r,s e PLC1[t„, to) and one of the inequalities (3.38)-(3.40) is strict, then (3.20) is oscillatory whenever any solution x of (3.19) is oscillatory.

As a consequence of Theorem 3.19, we have the following Leighton type comparison result which is analogous to Theorem 3.15.

Theorem 3.21 (Leighton type comparison [17]). Let r,s e PLC1[a,b]. If there exists a solution x e Q.rk of (3.19) such that

L[x] :=

q - p - - (s' - r') - -— ^ y 2v ' 4m

x2dt + (k - m) (x')2 idt

■ E \ qi - pi - 2(As -Ar) X > 0,

(3.41)

then every solution y of (3.20) must have at least one zero in (a, b).

As a consequence of Theorems 3.19 and 3.21, we have the following oscillation result.

Corollary 3.22. Suppose that for a given t\ > t0 there exists an interval (a,b) c [ti, to) for which conditions of either Theorem 3.19 or Theorem 3.21 are satisfied. Then (3.20) is oscillatory.

Moreover, it is possible to obtain results for (3.20) analogous to Theorem 3.12 and Corollary 3.13.

Theorem 3.23 (Wirtinger type inequality [17]). If there exists a solution y of (3.20) such that y / 0 on (a, b), then for s e PLC1 [a, b] and for all n e Q.sm,

Win] ^¡"{(q - 2Sm - C) n2 - m{n')2} dt + ^ (qi - 1 As)n2 ^ 0- (3.42)

Corollary 3.24. If there exists an n e Qsm with s e PLC1 [a, b] such that > 0, then every solution y of (3.20) must have at least one zero in (a, b).

As an immediate consequence of Corollary 3.24, we have the following oscillation

result.

Corollary 3.25. Suppose that for a given t1 > t0 there exists an interval (a, b) c [t1, to) and a function n e Q.sm with s e PLC1 (I) for which W[n] > 0. Then (3.20) is oscillatory.

3.3. Oscillation of Second-Order Linear Equations

The oscillation theory of second-order impulsive differential equations has developed rapidly in the last decade. For linear equations, we refer to the papers [11,19-21].

Let us consider the second-order linear differential equation with impulses

x" + p(t)x = 0, t /tk,

(3.43)

x(t+) = bkx(tk), x'(t+) = ckx'(tk), k e N, where p e C(J,R), {bk} and {ck} are two known sequences of real numbers, and

Atk) = lim x(tk + h) - x(tk), x'(t+) = lim x(tk + h) - . (3.44)

h^0- h V k' h^0+ h

For (3.43), it is clear that if bk < 0 for all large k, then (3.43) is oscillatory. So, we assume that bk > 0, k = 1,2, — The following theorem gives the relation between the existence of oscillatory solutions of (3.43) and the existence of oscillatory solutions of second-order linear nonimpulsive differential equation:

y" + p(t)y = 0. (3.45)

Theorem 3.26 (see [19]). Assume that bk = ck, k = 1,2, — Then the oscillation of all solutions of (3.43) is equivalent to the oscillation of all solutions of (3.45).

Proof. Let x(t) be any solution of (3.43). Set y(t) = x(t)nt0<tk<tb-1 for t > t0. Then, for all n> 1, we have

y(t+n) = x(tn)bnYI b-1 = x(tn) n b-1 = y(tn).

t0 <tk <t

Thus, y(t) is continuous on J. Furthermore, for t = tn, n e N, we have

-1\u- x(t + h) - x(t) - TT 1,-1

(3.46)

y'(t) =( n bA Hm0= x'(t) H b-1. (3.47)

\t0<tk<t /h ^0 h t0<tk<t

For t = tn, it can be shown that

y'(tn) = y'(t+n)= x\tn) n bk-

t0<tk<t n

Thus, y'(t) is continuous if we define the value of y'(t) at t = tn as

(3.48)

y'(tn) = x'(tn) n b- . (3.49)

t0<tk<t

Now, we have for t / tn,

y"(t) = x"(t) H b- = -p(t)y(t), (3.50)

t0<tk<t

Advances in Difference Equations 15 and for t = tn,

y'\t-n) = v"tn) = -p(tn)y(tn). (3.51)

Thus, we obtain

y"(t) n p(t)y(t) = 0, t > t0. (3.52)

This shows that y(t) is the solution of (3.45).

Conversely, if y(t) is the continuous solution of (3.45), we set x(t) = y(t)Y\t0<tk<tbk for t > t0. Then, x(t-) = x(tn), and x(tn) = y(tn)Ylt0<tk<tnbk = bnx(tn). Furthermore, for t / tn, we have

and so

'(t) = y'(t)H bk, x"(t) = y"(t) n bk,

t0<tk<t t0<tk<t

x'(tn) = bny\tn) Y[ bk = Cnx'(tn),

t0<tk<t n

x"(t)n p(t)x(t) = ( n bk) [y"(t) n p(t)y(t)] = 0, t = tn.

\t0<tk<t /

(3.53)

(3.54)

Thus, x(t) is the solution of (3.43). This completes the proof. □

By Theorems 3.26 and 2.3, one may easily get the following corollary. Corollary 3.27. Assume that p(t) > 0, bk = ck, k e N. Then, (3.43) is oscillatory if

lim inf t p(s)ds > - (3.55)

t Jt 4

and nonoscillatory if

t p(s)ds < -, for large t. (3.56)

When bk = ck, and bk,ck > 0, k e N, oscillation criteria for (3.43) can be obtained by means of a Riccati technique as well. First, we need the following lemma.

Lemma 3.28. Assume that p(t) > 0, p(t) ± 0 on any interval (tk,tkn1\, andlet x(t) be an eventually positive solution of (3.43). If

™ / n c.\

ZiTIr )(tnn1 - tn) = (3.57)

n=0\ i=0 bi /

where b0 = c0 = 1, then, eventually x'(t) > 0.

Now, let x(t) be an eventually positive solution of (3.43) such that x(t) > 0 and x'(t) > 0 for t > T > t0. Under conditions of Lemma 3.28, let u(t) = x'(t)/x(t), for t > T. Then, (3.43) leads to an impulsive Riccati equation:

u'(t) + u2(t)+ p(t) = 0, t > T, t/tk, u{tl) = dku(tk), tk > T,

(3.58)

where dk = ck/bk, k e N.

Theorem 3.29 (see [19]). Equation (3.43) is oscillatory if the second-order self-adjoint differential equation

n dk1 )y'(t)

t <tk<t

+ V(t)[ I! dk' )y(t) = 0, t>T > tc

kt <tk<t

(3.59)

is oscillatory, where dk = ck/bk, k e N.

Proof. Assume, for the sake of contradiction, that (3.43) has a nonoscillatory solution x(t), such that x(t) > 0, for t > T > Now, define

()=( n dAu(t), t>T.

\T <tk<t /

(3.60)

Then, it can be shown that v(t) is continuous and satisfies

(t) + ( n dk)v2(t) + ( n dA V(t) = 0, t>T. (3.61)

T<tk<t T<tk<t

Next, we define

w(t) = exp

( n dk )v(s)ds

yT\T <tk<s

(3.62)

Then w(t) > 0 is a solution of (3.59). This completes the proof.

By Theorems 3.29 and 2.4, we have the following corollary. Corollary 3.30. Assume that

( n dk)dt = TO, [ (n dk1)V(t)dt

T T<tk<t T T<tk<t k

where dk = ck/bk, k e N. Then, (3.43) is oscillatory.

Advances in Difference Equations Example 3.31 (see [19]). Consider the equation

1 1 x"(t) + 42 x(t) = 0, t > 2 ,t = k,

x(k+) = x(k), x'(k+) = k^YX(k), k e N.

If T e (m, m + 1] for some integer m > 0, then it is easy to see that

(3.64)

r(n*)*>r(JU, k+Ty=¡:* -», <36=)

where [•] denotes the greatest integer function, and

£( Ud)pm' >(t Uk+r ) 4ti dt=m+i r 4^ - <366»

Thus, by Corollary 3.30, (3.64) is oscillatory. We note that the corresponding differential equation without impulses

y" + ^ y = 0 (3.67)

is nonoscillatory by Theorem 2.3.

In [20], Luo and Shen used the above method to discuss the oscillation and nonoscillation of the second-order differential equation:

(a(t)x' )' + p(t)x = 0, t /tk,

(3.68)

x(t++) = bkx(tk), x'(t+) = CkX(tk), k e N,

where a e C(J, (0, :)), p e C(J,R).

In [21], the oscillatory and nonoscillatory properties of the second-order linear impulsive differential equation

u" = -p(t)u, t > 0 (3.69)

is investigated, where

p(t) = £anô(t - tn), (3.70)

an> 0 for all n e N, and 6(t) is the 6-function, that is,

ATO AS

6(t)y(t)dt = 6(t)y(t)dt = y>(0) (3.71)

J — to J —S

for all y(t) being continuous at t = 0. Before giving the main result, we need the following lemmas. For each n e N, define the sequence (P^TO^ inductively by

P1 = an+1(tn+1 — tn),

a / Pn \ (3.72)

an+k+1 f Pk x

Pn+1 = ( i pn + an+k(tn+k+1 tn+k)

an+k \ 1 — Pk

n n n n

where P^+1 = to provided P£ = 1 and P^+1 = 0 provided P£ = to. Let Sn = sup{P£ : k e N}.

