Bounded lattice fuzzy coincidence theorems with applications

1 Introduction

In our daily life, problems in economics, engineering, social sciences, and medical sciences do not always involve crisp data. Traditional mathematical methods cannot be frequently used because a large number of uncertainties exists in these practical problems. To overwhelmed these reservations, concepts like fuzzy set, intuitionistic fuzzy set, soft set, rough set, and bipolar fuzzy set have been prepared (see [18, 35]). Scholars should continue to explore the abstract algebraic concepts and results in the wider context of the fuzzy setting. One structure that’s mathematical applications are most generally argued is lattice theory. In an extensive variety of mathematical problems, the existence of a solution is corresponding to the existence of a fixed point for an appropriate map. The theory of fixed points has been evolved as an active and very useful area of research due to its applications in mathematics and related disciplines. The fixed point theorem based on the contraction principle is applied usually in proving the existence and uniqueness of the solutions of some scalar and mainly functional equations (for example differential and integral equations). In brief, fixed point theory is a powerful tool to determine uniqueness of solutions to dynamical systems and is widely used in theoretical and applied analysis. One such application is to feasibility and inverse optimization problems arising in the context of finding good radiation dose plans for the cure of cancer. These problems can often be formulated as inverse or feasibility problems, for which fixed-point theory provides algorithms, or arguments for the convergence of various iterative methods.

Fuzzy sets were introduced by Zadeh [33] in 1965 to represent/manipulate data and information possessing nonstatistical uncertainties. It was specifically designed to represent uncertainty and vagueness and to provide formalized tools for dealing with the imprecision intrinsic to many problems. Later in 1967, Goguen [14] extended this idea to L-fuzzy set theory by replacing the interval [0, 1] with a completely distributive lattice L. Ma et al. [18, 19] did a survey of decision making methods based on certain hybrid soft set models and two classes of hybrid soft set models. Since the concept of distance function plays an important role in approximation theory, therefore fuzzy sets and L-fuzzy sets have further been applied to the classical notion of metric.

In continuation of many investigations in fixed point theory, Abu-Donia [1], established common fixed points theorems for fuzzy mappings in metric spaces under φ contraction condition and Shatanawi [26] obtained some fixed point results for generalized ψ-weak contraction mappings in orbitally metric spaces. Azam et al. [5] established common fixed point of fuzzy mappings under a contraction condition. Tripathy et al. [31] proved a fixed point theorem in a generalized fuzzy metric space. Tripathy et al. [28–30] also has done work on convergence of series of fuzzy real number, fixed point results in fuzzy 2-metric spaces, fixed and periodic point theorems in fuzzy metric spaces. Das [12] established and proved theorems on fuzzy normed linear space valued statistically convergent sequences. In 1981, Heilpern [13] gave a fuzzy extension of Banach contraction principle and Nadler [20] fixed point theorems by introducing the concept of fuzzy contraction mappings in connection with d_∞-metric for fuzzy sets. Subsequently several other authors (e.g., [6–11, 27]) generalized this result and studied the existence of fixed points and common fixed points of fuzzy approximate quantity-valued mappings satisfying a contractive type condition in metric linear spaces. Recently Kamran [16] Azam et al. [4, 11] and some other authors (e.g., [15, 17]) continue to study fixed points of fuzzy set valued mappings in connection with d_∞-metric and Hausdorff metric for fuzzy subsets. Zhou et al. [41] worked on properties of the cut-sets of intuitionistic fuzzy relations.

Hausdorff distance for α-cut sets of L-fuzzy mapps was introduced by Rashid et al. [23, 24], to study fixed point theorems for L-fuzzy set valued-mappings and coincidence theorems for a crisp mapping and a sequence of L-fuzzy mappings. Many researchers have studied fixed point theory in the fuzzy context of metric spaces and normed spaces. Recently, Rashid et al. [25] established fixed point theorems for L-fuzzy mappings in quasi-pseudo metric spaces. Subsequently, Azam et al. [2] worked on coincidence point of L-fuzzy sets endowed with graph. In 2016, Azam et al. [3] obtained L-fuzzy fixed points in cone metric spaces.

In the present work, common coincidence point theorems for a crisp and a pair of L-fuzzy mappings satisfying generalized φ-contractive condition have been formulated and proved. A generalized common coincidence point theorem with the d L ∞ metric on 1- cuts of L-fuzzy sets is established, from which some corollaries has been deduced as well. Some applications regarding the existence of solution for non-linear integral equations and implicit function theorem of mathematical analysis have been achieved. Moreover, some interesting and non-trivial examples are presented.

