Coupled fixed points and coupled best proximity points for cyclic Ćirić type operators

Adrian Magdaş (Faculty of Mathematics and Computer Science, Babeş Bolyai University, Cluj-Napoca, Romania)

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Article publication date: 22 May 2019

Abstract

The purpose of this paper is to study the coupled fixed point problem and the coupled best proximity problem for single-valued and multi-valued contraction type operators defined on cyclic representations of the space. The approach is based on fixed point results for appropriate operators generated by the initial problems.

Keywords

Citation

Magdaş, A. (2019), "Coupled fixed points and coupled best proximity points for cyclic Ćirić type operators", Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 179-196. https://doi.org/10.1016/j.ajmsc.2019.05.002

Publisher

:

Emerald Publishing Limited

Copyright © 2019, Adrian Magdaş

License

Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode


1. Introduction

One of the most important metrical fixed point theorem, Banach contraction principle, has been generalized in several directions, see for example [1]. The concept of coupled fixed point was introduced by Guo and Lakshmikantham (see [2]). A new research direction for the theory of coupled fixed points was developed by many authors (see [3–9]) using contractive type conditions.

Definition 1.1 ([10]). Let X be a nonempty set. A pair (x,y)X×X is called coupled fixed point of the operator F:X×XX if F(x,y)=x and F(y,x)=y. If F(x,x)=x then x is called a strong coupled fixed point of F (or, in several papers, a fixed point of F).

Another generalization of the Banach principle was given by Kirk, Srinivasan and Veeramani using the concept of cyclic operators.

Definition 1.2 ([11]). Let A and B be nonempty subsets of a given set X. An operator T:ABAB is called cyclic if T(A)B and T(B)A.

More recently, Choudbury and Maity formulated the following definition.

Definition 1.3 ([12]). Let A and B be nonempty subsets of a given set X. An operator F:X×XX having the property that for any xA and yB, F(x,y)B and F(y,x)A, is called a cyclic operator with respect to A and B.

Definition 1.4 ([13]). Let A and B be nonempty subsets of a metric space (X,d). An operator F:X×XX is called a cyclic Ćirić operator with respect to A and B if F is cyclic with respect to A and B and for some constant q(0,1), F satisfies the following condition:

d(F(x,y),F(u,v))qM(x,v,y,u),
where x,vA, y,uB, and
M(x,v,y,u)=max{d(x,u),12d(u,F(x,y)),12d(x,F(u,v)),12[d(x,F(x,y))+d(u,F(u,v))]}.

Theorem 1.1 ([13]). Let A and B be nonempty closed subsets of a complete metric space (X,d), F:X×XX a cyclic Ćirić type operator with respect to A and B, with ABØ . Then F has a strong coupled fixed point in AB.

The first aim of this paper is to generalize the above theorem, weakening the contractive condition and excluding the condition ABØ. We prove the uniqueness of the strong coupled fixed point and we provide an iterative method for approximating the strong coupled fixed point.

We also present coupled fixed point and coupled best proximity point results for cyclic coupled Ćirić-type multivalued operators.

On the other hand, some qualitative properties of the coupled fixed point set, such as data dependence, generalized Ulam–Hyers stability and well-posedness are studied.

Our approach is based on the following idea: we transform the coupled fixed point/ best proximity point problem into a fixed point/ best proximity point problem for an appropriate operator defined on a cartesian product of the spaces. In this way, many coupled fixed point/ best proximity point results can be obtained using classical fixed point/ best proximity point theorems.

2. Preliminaries

The standard notations and terminologies in nonlinear analysis will be used throughout this paper.

Let (X,d) be a metric space. We denote:

P(X):={YX|Y is nonempty};Pb(X):={YP(X)|Y is bounded};
Pcl(X):={yP(X)|Y is closed};Pcp(X):={YP(X)|Y is compact}.

Let us define the following (generalized) functionals used in this paper:

  • • The gap functional

D:P(X)×P(X)+,D(A,B)=inf{d(a,b)|aA,bB};
  • • The generalized excess functional

ρ:P(X)×P(X)+{+},ρ(A,B)=sup{D(a,B)|aA};
  • • The generalized Pompeiu–Hausdorff functional

H:P(X)×P(X)+{+},H(A,B)=max{ρ(A,B),ρ(B,A)}.

There are several conditions upon the comparison function that have been considered in literature. In this paper we shall refer only to:

Definition 2.1 ([14]). A function ϕ:++ is called a comparison function if it satisfies:

  • (i)ϕ is increasing;

  • (ii)(ϕn(t))n converges to 0 as n, for all t+.

If the condition (ii) is replaced by the condition:

  • (iii)k=0ϕk(t)<, for any t>0, then ϕ is called a strong comparison function.

Lemma 2.1 ([1]). If ϕ:++ is a comparison function, then ϕ(t)<t , for any t>0 , ϕ(0)=0 and ϕ is continuous at 0.

