# Fixed Points ofL-Fuzzy Mappings in Ordered b-Metric Spaces

Authors: Ismat Beg; Mohamed A. Ahmed; Hatem A. Nafadi
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We prove common fixed point theorems for weakly commuting and occasionally coincidentally idempotentL-fuzzy mappings in ordered b-metric spaces. We also obtain common fixed point for pair of mapping satisfying (JCLR) property. An application to integral type and usual contractive condition is given.


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## Body

## 1. Introduction

In 1967, Goguen [1] introduced the notion of L-fuzzy sets as a generalization of fuzzy sets [2]. Afterward, Heilpern [3] gave the concept of fuzzy mappings and proved fixed point theorems for fuzzy contractive mappings in metric linear spaces as a generalization of Nadler [4] contraction principle. Subsequently, several authors studied the generalizations/extensions and applications of these results; in many papers sufficient conditions for the existence of fixed point for fuzzy and L-fuzzy contractive mappings in metric spaces and b-metric spaces are obtained (see [5–8]). Rashid et al. [9] proved common L-fuzzy fixed point theorem in complete metric spaces. They extended the results of Heilpern [3] into L-fuzzy mappings in metric spaces. In this paper, we prove some common fixed point theorems for L-fuzzy mappings in ordered b-metric spaces. An application to integral type and usual contractive condition is also given, we will generalize and extend results [10, 11].


## 2. Preliminaries

Definition 1 (see [12]).
LetX be a nonempty set. A mapping d:X×X→[0,∞) is called b-metric if there exists a real number b≥1 such that, for every x,y,z∈X, we have(d1)
d(x,y)=0⇔x=y,(d2)
d(x,y)=d(y,x),(d3)
d(x,z)≤b{d(x,y)+d(y,z)}.
In this case, the pair(X,d) is called a b-metric space.Definition 2 (see [13]).
Let(X,d) be a b-metric space. A sequence {xn} in X is called(i)
convergent if and only if there existsx∈X such that d(xn,x)→0 as n→∞,(ii)
Cauchy if and only ifd(xn,xm)→0 as m,n→∞.
Ab-metric space is said to be complete if and only if each Cauchy sequence in this space is convergent.Let(X,d) be a b-metric space, let CL(X) denote the collection of all nonempty closed subsets of X, let W(X) be the set of all fuzzy sets of X, where its α-level sets are nonempty compact subsets of X, and let R+ be the set of nonnegative real numbers; define the metrics d:X×CL(X)→R+ and H:CL(X)×CL(X)→R+ as d(x,A)=infy∈A⁡d(x,y) and H(A,B)=max⁡{supa∈A⁡d(a,B),supb∈B⁡d(A,b)}. Notice that d(a,B)≤d(a,b) and d(a,B)≤H(A,B), for each a∈A and b∈B. Let f:X→X, F:X→CL(X), and C(f,F) be the set of the coincidence points of (f,F).Lemma 3 (see [14]).
Let(X,d) be a b-metric space, A,B∈CL(X); then d(a,B)≤H(A,B) for all a∈A.Definition 4 (see [15]).
A partially ordered set consists of a setX and a binary relation ⪯ on X which satisfies the following conditions for all x,y,z∈X:(1)
x⪯x (reflexivity).(2)
Ifx⪯y and y⪯x, then x=y (antisymmetry).(3)
Ifx⪯y and y⪯z, then x⪯z (transitivity).
A set with a partial order⪯ is called a partially ordered set. Let (X,⪯) be a partially ordered set and x,y∈X. Elements x and y are said to be comparable elements of X if either x⪯y or y⪯x.Definition 5.
LetA and B be two nonempty subsets of (X,⪯); the relation ⪯1 between A and B is defined as A⪯1B: if for every a∈A there exists b∈B such that a⪯b.Definition 6 (see [1]).
A functionA:X→L is said to be an L -fuzzy set if A is a mapping from a nonempty set X into a complete distributive lattice L.Definition 7 (see [9, 10, 16]).
LetA be an L-fuzzy set in X. The αL-level set of A (denoted by AαL) is defined as (1)AαL=x∈X:αL⪯LAxifαL∈L∖0L,A0L=x∈X:0L⪯LAx¯,where B¯ denotes the closure of the set B.Definition 8 (see [9]).
LetX and Y be two arbitrary nonempty sets and IL(Y) denote the collection of all L-fuzzy sets in Y. A mapping F is called L-fuzzy mapping if F is a mapping from X into IL(Y). An L-fuzzy mapping F is an L-fuzzy subset on X×Y with membership function F(x)(y). The function F(x)(y) is the grade of membership of y in F(x).Definition 9 (see [9]).
LetF,G be L -fuzzy mappings from an arbitrary nonempty set X into IL(X). A point z∈X is called an L-fuzzy fixed point of F if z∈{Fz}αL, where αL∈L∖{0L}. The point z∈X is called a common L-fuzzy fixed point of F and G if z∈{Fz}αL∩{Gz}αL.Remark 10 (see [10, 16]).
IfαL=1L then the L-fuzzy fixed point is just the fixed point for the L-fuzzy mapping.Definition 11 (see [17]).
Let(X,d) be a metric space, f:X→X, and F:X→CL(X). Then the pair (f,F) is said to have (CLR) property if there exists a sequence {xn} in X and A∈CL(X) such that limn→∞⁡fxn=u∈A=limn→∞⁡Fxn, with u=fv, for some u,v∈X.Definition 12 (see [18]).
Let(X,d) be a metric space, Y⊆X, f:X→X, and F:X→CL(X). Two mappings (f,F) are said to be occasionally coincidentally idempotent if f2x=fx for some x∈C(f,F).Definition 13 (see [19]).
Let(X,d) be a metric space, f,g:X→X, and F,G:X→W(X). Let x0∈X; if there exists a sequence {yn} in X such that (2)y2n+1=gx2n+1⊂Fx2n,y2n+2=fx2n+2⊂Gx2n+1, then O(F,G,f,g,x0) is called the orbit for the mappings (F,G,f,g).Definition 14 (see [19]).
Metric spaceX is called x0 joint orbitally complete, if every Cauchy sequence of each orbit at x0 is convergent in X.Definition 15 (see [20]).
Letf be a mapping from a subset Y of a metric space (X,d) into X and F:Y→CL(X); then f is said to be F-weakly commuting at x∈Y if f2x∈Ffx.