Lemma 3.32. If Sn0 < 1 for some n0 e N, then Sn0+1 < 1, and Sn < 1 for all n > n0.

Proof. By induction and in view of the fact that the function f (x) = x/(1 — x) is increasing in x e [0,1), it can be seen that

0 < Pl°+l < Pn+1 < 1, Vk e N. (3.73)

Hence, Sn0+i < Sn0 < 1. □

The next lemma can also be proved by induction. Lemma 3.33. Suppose that 0 < ak < "k and Xk > 0 for all k e N. Define, by induction,

pi = aiXi, q1 = P1X1,

ak+1 ( Pk . \

Pk+1 = -l 1- + ak^k+1 ), /n7«

ak \1 — pk / (3"4)

fa+1 f qk _ , +i

qk+1 = T+Kt—* + 1), k e N

pk \ 1 — qk If 0 <qk < 1 for all k e N, then

0 <pk < ~j"~qk < 1, Vk e N. (3.75)

The following theorem is the main result of [21]. The proof uses the above two lemmas and the induction principle.

Theorem 3.34. The following statements are equivalent. (i) There is n0 e N such that Sn0 < 1.

(ii) There is n0 e N such that Sn < 1 for all n > n0.

(iii) Equation (3.69) is nonoscillatory.

(iv) Equation (3.69) has a nonoscillatory solution.

Applying Theorem 3.34, the nonoscillation and oscillation of (3.69), in the case of tn = t0 + Xn-1 T, X> 1, T > 0, and tn = t0 + nT, are investigated in [21].

In all the publications mentioned above, the authors have considered differential equations with fixed moments of impulse actions. That is, it is assumed that the jumps happen at fixed points. However, jumps can be at random points as well. The oscillation of impulsive differential equations with random impulses was investigated in [11]. Below we give the results obtained in this case.

Let Tk be a random variable defined in Dk = (0,dk), k e N, 0 < dk <g, and let t e R be a constant. Consider the second-order linear differential equation with random impulses:

y" + a(t)y' + p(t)y = 0, t e rt, t / Ik,

(3.76)

Ay(lk) = bk(Tk)y(lk), Ay'(Ik) = bk(Tk)y'{lk), k e N,

where RT = [t, <x>), a,p : RT ^ R are Lebesque measurable and locally essentially bounded functions, I0 = t0 e Rt, lk = lk-1 + Tk for all k e N, Ay(lk) = y(lk) - y(l-), Ay'dk) = y'(ik) - y^-h and y(l-) = limt^¿k-0y(t)■

Definition 3.35. Let X be a real-valued random variable in the probability space (Q,F,P), where Q is the sample space, F is the c-field, and P(■) is the probability measure. If JQ jxjP(dx) < go, then JQ xP(dx) is called the expectation of X and is denoted by E(X), that is,

E(X) = xP (dx). (3.77)

In particular, if X is a continuous random variable having probability density function f (x), then

E(X) = xf (x)dx. (3.78)

Definition 3.36. A stochastic process y(t) is said to be a sample path solution to (3.76) with the initial condition yt0 = y0 if for any sample value t1 < t2 < ■■■ < tk < ••• of {¿,k}k>1, then y(t) satisfies

y" + a(t)y' + p(t)y = 0, t e RT, t / tk,

Ay(tk) = bk(tk - tk-i)y(tk), Vk e N, (3.79)

Ay'(tk ) = bk (tk - tk-iWÇt-), Vk e N.

Definition 3.37. The exponential distribution is a continuous random variable with the probability density function:

f \e-Xx, x> 0, f (x) = ^ (3.80)

[0, elsewhere,

where 1> 0 is a parameter.

Definition 3.38. A solution y(t) of (3.76) is said to be nonoscillatory in mean if E[y(t)] is either eventually positive or eventually negative. Otherwise, it is called oscillatory.

Consider the following auxiliary differential equation:

x" + a(t)X + p(t)x = 0, t e RT. (3.81)

Lemma 3.39. The function y(t) is a solution of (3.76) if and only if

to / k \

y(t) = X( E[[1 + bi(Ti)]XttkA+1)(t) )x(t), (3.82)

k=0\ i=1 /

where x(t) is a solution of (3.81) with the same initial conditions for (3.76), and x is the index function, that is,

f1, if Ik < t<ik+1, X&M,1)(t) = \ (3.83)

[0, otherwise.

Proof. If x(t) is a solution of system (3.81), for any t0 < ti < ■■■ <tk< ■■■ , we have

to / k \

y(t) = M E[[1 + bi(ti - ti-1)]X[tk,tk+1)(t) )x(t). (3.84)

k=0 \ i=1 /

It can be seen that

y"(t) + a(t)y'(t) + p(t)y(t) = 0, t = tk,

(3.85)

y(tk) = [1 + bk(tk - tk-t)]y(tk), y1 (tk) = [1 + bk(tk - tk-1)]y'(t-),

which imply that y(t) satisfies (3.76), that is, y(t) is a sample path solution of (3.76). If y(t) is a sample path solution of (3.76), then it is easy to check that x(t) is a solution of (3.81). This completes the proof. □

Theorem 3.40 (see [11]). Let the following condition hold.

(C) Let Tk be exponential distribution with parameter 1 > 0, k e N, and let Ti be independent of Tjif i / j.

If there exists T e RT such that

O f f k

J' • • J ntl + bi (ui)]dvi • • • dvk ^3.86)

V1+- • • +Vk <t-to

does not change sign for all t > T, then all solutions of (3.76) are oscillatory in mean if and only if all solutions of (3.81) are oscillatory.

Proof. Let y(t) be any sample path solution of (3.76); then Lemma 3.39 implies

o / k \

y(t) = X( E[[1 + bi(Ti)]xukA+i)(t) )x(t), (3.87)

k=0\ i=1 /

where x(t) is a solution of (3.81). Hence

E[m] = e

œ / k

E(n[i+bimx^Mt) )x(t) k=0\ i=1

= X[1 + bi(Ti)]xuak+i)(t) )x(t). k=0 \ i=1

Further, it can be seen that

Em [1 + bi(Ti)]x&A+i)(t)

= xk+1 {• ■ ■ j ^T[1 + bi(vi)]e-lV1 ■ ■ ■ e-lVke-lVk+1 dvi- ■ ■ dvM

xk=i vi<t-t0<zk+1 Vi

= Xke-i(t-to) !■ ■ ■ I [1 + bi(vi)]dv^ ■ ■ dvk.

V1+ ■ ■ ■ + Vk <t-to

E[y(t)] = e-X(t-to^x(t)^Xk ■ ■ ■ n[1 + bi(Ui)]dV1■ ■ ■ dvk.

lr=0 J J i=1

k=0 i=1

V1+ ■ ■ ■+Vk <t-to

(3.88)

(3.89)

(3.90)

By assumption, E[y(t)] has the same sign as x(t) for all t > T. That is, all solutions of (3.76) are oscillatory in mean if and only if all solutions of (3.81) are oscillatory. This completes the proof. □

When bk(Tk) = bk is finite, k e N, then the following result can be proved.

Theorem 3.41 (see [11]). Let condition (C) hold, and let bk (Tk) = bk be finite for all k e N. Further assume that there are a finite number of bk such that bk < -1. Then all solutions of (3.76) are oscillatory in mean if and only if all solutions of (3.81) are oscillatory.

3.4. Oscillation of Higher-Order Linear Equations

Unlike the second-order impulsive differential equations, there are only very few papers on the oscillation of higher-order linear impulsive differential equations. Below we provide some results for third-order equations given in [22]. For higher-order liner impulsive differential equations we refer to the papers [23, 24].

Let us consider the third-order linear impulsive differential equation of the form

x'" + p(t)x = 0, t /tk, x(t+) = akx(tk), x'(t+) = bkx'(tk), x"(t+) = €kx"(tk), k e N,

(3.91)

x(t+) = xo,

x (tk) = xo

x (tk) = xo

where ak > 0, bk > 0, ck > 0, tk+1 > tk (k e N) and p e C(J, R+) is not always zero in (tk, tk+i] (k e N) for sufficiently large k.

The following lemma is a generalization of Lemma 1 in [25].

Lemma 3.42 (see [22]). Assume that x(t) is a solution of (3.91) and there exists T > t0 such that for any t > T, x(t) > 0(< 0). Let the following conditions be fulfilled.

(C1) One has

(t1 - t0) + T (t2 - t1) + ••• +

C1 •••Cn

b1 ■■■bn

(tn+1 - tn) + ••• = GO.

(3.92)

(C2) One has

n b1 b1 ■■■bn

(t1 - t0) + — (t2 - t1) + ■■■ + -

a1 a1■■ ■ an

(tn+1 - tn) + ■■■ = g.

