# Fuzzy Fixed Point Results ForΦ Contractive Mapping with Applications

Authors: Humaira; Muhammad Sarwar; G. N. V. Kishore
Journal: Complexity
 (2018)

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## Abstract

In this paper, using rational type contractions, common fuzzy fixed point result forΦ contractive mappings involving control functions as coefficients of contractions in the setting of complex-valued metric space is established. The derived results generalizes some result in the existing literature. To show the validity of the derived results an appropriate example and applications are also discussed.


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## Body

## 1. Introduction

Fixed point theory is considered to be the most interesting and dynamic area of research in the development of nonlinear analysis. In this area, Banach contraction principal [1] is an initiative for researchers during last few decades. This principal plays an important and key role in investigating the existence and uniqueness of solution to various problems in mathematics, physics, engineering, medicines, and social sciences which leads to mathematical models design by system of nonlinear integral equations, functional equations, and differential equations. Banach contraction principal has been generalized in different directions by changing the condition of contraction or by the underlying space. For instance, we refer to [2–8]. Particularly Dass and Gupta [9] extended the Banach contraction principal for rational type inequality and obtained fixed point results in metric space, which is further extended to different spaces by many authors. In the meanwhile researchers realized that where division occurs in cone metric spaces, the concept of rational type contraction is not meaningful.To overcome this problem a new metric space was recently established by Azam et al. [10], known as complex-valued metric space, where the author obtained fixed point results via rational type contractive condition. This work was further extended by Sitthikul and Saejung [11]. Afterwards Rouzkard and Imdad [12] extended the aforementioned results of Azam et al. by obtaining common fixed point results which satisfies certain rational contractions in complex-valued metrics spaces. Consequently in [13, 14], the authors extended common fixed point results for multivalued mappings in complex-valued metric space. In addition, Sintunavarat and Kumam [15] derived common fixed point results by substituting the constant coefficients in contractive condition by control functions.Heilpern [16] established the concept of fuzzy mappings and obtained fixed point results in metric linear space. He generalized the results of [1, 17], under the consideration of fuzzy mappings in complete metric linear spaces. Several mathematicians extended the work of Heilpern in different metric spaces for linear contraction. For instance, we refer to [18–24]. While in [25], the author investigated for fuzzy common fixed point with rational contractive condition. The concept of fuzziness is helpful in solving such real world problems where uncertainty occurs and many authors solve such problems by mathematical modeling in terms of fuzzy differential equations. For instance in [26], the author investigated the existence of solution for fuzzy differential equations. Nieto [27] worked on Cauchy problems for continuous fuzzy differential equations. Song et al. studied the global existence of solutions to fuzzy differential equation [28]. Moreover, the existence of fuzzy solution of first order initial value problem was studied in [29], which is lately extended to integrodifferential equations [30]. Recently Long et al. [31] combined the matrix convergent to zero technique with calculations of fuzzy-valued functions, which is quite a new approach to study the system of differential and partial differential equations (PDE’s) in generalized fuzzy metric spaces. In [32] Long et al. improved different results existing in the literature on the existence of coincidence points for a pair of mappings and studied applications to partial differential equations with uncertainty. After the wide study of fuzziness in the system of differential equations, it has now been studied in fractional differential equations to obtain the existence and uniqueness of fuzzy solution under Caputo generalized Hukuhara differentiability; for instance, see [33].In the current work, using rational type contraction, common fuzzy fixed point results forΦ contractive mappings are studied. The established results generalizes some results from the exiting literature particularly the result of Joshi et al. [34] for fuzzy mappings. Applications and appropriate example are also provided.


## 2. Preliminaries

Definition 1 (see [10]).
AssumeC is the set of complex numbers. For ε1,ε2∈C we define a partial order ≾ on C as follows:(Ci)
ε1≾ε2⇔Re⁡ε1≤Re⁡ε2 and Im⁡ε1≤Im⁡ε2;(Cii)
ε1≺ε2⇔Re⁡ε1<Re⁡ε2 and Im⁡ε1<Im⁡ε2;(Ciii)
ε1⋨ε2⇔Re⁡ε1=Re⁡ε2 and Im⁡ε1<Im⁡ε2;(Civ)
ε1=ε2⇔Re⁡ε1=Re⁡ε2 and Im⁡ε1=Im⁡ε2.
Clearly ifa≤b, ⇒az≾bz, for all z∈C and for all a,b∈R. Note that if ε1≠ε2 and one of (Ci), (Cii) and (Ciii) is satisfied then ε1⋨ε2, and we write ε1=ε2 if only (Civ) is satisfied. Note that(i)
0≾ε1⋨ε2⇒|ε1|<|ε2|,∀ε1,ε2∈C;(ii)
ε1≾ε2 and ε2≺ε3⇒ε1≺ε3,∀ε1,ε2,ε3∈C.Definition 2 (see [10]).
LetX be a nonempty set and d:X×X→C be a mapping which satisfies the following conditions:(1)
0≾ρ(z,w), for all z,w∈X and ρ(z,w)=0 if and only if z=w;(2)
ρ(z,w)=ρ(w,z), for all z,w∈X;(3)
ρ(z,w)≾ρ(z,z1)+ρ(z1,w), for all z,z1,w∈X.
