Hussainetal. Fixed Point Theory and Applications (2015) 2015:78 DOI 10.1186/s13663-015-0331-4

0 Fixed Point Theory and Applications

a SpringerOpen Journal

RESEARCH

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Coincidence point theorems for generalized contractions with application to integral equations

Nawab Hussain1, Jamshaid Ahmad2, Ljubomir Ciric3* and Akbar Azam2

CrossMark

"Correspondence: lciric@rcub.bg.ac.rs 3Faculty of MechanicalEngineering, University of Belgrade, Kraljice Marije 16, Belgrade, 11 000, Serbia Fulllist of author information is available at the end of the article

Abstract

In this article, we introduce a new type of contraction and prove certain coincidence point theorems which generalize some known results in this area. As an application, we derive some new fixed point theorems for F-contractions. The article also includes an example which shows the validity of our main result and an application in which we prove an existence and uniqueness of a solution for a general class of Fredholm integral equations of the second kind.

MSC: 46S40; 47H10; 54H25

Keywords: coincidence point; F-contractions; integral equations

1 Introduction and preliminaries

The Banach contraction principle [1] is one of the earliest and most important results in fixed point theory. Because of its application in many disciplines such as computer science, chemistry, biology, physics, and many branches of mathematics, a lot of authors have improved, generalized, and extended this classical result in nonlinear analysis; see, e.g., [2-10] and the references therein. In 2012, Azam [3] obtained the existence of a coincidence point of a mapping and a relation under a contractive condition in the context of metric space. For coincidence point results see also [11]. Consistent with Azam, we begin with some basic known definitions and results which will be used in the sequel. Throughout this article, N, R+, R denote the set of all natural numbers, the set of all positive real numbers, and the set of all real numbers, respectively.

Let A and B be arbitrary nonempty sets. A relation R from A to B is a subset of A x B and is denoted R : A ^ B. The statement (x, y) e R is read 'x is R-related to y, and is denoted xRy. A relation R : A ^ B is called left-total if for all x e A there exists a y e B such that xRy, that is, R is a multivalued function. A relation R: A ^ B is called right-total if for all y e B there exists an x e A such that xRy. A relation R : A ^ B is known as functional, if xRy, xRz implies that y = z,for x e A and y, z e B. A mapping T: A ^ B is a relation from A to B which is both functional and left-total.

For R : A ^ B, E c A we define

R(E) = {y e B : xRy for some x e E},

ringer

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dom(R) = {x e A : R((x}) = 0},

Range(R) = { y e B : y e R((x}) for some x e dom(R)}.

For convenience, we denote R((x}) by R(x}. The class of relations from A to B is denoted by R(A,B). Thus the collection M(A,B) of all mappings from A to B is a proper subcollection of R(A,B). An element w e A is called a coincidence point of T : A ^ B and R : A ^ B if Tw e R(w}. In the following we always suppose that X is a nonempty set and (Y, d) is a metric space. For R : X ^ Y and u, v e dom(R), we define

D(R(u}, R(v}) = inf d(x,y).

uRx,vRy

Wardowski [12] introduced and studied a new contraction called an F-contraction to prove a fixed point result as a generalization of the Banach contraction principle.

Definition 1 Let F : R+ ^ R be a mapping satisfying the following conditions: (F1) F is strictly increasing;

(F2) for all sequence (an} ç R+, limn^œ an = 0 if and only if limn^œ F(an) = -œ; (F3) there exists 0 < k <1 such that limn^0+ akF(a) = 0.

Consistent with Wardowski [12], we denote by F the set of all functions F : R+ ^ R satisfying the conditions (F1)-(F3).

Definition 2 [12] Let (X, d) be a metric space. A self-mapping T on X is called an F -contraction if there exists t >0 such that for x, y e X

d(Tx, Ty)>0 t + F(d(Tx, Ty)) < F(d(x,y)),

where F e F.

Theorem 3 [12] Let (X, d) be a complete metric space and T:X ^ X be a self-mapping. If there exists t >0 such that for all x, y e X : d(Tx, Ty)>0 implies

t + F(d(Tx, Ty)) < F(d(x,y)),

where F e F, then T has a unique fixed point.

Abbas etal. [13] further generalized the concept of an F-contraction and proved certain fixed and common fixed point results. Hussain and Salimi [14] introduced some new type of contractions called a-GF-contractions and established Suzuki-Wardowski type fixed point theorems for such contractions. For more details on F-contractions, we refer the reader to [11,13-20].

In this paper, we obtain coincidence points of mappings and relations under a new type of contractive condition in a metric space. Moreover, we discuss an illustrative example to highlight the realized improvements.

