Scholarly article on topic 'On a multi point boundary value problem for a fractional order differential inclusion'

On a multi point boundary value problem for a fractional order differential inclusion Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — Aurelian Cernea

Abstract The existence of solutions for a multi point boundary value problem of a fractional order differential inclusion is investigated. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

Academic research paper on topic "On a multi point boundary value problem for a fractional order differential inclusion"

Arab Journal of Mathematical Sciences

Arab J Math Sci 19(1) (2013), 73-83

On a multi point boundary value problem for a fractional order differential inclusion

Aurelian Cernea

Faculty of Mathematics and Informatics, University of Bucharest, Academiei 14, 010014 Bucharest, Romania

Abstract. The existence of solutions for a multi point boundary value problem of a fractional order differential inclusion is investigated. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

Mathematics subject classification: 34A60; 34B18; 34B15

Keywords: Fractional derivative; Differential inclusion; Boundary value problem; Fixed point

Differential equations with fractional order have recently proved to be strong tools in the modeling of many physical phenomena; for a good bibliography on this topic we refer to [18]. As a consequence there was an intensive development of the theory of differential equations of fractional order [2,16,22] etc.. The study of fractional differential inclusions was initiated by El-Sayed and Ibrahim [13]. Very recently several qualitative results for fractional differential inclusions were obtained in [1,3,6-11,15,20] etc.

In this paper we study the following problem

Received 4 June 2012; accepted 12 July 2012 Available online 21 July 2012

1. Introduction

Dax(t) 2 F(t, x(t), x'{t)) a.e. [0, 1]

Tel.: +40 785810358.

E-mail address: acernea@fmi.unibuc.ro

Peer review under responsibility of King Saud University.

Production and hosting by Elsevier

ELSEVIER

1319-5166 © 2012 King Saud University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.Org/10.1016/j.ajmsc.2012.07.001

where Da is the standard Riemann-Liouville fractional derivative, a e(2,3], m P 1, 0 < 6 < ••• < ím < 1, Et^r1 < 1, k > 0, a, > 0, i = TTm and F: [0, 1] x R x R ! P(R) is a set-valued map.

The present paper is motivated by a recent paper of Nyamoradi [19], where it is considered problem (1.1) and (1.2) with F single valued and several existence results are provided.

The aim of our paper is to extend the study in [19] to the set-valued framework and to present some existence results for problem (1.1) and (1.2). Our results are essentially based on a nonlinear alternative of Leray-Schauder type, on Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values and on Covitz and Nadler set-valued contraction principle. The methods used are known ([1,8,9] etc.), however their exposition in the framework of problem (1.1) and (1.2) is new.

The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the sequel and in Section 3 we prove our main results.

2. Preliminaries

In this section we sum up some basic facts that we are going to use later.

Let (X, d) be a metric space with the corresponding norm I and let I c R be a compact interval. Denoted by L(I) the r-algebra of all Lebesgue measurable subsets of I, by P(X) the family of all nonempty subsets of X and by B(X) the family of all Borel subsets of X. If A c I then vA: I fi {0,1} denotes the characteristic function of A. For any subset A c X we denote by A the closure of A.

Recall that the Pompeiu-Hausdorff distance of the closed subsets A,B c X is defined

dH(A, B) = max{d*(A, B), d*(B, A)}, d*(A, B) = sup{d(a, B); a e A},

where d(x,B) = infyeBd(x,y).

As usual, we denote by C(I,X) the Banach space of all continuous functions x: I fi X endowed with the norm I x| C = supteI x(t)| and by L1(I,X) the Banach space of all (Bochner) integrable functions x: I fi X endowed with the norm |x|1 = J |x(t)|dt.

A subset D c L1(I,X) is said to be decomposable if for any m,v e D and any subset A e L(I) one has + vvB e D, where B = T\A.

