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Coincidence theorems via alpha cuts of Lfuzzy sets with applications
Fixed Point Theory and Applications volume 2014, Article number: 212 (2014)
Abstract
In the present paper, existence theorems of coincidence points of a crisp mapping and a sequence of Lfuzzy mappings have been established in a complete metric space under contractive type conditions in connection with newly defined notions of {D}_{{\alpha}_{L}} and {d}_{L}^{\mathrm{\infty}} distances on the class of Lfuzzy sets. Furthermore, we obtain some fixed point theorems for Lfuzzy setvalued mappings to extend a variety of recent results on fixed points for fuzzy mappings and multivalued mappings in the literature. As applications, first we obtain coincidence points of a sequence of multivalued mappings with a self mapping and next established an existence and uniqueness theorem of the solution for a generalized class of nonlinear integral equations.
MSC:46S40, 47H10, 54H25.
1 Introduction
Since his creation, man has always been making sincere efforts in understanding nature intelligently and then developing a powerful connection between life and its requirements. These efforts consist of three phases: understanding of the surrounding environment, acknowledgement of new things, and then planning for the future. In this search so many issues like linguistic interpretation, inaccurate judgment, characterization of interrelated phenomena into proper classifications, use of restricted techniques, vague analysis of results and many others, highly affect the accuracy of the results. The above mentioned hurdles related with interpretation of data can be tackled to a great extent by considering fuzzy sets (due to their flexible nature) in place of crisp sets.
After the discovery of fuzzy set by Zadeh [1] a great revolution arose in the field of analysis. The potential of the introduced notion was realized by research workers of different fields of science and technology. By introducing a contraction condition for fuzzy mappings Heilpern [2] generalized the Banach principle and established a fixed point theorem for fuzzy mappings in complete metric linear spaces. Afterwards, many authors, e.g., [3–16] generalized and extended this result in various directions. In [17] Edelstein extended the Banach contraction principle by using the idea of locally and globally contractive mappings. Subsequently, many authors, e.g., [18–22] utilized this concept to prove numerous results. In 1967, Goguen [23] initiated an interesting generalization of fuzzy sets namely called Lfuzzy sets. The concept of Lfuzzy set is superior to fuzzy sets as L is a lattice which is not necessarily a totally ordered set. Recently, Rashid et al. [24] introduced the concept of Lfuzzy mappings and proved a common fixed point theorem via {\beta}_{{F}_{L}}admissible pair of Lfuzzy mappings.
In this article we introduce the notions of {D}_{{\alpha}_{L}} and {d}_{L}^{\mathrm{\infty}} distances for Lfuzzy sets to identify a contractive relation between Lfuzzy mappings and crisp mappings. Making use of this contractive relation on a complete metric space a coincidence point is obtained of a sequence of Lfuzzy mappings and a single valued crisp mapping. Analogous coincidence theorems for fuzzy mappings and multivalued mappings have been obtained as corollaries. These corollaries regarding coincidence point of fuzzy mappings and multivalued mappings have not been seen in the literature and therefore most of them are still original and new results. However, some imaginative fixed point theorems [3, 9, 11, 13, 17, 18, 25, 26] in the literature can be obtained as corollaries.
We also present some applications of the main theorem in two directions, one for obtaining fixed points and coincidence points of formal multivalued mappings and the other is for solutions of a generalized class of nonlinear integral equations to enhance the validity of our result.
2 Preliminaries
This section lists some preliminary notions and results. Let (X,d) be a metric space, denote:
CB(X)=\{A:A\text{is nonempty closed and bounded subset of}X\},
C(X)=\{A:A\text{is nonempty compact subset of}X\}.
For \u03f5>0 and the sets A,B\in CB(X) define
Then the Hausdorff metric H on CB(X) induced by d is defined as
A fuzzy set in X is a function with domain X and values in [0,1]. If A is a fuzzy set and x\in X, then the function values A(x) is called the grade of membership of x in A. The αlevel set of A is denoted by {[A]}_{\alpha} and is defined as follows:
A fuzzy set A in a metric linear space V is said to be an approximate quantity if and only if {[A]}_{\alpha} is compact and convex in V for each \alpha \in [0,1] and {sup}_{x\in V}A(x)=1. The collection of all approximate quantities in V is denoted by W(V).