Lemma 2.2 ([14]). If ϕ:++ is a strong comparison function, then the following hold:

  • (i) ϕ is a comparison function;

  • (ii) the function s:++ , defined by

s(t)=k=0ϕk(t),
is increasing and continuous at 0.

Example 2.1 ([15]). (1)ϕ:++, ϕ(t)=at, where a[0,1), is a strong comparison function;

(2)ϕ:++, ϕ(t)=12t, for t[0,1] and ϕ(t)=t12, for t>1, is a strong comparison function;

(3)ϕ:++, ϕ(t)=at+12[t], where a(0,12), is a strong comparison function;

(4)ϕ:++, ϕ(t)=t1+t, is a comparison function, but is not a strong comparison function.

For more examples and considerations on comparison functions see [1] and the references therein.

3. Coupled fixed points of cyclic Ćirić type single valued operators

In this section we present some coupled fixed point results for cyclic Ćirić type operators on complete metric spaces.

We introduce now the following new concept.

Definition 3.1 Let (X,d) be a metric space, A,BPcl(X), Y=AB and ϕ:R+R+ a strong comparison function. An operator F:Y×YY is called a cyclic coupled ϕ-contraction of Ćirić type if the following statements hold:

  • (i) F is cyclic with respect to A and B;

  • (ii)

(3.1) d(F(x,y),F(u,v))ϕ(M(x,v,y,u)),
for any x,vA and y,uB, where
M(x,v,y,u)=max{d(x,u),d(v,y),d(x,F(x,y)),d(u,F(u,v)),d(v,F(v,u)),d(y,F(y,x)),12[d(x,F(u,v))+d(u,F(x,y))],12[d(y,F(v,u))+d(v,F(y,x))]}.

The following theorem (which is a particular case of Theorem 3.2 in [16]) will be used to prove our results presented in this section.

Theorem 3.1 ([16]). Let (X,d) be a complete metric space, A,BPcl(X) , ϕ:++ be a strong comparison function and f:ABAB be an operator such that f(A)B and f(B)A . If f is a cyclic ϕ -contraction of Ćirić type, that is

d(f(x),f(y))ϕ(max{d(x,y),d(x,f(x)),d(y,f(y)),12[d(x,f(y))+d(y,f(x))]}),
for any xA and yB , then the following statements hold:

(1)f has a unique fixed point xAB and the Picard iteration {xn}n0 defined by xn=f(xn1), n1 , converges to x for any starting point x0AB;

(2)the following estimates hold:d(xn,x)s(ϕn(d(x0,x1))),n1;d(xn,x)s(d(xn,xn+1)),n1;

(3)for any xAB, d(x,x)s(d(x,f(x))) , where s is given by Lemma 2.2.

The main result of this section is the following theorem.

Theorem 3.2. Let (X,d) be a complete metric space, A,BPcl(X), Y=AB and F:Y×YY a cyclic coupled ϕ -contraction of Ćirić type. Then:

  1. F has a unique strong coupled fixed point xAB;

  2. for any (x0,y0)A×B, there exists a sequence {(xn,yn)}nX×X defined by

{xn=F(yn1,xn1)yn=F(xn1,yn1),n1,
that converges to (x,x);
  • (3) the following estimates hold:

max{d(xn,x),d(yn,x)}s(ϕn(max{d(x0,F(x0,y0)),d(y0,F(y0,x0))})),n1,max{d(xn,x),d(yn,x)}s(max{d(xn,xn+1),d(yn,yn+1)}),n1;
  • (4) for any x,yY, d(x,x)s(max{d(x,F(x,y)),d(y,F(y,x))}) , where s is given by Lemma 2.2.

Proof. (1)(2) Changing the roles between x and v and similarly for y and u, the inequality (3.1) becomes:

(3.2) d(F(v,u),F(y,x))ϕ(M(v,x,u,y)), for x,vA and y,uB.

Obviously, M(x,v,y,u)=M(v,x,u,y). From the inequalities (3.1) and (3.2) we obtain

(3.3) max{d(F(x,y),F(u,v)),d(F(y,x),F(v,u))}ϕ(M(x,v,y,u)).

For z=(x,y)A×B, w=(u,v)B×A, denote

(3.4) d(z,w)=max{d(x,u),d(y,v)}.

Then (X×X,d) is a complete metric space.

Let T:Y×YY×Y be defined by T(x,y)=(F(x,y),F(y,x)). We have:

12[d(z,T(w))+d(w,T(z))]=12max{d(x,F(u,v)),d(y,F(v,u))}+12max{d(u,F(x,y)),d(v,F(y,x))}max{12[d(x,F(u,v))+d(u,F(x,y))],12[d(y,F(v,u))+d(v,F(y,x))]}.

Using the above relation, from (3.3) we get

(3.5) d(T(z),T(w))ϕ(max{d(z,w),d(z,T(z)),d(w,T(w)),12[d(z,T(w))+d(w,T(z))]}),
for any zA×B, wB×A.