Then(X,ρ) is called a complex-valued metric space.Definition 3 (see [10]).
A pointz∈X is known as an interior point of a set Z⊆X, if we find 0≺ϵ∈C such that(1)Bz,ϵ=w∈X:ρz,w≺ϵ⊆Z.A point z∈Z is known as the limit point of Z, if there exists an open ball B(z,ϵ) such that(2)Bz,ϵ∩Z∖z≠ϕ,where 0≺ϵ∈C. A subset Z of X is said to be open if each point of Z is an interior point of Z. Furthermore, Z is said to be closed if it contain all its limit points.
The family(3)B=Bz,ϵ:z∈X,0≺ϵis a subbasis for a Hausdorff topology T on X.Now recall some definitions from [13, 14].Let(X,ρ) be a complex-valued metric space. Throughout this paper we denoted the family of all nonempty closed bounded subsets of complex-valued metric space X by CB(X). For ν∈C we represent(4)sν=z∈C:ν⪯zand for w∈X and B∈CBX.(5)sw,B=⋃b∈Bsρw,b=⋃b∈Bz∈C:ρw,b⪯z.ForA,B∈CB(X), we denote(6)sA,B=⋂p∈Asp,B∩⋂q∈Bsq,A.Let τ be a multivalued mapping from X into CB(X); for z∈X and Q∈CB(X) we define(7)WzQ=ρz,q:q∈Q.Thus for z,w∈X(8)Wzτw=ρz,u:u∈τw.Lemma 4 (see [35]).
Let(X,ρ) be complex-valued metric space.(i)
Letz,w∈C. If z⪯w, then s(z)⊂s(w).(ii)
Letz∈X and D∈N(X). If δ∈s(z,D), then z∈D.(iii)
Letw∈C,P,Q∈CB(X) and p∈P. If v∈s(P,Q), then z∈s(p,Q) for all p∈P or z∈s(P,q) for all q∈Q.Definition 5 (see [10]).
Let{wr} be a sequence in complex-valued metric space (X,ρ) and w∈X; then(i)
w is a limit point of {wr} if for each 0≺ϵ∈C there exists r0∈N such that ρ(wr,w)⪯ϵ for all r⪰r0 and it is written as limr→∞wr=w.(ii)
{wr} is a Cauchy sequence if for any 0≺ϵ∈C there exists r0∈N such that ρ(wr,wr+t)≺ϵ for all r≻r0 where t∈N.(iii)
we say that(X,ρ) is complete complex-valued metric space if every Cauchy sequence in X converges to a point in X.Definition 6 (see [18]).
Let(V,ρ) be a metric space. A fuzzy set B is characterized by its membership function fB:V→[0,1]. A set of elements of V along with its grade of membership is called a fuzzy set. For simplicity we denote fB(u) by B(u). The α-level set of a fuzzy set B is mentioned by [B]α and is defined as follows:(9)Bα=u:Bu≥αifα∈0,1,B0=u:Bu>0¯.Definition 7 (see [18]).
LetL(X) be the family of all fuzzy sets in a metric space X. For G,H∈L(X),G⊂H means G(z)≤H(z) for each z∈X.Definition 8 (see [16]).
AssumeX is an arbitrary set and Y is a metric space. A mapping G is called a fuzzy mapping if G:X→L(Y). A fuzzy mapping G is a fuzzy subset on X×Y with a membership function G(x)(y). The function G(x)(y) is the grade of membership of y in G(x).Definition 9 (see [20]).
Assume that(X,ρ) is complex-valued metric space and G1,G2:X→L(X) are fuzzy mappings. A point w∈X is a fuzzy fixed point of G1 if w∈[G1w]α where α∈[0,1] and a common fuzzy fixed point of G1,G2 if w∈[G1w]α∩[G2w]α. If α=1 then w is known as common fixed point of fuzzy mappings.Definition 10 (see [14]).
Suppose(X,ρ) is complex-valued metric space; the fuzzy mapping G1:X→L(X) enjoys the greatest lower bound property (glb property) on (X,ρ), if, for any w∈X and α∈(0,1], the greatest lower bound of Ww([G1y]α) exists in C for all w,y∈X. Here we mention ρ(w,[G1y]α) by the glb of Ww([G1y]α). That is,(10)ρw,G1yα=inf⁡ρw,u:u∈G1yα.Remark 11 (see [13]).
Let(X,ρ) be a complex-valued metric space. If C=R, then (X,ρ) is a metric space. Furthermore H(A,B)=inf⁡s(A,B) is the Hausdorff distance induced by ρ, where A,B∈CB(X).Definition 12 (see [34]).
SupposeΨ is a collection of nondecreasing functions, Φ:C→C, such that Φ(0)=0 and Φ(t)≺t, when 0≺t.


## 3. Main Result

In this section we present our main results. To present the main results we need the lemmas given below.Lemma 13.