For A,B\in {I}^{X}, A\subset B means A(x)\le B(x) for each x\in X. If there exists an \alpha \in [0,1] such that {[A]}_{\alpha},{[B]}_{\alpha}\in CB(X) then define
If {[A]}_{\alpha},{[B]}_{\alpha}\in CB(X) for each \alpha \in [0,1] then define
Lemma 2.1 [26]
Let (X,d) be a metric space and A,B\in CB(X), then for each a\in A
Lemma 2.2 [26]
Let (X,d) be a metric space and A,B\in CB(X), then for each a\in A, \u03f5>0, there exists an element b\in B such that
Definition 2.3 [23]
A partially ordered set (L,{\precsim}_{L}) is called

(i)
a lattice, if a\vee b\in L, a\wedge b\in L for any a,b\in L;

(ii)
a complete lattice, if \vee A\in L, \wedge A\in L for any A\subseteq L;

(iii)
distributive if a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c), a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c) for any a,b,c\in L.
Definition 2.4 [23]
Let L be a lattice with top element {1}_{L} and bottom element {0}_{L} and let a,b\in L. Then b is called a complement of a, if a\vee b={1}_{L}, and a\wedge b={0}_{L}. If a\in L has a complement element, then it is unique. It is denoted by \stackrel{\xb4}{a}.
Definition 2.5 [23]
A Lfuzzy set A on a nonempty set X is a function A:X\to L, where L is complete distributive lattice with {1}_{L} and {0}_{L}.
Remark 2.6 The class of Lfuzzy sets is larger than the class of fuzzy sets as an Lfuzzy set is a fuzzy set if L=[0,1].
From now, the class of Lfuzzy subsets of X will be denoted by {F}_{L}(X). The {\alpha}_{L}level set of Lfuzzy set A, is denoted by {A}_{{\alpha}_{L}}, and is defined as follows:
Here \overline{B} denotes the closure of the set B.
Definition 2.7 [24]
Let X be an arbitrary set, Y be a metric space. A mapping T is called Lfuzzy mapping if T is a mapping from X into {F}_{L}(Y) (class of Lfuzzy subsets of Y). A Lfuzzy mapping T is a Lfuzzy subset on X\times Y with membership function T(x)(y). The function T(x)(y) is the grade of membership of y in T(x).
Definition 2.8 [24]
Let (X,d) be a metric space and S, T be Lfuzzy mappings from X into {F}_{L}(X). A point z\in X is called a Lfuzzy fixed point of T if z\in {[Tz]}_{{\alpha}_{L}}, for some {\alpha}_{L}\in L\mathrm{\setminus}\{{\mathit{0}}_{L}\}. The point z\in X is called a common Lfuzzy fixed point of S and T if z\in {[Sz]}_{{\alpha}_{L}}\cap {[Tz]}_{{\alpha}_{L}}. When {\alpha}_{L}={1}_{L}, it is called a common fixed point of Lfuzzy mappings.
Definition 2.9 Let \epsilon \in (0,\mathrm{\infty}], and \lambda \in (0,1). A metric space (X,d) is said to be εchainable if given u,v\in X, there exists an εchain from u to v (i.e., a finite set of points u={u}_{0}, {u}_{1},{u}_{2},\dots ,{u}_{l}=v such that d({u}_{t1},{u}_{t})<\epsilon for all t=1,2,\dots ,l).
Definition 2.10 [8]
A function \phi :[0,+\mathrm{\infty})\to [0,1) is said to be a MTfunction if it satisfies the following condition:
Clearly, if \phi :[0,+\mathrm{\infty})\to [0,1) is a nondecreasing function or a nonincreasing function, then it is a MTfunction. So the set of MTfunctions is a rich class.
Proposition 2.11 [8]
Let \phi :[0,+\mathrm{\infty})\to [0,1) be a function. Then the following statements are equivalent.

(i)
φ is a MTfunction.

(ii)
For each t\in [0,\mathrm{\infty}), there exist {r}_{t}^{(1)}\in [0,1) and {\epsilon}_{t}^{(1)}>0 such that \phi (s)\le {r}_{t}^{(1)} for all s\in (t,t+{\epsilon}_{t}^{(1)}).