## 3. Definitions inb-Metric Spaces

Let(X,d) be a b-metric space, Y⊆X, f,g:Y→X and F,G are two L-fuzzy mappings from Y into IL(X) such that, for each x∈Y, αL∈L∖{0L}, {Fx}αL and {Gx}αL are nonempty closed subsets of X. Similar to [11], first we rewrite some notions in b-metric spaces as follows.Definition 16.
The pairs(f,F) and (g,G) are said to have (JCLR) property if there exist two sequences {xn} and {yn} in Y and A,B∈CL(X) such that (3)limn→∞fxn=limn→∞gyn=u,limn→∞FxnαL=A,limn→∞GynαL=B,with u=fv=gw,u∈A∩B, for some u,v,w∈Y.Definition 17.
Two mappingsf and F are said to be occasionally coincidentally idempotent if f2x=fx for some x∈C(f,F).Definition 18.
Two mappingsf and F are said to be F-weakly commuting at x∈Y if ffx∈{Ffx}αL.LetΦ be the family of all continuous mappings ϕ:[0,∞)6→[0,∞) satisfying the following properties:(ϕ1)
F is nondecreasing in the 1st variable and nonincreasing in the 3rd, 4th, 5th, and 6th coordinate variables.(ϕ2)
There existsh∈(0,1) such that for every u,v≥0,b≥1 with ϕ(u,v,v,u,b(u+v),0)≤0 or ϕ(u,v,u,v,0,b(u+v))≤0 implies u≤hv.Example 19.
(i)ϕt1,t2,t3,t4,t5,t6=t1-hmax⁡{t2,(t3+t4)t5t6}.
(ii)ϕ(t1,t2,t3,t4,t5,t6)=t1-hmin⁡{t2,(t3+t4)/b,(t5+t6)/b2},b≥2.
(iii)ϕ(t1,t2,t3,t4,t5,t6)=t1-ht2.
LetΨ be the family of continuous mappings ψ:[0,∞)3→R, which is nondecreasing in the first coordinate and satisfying the following condition for each u,v≥0: if ψ(u,v,v)≤0 or ψ(u,v,u)≤0, then u≤v.Example 20.
(i)ψ(t1,t2,t3)=t1-min⁡{t2,(t2+t3)/n},n≥2.
(ii)ψ(t1,t2,t3)=t1-t2.
(iii)ψ(t1,t2,t3)=t1-(t2+t3)/n,n≥2.