Consider T: X ! P(X) a set-valued map. A point x e X is called a fixed point for T if x e T(x). T is said to be bounded on bounded sets if T(B):= UxeBT(x) is a bounded subset of X for all bounded sets B in X. T is said to be compact if T(B) is relatively compact for any bounded sets B in X. T is said to be totally compact if T(X) is a compact subset of X. T is said to be upper semicontinuous if for any open set D c X, the set {x e X: T(x) c D} is open in X. T is called completely continuous if it is upper semicon-tinuous and totally bounded on X.

It is well known that a compact set-valued map T with nonempty compact values is upper semicontinuous if and only if T has a closed graph.

We recall the following nonlinear alternative of Leray-Schauder type and its consequences.

Theorem 2.1 [21]. Let D and D be open and closed subsets in a normed linear space X such that 0 2 D and let T: D ! P(X) be a completely continuous set-valued map with compact convex values. Then either

(i) the inclusion x 2 T(x) has a solution, or

(ii) there exists x 2 OD (the boundary of D) such that kx 2 T(x) for some k > 1.

Corollary 2.2. Let Br(0) and Br(0) be the open and closed balls in a normed linear space X centered at the origin and of radius r and let T: Br(0) ! P(X) be a completely continuous set-valued map with compact convex values. Then either

(i) the inclusion x 2 T(x) has a solution, or

(ii) there exists x 2 X with I x| = r and kx 2 T(x) for some k > 1.

Corollary 2.3. Let Br(0) and Br(0) be the open and closed balls in a normed linear space X centered at the origin and of radius r and let T : Br(0)! X be a completely continuous single valued map with compact convex values. Then either

(i) the equation x = T(x) has a solution, or

(ii) there exists x 2 X with I x| = r and x = kT(x) for some k < 1.

We recall that a multifunction T : X ! P(X) is said to be lower semicontinuous if for any closed subset C c X, the subset {s 2 X: T(s) c C} is closed.

If F : I x R x R ! P(R) is a set-valued map with compact values and x 2 C(I,R) we define

Sf(x) = ff 2 L1(I, R) : f(t) 2 F(t, x(t), x'(t)) a.e. I}.

We say that F is of lower semicontinuous type if SF(.) is lower semicontinuous with closed and decomposable values.

Theorem 2.4 [4]. Let S be a separable metric space and G : S ! P(L1 (I, R)) be a lower semicontinuous set-valued map with closed decomposable values.

Then G has a continuous selection (i.e., there exists a continuous mapping g: S fi L1(I,R) such that g(s) 2 G(s) "s 2 S).

A set-valued map G : I! P(R) with nonempty compact convex values is said to be measurable if for any x 2 R the function t fi d(x,G(t)) is measurable.

A set-valued map F : I x R x R ! P(R) is said to be Caratheodory if t fi F(t,x,y) is measurable for all x,y 2 R and (x,y) fi F(t,x,y) is upper semicontinuous for almost all t 2 I.

F is said to be L1-Caratheodory if for any l > 0 there exists hi 2 L1(I,R) such that sup{ vl : v 2 F(t,x,y)} 6 h(t) a.e. I, 8x, y 2 Bl(0).

Theorem 2.5 [17]. Let Xbe a Banach space, let F : I x X ! P(X) be a L1-Caratheodory set-valued map with SF „ 0 and let f: L1 (I,X) fi C(I,X) be a linear continuous mapping.

Then the set-va/wed map C o SF : C(/, X) ! P(C(/, X)) defined by (r o Sf)(x) = T(Sf(x))

has compact convex va/wes and has a c/osed graph in C(/,X) X C(/,X).

Note that if dimX < 1 , and F is as in Theorem 2.5, then SF(x) „ 0 for any x € C(/,X)

Consider a set valued map T on X with nonempty values in X. T is said to be a k-contraction if there exists 0 < k <1 such that

dff(T(x), T(y)) 6 kd(x, y) 8x, y e X.

The set-valued contraction principle [12] states that if X is complete, and T: X ! P(X) is a set valued contraction with nonempty closed values, then T has a fixed point, i.e. a point z e X such that z e T(z).

Definition 2.6.