(3.93)

Then for sufficiently large t either (A) or (B) holds, where (A) one has

x"(t) > 0(< 0), x'(t) > 0(< 0);

(3.94)

(B) one has

x"(t) > 0(< 0), x'(t) < 0(> 0).

(3.95)

Theorem 3.43 (see [22]). Assume that conditions of Lemma 3.42 are fulfilled and for any n e N,an > 0, 1 > bn > 0, and 1 > cn > 0. Moreover, assume that the sequence of numbers (nil flt} has a positive lower bound, \an - 1| converges, and Jt2p(t)dt = to holds. Then every bounded solution of (3.91) either oscillates or tends asymptotically to zero with fixed sign.

Proof. Suppose that x(t) is a bounded nonoscillatory solution of (3.91) and x(t) > 0, t > T > t0. According to Lemma 3.42, either (A) or (B) is satisfied. We claim that (A) does not hold. Otherwise, x'(tk) > 0 for some tk > T0. Since x"(t) > 0, it follows that x'(t) is monotonically increasing for t e (tk+i-1, tk+i], i e N. For any t e (tk, tk+1], x'(t) > x'(t+) = j > 0. By induction, it can be seen that

x'(t) > bk+n-1 • • • bk+1j > 0, (3.96)

in particular,

x'(tk+n) > bk+n-1 • • • bk+1j > 0,

(3.97)

x'(t++n) = bk+nx(tk+n) > bk+n • • • bk+1j > 0.

Integrating x'(t) > j from tk to tk+1, we obtain

x(tk+1) > x(t+) + Y (tk+1 - tk ).

(3.98)

By induction, for any natural number n > 2, we have

x(tk +n ) > ak+n-1 ■ ■ ■ ak+1

x'(t+) + Y (tk+1 - tk ) + Y^11 (tk+2 - tk+1) . k ak+1

bk+n-1 ■ ■ ■ bk+1 ,, , s

+ ■■■ + Y-(tk+n - tk+n-1)

ak+n-1 ■ ■ ■ ak+1

(3.99)

Considering the condition (C2) in Lemma 3.42 and the sequence of numbers {J^rn=1ai) has a positive lower bound, we conclude that the inequality above leads to a contradiction that the right side tends to to while x(t) is bounded. Therefore, case (B) holds. x'(t) < 0 implies that x(t) is strictly monotonically decreasing. From the facts that the series ^TO=1 \ai - 1| converges and x(t) is bounded, it follows that ^TOTO1 [x(t+) - x(t) converges and there exists limit limt^TOx(t) = l, where 0 < l < to. Now, we claim that l = 0. Otherwise, l > 0 and there exists T1 > T0 such that x(t) > l/2 for t > T1. From (3.91) and the last inequality, we can deduce

x"'(t) = -p(t)x(t) < - l-p(t),

,t+n , rts+n (3100)

l 1 -2„

P" t2x"'(t)dt <-l~ \S+" t2p(t)dt.

Jts 2 Jts

Integrating by parts of the above inequality and considering bn < 1, cn < 1 (n e N), and x'(t) < 0, x"(t) > 0, we have the following inequality:

1 f ts+n n-1

- - t2p(t)dt > 2x(ts+n) + 2^ (1 - as+i)x(ts+i) -\t2sx"(t+s) - 2tsx'(t+s) + 2x(t+s)\. (3.101)

2 Jts i=1

Since \to t2p(t)dt = to and the series ^- ai)x(ti) converges, the above inequality contradicts the fact that x(t) is bounded, hence l = 0, and the proof is complete. □

The proof of the following theorem is similar.

Theorem 3.44 (see [22]). Assume that conditions of Lemma 3.42 hold and for any n e N, an > 0, 1 > bn > 0, 1 > cn > 0, and an/cn > tn+1/tn. Moreover, assume that the sequence of numbers illin=iai} is bounded above, ££=i |an - 1| converges, and \to tp(t)dt = to holds. Then every solution of (3.91) either oscillates or tends asymptotically to zero with fixed sign.

Some results similar to the above theorems have been obtained for fourth-order linear impulsive differential equations; see [24].

4. Nonlinear Equations

In this section we present several oscillation theorems known for super-liner, half-linear, super-half-linear, and fully nonlinear impulsive differential equations of second and higherorders. We begin with Sturmian and Leighton type comparison theorems for half-linear equations.

4.1. Sturmian Theory for Half-Linear Equations

Consider the second-order half linear impulsive differential equations of the form:

(k(t)p(x'))' + p(t)p(x) = 0, t /di,

Axlt=6t = 0, A(k(t)p(x')) ^ + pip(x) = 0,

(m(t)p(y'))' + q(t)p(y) = 0, t /di,

i (4.2)

AV\t=di = 0 HmVpy1))\t=dt + qipiy) = a

where p(s) = |s|a-1s, a> 0, di < di+1, {pi} and {qi} are real sequences, and k,m,p,q e PLC (J) with k(t) > 0 and m(t) > 0.

The lemma below can be found in [26].

Lemma 4.1. Let A,B e R and y > 0 be a constant; then

Apr (A) + jBp(B) - (y + 1)Ap(B) > 0, (4.3)

where equality holds if and only if A = B.

The results of this section are from [16].

Theorem 4.2 (Sturm-Picone type comparison). Let x(t) be a solution of (4.1) having two consecutive zeros a and b in J. Suppose that p(t) < q(t) and m(t) < k(t) are satisfied for all t e [a,b], and that pi < qi for all i e N for which 9t e [a,b]. If either p(t) /q(t) or k(t) /m(t) or pi^ qi, then any solution y(t) of (4.2) must have at least one zero in (a, b).

Proof. Assume that y(t) never vanishes on (a, b). Define

u(t) := —^[k(t)y{y)y{x') - m(t)p(x)^y')],

where the dependence on t of the solutions x and y is suppressed. It is not difficult to see that

u'(t) = {k(t) - m(t)]\x'\a+1 + {q(t) - p(t)}|x|a+1 + m(t)

x'P(x') + - (a + l)x'^xy)

, t /di,

&ult=ei = (qi- pi) Wft)!

Clearly, the last term of (4.5) is integrable over (a, b) if y(a) / 0 and y(b) = 0. Moreover, u(a+) = u(b~) = 0 in this case. Suppose that y(a+) = 0. The case y(b^) = 0 is similar. Since y'(a+)/ 0 and

,. x(t) x'(t)

lim —= lim —7-f < go, í — a+ y(t) t — a+ y'(t)

we get

lim P^) t — a+ \ y(t) J

and so

t — a+

P(x') + °(fHf) - (a + ^(f) x'p(x>) + - (a +1)x'

t — a+

Moreover,

lim u(t) = lim x

k(t)p(x') - m(t)y( -y

(4.10)

Integrating (4.5) from a to b and using (4.6), we see that C b

0 = \(k(t) - m(t))\x'(t)\a+1 + (q(t) - p(t))\x(t)\a+1]dt

+ C m(t) x'y(x') + ai^-) y(—)-(a + 1)x'v(—) dt + £ - pi)\x(di)\a+1,

J a \ y / \ y / \y/J a<0i<b

(4.11)

where we have used Lemma 4.1 with A = x', B = xy'/y, j = a, and ya(s) = y(s). It is clear that (4.11) is not possible under our assumptions, and hence y(t) must have a zero in (a,b). □

Corollary 4.3 (Separation Theorem). The zeros of two linearly independent solutions x(t) and y(t) of (4.1) separate each other.

Corollary 4.4 (Comparison Theorem). Suppose that p(t) < q(t) and m(t) < k(t) are satisfied for all t e [t„, to) for some t„ > t0, and that pi < qi for all i e N for which di > t„. If either p(t) ^ q(t) or k(t) / m(t) or pi / qi, then every solution y(t) of (4.2) is oscillatory whenever a solution x(t) of (4.1) is oscillatory.

Corollary 4.5 (Dichotomy Theorem). The solutions of (4.1) are either all oscillatory or all nonoscillatory.

Theorem 4.6 (Leighton-type Comparison). Let x(t) be a solution of (4.1) having two consecutive zeros a and b in J. Suppose that

Va[x] := \{q(t) - p(t)}\x(t)\a+1 + {k(t) - m(t)}\x'(t)\a+1] dt + £ (qi - p0 \x(9l)\a+1 > 0.

J a a<6i<b

(4.12)

Then any nontrivial solution y(t) of (4.2) must have at least one zero in (a, b).

Proof. Assume that y(t) has no zero in (a,b). Define the function u(t) as in (4.4). Clearly, (4.5) and (4.6) hold. It follows that

0 = u(b~) - u(a+) Cb

= [(k(t) - m(t))\x'\a+1 + (q(t) - p(t)) xr1] dt

+ [ m(t) x'y(x') + a(y) ?(—)- (a + 1)x'v(—) dt + £ \x(0i)\a+1 (q{ - p{)

J a \ y / \ y / \ y / a<6i<b

= Va[x]+j m(t) x'y(x') + a^y^V^y^ - (a + 1)x'^"y")

dt > 0,

(4.13)

which is a contradiction. Therefore, y(t) must have a zero on (a, b). □

If Va[x] > 0, then we may conclude that either y(t) has a zero in (a,b) or y(t) is a constant multiple of x(t).

