Coincidence and common fixed points of integral contractions for L-fuzzy maps with applications in fuzzy functional inclusions

Abstract

Branciari defined the integral contractions to generalize the Banach contraction principle. Moreover recently Phiangsungnoen proved a fixed point theorem to generalize the ordered structure and contractive conditions with admissible mappings. In this article, we prove some coincidence and common fixed point theorems for a pair of L-fuzzy mappings satisfying integral β-admissible and generalize integral contractions. These results generalize the above famous results and many others in the literature. We define two types of fuzzy functional inclusions and prove existence of their solutions as a consequence of our results. Some nontrivial examples are given to provide significance of ourresults.

Keywords

Integral contractions L-fuzzy mappings functional inclusions

1 Introduction

The integral type generalized contractions are introduced by Branciari [10]. Given (X, d) be a complete metric space. Branciari discussed the self mapping T on X, satisfying the contractive conditions of the following type, for x, y ∈ X and for some λ ∈ (0, 1), $\overset{d (Tx, Ty)}{\int_{0}} φ (t) dt \leq λ \overset{d (x, y)}{\int_{0}} φ (t) dt$ holds for any Lebesgue integrable function φ: [0, + ∞) → [0, + ∞) which is a summable on each compact subset of [0, + ∞) and satisfies $\overset{ɛ}{\int_{0}} φ (t) dt > 0,$ for all ɛ > 0. Many authors have generalized this result in different directions, for more details we refer the readers to ([1 , 27]) and the references therein.

Fuzzy sets were introduced by Zadeh [33] in 1965. A remarkable progress in fuzzy set theory has been made in last few decades. The fuzzy set theory has not only applications in physical and applied sciences, but also in mathematical analysis, decision making, clustering, data mining and soft sciences as well [21 , 35–39]. The concept of fuzzy mappings was introduced in [14]. Many articles regarding fuzzy mappings can be found in the literature [6–8 , 30].

A real generalization of fuzzy sets by replacing the interval [0, 1] by a complete distributive lattice was presented in [13] and named as L-fuzzy sets. In [31], it has been shown that L-fuzzy sets are also a generalization of the Intuitionistic fuzzy sets [4]. Fuzzy fixed points by β-admissible mappings were introduced by Phiangsungreon et al. [30]. This concept for L-fuzzy mappings was initiated in [26], the authors used L-fuzzy β-admissible mappings to find the common fixed points of two mappings. However the notion of β-admissible mappings was presented by Samet et.al. in [29].

In this article the notion of an integral β_{F
_L}-admissible contraction is introduced for a pair of L-fuzzy mappings, using this type of contraction a coincidence point theorem is proved. A generalized integral type theorem is also presented to find the common fixed points of a pair of L-fuzzy mappings. An example has been given to show that contractive conditions given in the main results of [6–8 , 30], are not satisfied, but our contractive condition is satisfied to obtain the common fixed points of the given mappings. This means the mentioned results are not applicable to find the common fixed points but our results are applicable. Many optimization problems have been solved for system of functional equations [11, 34]. It is well known that the physical problems with uncertainty can be handled more accurately by means of fuzzy sets approach. Keeping this in view, we introduce two types of systems of fuzzy functional inclusions, a hybridization of functional equations with fuzzy mappings. In the first type, level cuts of fuzzy mappings are used while in the second type, the level sets of given fuzzy mapping are used. To validate the significance of our results, we present the existence theorems for the solutions of these two types of systems of fuzzy functional inclusions. We remark that our results will be useful and significant in the modeling and solution of optimization problems in mathematical analysis.

Throughout this article we use these notations, for any complete distributive lattice L and any non-empty subset A of L, for a, b ∈ L, min { a, b } = a ∨ b, min(A) = ∨ A, max { a, b } = a ∧ b, and max(A) = ∧ A. Let (Y, d) be a metric space, by CB (Y) and K (Y) we denote respectively the set of all closed bounded, and compact subsets of Y. Recall that the Hausdorff distance of closed and bounded subsets A and B is defined as; $H (A, B) = max {sup_{a \in A} d (a, B), sup_{b \in B} d (b, A)} .$

2 Preliminaries

In this section, we recall some basic concepts, which will be used to prove our results.

2.1 Definition [13]

Let L be a nonempty set and ≾_L be a partial order relation on L. Then the partially ordered set (L, ≾ _L) is called

i) a lattice, if a ∨ b ∈ L, a ∧ b ∈ L for any a, b ∈ L,

ii) a complete lattice, if ∨A ∈ L, ∧A ∈ L for any A ⊆ L,

iii) distributive if a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c), a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) for any a, b, c ∈ L.

2.2 Definition [13]

Let L be a lattice with top (infinity) element 1_L and bottom (zero) element 0_L and let a, b ∈ L. Then b is called a complement of a, if a ∨ b = 1_L, and a ∧ b = 0_L. If a ∈ L has a complement element, then it is unique, and is denoted by $\overset{´}{a} .$

2.3 Definition [13]

An L-fuzzy set A on a non-empty set X is a function A: X → L, where L is a complete distributive lattice with 1_L and 0_L.

2.4 Remark

The class of L-fuzzy sets is larger than the class of fuzzy sets as an L-fuzzy set is a fuzzy set if L = [0, 1].

The α_L-level set and α_L-cut of an L-fuzzy set A, are denoted by [A] _{α
_L} and (A) _{α
_L} respectively, and are defined as; ${[A]}_{α_{L}} = {x \in X : α_{L} ≾_{L} A (x)} if α_{L} \in L ∖ {0_{L}},$ and ${(A)}_{α_{L}} = {x \in X : α_{L} ≺_{L} A (x)} if α_{L} \in L ∖ {0_{L}} .$

Let X be a non-empty set, denote the class of all L-fuzzy set on X by _L (X). F (X) denotes the set of all fuzzy sets on X. The characteristic function χ_{L
_A} of an L-fuzzy set A is defined in [26] by: $χ_{L_{A}} (x) : = {\begin{matrix} 0_{L} if x \notin A \\ 1_{L} if x \in A . \end{matrix}$

2.5 Definition [26]

Let X be an arbitrary set, Y be a metric space. A mapping T is called L-fuzzy mapping, if T is a mapping from X into ℱ_L (Y). An L-fuzzy mapping T is a L-fuzzy subset on X × Y with membership function T (x) (y). The function T (x) (y) is the grade of membership of y in T (x) with respect to L.

2.6 Definition [26]

Let (X, d) be a metric space and S, T be L-fuzzy mappings from X into ℱ_L(X). A point z ∈ X is called an L-fuzzy fixed point of T if z ∈ [Tz] _{α
_L}, where α_L ∈ L ∖ {0_L}. The point z ∈ X is called a common L-fuzzy fixed point of S and T if z ∈ [Sz] _{α
_L} ∩ [Tz] _{α
_L}.

2.7 Definition [26]

Let (X, d) be a metric space, β: X × X ⟶ [0, ∞) and S, T be L-fuzzy mappings from X into ℱ_L (X). The pair (S, T) is said to be β_{ℱ_L}-admissible if it satisfies the following conditions:

i) for each x ∈ X and y ∈ [Sx] _{α_L(x)}, where α_L (x) ∈ L ∖ {0_L}, with β (x, y) ≥ 1, we have β (y, z) ≥ 1 for all z ∈ [Ty] _{α_L(y)} ≠ φ, where α_L (y) ∈ L ∖ {0_L},

ii) for each x ∈ X and y ∈ [Tx] _{α_L(x)}, where α_L (x) ∈ L ∖ {0_L}, with β (x, y) ≥ 1, we have β (y, z) ≥ 1 for all z ∈ [Sy] _{α_L(y)} ≠ φ, where α_L (y) ∈ L ∖ {0_L}.

If S = T, then T is called β_{ℱ_L}-admissible.

Let Φ denotes the family of all mappings $φ : ℝ^{+} \to ℝ$ satisfying: φ is nonnegative, Lebesgue integrable, integrably subadditive, and for each ɛ > 0, $\underset{0}{\int^{ɛ}} φ (t) dt > 0 .$ Now, let us recall the following lemma.

2.8 Lemma [23]

If {a_n} is a sequence in [0, ∞), then $lim_{n \to \infty} \underset{0}{\int^{a_{n}}} φ (s) ds = 0$ if and only if a_n → 0, as n → ∞.