(a) The fractional integral of order a >0 of a Lebesgue integrable function f: (0, i) fi R is defined by

provided the right-hand side is pointwise defined on (0,1) and C is the (Euler's) Gamma function defined by C(a) = J0°° ta_1e~'dt. (b) The Riemann-LioMvi//e fractiona/ derivative of order a >0 of a continuous function f: (0,1) fi R is defined by

where n = [a] + 1, provided the right-hand side is pointwise defined on (0, i).By AC:([0,1],R) we denote the space of continuous real-valued functions whose first derivative exists and it is absolutely continuous on I. On AC1([0,1],R) we consider the norm

||x|| = max{ sup |x(t)|, sup |x'(t)|}.

Definition 2.7. A function x 2 AC1([0,1],R) is called a solution of problem (1.1) and (1.2) if there exists a function v G L1([0,1],R) with v(t) 2 F(t,x(t),x'(t)), a.e. [0,1] such that Dax(t) = v(t), a.e. [0,1] and conditions (1.2) are satisfied.

In what follows I = [0,1], a 2 (2,3], and A = E^nr1 2 (0, 1). Next we need the following technical result proved in [19].

Lemma 2.8 19. For any h 2 L1(I,R) the problem

Dax(t) = h(t) a.e. [0, 1],

(e.g., [17]).

í6[0,1

í6[0,1

x(0) = x'(0) = 0, x(1)^aix(ni) = k

has a unique solution given by

ta-i i-1 1 m r i

kt™ I t™ m i'I

x(t)=I—T + G(t,s)h{s)ds + -—^^ / G(£i,s)h(s)ds, t 2 [0, I], <j 0 i=i J 0

G{t s) ,= [t(I - s)]™-1 -(t - s)™-\ if 0 6 s < t 6 I,

1 ' ' r(ai)\ [t(I - s)]™-1, if 0 6 t < s 6 I.

Note that G(t,s) > 0 "t,s 2 I and G(t, s) 6 f^y, (e.g., Lemma 5 in [ 19]). If we denote Gi(t,s) = G{t,s) + Ph a-DG(t„s) one has \Gi{t,s)| 6 jfe (1 + and

|@Gi (t 6 2(az1l{1 + ZXiOi)

I @t (t; s)l 6 r(a) I 1 + 1-a j.

Let K1\=suptjs2^ Gi(t,s)\ and K2 := supt,s2/\ ^ (t, s)\. Finally, we denote z(t) = f^ and C1:=supt2I\\z(t)\\.

3. The main results

Now we are able to present the existence results for problem (1.1) and (1.2). We consider first the case when F is convex valued.

Hypothesis 3.1.

(i) F : I x R x R ! P(R) has nonempty compact convex values and is Caratheodory.

(ii) There exist u 2 L1(I,R) with u (t) > 0 a.e. I and there exists a nondecreasing function W:[0,1) fi (0,1) such that

sup{\v\, v 2 F(t, x,y)} 6 u(t)W(max{\x\, \y\}) a.e. I, 8x,y 2 R.

Theorem 3.2. Assume that Hypothesis 3.1 is satisfied and there exists r > 0 such that r > C1 + max{K1,K2}\u\1W(r). (3.1)

Then problem (1.1) and (1.2) has at least one solution x such that \\x\\ < r.

Proof. Let X = AC1(I,R) and consider r > 0 as in (3.1). It is obvious that the existence of solutions to problem (1.1) and (1.2) reduces to the existence of the solutions of the integral inclusion

x(t)2z(t)+[ G1 (t, s)F(s, x(s), x'(s))ds, t 2 I. (3.2)

Consider the set-valued map T: Br(0) ! P(AC1(/, R)) defined by

T(x) :=j v 2 AC1 (I, R); v(t)= z(t) +jf G(t, s)/(s)ds, / 2 SF(x)J. (3.3) We show that T satisfies the hypotheses of Corollary 2.2.