As a consequence of Theorems 4.2 and 4.6, we have the following oscillation result.

Corollary 4.7. Suppose for a given T > f„ there exists an interval (a, b) c [T, to) for which either conditions of Theorem 4.2 or Theorem 4.6 are satisfied, then every solution y(t) of (4.2) is oscillatory.

4.2. Oscillation of Second-Order Superlinear and Super-Half-Linear Equations

Let us consider the forced superlinear second-order differential equation of the following form:

(r(t)x')' + p(t)\x\a-1 x = q(t), t /Tk,

x(t+) = akxT), x'(r+) = bkx'T), k e N, (4.14)

x(fo^ = ^ x (t0) = x0,

where {rk} denotes the impulse moments sequence with 0 < t0 = t0 < t1 < ••• < Tk < ■■■, limk^œTk = to.

Assume that the following conditions hold.

(A1) a> 1 is a constant, r : J ^ (0, to) is a continuous function, p,q e C(J). (A2) bk > ak> 0 are constants, k e N.

(A3) [c1,d1] and [c2,d2] are two intervals such that Cj,dj e {Tk} (j = 1,2) with c1 < d1 < c2 < d2, p(t) > 0 for t e [c1,d1] u [c2,d2], and q(t) < 0 for t e [c1,d1],q(t) > 0 for t e [c2,d2].

Interval oscillation criteria for (4.14) are given in [27]. Denote k(s) = max{i : t0 < Ti < s}, and for j = 1,2, let n = max{r(t) : t e [cj,dj]},

Qw(cj,dj) = e C1 [cj,dj], w(t) ^0,w(cj) = w(dj) = 0},

i /-' (4.15)

Q.G(cj,dj) = \ G e C1 [cj,dj], G(t) > 0, ±0, G(cj) = G(dj) = 0, G(t) = 2g(t)^G(t)

Theorem 4.8 (see [27]). Assume that conditions (A1)-(A3) hold, and that there exists w(t) e Qw (cj,dj) such that

{[p(t)] 1/a\q(t)\1-1/aw2(t) - r(t)w2(t)} dt > Q(w,Cj,dj), (4.16)

where Q(w,cj,dj) = 0 for k(cj) = k(dj), and

f U rt k(dj ) , ^

I it \ bk(cj )+1 - ak(c> )+1 ¿4 9/ , bi - ai |

Q{w,cj,dj) = nAw\Tk(c, )+1)-^-+ X w(T) a JJ A (4.17)

[ ak(j )+1\Tk(c] )+1 - j i=k(c, )+2 ai(Ti Ti-1) J

for k(cj) < k(dj), j = 1,2. Then every solution of (4.14) has at least one zero in [c1,d1] u [c2,d2].

Proof. Let x(t) be a solution of (4.14). Suppose that x(t) does not have any zero in [c1,d1] U [c2,d2 ]. Without loss of generality, we may assume that x(t) > 0 for t e [c1,d1]. Define

u(t) = -t®^, t e [ci,di]. (4.18)

Then, by Holder's inequality, for t e [c1,d1 ] and t = Tk, we have

u'(t)= p(t)[x(t)]a-1 - x$)+ u*(t)

> [p(t)] 1/a\q(t)\1-1/a + ^u2(t), t e (ci,di), t = Tk.

(4.19)

For t = Tk, k e N, we obtain

u(t+) = —u(Tk). (4.20)

If k(c1) < k(d1), then all impulsive moments are in [c1,d1] : Tk(ci)+1,Tk(ci)+2,...,Tk(dl). Multiplying both sides of (4.19) by w2(t) and integrating on [c1,d1] and using the hypotheses, we get

k(U1) .J _ U fU1

£ w2(Tl)u(Tl) > { [p(t)]1/a\q(t)\1-1/aw2(t) - r(t)w'2(t)}dt. (4.21)

i=k(c1) + 1 ai Jci

On the other hand, for t e (c1,Tk(c1)+1], it follows that

(r(t)x'(t)y = q(t) - p(t)\x(t)\a-1x(t) < 0, (4.22)

which implies that r(t)x'(t) is nonincreasing on (c1,Tk(c1)+1]. So, for any t e (c1 ,Tk(c1)+1 ], one has

x(t) - x(c1) > r(t)*'(t) (t - d), £ e (c1,t). (4.23) r (£)

It follows from the above inequality that

u(Tk(C1)+1) >--—->----. (4.24)

Tk(d)+1 - c1 Tk(d)+1 - c1

Making a similar analysis on (Ti-1 ,Ti], i = k(c1) + 2,..., k(d1), we obtain

u(t)= -r(Ti]x'[Ti) >--, i = k(c1)+ 2.....k(d1). (4.25)

x(Ti) Ti - Ti-1

Advances in Difference Equations From (4.21)-(4.25) and (A2), we get

1 [[p(t)]1/a|q(t) |(t) - r(t)w2(t)] dt < Q(w,ci,di), (4.26)

•'ci

which contradicts (4.16). If k(c1) = k(d1), then Q(w,c1,d1) = 0, and there are no impulse moments in [c1,d1]. Similarly to the proof of (4.21), we get

0 > I"1 {[p(t)]1/a|q(t)|1-1/aw2(t) -r(t)w'2(t)]dt, (4.27)

which again contradicts (4.16).

In the case x(t) < 0, one can repeat the above procedure on the subinterval [c2,d2] in place of [c1 ,d1]. This completes the proof. □

Corollary 4.9. Assume that conditions (A1) and (A2) hold. If for any T > 0 there exist Cj,dj satisfying (A3) with T < c1 < d1 < c2 < d2, and w(t) e Qw (cj,dj) satisfying (4.16), j = 1,2, then (4.14) is oscillatory.

The proof of following theorem is similar to that of Theorem 4.8.

Theorem 4.10 (see [27]). Assume that conditions (A1)-(A3) hold, and there exists a G e QG(cj,dj) such that

{ [p(t)] 1/aiq(t)111/aG(t) - r(t)h2(t)}dt > R(G,Cj,dj), (4.28)

where R(G,Cj,dj) = 0 for k(cj) = k(dj), and

R(G,Cj,dj) = nj

bk(Cj) + 1 - ak(cj) + 1 ) bi - Ui

/ \ bk(Cj) + 1 - Uk(Cj) + 1 ^

G(Tk(c,)+i)---^ G(Ti) ( i)

Uk(cj )+l{Tk(Cj ) + 1 - Cj) i=k(Cj )+2 Ui(T li-1)

(4.29)

for k(Cj) < k(dj), j = 1,2. Then every solution of (4.14) has at least one zero in [C1,d1] u [C2,d2].

Corollary 4.11. Assume that conditions (A1) and (A2) hold. If for any T > 0 there exist Cj,dj satisfying (A3) with T < c1 < d1 < c2 < d2, and G e Q.G(Cj,dj) satisfying (4.28), j = 1,2, then (4.14) is oscillatory.

Example 4.12. Consider the following superlinear impulsive differential equation: ((1 + sin2 t)x+ (ß cos t)\x\a-1x = sin t, t = 2kn ± ^r,

x(r+) = Ukx(Tk), x'(rk+) = bkx'(rk), Tk = 2kn ± —, k e N.

It can be seen that if

p/a E(3 + l/a)r(4 + Ua) > 3x + 2(bk - ak) , fc g N, (4.31)

then, conditions of Corollary 4.9 are satisfied; here a > 1, r is the gamma function, and ak and bk satisfy (A2). So, every solution of (4.30) is oscillatory.

In [28-30], the authors have used an energy function approach to obtain conditions for the existence of oscillatory or nonoscillatory solutions of the half-linear impulsive differential equations of the following form:

(fop (X))' + fop(x) = 0, t /tn,

(4.32)

x(tn + 0) = x(tn), x'(tn + 0) = bnx'(tn),

where bn > 1 for n e N.

Define the energy functional

V(x,y) = yfop(y) - f fop(s)ds + f fy(s)ds =: Op(y) + (x), (4.33)

where in explicit form Op(y) = (p/(p + 1))|y|p+1 and Fp(x) = (1/(p + 1))|x|p+1. The functions Op and Fp are both even and positive definite.

The function V(t) = V(x(t),x'(t)) is constant along the solutions of the nonimpulsive equation

(fop (x'))' + fop (x)= 0. (4.34)

The change in the energy along the solutions of (4.32) is given by

V(tn+1) - V(tn) = V(tn + 0) - V(tn)

= Op(x'(tn + 0)) + Fp(x(tn + 0)) - Op(x'(tn)) - Fp(x(tn)) (4.35)

= bn+1Op(x'(tn)) - Op (x'(tn)) = Op (x'(tn))( bn+ - 1) .