## 4. Common Fixed Points in Orderedb-Metric Spaces

Theorem 21.
Let(X,d) be a b-metric space, Y⊆X, and ⪯ be a partial order defined on Y, f,g:Y→X such that f(Y) and g(Y) are closed. Suppose that F,G are two L-fuzzy mappings from Y into IL(X) such that for each x∈Y and αL∈L∖{0L}, {Fx}αL and {Gx}αL are nonempty closed subsets of X which satisfy the following conditions:(1)
{Fx}αL⪯1g(Y) and {Gx}αL⪯1f(Y).(2)
gy∈{Fx}αL or fy∈{Gx}αL implies x⪯y.(3)
Ifyn→y, then yn⪯y for all n∈N.(4)
(f,F) and (g,G) are weakly commuting and occasionally coincidentally idempotent.(5)
For all comparable elementsx,y∈Y and E>0, there exists ϕ∈Φ such that(4)ϕQ+Emin⁡Q≤0,where(5)Q=HFxαL,GyαL,dfx,gy,dfx,FxαL,dgy,GyαL,dfx,GyαL,dgy,FxαL,(6)
One off(Y) or g(Y) is x0 joint orbitally complete for some x0∈Y;
thenf and g have a common fixed point, while F and G have a common L-fuzzy fixed pointProof.
Letx0∈X and y0=fx0; then by (1), there exist x1,x2∈Y such that y1=gx1∈{Fx0}αL and y2=fx2∈{Gx1}αL, from (2), x0⪯x1⪯x2. Now, d(y1,y2)=d(gx1,fx2)≤H({Fx0}αL,{Gx1}αL); then (6)ϕHFx0αL,Gx1αL,dfx0,gx1,dfx0,Fx0αL,dgx1,Gx1αL,dfx0,Gx1αL,0≤0,but (7)ϕdy1,y2,dy0,y1,dy0,y1,dy1,y2,bdy0,y1+dy1,y2,0≤ϕHFx0αL,Gx1αL,dfx0,gx1,dfx0,Fx0αL,dgx1,Gx1αL,dfx0,Gx1αL,0≤0;then from the property ϕ(u,v,v,u,b(u+v),0)≤0 which implies u≤hv, there exists h∈(0,1) such that d(y1,y2)≤hd(y0,y1). Again, y2=fx2∈{Gx1}αL, y3=gx3∈{Fx2}αL, x2⪯x3, and d(y3,y2)=d(gx3,fx2)≤H({Fx2}αL,{Gx1}αL); then (8)ϕHFx2αL,Gx1αL,dfx2,gx1,dfx2,Fx2αL,dgx1,Gx1αL,0,dgx1,Fx2αL≤0.Also (9)ϕdy2,y3,dy1,y2,dy2,y3,dy1,y2,0,bdy1,y2+dy2,y3≤ϕHFx2αL,Gx1αL,dfx2,gx1,dfx2,Fx2αL,dgx1,Gx1αL,0,dgx1,Fx2αL≤0;then from the property ϕ(u,v,u,v,0,b(u+v))≤0 which implies u≤hv, there exists h∈(0,1) such that d(y2,y3)≤hd(y1,y2)≤h2d(y0,y1). By induction we obtain d(yn,yn+1)≤hnd(y0,y1). Continuing in this way, we obtain an orbit O(F,G,f,g,x0) such that y2n+1=gx2n+1∈{Fx2n}αL and y2n+2=fx2n+2∈{Gx2n+1}αL. Further, (10)dyn,ym≤bdyn,yn+1+b2dyn+1,yn+2+⋯+bm-n-1dym-1,ym≤bdyn,yn+1+b2dyn+1,yn+2+⋯+bm-ndym-1,ym≤bhndy0,y1t+b2hn+1dy0,y1+⋯+bm-nhm-1dy0,y1=bhn1-bhdy0,y1.Therefore, limn,m→∞⁡d(yn,ym)=0. Hence, {yn} is