(iii)
For each t\in [0,\mathrm{\infty}), there exist {r}_{t}^{(2)}\in [0,1) and {\epsilon}_{t}^{(2)}>0 such that \phi (s)\le {r}_{t}^{(2)} for all s\in [t,t+{\epsilon}_{t}^{(2)}].

(iv)
For each t\in [0,\mathrm{\infty}), there exist {r}_{t}^{(3)}\in [0,1) and {\epsilon}_{t}^{(3)}>0 such that \phi (s)\le {r}_{t}^{(3)} for all s\in (t,t+{\epsilon}_{t}^{(3)}].

(v)
For each t\in [0,\mathrm{\infty}), there exist {r}_{t}^{(4)}\in [0,1) and {\epsilon}_{t}^{(4)}>0 such that \phi (s)\le {r}_{t}^{(4)} for all s\in [t,t+{\epsilon}_{t}^{(4)}).

(vi)
For any nonincreasing sequence {\{{x}_{n}\}}_{n\in \mathbb{N}} in [0,\mathrm{\infty}), we have 0\le {sup}_{n\in \mathbb{N}}\phi ({x}_{n})<1.

(vii)
φ is a function of contractive factor [27], that is, for any strictly decreasing sequence {\{{x}_{n}\}}_{n\in \mathbb{N}} in [0,\mathrm{\infty}), we have 0\le {sup}_{n\in \mathbb{N}}\phi ({x}_{n})<1.
3 Coincidence theorems for Lfuzzy mappings
In this section the notion of {D}_{{\alpha}_{L}} distance is used to study coincidence theorems concerning Lfuzzy mappings. For a metric space (X,d), we define
and
whenever A,B\in {F}_{L}(X) and {A}_{{\alpha}_{L}},{B}_{{\alpha}_{L}}\in CB(X) for each {\alpha}_{L}\in L\mathrm{\setminus}\{{\mathit{0}}_{L}\}.
Definition 3.1 A mapping T:X\to X is called an (\epsilon ,\lambda ) uniformly locally contractive mapping if u,v\in X and 0<d(u,v)<\epsilon implies d(Tu,Tv)\le \lambda d(u,v). A mapping T:X\to {F}_{L}(X) is called an (\epsilon ,\lambda ) uniformly locally contractive Lfuzzy mapping if u,v\in X and 0<d(u,v)<\epsilon, then {d}_{L}^{\mathrm{\infty}}(T(u),T(v))\le \lambda d(u,v).
Theorem 3.2 Let \epsilon \in (0,\mathrm{\infty}], (X,d) be a complete εchainable metric space, {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} a sequence of mappings from X into {F}_{L}(X), and S:X\to X a surjection such that for each u\in X and q\in \mathbb{N}, {[{T}_{q}(u)]}_{{\alpha}_{L}}\in CB(X), for some {\alpha}_{L}\in L\mathrm{\setminus}\{{0}_{L}\}. If u,v\in X such that 0<d(Su,Sv)<\epsilon implies
for all q,r\in \mathbb{N}, where \mu :[0,\epsilon )\to [0,1) is a MTfunction, then S and the sequence {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} have a coincidence point, i.e., there exists {v}^{\ast}\in X such that S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{{\alpha}_{L}}.
Proof Let {v}_{0} be an arbitrary, but fixed element of X. Find {v}_{1}\in X such that S{v}_{1}\in {[{T}_{1}({v}_{0})]}_{{\alpha}_{L}}. Let
be an arbitrary εchain from S{v}_{0} to S{v}_{1}. (Without any loss of generality, we assume that S{u}_{(1,q)}\ne S{u}_{(1,r)} for each q,r\in \{0,1,2,\dots ,l\} with q\ne r.)
Since 0<d(S{u}_{(1,0)},S{u}_{(1,1)})<\epsilon, we get
Rename S{v}_{1} as S{u}_{(2,0)}. Since S{u}_{(2,0)}\in {[{T}_{1}({u}_{(1,0)})]}_{{\alpha}_{L}}, using Lemma 2.1 we find S{u}_{(2,1)}\in {[{T}_{2}({u}_{(1,1)})]}_{{\alpha}_{L}} such that
Since 0<d(S{u}_{(1,1)},S{u}_{(1,2)})<\epsilon, we deduce that
Similarly to S{u}_{(2,1)}\in {[{T}_{2}({u}_{(1,1)})]}_{{\alpha}_{L}}, again using Lemma 2.1 we find S{u}_{(2,2)}\in {[{T}_{2}({u}_{(1,2)})]}_{{\alpha}_{L}} such that
Thus we obtain a set \{S{u}_{(2,0)},S{u}_{(2,1)},S{u}_{(2,2)},\dots ,S{u}_{(2,l)}\} of l+1 points of X such that S{u}_{(2,0)}\in {[{T}_{1}({u}_{(1,0)})]}_{{\alpha}_{L}} and S{u}_{(2,t)}\in {[{T}_{2}({u}_{(1,t)})]}_{{\alpha}_{L}} for t=1,2,\dots ,l, with
for t=0,1,2,\dots ,l1.