We see that these impulsive perturbations increase the energy. If the energy increases slowly, then we expect the solutions to oscillate. On the other hand, if the energy increases too fast, the solutions become nonoscillatory. Let x(t) be a solution of (4.32), rn = V(tn-1 +0) = V(tn-0), and Fp(x(tn)) = anrn. Calculating Fp(x(tn)) in terms of rn+1 = V(tn + 0), we obtain

Fp (x(tn)) = p+1 -rn+1. (4.36)

bn (1 - On) + On

To simplify the notation, we introduce the function

0(u,b) := U -. (4.37)

bP+l(1 - u) + u v y

The function 0 gives the jump in the quantity Fp(x(t))/V(t). Note that 0(0,b) = 0, 0 < 0(u, b) <u for 0 <u< 1 and b > 1; 0 is monotone increasing with respect to u and decreasing with respect to b.

Theorem 4.13 (see [29]). Assume that there exist a constant N > 0 and a sequence [j„} with 0(jn, bn) < jn+i < 1 such that

fJn+1 dv

tn+1 - tn < —t-r--(4 38)

j0(jnA) fo(F-1(v)p--1(1 - v) J

holds for every n> N. Then every solution of (4.32) is nonoscillatory.

Theorem 4.14 (see [29]). Assume that there exist a constant N > 0 and a sequence {Xn} with Xn > 0 and^ n=i Xn = n such that for every y e [0,1]

fY+Xn dv

tn+1 - tn > —-7—-T—---(4.39)

J0(rM fo^F-1(v))0-1(1 - v)

holds for every n> N. Then every solution of (4.32) is oscillatory.

Proof. Let x(t) be a nontrivial solution of (4.32). It suffices to show that x(t)x'(t) > 0 cannot hold on any interval [T, n). Assume that to the contrary, x(t) > 0, x'(t) > 0 for t e [tN, n). Let on be defined by Fp(x(tn)) = onrn, where rn = V(tn - 0). It follows from (4.39) that an+i > on + Xn. Hence,

On+1 > on + ^Xn. (4.40)

Since on = Fp(x(tn))/rn < 1, and the right side of the above inequality tends to infinity as n ^ n, we have a contradiction. □

Now, assume that bn > b> 1 for every n> N. It can be shown that the integral

cr _dv_

i0(rb) fop(F-1(v))®--1(1 - v) (441)

takes its maximum in [0,1] at

Y = bP(b - 1) (4.42)

Y bP+^1

In the special case bn = b, tn+1 - tn = d (n e N), we have the following necessary and sufficient condition.

Theorem 4.15 (see [29]). Assume that bn = b, tn+1 - tn = d (n e N). Then every solution of (4.32) is nonoscillatory if and only if

„ (Y dv

d - , ^ w ^ ^-:. (4.43)

'Q(rb) fy(F-1(v))®-1(1 - v)

Remark 4.16. Equation (4.32) with 0 - bn - 1, was studied in [30].

Finally, we consider the second-order impulsive differential equation of the following

(r(t)fa(x'))' + p(t)fa(x') + q(t)yp(x) = e(t), t /di,

(4.44)

M r(t)ya{x')) + qtyp(x) = ei, t = di,

where yr(s) = |s|r-1 s, ¡5 > a > 0 are real constants, (d^ is a strictly increasing unbounded sequence of real numbers, (qi} and (ei} are real sequences, r,p,q,e e PLC(J), and r(t) > 0. All results given in the remainder of this section are from [31].

Theorem 4.17. Suppose that for any given t, > t0, there exist intervals I1 = [s1, t1], I2 = [s2, t2] c [t,, to), such that

(a) q(t) > 0 for all t e(I1 U I2}\ (di} and qi > 0 for all i e N for which di e I1 UI2;

(b) e(t) - 0, t e Ii \ (di}, e(t) > 0, t e I2 \ (di}; ei - 0, di e Ii, ei > 0, di e h for all i e N.

If there exists H e D(sk,tk) such that

Hq(t)IH(t)la+1 - (a + 1)-

(a + 1)H'(t) - ^H(t) r(t)

(4.45)

+ £ qilH(di)Ia+1 > 0, (k = 1,2),

sk-di<tk

q(t) = ¡a-aH{p - a)a/5-1[q(t)]a/5Ie(t)I1-a/¡¡, (4.46)

qi = ¡¡a-a/5H - a)a/¡¡-1 [q,]a/5\el\1-a/5/ (4.47)

then (4.44) is oscillatory.

Proof. Suppose that there exists a nonoscillatory solution x(t) of (4.44) so that x(t) / 0 for all t > t. for some t.. Let

*):- «T' t > t.. (4.48)

pa(x(t))

It follows that for t>U

p -v...... - K" -'=*

Wa(x),

v = —v- a

r rl/a V fa(x)

Av = -(qi| x | ¡¡-a -t^Y t = 0i

(4.49)

where t dependence is suppressed for clarity.

Define a function E(z) : (0, to) ^ (0, to) by

E(z) := A^ + A2z^a, Ai > 0, A2 > 0. (4.50)

It is not difficult to see that if ¡5 > a, then

min E(z) = ¡a-a/5(5 - a)(a-5)/5Aa1/5Ai25-a)/¡. (4.51)

zg(0,to)

Clearly, if 5 = a, then we have E(z) > A1. Thus, with our convention that 00 = 1, (4.51) holds for ¡5 > a.

Suppose that x(t) > 0 for all t > t. .Choose s1 > t. and consider the interval I1 = [s1,t1]. From (b), we see that e(t) < 0 on I1 \ [di} and ei < 0 for all i e N for which di e I1. Applying (4.51) to the terms in the parenthesis in (4.49) we obtain

p VI 1+1/a q <-v'-pv - a1 ' , t e I1 \{0i}; ' r r 1/a

Av(di) + qi < 0, di e I1,

where q and qi are defined by (4.46) and (4.47), respectively.

Let H e D(s1, t1). Multiplying (4.52) by |H|a+1 and integrating over I1 give

p q'|H|a+1dt <- p P |H|a+1v dt - a p ^^|H|a+1dt - p |H|a+1v'dt.

J Si J Si r J Si r J Si

(4.52)

In view of (4.52) and the assumption H(s1) = H(i1) = 0, employing the integration by parts formula in the last integral we have

r q\H| a+idt + X q I H| a+1

•'S! Sl<6i<tl

fi fi V fi Ivl 1 + 1/a

< (a + in 9a (H)H'v dt - - |H\a+1v dt - a ' \/a \H\a+1dt

J si J si r J si r

(4.54)

(a + i)9a(H)H'- V\H\a+i r

\ V \ dt - a Hp! - \ i+i/a

We use Lemma 4.i with

Y = 1 A = aa/(a+i) \ H\ a .I

Y a' A a r 1/(a+i) \ V \ '

a(a + i)-(a+i)r\a/(a+i)

a(a+i)

(a + i)9a(H)H'- V I H\a+i

(4.55)

to obtain

T q\H\a+idt + X qi\H\a+i < (a + i)-(a+i) T r

•^i si<9i<ti 'si

(a + i)H '-*- H r

(4.56)

which obviously contradicts (4.45).

If x(t) is eventually negative then we can consider I2 and reach a similar contradiction. This completes the proof. □

Example 4.18. Consider

( \ x \ ~2/3x')' + (cos 3t) \ x' \ -2/3x' + (cos t) \ x \-i/2x = sin 3t, t /

in i2 '

( \ X\ ~2/3x') + y(sec3/2t)\ x \-i/2x = cos3t sin3t,

in i2 '

(4.57)

where j is a positive real number.

Let H(i) = sin23t, I1 = [(2n + 5/3)n, (2n + 2)n], and I2 = [(2n + 2)n, (2n + 7/3)n]. For any given i» > 0 we may choose n e N sufficiently large so that 2nn > i». Then conditions

(a)-(b) are satisfied. It is also easy to see that, for j = 2n + 5/3 and j = 2n + 2,

Ai+1/3)n . . 34/3

3.2-2/3cos2/3t I sin

(2r 2 j sin330,

;n<0i<(;+1/3)n y\ 2/3

(4.58)

= a+3(2) ...2.

12;<i<12;+4

3 / in Sin^T

= A+3( i)2/ïi + ^ -

where A2n+5/3 a -2.17024 and A2n+2 a -1.58213. It follows from Theorem 4.17 that (4.57) is oscillatory if

3 (V2 - 2)A2n+5/3

■ 0.55172.

(4.59)

Note that if there is no impulse then the above integrals are negative, and therefore no conclusion can be drawn.

When ¡5 = a, then (4.44) reduces to forced half-linear impulsive equation with damping

(r (t)fa(x')) ' + p(t)fa{x') + q(t)fa(x) = e(t), t /0,-M r (t) Pa(x')) + qi^a (x) = e,- t = 0i.

(4.60)

Corollary 4.19. Suppose that for any given t„ > t0, there exist intervals I1 = [s1, t1], I2 = [s2, t2] c [t„, to) for which (a)-(b) hold.

If there exists H e D(sk,tk) such that

Ç\q(t)\H(t)\a+1 - (a + 1)-(a+Vr(t)

(a + 1)H'(t) - ^H(t) r (t)

+ £ ql\H(0i)\a+1 > 0 (k = 1- 2)-

Sk<0i<tk

(4.61)

then (4.60) is oscillatory.

Taking a = 1 in (4.44), we have the forced superlinear impulsive equation with dam-

(r(t)x')' + p(t)x' + q(t)\x\5-1x = e(t)- t / 0h M(r (t)x') + qi\x\5-1x = ei, t = 0.