Because F(A×B)B and F(B×A)A, we have

(3.6)T(A×B)B×A and T(B×A)A×B.
(3.5) and (3.6) means that the operator T is a cyclic ϕ-contraction of Ćirić type. Applying Theorem 3.1, there exists a unique z=(x,y)(A×B)(B×A) such that T(z)=z and the Picard iteration zn=T(zn1) converges to z for any starting point z0Y. So
(3.7) {F(x,y)=xF(y,x)=y
where x,yAB.

From unicity of the pair (x,y) and the symmetry with respect to x and y of the system (3.7) we conclude x=y.

Then F has a unique strong coupled fixed point xAB and for any starting point (x0,y0)A×B there exists a sequence {(xn,yn)}nY×Y with

{xn=F(yn1,xn1)yn=F(xn1,yn1),n1
that converges to (x,x).

(3) By the second conclusion of Theorem 3.1,

d(zn,(x,x))s(ϕn(d(z0,z1)))
and
d(zn,(x,x))s(d(zn,zn+1)),n1.

Hence

max{d(xn,x),d(yn,x)}s(ϕn(max{d(x0,F(x0,y0)),d(y0,F(y0,x0))}))max{d(xn,x),d(yn,x)}s(max{d(xn,xn+1),d(yn,yn+1})),n1.

(4) Using (3) from Theorem 3.1, for any (x,y)Y×Y,

d((x,y),(x,x))s(d((x,y),T(x,y))).

Hence

max{d(x,x),d(y,x)}s(max{d(x,F(x,y)),d(y,F(y,x))}).

Example 3.1. Let X=,d(x,y)=|xy|,for anyx,y, A=[0,2], B=[0,1], Y=AB, F:Y×YY, F(x,y)=x+3y9.

It is easy to verify that F is cyclic with respect to A and B.

For  any x, v ∈ A and y, u ∈ B

d(F(x,y),F(u,v))=|x+3y9u+3v9|=|xu9+yv3||19(xu)+1027(yv)|=13|yv+3u9+y+3x9v|13(|yF(v,u)|+|vF(y,x)|)2312[d(y,F(v,u))+d(v,F(y,x))].

Then F is a cyclic coupled ϕ-contraction of Ćirić type, where ϕ(t)=23t.

The hypotheses of Theorem 3.2 are satisfied, so by Theorem 3.2, F has a unique strong coupled fixed point xAB. By calculation we get:

F(x,x)=xx=0.

Our next theorem gives the well-posedness property for the coupled fixed point problem. For the concept of well-posedness for the fixed point problems see [17].

Theorem 3.3. Let F:Y×YY be as in Theorem 3.2. Then the coupled fixed point problem is well posed, that is, if there exists a sequence {(an,bn)}nY×Y such that

{d(an,F(an,bn))0d(bn,F(bn,an))0as n,
then anx and bnx, as n.

Proof. Using the inequality

d(x,x)s(max{d(x,F(x,y)),d(y,F(y,x))})
from Theorem 3.2 for x:=an and next for x:=bn, we have:
{d(an,x)s(max{d(an,F(an,bn)),d(bn,F(bn,an))})d(bn,x)s(max{d(bn,F(bn,an)),d(an,F(an,bn))}),n,
and letting n we obtain
{d(an,x)0d(bn,x)0,n.

For the data dependence problem we have the following result.

Theorem 3.4. Let F:Y×YY be as in Theorem 3.2. Let G:Y×YY be such that:

  • (i) G has at least one strong coupled fixed point xG;

  • (ii) there exists η>0 such that

d(F(x,x),G(x,x))η, for any xY.

Then d(xF,xG)s(η) , where xF is the unique strong coupled fixed point of F and

s(t)=k=0ϕk(t),t+.

Proof.

By letting x:=xG and y:=xG in the inequality

d(x,x)s(max{d(x,F(x,y)),d(y,F(y,x))}),
we have
d(xG,xF)s(d(xG,F(xG,xG)))=s(d(G(xG,xG),F(xG,xG))),
and using the monotonicity of s we obtain
d(xF,xG)s(η).

Theorem 3.5. Let F:Y×YY be as in Theorem 3.2 and Fn:Y×YY , nN , be such that:

  • (i) for each n there exists a strong coupled fixed point xn of Fn ;

  • (ii) {Fn}n converges uniformly to F.

Then xnx as n , where x is the unique strong coupled fixed point of F.

Proof. The sequence {Fn}n converges uniformly to F. Then there exist ηn+, n such that ηn0 as n and

d(Fn(x,y),F(x,y))ηn for any (x,y)Y×Y.

Using Theorem 3.3 for G:=Fn, nN, we have

d(xn,x)s(ηn) as n.

We will discuss Ulam–Hyers stability for the coupled fixed point problem corresponding to a cyclic operator.