The following definition will be crucial for our main results, in which a new contractive condition of integral type is defined.

2.9 Definition

Let (X, d) be a complete metric space, β: X × X ⟶ [0, ∞), φ ∈ Φ and S, T be L-fuzzy mappings (mappings from X into F_L (X)). The pair (S, T) is said to be an integral β_{F
_L}-admissible contraction if the following conditionshold:

a) For each x ∈ X, there exists α_L (x) ∈ L ∖ {0_L} such that [Sx] _{α_L(x)}, [Tx] _{α_L(x)} are non-empty closed bounded subsets of X and for x₀ ∈ X, there exists x₁ ∈ [Sx₀] _{α_L(x₀)} such that β (x₀, x₁) ≥ 1,

b) For all x, y ∈ X, we have $\begin{matrix} \int_{0}^{M} φ (t) dt & < & a_{1} \int_{0}^{d (x, {[Sx]}_{α_{L} (x)})} φ (t) dt \\ + & a_{2} \int_{0}^{d (y, {[Ty]}_{α_{L} (y)})} φ (t) dt \\ + a_{3} \int_{0}^{d (x, {[Ty]}_{α_{L} (y)})} φ (t) dt \\ + & a_{4} \int_{0}^{d (y, {[Sx]}_{α_{L} (x)})} φ (t) dt \\ + a_{5} \int_{0}^{d (x, y)} φ (t) dt + ϵ, \end{matrix}$ where $M = max {β (x, y), β (y, x)} [H ({[Sx]}_{α_{L} (x)}, {[Ty]}_{α_{L} (y)})] + ϵ,$ for all x, y ∈ X, where a₁, a₂, a₃, a₄, a₅ are nonnegative real numbers and $\sum_{i = 1}^{5} a_{i} < 1$ and either a₁ = a₂ or a₃ = a₄,

c) (S, T) is β_{F
_L}-admissible pair,

d) If {x_n } ∈ X, such that β (x_n, x_n+1) ≥ 1 and x_n ⟶ x implies β (x_n, x) ≥ 1.

The following is the example of an integral β_{F
_L}-admissible contraction.

2.10 Example

Let X ={ 1, 2, 3 } and define a metric $d : X \times X \to ℝ$ by $d (x, y) = {\begin{matrix} 0, if x = y \\ 1, if x \neq y and x, y \in {1, 3} \\ 2, if x \neq y and x, y \in {1, 2} \\ 3, if x \neq y and x, y \in {2, 3} \end{matrix}$ It is easy to see that (X, d) is complete metric space. Let L = {a, b, c, d} with a ≾ _Lb ≾ _Ld, a ≾ _Lc ≾ _Ld, b and c are not comparable. Define L-fuzzy mappings S, T: X → F_L (X) by $(T 1) (t) = {\begin{matrix} d if t = 1 \\ a if t = 2, 3 \end{matrix}$ $(T 2) (t) = {\begin{matrix} c if t = 1, 2 \\ d if t = 3 \end{matrix}$ $(T 3) (t) = {\begin{matrix} b if t = 1, 3 \\ d if t = 2 \end{matrix}$ and $(S 1) (t) = (S 2) (t) = (S 3) (t) = {\begin{matrix} d if t = 1 \\ b if t = 2, 3 \end{matrix}$ Take α = d, then we get ${[Tx]}_{α} = {\begin{matrix} {1} if x = 1 \\ {3} if x = 2 \\ {2} if x = 3, \end{matrix}$ and ${[Sx]}_{α} = {1} for all x \in X .$ We define β: X × X → [0, ∞) by $β (x, y) = {\begin{matrix} 1, if x, y \in {1.2} \\ 0, otherwise . \end{matrix}$ First we show that (S, T) is a β_{F
_L}-admissible pair. For x ∈ X, y∈ [Sx] _α = { 1 } with β (x, y) ≥ 1, we have either x = 1, y = 1 or x = 2, y = 1. If x = 1, y = 1, then β (y, z) ≥ 1 for all z ∈ [Ty] _α = { 1 }. If x = 2, y = 1, then β (y, z) ≥ 1 for all z ∈ [Ty] _α = { 1 }.

Now for x ∈ X and y ∈ [Tx] _α with β (x, y) ≥ 1, we get only x = y = 1, therefore β (y, z) ≥ 1 for all z ∈ [Sx] _α = { 1 }. Hence the pair (S, T) is β_{F
_L}-admissible.

2.11 Remark

For L = [0, 1], the above definition will be reduced to the pair (S, T), an integral β_F-admissible contraction of two pairs of fuzzy mappings S and T.

The following Lemmas will be very useful to prove our main results.

2.12 Lemma [25]

Let (X, d) be a metric space and A, B ∈ CB (X). Then for each a ∈ A $d (a, B) \leq H (A, B) .$

2.13 Lemma [25]

Let (X, d) be a metric space and A, B ∈ CB (X). Then for each a ∈ A, and for each ϵ > 0, there exists an element b ∈ B such that $d (a, b) \leq H (A, B) + ϵ .$

In [30], the authors have generalized the results of [3 , 29], in the following sense:

2.14 Theorem [30]

Let (X, d) be a complete metric space, S and T be fuzzy mappings from X into F (X), α: X → (0, 1] and β: X × X → [0, ∞) satisfy the following conditions:

(a) For each x, y ∈ X, [Sx] _α and [Tx] _α are nonempty closed and bounded subsets of X with $\begin{matrix} max {β (x, y), β (y, x)} [H ({[Sx]}_{α (x)}, {[Ty]}_{α (y)})] \\ \leq & a_{1} d (x, {[Sx]}_{α (x)}) + a_{2} d (y, {[Ty]}_{α (y)}) \\ + a_{3} d (x, {[Ty]}_{α (y)}) + a_{4} d (y, {[Sx]}_{α (x)}) \\ + a_{5} d (x, y), \end{matrix}$ where a₁, a₂, a₃, a₄, a₅ are nonnegative real numbers and $\sum_{i = 1}^{5} a_{i} < 1$ and either a₁ = a₂ or a₃ = a₄,

(d) If {x_n } ∈ X, such that β (x_n, x_n+1) ≥ 1 and x_n ⟶ x implies β (x_n, x) ≥ 1. Then there exists z ∈ X such that z ∈ [Sz] _α(z) ∩ [Tz] _α(z).

3 Main results

The ordering of ordered metric spaces have been generalized in [16], by using the graphs with graphic contractive mappings. Afterward, in [3 , 30], this graphic structure is further generalized. In the next theorem, we generalize Theorem 2.14 of [30] using the L-fuzzy mappings and β - admissible integral type contractive conditions. Conditions for the existence of common fixed points of a pair of L-fuzzy mappings are discussed. In this theorem contractive condition, mappings and structure of the space are generalized.

3.1 Theorem

Let (X, d) be a complete metric space, β: X × X ⟶ [0, ∞), φ ∈ Φ and S, T be L-fuzzy mappings from X into F_L (X) such that (S, T) is an integral β_{F
_L}-admissible contraction. Then there exists z ∈ X such that z ∈ [Sz] _α(z) ∩ [Tz] _α(z).

Proof. Clearly the conditions (a) - (d) of Definition 2.9 are satisfied. We consider the following three possible cases:

(i) a₁ + a₃ + a₅ = 0,

(ii) a₂ + a₄ + a₅ = 0,

(iii) a₁ + a₃ + a₅ ≠ 0, a₂ + a₄ + a₅ ≠ 0. Case (i) For x₀ ∈ X in condition (a), there exist α_L (x₀) ∈ L ∖ {0_L} and x₁ ∈ [Sx₀] _{α_L(x₀)} such that β (x₀, x₁) ≥ 1, and also there exists α_L (x₁) ∈ L ∖ {0_L} such that [Sx₀] _{α_L(x₀)} and [Tx₁] _{α_L(x₁)} are non-empty closed bounded subsets of X. From Lemma 2.12, we obtain that $\begin{matrix} d (x_{1}, {[{Tx}_{1}]}_{α_{L} (x_{1})}) \\ \leq & H ({[{Sx}_{0}]}_{α_{L} (x_{0})}, {[{Tx}_{1}]}_{α_{L} (x_{1})}) \\ \leq & β (x_{0}, x_{1}) [H ({[{Sx}_{0}]}_{α_{L} (x_{0})}, {[{Tx}_{1}]}_{α_{L} (x_{1})})] \\ \leq & max {β (x_{0}, x_{1}), β (x_{1}, x_{0})} \\ \times [H ({[{Sx}_{0}]}_{α_{L} (x_{0})}, {[{Tx}_{1}]}_{α_{L} (x_{1})})] + ϵ . \end{matrix}$ Now (b) implies that $\begin{matrix} \int_{0}^{d (x_{1}, {[{Tx}_{1}]}_{α_{L} (x_{1})})} φ (t) dt \\ < & a_{1} \int_{0}^{d (x_{0}, {[{Sx}_{0}]}_{α_{L} (x_{0})})} φ (t) dt \\ + a_{2} \int_{0}^{d (x_{1}, {[{Tx}_{1}]}_{α_{L} (x_{1})})} φ (t) dt \\ + a_{3} \int_{0}^{d (x_{0}, {[{Tx}_{1}]}_{α_{L} (x_{1})})} φ (t) dt \\ + & a_{4} \int_{0}^{d (x_{1}, {[{Sx}_{0}]}_{α_{L} (x_{0})})} φ (t) dt \\ + a_{5} \int_{0}^{d (x_{0}, x_{1})} φ (t) dt + ϵ . \end{matrix}$ Using a₁ + a₃ + a₅ = 0 together with the fact that d (x₁, [Sx₀] _{α_L(x₀)}) = 0, we get $\int_{0}^{d (x_{1}, {[{Tx}_{1}]}_{α_{L} (x_{1})})} φ (t) dt < a_{2} \int_{0}^{d (x_{1}, {[{Tx}_{1}]}_{α_{L} (x_{1})})} φ (t) dt + ϵ .$ It follows that x₁ ∈ [Tx₁] _{α_L(x₁)}, which implies that $d (x_{1}, {[{Sx}_{1}]}_{α_{L} (x_{1})}) \leq H ({[{Tx}_{1}]}_{α_{L} (x_{1})}, {[{Sx}_{1}]}_{α_{L} (x_{1})}) .$ Using condition (c), for x₀ ∈ X and x₁ ∈ [Sx₀] _{α_L(x₀)} such that β (x₀, x₁) ≥ 1, we have β (x₁, z) ≥ 1 for all z ∈ [Tx₁] _{α_L(x₁)}. Since x₁ ∈ [Tx₁] _{α_L(x₁)}, therefore β (x₁, x₁) ≥ 1 and hence for arbitrary ϵ > 0, we have $\begin{matrix} d (x_{1}, {[{Sx}_{1}]}_{α_{L} (x_{1})}) \\ \leq β (x_{1}, x_{1}) [H ({[{Sx}_{1}]}_{α_{L} (x_{1})}, {[{Tx}_{1}]}_{α_{L} (x_{1})})] + ϵ . \end{matrix}$ Again, using (b), and the fact that a₁ + a₃ + a₅ = 0 and d (x₁, [Tx₁] _{α_L(x₁)}) = 0, we get $\int_{0}^{d (x_{1}, {[{Sx}_{1}]}_{α_{L} (x_{1})})} φ (t) dt < a_{4} \int_{0}^{d (x_{1}, {[{Sx}_{1}]}_{α_{L} (x_{1})})} φ (t) dt + ϵ,$ which implies that x₁ ∈ [Sx₁] _{α_L(x₁)} and hence $x_{1} \in {[{Sx}_{1}]}_{α_{L} (x_{1})} \cap {[{Tx}_{1}]}_{α_{L} (x_{1})} .$ Case (ii) For x₀ ∈ X in condition (a), there exist α_L (x₀) ∈ L ∖ {0_L} and x₁ ∈ [Sx₀] _{α_L(x₀)} such that β (x₀, x₁) ≥ 1 and also there exists α_L (x₁) ∈ L ∖ {0_L} such that [Sx₀] _{α_L(x₀)} and [Tx₁] _{α_L(x₁)} are non-empty closed bounded subsets of X. By condition (c), we have β (x₁, x₂) ≥ 1 for all x₂ ∈ [Tx₁] _{α_L(x₁)}. From Lemma 2.12, we obtain that $\begin{matrix} \int_{0}^{d (x_{2}, {[{Sx}_{2}]}_{α_{L} (x_{2})})} φ (t) dt \\ \leq & \int_{0}^{H ({[{Tx}_{1}]}_{α_{L} (x_{1})}, {[{Sx}_{2}]}_{α_{L} (x_{2})})} φ (t) dt \\ \leq & \int_{0}^{β (x_{1}, x_{2}) H ({[{Sx}_{2}]}_{α_{L} (x_{2})}, {[{Tx}_{1}]}_{α_{L} (x_{1})})} φ (t) dt \\ \leq & \int_{0}^{max {β (x_{1}, x_{2}), β (x_{2}, x_{1})} H ({[{Sx}_{2}]}_{α_{L} (x_{2})}, {[{Tx}_{1}]}_{α_{L} (x_{1})}) + ϵ} φ (t) dt \end{matrix}$ Using (b), and the fact that a₂ + a₄ + a₅ = 0 and d (x₂, [Tx₁] _{α_L(x₁)}) = 0, we get $\int_{0}^{d (x_{2}, {[{Sx}_{2}]}_{α_{L} (x_{2})})} φ (t) dt < a_{1} \int_{0}^{d (x_{2}, {[{Sx}_{2}]}_{α_{L} (x_{2})})} φ (t) dt + ϵ .$ It follows that x₂ ∈ [Sx₂] _{α_L(x₂)}, which further implies that $d (x_{2}, {[{Tx}_{2}]}_{α_{L} (x_{2})}) \leq H ({[{Sx}_{2}]}_{α_{L} (x_{2})}, {[{Tx}_{2}]}_{α_{L} (x_{2})}) .$ Using condition (c), we have β (x₂, x₂) ≥ 1, and hence $\begin{matrix} \int_{0}^{d (x_{2}, {[{Tx}_{2}]}_{α_{L} (x_{2})})} φ (t) dt \\ \leq & \int_{0}^{β (x_{2}, x_{2}) H ({[{Sx}_{2}]}_{α_{L} (x_{2})}, {[{Tx}_{2}]}_{α_{L} (x_{2})}) + ϵ} φ (t) dt . \end{matrix}$ Again using (b), with a₂ + a₄ + a₅ = 0 and d (x₂, [Sx₂] _{α_L(x₂)}) = 0, we get $\int_{0}^{d (x_{2}, {[{Tx}_{2}]}_{α_{L} (x_{2})})} φ (t) dt < a_{3} \int_{0}^{d (x_{2}, {[{Tx}_{2}]}_{α_{L} (x_{2})})} φ (t) dt + ϵ,$ which implies that x₂ ∈ [Tx₂] _{α_L(x₂)} and hence $x_{2} \in {[{Sx}_{2}]}_{α_{L} (x_{2})} \cap {[{Tx}_{2}]}_{α_{L} (x_{2})} .$ Case (iii) Let $λ = (\frac{a_{1} + a_{3} + a_{5}}{1 - a_{2} - a_{3}})$ and $μ = (\frac{a_{2} + a_{4} + a_{5}}{1 - a_{1} - a_{4}}) .$ Next, we show that if a₁ = a₂ or a₃ = a₄, then 0 < λμ < 1.