2 Main results

Now we state and prove the main results of this section.

Theorem4 Let X be a nonempty set and (Y, d) be a metric space. LetT: X ^ Y, R: X ^ Y be such thatR is left-total, Range(T) c Range(R) and Range(T) or Range(R) is complete. If there exist a mapping F: R+ ^ R and t > 0 such that

d(Tx, Ty)> 0 t + F(d(Tx, Ty)) < F(D(R{x}, Rjy})) (2.1)

for all x, y e X, then there exists w e X such that Tw e R{w}.

Proof Let x0 e X be an arbitrary but fixed element. We define the sequences {xn} c X and {yn} c Range(R). Let y1 = Tx0, Range(T) c Range(R). We may choose x1 e X such that x1Ry1, since R is left-total. Let y2 = Tx1, since Range(T) c Range(R). If Tx0 = Tx1, then we have x1Ry2. This implies that x1 is the required point that is Tx1 e R{x1}. So we assume that Tx0 = Tx1, then from (2.1) we get

t + F(d(yl,y2)) = t + F(d(Tx0, Tx 1)) < F(D(R{x0},R{»i})). (2.2)

We may choose x2 e X such that x2Ry2, since R is left-total. Let y3 = Tx2, since Range(T) c Range(R). If Ix1 = Tx2, then we have x2Ry3. This implies that Tx2 e R{x2} and x2 is the coincidence point. So Tx1 = Tx2, then from (2.1), we get

t + F(d(y2,ya)) = t + F(d(Tx1, Tx2)) < F(D(R{*1},R{x2})). (2.3)

By induction, we can construct sequences {xn} c X and {yn} c Range(R) such that

yn = Txn-1 and xnRyn (2.4)

for all n e N.If there exists n0 e N for which Txn0-1 = Txn0. Then xn0Ryn0+1. Thus Txn0 e R{xn0} and the proof is finished. So we suppose now that Txn-1 = Txn for every n e N. Then from (2.2), (2.3), and (2.4), we deduce that

t + F(d(yn,yn+1)) = t + F(d(Txn-1, Txn)) < F(D(R{xn-1},R{xn})) (2.5)

for all n e N. Since xnRyn and xn+1Ryn+1, by the definition of D, we get D(R{xn-1}, R{xn}) < d(yn-1,yn). Thus from (2.5), we have

t + F(d(yn,yn+1^ < F(d(yn_1,yn)), (.6)

which further implies that

F{d(yn>yn+i)) < F(d(yn-i,yn)) - t < F(d(yn-2,yn-i^ - 2t < < F(d(y0,yi^ - nx.

From (2.7), we obtain

lim F(d(ymyn+i)) = -co. (.8)

n—XX)

Then from (F2), we get

lim d(yn,yn+i) = 0. (2.9)

n—oo

Now from (F3), there exists 0 < k < 1 such that

—n [d(yn,yn+i)]kF(d(yn,yn+i)) = 0. (2.10)

By (2.7), we have

d(yn,yn+i)kF(d(yn,yn+i)) - d(yn,yn+i)kF(d(yo,yi)) < d(yn,yn+i)k[F(d(yo,yi)-n^ -F(d(yo,yi))]

= —nT [d(yn,yn+i)]k < 0. (2.ii)

By taking the limit as n — c in (2.ii) and applying (2.9) and (2.i0), we have

lim n[d(yn,yn+i)]k = 0. (2.i2)

n—oo L

It follows from (2.i2) that there exists ni e N such that

n[d(yn,yn+i)]k < i (2.i3)

for all n > ni. This implies

d(yn,yn+i) < -L; (2.i4)

for all n > ni. Now we prove that (yn} is a Cauchy sequence. For m > n > ni we have

m-i m-i i

d(yn,ym) d(yi,yi+i) Irt • (2.i5)

y ni/mt . .

i=n i=n

Since 0 < k < i, Jfc converges. Therefore, d(yn,ym) — 0 as m, n — oo . Thus we proved that (yn} is a Cauchy sequence in Range(R). Completeness of Range(R) ensures that there exists z e Range(R) such that yn — z as n — oo . Now since R is left-total, wRz for some w e X. Now

F(d(ym Tw)) = F(d(Txn-i, Tw)) < F(D(R(Xn-i}, R{w})) - t < F(d(yn-i, z)) - t.