2 Preliminaries

In this section, we recall the concepts and definitions related to bounded lattice, fuzzy sets, hausdorff metric spaces, metric linear spaces etc.

Let (X, d)  be a metric space, C (2 ^X ) be the family of nonempty compact subsets of X and A, B ∈ C (2 ^X ). d ( x , A ) = inf y ∈ A d ( x , y ) , d ( A , B ) = inf x ∈ A , y ∈ B d ( x , y ) . Then the Hausdorff metric H on C (2 ^X ) induced by d is defined as: H ( A , B ) = max { sup a ∈ A d ( a , B ) , sup b ∈ B d ( A , b ) } .

For crisp subset  A  of  X,  we denote the characteristic function of by χ _A  .

Definition 1. [18, 32] A fuzzy set in X is a function with domain X and values in [0, 1]. Let F ( X ) be the collection of all fuzzy sets in X . If A is a fuzzy set and x ∈ X, then the function values A (x)  is called the grade of membership of x in A.

The α -level set of a fuzzy set A, denoted by  [A]  _α , is defined as follows [ A ] α = { x : A ( x ) ≥ α } if α ∈ ( 0 , 1 ] , [ A ] 0 = { x : A ( x ) > 0 } , ¯ A ˆ = { x : A ( x ) = max y ∈ X A ( y ) } . Definition 2. A partially ordered set (L,   ⪯  _L ) is called a lattice; if a ∨ b ∈ L, a ∧ b ∈ L for any a,  b ∈ L. A lattice L is said to be

Example. Let S = {a, b, c} be an arbitrary set, then P (S)(power set of S) with partial order ⊆ shown in figure 1 is a distributive lattice. A lattice shown in figure 2 is non-distributive because a ∧ (b ∨ c) = a ∧ I = a and (a ∧ b) ∨ (a ∧ c) =0 ∨ 0 =0, i.e. distributive law does not hold.

OPEN IN VIEWER

Fig.1

Distributive Lattice.

OPEN IN VIEWER

Fig.2

Non-distributive Lattice.

Definition 3. (Goguen [14]) Let L be a lattice, the top and bottom elements of L are 1 _L and 0 _L respectively, and if a,  b ∈ L, a ∨ b = 1 _L and a ∧ b = 0 _L then b is a unique complement of a denoted by a ´ .

Definition 4. [14] An L-fuzzy set in X is a function with domain X and values in L, where L is a complete distributive lattice with 1 _L and 0 _L , i.e if A : X → L, then A is an L-fuzzy set. F L ( X ) is the collection of all L-fuzzy sets in X . If A is an L-fuzzy set and x ∈ X, then the function values A (x)  is called the grade of membership of x in A.

Remark. If L = [0, 1], then the L-fuzzy set is the special case of fuzzy sets in the original sense of Zadeh [33], which shows that the concept of L-fuzzy set is larger.

The α _L -level set of an L-fuzzy set A, denoted by  [A]  _{α L} , is defined as follows: [ A ] α L = { x : α L ⪯ L A ( x ) } for α L ∈ L - { 0 L } , [ A ] 0 L = { x : 0 L ⪯ L A ( x ) } . ¯

If  max y ∈ X A ( y ) exists, then A ˆ = { x : A ( x ) = max y ∈ X A ( y ) } .

An L-fuzzy set A in a metric linear space V is said to be an approximate quantity if and only if  [A]  _{α L} is compact and convex in V for each α _L  ∈ L - {0 _L } and sup x ∈ V A ( x ) = 1 .

Define some sub-collections of F L ( X ) and F L ( V ) as follows: W L ( V ) = { A ∈ F L ( V ) : × A is an approximate quantity in V } , K ( X ) = { A ∈ F L ( X ) : max y ∈ X A ( y ) existed , × A ˆ ∈ C ( 2 X ) } , X ) = { A ∈ F L ( X ) : [ A ] α L ∈ C ( 2 X ) , × for each α L ∈ L } .

For A , B ∈ F L ( X ) , A ⊂ B means A (x) ⪯  _L B (x)  for each x ∈ X. If there exists an α _L  ∈ L - {0 _L } such that [A]  _{α L} , [B]  _{α L}  ∈ C (2 ^X )  then define p α L ( A , B ) = inf x ∈ [ A ] α L , y ∈ [ B ] α L d ( x , y ) , D α L ( A , B ) = H ( [ A ] α L , [ B ] α L ) . If  [A]  _{α L} , [B]  _{α L}  ∈ C (2 ^X ) for each α _L  ∈ L then define p ( A , B ) , d L ∞ ( A , B ) : C L ( X ) × C L ( X ) → ℝ , (induced by the Hausdorff metric H) as follows: p ( A , B ) = sup α L p α L ( A , B ) , d L ∞ ( A , B ) = sup α L D α L ( A , B ) .