Definition 3.2. Let (X,d) be a metric space, YP(X) and F:Y×YY be an operator. The coupled fixed point problem

(3.8) {F(x,y)=xF(y,x)=y,x,yY
is called generalized Ulam–Hyers stable if there exists ψ:++ increasing, continuous at 0 and ψ(0)=0 such that for any ε1>0,ε2>0 and for any solution (x,y)Y×Y of the system
{d(x,F(x,y))ε1d(y,F(y,x))ε2

there exists a solution (x,y) of the coupled fixed point problem such that

{d(x,x)ψ(ε)d(y,y)ψ(ε),where ε=max{ε1,ε2}.

In particular, if x=y, then we have generalized Ulam–Hyers stability for the strong coupled fixed point problem F(x,x)=x,xY.

Theorem 3.6. Suppose that all the hypotheses of Theorem 3.2 hold. Then the coupled fixed point problem (3.8) is generalized Ulam–Hyers stable.

Proof. By Theorem 3.2 we have a unique xY such that F(x,x)=x.

Let ε1>0,ε2>0 and (x˜,y˜)Y×Y such that

{d(x˜,F(x˜,y˜))ε1d(y˜,F(y˜,x˜))ε2.

We know that

d(x,x)s(max{d(x,F(x,y)),d(y,F(y,x))}),(x,y)Y×Y.

Then for

(x:=x˜y:=y˜)
and next for
(x:=y˜y:=x˜)
using the monotonicity of s, we obtain that
max{d(x˜,x),d(y˜,x)}s(max{d(x˜,F(x˜,y˜)),d(y˜,F(y˜,x˜))})s(max{ε1,ε2}).

As a conclusion, the coupled fixed point problem (3.8) is generalized Ulam–Hyers stable with ψ=s.

4. Coupled fixed points and coupled best proximity points of cyclic Ćirić type multivalued operators

The purpose of this section is to consider the above problems in the multi-valued setting. We present first a new concept of cyclic multi-valued operator.

Definition 4.1. Let (X,d) be a metric space, A,BP(X), Y=AB and ϕ:++ a strong comparison function. A multivalued operator F:Y×YP(Y) is called a cyclic coupled ϕ-contraction of Ćirić type multivalued operator if the following statements hold:

  • (i) F is cyclic with respect to A and B, that is

F(A×B)B and F(B×A)A;
  • (ii)

(4.1) H(F(x,y),F(u,v))ϕ(M˜(x,v,y,u)), for any x,vA,y,uB
where
M˜(x,v,y,u)=max{d(x,u),d(v,y),D(x,F(x,y)),D(u,F(u,v)),D(v,F(v,u)),D(y,F(y,x)),12[D(x,F(u,v))+D(u,F(x,y))],12[D(y,F(v,u))+D(v,F(y,x))]}.

Definition 4.2. Let (X,d) be a metric space. Then YP(X) is called proximinal if for any xX, there exists yY such that

d(x,y)=D(x,Y).

We denote Pprox={yP(X)|Y is proximinal}.

Remark 4.1. Let (X,d) be a metric space. Then

Pcp(X)Pprox(X)Pcl(X).

Remark 4.2. Every closed convex subset of a uniformly Banach space is proximinal, see [18].

For details concerning the above notions see [1,19] and [20].

The following theorem (which is a particular case of Theorem 2.7 in [21]) will be used to prove the first result in this section.

Theorem 4.1. ([21]). Let (X,d) be a complete metric space, A,BPcl(X) and T:ABPprox(AB) a multivalued cyclic ϕ -contraction of Ćirić type, that is:

  • (i) T(A)B and T(B)A;

  • (ii) there exists a strong comparison function ϕ:++ such that

H(T(x),T(y))ϕ(max{d(x,y),D(x,T(x)),D(y,T(y)),12[D(x,T(y))+D(y,T(x))]}),
for any xA and yB .

Then the following statements hold:

  1. there exists xAB such that xT(x);

  2. for any xA and yT(x) , there exists a sequence (xn)n with x0=x , x1=y and xnT(xn1), n1 , that converges to a fixed point xAB of T.

The following lemma presents a well-known result (see for example [22]).

Lemma 4.1. Let (X,d) be a metric space, d the metric defined on X×X by (3,4) and D the gap functional, respectively H the generalized Pompeiu–Hausdorff functional generated by d . Then for any a,bX and any A,B,C,DPprox(X), the following statements hold:

  1. D((a,b),C×D)=max(D(a,C),D(b,D));

  2. D(A×B,C×D)=max(D(A,C),D(B,D));

  3. H(A×B,C×D)=max{H(A,C),H(B,D)};

  4. D(A×B,B×A)=D(A,B).

Proof. (1)+(2) Since the sets C and D are proximinal then there exists c0C,d0D such that D(a,C)=d(a,c0) and D(b,D)=d(b,d0).

