# Existence of Strong Coupled Fixed Points for Cyclic Coupled Ciric-Type Mappings

Authors: Xavier Udo-utun
Journal: Journal of Operators
 (2014)

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## Abstract

In this short communication the concept of cyclic coupled Kannan-type contractions is generalized using a certain class of Ciric-type mappings.


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## Body

## 1. Introduction and Preliminaries

The Banach contraction condition in a metric space(X,d) given by d(Tx,Ty)≤Ld(x,y),0≤L<1, has so many significant generalizations which include the class of generalized contractions defined by Ciric [1] as follows. A mapping T:X→X is called a generalized contraction if and only if there exist nonnegative numbers q.  r, s, and t such that
(1)dTx,Ty≤qd(x,y)+rd(x,Tx)+sd(y,Ty)+tdx,Ty+dy,Tx;sup⁡q+r+s+2s<1.Tis calledcontractive if
(2)dTx,Ty<dx,y.
It is worth mentioning that the contractive condition (2) restricts applications only to the class of continuous operators while the contractive conditions (1) accommodate discontinuous operators as well. The search for contractive conditions that do not require continuity of operators culminated in 1969 with the appearance of the Kannan [2] contractive condition below:
(3)d(Tx,Ty)≤ad(x,Tx)+d(y,Ty),0≤a<12.
The Chatterjea [3] contractive condition which followed is independent of both the contractive condition (2) and the Kannan condition (3) which in turn is independent of (2). Consequently, unlike condition (2) the Kannan condition (3) does not generalize the well-known Banach condition above. In a first attempt, the three contractive conditions were combined by Zamfirescu [4] in one theorem to generalize and extend the Banach fixed point theorem. Following Zamfirescu Ciric unified contractive conditions mentioned above by introducing the larger and unifying class of operators called quasi-contractions. T is called aquasi-contraction (Ćirić [5]) if there exists L∈(0,1) such that
(4)d(Tx,Ty)≤Lmax⁡dx,y,dx,Tx,dy,Ty,lllllllllllllllllllllllllllllldx,Ty,dy,Tx.Ćirić [5] observed that the class of quasi-contractions contains the class of generalized contractions as a proper subclass. Rhoades [6] noted that the Zamfirescu result is generalized by the Ciric contractive condition (4).There have been numerous generalizations and extensions of the Banach fixed point theorem in literature and they are, basically, modifications of those mentioned above. Very recently Choudhury and Maity [7] introduced the concept ofcyclic coupled Kannan-type contractions and established astrong cyclic coupled fixed point result below. We recall the following definition. Let A and B be two nonempty subsets of a given set X. A mapping F:X×X→X, such that F(x,y)∈B if x∈A and y∈B and F(x,y)∈A if x∈B and y∈A, is called a cyclic mapping with respect to A and B.Definition 1 (see [8]).
Let(X,d) be a metric space and A,B⊂X nonempty subsets. T:A∪B→A∪B is called a cyclic (or 2-cyclic) q-contraction if TA⊂B and TB⊂A and the following condition is satisfied:
(5)d(Tx,Ty)≤qd(x,y)+(1-q)dist⁡(A,B)
for all x∈A,y∈B.Definition 2 (see [7]).
LetA and B be two nonempty subsets of a metric space (X,d). A mapping F:X×X→X is called a cyclic coupled Kannan-type contraction with respect to A and B if F is cyclic with respect to A and B and, for some k∈(0,1/2), F satisfies the following condition:
(6)dFx,y,Fu,v≤kdx,F(x,y)+du,Fu,v,
where x,v∈A and y,u∈B.Definition 3 (see [7]).
LetX be a nonempty set. An element (px,py)∈X×X is called a coupled fixed point of a mapping F:X×X→X if px=F(px,py) and py=F(py,px). A point (p,p)∈X×X is called a strong coupled fixed point if F(p,p)=p.Theorem 4 (see [7]).
LetA and B be two nonempty closed subsets of a complete metric space (X,d), F:X×X→X a cyclic coupled Kannan contraction with respect to A and B with A∩B≠∅. Then F has a strong coupled fixed point in A∩B.The aim of this paper is to obtain a generalization of Theorem4 in the context of Ciric-type contractions.