Let(X,ρ) be complex-valued metric space and G1,G2:X→L(X) be fuzzy mappings, such that for each w∈X and some α∈(0,1] there exists [G1w]α,[G2w]α, nonempty closed and bounded subsets of X. Let w0∈X and define the sequence {wk} by(11)w2k+1∈G1w2kα,w2k+2∈G2w2k+1α,∀k=0,1,2,….Assume that there exists a mapping ϕ:X→[0,1) such that ϕ(u)⪯ϕ(w) for all u∈[G1w]α and ϕ(v)⪯ϕ(w) for all v∈[G2w]α. Then ϕ(w2k)⪯ϕ(w0) and ϕ(w2k+1)⪯ϕ(w1).Proof.
Supposew∈X and k=0,1,2,…. Then we have(12)ϕw2k⪯ϕw2k-2forw2k-1∈G1w2k-2α,⪯ϕw2k-4forw2k-2∈G1w2k-4α,⪯⋯⪯ϕw0.Similarly we have(13)ϕw2k+1⪯ϕw1.Theorem 14.
Suppose(X,ρ) is a complete complex-valued metric space and G1,G2:X→L(X) are fuzzy mappings satisfying glb property. Assume that for each y∈X and some α∈(0,1] there exist [G1y]α,[G2y]α which are nonempty closed bounded subsets of X. Suppose there exist mappings ψi:X→[0,1),i=1,…,7 such that(i)
ψi(u)⪯ψi(y),i=1,…,7 for all u∈G1yα and y∈X;(ii)
ψi(v)⪯ψi(y),i=1,…,7 for all v∈G2yα and y∈X;(iii)
∑ψi(y)+2ψ4(y)<1,i=1,2,3,6,7∀y∈X; and(14)Φψ1yρy,w+ψ2yρy,G1yα+ψ3yρw,G2wα+ψ4yρy,G1yαρy,G2wα1+ρy,w+ψ5yρw,G1yαρw,G2wα1+ρy,w+ψ6yρy,G1yαρw,G2wα1+ρy,w+ψ7yρw,G1yαρy,G2wα1+ρy,w∈sG1yα,G2wαfor some Φ∈Ψ and for all y,w∈X. Then G1 and G2 have a common fuzzy fixed point.Proof.
Lety0∈X and y1∈[G1y0]α. Using (14) with y=y0 and w=y1 we get(15)Φψ1y0ρy0,y1+ψ2y0ρy0,G1y0α+ψ3y0ρy1,G2y1α+ψ4y0ρy0,G1y0αρy0,G2y1α1+ρy0,y1+ψ5y0ρy1,G1y0αρy1,G2y1α1+ρy0,y1+ψ6y0ρy0,G1y0αρy1,G2y1α1+ρy0,y1+ψ7y0ρy1,G1y0αρy0,G2y1α1+ρy0,y1∈sG1y0α,G2y1α.By Lemma 4(iii) we have(16)Φψ1y0ρy0,y1+ψ2y0ρy0,G1y0α+ψ3y0ρy1,G2y1α+ψ4y0ρy0,G1y0αρy0,G2y1α1+ρy0,y1+ψ5y0ρy1,G1y0αρy1,G2y1α1+ρy0,y1+ψ6y0ρy0,G1y0αρy1,G2y1α1+ρy0,y1+ψ7y0ρy1,G1y0αρy0,G2y1α1+ρy0,y1∈sy1,G2y1α.By definition there exists some y2∈[G2y1]α, such that(17)Φψ1y0ρy0,y1+ψ2y0ρy0,G1y0α+ψ3y0ρy1,G2y1α+ψ4y0ρy0,G1y0αρy0,G2y1α1+ρy0,y1+ψ5y0ρy1,G1y0αρy1,G2y1α1+ρy0,y1+ψ6y0ρy0,G1y0αρy1,G2y1α1+ρy0,y1+ψ7y0ρy1,G1y0αρy0,G2y1α1+ρy0,y1∈sρy1,y2.Therefore(18)ρy1,y2⪯Φψ1y0ρy0,y1+ψ2y0ρy0,G1y0α+ψ3y0ρy1,G2y1α+ψ4y0ρy0,G1y0αρy0,G2y1α1+ρy0,y1+ψ5y0ρy1,G1y0αρy1,G2y1α1+ρy0,y1+ψ6y0ρy0,G1y0αρy1,G2y1α1+ρy0,y1+ψ7y0ρy1,G1y0αρy0,G2y1α1+ρy0,y1≺ψ1y0ρy0,y1+ψ2y0ρy0,G1y0α+ψ3y0ρy1,G2y1α+ψ4y0ρy0,G1y0αρy0,G2y1α1+ρy0,y1+ψ5y0ρy1,G1y0αρy1,G2y1α1+ρy0,y1+ψ6y0ρy0,G1y0αρy1,G2y1α1+ρy0,y1+ψ7y0ρy1,G1y0αρy0,G2y1α1+ρy0,y1.Using the glb property of G1 and G2 we