Then

D((a,b),C×D)=inf{d((a,b),(c,d))|cC,dD}=inf{max{d(a,c),d(b,d)}|cC,dD}=max{d(a,c0),d(b,d0)}.

Similarly, we can prove (2).

(3) H(A×B,C×D)=

max{sup(a,b)A×B{D((a,b),C×D)},sup(c,d)C×D{D((c,d),A×B)}}.

Using statement (1), we have

H(A×B,C×D)=max{sup(a,b)A×B{D(a,C),D(b,D)},sup(c,d)C×D{D(c,A),D(d,B)}}
=max{H(A,C),H(B,D)}

(4) We use statement (2) for C=A,D=B.

Lemma 4.2. Let (X,d) be a metric space, d the metric defined on X×X by (3.4) . If a multivalued operator F:X×XP(X) takes proximinal values with respect to d then the multivalued operator T:X×XP(X×X), T(x,y)=(F(x,y),F(y,x)) takes proximinal values with respect to d.

Proof. For any pair (a,b)X×X,F(a,b) is a proximinal set, which means that for any xX, there exists cF(a,b) such that

d(x,c)=D(x,F(a,b)).

In a similar way, for any yX, there exists dF(b,a) such that

d(y,d)=D(y,F(b,a)).

Then for any (x,y)X×X, there exists (c,d)T(a,b) such that

d((x,y),(c,d))=max{d(x,c),d(y,d)}=max{D(x,F(a,b)),D(y,F(b,a))}=D((x,y),T(a,b)).

The first result in this section is the following theorem.

Theorem 4.2. Let (X,d) be a complete metric space, A,BPcl(X), Y=AB and F:Y×YPprox(Y) a cyclic coupled ϕ-contraction of Ćirić type multivalued operator.

Then the following statements hold:

  1. there exist x,yAB such that

xF(x,y),yF(y,x),
(that is the pair (x,y) is a coupled fixed point of F );
  • (2) for each (a,b)A×B there exists a sequence (an,bn)nY×Y with a0=a, b0=b and

anF(bn1,an1),bnF(an1,bn1) for n1
that converges to a coupled fixed point (x,y)AB of F .

Proof.

It is easy to observe that

M˜(x,v,y,u)=M˜(v,x,u,y), for any x,vA,y,uB.

If we change the roles between x and v and similarly for y and u, then the inequality (4.1) becomes

(4.2) H(F(v,u),F(y,x))ϕ(M˜(x,v,y,u)).

From (4.1) and (4.2) we obtain

max{H(F(x,y),F(u,v)),H(F(y,x),F(v,u))}ϕ(M˜(x,v,y,u)).

Let T:Y×YP(Y×Y), T(x,y)=(F(x,y),F(y,x)).

We consider on Y×Y the metric d defined by (3.4), using the same functionals D and H as in Lemma 4.1.

For z=(x,y)A×B, w=(u,v)B×A, using Lemma 4.1,

(4.3) H(T(z),T(w))=H((F(x,y),F(y,x)),(F(u,v),F(v,u)))=max{H(F(x,y),F(u,v)),H(F(y,x),F(v,u))}ϕ(M˜(x,v,y,u)).

By Lemma 4.1,

D(z,T(z))=max{D(x,F(x,y)),D(y,F(y,x))},D(w,T(w))=max{D(u,F(u,v)),D(v,F(v,u))},
12[D(w,T(z))+D(z,T(w))]=12[max{D(u,F(x,y)),D(v,F(y,x))}+max{D(x,F(u,v)),D(y,F(v,u))}]max{12[D(u,F(x,y))+D(x,F(u,v))],12[D(v,F(y,x))+D(y,F(v,u))]}.

Using the monotonicity of ϕ, (4.3) becomes

H(T(z),T(w))ϕ(max{d(z,w),D(z,T(z)),D(w,T(w)),12[D(w,T(z))+D(z,T(w))]}), for any zA×B,wB×A,
and because T satisfies the cyclic condition
T(A×B)=(F(A×B),F(B×A))B×A,T(B×A)A×B,
where A×B,B×APcl(Y×Y), we conclude that T is a multivalued cyclic ϕ-contraction of Ćirić type.

By Lemma 4.2, the property of the operator F to have proximinal values is transferred to the operator T, so we are in the conditions of Theorem 4.1.

Then there exists (x,y)(A×B)(B×A) such that (x,y)(F(x,y),F(y,x)) and for each (a,b)A×B there exists a sequence (an,bn)nY×Y with a0=a, b0=b and

(an,bn)(F(bn1,an1),F(an1,bn1)),n1
that converges to (x,y).

Hereinafter we define and study the generalized Ulam–Hyers stability of the following coupled fixed point problem.

Definition 4.3.