a Cauchy sequence. As {y2n+2} is a Cauchy sequence in f(Y), and f(Y) is joint orbitally complete, therefore, there exists z∈X such that y2n+2→z=fu, for u∈Y. Next, we show that z∈{Fu}αL. Since (11)ϕQ+Emin⁡Q≤0,where (12)Q=HFuαL,Gx2n+1αL,dfu,gx2n+1,dfu,FuαL,dgx2n+1,Gx2n+1αL,dfu,Gx2n+1αL,dgx2n+1,FuαL,then (13)ϕHFuαL,Gx2n+1αL,dfu,gx2n+1,dfu,FuαL,dgx2n+1,Gx2n+1αL,dfu,Gx2n+1αL,dgx2n+1,FuαL≤0,ϕdy2n+2,FuαL,dz,y2n+1,dz,FuαL,dy2n+1,y2n+2,dz,y2n+2,bdy2n+1,FuαL≤ϕHFuαL,Gx2n+1αL,dfu,gx2n+1,dfu,FuαL,dgx2n+1,Gx2n+1αL,dfu,Gx2n+1αL,dgx2n+1,FuαL≤0.As n→∞(14)ϕdz,FuαL,0,dz,FuαL,0,0,bdz,FuαL≤0.By ϕ(u,v,u,v,0,b(u+v))≤0 which implies u≤hv, we have d(z,{Fu}αL)≤h.0=0. Thus, z∈{Fu}αL. As z=fu∈{Fu}αL∈g(Y), therefore, there exists υ∈Y such that z=gυ. Similarly, z=gυ∈{Gυ}αL. Since the pair (F,f) are weakly commuting and occasionally coincidentally idempotent and z=fu∈{Fu}αL, then z=fu=ffu=fz; therefore, z=fz=ffu∈{fFu}αL⊆{Ffu}αL={Fz}αL. Also z=gυ∈{Gυ}αL; then z=gυ=ggυ=gz; therefore, gz=ggυ∈{gGυ}αL⊆{Ggυ}αL={Gz}αL.Example 22.
LetY⊆X=[0,1] and d(x,y)=x-y2; then (X,d) is a b-metric space with b=2. Define the partial order x⪯y as x≤y for each x,y∈X and X⪯1Y as follows: for each x∈X, there exists y∈Y such that x≤y. Let L=[0,1] and define the maps f,g,F,G on X as gx=x/2, fx=x/3, (15)Fxy=0,if0≤y≤x5,13,ifx5<y<x4,23,ifx4≤y≤1,Gxy=0,if0≤y≤x7,16,ifx7<y<x6,14,ifx6≤y≤1.Define the sequences x2n, x2n+1, and x2n+2 in Y as x2n={1/(2n+1)}, x2n+1={1/(2n+2)}, and x2n+2={1/(2n+3)}, n∈N; then y2n+1=gx2n+1=1/2(2n+2) and y2n+2=fx2n+2=1/3(2n+3). Now, {Fx2n}2/3=[1/4(2n+1),1] and {Gx2n+1}1/4=[1/6(2n+2),1]; then gx2n+1∈{Fx2n}2/3 and fx2n+2∈{Gx2n+1}1/4. Further, limn→∞⁡fx2n+2=limn→∞⁡gx2n+1=0∈[0,1]=limn→∞⁡{Fx2n}2/3=limn→∞⁡{Gx2n+1}1/4. Let ϕ(t1,…,t6)=t1-ht2; then we have (16)ϕQ+Emin⁡Q=0,where (17)Q=HFx2nαL,Gx2n+1αL,dfx2n+2,gx2n+1,dfx2n+2,Fx2nαL,dgx2n+1,Gx2n+1αL,dfx2n+2,Gx2n+1αL,dgx2n+1,Fx2nαL.Finally, f0=ff0∈[0,1]=Ff0 and gg0=g0∈[0,1]=Gg0; that is, (f,F) and (g,G) are weakly commuting and occasionally coincidentally idempotent (f,F) and (g,G) are weakly commuting and occasionally coincidentally idempotent. Now, f,g,F,G satisfy all conditions of Theorem 21 and f0=g0=0∈[0,1]={F0}2/3={G0}1/4 is a common fixed point.Corollary 23.