have(19)ρy1,y2⪯ψ1y0ρy0,y1+ψ2y0ρy0,y1+ψ3y0ρy1,y2+ψ4y0ρy0,y1ρy0,y21+ρy0,y1+ψ5y0ρy1,y1ρy1,y21+ρy0,y1+ψ6y0ρy0,y1ρy1,y21+ρy0,y1+ψ7y0ρy1,y1ρy0,y21+ρy0,y1.It implies that(20)ρy1,y2⪯ψ1y0ρy0,y1+ψ2y0ρy0,y1+ψ3y0ρy1,y2+ψ4y0ρy0,y2+ψ6y0ρy1,y2⪯ψ1y0ρy0,y1+ψ2y0ρy0,y1+ψ3y0ρy1,y2+ψ4y0ρy0,y1+ψ4y0ρy1,y2+ψ6y0ρy1,y2.Finally we get(21)ρy1,y2⪯μρy0,y1ρy1,y2≤μρy0,y1,where(22)μ=ψ1y0+ψ2y0+ψ4y01-ψ3y0+ψ4y0+ψ6y0<1.Now for y2∈[G2y1]α, consider(23)Φψ1y2ρy2,y1+ψ2y2ρy2,G1y2α+ψ3y2ρy1,G2y1α+ψ4y2ρy2,G1y2αρy2,G2y1α1+ρy2,y1+ψ5y2ρy1,G1y2αρy1,G2y1α1+ρy2,y1+ψ6y2ρy2,G1y2αρy1,G2y1α1+ρy2,y1+ψ7y2ρy1,G1y2αρy2,G2y1α1+ρy2,y1∈sG1y2α,G2y1α.Using Lemma 4(iii) we get(24)Φψ1y2ρy2,y1+ψ2y2ρy2,G1y2α+ψ3y2ρy1,G2y1α+ψ4y2ρy2,G1y2αρy2,G2y1α1+ρy2,y1+ψ5y2ρy1,G1y2αρy1,G2y1α1+ρy2,y1+ψ6y2ρy2,G1y2αρy1,G2y1α1+ρy2,y1+ψ7y2ρy1,G1y2αρy2,G2y1α1+ρy2,y1∈sG1y2α,y2.By definition there exists y3∈[G1y2]α, such that(25)Φψ1y2ρy2,y1+ψ2y2ρy2,G1y2α+ψ3y2ρy1,G2y1α+ψ4y2ρy2,G1y2αρy2,G2y1α1+ρy2,y1+ψ5y2ρy1,G1y2αρy1,G2y1α1+ρy2,y1+ψ6y2ρy2,G1y2αρy1,G2y1α1+ρy2,y1+ψ7y2ρy1,G1y2αρy2,G2y1α1+ρy2,y1∈sρy3,y2.Therefore(26)ρy3,y2⪯Φψ1y2ρy2,y1+ψ2y2ρy2,G1y2α+ψ3y2ρy1,G2y1α+ψ4y2ρy2,G1y2αρy2,G2y1α1+ρy2,y1+ψ5y2ρy1,G1y2αρy1,G2y1α1+ρy2,y1+ψ6y2ρy2,G1y2αρy1,G2y1α1+ρy2,y1+ψ7y2ρy1,G1y2αρy2,G2y1α1+ρy2,y1≺ψ1y2ρy2,y1+ψ2y2ρy2,G1y2α+ψ3y2ρy1,G2y1α+ψ4y2ρy2,G1y2αρy2,G2y1α1+ρy2,y1+ψ5y2ρy1,G1y2αρy1,G2y1α1+ρy2,y1+ψ6y2ρy2,G1y2αρy1,G2y1α1+ρy2,y1+ψ7y2ρy1,G1y2αρy2,G2y1α1+ρy2,y1.Again utilizing the greatest lower bound property of G1 and G2 we get(27)ρy3,y2⪯ψ1y2ρy2,y1+ψ2y2ρy2,y3+ψ3y2ρy1,y2+ψ4y2ρy2,y3ρy2,y21+ρy2,y1+ψ5y2ρy1,y3ρy1,y21+ρy2,y1+ψ6y2ρy2,y3ρy1,y21+ρy2,y1+ψ7y2ρy1,y3ρy2,y21+ρy2,y1.It implies that(28)ρy3,y2⪯ψ1y2ρy2,y1+ψ2y2ρy2,y3+ψ3y2ρy1,y2+ψ5y2ρy1,y3+ψ6y2ρy2,y3⪯ψ1y2ρy2,y1+ψ2y2ρy2,y3+ψ3y2ρy1,y2+ψ5y2ρy1,y2+ψ5y2ρy2,y3+ψ6y2ρy2,y3.Applying Lemma 13 we get(29)ρy3,y2⪯ψ1y0ρy2,y1+ψ2y0ρy2,y3+ψ3y0ρy1,y2+ψ5y0ρy1,y2+ψ5y0ρy2,y3+ψ6y0ρy2,y3.Finally we get(30)ρy3,y2⪯νρy2,y1ρy2,y3≤νρy1,y2,where(31)ν=ψ1y0+ψ3y0+ψ5y01-ψ2y0+ψ5y0+ψ6y0<1.Inductively we can obtain a sequence {yn} in X such that y2r+1∈[G1x2r]α,y2r+2∈[G2y2r+1]α for r=0,1,2….(32)ρy2r+1,y2r+2≤μρy2r,y2r+1ρy2r+2,y2r+3≤νρy2r+1,y2r+2.It implies that(33)ρy2r+1,y2r+2≤μρy2r,y2r+1≤μνρy2r-1,y2r≤μνμρy2r-2,y2r-1≤⋯≤μμνrρy0,y1,ρy2r+2,y2r+3≤νρy2r+1,y2r+2≤⋯≤μνr+1ρy0,y1.Then for s<t, we have(34)ρy2s+1,y2t+1⪯ρy2s+1,y2s+2+ρy2s+2,y2s+3+ρy2s+3,y2s+4+⋯+ρy2t,y2t+1,which implies that(35)ρy2s+1,y2t+1≤ρy2s+1,y2s+2+ρy2s+2,y2s+3+ρy2s+3,y2s+4+⋯+ρy2t,y2t+1≤μ∑k=st-1μνk+∑k=s+1tμνkρy0,y1.Similarly