Corollary 4.20. Let ¡¡0 = ¡¡(p-1)1/p-1, ¡¡> 1. Suppose that for any given t„ > t0, there exist intervals I1 = [s1, t1], I2 = [s2, t2] c [t„, to), such that (a)-(b) hold for all i e N. If there exists H e D(sk,tk) such that

f U [q(t)]

\e(t)\1-1/pH2(t) - r(t)

H'(t) -

P(t) 2r(t)

+ E ¡oq1/P\el\1-1/pH2(ei) > 0 (k = 1,2),

Sk <0i<tk

(4.63)

then (4.62) is oscillatory.

Let p = 1 in (4.62). Then we have the forced linear equation:

(r(t)x')' + p(t)x' + q(t)x = e(t), t / Q, A(r (t)x') + qix = ei, t = Qi.

(4.64)

Corollary 4.21. Suppose that for any given t„ > t0, there exist intervals I1 = [s1, t1], I2 = [s2, t2] c [t„, to), such that (a)-(b) hold for all i e N. If there exists H e D(sk,tk) such that

q(t)H2(t) - r(t)

H '(t) - 2Ti H (t)

dt + Z qiH2(Qi) > 0 (k = 1,2), (4.65)

sk <Qi<tk

then (4.64) is oscillatory. Example 4.22. Consider

x" - (sin t)x'- (cos t)x = e cos sin t, t /

in 18,

(4.66)

Ax' + (o cos3t)x tan t, t = —

where o is a positive real number.

Let H = sin 6t. For any T > 0, choose n e N sufficiently large so that nn > T and set I1 = [(2n-5/6)n, (2n-2/3)n] and I2 = [(2n+2/3)n, (2n+5/6)n]. Clearly, (a)-(b) are satisfied for all n. It is easy to see that for j = 2n - 5/6 and j = 2n + 2/3,

(j + 1/6)n

- cos t sin26t - ^6 cos 6t + 2 sin t sin6t 2 o cos(3Qi)sin2(6Qi)

jn<Qi<(]+1/6)n

(-1 + V3)-

289n 3o

Advances in Difference Equations Thus (4.65) holds if

- ^-O - HM (4.68)

which by Corollary 4.21 is sufficient for oscillation of (4.66).

Note that if the impulses are removed, then (4.66) becomes nonoscillatory with a nonoscillatory solution x(t) = te~cos

Finally we state a generalization of Theorem 4.17 for a class of more general type impulsive equations. Let a, ¡5, q(t), e(t), di, [ei], and [qi] be as above, and consider

(r(t)ya(x')) ' + p(t)pa (x') + f (t,x)= e(t), t / di, A(r (t)pa(x')) + g(t,x) = ei, t = di,

(4.69)

where the functions f and g satisfy

sf (t, s) > q(t)\s\5+1 for t e II u I2, s e R, sg(di, s) > qi\s\5+1 for 0i e Ii u I2, s e R.

(4.70)

Theorem 4.23. In addition to conditions of Theorem 4.17, if (4.70) holds then (4.69) is oscillatory. Example 4.24. Consider

(|x'|-2/3x'y + (cos3t)\x'\-2/3 x + f (t,x)= sin 3t, t = 12, A (\x'\-2/3x^ + g(t, x) = cos3t sin 3t, t = 12,

(4.71)

f (t,x) = cos2(^2) [2 + sin(x + t)]\x\1/2x,

(4.72)

g (t,x) =

2 / x + t Y + 2cos2

(sec3/2t) \x\1/2x, (y> 0).

Clearly if we take Ii = [(2n + 5/3)n, (2n + 2)n] and I2 = [(2n + 2)n, (2n + 7/3)n], then (4.70) holds with ¡5 = 1/2, q(t) = cos t and qi = ysec3/2t. Further, we see that all conditions of Theorem 4.17 are satisfied if H(t) = sin23t and y > 0.552; see Example 4.18. Therefore we may deduce from Theorem 4.23 that (4.71) is oscillatory if y > 0.552.

4.3. Oscillation of Second-Order Nonlinear Equations

In this section, we first consider the second-order nonlinear impulsive differential equations of the following form:

x"(t)+ f (t,x) = 0, t /tk, x(tO) = gk(x(tk)), x'(t+k) = hk(x'(tk)), k e N, (4.73)

x(to ) = xo, x (tk) = x0'

Assume that the following conditions hold.

(i) f (t,x) e C(J x R), xf (x) > 0 (xf 0), and f (t,x)/y(x) > p(t) (x / 0), where p e C(J), p(t) > 0, and xy(x) > 0 (x / 0), y'(x) > 0'

(ii) gk, hk e C(R), and there exist positive numbers ak, a*k, bk, b*k such that

* . gk(x) . , * . hk (x) (4 74) ak <2-< ak, bk <-< bk' (4.74)

In most of the investigations about oscillation of nonlinear impulsive differential equations, the following lemma is an important tool.

Lemma 4.25 (see [25]). Let x(t) be a solution of (4.73). Suppose that there exists some T > t0 such that x(t) > 0 for t > T. If conditions (i) and (ii) are satisfied, and

(iii) (ti - to) + (b\/ai)(t2 - ti) + (b*tb*2/aia2)(t3 - t2) + ■■■ + (b\b*2 ■■■ b*n/aia2 ••• a„)(t„+i -tn) + ■■■ = holds, then x'(t+) > 0 and x'(t) > 0 for t e (tk, tk+1], where tk > T.

Theorems 4.26-4.32 are obtained in [25]. For some improvements and/or generalizations, see [32-35].

Theorem 4.26. Assume that conditions (i), (ii), and (iii) of Lemma 4.25 hold, and there exists a positive integer k0, such that a*k > 1 for k > k0. If

ÇH 1 çt2 1 Çt3 1 çtnol

p(t)dt ^ p(t)dt ^^ p(t)dt o ---o—-— p(t)dt o ■■■ = œ, (4.75)

Jt0 b1 Jh b1b2 J t2 b1b2 ■■■bnJt„

then every solution of (4.73) is oscillatory.

Proof. Without loss of generality, we can assume k0 = 1. If (4.73) has a nonoscillatory solution x(t), we might as well assume x(t) > 0, t > t0. From Lemma 4.25, x'(t) > 0 for t e (tk,tk+1], where k e N. Let

u(t) = igwr (476)

Then, u(t^) > 0 (k e N), u(t) > 0 (t > t0). Using condition (i) in (4.73), we get for t / tk,

u'(t) = -

f (t,x(t)) y(x(t))

f(x(t))

y'(x(t)) < -p{t).

Using condition (ii) and a*k > 1, p'(x) > 0 yield

u(&) =

x,(tj) p(x(t+k))

< bku(tk), k e N.

From the above inequalities, we have

u(si) < u(s) -

p(t)dt,

where t0 < s < s1 < t1. Taking s ^ and s1 ^ t1, we get

u(tj) < biu(ti) < bi

By induction, for any natural number n, we obtain

u(t+n) < bib2 •••bn

rh i m

<tk) p(t)dt - - J^ p(t)dt

Ç tn-1

p(t)dt -

J in-2

bib2 ■■■bn-2 Jtn.

bib2 ■■■bn-1

p(t)dt

J tn-i

(4.77)

(4.78)

(4.79)

Ç ti çti

(tj) - p(t)dt < bi^tj) - bi p(t)dt. (4.80)

Jtn J to

(4.8i)

Since, u(t^) > 0 (k e N), above inequality and the hypothesis lead to a contradiction. So, every solution of (4.73) oscillatory.

From Theorem 4.26, the following corollary is immediate.

Corollary 4.27. Assume that conditions (i)-(iii) of Lemma 4.25 hold and there exists a positive integer k0 such that a*k > 1, bk < 1 for k > k0. If

p(t)dt = œ,

then every solution of (4.73) is oscillatory.

The proof of the following theorem is similar to that of Theorem 4.26.

Theorem 4.28. Assume that conditions (i), (ii), and (iii) of Lemma 4.25 hold and y(ab) > y(a)y(b) for any ab> 0. If

r V(al) ( Vial)v(a2) f3

p(t)dt + p(t)dt + yv Y\v 2 p(t)dt + ■■■

Jto b1 Jti b1b2 Jt2

(4.83)

, vte) ■■■V(an) r^ p(t)dt_

blb2 ■■■bn

then every solution of (4.73) is oscillatory.

■ + ■■■ = to,

Corollary 4.29. Assume that conditions (i), (ii), and (iii) of Lemma 4.25 hold and y(ab) > y(a)y(b) for any ab > 0. Suppose that there exist a positive integer k0 and a constant a> 0 such that

V«) ^ f tk+i bk

> for k > k0.