Since limn—o d(yn-i, z) = 0, by (F2), we have limn—o F(d(yn-i, z)) = - oo. This implies that limn—oo F(d(ym Tw)) = - , which further implies that limn— d(yn, Tw) = 0. Hence

d(z, Tw) = 0. It follows that z = Tw. Hence Tw e R{w}. In the case when Range(T) is complete. Since Range(T) c Range(R), there exists an element z* e Range(R) such that yn ^ z*. The remaining part of the proof is the same as in previous case. □

Example 5 Let X = Y = R, d(x,y) = |x -y|. Define T: R ^ R, R: R — R as follows:

Tx =(1 if x e Q, x [0 ifx e Q',

R = (Qx[0,4j) U (Q'x[7,9]).

Then Range(T) = {0,1} c Range(R) = [0,4] U [7,9]. Let F(t) = ln(t) and t = 1.

For x e Q, y e Q' or y e Q, x e Q', d(Tx, Ty)>0 implies that

t + F(d(Tx, Ty)) < F(D(R{x},R{y})).

Thus all conditions of the above theorem are satisfied and 1 is the coincidence point of T and R.

From Theorem 4 we deduce the following result immediately.

Theorem 6 LetX be a nonempty set and (Y, d) be a metric space. Let T, R: X ^ Y be two mappings such that Range(T) c Range(R) and Range(T) or Range(R) is complete. If there exist a mapping F: R+ ^ R and t >0 such that

t + F(d(Tx, Ty)) < F(d(Rx,Ry))

for all x, y e X, then T and R have a coincidence point in X. Moreover, if either T or R is injective, then R and T have a unique coincidence point in X.

Proof By Theorem 4, we see that there exists w e X such that Tw = Rw, where

Rw = lim Rxn = lim Txn-1, x0 e X.

For uniqueness, assume that w1, w2 e X, w1 = w2, Tw1 = Rw1, and Tw2 = Rw2. Then t + F(d(Tw1, Tw2)) < F(d(Rw1,Rw2)) for some t > 0. If R or T is injective, then

d(Rw1, Rw2) > 0

t + F(d(Rw1,Rw2^ = t + F(d(Tw1, Tw2)) < F{d(RwitRw2)), a contradiction to the fact that t >0. □

Remark 7 If in the above theorem we choose X = Y, R = I (the identity mapping on X), we obtain Theorem 3, which is Theorem 3.1 of Wardowski [12].

Corollary 8 Let T : X — Y, R : X — Y be such that R is left-total, Range(T) c Range(R) and Range(T) or Range(R) is complete. If there exists X e [0, i) such that for all x, y e X

d(Tx, Ty) < XD(R(x},R(y}),

then there exists w e X such that Tw e R(w}.

Proof Consider the mapping F(t) = ln(t), for t >0. Then obviously F satisfies (Fi)-(F3). From Theorem 4, we obtain the desired conclusion. □

Corollary 9 Let X be nonempty set and (Y, d) be a metric space. T, R : X — Y be two mappings such that Range(T) c Range(R) and Range(T) or Range(R) is complete. If there exists a X e [0, i) such that for all x, y e X

d(Tx, Ty) < Xd(Rx, Ry),

then R and T have a coincidence point in X. Moreover, if either T or R is injective, then R and T have a unique coincidence point in X.

Proof Consider the mapping F(t) = ln(t), for t >0. Then obviously F satisfies (Fi)-(F3). From Theorem 6, we obtain the desired conclusion. □

Remark 10 If in the above corollary we choose X = Y and R = I (the identity mapping on X), we obtain the Banach contraction theorem.

In this way, we recall the concept of F-contractions for multivalued mappings and proved Suzuki-type fixed point theorem for such contractions. Nadler [i0] invented the concept of a Hausdorff metric H induced by metric d on X as follows:

for every A, B e CB(X). He extended the Banach contraction principle to multivalued mappings. Since then many authors have studied fixed points for multivalued mappings. Very recently, Sgroi and Vetro extended the concept of the F-contraction for multivalued mappings (see also [2i]).

Theorem 11 Let (X, d) be a metric space and letT: X — CB(X). Assume that there exist a function F e F that is continuous from the right and t e R+ such that

for all x, y e X. Then T has a fixed point.

Proof Let xo e X be an arbitrary point of X and choose xi e Txo. If xi e Txi, then xi is a fixed point of T and the proof is completed. Assume that xi e Txi, then Txo = Txi. Now

-d(xo, Txo) < -d(xo,xi) < d(xo,xi).

(2.16)

From the assumption, we have

2t + F(H(Tx0, Tx\)) < F(d(x0,x^).

Since F is continuous from the right, there exists a real number h >1 such that

F(hH(Tx0, Tx1)) < F(H(Tx0, Tx1)) + t.