we obtain(36)ρy2s,y2t+1≤∑k=stμνk+μ∑k=st-1μνkρy0,y1,ρy2s,y2t≤∑k=st-1μνk+μ∑k=st-1μνkρy0,y1,ρy2s+1,y2t≤μ∑k=st-1μνk+∑k=s+1t-1μνkρy0,y1.Since (μν)<1, therefore {yr} is a Cauchy sequence in X. Since X is complete so there exists l∈X such that yr→l when r→∞. Now we have to prove that l∈[G1l]α and l∈[G2l]α. From (14) with y=y2r and w=l we get(37)Φψ1y2rρy2r,l+ψ2y2rρy2r,G1y2rα+ψ3y2rρl,G2lα+ψ4y2rρy2r,G1y2rαρy2r,G2lα1+ρy2r,l+ψ5y2rρl,G1y2rαρl,G2lα1+ρy2r,l+ψ6y2rρy2r,G1y2rαρl,G2lα1+ρy2r,l+ψ7y2rρl,G1y2rαρy2r,G2lα1+ρy2r,l∈sG1y2rα,G2lα.Since y2r+1∈[G1y2r]α, so by Lemma 4(iii) we have(38)Φψ1y2rρy2r,l+ψ2y2rρy2r,G1y2rα+ψ3y2rρl,G2lα+ψ4y2rρy2r,G1y2rαρy2r,G2lα1+ρy2r,l+ψ5y2rρl,G1y2rαρl,G2lα1+ρy2r,l+ψ6y2rρy2r,G1y2rαρl,G2lα1+ρy2r,l+ψ7y2rρl,G1y2rαρy2r,G2lα1+ρy2r,l∈sy2r+1,G2lα.By definition there exists some wr∈[G2l]α, such that(39)Φψ1y2rρy2r,l+ψ2y2rρy2r,G1y2rα+ψ3y2rρl,G2lα+ψ4y2rρy2r,G1y2rαρy2r,G2lα1+ρy2r,l+ψ5y2rρl,G1y2rαρl,G2lα1+ρy2r,l+ψ6y2rρy2r,G1y2rαρl,G2lα1+ρy2r,l+ψ7y2rρl,G1y2rαρy2r,G2lα1+ρy2r,l∈sρy2r+1,wr.Therefore(40)ρy2r+1,wr⪯Φψ1y2rρy2r,l+ψ2y2rρy2r,G1y2rα+ψ3y2rρl,G2lα+ψ4y2rρy2r,G1y2rαρy2r,G2lα1+ρy2r,l+ψ5y2rρl,G1y2rαρl,G2lα1+ρy2r,l+ψ6y2rρy2r,G1y2rαρl,G2lα1+ρy2r,l+ψ7y2rρl,G1y2rαρy2r,G2lα1+ρy2r,l.By using the greatest lower bound property of G1 and G2, we have(41)ρy2r+1,wr⪯Φψ1y2rρy2r,l+ψ2y2rρy2r,y2r+1+ψ3y2rρl,wr+ψ4y2rρy2r,y2r+1ρy2r,wr1+ρy2r,l+ψ5y2rρl,y2r+1ρl,wr1+ρy2r,l+ψ6y2rρy2r,y2r+1ρl,wr1+ρy2r,l+ψ7y2rρl,y2r+1ρy2r+1,wr1+ρy2r,l⪯ψ1y2rρy2r,l+ψ2y2rρy2r,y2r+1+ψ3y2rρl,wr+ψ4y2rρy2r,y2r+1ρy2r,wr1+ρy2r,l+ψ5y2rρl,y2r+1ρl,wr1+ρy2r,l+ψ6y2rρy2r,y2r+1ρl,wr1+ρy2r,l+ψ7y2rρl,y2r+1ρy2r+1,wr1+ρy2r,l.Now by using triangular inequality, we get(42)ρl,wr⪯ρl,y2r+1+ρy2r+1,wr⪯ρl,y2r+1+ψ1y2rρy2r,l+ψ2y2rρy2r,y2r+1+ψ3y2rρl,wr+ψ4y2rρy2r,y2r+1ρy2r,wr1+ρy2r,l+ψ5y2rρl,y2r+1ρl,wr1+ρy2r,l+ψ6y2rρy2r,y2r+1ρl,wr1+ρy2r,l+ψ7y2rρl,y2r+1ρy2r+1,wr1+ρy2r,l.Applying Lemma 13 we get(43)ρl,wr⪯ρl,y2r+1+ρy2r+1,wr⪯ρl,y2r+1+ψ1y0ρy2r,l+ψ2y0ρy2r,y2r+1+ψ3y0ρl,wr+ψ4y0ρy2r,y2r+1ρy2r,wr1+ρy2r,l+ψ5y0ρl,y2r+1ρl,wr1+ρy2r,l+ψ6y0ρy2r,y2r+1ρl,wr1+ρy2r,l+ψ7y0ρl,y2r+1ρy2r+1,wr1+ρy2r,l,which, on r→∞, reduced to(44)ρl,wr⪯ψ3y0ρl,wrρl,wr≤ψ3y0ρl,wr.Since ψ3(y0)<1, so |ρ(l,wr)|→0 as r→∞. so we have wr→l as r→∞. Since [G2l]α is closed, so l∈[G2l]α. Similarly, it follows that l∈[G1l]α. Thus we obtain that G1 and G2 have common fixed points.Corollary 15.