Let(X,d) be a complete b-metric space, Y⊆X, and ⪯ be a partial order defined on Y, f,g:Y→X such that f(Y) and g(Y) are closed. Suppose that F,G are two L-fuzzy mappings from Y into IL(X) such that, for each x∈Y and αL∈L∖{0L}, {Fx}αL and {Gx}αL are nonempty closed subsets of X which satisfy conditions ((1)–(5)); then f and g have a common fixed point, while F and G have a common L-fuzzy fixed point.Remark 24.
(
1
) Theorem 21 is a generalization of Theorem 2.2 [6], Theorem 14 [9], and Theorem 3.1 [8].
(
2
) In Theorem 21, one may put one of f=g or F=G or both; in this case, condition (4) has the following versions: (18)ϕHFxαL,GyαL,dfx,fy,dfx,FxαL,dfy,GyαL,dfx,GyαL,dfy,FxαL+Emin⁡HFxαL,GyαL,dfx,fy,dfx,FxαL,dfy,GyαL,dfx,GyαL,dfy,FxαL≤0,ϕHFxαL,FyαL,dfx,gy,dfx,FxαL,dgy,FyαL,dfx,FyαL,dgy,FxαL+Emin⁡HFxαL,FyαL,dfx,gy,dfx,FxαL,dgy,FyαL,dfx,FyαL,dgy,FxαL≤0,ϕHFxαL,FyαL,dfx,fy,dfx,FxαL,dfy,FyαL,dfx,FyαL,dgy,FxαL+Emin⁡HFxαL,FyαL,dfx,gy,dfx,FxαL,dfy,FyαL,dfx,FyαL,dfy,FxαL≤0.Theorem 25.
Let(X,d) be a b-metric space, Y⊆X, and ⪯ be a partial order defined on Y, f,g:Y→X such that f(Y) and g(Y) are closed. Suppose that Fn is a sequence of L-fuzzy mappings from Y into IL(X) such that, for each x∈Y and αL∈L∖{0L}, {Fnx}αL are nonempty closed subsets of X which satisfy the following conditions:(1)
{Fkx}αL⪯1g(Y) and {Flx}αL⪯1f(Y).(2)
gy∈{Fkx}αL or fy∈{Flx}αL implies x⪯y.(3)
Ifyn→y, then yn⪯y for all n∈N.(4)
(f,Fk) and (g,Fl) are weakly commuting and occasionally coincidentally idempotent.(5)
For all comparable elementsx,y∈Y, k=2n+1 and l=2n+2, n∈N and E>0, there exists ϕ∈Φ such that(19)ϕQ+Emin⁡Q≤0,where(20)Q=HFkxαL,FlyαL,dfx,gy,dfx,FkxαL,dgy,FlyαL,dfx,FlyαL,dgy,FkxαL.(6)
One off(Y) or g(Y) is x0 joint orbitally complete for some x0∈Y,
thenf and g have a common fixed point while Fk and Fl have a common L-fuzzy fixed point.Proof.
PutF=Fk and G=Fl in Theorem 21. This concludes the proof.