First, we show that T(x) c AC1(/,R) is convex for any x 2 AC1(/,R). If v1,v2 2 T(x) then there exist /1/2 2 SF(x) such that for any t 2 I one has

v,(0=z(t)+/ G1(t, s)/-(s)ds, i = 1,2. Jo

Let 0 6 a 6 1. Then for any t 2 I we have

(av1 + (1 - a)v,)(t)=z(t) + [ G1(t,s)[a/(s) + (1 - a)/,(s)]ds.

The values of F are convex, thus SFx) is a convex set and hence

av1 + (1 — a)v2 2 T(x).

Second, we show that T is bounded on bounded sets of AC1(/,R). Let B c AC1(/,R) be a bounded set. Then there exists m >0 such that ||x|| 6 m "x 2 B. If v 2 T(x) there exists /2 SF(x) such that v(t) = jJ G1(t, s)/(s)ds. One may write for any t 2 I

|v(t)| 6 |z(t)|+ f |G1(t,s)| • f(s)|ds Jo

6 |z(t)|+ f |G1(t,s)|u(s)W(max{|x(s)|, |x'(s)|})ds. Jo

On the other hand,

n1 , @G1

I (t, s)H

/»l -G

6 |z'(t)|+/ I"G(t,s)|u(s)W(max{|x(s)|, |x'(s)|})ds.

and therefore

|vk=niax{|v(t)|, |v'(t)|}

fi — g

6 maxmax{|z(t)|, |z'(t)|}+ / max{|Gi(t,s)|, |--1 teJ _/0 í,se/ — t

x(t,s)|}u(s)W(max{|x(s)|, |x'(s)|})ds 6 C1 + max{K1,K,}|u|1W(m) "v 2 T(x), i.e., T(B) is bounded.

We show next that T maps bounded sets into equi-continuous sets. Let B c AC1(I,R) be a bounded set as before and v 2 T(x) for some x 2 B. There exists f 2 SF(x) such that v(t) = z(t) + jJ G1 (t, s)f(s)ds. Then for any t,s 2 I we have

\v(t)- v(s)\ 6 \z(t)-z(s)\ + \ f1 G1(t, s)f(s)ds - f G1 (s, s)f(s)ds\

6 \z(t)-z(s)\+ f \G1(t, s) Jo

- G1(s,s)\u(s)W(max{\x(s)\, \x'(s)\})ds

6 \z(t)-z(s)\+ f \G1(t, s)-G1(s, s)\u(s)W(m)ds. J0

Similarly, we have

/1 —G —G

\ -G- (t, s)--G (s, s)\u(s)W(m)ds.

It follows that \ v(t) - v(s)\ fi 0 as t fi s . Therefore, T(B) is an equi-continuous set in AC1(I,R). We apply now Arzela-Ascoli's theorem we deduce that T is completely continuous on AC1(I,R).

In the next step of the proof we prove that T has a closed graph. Let xn 2 AC1(I,R) be a sequence such that xn fi x and vn 2 T(xn) "n 2 N such that vn fi v . We prove that v 2 T(x ). Since vn 2 T(xn), there exists fn 2 SF(xn) such that vn(t) = z(t)+ /1 G1(t,s)fn(s)ds. Define f L1(I,R) fi AC1(I,R) by (rf))(t) := /J G1(t,s)f(s)ds. One has

niax{\v„(t)-z(t)-(v*(t)-z(t))\,\v'n(t)-z' (t)-((v*)'(t)-z' (t))\ = niax{\v„(t)- v*(t)\,\v'n(t)-(v*)'(t)\} = kv„ - v*\\ ! 0 as n fi 1.

We apply Theorem 2.5 to find that roSF has closed graph and from the definition of r we get vn 2 ro SF(xn). Since xn fi x*, vn fi v* it follows the existence of f 2 SF(x*) such that v* (t)- z(t) = f! G1(t, s)f*(s)ds. Therefore, T is upper semicontinuous and compact on Br (0).

We apply Corollary 2.2 to deduce that either (i) the inclusion x 2 T(x) has a solution in Br(0), or (ii) there exists x 2 X with \\x\\ = r and kx 2 T(x) for some k >1.