Let(X,ρ) be a complete complex-valued metric space and G1:X→L(X) be fuzzy mapping with glb property. For each y∈X there exists α∈(0,1] such that [G1y]α is nonempty closed bounded subset of X. Then there exists mappings ψi:X→[0,1),i=1,…,7 with(i)
ψi(u)⪯ψi(y),i=1,2,…,7 for all u∈[G1y]α and ∀y∈X;(ii)
∑ψi(y)+2ψ4(y)<1, where i=1,2,3,6,7∀y∈X; and(45)Φψ1yρy,w+ψ2yρy,G1yα+ψ3yρw,G1wα+ψ4yρy,G1yαρy,G1wα1+ρy,w+ψ5yρw,G1yαρw,G1wα1+ρy,w+ψ6yρy,G1yαρw,G1wα1+ρy,w+ψ7yρw,G1yαρy,G1wα1+ρy,w∈sG1yα,G1wα,for some Φ∈Ψ and for all y,w∈X. Then G1 has a fuzzy fixed point.Proof.
Proof is immediate on settingG1=G2 in Theorem 14.Corollary 16.
Let(X,ρ) be a complete complex-valued metric space and G1,G2:X→L(X) be fuzzy mappings with glb property. For each y∈X there exists some α∈(0,1] such that [G1y]α,[G2y]α, nonempty closed bounded subsets of X. Then there exist mappings ϱ,σ,γ,ξ,ζ,λ,η:X→[0,1) with(i)
ϱ(u)⪯ϱ(y), σ(u)⪯σ(y), γ(u)⪯γ(y), ξ(u)⪯ξ(y), ζ(u)⪯ζ(y), λ(u)⪯λ(y), η(u)⪯η(y) for all u∈[G1y]α and y∈X;(ii)
ϱ(v)⪯ϱ(w), σ(v)⪯σ(w), γ(v)⪯γ(w), ξ(v)⪯ξ(w), ζ(v)⪯ζ(w), λ(v)⪯λ(w), η(v)⪯η(w) for all v∈[G2w]α and w∈X;(iii)
ϱ(y)+σ(y)+γ(y)+2ξ(y)+ζ(y)+λ(y)+η(y)⪯1∀y∈X; and(46)ϱyρy,w+σyρy,G1yα+γyρw,G2wα+ξyρy,G1yαρy,G2wα1+ρy,w+ζyρw,G1yαρw,G2wα1+ρy,w+λyρy,G1yαρw,G2wα1+ρy,w+ηxρw,G1yαρy,G2wα1+ρy,w∈sG1yα,G2wα,∀y,w∈X; then G1 and G2 have a common fuzzy fixed point.Proof.
It can be easily proven by lettingΦ(t)=pt where p∈(0,1) in Theorem 14 with ϱ(y)=pψ1(y),σ(y)=pψ2(y),γ(y)=pψ3(y),ξ(y)=pψ4(y),ζ(y)=pψ5(y),λ(y)=pψ6(y),η(y)=pψ7(y).Corollary 17.
Suppose(X,ρ) is a complete complex-valued metric space and G1,G2:X→L(X) are fuzzy mappings enjoying glb property. For each y∈X there exists some α∈(0,1] such that [G1y]α,[G2y]α, nonempty closed bounded subsets of X with(47)ϱρy,w+σρy,G1yα+γρw,G2wα+ξρy,G1yαρy,G2wα1+ρy,w+ζρw,G1yαρw,G2wα1+ρy,w+λρy,G1yαρw,G2wα1+ρy,w+ηρw,G1yαρy,G2wα1+ρy,w∈sG1yα,G2wα,for all y,w∈X and ϱ,σ,γ,ξ,ζ,λ,η are nonnegative reals with ϱ+σ+γ+2ξ+ζ+λ+η<1. Then G1 and G2 have a common fuzzy fixed point.Proof.
It can be easily proven by settingϱ(y)=ϱ, σ(y)=σ, γ(y)=γ, ξ(y)=ξ, ζ(y)=ζ, λ(y)=λ, η(y)=η in Corollary 16 with ϱ,σ,γ,ξ,ζ,λ,η being nonnegative reals such that ϱ+σ+γ+2ξ+ζ+λ+η<1.Using Remark11 we get the following corollaries from Theorem 14.Corollary 18.
Suppose(X,ρ) is a complete metric space and G1,G2:X→L(X) are fuzzy mappings with glb property. For each y∈X related to some α∈(0,1] there exists [G1y]α,[G2y]α, nonempty closed bounded subsets of X. Then there exist mappings ψi:X→[0,1),i=1,2,…,7 such that(i)
ψi(u)≤ψi(y), for all u∈[G1y]α and ∀y∈X;(ii)
ψi(v)≤ψi(y), for all v∈[G2y]α and ∀y∈X;(iii)
∑ψiy+2ψ4(y)<1,i=1,2,3,5,6,7∀y∈X; and(48)HG1yα,G2wα≤Φψ1yρy,w+ψ2yρy,G1yα+ψ3yρw,G2wα+ψ4yρy,G1yαρy,G2wα1+ρy,w+ψ5yρw,G1yαρw,G2wα1+ρy,w+ψ6yρy,G1yαρw,G2wα1+ρy,w+ψ7yρw,G1yαρy,G2wα1+ρy,w,for some Φ∈Ψ and for all y,w∈X. Then G1 and G2 have a common fuzzy fixed point.Corollary 19.
Suppose(X,ρ) is complete metric space and G1,G2:X→L(X) is fuzzy mappings with glb property. For each y∈X there exists some α∈0,1 such that [G1y]α,[G2y]α, nonempty closed bounded subsets of X such that(49)HG1yα,G2wα≤ϱρy,w+σρy,G1yα+γρw,G2wα+ξρy,G1yαρy,G2wα1+ρy,w+ζρw,G1yαρw,G2wα1+ρy,w+λρy,G1yαρw,G2wα1+ρy,w+ηρw,G1yαρy,G2wα1+ρy,w,for all y,w∈X and ϱ,σ,γ,ξ,ζ,λ,η are nonnegative reals with ϱ+σ+γ+2ξ+ζ+λ+η<1. Then G1 and G2 have a common fuzzy fixed point.Proof.
By puttingϱ(y)=ψ1(y), σ(y)=ψ2(y),γ(y)=ψ3(y),ξ(y)=ψ4(y),ζ(y)=ψ5(y),λ(y)=ψ6(y),η(y)=ψ7(y) in Corollary 18, it can be easily proven.


## 4. Application

Theorem 20.
Let(X,ρ) be a complete complex-valued metric space and G1,G2:X→CB(X) be multivalued mapping with glb property. If there exist mappings ψi:X→[0,1),i=1,2,…,7 such that(i)
ψi(u)⪯ψi(y) for all u∈G1y and ∀y∈X;(ii)
ψi(v)⪯ψi(y) for all v∈G2y and ∀y∈X;(iii)
ψi(y)+2ψ4(y)<1,i=1,2,3,5,6,7∀y∈X; and(50)Φψ1yρy,w+ψ2yρy,G1y+ψ3yρw,G2w+ψ4yρy,G1yρy,G2w1+ρy,w+ψ5yρw,G1yρw,G2w1+ρy,w+ψ6yρy,G1yρw,G2w1+ρy,w+ψ7yρw,G1yρy,G2w1+ρy,w∈sG1y,G2w,for some Φ∈Ψ and for all y,w∈X, then G1 and G2 have a common fixed point.Proof.
Let the fuzzy mappingS,T:X→L(X) be defined by(51)Sy=αify∈G1y0ify∉G1y.Ty=αify∈G2y0ify∉G2y.Then for any α∈(0,1,Sy]α=G1y and [Ty]α=G2y.
Since for everyy,w∈X, s([Sy]α,[Tw]α)=s(G1y,G2w), therefore Theorem 14 can be applied to obtain some points u∈X such that u∈G1(u)∩G2(u).Corollary 21.
Let(X,ρ) be a complete complex-valued metric space and G1,G2:X→CB(X) be multivalued mapping with glb property. Suppose there exist mappings ψi:X→[0,1),i=1,2,…,7 such that(i)
ψi(u)⪯ψi(y) for all u∈G1y and ∀y∈X;(ii)
ψi(v)⪯ψi(y) for all v∈G2y and ∀y∈X;(iii)
ψi(y)+2ψ4(y)<1,i=1,2,3,5,6,7∀y∈X; and(52)ψ1yρy,w+ψ2yρy,G1y+ψ3yρw,G2w+ψ4yρy,G1yρy,G2w1+ρy,w+ψ5yρw,G1yρw,G2w1+ρy,w+ψ6yρy,G1yρw,G2w1+ρy,w+ψ7yρw,G1yρy,G2w1+ρy,w∈sG1y,G2w.Then G1 and G2 have a common fixed point.Proof.
It can be proven by the same way as Corollary16.Remark 22.