Let (X,d) be a metric space, YP(X), F:Y×YP(Y) be a multivalued operator. By definition, the coupled fixed point problem

(4.4) {xF(x,y)yF(y,x),x,yY
is said to be generalized Ulam–Hyers stable if there exists an increasing function ψ:++, continuous at 0, with ψ(0)=0 such that for each ε>0 and for each solution (x,y)Y×Y of the inequality
max{D(x,F(x,y)),D(y,F(y,x))}ε,
there exists a solution (x,y)Y×Y of the coupled fixed point problem such that
max{d(x,x),d(y,y)}ψ(ε).

Our stability result is a consequence of the following theorem.

Theorem 4.3

([21]). Let T:YPprox(Y) be as in Theorem 4.2, ε>0 and xY be such that D(x,T(x))ε . Then there exists x a fixed point of T such that d(x,x)s(ε), where s is given by Lemma 2.2.

Theorem 4.4.

If all the hypotheses of Theorem 4.2 hold, then the coupled fixed point problem (4.4) is generalized Ulam–Hyers stable.

Proof.

Let any ε>0 and let (x¯,y¯)Y×Y such that

{D(x¯,F(x¯,y¯))εD(y¯,F(y¯,x¯))ε.

As before, we consider T:Y×YP(Y×Y),

T(x,y)=(F(x,y),F(y,x)).

For z=(x¯,y¯),

D(z,T(z))=max{D(x¯,F(x¯,y¯)),D(y¯,F(y¯,x¯))}ε.

Applying Theorem 4.3, there exists a fixed point z=(x,y) of T such that d(z,z)s(ε), that is there exists a solution (x,y) of the coupled fixed point problem (4.4) such that

max{d(x¯,x),d(y¯,y)}s(ε).

In the last part of this section we will consider the following best proximity problem for a cyclic coupled multivalued operator:

If (X,d) is a metric space, A,BP(X), Y=AB, F:Y×YP(Y) is a coupled multivalued operator satisfying the cyclic condition F(A×B)B, F(B×A)A, then we are interested in finding (x,y)A×B such that

(4.5) D(x,F(x,y))=D(y,F(y,x))=D(A,B).
(x,y) is said to be a coupled best proximity point of F.

Notice that, in particular, if ABØ then (x,y) is a coupled fixed point of F.

Definition 4.4.

Let (X,d) be a metric space, A,BP(X), Y=AB. A multivalued operator F:Y×YP(Y) is called a cyclic coupled Ćirić type multivalued operator if:

  • (i) F(A×B)B and F(B×A)A;

  • (ii) there exists a comparison function ϕ:++ such that

H(F(x,y),F(u,v))ϕ(M˜(x,v,y,u)D(A,B))+D(A,B),

for any x,vA, y,uB.

In 2009, Suzuki, Kikkawa and Vetro introduced the following property.

Definition 4.5.

[23] Let A and B be nonempty subsets of a metric space (X,d). Then (A,B) is said to satisfy the property UC if for (xn)n and (zn)n sequences in A and (yn)n a sequence in B such that d(xn,yn)D(A,B) and d(zn,yn)D(A,B) as n, then d(xn,zn)0 as n.

Example 4.1.

[24] [23] (1) Any pair of nonempty subsets (A,B) of a metric space (X,d) with D(A,B)=0 satisfies the property UC;

(2)Any pair of nonempty subsets (A,B) of a uniformly convex Banach space with A convex satisfies the property UC.

Lemma 4.3.

Let A and B be nonempty subsets of a metric space (X,d) , and d be the metric defined on X×X by (3.4). If (A,B) and (B,A) satisfy the property UC with respect to d then (A×B,B×A) satisfy the property UC with respect to d.

Proof.

We denote D(A×B,B×A)=D(A,B)=D. Let xn=(an,bn),zn=(an,bn)A×B,yn=(βn,αn)B×A such that d(xn,yn)D and d(zn,yn)D as n.

Then

max{d(an,βn),d(bn,αn)}D and 
max{d(an,βn),d(bn,αn)}D as n.

It is obvious that d(an,βn)D,d(an,βn)D and because (A,B) satisfies the property UC we get d(an,an)0.

From d(bn,αn)D,d(bn,αn)D as n and using (B,A) satisfies the property UC we get d(bn,bn)0.

Finally,

d(xn,zn)=max{d(an,an),d(bn,bn)}0 as n.

We recall the following result.

Theorem 4.5

([25]). Let (X,d) be a complete metric space, APcl(X),BP(X) such that (A,B) satisfies the property UC. Let T:ABPprox(X) be a multivalued Ćirić type cyclic operator that is:

  • (i) T(A)B and T(B)A;

  • (ii) there exists a comparison function ϕ:++ such that

H(T(x),T(y))ϕ(M(x,y)D(A,B))+D(A,B), where 
M(x,y)=max{d(x,y),D(x,T(x)),D(y,T(y)),12[D(x,T(y))+D(y,T(x))]}.

Then the following statements hold:

  • (1) T has a best proximity point xAA ;

  • (2) there exists a sequence (xn)n with x0A , and xn+1T(xn) , n0 , such that (x2n)n converges to xA.