Assume that (ii) is true. With the same arguments as in the second step of our proof we get r = \\x\\ 6 C1 + max{K1,K2}\ u\ 1W (r) which contradicts (3.1). Hence only (i) is valid and theorem is proved.

We consider now the case when F is not necessarily convex valued. Our first existence result in this case is based on the Leray-Schauder alternative for single valued maps and on Bressan Colombo selection theorem. □

Hypothesis 3.3.

(i) F : I x R x R ! P(R) has compact values, F is L(I)< B(R)< B(R) measurable and (x,y) fi F(t,x,y) is lower semicontinuous for almost all t 2 I.

(ii) There exist u 2 L1(I,R) with u (t) > 0 a.e. I and there exists a nondecreasing function W :[0,1) fi (0,1) such that

sup{|v|, v 2 F(t, x,y)} 6 u(t)W(max{|x|, |y|}) a.e. I, Vx,y 2 R.

Theorem 3.4. Assume that Hypothesis 3.3 is satisfied and there exists r > 0 sMch that conditio« (3.1) is satisfied. Then problem (1.1) and (1.2) has at least one solMtion on I.

Proof. We note first that if Hypothesis 3.3 is satisfied then F is of lower semicontinuous type (e.g., [14]). Therefore, we apply Theorem 2.4 to deduce that there exists f: AC1(I,R) fi L1(I,R) such that fx) 2 SF(x) "x 2 AC1(I,R).

We consider the corresponding problem

x(t)=z(t)+/ G1(t,s)f(x(s))ds, t 2 I (3.4)

in the space X = AC1(I,R). It is clear that if x 2 AC1(I,R) is a solution of the problem (3.4) then x is a solution to problem (1.1) and (1.2).

Let r >0 that satisfies the condition (3.1) and define the set-valued map T : Br(0)!p(AC!(I, R)) by

(T(x))(t) := z(t)+ / G1(t,s)f(x(s))ds. Jo

Obviously, the integral Eq. (3.4) is equivalent with the operator equation

x(t) = (T(x))(t), t 2 I. (3.5)

It remains to show that T satisfies the hypotheses of Corollary 2.3.

We show that T is continuous on Br(0). From Hypotheses 3.3. (ii) we have

|f(x(t))| 6 u(t)W(max{|x(t)|, |x'(t)|}) a.e. I

for all x 2 AC1(I,R). Let xn, x 2 Br(0) such that xn fi x. Then

f(x„(t))| 6 u(t)W(r) a.e. I.

From Lebesgue's dominated convergence theorem and the continuity off we obtain, for all t 2 I

lim (T(x„))(t)= z(t) + lim / G1(t,s)f(x„(s))ds = z(t)+ / G1(t,s)f(x(s))ds

n!1 o Jo

= (T(x))(t)

f i —G

lim (r(x„))'(t) = z'(t) + lim / —1 (t,s)/(x„(s))ds

n!1 J 0 —t

/»1 —g

= Z(t)+jf —G (t, s)f(x(s))ds =(T(x))'(t)

i.e., T is continuous on Br(0).

Repeating the arguments in the proof of Theorem 3.2 with corresponding modifications it follows that T is compact on Br(0). We apply Corollary 2.3 and we find that either (i) the equation x = T(x) has a solution in Br(0), or (ii) there exists x 2 X with ||x|| = r and x = kT(x) for some k <1.

As in the proof of Theorem 3.2 if the statement (ii) holds true, then we obtain a contradiction to (3.1). Thus only the statement (i) is true and problem (1.1) has a solution x 2 AC1(I,R) with ||x|| < r.

In order to obtain an existence result for problem (1.1) and (1.2) by using the set-valued contraction principle we introduce the following hypothesis on F. □

Hypothesis 3.5.

(i) F : I x R x R ! P(R) has nonempty compact values, is integrably bounded and for every x,y 2 R, F(.,x,y) is measurable.