Let S{u}_{(2,l)}={v}_{2}. Thus the set of points S{v}_{1}=S{u}_{(2,0)},S{u}_{(2,1)},S{u}_{(2,2)},\dots ,S{u}_{(2,l)}=S{v}_{2}\in {[{T}_{2}({v}_{1})]}_{{\alpha}_{L}} is an εchain from S{v}_{1} to S{v}_{2}. Rename S{v}_{2} as S{u}_{(3,0)}. Then by the same procedure we obtain an εchain
from S{v}_{2} to S{v}_{3}. Inductively, we obtain
with
for t=0,1,2,\dots ,l1.
Consequently, we construct a sequence {\{S{v}_{h}\}}_{h=1}^{\mathrm{\infty}} of points of X with
for all h\in \mathbb{N}.
For each t\in \{0,1,2,\dots ,l1\}, we deduce from (2) that {\{d(S{u}_{(h,t)},S{u}_{(h,t+1)})\}}_{h=1}^{\mathrm{\infty}} is a decreasing sequence of nonnegative real numbers and therefore there exists {l}_{t}\ge 0 such that
By assumption, lim{sup}_{t\to {l}_{t}^{+}}\mu (t)<1, so there exists {h}_{t}\in \mathbb{N} such that \mu (d(S{u}_{(h,t)},S{u}_{(h,t+1)}))<\omega ({l}_{t}) (a nonnegative real number) for all h\ge {h}_{t} where lim{sup}_{t\to {l}_{t}^{+}}\mu (t)<\omega ({l}_{t})<1.
Let
Then, for every h>{h}_{t}, we obtain
Putting N=max\{{h}_{t}:t=0,1,2,\dots ,l1\}, we have
for all h>N+1. Hence
whenever p>h>N+1.
Since {\mathrm{\Theta}}_{t}<1 for all t\in \{0,1,2,\dots ,l1\}, it follows that \{S{v}_{h}=S{u}_{(h,l)}\} is a Cauchy sequence. Since (X,d) is complete, there is {v}^{\ast}\in X such that S{v}_{h}\to S{v}^{\ast}. Hence there exists an integer M>0 such that h>M implies d(S{v}_{h},S{v}^{\ast})<\epsilon. This from the point of view of inequality (1) implies {D}_{{\alpha}_{L}}({T}_{h+1}({v}_{h}),{T}_{q}({v}^{\ast}))<\epsilon for all q\in \mathbb{N}.
Now consider for all q\in \mathbb{N},
Letting h\to \mathrm{\infty} in the above inequality, we get d(S{v}^{\ast},{[{T}_{q}({v}^{\ast})]}_{{\alpha}_{L}})\to 0, which implies S{v}^{\ast}\in {[{T}_{q}({v}^{\ast})]}_{{\alpha}_{L}} for all q\in \mathbb{N}. Hence, S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{{\alpha}_{L}}. □
Corollary 3.3 Let \epsilon \in (0,\mathrm{\infty}], (X,d) a complete εchainable metric space, {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} a sequence of mappings from X into {F}_{L}(X) and S:X\to X a surjection such that for each u\in X and q\in \mathbb{N}, {[{T}_{q}(u)]}_{{\alpha}_{L}}\in CB(X), for some {\alpha}_{L}\in L\mathrm{\setminus}\{{0}_{L}\}. If u,v\in X such that 0<d(Su,Sv)<\epsilon, implies
for all q,r\in \mathbb{N}, where \rho \in (0,1), then S and sequence {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} have a coincidence point, i.e., there exists {v}^{\ast}\in X such that S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{{\alpha}_{L}}.