## 2. Main Results

Following Definition2 we formulate the following definition.Definition 5.
LetA and B be two nonempty subsets of a metric space (X,d). A mapping F:X×X→X is called a cyclic coupled Ciric-type mapping with respect to A and B if F is cyclic with respect to A and B and, for some constants q∈(0,1), F satisfies the following condition:
(7)dFx,y,Fu,v≤qMx,u,
where x,v∈A and y,u∈B and M(x,u) is given by
(8)Mx,u=max⁡⁡d(x,u),12du,Fx,y,12dx,Fu,v,lllllllllllll12dx,Fx,y+du,Fu,v.Examples of coupled cyclic Ciric-type mappings include coupled cyclic Kannan-type mappings studied in [7] since the later imply the former. We now prove the main theorem of this work.Theorem 6.
LetA and B be two nonempty closed subsets of a complete metric space (X,d), F:X×X→X a cyclic coupled Kannan contraction with respect to A and B with A∩B≠∅. Then F has a strong coupled fixed point in A∩B.Proof.
Letx0∈A and y0∈B; we put x1=F(y0,x0), y1=F(x0,y0), x2=F(y1,x1), y2=F(x1,x1), and so forth. Then we obtain the following as estimates for respective displacements:
(9)dx1,y1=dFy0,x0,Fx0,y0≤qmax⁡dy0,x0,12dy0,Fx0,y0,lllllllllllllll12dx0,Fy0,x0,llllllllllllll12dy0,Fy0,x0+dx0,Fx0,y0≤qmax⁡dx0,y0,12dy0,y1,12dx0,x1,lllllllllllllll12dy0,x1+dx0,y1≤qmax⁡dx0,y0,dx0,x1,dy0,y1,lllllllllllllldy0,x1,dx0,y1=qτ,
where τ=max⁡{d(x0,y0),d(x0,x1),d(y0,y1),d(y0,x1),d(x0,y1)}.
Similarly, using the fact thatdy1,x2≰qd(y1,x2) we obtain
(10)dy1,x2=dFx0,y0,Fy1,x1≤qmax⁡dx0,y1,12dy1,Fx0,y0,llllllllllllllll12dx0,Fy1,x1,llllllllllllllll12dx0,Fx0,y0lllllllllllllllllll+dy1,Fy1,x112≤qmax⁡dx0,y1,12dx0,x2,lllllllllllllll12dx0,y1+dy1,x2≤qmax⁡d(x0,y1),12dx0,y1+dy1,x2(11)yieldingdy1,x2≤qdx0,y1≤qτ,(12)llllllllllllld(x1,y2)≤qd(y0,x1)≤qτ.
Next, we will use (9), (11), and (12) to obtain the following estimates for the displacements d(x1,x2) and d(y1,y2):
(13)dx1,x2=dFy0,x0,Fy1,x1≤qmax⁡dy0,y1,12dy1,Fx0,y0,lllllllllllllll12dy0,Fy1,x1,llllllllllllll12dy0,Fx0,y0+dy1,Fy1,x1≤qmax⁡d(y0,y1),12d(y0,x2),llllllllllllll12[d(y0,y1)+d(y1,x2)]≤qmax⁡d(y0,y1),12[d(y0,y1)+d(y1,x2)],llllllllllllll12[d(y0,y1)+qd(x0,y1)]≤qmax⁡dy0,y1,12dy0,y1+qdx0,y1(14)Yieldingdx1,x2≤qmax⁡dy0,y1,qdx0,y1≤qτ,(15)lllllllllllllldy1,y2≤qmax⁡dx0,x1,qdy0,x1≤qτ.
Similarly,
(16)dy2,x3≤dFx1,y1,Fy2,x2≤qmax⁡dx1,y2,12dy2,Fx1,y1,lllllllllllllll12dx1,Fy2,x2,llllllllllllll12dx1,Fx1,y1+dy2,Fy2,x2≤qmax⁡d(x1,y2),12d(x1,y3),llllllllllllll12[d(x1,y2)+d(y2,x3)];thatis,dy2,x3≤qdx1,y2≤q2τ,by11,lllllllllllldx2,y3≤qdy1,x2≤q2τ,by12.
