Fuzzy Fixed Point Results For Contractive Mapping with Applications

Abstract

In this paper, using rational type contractions, common fuzzy fixed point result for 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.

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 for 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]). Assume is the set of complex numbers. For we define a partial order on as follows:(Ci) and ;(Cii) and ;(Ciii) and ;(Civ) and . Clearly if , , for all and for all . Note that if and one of (Ci), (Cii) and (Ciii) is satisfied then , and we write if only (Civ) is satisfied. Note that(i);(ii) and .

Definition 2 (see [10]). Let be a nonempty set and be a mapping which satisfies the following conditions:(1), for all and if and only if ;(2), for all ;(3), for all . Then is called a complex-valued metric space.

Definition 3 (see [10]). A point is known as an interior point of a set , if we find such thatA point is known as the limit point of , if there exists an open ball such thatwhere . A subset of is said to be open if each point of is an interior point of . Furthermore, is said to be closed if it contain all its limit points.
The familyis a subbasis for a Hausdorff topology on .

Now recall some definitions from [13, 14].

Let be a complex-valued metric space. Throughout this paper we denoted the family of all nonempty closed bounded subsets of complex-valued metric space by . For we representand for and .

For , we denoteLet be a multivalued mapping from into ; for and we defineThus for

Lemma 4 (see [35]). Let be complex-valued metric space.(i)Let If , then .(ii)Let and If , then .(iii)Let and If , then for all or for all .

Definition 5 (see [10]). Let be a sequence in complex-valued metric space and ; then(i) is a limit point of if for each there exists such that for all and it is written as .(ii) is a Cauchy sequence if for any there exists such that for all where .(iii)we say that is complete complex-valued metric space if every Cauchy sequence in converges to a point in .

Definition 6 (see [18]). Let be a metric space. A fuzzy set is characterized by its membership function . A set of elements of along with its grade of membership is called a fuzzy set. For simplicity we denote by . The -level set of a fuzzy set is mentioned by and is defined as follows:

Definition 7 (see [18]). Let be the family of all fuzzy sets in a metric space . For means for each .

Definition 8 (see [16]). Assume is an arbitrary set and is a metric space. A mapping is called a fuzzy mapping if . A fuzzy mapping is a fuzzy subset on with a membership function . The function is the grade of membership of in .

Definition 9 (see [20]). Assume that is complex-valued metric space and are fuzzy mappings. A point is a fuzzy fixed point of if where and a common fuzzy fixed point of if If then is known as common fixed point of fuzzy mappings.

Definition 10 (see [14]). Suppose is complex-valued metric space; the fuzzy mapping enjoys the greatest lower bound property (glb property) on , if, for any and , the greatest lower bound of exists in for all . Here we mention by the glb of That is,

Remark 11 (see [13]). Let be a complex-valued metric space. If , then is a metric space. Furthermore is the Hausdorff distance induced by , where .

Definition 12 (see [34]). Suppose is a collection of nondecreasing functions, , such that and , when .

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 be complex-valued metric space and be fuzzy mappings, such that for each and some there exists , nonempty closed and bounded subsets of . Let and define the sequence byAssume that there exists a mapping such that for all and for all . Then and .

Proof. Suppose and . Then we haveSimilarly we have

Theorem 14. Suppose is a complete complex-valued metric space and are fuzzy mappings satisfying glb property. Assume that for each and some there exist which are nonempty closed bounded subsets of . Suppose there exist mappings such that(i) for all and ;(ii) for all and ;(iii); andfor some and for all . Then and have a common fuzzy fixed point.

Proof. Let and Using (14) with and we getBy Lemma 4(iii) we haveBy definition there exists some , such thatThereforeUsing the glb property of and we haveIt implies thatFinally we getwhereNow for , considerUsing Lemma 4(iii) we getBy definition there exists , such thatThereforeAgain utilizing the greatest lower bound property of and we getIt implies thatApplying Lemma 13 we getFinally we getwhereInductively we can obtain a sequence in such that for .It implies thatThen for , we havewhich implies thatSimilarly we obtainSince , therefore is a Cauchy sequence in . Since is complete so there exists such that when Now we have to prove that and . From (14) with and we getSince , so by Lemma 4(iii) we haveBy definition there exists some , such thatThereforeBy using the greatest lower bound property of and , we haveNow by using triangular inequality, we getApplying Lemma 13 we getwhich, on , reduced toSince , so as so we have as . Since is closed, so Similarly, it follows that Thus we obtain that and have common fixed points.

Corollary 15. Let be a complete complex-valued metric space and be fuzzy mapping with glb property. For each there exists such that is nonempty closed bounded subset of . Then there exists mappings with(i) for all and ;(ii), where ; andfor some and for all . Then has a fuzzy fixed point.

Proof. Proof is immediate on setting in Theorem 14.

Corollary 16. Let be a complete complex-valued metric space and be fuzzy mappings with glb property. For each there exists some such that , nonempty closed bounded subsets of . Then there exist mappings with(i), , , , , , for all and ;(ii), , , , , , for all and ;(iii); and; then and have a common fuzzy fixed point.

Proof. It can be easily proven by letting where in Theorem 14 with .

Corollary 17. Suppose is a complete complex-valued metric space and are fuzzy mappings enjoying glb property. For each there exists some such that , nonempty closed bounded subsets of withfor all and are nonnegative reals with . Then and have a common fuzzy fixed point.

Proof. It can be easily proven by setting , , , , , , in Corollary 16 with being nonnegative reals such that .

Using Remark 11 we get the following corollaries from Theorem 14.

Corollary 18. Suppose is a complete metric space and are fuzzy mappings with glb property. For each related to some there exists , nonempty closed bounded subsets of . Then there exist mappings such that(i), for all and ;(ii), for all and ;(iii); andfor some and for all . Then and have a common fuzzy fixed point.

Corollary 19. Suppose is complete metric space and is fuzzy mappings with glb property. For each there exists some such that , nonempty closed bounded subsets of such thatfor all and are nonnegative reals with . Then and have a common fuzzy fixed point.

Proof. By putting , in Corollary 18, it can be easily proven.

4. Application

Theorem 20. Let be a complete complex-valued metric space and be multivalued mapping with glb property. If there exist mappings such that(i) for all and ;(ii) for all and ;(iii); andfor some and for all , then and have a common fixed point.

Proof. Let the fuzzy mapping be defined byThen for any and .
Since for every , , therefore Theorem 14 can be applied to obtain some points such that .

Corollary 21. Let be a complete complex-valued metric space and be multivalued mapping with glb property. Suppose there exist mappings such that(i) for all and ;(ii) for all and ;(iii); andThen and have a common fixed point.

Proof. It can be proven by the same way as Corollary 16.

Remark 22. (i) Theorem 20 is actually Theorem of [34].
(ii) By setting in Corollary 21, we get Theorem of [13].
(iii) By setting in Corollary 21 we obtain Theorem of [13].
(iv) By setting in Corollary 21 we get Theorem of [36].
(v) By setting in Corollary 17, we get Theorem of [25].
(vi) By setting in Corollary 17, we get Theorem of [25].

Example 23. Let and be complex-valued metric space defined by , for all .
Let and be fuzzy mappings defined byif ,Let be defined by .
Then for , and and .
And thenLet and be the greatest lower bound of and . Thenalso , and .
It can be easily verified that and . Moreover if such thatthenConsiderthen clearly for and .which can be easily calculated byThe remaining terms of (59) are nonzero reals. Consequently we can obtainThereforeHence all conditions of Theorem 14 are satisfied by ; therefore there exists such that .

Conflicts of Interest

The authors declare that they have no conflicts of interest.