If a₃ = a₄, then λ, μ < 1 and so 0 < λμ < 1. Now if a₁ = a₂, then $\begin{matrix} 0 & < & λ μ = (\frac{a_{1} + a_{3} + a_{5}}{1 - a_{2} - a_{3}}) (\frac{a_{2} + a_{4} + a_{5}}{1 - a_{1} - a_{4}}) \\ = & (\frac{a_{1} + a_{3} + a_{5}}{1 - a_{1} - a_{3}}) (\frac{a_{1} + a_{4} + a_{5}}{1 - a_{1} - a_{4}}) \\ = & (\frac{a_{1} + a_{3} + a_{5}}{1 - a_{1} - a_{4}}) (\frac{a_{1} + a_{4} + a_{5}}{1 - a_{1} - a_{3}}) < 1 . \end{matrix}$ Using condition (a), for x₁ ∈ X, there exists α_L (x₁) ∈ L ∖ {0_L} such that [Tx₁] _{α_L(x₁)} is a non-empty closed bounded subset of X. Since a₁ + a₃ + a₅ > 0, by Lemma 2.13, there exists x₂ ∈ [Tx₁] _{α_L(x₁)} such that $\begin{matrix} d (x_{1}, x_{2}) \\ \leq & H ({[{Sx}_{0}]}_{α_{L} (x_{0})}, {[{Tx}_{1}]}_{α_{L} (x_{1})}) \\ + a_{1} + a_{3} + a_{5} \\ \leq & β (x_{0}, x_{1}) H ({[{Sx}_{0}]}_{α_{L} (x_{0})}, {[{Tx}_{1}]}_{α_{L} (x_{1})}) \\ + a_{1} + a_{3} + a_{5} \\ \leq & max {β_{(x_{0}, x_{1})}, β_{(x_{1}, x_{0})}} H ({[{Sx}_{0}]}_{α_{L} (x_{0})}, {[{Tx}_{1}]}_{α_{L} (x_{1})}) \\ + (a_{1} + a_{3} + a_{5}) . \end{matrix}$ So using condition (b), and subadditivity of the integral, we have $\begin{matrix} \int_{0}^{d (x_{1}, x_{2})} φ (t) dt \\ < & (a_{1} + a_{3} + a_{5}) \int_{0}^{d (x_{0}, x_{1})} φ (t) dt \\ + (a_{2} + a_{3}) \int_{0}^{d (x_{1}, x_{2})} φ (t) dt \\ + (a_{1} + a_{3} + a_{5}) . \end{matrix}$ This implies that $\int_{0}^{d (x_{1}, x_{2})} φ (t) dt < λ \int_{0}^{d (x_{0}, x_{1})} φ (t) dt + λ .$ Using the same argument, for x₂ ∈ X, there exists α_L (x₂) ∈ L ∖ {0_L} such that [Sx₂] _{α_L(x₂)} is a non-empty closed bounded subset of X. Since a₂ + a₄ + a₅ > 0, by Lemma 2.13, there exists x₃ ∈ [Sx₂] _{α_L(x₂)} such that d (x₂, x₃) ≤ H ([Tx₁] _{α_L(x₁)}, [Sx₂] _{α_L(x₂)}) + λ (a₂ + a₄ + a₅). Using condition (c), for x₀ ∈ X and x₁ ∈ [Sx₀] _{α_L(x₀)} such that β (x₀, x₁) ≥ 1, we have β (x₁, x₂) ≥ 1 for x₂ ∈ [Tx₁] _{α_L(x₁)}. So we have $\begin{matrix} d (x_{2}, x_{3}) \leq H ({[{Tx}_{1}]}_{α_{L} (x_{1})}, {[{Sx}_{2}]}_{α_{L} (x_{2})}) \\ + λ (a_{2} + a_{4} + a_{5}) \\ \leq β_{(x_{1}, x_{2})} H ({[{Sx}_{2}]}_{α_{L} (x_{2})}, {[{Tx}_{1}]}_{α_{L} (x_{1})}) \\ + λ (a_{2} + a_{4} + a_{5}) \\ \leq max {β_{(x_{1}, x_{2})}, β_{(x_{2}, x_{1})}} H ({[{Sx}_{2}]}_{α_{L} (x_{2})}, {[{Tx}_{1}]}_{α_{L} (x_{1})}) \\ + λ (a_{2} + a_{4} + a_{5}) . \end{matrix}$ Using the given condition (b), and subadditivity of the integral we get $\begin{matrix} \int_{0}^{d (x_{2}, x_{3})} φ (t) dt \\ < & (a_{2} + a_{4} + a_{5}) \int_{0}^{d (x_{1}, x_{2})} φ (t) dt \\ + (a_{1} + a_{4}) \int_{0}^{d (x_{2}, x_{3})} φ (t) dt \\ + & λ (a_{2} + a_{4} + a_{5}), \end{matrix}$ which implies that $\int_{0}^{d (x_{2}, x_{3})} φ (t) dt < μ \int_{0}^{d (x_{1}, x_{2})} φ (t) dt + λ μ .$ Repeating the above process, for x₃ ∈ X, there exists α_L (x₃) ∈ L ∖ {0_L} such that [Tx₃] _{α_L(x₃)} is a non-empty closed bounded subset of X. From Lemma 2.13, and using conditions (c), and (b), we have $\begin{matrix} \int_{0}^{d (x_{3}, x_{4})} φ (t) dt & < & λ \int_{0}^{d (x_{2}, x_{3})} φ (t) dt \\ + λ (λ μ) . \end{matrix}$ By induction, we produce a sequence {x_n} in X such that $\begin{matrix} x_{2 k + 1} & \in & {[{Sx}_{2 k}]}_{α_{L} (x_{2 k})}, \\ x_{2 k + 2} & \in & {[{Tx}_{2 k + 1}]}_{α_{L} (x_{2 k + 1})}, k = 0, 1, 2, . . ., \end{matrix}$ and $β (x_{n - 1}, x_{n}) \geq 1, \forall n \in ℕ .$ Now, we have $\begin{matrix} d (x_{2 k + 1}, x_{2 k + 2}) \\ \leq H ({[{Sx}_{2 k}]}_{α_{L} (x_{2 k})}, {[{Tx}_{2 k + 1}]}_{α_{L} (x_{2 k + 1})}) \\ + {(λ μ)}^{k} (a_{1} + a_{3} + a_{5}) \\ \leq β (x_{2 k}, x_{2 k + 1}) \\ H ({[{Sx}_{2 k}]}_{α_{L} (x_{2 k})}, {[{Tx}_{2 k + 1}]}_{α_{L} (x_{2 k + 1})}) \\ + {(λ μ)}^{k} (a_{1} + a_{3} + a_{5}) \\ \leq max {β_{(x_{2 k}, x_{2 k + 1})}, β_{(x_{2 k + 1}, x_{2 k})}} \\ \cdot H ({[{Sx}_{2 k}]}_{α_{L} (x_{2 k})}, {[{Tx}_{2 k + 1}]}_{α_{L} (x_{2 k + 1})}) \\ + {(λ μ)}^{k} (a_{1} + a_{3} + a_{5}) . \end{matrix}$ By the given condition and the subadditivity of the integral we get $\begin{matrix} \int_{0}^{d (x_{2 k + 1}, x_{2 k + 2})} φ (t) dt \\ < λ \int_{0}^{d (x_{2 k}, x_{2 k + 1})} φ (t) dt + λ {(λ μ)}^{k} . \to (3.1) \end{matrix}$ Similarly, $\begin{matrix} d (x_{2 k + 2}, x_{2 k + 3}) \\ \leq H ({[{Sx}_{2 k + 2}]}_{α_{L} (x_{2 k + 2})}, {[{Tx}_{2 k + 1}]}_{α_{L} (x_{2 k + 1})}) \\ + {(λ μ)}^{k} λ (a_{2} + a_{4} + a_{5}) \\ \leq β (x_{2 k + 1}, x_{2 k + 2}) \\ \cdot H ({[{Sx}_{2 k + 2}]}_{α_{L} (x_{2 k + 2})}, {[{Tx}_{2 k + 1}]}_{α_{L} (x_{2 k + 1})}) \\ + {(λ μ)}^{k} λ (a_{2} + a_{4} + a_{5}) \\ \leq max {β_{(x_{2 k + 1}, x_{2 k + 2})}, β_{(x_{2 k + 2}, x_{2 k + 1})}} \\ \cdot H ({[{Sx}_{2 k + 2}]}_{α_{L} (x_{2 k + 2})}, {[{Tx}_{2 k + 1}]}_{α_{L} (x_{2 k + 1})}) \\ + {(λ μ)}^{k} λ (a_{2} + a_{4} + a_{5}) . \end{matrix}$ Again using the given condition and the subadditivity of the integral, we get $\int_{0}^{d (x_{2 k + 2}, x_{2 k + 3})} φ (t) dt 3.2$ (1) $\begin{matrix} < & μ \int_{0}^{d (x_{2 k + 1}, x_{2 k + 2})} φ (t) dt \\ + & {(λ μ)}^{k + 1} . \end{matrix}$ From (3.1) and (3.2), it follows that for each k = 0, 1, 2, …, $\begin{matrix} \int_{0}^{d (x_{2 k + 1}, x_{2 k + 2})} φ (t) dt \\ < & λ \int_{0}^{d (x_{2 k}, x_{2 k + 1})} φ (t) dt + λ {(λ μ)}^{k} \\ < & λ [μ \int_{0}^{d (x_{2 k - 1}, x_{2 k})} φ (t) dt + {(λ μ)}^{k}] + λ {(λ μ)}^{k} \\ = & (λ μ) \int_{0}^{d (x_{2 k - 1}, x_{2 k})} φ (t) dt + 2 λ {(λ μ)}^{k} \\ < & (λ μ) [λ \int_{0}^{d (x_{2 k - 2}, x_{2 k - 1})} φ (t) dt + λ {(λ μ)}^{k - 1}] + 2 λ {(λ μ)}^{k} \\ = & (λ μ) λ \int_{0}^{d (x_{2 k - 2}, x_{2 k - 1})} φ (t) dt + 3 λ {(λ μ)}^{k} \\ ⋮ \\ \leq & λ {(λ μ)}^{k} \int_{0}^{d (x_{0}, x_{1})} φ (t) dt + (2 k + 1) λ (λ μ)^{k}, \end{matrix}$ and $\begin{matrix} \int_{0}^{d (x_{2 k + 2}, x_{2 k + 3})} φ (t) dt \\ \leq & μ \int_{0}^{d (x_{2 k + 1}, x_{2 k + 2}) + {(λ μ)}^{k + 1}} φ (t) dt \\ ⋮ \\ < & {(λ μ)}^{k + 1} \int_{0}^{d (x_{0}, x_{1})} φ (t) dt + (2 k + 2) {(λ μ)}^{k + 1} . \end{matrix}$ Then for m < n, we have $\begin{matrix} \int_{0}^{d (x_{2 m + 1}, x_{2 n + 1})} φ (t) dt \\ < & [λ \sum_{i = m}^{n - 1} {(λ μ)}^{i} + \sum_{i = m + 1}^{n} {(λ μ)}^{i}] \int_{0}^{d (x_{0}, x_{1})} φ (t) dt \\ + & λ \sum_{i = m}^{n - 1} (2 i + 1) {(λ μ)}^{i} + \sum_{i = m + 1}^{n} 2 i {(λ μ)}^{i} . \end{matrix}$ Similarly, we obtain that $\begin{matrix} \int_{0}^{d (x_{2 m}, x_{2 n + 1})} φ (t) dt \\ < & [\sum_{i = m}^{n} {(λ μ)}^{i} + λ \sum_{i = m}^{n - 1} {(λ μ)}^{i}] \int_{0}^{d (x_{0}, x_{1})} φ (t) dt \\ + & \sum_{i = m}^{n} 2 i {(λ μ)}^{i} + λ \sum_{i = m}^{n - 1} (2 i + 1) {(λ μ)}^{i}, \end{matrix}$ and $\begin{matrix} \int_{0}^{d (x_{2 m}, x_{2 n})} φ (t) dt \\ < & [\sum_{i = m}^{n - 1} {(λ μ)}^{i} + λ \sum_{i = m}^{n - 1} {(λ μ)}^{i}] \int_{0}^{d (x_{0}, x_{1})} φ (t) dt \\ + & \sum_{i = m}^{n - 1} 2 i {(λ μ)}^{i} + λ \sum_{i = m}^{n - 1} (2 i + 1) {(λ μ)}^{i}, \end{matrix}$ $\begin{matrix} \int_{0}^{d (x_{2 m + 1}, x_{2 n})} φ (t) dt \\ < & λ [\sum_{i = m}^{n - 1} {(λ μ)}^{i} + \sum_{i = m + 1}^{n - 1} {(λ μ)}^{i}] \int_{0}^{d (x_{0}, x_{1})} φ (t) dt \\ + & λ \sum_{i = m}^{n - 1} (2 i + 1) {(λ μ)}^{i} + \sum_{i = m + 1}^{n - 1} 2 i {(λ μ)}^{i} . \end{matrix}$ Since 0 < λμ < 1, so by Cauchy’s root test, we get ∑ (2i + 1) (λμ) ⁱ and ∑2i (λμ) ⁱ are convergent series. Therefore, {x_n} is a Cauchy sequence in X. Now, from the completeness of X, there exists z ∈ X such that x_n ⟶ z as n ⟶ ∞. By condition (d), we have β (x_n-1, z) ≥ 1 for all $n \in ℕ .$ Now,we have $\begin{matrix} d (x_{2 n}, {[Sz]}_{α (z)}) \\ \leq & H ({[{Tx}_{2 n - 1}]}_{α_{L} (x_{2 n - 1})}, {[Sz]}_{α (z)}) \\ = & H ({[Sz]}_{α (z)}, {[{Tx}_{2 n - 1}]}_{α_{L} (x_{2 n - 1})}) \\ \leq & β (x_{2 n - 1}, z) H ({[Sz]}_{α (z)}, {[{Tx}_{2 n - 1}]}_{α_{L} (x_{2 n - 1})}) \\ \leq & max {β_{(x_{2 n - 1}, z)}, β_{(z, x_{2 n - 1})}} \\ \cdot H ({[Sz]}_{α (z)}, {[{Tx}_{2 n - 1}]}_{α_{L} (x_{2 n - 1})}) \\ + ϵ, \end{matrix}$ which implies $\begin{matrix} \int_{0}^{d (x_{2 n}, {[Sz]}_{α (z)})} φ (t) dt \\ < & a_{1} \int_{0}^{d (z, {[Sz]}_{α_{L} (z)})} φ (t) dt \\ + a_{2} \int_{0}^{d (x_{2 n - 1}, {[{Tx}_{2 n - 1}]}_{α_{L} (x_{2 n - 1})})} φ (t) dt \\ + a_{3} \int_{0}^{d (z, {[{Tx}_{2 n - 1}]}_{α_{L} (x_{2 n - 1})})} φ (t) dt \\ + a_{4} \int_{0}^{d (x_{2 n - 1}, {[Sz]}_{α_{L} (z)})} φ (t) dt \\ + & a_{5} \int_{0}^{d (z, x_{2 n - 1})} φ (t) dt + ϵ \end{matrix}$ We have, $\begin{matrix} \int_{0}^{d (z, {[Sz]}_{α_{L} (z)})} φ (t) dt \\ < & (\frac{1 + a_{3}}{1 - a_{1} - a_{4}}) \int_{0}^{d (z, x_{2 n})} φ (t) dt \\ + (\frac{a_{4} + a_{5}}{1 - a_{1} - a_{4}}) \int_{0}^{d (z, x_{2 n - 1})} φ (t) dt \\ + & (\frac{a_{2}}{1 - a_{1} - a_{4}}) \int_{0}^{d (x_{2 n - 1}, x_{2 n})} φ (t) dt \\ + \frac{ϵ}{1 - a_{1} - a_{4}} . \end{matrix}$ Letting n ⟶ ∞, we have d (z, [Sz] _{α_L(z)}) = 0. It implies that z ∈ [Sz] _{α_L(z)}. Similarly, by using $d (z, {[Tz]}_{α_{L} (z)}) \leq d (z, x_{2 n + 1}) + d (x_{2 n + 1}, {[Tz]}_{α_{L} (z)}),$ it can be shown that z ∈ [Tz] _{α_L(z)}. Therefore, z ∈ [Sz] _{α_L(z)} ∩ [Tz] _{α_L(z)}. This proves the theorem.■