Definition 5. Let W be an arbitrary set, X be a metric space. A mapping T is called L-fuzzy mapping if T is a mapping from W into F L ( X ) . An L-fuzzy mapping T is an L-fuzzy subset on W × X with membership function T (x) (y). The function T (x) (y)  is the grade of membership of y in T (x) . The family of all mappings from W into F L ( X ) is denoted by ( F L ( X ) ) W .

Definition 6. Let (X, d) be a metric space and T : X → F L ( X ) . A point z ∈ X is said to be an L-fuzzy fixed point of T if z ∈ [Tz]  _{α L} , for some α _L  ∈ L - {0 _L }.

Definition 7. A point u ∈ W is called L-fuzzy coincidence point of F : W → X, T : W → F L ( X ) if F (u) ∈ [Tu]  _{α L} . The set {u ∈ W: F (u) ∈ [Tu]  _{α L} , α _L ∈ L - {0 _L } of all coincidence points of F and T is denoted by C _(F,T)  . A point x∈ C _(I,T)  (where I is identity map) is a fixed point of T.

Lemma 1. Let A and B be nonempty closed and bounded subsets of a metric space (X, d) .  If a ∈ A, then d (a, B) ≤ H (A, B) .

Lemma 2. Let (V, d _v )  be a complete metric linear space, T : V ⟶ W L ( V ) be a fuzzy mapping and x₀ ∈ V . Then there exists x₁ ∈ V such that   χ{x₁} ⊂ T (x₀) .

Lemma 3. Let φ : ℝ + → ℝ + be a nondecreasing function satisfying the following conditions:φ is continuous from right and ∑ i = 0 ∞ φ i ( t ) < ∞ for all t > 0 (φ ⁱ  denote the ith iterative function of φ).  Then φ (t) < t .

In the following we always assume that Δ^∗ be the set of all continuous functions satisfying the condition of Lemma 3.

Theorem 4. Let Y be a nonempty set,  (X, d)  be a metric space and Φ , Ψ : Y → F L ( X ) , F : Y → X . Suppose that for x ∈ Y there exists α_{Φ L (x)}, α_{Ψ L (x)} ∈ L - {0 _L } such that [Φx]  _{α Φ L (x)} ,   [Ψx]  _{α Ψ L (x)}  ∈ C (2 ^X ) ,   ( 1 ) ⋃ x ∈ Y [ [ Φ x ] α Φ L ( x ) ∪ [ Ψ x ] α Ψ L ( x ) ] ⊆ FY .(1) Suppose that there exists φ ∈ Δ^∗ such that for all x, y ∈ Y

( 2 ) H ( [ Φ x ] α Φ L ( x ) , [ Ψ x ] α Ψ L ( x ) ) ≤ φ ( max { d ( Fx , Fy ) , d ( Fx , [ Φ x ] α Φ L ( x ) ) , d ( Fy , [ Ψ x ] α Ψ L ( x ) ) , 1 2 [ d ( Fx , [ Ψ x ] α Ψ L ( x ) ) + d ( Fy , [ Φ x ] α Φ L ( x ) ) ] } ) .(2)

If F is injective and either FY or   ⋃ _x∈Y [[Φx]  _{α Φ L (x)}  ∪ [Ψx]  _{α Ψ L (x)} ] is complete. Then C ( F , Φ ) ⋂ C ( F , Ψ ) ≠ φ . Moreover, if [Φx]  _{α Φ L (x)} , [Ψx]  _{α Ψ L (x)} are singleton subsets of X for all x, y ∈ X, then C _(F,Φ)  ⋂ C _(F,Ψ)  is singleton.

Proof. Choose x₀ ∈ Y,  then by hypotheses [Φx₀]  _{α Φ L (x0)}  ∈ C (2 ^X ) .  Using (1) and the compactness of [Φx₀]  _{α Φ L (x0)} , we obtain x₁ ∈ Y such that Fx₁ ∈ [Φx₀]  _{α Φ L (x0)} and  d ( Fx 0 , Fx 1 ) = d ( Fx 0 , [ Φ x 0 ] α Φ L ( x 0 ) ) .