Proof Apply Theorem 3.2 where μ is the MTfunction defined as \mu (t)=\rho for all t\in [0,\epsilon ). □
In the following we furnish an example to support Theorem 3.2.
Example 3.4 Let \epsilon \in (0,\mathrm{\infty}], X=[0,1], and d(u,v)=uv, whenever u,v\in X, then (X,d) is a complete εchainable metric space. Let L=\{\zeta ,\eta ,\xi ,\varsigma \} with \zeta {\precsim}_{L}\eta {\precsim}_{L}\varsigma, \zeta {\precsim}_{L}\xi {\precsim}_{L}\varsigma, η and ξ are not comparable, then (L,{\precsim}_{L}) is a complete distributive lattice. Suppose {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} to be a sequence of mappings defined from X into {F}_{L}(X) as
and S:X\to X be a surjective self mapping defined as Sx={x}^{3}, for all x\in X. Now for x,y\in X\mathrm{\setminus}\{0\}, suppose \epsilon =\frac{1}{xy}\in (0,\mathrm{\infty}],
Assume {\alpha}_{L}=\eta then {[{T}_{q}(u)]}_{\eta}=[0,\frac{{u}^{3}}{11\beta q}]. For q,r\in \mathbb{N} with q\le r, u,v\in X, and \mu (t)=\frac{1}{\beta} for all t\in (0,\epsilon ] consider
Since all the conditions of Theorems 3.2 are satisfied, there exist a coincidence point of S and the sequence {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}}, i.e.
for some {\alpha}_{L}\in L\mathrm{\setminus}\{{0}_{L}\}.
4 Coincidence theorems for Lfuzzy mappings via {d}_{L}^{\mathrm{\infty}}distance
This section deals with the study of coincidence theorems in connection with the notion of {d}_{L}^{\mathrm{\infty}}distance. The results proved in this section are also new.
Theorem 4.1 Let \epsilon \in (0,\mathrm{\infty}], (X,d) a complete εchainable metric space, {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} a sequence of Lfuzzy mappings from X into {F}_{L}(X) and S:X\to X a surjection such that for each u\in X and q\in \mathbb{N}, {[{T}_{q}(u)]}_{{\alpha}_{L}}\in CB(X), for some {\alpha}_{L}\in L\mathrm{\setminus}\{{0}_{L}\}. If u,v\in X such that 0<d(Su,Sv)<\epsilon implies
for all q,r\in \mathbb{N}, where \mu :[0,\epsilon )\to [0,1) is a MTfunction, then S and sequence {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} have a coincidence point, i.e., there exists {v}^{\ast}\in X such that S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{{\alpha}_{L}}.
Proof Since {D}_{{\alpha}_{L}}({T}_{q}(u),{T}_{r}(v))\le {d}_{L}^{\mathrm{\infty}}({T}_{q}(u),{T}_{r}(v)) for all q,r\in \mathbb{N}, the result follows immediately from Theorem 3.2. □
By taking S=I and in Theorem 4.1 we obtain the following result.
Corollary 4.2 Let \epsilon \in (0,\mathrm{\infty}], (X,d) a complete εchainable metric space and {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} a sequence of Lfuzzy mappings from X into {F}_{L}(X) such that for each u\in X and q\in \mathbb{N}, {[{T}_{q}(u)]}_{{\alpha}_{L}}\in CB(X), for some {\alpha}_{L}\in L\mathrm{\setminus}\{{0}_{L}\}. If u,v\in X such that 0<d(u,v)<\epsilon, implies
for all q,r\in \mathbb{N}, where \mu :[0,\epsilon )\to [0,1) is a MTfunction, then the sequence {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} has a common fixed point, i.e., there exists {v}^{\ast}\in X such that {v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{\alpha}.
5 Coincidence theorems for fuzzy mappings
In the present section, by considering L=[0,1] in Theorem 3.2, some further new results for fuzzy mappings are obtained.
Theorem 5.1 Let \epsilon \in (0,\mathrm{\infty}], (X,d) a complete εchainable metric space, {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} a sequence of fuzzy mappings from X into {I}^{X} and S:X\to X a surjection such that for each u\in X and q\in \mathbb{N}, {[{T}_{q}(u)]}_{\alpha}\in CB(X), for some \alpha \in (0,1]. If u,v\in X such that 0<d(Su,Sv)<\epsilon, implies
for all q,r\in \mathbb{N}, where \mu :[0,\epsilon )\to [0,1) is a MTfunction, then S and sequence {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} have a coincidence point, i.e., there exists {v}^{\ast}\in X such that S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{\alpha}.