Now, from

d(x1, Tx1) < H (Tx0, Tx1) < hH (Tx 0, Tx1),

we deduce that there exists x2 e Tx1 such that

d(x1, x2) < hH (Tx 0, Tx1).

Consequently, we get

F(d(x1,x2)) < F{hH(Tx0, Tx 0) < F{H(Tx0, Tx\)) + t,

which implies that

2t + F(d(x1,x2)) < 2t + F(H(Tx0, Tx\)) + t < F(d(x0,x^) + t.

t + F(d(x1,x2)) < F(d(x0,x^).

Continuing in this manner, we can define a sequence {xn} c X such that xn e Txn, xn+1 e Txn and

t + F(d(xn,xn+0) < F(d(xn-1,xn))

for all n e N U {0}. Therefore

F(d(xn,xn+0) < F(d(xn-1,xn)) - t < F(d(xn-2,xn-1)) - 2t <• • •

< F(d(x0,x^)- nT (2.17)

for all n e N. Since F e F,by taking the limit as n ^x in (2.17) we have

lim F(d(xn,xn+0) = -x ^^ lim d(xn,xn+1) = 0. (2.18)

Now from (F3), there exists 0 < k <1 such that

lim [d(xn,xnri)]kFiKd(xn,xn+1)) = 0. (2.19)

By (2.17), we have

d(xn,xn+1)kF(d(x„,xn+1)) - d(xn,xn+OkF(d(x0,x1)) < d(xn,xn+1)k[F(d(xo,x1) -nr) - F(d(xo,x1))]

= -nr [d(xn,xn+1)\k < o. (2.20)

By taking the limit as n ^roin (2.20) and applying (2.18) and (2.19), we have

lim n[d(xn,xn+0]k = 0. (2.21)

n^ro L J

It follows from (2.21) that there exists n1 e N such that

[d(xn,xn+1)]k < 1 (.22)

for all n > n1. This implies

d(xn,xn+1) < njk (2.23)

for all n > n1. Now we prove that {xn} is a Cauchy sequence. For m > n > n1 we have

m-1 m-1 1

d(xn,xm) d(xi,xi+1) < ^ Uk. (2.24)

l=n l=n

Since 0 < k <1, (i=1 J!k converges. Therefore, d(xn,xm) ^ 0 as m, n ^ro. Thus {xn} is a Cauchy sequence. Since X is a complete metric space, there exists z e X such that such that xn ^ z as n ^ +ro. Now, we prove that z is a fixed point of T. If there exists an increasing sequence {nk}c N such that x„k e Tz for all k e N. Since Tz is closed and xn ^ z as n ^ +ro, we get z e Tz and the proof is completed. So we can assume that there exists n0 e N such that xn0 e Tz for all n e N with n > n0. This implies that Txn-1 = Tz for all n > n0. We first show that

d(z, Tx) < d(z, x)

for all x e X\{z}. Since xn ^ z, there exists n0 e N such that

d(z, xn) < 1 d(z, x)

for all n e N with n > n0. Then we have

-d(xm Txn) < d(xn, Txn) << d(xn, xn+1)

< d(xn, z) + d(z,xn+1)

< - d(x, z) = d(x, z) - - d(x, z)

< d(x, z) - d(z,xn) < d(x,xn).

Thus, by assumption, we get

2t + F(H(Txn, Tx)) < F(d(x„,x)).

Since F is continuous from the right, there exists a real number h >1 such that

F(hH(Txn, Tx)) < F(H(Txn, Tx)) + r. Now, from

d(xn+1, Tx) < H(Txn, Tx) < hH(Txn, Tx), we obtain

F(d(xn+1, Tx)) < F{hH(Txn, Tx)) < F(H(Txn, Tx)) + r. Thus we have

2r + F(d(xn+1, Tx)) < 2r + F[H(Txn, Tx)) + r

Since F is strictly increasing, we have

d(xn+i, Tx) < d(xn,x).

Letting n tend to we obtain

d(z, Tx) < d(z, x)

for all x e X\{z}. We next prove that

2r + F(H(Tz, Tx)) < F(d(z,x))

for all x e X. Since F e F,we take x = z. Then for every n e N, there exists yn e Tx such that

d(z, yn) < d(z, Tx) + 1 d(z, x).

< F(d(xn, x)) + t .

So we have

d(x, Tx) < d(x, yn)

< d(x, z) + d(z,yn)

< d(x, z) + d(z, Tx) + - d(z, x)

< d(x, z) + d(x, z) + — d(z, x)

for all n e N and hence 2d(x, Tx) < d(x,z). Thus by assumption, we get

2t + F(H(Tz, Tx)) < F(d(z,x)).