In the next example, we show that the above result is in fact a real generalization of many results in the literature.

3.2 Example:

Consider X ={ 1, 2, 3 } with metric d defined in Example 2.10 and same lattice L and L-fuzzy mappings S, T and β. Then clearly (S, T) is an integral β_F-admissible pair. With usual calculations, we have $H ({[Sx]}_{α}, {[Ty]}_{α}) = {\begin{matrix} 0, if y = 1 and for all x \in X \\ 1, if y = 2 and for all x \in X \\ 2, if y = 3 and for all x \in X, \end{matrix}$ where α = d. It can be seen that with $a_{1} = a_{2} = \frac{1}{5},$ a₃ = a₅ = 0, $a_{4} = \frac{1}{4},$ and $φ (t) = {\begin{matrix} t^{\frac{1}{t - 2}} [1 - ln (t)], if t > 0 \\ 0, if t = 0 . \end{matrix}$ the integral contractive condition in (b), of Definition 2.9, is satisfied. Also there exists x₀ = 2 ∈ X and x₁ = 1∈ [Sx₀] _α = { 1 }, such that β (x₀, x₁)≥1.

Now if {x_n} is a sequence in X and x ∈ X is such that β (x_n, x) ≥ 1 for all $n \in ℕ .$ Clearly x_n ∈ { 1, 2 }, for all $n \in ℕ$ therefore x∈ { 1, 2 } hence β (x_n, x) ≥ 1 for all $n \in ℕ .$ Then all the conditions of Theorem 3.1 are satisfied to obtain 1 ∈ X such that 1 ∈ [S1] _α ∩ [T1] _α = { 1 }.

3.3 Remark

With the same values used in above example, consider $β (1, 2) H ({[S 1]}_{α}, {[T 2]}_{α}) = (1) H ({1}, {3}) = 1,$ but $\begin{matrix} a_{1} d (1, {[S 1]}_{α}) + a_{2} d (2, {[T 2]}_{α}) + a_{4} d (2, {[S 1]}_{α}) \\ < & β (1, 2) H ({[S 1]}_{α}, {[T 2]}_{α}) . \end{matrix}$ This shows that the contractive condition in [30] is not satisfied, but S and T has coincidence point. Moreover for any λ ∈ (0, 1], if X is not a linear space, consider $\begin{matrix} H ({[S 1]}_{α}, {[T 3]}_{α}) & = & H ({1}, {2}) = 2 \\ > & \frac{λ}{2} d (1, 3) = \frac{λ}{2} . \end{matrix}$ This shows that the contractive conditions of the main results of [6–8 , 14] are also not satisfied.

Since every L-fuzzy mapping is a fuzzy mapping. Using Remark 2.4, we deduce the following corollary.

3.4 Corollary

Let (X, d) be a complete metric space, β: X × X ⟶ [0, ∞), φ ∈ Φ and S, T be a pair of fuzzy mappings from X into F (X) such that (S, T) is an integral β_F-admissible contraction. Then there exists z ∈ X such that z ∈ [Sz] _α ∩ [Tz] _α, for α ∈ (0, 1].

3.5 Theorem

Let (X, d) be a complete metric space, β: X × X ⟶ [0, ∞), φ ∈ Φ and S, T: X → CB (Y) be a pair of set-valued mappings such that:

(a) For some x₀ ∈ X, there exists x₁ ∈ S (x₀) such that β (x₀, x₁) ≥ 1,

(b) For all x, y ∈ X, we have $\begin{matrix} \int_{0}^{M} φ (t) dt & < & a_{1} \int_{0}^{d (x, S (x))} φ (t) dt \\ + & a_{2} \int_{0}^{d (y, T (y))} φ (t) dt \\ + a_{3} \int_{0}^{d (x, T (y))} φ (t) dt \\ + & a_{4} \int_{0}^{d (y, S (x))} φ (t) dt \\ + a_{5} \int_{0}^{d (x, y)} φ (t) dt + ϵ, \end{matrix}$ where M is same as given in Definition 2.9, and also (c), (d) are satisfied. Then there exists u ∈ X such that S (u) ∩ T (u) ¬ = φ.

Proof. Consider two fuzzy mappings U, V: X → F (Y) which are defined by $U (x) = χ_{Sx}, V (x) = χ_{Tx} .$ Then for α ∈ (0, 1] $\begin{matrix} {[Ux]}_{α} & = & Sx, \end{matrix}$ and ${[Vx]}_{α_{V} (x)} = Fx .$ Clearly $H ({[Ux]}_{α_{U} (x)}, {[Vy]}_{α_{V} (y)}) = H (Tx, Ty),$ the above corollary can be applied to obtain a point w ∈ X such that T (w) ∩ F (w) ≠ φ, and hence the proof is completed.■

In the next theorem, conditions for existence of coincidence points of a pair of L-fuzzy mappings has been discussed with an integral type of contractive condition. This result will improve and generalize many results regarding coincidence points of fuzzy mappings, and the coincidence points of set-valued mappings present in theliterature.

3.6 Theorem

Let X be a non-empty set, (Y, d) be a metric space and T, F a pair of L-fuzzy mappings from X into F_L (X). Suppose that for each x ∈ X there exists α_T (x), α_F (x) ∈ L ∖ {0_L} such that [Tx] _{α_T(x)}, [Fx] _{α_F(x)} ∈ CB (Y), ⋃_x∈X [Tx] _{α_T(x)} ⊆ ⋃ _x∈X[Fx] _{α_F(x)} and one of ⋃_x∈X [Tx] _{α_T(x)} or ⋃ _x∈X [Fx] _{α_F(x)} is complete. If there exist λ ∈ [0, 1) and ɛ > 0 such that for all x, y ∈ X, $\int_{0}^{H ({[Tx]}_{α_{T} (x)}, {[Ty]}_{α_{T} (y)}) + ϵ} φ (t) dt 3.6$ (2) $\leq λ \int_{0}^{d ({[Fx]}_{α_{F} (x)}, {[Fy]}_{α_{F} (y)}) + \frac{ϵ}{λ}} φ (t) dt,$ then there exists a pointu ∈ X such that [Tu] _{α_T(u)} ∩ [Fu] _{α_F(u)} ≠ φ.

Proof. Let x_o be an arbitrary element of X, we shall construct sequences {x_n} ⊂ X and {y_n} ⊂ ⋃ _x∈X [Fx] _{α_F(x)}. Let y₁ ∈ [Tx₀] _{α_T(x₀)}. As ⋃_x∈X [Tx] _{α_T(x)} ⊆ ⋃ _x∈X [Fx] _{α_F(x)}, we may choose x₁ ∈ X such that y₁ ∈ [Fx₁] _{α_F(x₁)} and it further implies that ${[{Tx}_{0}]}_{α_{T} (x_{0})} \cap {[{Fx}_{1}]}_{α_{F} (x_{1})} \neq φ .$ If λ = 0, then by assumptions [Tx₀] _{α_T(x₀)} = [Tx₁] _{α_T(x₁)}. It implies that $y_{1} \in {[{Tx}_{1}]}_{α_{T} (x_{1})} \cap {[{Fx}_{1}]}_{α_{F} (x_{1})},$ which means that x₁ is the required point. Thus in the sequel, we may assume that λ ≠ 0. If $d ({[{Fx}_{0}]}_{α_{F} (x_{0})}, {[{Fx}_{1}]}_{α_{F} (x_{1})}) = 0,$ then by the same argument the conclusion is obtained. Now, if $d ({[{Fx}_{0}]}_{α_{F} (x_{0})}, {[{Fx}_{1}]}_{α_{F} (x_{1})}) \neq 0 .$ By Lemma 2.13, we can choose y₂ ∈ [Tx₁] _{α_T(x₁)} such that $d (y_{1}, y_{2}) \leq H ({[{Tx}_{0}]}_{α_{T} (x_{0})}, {[{Tx}_{1}]}_{α_{T} (x_{1})}) + λ^{2}$ then, using Inequality (3.6), we have $\begin{matrix} \int_{0}^{d (y_{1}, y_{2})} φ (t) dt \\ \leq & \int_{0}^{H ({[{Tx}_{0}]}_{α_{T} (x_{0})}, {[{Tx}_{1}]}_{α_{T} (x_{1})}) + λ^{2}} φ (t) dt \\ \leq & λ \int_{0}^{d ({[{Fx}_{0}]}_{α_{F} (x_{0})}, {[{Fx}_{1}]}_{α_{F} (x_{1})}) + λ} φ (t) dt \\ \leq & λ \int_{0}^{d (y_{0}, y_{1}) + λ} φ (t) dt . \end{matrix}$ As y₂ ∈ [Tx₁] _{α_T(x₁)}, we may use the fact that ⋃_x∈X [Tx] _{α_T(x)} ⊆ ⋃ _x∈X [Fx] _{α_F(x)} to obtain x₂ ∈ X such that y₂ ∈ [Fx₂] _{α_F(x₂)}. If $d ({[{Fx}_{1}]}_{α_{F} (x_{1})}, {[{Fx}_{2}]}_{α_{F} (x_{2})}) = 0,$ then by a similar argument the conclusion is obtained. Continuing this process, having chosen x_n ∈ X, y_n ∈ [Tx_n-1] _{α_T(x_n-1)} ∩ [Fx_n] _{α_F(x_n)} we obtain $x_{n + 1} \in X, y_{n + 1} \in {[{Tx}_{n}]}_{α_{T} (x_{n})} \cap {[{Fx}_{n + 1}]}_{α_{F} (x_{n + 1})}$ such that $\begin{matrix} \int_{0}^{d (y_{n}, y_{n + 1})} φ (t) dt \\ \leq & λ^{n} \int_{0}^{d (y_{0}, y_{1}) + \frac{λ}{(1 - λ)}} φ (t) dt . \end{matrix}$ Now for m < n, as integral is subadditive, consider $\begin{matrix} \int_{0}^{d (y_{m}, y_{n})} φ (t) dt \\ \leq & \int_{0}^{d (y_{m}, y_{m + 1})} φ (t) dt \\ + \int_{0}^{d (y_{m + 1}, y_{m + 2})} φ (t) dt + \\ . . . + \int_{0}^{d (y_{n - 1}, y_{n})} φ (t) dt \\ \leq & [\frac{λ^{m}}{1 - λ}] \int_{0}^{d (y_{0}, y_{1}) + \frac{λ}{(1 - λ)}} φ (t) dt as λ < 1 . \end{matrix}$ $\int_{0}^{d (y_{m}, y_{n})} φ (t) dt \to 0, as m, n \to \infty$ . Hence {y_n} is a Cauchy sequence in ⋃ _x∈X [Fx] _{α_F(x)}. By completeness, there exists an element z ∈ ⋃ _x∈X [Fx] _{α_F(x)} such that y_n ⟶ z (this holds also if ⋃_x∈X [Tx] _{α_T(x)} is complete with z ∈ ⋃ _x∈X [Tx] _{α_T(x)} ⊆ ⋃ _x∈X [Fx] _{α_F(x)}). It further implies that z ∈ [Fw] _{α_T(w)} for some w ∈ X. Suppose on contrary that d (z, [Tw] _{α_T(w)}) = ρ > 0. Now, $\begin{matrix} d (z, {[Tw]}_{α_{T} (w)}) \\ \leq & d (z, y_{n}) + d (y_{n}, {[Tw]}_{α_{T} (w)}) . \end{matrix}$ Consider $\begin{matrix} \int_{0}^{d (y_{n}, {[Tw]}_{α_{T} (w)})} φ (t) dt \\ \leq & \int_{0}^{H ({[{Tx}_{n - 1}]}_{α_{T} (x_{n - 1})}, {[Tw]}_{α_{T} (w)}) + λ ρ} φ (t) dt \\ < & λ \int_{0}^{d (y_{n - 1}, z) + ρ} φ (t) dt . \end{matrix}$ Letting n⟶ ∞, we have $\int_{0}^{ρ} φ (t) dt < λ \int_{0}^{ρ} φ (t) dt,$ which contradicts our assumption. Hence d (z, [Tw] _α) =0. It follows that z ∈ [Tw] _α. Hence [Tw] _α ∩ [Fw] _α ≠ φ.■