Corollary 5.2 Let \epsilon \in (0,\mathrm{\infty}], (X,d) a complete εchainable metric linear space, {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} a sequence of fuzzy mappings from X into W(X) and S:X\to X a surjection such that for each u\in X and q\in \mathbb{N}, {[{T}_{q}(u)]}_{\alpha}\in CB(X), for some \alpha \in (0,1]. If u,v\in X such that 0<d(Su,Sv)<\epsilon, implies
for all q,r\in \mathbb{N}, where \mu :[0,\epsilon )\to [0,1) is a MTfunction, then S and sequence {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} have a coincidence point, i.e., there exists {v}^{\ast}\in X such that S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{\alpha}.
Proof Since W(X)\subseteq CB(X) and {D}_{\alpha}({T}_{q}(u),{T}_{r}(v))\le {d}_{\mathrm{\infty}}({T}_{q}(u),{T}_{r}(v)) for all q,r\in \mathbb{N}, the result follows immediately from Theorem 5.1. □
6 Fixed point theorems for Lfuzzy mappings
In this section some new fixed point results are deduced from the above mentioned coincidence results. If we take S=I in Theorem 3.2 we get the following result.
Theorem 6.1 Let \epsilon \in (0,\mathrm{\infty}], (X,d) a complete εchainable metric space and {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} a sequence of mappings from X into {F}_{L}(X) such that for each u\in X and q\in \mathbb{N}, {[{T}_{q}(u)]}_{{\alpha}_{L}}\in CB(X), for some {\alpha}_{L}\in L\mathrm{\setminus}\{{0}_{L}\}. If u,v\in X such that 0<d(u,v)<\epsilon implies
for all q,r\in \mathbb{N}, where \mu :[0,\epsilon )\to [0,1) is a MTfunction, then the sequence {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} has a common fixed point, i.e., there exists {v}^{\ast}\in X such that {v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{{\alpha}_{L}}.
If we take S=I in Theorem 5.1 we get the following result.
Corollary 6.2 Let \epsilon \in (0,\mathrm{\infty}], (X,d) a complete εchainable metric space and {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} a sequence of mappings from X into {I}^{X} such that for each u\in X and q\in \mathbb{N}, {[{T}_{q}(u)]}_{\alpha}\in CB(X), for some \alpha \in (0,1]. If u,v\in X such that 0<d(u,v)<\epsilon, implies
for all q,r\in \mathbb{N}, where \mu :[0,\epsilon )\to [0,1) is a MTfunction, then the sequence {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} has a common fixed point, i.e., there exists {v}^{\ast}\in X such that {v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{\alpha}.
By considering \alpha =1 in the above corollary we deduce the main result and hence all the corollaries of [18].
7 Applications to multivalued maps
Multivalued mapping is a lefttotal relation, arise in optimal control theory and game theory. In mathematics, multivalued mappings play an increasingly important role. For example fixed point results for multivalued mappings have been applied to prove existence of Nash equilibrium, the solutions of integral and differential inclusions etc. In this section, we will apply our main result to prove some coincidence results for multivalued mappings and then obtain some practical fixed point theorems in the existing literature.
Theorem 7.1 Let \epsilon \in (0,\mathrm{\infty}), (X,d) a complete εchainable metric space, {\{{J}_{q}\}}_{q=1}^{\mathrm{\infty}} be a sequence of multivalued mappings from X into CB(X) and S:X\to X a surjection such that 0<d(Su,Sv)<\epsilon, implies
u,v\in X, where \mu :[0,\epsilon )\to [0,1) is a MTfunction, then S and the sequence {\{{J}_{q}\}}_{q=1}^{\mathrm{\infty}} have a coincidence point, i.e., there exists {v}^{\ast}\in X such that S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}\{{J}_{q}({v}^{\ast})\}.