Now, to estimated(xn,xm) and d(xn,yn) we proceed as follows: given that dx1,x2≤qτ,  d(y1,y2)≤qτ,  d(y2,x3)≤qd(x1,y2)≤q2τ , and  dx2,y3≤qdy1,x2≤q2τ, we have the following:
(17)dx2,x3=dFy1,x1,Fy2,x2≤qmax⁡dy1,y2,12dy2,Fy1,x1,lllllllllllllll12dy1,Fy2,x2,lllllllllllllll12dy1,Fy1,x1lllllllllllllllllll+dy2,Fy2,x212≤qmax⁡d(y1,y2),12d(y2,x2),12d(y1,x3),llllllllllllll12dy1,x2+dy2,x3,by(14)yieldingd(x2,x3)≤qmax⁡qτ,q2τ=q2τ,llldy2,y3≤q2τ,by(15).
Inductively, we assume d(xk,xk+1)≤qkτ and d(yk,yk+1)≤qkτ; k≥1. Also, we assume d(xk,yk+1)≤qkτ and d(yk,xk+1)≤qkτ; k≥1. Then
(18)dyk+1,xk+2=dFxk,yk,Fyk+1,xk+1≤qmax⁡dxk,yk+1,12dyk+1,Fxk,yk,lllllllllllllll12dxk,Fyk+1,xk+1,lllllllllllllll12dxk,Fxk,ykllllllllllllllllll+dyk+1,Fyk+1,xk+112≤qmax⁡d(xk,yk+1),12d(xk,xk+2),llllllllllllll12dxk,yk+1+dyk+1,xk+2≤qmax⁡12dxk,yk+1,llllllllllllll12dxk,yk+1+dyk+1,xk+2(19)yieldingdyk+1,xk+2≤qdxk,yk+1≤qk+1τ,(20)lllllllllllllld(xk+1,yk+2)≤qd(yk,xk+1)≤qk+1τ.
It follows from (19) and (20) that
(21)dxk+1,xk+2=dFyk,xk,Fyk+1,xk+1≤qmax⁡d(yk,yk+1),12dyk+1,Fxk,yk,llllllllllllllll12dyk,Fyk+1,xk+1,llllllllllllllll12dyk,Fxk,yklllllllllllllllllll+dyk+1,Fyk+1,xk+112≤qmax⁡d(yk,yk+1),12d(yk,xk+2),lllllllllllllll12dyk,yk+1+dyk+1,xk+2≤qmax⁡dyk,yk+1,dyk,xk+1,llllllllllllllldyk+1,xk+2(22)sowehavedxk+1,xk+2≤qmax⁡qkτ,qk+1τ≤qk+1τ,(23)d(yk+1,yk+2)≤qk+1τ.lllllllllllll
Therefore, using (19)–(23) we conclude that for all n≥1 we have d(xn,xn+1)≤qnτ and d(xn,xn+1)≤qnτ. Also, using the fact that d(xn,yn)≤d(xn,yn+1)+d(yn,yn+1) we conclude that d(xn,yn)≤qn2τ for all n≥1 implying that if px=lim⁡xn and py=lim⁡yn then py=px=p and (p,p)∈Fix⁡(F). To prove that the sequences {xn} and {yn} are convergent it suffices to verify that they are Cauchy sequences.
To this end, we observe that ifm≥n then m=n+l for some l≥0 so that
(24)dxn,xm=dxn,xn+p≤dxn,xn+1+dxn+1,xn+2+⋯+d(xm-1,xn+l)≤qn1+q+q2+⋯+qlτ≤qn1-qτ.
Therefore, the sequences {xn} and {yn} are Cauchy sequences and since A and B are closed sets with A∩B≠∅ we submit that the strong cyclic coupled fixed point (p,p) of F so derived satisfies p∈A∩B.


## 3. Conclusion

Further studies include proving our main result and Theorem4 without the restriction A∩B≠∅. This can be studied as a best proximity point problem.



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*Source: 381685-2014-11-23.xml*