3.7 Remark

The above result holds for a pair of fuzzy mappings satisfying all conditions of Theorem 3.6.

The next theorem is the consequence of above corollary for finding the coincidence point of two multivalued mappings.

3.8 Theorem

Let X be a non-empty set, (Y, d) be a metric space and T, F: X → CB (Y) be two set-valued mappings. Suppose that ⋃_x∈XT (x) ⊆ ⋃ _x∈XF (x) and one of ⋃_x∈XT (x) or ⋃ _x∈XF (x) is complete. If there exist λ ∈ [0, 1) and ɛ > 0 such that for all x, y ∈ X, $\begin{matrix} \int_{0}^{H (T (x), T (y)) + ϵ} φ (t) dt \\ \leq & λ \int_{0}^{d (F (x), F (y)) + \frac{ϵ}{λ}} φ (t) dt, \end{matrix}$ then there exists a point u ∈ X such that T (u) ∩ F (u) ≠ φ.

Proof. Similar to the proof of Theorem 3.5.■

4 Applications

Mathematical modeling of physical phenomenons has never been as accurate as demanded. The incomplete and uncertain systems can be modeled in a better way by means of fuzzy sets approach. Optimization problems are very important in every field of life. To improve the accuracy of the functional equations many tools have been introduced. In the literature and according to our best knowledge, the fuzzy functional inclusions have not been discussed yet. Using the important and useful properties of fuzzy mappings and their level sets [19], hybrid types of functional inclusions have been introduced. The existence of the solution by using Theorem 3.6, and its consequences has been obtained.

The following proposition and lemma will be useful to find the solution of the functional inclusions (4.3) and (4.4) given below.

4.1 Proposition [32]

Let D be a paracompact Hausdorff topological space, Y be a topological vector space and F: D → 2^Y be a multivalued non-empty convex valued function. If F has open lower sections, that is; for any y ∈ Y, F^-1 (y) is open in D, then there exists a continuous selection f: D → Y such that f (x) ∈ F (x) for any x ∈ D.

4.2 Lemma [5]

Let Ω ⊂ R × Rⁿ × R^m be an open set, (t, x, y) ∈ Ω and T: Ω → K (Rⁿ × R^m) an upper semicontinuous set-valued operator. Then there exists an open interval J of R, for a > 0, M > 0, such that;

(i) $J \times S_{aM}^{(x, y)} \subset Ω .$ (Here $S_{aM}^{(x, y)}$ denotes a closed ball with radius aM and center at (x, y) ⊂ Ω)

(ii) ||T (t, x, y) || ≤ M on $J \times S_{aM}^{(x, y)} .$

4.3 Problem

Consider the following type of system of fuzzy functional inclusions: $\begin{array}{l} (\sup_{y \in D} H (x, y, F (τ (x, y)))) (F (x) \geq α (x, y), \\ (\sup_{y \in D} K (x, y, G (τ (x, y)))) (G (x)) \geq β (x, y), \end{array}$ for all x ∈ W, where τ: W × D → W, H, and $K : W \times D \times R \to F (ℝ) .$ Define two set-valued functions $\tilde{F} : W \to 2^{ℝ}$ and $\tilde{G} : W \to 2^{ℝ}$ by $\begin{matrix} \tilde{F} (x) & = & {[sup_{y \in D} H (x, y, F (τ (x, y)))]}_{α (x, y)}, \\ \tilde{G} (x) & = & {[sup_{y \in D} K (x, y, G (τ (x, y)))]}_{β (x, y)}, \\ for each x & \in & W . \end{matrix}$ To show that, both $\tilde{F}$ and $\tilde{G}$ are upper semicontinuous. Here it has been considered the case for $\tilde{F}$ only and then the similar arguments will be assumed for $\tilde{G} .$ Now for a given $\hat{x} \in W,$ consider the neighborhood of $\tilde{F} (\hat{x})$ as follows; $S_{r}^{\tilde{F} (\hat{x})} = {w \in ℝ^{n} : d (w, \tilde{F} (\hat{x})) < r} .$ For x ∈ W and $δ \in \tilde{F} (x),$ we have $\begin{matrix} d (δ, \tilde{F} (\hat{x})) \\ \leq & H (\tilde{F} (x), \tilde{F} (\hat{x})) \\ \leq & H (\tilde{F} (\hat{x}), \tilde{F} (x)) + \\ H ({[F (x)]}_{α (\hat{u}, \hat{v})}, {[F (x)]}_{α (u, v)}) . \end{matrix}$ Since [F (w, x, u, v)] _α(u,v) is uniformly continuous for each α (u, v) and α is also uniformly continuous, and using above inequality, we can find a small enough neighborhood $\hat{W}$ of $\hat{x}$ in W [40], such that for all x ∈ W and $z \in \tilde{F} (x)$ $d (z, \tilde{F} (\hat{x})) < r,$ thus $\tilde{F} (\hat{W}) \subset S_{r}^{\tilde{F} (\hat{x})},$ which shows $\tilde{F} (x)$ is upper semicontinuous. So by using the Lemma 4.2, there exist real constants ρ₁, ρ₂ > 0 such that $max_{x \in W} | | \tilde{F} (x) | | \leq ρ_{1} and max_{x \in W} | | \tilde{G} (x) | | \leq ρ_{2} .$ Further assume that $ρ_{1} < ρ_{2} . H$ (3) Let $\bar{U} = {u \in C (W, ℝ) : | | u (w) | | \leq ρ_{1}, all w \in W}$ with a metric $d : \bar{U} \times \bar{U} \to ℝ$ defined by; $d (u_{1}, u_{2}) = sup_{w \in W} {| | u_{1} (w) - u_{2} (w) | |},$ then $(\bar{U}, d)$ is a complete metric space. Define $S, T : \bar{U} \to 2^{\bar{U}}$ by $T (h) = {\bar{h} (x) \in [sup_{y \in D} H (x, y, F (τ (x, y)))]_{α (x, y)} a . e W},$ and $S (k) = {\bar{k} (x) \in [sup_{y \in D} K (x, y, G (τ (x, y)))]_{β (x, y)} a . e W},$ Clearly T (h) ¬ = φ and S (h) ¬ = φ for all $h \in \bar{U} .$ Since the operator $\tilde{F} (x)$ is upper semicontinuous with compact values. So by the well known selection Theorem of Kuratowski-Ryll-Nardzewski [18], $\tilde{F} (x)$ has continuous selection $h_{0} (x) \in \tilde{F} (x)$ for all x ∈ W. Let h₀ (x): = $sup_{y \in D} h_{0} (x, y, F (τ (x, y))) \in T (h)$ , thus T (h) ¬ = φ.