Proof Define a sequence of Lfuzzy mappings {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}} from X into {F}_{L}(X) as, for some {\alpha}_{L}\in L\mathrm{\setminus}\{{0}_{L}\}, {T}_{q}(v)(t)={\alpha}_{L} if t\in {J}_{q}(v) and {T}_{q}(v)(t)={0}_{L}, otherwise. Then {[{T}_{q}(v)]}_{{\alpha}_{L}}={J}_{q}(v) for all v\in X, so {[{T}_{q}(v)]}_{{\alpha}_{L}}\in CB(X) for all v\in X. Since
for all u,v\in X, we deduce that condition (1) of Theorem 3.2 is satisfied for {\{{T}_{q}\}}_{q=1}^{\mathrm{\infty}}. Hence there exists a point {v}^{\ast} in X, such that S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}{[{T}_{q}({v}^{\ast})]}_{{\alpha}_{L}}. From this we conclude that S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}\{{J}_{q}({v}^{\ast})\}. This completes the proof. □
Corollary 7.2 Let \epsilon \in (0,\mathrm{\infty}), (X,d) a complete εchainable metric space, {\{{J}_{q}\}}_{q=1}^{\mathrm{\infty}} be a sequence of multivalued mappings from X into CB(X) and S:X\to X a surjection such that 0<d(Su,Sv)<\epsilon, implies
u,v\in X, where \rho \in (0,1), then S and the sequence {\{{J}_{q}\}}_{q=1}^{\mathrm{\infty}} have a coincidence point, i.e., there exists {v}^{\ast}\in X such that S{v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}\{{J}_{q}({v}^{\ast})\}.
By taking S=I in Theorem 7.1 we get the following.
Corollary 7.3 [25]
Let \epsilon \in (0,\mathrm{\infty}], (X,d) be a complete εchainable metric space, and {\{{J}_{q}\}}_{q=1}^{\mathrm{\infty}} be a sequence of multivalued mappings from X into CB(X) such that 0<d(u,v)<\epsilon, implies
u,v\in X, where \mu :[0,\epsilon )\to [0,1) is a MTfunction, then the sequence {\{{J}_{q}\}}_{q=1}^{\mathrm{\infty}} has a common fixed point, i.e., there exists {v}^{\ast}\in X such that {v}^{\ast}\in {\bigcap}_{q\in \mathbb{N}}\{{J}_{q}({v}^{\ast})\}.
Let (X,d) be a complete metric space, J a multivalued mapping from X into CB(X), and \mu :[0,\mathrm{\infty})\to [0,1) a MTfunction such that
for all u,v\in X. Then J has a fixed point in X.
Proof Taking q=r=1 with \epsilon =\mathrm{\infty}, in the above corollary we get the required result. □
Corollary 7.5 [26]
Let \epsilon \in (0,\mathrm{\infty}], (X,d) a complete εchainable metric space and J be a multivalued mapping from X into CB(X) such that 0<d(u,v)<\epsilon, implies
u,v\in X, where \rho \in (0,1). Then J has a fixed point.
By considering J to be a single valued mapping in the above corollary we deduce the following result.
Corollary 7.6 [17]
Let \epsilon \in (0,\mathrm{\infty}], (X,d) a complete εchainable metric space and T:X\to X be a (\epsilon ,\lambda ) uniformly locally contractive single valued mapping. Then T has a fixed point.
8 Applications to integral and differential equations
The theory of differential inclusions was scientifically recognized by Aubin and Cellina [28]. They studied the existence and properties of solutions to differential inclusions of the form \frac{d}{dt}x(t)\in L(t,x(t)). Theorem 3.2 can deal with the existence of the solutions of differential inclusions of form \frac{d}{dt}f(x(t))\in K(t,x(t)). However, to identify it we have to explore some extra material concerned with a version of measurable selection theorem for continuous multivalued functions with nonempty convex closed (or compact) values on a Banach space, which may be problematic for a common reader. Therefore in this section, let us restrict our research area. We shall mainly consider the nonlinear differential equations of form \frac{d}{dt}f(x(t))=K(t,x(t)). The main objective of this section is to study the existence and uniqueness of the solution of a general class of Volterra integral equations arising from differential equations of the form \frac{d}{dt}f(x(t))=K(t,x(t)) under various assumptions on the functions involved. Theorem 3.2 together with a function space (C[a,b],\mathbb{R}), and a contractive inequality are used to establish the result. Consider the integral equation:
where x:[a,b]\to \mathbb{R} is unknown, and h:\mathbb{R}\to \mathbb{R} is given, η is a parameter. If h=I (the identity mapping on ℝ), then (3) is known as the Volterra integral equation.