## 5. Common Fixed Points with (JCLR) Property

Theorem 26.
Let(X,d) be a b-metric space, Y⊆X, and ⪯ be a partial order defined on Y, f,g:Y→X such that f(Y) and g(Y) are closed. Suppose that F,G are two L-fuzzy mappings from Y into IL(X) such that, for each x∈Y and αL∈L∖{0L}, {Fx}αL and {Gx}αL are nonempty closed subsets of X which satisfy the following conditions:(1)
(f,F) and (g,G) satisfy (JCLR) property.(2)
(f,F) and (g,G) are weakly commuting and occasionally coincidentally idempotent.(3)
For all comparable elementsx,y∈Y and E>0 there exists ϕ∈Φ such that(21)ϕQ+Emin⁡Q≤0,where(22)Q=HFxαL,GyαL,dfx,gy,dfx,FxαL,dgy,GyαL,dfx,GyαL,dgy,FxαL;then f and g have a common fixed point, while F and G have a common L-fuzzy fixed point.Proof.
Since(f,F) and (g,G) satisfy (JCLR) property, there exist two sequences {xn},{yn} in Y and A,B∈CL(X) such that (23)limn→∞⁡fxn=limn→∞⁡gyn=u,limn→∞⁡FxnαL=A,limn→∞⁡GynαL=B.With u=fv=gw and u∈A∩B, for some u,v,w∈Y, we show that A=B, since (24)ϕQ+EQ≤0,where (25)Q=HFxnαL,GynαL,dfxn,gyn,dfxn,FxnαL,dgyn,GynαL,dfxn,GynαL,dgyn,FxnαL.As n→∞, (26)ϕHA,B,0,dfv,A,dfv,B,dfv,B,dfv,A≤0,which gives (27)ϕHA,B,0,0,HA,B,bHA,B+0,0≤ϕHA,B,0,dfv,A,dfv,B,dfv,B,dfv,A≤0.From the property ϕ(u,v,u,v,b(u+v),0)≤0 which implies u≤hv, there exists h∈(0,1) such that H(A,B)≤h.0=0, so that A=B. Now, fv∈{Fv}αL; to prove this, since fv∈A, we show that A={Fv}αL. As (28)ϕQ+EQ≤0,where (29)Q=HFvαL,GynαL,dfv,gyn,dfv,FvαL,dgyn,GynαL,dfv,GynαL,dgyn,FvαL,when n→∞, we have that (30)ϕHFvαL,A,0,dfv,FvαL,dfv,A,dfv,A,dfv,FvαL≤0.So that (31)ϕHFvαL,A,0,HA,FvαL,0,0,b0+HA,FvαL≤ϕHFvαL,A,0,dfv,FvαL,dfv,A,dfv,A,dfv,FvαL≤0.From the property ϕ(u,v,u,v,0,b(u+v))≤0 which implies u≤hv, there exists h∈(0,1) such that H({Fv}αL,A)≤h.0=0; this gives H(A,{Fv}αL)=0; then A={Fv}αL, and fv∈{Fv}αL. By a similar way, one can find gw∈{Gw}αL for w∈Y. Further, ffv=fv and ffv∈{Ffv}αL so that u=fu∈{Fu}αL. Also, ggw=gw and ggw∈{Ggw}αL imply u=gu∈{Gu}αL. Then f,g,F,G have a common fixed point.Theorem 27.
In Theorem26, we may replace condition (3) by another: for all x,y∈Y and E>0 there exist ψ∈Ψ such that (32)ψHFxαL,GyαL,dfx,gy,dfx,FxαL+Emin⁡HFxαL,GyαL,dfx,gy,dfx,FxαL≤0;then f and g have a common fixed point, while F and G have a common L-fuzzy fixed point.Proof.
Since(f,F) and (g,G) satisfy (JCLR) property, there exist two sequences {xn},{yn} in Y and A,B∈CL(X) such that limn→∞⁡fxn=limn→∞⁡gyn=u,limn→∞⁡{Fxn}αL=A,limn→∞⁡{Gyn}αL=B, with u=fv=gw and u∈A∩B, for some u,v,w∈Y. Now, we show that fv∈{Fv}αL; otherwise, since(33)ψHFvαL,GynαL,dfv,gyn,dfv,FvαL+Emin⁡HFvαL,GynαL,dfv,gyn,dfv,FvαL≤0.When n→∞, we have ψ(H({Fv}αL,B),0,d(fv,{Fv}αL))≤0. But fv∈B implies d(fv,{Fv}αL)=d({Fv}αL,fv)≤H({Fv}αL,B); then (34)ψdFvαL,fv,0,dfv,FvαL≤ψHFvαL,B,0,dfv,FvαL≤0.By ψ(u,v,u)≤0⇒u≤v, this gives d({Fv}αL,fv)≤0 which is a contradiction of d(x,y)≥0,∀x,y∈Y; then d(fv,{Fv}αL)=0 and fv∈{Fv}αL. Also, gw∈{Gw}αL for w∈Y. We prove this by contradiction; otherwise, since(35)ψHFxnαL,GwαL,dfxn,gw,dfxn,FxnαL+Emin⁡HFxnαL,GwαL,dfxn,gw,dfxn,FxnαL≤0.Letting n→∞, we have ψ(H(A,{Gw}αL),0,d(fv,A))≤0, but gw∈A implies d(gw,{Gw}αL)≤H(A,{Gw}αL); then (36)ψdgw,GwαL,0,0≤ψHA,GwαL,0,0≤0.From ψ(u,v,v)≤0⇒u≤v, this gives d(gw,{Gw}αL)≤0 which is a contradiction; then gw∈{Gw}αL. Further, being weakly commuting and occasionally coincidentally idempotent imply that, respectively, u=fv=ffv=fu and ffv∈{Ffv}αL, so that u=fu∈{Fu}αL. Also, ggw=gw and ggw∈{Ggw}αL imply u=gu∈{Gu}αL. Then f,g,F,G have a common fixed point.