Next we show that T (h) is closed for all $h \in \bar{U} .$ Let h_n be a sequence in T (h) converges to $\hat{h} \in \bar{U} .$ Since $\tilde{F} (x)$ is closed, so we have $\hat{h} \in T (h) .$ Similar arguments for S (v) are valid. Assume that S and T satisfies $\int_{0}^{H (Tx, Ty) + ϵ} φ (t) \leq λ \int_{0}^{d (Sx, Sy) + \frac{ϵ}{λ}} φ (t)$ for some ϵ > 0 and λ ∈ (0, 1). Clearly from (H), ∪T (h) ⊆ ∪ S (h), and assumed to be closed, let k₁ ∈ T (h₀), which implies there exists $h_{1} \in \bar{U}$ , we can choose k₁ ∈ S (h₁) such that for λ ¬ =0, using Lemma 2.13, we may choose k₂ ∈ T (h₁) such that $d (k_{1}, k_{2}) \leq H (T (h_{0}), T (h_{1})) + λ^{2}$ Consequently, using the procedure in Theorem 3.6, we can find a coincidence point of T and S, which will be the solution of the Problem (4.3).

4.4 Problem

Now consider the following system of fuzzy functional equations arising in fuzzy dynamic programming: $\begin{matrix} (sup_{y \in D} H (x, y, F (τ (x, y)))) (F (x)) & > & α (x, y), \\ (sup_{y \in D} K (x, y, G (τ (x, y)))) (G (x)) & > & β (x, y), \end{matrix}$ for all x ∈ W, where τ: W × D → W, $H, K : W \times D \times R \to F (ℝ) .$

The next lemma describes the conditions on fuzzy mappings for existence of a continuous selection.

4.5 Lemma

Let Ω be a an open subset in $W \times D \times ℝ$ with (x, y, s) ∈ Ω. Suppose that $H : Ω \to F (ℝ)$ is bounded convex and lower open fuzzy mapping and α: W × D → [0, 1) is an upper semicontinuous function such that ${(sup_{y \in D} H_{(x, y, F (τ (x, y)))})}_{α (x, y)}$ is non-empty for each x ∈ W. Then for a given continuous surjection $f : W \to ℝ$ we have a continuous $h \in {(sup_{y \in D} H_{(x, y, F (τ (x, y)))})}_{α (x, y)}$ such that; $f (x) = sup_{y \in D} h (x, y, F (τ (x, y))) for x \in W .$

Proof. Using the concepts of Zadeh’s extension principle we define a function $\tilde{H} : Ω \to 2^{R^{n}}$ as; $\tilde{H} (x, y, s) = {(sup_{y \in D} H_{(x, y, F (τ (x, y)))})}_{α (x, y)}, for each x \in W .$ Clearly $\tilde{H} (x, y, s)$ is non-empty and convex. It is enough to prove that the complement of ${\tilde{H}}^{- 1} (z),$ the set, ${({\tilde{H}}^{- 1} (z))}^{c}$ given by; ${(x, y, s) \in Ω : (sup_{y \in D} H_{(x, y, F (τ (x, y)))}) (z) \leq α (x, y)},$ is closed. Let (x_n, y_n, s_n) be a sequence in ${({\tilde{H}}^{- 1} (z))}^{c}$ such that (x_n, y_n, s_n) → (x, y, s). Taking limit as n→ ∞ we have $(sup_{y \in D} H_{(x, y, F (τ (x, y))}) (x, y, s) \leq α (x, y),$ since H is lower open and α is upper semicontinuous. Thus $(x, y, s) \in {({\tilde{H}}^{- 1} (z))}^{c}$ and hence $\tilde{H}$ has open lower sections. Thus by Proposition 4.1, there exists a continuous selection $\tilde{H} (x) \in \tilde{H} (x, y, s)$ for each x ∈ W. As F is a surjection and $\tilde{H} (x, y, s)$ is bounded we get the problem $f (x) = sup_{y \in D} \tilde{H} (x, y, F (τ (x, y))) for x \in W .$

4.6 Theorem

Let Ω be a an open subset in $W \times D \times ℝ$ with (x, y, s) ∈ Ω. Suppose that $H, K : Ω \to F (ℝ)$ are bounded convex and lower open fuzzy mapping and α, β: W × D → [0, 1) are upper semicontinuous function such that open level sets are non-empty.

Moreover let $A, T : C (W, ℝ) \to C (W, ℝ)$ be two bounded operators. Assume that: $\underset{}{underset 0 \int^{| h (x, y, F (τ (x, y))) - k (x, y, G (τ (x, y))) |} φ (s) d s} 4.6$ (4) $\leq α \underset{0}{\int^{max {| Ah - Ak |, | Th - Tk |, Ah - Tk}}} φ (s) ds,$ for all $h, k \in C (W, ℝ), 0 \leq α < 1$ and $φ : ℝ^{+} \to ℝ^{+}$ is a non-negative summable Lebesgue-integrable function such that $\overset{ɛ}{\int_{0}} φ (s) ds > 0$ for each ɛ > 0. Then the system of functional inclusions (4.4) has a bounded solution.

Proof. Define the functions $\tilde{K}, \tilde{H} : Ω \to 2^{R^{n}}$ by; $\begin{matrix} \tilde{H} (x, y, s) & = & {(sup_{y \in D} H_{(x, y, F (τ (x, y)))})}_{α (x, y)}, for each x \in W, \\ \tilde{K} (x, y, s) & = & {(sup_{y \in D} K_{(x, y, F (τ (x, y)))})}_{β (x, y)}, for each x \in W \end{matrix}$ Using Lemma 4.5, we can find continuous bounded selections $\tilde{H} \in \tilde{H} (x, y, s)$ and $\tilde{k} \in \tilde{K} (x, y, s)$ such that, $Ah (x) = sup_{y \in D} {\tilde{H} (x, y, F (τ (x, y)))}, 4.61$ (5) $Tk (x) = sup_{y \in D} {\tilde{k} (x, y, G (τ (x, y)))},_$ for all h, k ∈ C (W) and x ∈ W. The unique bounded solution of system of functional equations (4.61) with condition like (4.6) can be found in [11, 34].■

4.7 Conclusion

Coincidence and common fixed points of L-fuzzy mappings are obtained. In Theorem 3.1, the contractive conditions, mappings and the structure of the space has been generalized. Our theorem can be applied to obtain the coincidence points in ordered metric spaces. Using Theorems 3.1 and 3.6 with its consequences for single valued mappings like (4.6), two types of fuzzy differential inclusions (4.3) and (4.4) have been solved. Our applications would constitute a base for fuzzy optimization problems arises in many fields. It is remained to solve these types of the functional inclusions for L-fuzzy mappings. Because all the properties of fuzzy mappings and L-fuzzy mappings are not same [15, 19]. We initiate this study to open these types of problems for researchers and scientists. These problems can also be extended to more generalize structures like neutrosophic soft mappings [28], and their applications in decision makings problems as discussed in [21 , 35–39].

Footnotes

Acknowledgement

The first author gratefully acknowledges with thanks the Department of Research Affairs at UAEU. This article is supported by the grant: UPAR (11) 2016, Fund No. 31S249 (COS). The paper has been prepared during the second named author visit as a post doc researcher to the Department of Mathematical Science, UAEU, and would like to thanks the Department of Mathematical Science, UAEU for their support.

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