(4.84)

If J^ tap(t)dt = to, then every solution of (4.73) is oscillatory. Example 4.30. Consider the superlinear equation:

x" + ( i)x2n-1 = 0, t > I,t / k,

x(k+) = ^^kr)x(k), x'(k+) = X(k), k e N, (4.85)

x( 2) = xo, m 2 I = ^

where n > 1 is a natural number. Since ak = a*k = (k + 1)/k, bk = b*k = 1, p(t) = 1/t3, tk = k, and y(x) = x2n-1. It is easy to see that conditions (i), (ii), and (iii) are satisfied. Moreover

V(a;) = f k + 1 \2n-1 > f tk+^3

bk k tk

(4.86)

ftt /"O

tap(t)dt =

^ U)A,= I t3( -3 )dt = 00.

Hence, by Corollary 4.29, we find that every solution of (4.85) is oscillatory. On the other hand, by Theorem 2.6, it follows that (4.85) without impulses is nonoscillatory.

Theorem 4.31. Assume that conditions (i), (ii), and (iii) of Lemma 4.25 hold, and there exists a positive integer k0 such that a*k > 1 for k > k0. If

C™ du t

< for some e > 0,

& ftk+1 ftk+1 1 ftk+2 1 ftk+3

£ p(t)dt +— p(t)dt + p(t)dt ■ t=0J tk Js bk+1 Jtk+1 bk+1bk+2 Jtk+2

1 ftk+n+1

p(t)dt ■

(4.87)

+n Jtk+

bk+ibk+2 ■■■bk then every solution of (4.73) is oscillatory.

Proof. Without loss of generality, let k0 = 1. If (4.1) has a nonoscillatory solution x(t), assume x(t) > 0 (t > to). By Lemma 4.25, x'(t^) > 0 (k e N), x'(t) > 0 (t > to). Since a*k > 1 (k e N), we have

x(tk) < x(ti) < x(t\) < x(t2) < x(ti,) <••• . (4.88)

It is easy to see that x(t) is monotonically nondecreasing in J. Now (4.73) yields

x"(t) = -f (t,x(t)) < -p(t)y(x(t)), t = tk. (4.89)

Hence, from the above inequality and condition (ii), we find that

f t1 f t1 x'( t+)

(t+) > p(t)y(x(t))dt + x'(h) > p(t)y(x(t))dt + —^ • (4.90)

Jto J to b1

Generally, for any natural number n, we have

f tn+1

x'(t+) > p(t)y(x(t))dt + x'(tn+1)

f tn+1

> p(t)y(x(t))dt-

(4.91)

tn bn+1

By (4.89) and (4.91), noting x(tk) > 0 (k e N) and bi > 0 (i e N), for any natural number n, we obtain

tk+1 1 tk+2 (s) > p(t)y(x(t))dt + -- p(t)y(x(t))dt

Js bk+1 J tt+1

s tk+1 (4.92)

bk+1bk+2 ■■■bk-

1 ftk+n+1

p(t)f(x(t))dt-

+n tk+

Note that xy(x) > 0 (xf 0) and y(x) is nondecreasing. Dividing the above inequality by y(x(s)) and then integrating from tk to tk+1, we get

Çtk+1 Çtk+1 1 rtk+2 1

Jtk J s p(tt>dt + bk+1 Jtk+1 p(tt>dt + + bk+1 bk+2 • • • bk+n J tk.

< ftk+1 x'(s) ds _ p " *

Jik '(x(s)) s Jx(i+) '(u)

p tk+n+l

p(t)dt ■

Since (4.88) holds, the above inequality yields

TO p tk+1 ptk+1 1 ptk+2

X p(t)dt + — p(t)dt

k_0J tk J s bk+1 J tk+1

p tk+n+l tk+n

TO p x(tk+1) 1

< k_0Jx(t+) i(u)du < Jx(t+) i(u)

bk+1bk+2 •••bk+n J tk-1

p(t)dt -

(4.93)

(4.94)

The above inequality and the hypotheses lead to a contradiction. So, every solution of (4.73) is oscillatory. □

The proof of the following theorem is similar to that of Theorem 4.31.

Theorem 4.32. Assume that conditions (i), (ii), and (iii) of Lemma 4.25 hold, and there exists a positive integer k0 such that a*k > 1 for k > k0. Suppose that '(ab) > '(a)'(b) for any ab> 0 and

(±TO du ,

—— < to, for some e > 0,

TO tk+1 tk+1 *

k_0J tk J s

TO ftk+1 ftk+1 ml a * I ftk+2

* p(t)dt+'p^ p(t)dt

k_0J tk J s bk+1 Jtk+1

'K+1 MO (tk+3 p(t)dtH tk+2

(4.95)

bk+1bk+1

y(al+1) vK+2) •••'(ak+n)

bk+1bk+1 • ••bk+n

ptk+n+l

p(t)dt

Then, every solution of (4.73) is oscillatory.

In [36], the author studied the second-order nonlinear impulsive differential equations of the following form:

(r (t)f(x)ya(x')) ' + p(t)f(x)ya(x') + q(t)f (x) = 0, t /6i;

(4.96)

A( rf(x)ya(x')) + qif (x) = 0, t = dt,

where ya(s) := |s|a-1s and Az(t) := z(t+) - z(t-), z(t±) = limT^t±z(t).

Sufficient conditions are obtained for oscillation of (4.96) by using integral averaging technique. In particular the Philos type oscillation criteria are extended to impulsive differential equations.

It is assumed that

(i) a> 0 is a constant;

(ii) {0t} is a strictly increasing unbounded sequence of real numbers; {qt} is a real sequence;

(iii) r,p,q e PLC (J ), r (t) > 0;

(iv) f e C(R), f e C1 (R) with f (s) > 0, sf (s) > 0 for s = 0 and

f'(s) > K

-Ka (4.97)

(f(s)\f (s)|a-1)

is satisfied; Ka> 0 is a constant. In order to prove the results the following well-known inequality is needed [26]. Lemma 4.33. If A, B are nonnegative numbers, then

AY - jABY-1 + (y - 1)BY > 0, y> 1, (4.98)

and the equality holds if and only if A = B.

The following theorem is one of the main results of this study.

Theorem 4.34 (see [36]). Let D0 = {(t,s) : t > s > t0} and D = {(t,s) : t > s > t0}. Assume that H(t,s) e C1(D, (0, <x>)), h(t,s) e C(D0, R) satisfy the following conditions.

(i) H(t, t) = 0 for t > t0 and H(t, s) > 0 on D0.

(ii) H has a continuous and nonpositive partial derivative on D0 with respect to the second variable.

(iii) One has

(trs) + p(s) H (t, s) = h(t, s)Ha + 1 (t,s), (t, s) e D0. (499)

os r (s)

limsuo 1 J I" \h(t,s)q(s) - rar(s)\h(t, s)\a+1]ds + V H(t,di)qA = to, (4.100) t^ h(t,t0) [)t0v J t0<e!<t J

where ra = (a + 1y(a+1)aaK-a, then (4.96) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of (4.96). We assume that x(t) / 0 on [T, to) for some sufficiently large T > t0. Define

=rmwtpM t > t. (4.101)

Differentiating (4.101) and making use of (4.96) and (4.97), we obtain

w'(t) < -pf w(t) - q(t) - Ka t /0, (4102)

Aw(t) = -qi, t = 0i. (4.103)

Replacing t by s in (4.102) and multiplying the resulting equation by H(t,s) and integrating from T to t, we get

f H(t,s)q(s)ds <- f H(t,s)w(s)ds Jt Jt r(s)

ft iws) \1+1/a ft

- Ka\ H(t,s) -ds - H(t,s)w'(s)ds.

Jt r1/a(s) Jt

(4.104)

Integrating by parts and using (4.103), we find

I" H(t,s)w'(s)ds = I"

A A 7-T

— {H(t,s)w(s)}- w(s) — (t,s)

f t dH

= -H(t,T)w(T)+ V H(t,0i)qi - w(s) — (t,s)ds.

T<0<t JT ds

(4.105)

Combining (4.104) and (4.105), we obtain f H(t,s)q(s)ds + V H(t/di)qi

JT T <0i<t

ft f |w(s)l1+1/a < H(t,T)w(T) KaH(t,s)wkasr - \h(t,s)\Ha/(a+1}(t,s)\w(s)\

Using inequality (4.98) with j = 1 + 1/a,

(K H)a/(a+1)\w\ / a \«/ r \a/(a+1

A = (KaH)-\w\ B =/ h\a (4.107)

A r1/(a+1) ' B \a +1) \KaJ |h| ' V ^

we find

W(s) 11+1 /a

KaH(t,s) , - \h(t,s)\Ha/(a+1^(t,s)\w(s)\ > -rar(s)\h(t/s)\a+1. (4.108)

r 1/a(s)

From (4.106) and (4.108), we obtain C t

\h(t,s)q(s) - Yar(s)\h(t,s)\a+!]ds + £ H(t,9l)ql Jt r<ei<t (4.109)

< H(t,T)w(T) < H(t,T)\w(T)\ < H(t,t0)\w(T)\

for all t > T > t0. In the above inequality we choose T = T0, to get Ct

[H(t,s)q(s) - Tar(s)\h(t,s)\a+!]ds + £ H(t,9l)ql

'tn t0<0i<t

< H(t,t0)\i°\q(s)\ds + £ |qt|[ + H(t,t0)\w(T0)\.