## 6. An Application to Integral Contractive Condition

Suppose thatα,β:[0,∞)→[0,∞) are summable nonnegative Lebesgue integrable functions such that for each ϵ>0, ∫0ϵα(s)ds≥0 and ∫0ϵβ(e)de≥0. As a simple application of Theorems 21, 25, and 26, we give the following result.Theorem 28.
Let(X,d) be a b-metric space, Y⊆X, and let ⪯ be a partial order defined on Y, f,g:Y→X such that f(Y) and g(Y) are closed. Suppose that F,G are two L-fuzzy mappings from Y into IL(X) such that for each x∈Y and αL∈L∖{0L}, {Fx}αL and {Gx}αL are nonempty closed subsets of X and satisfying the following conditions(1)
(f,Fk) and (g,Fl) satisfy (JCLR) property.(2)
(f,Fk) and (g,Fl) are weakly commuting and occasionally coincidentally idempotent.(3)
For all comparable elementsx,y∈Y and E>0, there exist ϕ∈Φ such that(37)∫0ϕQαsds+E∫0min⁡Qβede≤0,where(38)Q=HFkxαL,FlyαL,dfx,gy,dfx,FkxαL,dgy,FlyαL,dfx,FlyαL,dgy,FkxαL,where k=2n+1, l=2n+2, n∈N; then f and g have a common fixed point, while Fk and Fl have a common L-fuzzy fixed point.Proof.
Since(f,Fk) and (g,Fl) satisfy the property (JCLR), there exist two sequences {xkn},{ykn} in Y and Ak,Bk∈CL(X) such that (39)limn→∞⁡fxkn=limn→∞⁡gykn=uk,limn→∞⁡FkxknαL=Ak,limn→∞⁡FlyknαL=Bk,with uk=fvk=gwl, uk∈Ak∩Bk, for some uk,vk,wk∈Y. Now, we show that Ak=Bk. As (40)∫0ϕQαsds+E∫0min⁡Qβede≤0,where (41)Q=HFkxknαL,FlyknαL,dfxkn,gykn,dfxkn,FkxknαL,dgykn,FlyknαL,dfxkn,FlyknαL,dgykn,FkxknαL.As n→∞, we find that (42)∫0ϕHAk,Bk,0,0,dfvk,Bk,dfvk,Bk,0αsds≤0.So that (43)ϕHAk,Bk,0,0,dfvk,Bk,dfvk,Bk,0≤0,which give (44)∫0ϕHAk,Bk,0,0,HAk,Bk,bHAk,Bk+0,0αsds≤∫0ϕHAk,Bk,0,dfvk,Ak,dfvk,Bk,dfvk,Bk,dfvk,Akαsds≤0.By (ϕ), we have H(Ak,Bk)=0; that is, Ak=Bk. As uk∈Ak, we show that {Fkvk}αL=Ak. As (45)∫0ϕQαsds+E∫0min⁡Qβede≤0,where (46)Q=HFkvkαL,FlyknαL,dfvk,gykn,dfvk,FkvkαL,dgykn,FlyknαL,dfvk,FlyknαL,dgykn,FkvkαL,which on making n→∞ reduces to (47)∫0ϕHFkvkαL,Ak,0,0,dFkvkαL,Ak,dFkvkαL,Ak,0αsds≤0.By a similar way, we have {Fkvk}αL=Ak. The remaining parts are easy to prove. This concludes the proof.Now, we may apply Theorem28 with the following example.Example 29.