2r + F(d(xn+i, Tz)) < 2r + F{H(Txn, Tz)) < F(d(xm z)).

Since F is strictly increasing, we have d(xn+1, Tz) < d(xn,z). Letting n we get

3 Applications

Fixed point theorems for contractive operators in metric spaces are widely investigated and have found various applications in differential and integral equations (see [9,15, 22, 23] and references therein). In this section we discuss the existence and uniqueness of solution of a general class of the following Volterra type integral equations under various assumptions on the functions involved. Let C[0, ©] denote the space of all continuous functions on [0, ©], where © >0 and for an arbitrary \\x\\r = supte[00]{|x(t)|e-Tt}, where r > 0 is taken arbitrary. Note that \\ • \\r is a norm equivalent to the supremum norm, and (C([0, ©], R), \\ • \\r) endowed with the metric dr defined by

dr (x, y)= sup {|x(t)-y(t)|e-rt}

te[0,©]

for all x, y e C([0, ©], R) is a Banach space; see also [19]. Consider the integral equation

where x :[0, ©] ^ R is unknown, g : [0, ©] ^ R, and h,f : R ^ R are given functions. The kernel K of the integral equation is defined on [0, ©] x [0, ©].

Theorem 12 Assume that the following conditions are satisfied:

(i) K: [0, ©] x [0, ©] x R ^ R, g: [0, ©] ^ R andf: R ^ R are continuous,

(ii) /0 K(t, s, •): R ^ R is increasing, for all t, s e [0, ©],

(iii) there exists r e (0, such that

|K(t, s, hx(s)) - K(t, s, hy(s)) | < r |hx(s) - hy(s) |

for all t, s e [0, ©] and hx, hy e R,

(iv) iff is injective, then for r >0 there exists e— e R+ such that for all x, y e R;

|hx - hy| < e-r [fx -fyl

d(z, Tz) < 0. Since Tz is closed, we obtain z e Tz. Thus z is fixed point of T.

and {fx: x e C([0, ©], R)} is complete. Then there exists w e C([0, ©], R) such that for xo e C([0, ©], R) andxn(t) =fxn-i(t)

g(t)+ / K(t,s, hx„-i(s)) ds Jo

fw(t) = lim fxn(t) = lim

and w is the unique solution of (3.1). Proof Let X = Y = C([0, ©], R) and dT (x, y)= sup {|x(t)-y(t)|e-Tt}

te[0,®]

for all x,y e X. Let T, R: X ^ X be defined as follows:

(Tx)(t)=g(t)+ / K(t,s, hx(s)) ds and Rx = fx. Jo

Then by assumptions RX = {Rx: x e X} is complete. Let x* e TX, then x* = Tx for x e X and x*(t) = Tx(t).Bythe assumptions there existsy e X such that Tx(t) =fy(t), hence RX c TX. Since

|(rx)(t)-(2»(t)| =

i [K(t,s, hx(s))] ds - i [K(t,s, hy(s))\ds

i IK (t, s, hx(s)) - K (t, s, hy(s)) | ds

< t |hx(s) - hy(s) I ds

< i t e- \fx(s) -fy(s)1 ds

= i t e-T|(Rx)(s)-(Ry)(s)|e-TseTs ds

< te-T \\Rx - Ry\\x i eTs ds

„t t

< te-T \\Rx - Ry\\r — = e-T \\Rx - Ry\\T eT t.

This implies that

|(Tx)(t) - (Ty)(t)|eTt < e-T\\Rx - Ryh, or equivalently

dT (Tx, Ty) < e-tdT (Rx, Ry). Taking logarithms, we have

ln(dT (Tx, Ty)) < ln(e-TdT (Rx, Ry)).

After routine calculations, one can easily get

r + ln(dr(Tx, Ty)) < ln(dr(Rx,Ry)).

Now, we observe that the function F : R+ ^ R defined by F(t) = ln(t) for each t e C([0, ©], R) and r > 0 isin F. Thus all conditions of Theorem 6 are satisfied. Hence, there exists a unique w e X such that

fw(t) = lim Rxn(t) = lim Txn-1(t) = T(w)(t), x0 e X,

n^TO n^TO

for all t, which is the unique solution of(3.1). □

Competing interests

The authors declare that they have no competing interests. Authors' contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper. Author details

1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 2Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Islamabad, 44000, Pakistan. 3Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade, 11 000, Serbia.

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, first author acknowledges with thanks DSR, KAU for financial support.

Received: 21 November 2014 Accepted: 17 May 2015 Published online: 05 June 2015 References

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