(i) Theorem20 is actually Theorem 2.3 of [34].
(ii) By settingψ2(y)=ψ3(y)=ψ4(y)=ψ5(y)=0,ψ1(y)=ϱ,ψ6(y)=λ,ψ7(y)=η in Corollary 21, we get Theorem 9 of [13].
(iii) By settingψ1(y)=ψ4(y)=ψ5(y)=ψ7=0,ψ2(y)=σ,ψ3(y)=γ,ψ6(y)=λ in Corollary 21 we obtain Theorem 15 of [13].
(iv) By settingψ2(y)=ψ3(y)=0,ψ1(y)=ϱ,ψ4(y)=ξ,ψ5(y)=ζ,ψ6(y)=λ,ψ7(y)=η in Corollary 21 we get Theorem 9 of [36].
(v) By settingσ=γ=ξ=ζ=0 in Corollary 17, we get Theorem 12 of [25].
(vi) By settingϱ=ξ=ζ=η=0 in Corollary 17, we get Theorem 19 of [25].Example 23.
LetX=[0,1] and ρ:X×X→C be complex-valued metric space defined by ρ(y,w)=|y-w|eι˙(π/12), for all y,w∈X.
Letα∈(0,1] and G1,G2:X→L(X) be fuzzy mappings defined by(53)G10r=1ifr=012if0<r≤y500ify50<r≤1,G20r=1ifr=016if0<r≤y1500ify150<r≤1,if y≠0,(54)G1yr=αif0≤r≤y75α3ify75<r≤y10α4ify10<r≤1,G2yr=αif0≤r≤y40α2ify40<r≤y20α5ify20<r≤1.Let ψ1,ψ2,ψ3,ψ4,ψ5,ψ6,ψ7:X→[0,1) be defined by ψ1(y)=(y+1)/74,ψ2(y)=y/10,ψ3(y)=y/20,ψ4(y)=y/30,ψ5(y)=y/60,ψ6(y)=y/40.
Then fory=0, [G10]1=[G20]1={0} and ∀y,w≠0[G1y]α=[0,y/75] and [G2y]α=[0,y/40].
And then(55)WwG1yα=ρw,u:u∈0,y75WwG2yα=ρw,u:u∈0,y40.Let ρ(w,[G1w]α) and ρ(w,[G2w]α) be the greatest lower bound of Ww([G1y]α) and Ww([G2y]α). Then(56)ρw,G1yα=0ifw≤y75w-y75eι˙π/12ifw>y75,ρy,G2wα=0ify≤y40y-w40eι˙π/12ifx>w40;also ρ(y,[G1y]α)=(74y/75)eι˙(π/12), and ρ(w,[G2w]α)=(39w/40)eι˙(π/12).
It can be easily verified thatψi(u)≤ψi(w),∀u∈[G1y]α and ψi(v)≤ψi(w),∀v∈[G2y]α. Moreover if ϖyw∈C such that(57)ϖyw=y75-w402eι˙π/12,then(58)sG1yα,G2wα=ω∈C:ϖyw⪯ω.Consider(59)Φψ1yρy,w+ψ2yρy,G1yα+ψ3yρw,G2wα+ψ4yρy,G1yαρy,G2wα1+ρy,w+ψ5yρw,G1yαρw,G2wα1+ρy,w+ψ6yρy,G1yαρw,G2wα1+ρy,w+ψ7yρw,G1yαρy,G2wα1+ρy,w;then clearly for ψ1(y)=(y+1)/74,ψ2(y)=y/10,ψ3(y)=y/20,ψ4(y)=y/30,ψ5(y)=y/60,ψ6(y)=y/40,ψ7=y/25 and Φ(t)=74t/75.(60)y75-w40≤7475y+174y-w+y1074y75+y2039w40+y3074y/75y-w/401+y-w+y60w-y/7539w/401+y-w+y4074y/7539w/401+y-w+y75w-y/75y-w/401+y-wwhich can be easily calculated by(61)7475y+174y-w=7425174y2+y-w-yw≥175y-w-yw=y75-w1+y75≥y75-w75≥y75-w40.The remaining terms of (59) are nonzero reals. Consequently we can obtain(62)Φψ1yρy,w+ψ2yρy,G1yα+ψ3yρw,G2wα+ψ4yρy,G1yαρy,G2wα1+ρy,w+ψ5yρw,G1yαρw,G2wα1+ρy,w+ψ6yρy,G1yαρw,G2wα1+ρy,w+ψ7yρw,G1yαρy,G2wα1+ρy,w⪰ϖyw.Therefore(63)Φψ1yρy,w+ψ2yρy,G1yα+ψ3yρw,G2wα+ψ4yρy,G1yαρy,G2wα1+ρy,w+ψ5yρw,G1yαρw,G2wα1+ρy,w+ψ6yρy,G1yαρw,G2wα1+ρy,w+ψ7yρw,G1yαρy,G2wα1+ρy,w∈sG1yα,G2wα.Hence all conditions of Theorem 14 are satisfied by G1,G2; therefore there exists 0∈X such that 0∈[G10]α∩[G20]α.



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*Source: 5303815-2018-01-09.xml*