LetX=[0,1], Y⊆X, and d(x,y)=x-y2; then (X,d) is a b-metric space with b=2. Define the partial order x⪯y as x≤y for each x,y∈X and X⪯1Y as follows: for each x∈X, there exist y∈Y such that x≤y. Let L=[0,1] and define the maps f,g,F,G on X as(48)fx=16,if0≤x<12,x3,if12≤x≤1,gx=16,if0≤x<13,x2,if13≤x≤1,Fxy=56,if0≤y≤110,45,if110<y<x+19,23,ifx+19≤y≤1,Gxy=34,if0≤y<19,12,if19≤y<x+18,14,ifx+18≤y≤1,for all x,y∈X. Define two sequences {xn} and {yn} in Y such that {xn}={1/2+1/4n},{yn}={1/3+1/6n},n∈N. Now, we show that for x,y∈Y there exist ϕ∈Φ such that (49)∫0ϕQαsds+E∫0min⁡Qβede≤0,where (50)Q=HFxαL,GyαL,dfx,gy,dfx,FxαL,dgy,GyαL,dfx,GyαL,dgy,FxαL.Since limn→∞⁡fxn=limn→∞⁡gyn=1/6∈[1/6,1]=limn→∞⁡Fxn=limn→∞⁡Gyn, the hybrid pairs (f,F) and (g,G) satisfy the property (JCLR). Also f(1/6)=ff(1/6)=1/6 and g(1/6)=gg(1/6)=1/6 (occasionally coincidentally idempotent). Finally, ff(1/6)=1/6∈[7/54,1]={Ff(1/6)}2/3 and gg(1/6)=1/6∈[7/48,1]={Gg(1/6)}2/3 (weakly commuting). Let ϕ(t1,…,t6)=t1-hmax⁡{t2,(t3+t4)t5t6}. Further, (51)ϕHFxnαL,GynαL,dfxn,gyn,dfxn,FxnαL,dgyn,GynαL,dfxn,GynαL,dgyn,FxnαL=0,which gives (52)∫0ϕHFxnαL,GynαL,dfxn,gyn,dfxn,FxnαL,dgyn,GynαL,dfxn,GynαL,dgyn,FxnαLαsds=0.Then (53)∫0ϕQαsds+E∫0min⁡Qβede≤0,where (54)Q=HFxnαL,GynαL,dfxn,gyn,dfxn,FxnαL,dgyn,GynαL,dfxn,GynαL,dgyn,FxnαL.Thus, the maps f,g,F,G satisfy the conditions in Theorem 28. Now, the maps f,g,F,G have a common fixed point.Remark 30.
In Theorem28, condition (37), one may put α(s)=1 or β(e)=1 or both, which give the following: (55)ϕQ+E∫0min⁡Qβede≤0,∫0ϕQαsds+Emin⁡Q≤0,ϕQ+Emin⁡Q≤0.Theorem 31.
Let(X,d) be a b-metric space, Y⊆X, f,g:Y→X and F,G are two L-fuzzy mappings from Y into IL(X) such that for each x∈Y, αL∈L∖{0L}, {Fx}αL and {Gx}αL are nonempty closed subsets of X which satisfy the following condition: for all x,y∈Y and E>0, there exists ϕ∈Φ such that (56)∫0ψQαsds+E∫0min⁡Qβede≤0,where (57)Q=HFxαL,GyαL,dfx,gy,dfx,FxαL.If (f,F) and (g,G) satisfy (JCLR) property, weakly commuting and occasionally coincidentally idempotent, then f and g have a common fixed point, while F and G have a common L-fuzzy fixed point.Proof.
It is similar to Theorem27.



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*Source: 5650242-2018-03-01.xml*