(ii) There exists l1,l2 2 L1(I,R+) such that for almost all t 2 I,

dff(F(t,x1,yj,F(t,x2,y2)) 6 l1(t)|x1 - x2|+l2(t)|y! -y2| "x1,x2,y1,y2 2 R.

Theorem 3.6. Assume that Hypothesis 3.5. is satisfied and (I l7| 7 + I l2I 7)-max{K7,K2} < J. Then problem (1.1) and (1.2) has a solution.

Proof. We transform the problem (1.1) and (1.2) into a fixed point problem. Consider the set-valued map T: AC1 (I, R) ! p(aC1(I, R)) defined by

T(x) := {v 2 AC1 (I,R); v(t) = z(t)+ / G^t,s)f(s)ds, f 2 Sf(x)}.

Note that since the set-valued map F(.,x(.)) is measurable with the measurable selection theorem (e.g., Theorem III. 6 in [5]) it admits a measurable selection f: I fi R. Moreover, since F is integrably bounded, f 2 L1(I,R). Therefore, SF,x „ 0.

It is clear that the fixed points of T are solutions of problem (1.1) and (1.2). We shall prove that T fulfills the assumptions of Covitz Nadler contraction principle.

First, we note that since SF,x „ 0, T(x) „ 0 for any x 2 AC1(I,R).

Second, we prove that T(x) is closed for any x 2 AC1(I,R). Let {xn} np0 2 T(x) such that xn fi x* in AC1(I,R). Then x* 2 AC1(I,R) and there exists fn 2 SFx such that

xn(t)=z(t) + G1(t, s)fn(s)ds. J0

Since F has compact values and Hypothesis 3.5 is satisfied we may pass to a subsequence (if necessary) to get that fn converges to f 2 L1(I,R) in L1(I,R). In particular, f2 SF,x and for any t 2 I we have

xn(t) ! x*(t) = z(t)+ i G1(t, s)f(s)ds, J0

i.e., x* 2 T(x) and T(x) is closed.

Finally, we show that T is a contraction on AC1(I,R). Let x1,x2 2 AC1(I,R) and v1 2 T(x1). Then there exist f1 2 SF,x1 such that

v1 (t) = z(t)+ i G(t, s)f (s)ds, t 2 I. J0

Consider the set-valued map

H(t) := F(t, x2(t), x'2(t)) \{x 2 R; f (t)-x\

6 l1(t)\x1(t)- x2(t)\ + h(t)\x1(t)- x2(t)\}, t 2 I. From Hypothesis 3.5 one has

dH(F(t, x1(t), x1 (t)), F(t, x2(t), x2(t))) 6 l1(t)\x1(t)- x2(t)\+ l2(t)\x!1 (t)-x,2(t)\,

hence H has nonempty closed values. Moreover, since H is measurable, there exists f2 a measurable selection of H. It follows that f2 2 SF;x2 and for any t 2 I

\f1(t)-f2(t)\ 6 h(t)\x1(t)-x2(t)\ + l2(t)\x\(t)- x'2(t)\.

Define

v2(t)=z(t)+ ( G1 (t,s)f2(s)ds, t 2 I J0

and we have

\v 1 (t) v2(t)\ 6 f \G1(t, s)\-f1(s)-f2(s)\ds J0

6 f G1(t,s)[l1(s)\x1(s)-x2(s)\+ l2(s)\x[(s)- x2(s)\]ds J0

6 K1(\l1\1 + \l2\1)\\x1 - x2\. Similarly, we have

\v[(t)-v2 (t)\ 6 K2(\l1\1 + \l2\1)\x1 - x2\\.

So, |v1 — v2|| 6 (I lj 1 + I l2I 1)max{K1,K2}||x1 — x2||. From an analogous reasoning by interchanging the roles of x1 and x2 it follows

dH(T(x1), T(x2)) 6 (|l111 I 11211) max{K1, K2}kx1 — x2||.

Therefore, T admits a fixed point which is a solution to problem (1.1) and (1.2). □

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