1^0 t0<ei<T0

Thus, it follows that

t —> to

H (t,t0)

limsupH(TT) \ \ [H(t,s)q(s) - Far(s)\h(t,s)\a+^ds + £ H(t,6i)ql

< \q(s)\ds + ^ \qi\ + \w(T0)\< to,

limsup 1 J | H(t,s)q(s)ds + V H(t,di)qA = to,

t — to H(t,t0) [ Jt0 tof£<t 1

limsup ^ r(s)\h(t,s)\a+1ds < to,

t — to H(t,t0) J t0

then (4.96) is oscillatory.

(4.110)

(4.111)

which contradicts (4.100). This completes the proof. □

As a corollary to Theorem 4.34, we have the following result. Corollary 4.35. Let condition (4.100) in Theorem 4.34 be replaced by

Note that in the special case of half-linear equations, for y(s) = 1 and f (s) = ya(s), the condition (4.97) is satisfied with Ka = a.

The proof of the following theorem can be accomplished by using the method developed for the nonimpulsive case and similar arguments employed in the proof of Theorem 4.34.

Theorem 4.36 (see [36]). Let the functions H and h be defined as in Theorem 4.34. Moreover, Suppose that

0 < infilim inf H<œ,

s>fo [ tsœ H(t,t0) J

(4.113)

lim sup r (s)\h(t,s)\a+1ds < œ.

t sœ H(t,t0) J t0

If there exists a function A e C(J, R) such that

ft 41+1/«(s)

''ms^l ds =œ (4114)

and for every T > t0

lim.SœUP Hit/Tj

H(t, s)q(s) - rar(s)\h(t,s)\a+1] ds + £ H(t,di)qA > A(T), (4.115)

t <ei<t

where A+(s) = max{A(s),0}, then (4.96) is oscillatory.

4.4. Higher-Order Nonlinear Equations

There are only a very few works concerning the oscillation of higher-order nonlinear impulsive differential equations [37-40].

In [37] authors considered even order impulsive differential equations of the following

+ f (t, x) = 0, t /tk,

x(i) (t+) = gk(i)(x(i)(tk)),

i = 0,1,

,2n- 1, ke N,

(4.116)

x(i) t0+ = x

Advances in Difference Equations where

XHtk) = lim ^ - h) - ),

h ^ 0- h

(4.117)

¿CO = lim X('-1)('k - h) -«

N 7, , (V

h 0- h

Let the following conditions hold.

(A) f (t,x) e C(J x R); xf (t,x) > 0 for x / 0; f (t,x)/y(x) > p(t) for x = 0, where p(t) is positive and continuous on J; xy(x) > 0 for xf 0; y'(x) > 0.

(B) gk(i) e C(R), and there exist positive numbers a^, bk, i = 0,1,...,2n - 1 such that

(i) , gk(i)(x) , ak < x < bk(i),

(i) (i) (i) (i) (i) (i) ax ai a2 ai a2 ••• a^^

(ti -10)+b^(t2 - ti)+b(i-i)b2i-i)(t3 -12)+•••+h(-i) b«-i) • • • bi-i)(tm+i - tm)+•••=

(4.118)

A function x : [t0,t0 + a) ^ R, t0 > 0, a > 0 is said to be a solution of (4.116), if (i) x(i)(t++) = x(li), i = 0,1,...,2n - 1; (ii) for t e [t0,t0 + a) and t/tk, x(t) satisfies x(2n)(t) + f (t,x(t)) = 0; (iii) x(i)(t) is left continuous on tk e [to, to + a), and x(i)(t+) = gk(i)x(i) (tk), i = 0,1,...,2n -1.

The first two theorems can be considered as modifications of Theorems 3.43 and 4.26, respectively.

Theorem 4.37 (see [37]). If conditions (A) and (B) hold, b® < 1, af) > 1, b{°} > 1 (i = 1,2,...,2n - 1; k = 1,2,...), and if

t2n-1p(t)dt = to, (4.119)

then every bounded solution of (4.116) is oscillatory.

Theorem 4.38 (see [37]). If conditions (A) and (B) hold, af) > 1 and

ft1 1 ft2 1 ft3 Lpm + o-i l„pm'+b^n^J ,p(t^

v b^" h, b^"bf " Jh

b(2n-1)b(2n-1) b(2n-1) j b, b2 ■ ■ ■ &m '

p(t)dt ■

■ ■ ■ = TO,

then every solution of (4.116) is oscillatory.

Theorem 4.39 (see [37]). If conditions (A) and (B) hold, Uk=iak) >b> 0 (m = 12,...), bfn-1) < 1, and for any 6 > 0,

inf f(t,x)dt

J 6<|x|<o

then every solution of (4.116) is oscillatory.

Corollary 4.40. Assume that conditions (A) and (B) hold, and that a(0) > 1, b(2n-1) < 1. If

(4.121)

¡•œ

p(t)dt = o, (4.122)

then every solution of (4.116) is oscillatory.

Corollary 4.41. Assume that conditions (A) and (B) hold, and that there exists a positive number a > 0, such that a(°) > 1,1/b[2n-1) > (tk+\/tk)a. If

/•CO

tap(t)dt = o, (4.123)

then every solution of (4.116) is oscillatory.

Example 4.42. Consider the impulsive differential equation:

x(2n) + —x3 = 0, t > -, t = k, k e N,

x(k+) = x(k), x(i)(k+) = x(i)(k), i = 1,...,2n - 1, (4.124)

1 ^ i)( 1 \ =

x( 2J = xo, xw ^J = x0',

where af = bf) = (k + 1)/k > 1, af = bf = 1, i = 1,2,...,2n - 1, p(t) = 1/4t, y(x) = x3, f (t,x) = (1/4t)x3, tk = k, to = 1/2.

It is easy to verify conditions of Theorem 4.38. So every solution of (4.124) is oscillatory.

Example 4.43. Consider the impulsive differential equation:

x(2n) + -x1/3 = 0, t >-, t = k, k e N, 12 - 2' T

x(k+) = x(k), x(i)(k+) = k+1 x(i)(k), i = 1,...,2n - 1, (4.125)

1 ^ i)( 1 \ =

x 2J = x0, x = x0

where a^ = b(0) = 1, a= b= k/(k + 1), i = 1,2,...,2n - 1, p(t) = 1/t2, tk = k, q>(x) = x1/3, f (t,x(t)) = (1/t2)x1/3(t), to = 1/2.

In this case, it can be show that conditions of Corollary 4.41 are satisfied. Thus, every solution of (4.125) is oscillatory.

In [40], the authors considered the impulsive differential equations with piecewise constant argument of the following form:

x(m)(t) + a(t)(x(r)([t]))2P + e(t) sign x(t) = 0, t / n, t > 0, n e Z+

x(j)(n) = ßd)x(i) (n),

(4.126)

x(j)([ta]) = xj,

where Z+ is the set of all positive integers, m > 1, j and p are given positive constants, [■] denotes the set of maximum integers, and xj sp^ > 0 for all n,j. It is assumed that

(i) a,e e C(J,R+);

(ii) for any n> 0 and all i = 1,2,...,m,

a(m-i) ßn+1 Mm-i-1) ßn+1

ß(m-i) ß(m-i) ßn+2 ßn+1 ß(m-i-1) ß(m-i-1) ßn+2 ßn+1

nj-1 ß(m-i)

1 !s=1ßn+1

nj-1 ß(m-i-1)

s=1ßn+s

<X>, j

(4.127)

(iii) there exists C > 0 such that

inf et) > c

t>0 a(t)

(4.128)

Theorem 4.44 (see [40]). Assume that conditions (i), (ii), and (iii) hold. Moreover, suppose that for any n> 0, there exists a j > 0 such that

f a(s)ds+(ßim-»)-,i f>*+A* ds+...

Jn Jn+1 An+1 Jn+2 An+2

(4.129)

■j-t (a (m-rn-1 ( n+j a(s)

TT (a(m-,)\-T r+j a(s) ^

l(ßn+s) —t~ds j

s=1X jn+j-1

where An = n!/k!. Then every solution of (4.126) is oscillatory.

Theorem 4.45 (see [40]). Assume that conditions (i) and (ii) hold. Moreover, suppose that for any n> 0, there exists a j > 0 such that

n i rn+l n+1 i z^n+2

Tl(p{m-1)) e(s)ds + H(^V e(s)ds + ■■■

s=1 Jn s=1 n+1

(4.130)

n+j-1 1 fn+j

+ E[ e(s)ds>x0r-1).

s=1 n+j-1

Then every solution of (4.126) is oscillatory.

Example 4.46. Consider the impulsive differential equation:

x(m) + ln(1 + t)(x(m-1)([t]))2 + e- signx(t) = 0, t/n, t > 0, n e Z+,

xj)(n) = ( 3) x(j)(n-), (4.131)

x(j) (0) = 2.

It is easy to verify that conditions of Theorem 4.45 are all satisfied. Therefore every solution of (4.131) is oscillatory.

Acknowledgments

This work was done when the second author was on academic leave, visiting Florida Institute of Technology. The financial support of The Scientific and Technological Research Council of Turkey (TUBITAK) is gratefully acknowledged.

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