The next result is a consequence of the above theorem.

Theorem 4.6.

Let (X,d) be a complete metric space, A,BPcl(X) such that (A,B) and (B,A) satisfy the property UC, and Y=AB . If F:Y×YPprox(Y) is a cyclic coupled Ćirić type multivalued operator, then the following statements hold:

  • (i) F has a coupled best proximity point (x,y)A×B ;

  • (ii) there exist two sequences (xn)n , (yn)n with

(x0,y0)A×B,xn+1F(xn,yn),yn+1F(yn,xn),
such that ((x2n,y2n))n converges to (x,y) .

Proof.

Considering again on Y×Y the metric d defined by (3.4), in a similar manner as in Theorem 4.2, we obtain that the operator T:Y×YP(Y×Y),

T(x,y)=(F(x,y),F(y,x)).
is a multivalued Ćirić type cyclic operator which takes proximinal values.

Using Lemma 4.1, the pair (A×B,B×A) satisfies the property UC with respect to d.

Consequently, we are in the conditions of Theorem 4.5, so T has a best proximity point (x,y)A×B and there exists a sequence (xn,yn)n with (x0,y0)A×B and (xn+1,yn+1)T(xn,yn) such that (x2n,y2n)n converges to (x,y) with respect to d.

5. An application to a system of integral equations

We apply the results given by Theorem 3.2 to study the existence and the uniqueness of solutions of the following system of integral equations:

(5.1){x(t)=abG(t,s)f(s,x(s),y(s))dsy(t)=abG(t,s)f(s,y(s),x(s))ds,t[a,b]
where a,b, a<b,
GC([a,b]×[a,b],[0,)),
fC([a,b]××,).
Theorem 5.1.

We suppose that:

  • (i) there exist α,βC([a,b],) , with α(t)β(t) , for any t[a,b] , such that

(5.2) {α(t)abG(t,s)f(s,β(s),α(s))dsβ(t)abG(t,s)f(s,α(s),β(s))dsfor any t[a,b];
  • (ii) there exists a strong comparison function ϕ:++ such that

|f(s,u1,u2)f(s,v1,v2)|ϕ(max{|u1v1|,|u2v2|}),
for any s[a,b] and u1,u2,v1,v2;
  • (iii) supt[a,b]abG(t,s)ds1 ;

  • (iv) f(s,,y) is monotone decreasing for any s[a,b] and any y;

  • (v) f(s,x,) is monotone increasing for any s[a,b] and any x .

Then the system (5.1) has a unique solution (x,x)C([a,b],2) , with αxβ .

Proof.

Let us consider

X:=C([a,b],),and the Chebyshev norm |x|=maxt[a,b]|x(t)|.

Then (X,||) is a Banach space. We consider the following closed subsets of X:

A={xX|xβ},B={xX|xα},

Y=AB and the operator F:Y×YY,

F(x,y)(t):=abG(t,s)f(s,x(s),y(s))ds.

The system (5.1) is equivalent to

{F(x,y)=xF(y,x)=y,x,yY.

We will prove that F is cyclic with respect to A and B, that is

F(A×B)B and F(B×A)A.

Let xA and yBx(s)β(s),y(s)α(s),s[a,b].

Using the monotonicity of f we have

G(t,s)f(s,x(s),y(s))G(t,s)f(s,β(s),α(s)),
and from (i), by integration,
abG(t,s)f(s,x(s),y(s))dsα(t),
which means that
F(x,y)(t)α(t),t[a,b]F(x,y)B.

So F(A×B)B. In a similar way we have F(B×A)A.

Using the conditions (ii) and (iii), and the monotonicity of ϕ, for any x,vA and y,uB, we have

|f(s,x(s),y(s))f(s,u(s),v(s))|ϕ(maxs[a,b]{|x(s)u(s)|,|y(s)v(s)|})ϕ(max{|xu|,|yv|})
|F(x,y)(t)F(u,v)(t)|abG(t,s)|f(s,x(s),y(s))f(s,u(s),v(s))|dsϕ(max{|xu|,|yv|})abG(t,s)dsϕ(max{|xu|,|yv|}),t[a,b].

We have

|F(x,y)F(u,v)|ϕ(max{|xu|,|yv|}) for any x,vA and y,uB,

so the operator F is a cyclic coupled ϕ-contraction of Ćirić type.

All the conditions of Theorem 3.2 are satisfied, so T has a unique strong coupled fixed point (x,x)AB, with α(t)x(t)β(t), for any t[a,b].

Definition 5.1.

The system (5.1) is said to be generalized Ulam–Hyers stable if there exists ψ:++ increasing, continuous at 0 and ψ(0)=0 such that for any ε1>0,ε2>0 and for any solution (x,y)C([a,b],2), of the system

{|x(t)abG(t,s)f(s,x(s),y(s))ds|ε1|y(t)abG(t,s)f(s,y(s),x(s))ds|ε2
there exists a solution (x,y)C([a,b],2) of the system (5.1) such that for any t[a,b],
{|x(t)x(t)|ψ(ε)|y(t)y(t)|ψ(ε), where ε=max(ε1,ε2).