Theorem 8.1 Let {L}_{0}:[a,b]\to \mathbb{R}, L:[a,b]\times \mathbb{R}\to \mathbb{R} be continuous mappings and h:\mathbb{R}\to \mathbb{R} a continuous surjection. If there exists K<\frac{1}{ba} such that for r,s\in \mathbb{R},
then the integral equation
has a solution in (C[a,b],\mathbb{R}).
Proof Let X=(C[a,b],\mathbb{R}); then X is a complete εchainable metric space for \epsilon \in (0,\mathrm{\infty}). Let {\phi}_{L}:X\to L\mathrm{\setminus}\{{0}_{L}\} be an arbitrary mapping. Define d:X\times X\to \mathbb{R} as d(x,y)={max}_{t\in [a,b]}x(t)y(t). Assume that, for x\in X,
Define the mappings T:X\to {F}_{L}(X) and S:X\to X as follows:
Note that hx(t)hy(t)<\epsilon for all t\in [a,b]\u27fad(Sx,Sy)<\epsilon.
Take {\alpha}_{L}={\phi}_{L}(x). Moreover, for some f\in {[Tx]}_{{\alpha}_{L}}, we obtain, T(x)(f)={\phi}_{L}(x). Then, by the assumptions, for every f\in X there exists y\in X such that f=Sy=h\circ y.
Moreover, we obtain
If hx(t)hy(t)<\epsilon for all t\in [a,b], by assumptions, we have
It implies that
Hence, if for a MTfunction \mu :[0,\epsilon )\to [0,1), \mu (d(Sx,Sy))=K(ba), all conditions of Theorem 3.2 are satisfied to find a continuous function u:[a,b]\to \mathbb{R} such that Su\in {[T(u)]}_{{\alpha}_{L}}. That is, h\circ u={\tau}_{u} and u will be a solution of the integral equation (4). □
Corollary 8.2 Let {K}_{0}\in \mathbb{R}, L:[a,b]\times \mathbb{R}\to \mathbb{R} are a continuous mapping and h:\mathbb{R}\to \mathbb{R} a continuous surjection. If there exists K<\frac{1}{ba} such that for r,s\in \mathbb{R},
then the initial value problem
has a solution in (C[a,b],\mathbb{R}).
Proof Considering the integral equation:
we get the required result by Theorem 8.1 for {L}_{0}(t)={K}_{0}. □
9 An illustrative example
In this section we provide a simple but practical example to illustrate the theory developed in the above section. The problem under consideration is a solution of the nonlinear integral equation:
Note that, for L(t,s)=[{x}^{7}(s)+17s+5]s, {L}_{0}(t)=tant, hs={s}^{7},
for all t\in [0,c] and all conditions of Theorem 8.1 are satisfied (for K=c, a=0). Let X=(C[0,c],\mathbb{R}). Define the mappings T:X\to {F}_{L}(X) and S:X\to X as follows:
In the following we approximate the value of u, by constructing the iterative sequences:
Suppose
Note that
Let {x}_{0}:[0,c]\to \mathbb{R} be defined as {x}_{0}(t)=0 for all t\in [0,c]. Then
Thus, S{x}_{1}={\tau}_{{x}_{0}}, where
Now,
where
and
Similarly,
and {x}_{3}(t)={(tant+\frac{17{t}^{3}}{3}+\frac{5{t}^{2}}{2}+(\frac{17{t}^{5}}{3.5}+\frac{5{t}^{4}}{2.4}+\frac{(tant){t}^{2}}{2})+(\frac{17{t}^{7}}{3.5.7}+\frac{5{t}^{6}}{2.4.6}+\frac{(tant){t}^{4}}{2.4}))}^{\frac{1}{7}}.
It follows that
Hence,
is a solution of integral equation (6).
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Rashid, M., Kutbi, M.A. & Azam, A. Coincidence theorems via alpha cuts of Lfuzzy sets with applications. Fixed Point Theory Appl 2014, 212 (2014). https://doi.org/10.1186/168718122014212
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DOI: https://doi.org/10.1186/168718122014212
Keywords
 coincidence point
 fixed point
 multivalued mapping
 Lfuzzy mapping
 fuzzy mapping
 integral equation