Theorem 5.2.

Suppose that the hypotheses of Theorem 5.1 hold. Then the system (5.1) is generalized Ulam–Hyers stable.

Proof.

By Theorem 5.1, the system (5.1) has a unique solution (x,x)C([a,b],2), with αxβ. Applying Theorem 3.6 to the operator F:Y×YY,

F(x,y)(t):=abG(t,s)f(s,x(s),y(s))ds,
in the same setting as in the proof of Theorem 5.1, we get the conclusion.

References

[1]I.A. Rus, A. Petruşel, G. Petruşel, Fixed Point Theory, Cluj University Press, Cluj-Napoca, 2008.

[2]D. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. Theory Methods Appl. 11 (1987) 623632.

[3]J.G. Kadwin, M. Marudai, Fixed point and best proximity point results for generalised cyclic coupled mappings, Thai J. Math. 14 (2) (2016) 431441.

[4]V. Lakshmikantham, L. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009) 43414349.

[5]A. Petruşel, G. Petruşel, B. Samet, J.-C. Yao, Coupled fixed point theorems for symmetric contractions in b-metric spaces with applications to operator equation systems, Fixed Point Theory 17 (2) (2016) 457476.

[6]A. Petruşel, G. Petruşel, B. Samet, A study of the coupled fixed point problem for operators satisfying a max-symmetric condition in b-metric spaces with applications to a boundary value problem, Miskolc Math. Notes 17 (1) (2016) 501516.

[7]B. Samet, C. Vetro, Coupled fixed point, F-invariant set and fixed point of N-order, Ann. Funct. Anal. 1 (2010) 4656.

[8]B. Samet, C. Vetro, Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 42604268.

[9]B. Samet, E. Karapinar, H. Aydi, V.C. Rajić, Discussion on some coupled fixed point theorems, 2013:50, 12 pp. (2013).

[10]T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 13791393.

[11]W.A. Kirk, P.S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (1) (2003) 7989.

[12]B.S. Choudbury, P. Maity, Cyclic coupled fixed point result using Kannan type contractions, J. Oper. 2014 (2014) 876749, 5.

[13]X. Udo-utun, Existence of strong coupled fixed points for cyclic coupled Ćirić-type mappings, J. Oper. 2014 (2014) 381685, 4.

[14]I.A. Rus, M.A. Şerban, Some generalizations of a Cauchy lemma and applications, in: Topics in Mathematics, Computer Science and Philosophy (Şt. Cobzaş Ed.), Cluj University Press, 2008, pp. 173181.

[15]I.A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, 2001.

[16]A. Magdaş, Fixed point theorems for generalized contractions defined on cyclic representations, J. Nonlinear Sci. Appl. 8 (2015) 12571264.

[17]S. Reich, Zaslavski, Well-posedness of fixed point problems, Far East J. Math. Sci. III (2001) 393401.

[18]J. Fletcher, W.B. Moors, Chebyshev Sets, J. Aust. Math. Soc. 98 (2015) 161231.

[19]G. Petruşel, Cyclic representations and periodic points, Stud. Univ. Babeş-Bolyai Math. 50 (3) (2005) 107112.

[20]S.P. Singh, B. Watson, P. Srivastava, Fixed Point Theory and Best Approximation: The KKM-Map Principle, Kluwer Academic Publishers, Dordrecht, 1997.

[21]A. Magdaş, A fixed point theorem for Ćirić type multivalued operators satisfying a cyclical condition, J. Nonlinear Convex Anal. 17 (6) (2015) 11091116.

[22]O. Mleşniţe, A. Petruşel, Existence and Ulam-Hyers stability results for multivalued coincidence problems, Filomat 26 (5) (2012) 965976.

[23]T. Suzuki, M. Kikkawa, C. Vetro, The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal. 71 (2918) (2009) 7–8–2926.

[24]A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2) (2006) 10011006.

[25]A. Magdaş, Best proximity problems for Ćirić type multivalued operators satisfying a cyclic condition, Stud. Univ. Babeş -Bolyai Math. 62 (3) (2017) 395405.

Acknowledgements

The author is thankful to the referees for their useful suggestions. Declaration of Competing Interest: No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work.The publisher wishes to inform readers that the article “Coupled fixed points and coupled best proximity points for cyclic Ćirić type operators” was originally published by the previous publisher of the Arab Journal of Mathematical Sciences and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was necessary for the journal to transition from the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To access and cite this article, please use Magdaş, A. (2019), “Coupled fixed points and coupled best proximity points for cyclic Ćirić type operators”, Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 179-196. The original publication date for this paper was 22/05/2019.

Corresponding author

Adrian Magdaş and can be contacted at: amagdas@yahoo.com