Similarly, we can choose x₂ ∈ Y such that Fx₂ ∈ [Ψx₁]  _{α Ψ L (x1)} and d ( Fx 1 , Fx 2 ) = d ( Fx 1 , [ Ψ x 1 ] α Ψ L ( x 1 ) ) .

By induction we produce sequences {x _n  }  and { Fx _n  } of points of Y and FY respectively such that Fx 2 k + 1 ∈ [ Φ x 2 k ] α Φ L ( x 2 k ) , Fx 2 k + 2 ∈ [ Ψ x 2 k + 1 ] α Ψ L ( x 2 k + 1 ) , k = 0, 1, 2, ⋯ such that d ( Fx 2 k , Fx 2 k + 1 ) = d ( Fx 2 k , [ Φ x 2 k ] α Φ L ( x 2 k ) ) , d ( Fx 2 k + 1 , Fx 2 k + 2 ) = d ( Fx 2 k + 1 , × [ Ψ x 2 k + 1 ] α Ψ L ( x 2 k + 1 ) ) , k = 0, 1, 2, ⋯ .

Using Lemma 1 along with the above equations, we have d ( Fx 2 k , Fx 2 k + 1 ) ≤ H ( [ Ψ x 2 k - 1 ] α Ψ L ( x 2 k - 1 ) , [ Φ x 2 k ] α Φ L ( x 2 k ) ) , d ( Fx 2 k + 1 , Fx 2 k + 2 ) ≤ H ( [ Φ x 2 k ] α Φ L ( x 2 k ) , [ Ψ x 2 k + 1 ] α Ψ L ( x 2 k + 1 ) ) , where k = 0, 1, 2, ⋯ .

Assume that Fx_2k = Fx_2k+1 for some k ≥ 0,  then as F is injective, x 2 k = x 2 k + 1 , [ Ψ x 2 k ] α Ψ L ( x 2 k ) = [ Ψ x 2 k + 1 ] α Ψ L ( x 2 k + 1 ) and d ( Fx 2 k + 1 , Fx 2 k + 2 ) ≤ H ( [ Φ x 2 k ] α Φ L ( x 2 k ) , [ Ψ x 2 k + 1 ] α Ψ L ( x 2 k + 1 ) ) ≤ φ ( max { d ( Fx 2 k , Fx 2 k + 1 ) , d ( Fx 2 k , [ Φ x 2 k ] α Φ L ( x 2 k ) ) , d ( Fx 2 k + 1 , [ Ψ x 2 k + 1 ] α Ψ L ( x 2 k + 1 ) ) , 1 2 [ d ( Fx 2 k , [ Ψ x 2 k + 1 ] α Ψ L ( x 2 k + 1 ) ) + d ( Fx 2 k + 1 , [ Φ x 2 k ] α Φ L ( x 2 k ) ) ] } ) ≤ φ ( max { d ( Fx 2 k + 1 , Fx 2 k + 2 ) , 1 2 [ d ( Fx 2 k , Fx 2 k + 2 ) + d ( Fx 2 k + 1 , Fx 2 k + 1 ) ] } ) ≤ φ ( d ( Fx 2 k + 1 , Fx 2 k + 2 ) ) .

Since, φ (t) < t for all t > 0, therefore, the above inequality implies that d (Fx_2k+1, Fx_2k+2) = 0,  which further implies thatFx_2k = Fx_2k+1 ∈   [Φx_2k]  _{α Φ L (x2k)} , Fx 2 k = Fx 2 k + 1 = Fx 2 k + 2 ∈ [ Ψ x 2 k + 1 ] α Ψ L ( x 2 k + 1 ) = [ Ψ x 2 k ] α Ψ L ( x 2 k ) .

It follows that Fx 2 k ∈ [ Φ x 2 k ] α Φ L ( x 2 k ) ∩ [ Ψ x 2 k ] α Ψ L ( x 2 k ) . Then we get the required result. Thus in this sequel of the proof we can suppose that Fx_n+1 ≠ Fx _n  for n = 0, 1, 2 ⋯ .  Again by using inequality(2), we have d ( Fx 2 k + 1 , Fx 2 k + 2 ) ≤ H ( [ Φ x 2 k ] α Φ L ( x 2 k ) , [ Ψ x 2 k + 1 ] α Ψ L ( x 2 k + 1 ) ) ≤ φ ( max { d ( Fx 2 k , Fx 2 k + 1 ) , d ( Fx 2 k + 1 , Fx 2 k + 2 ) , 1 2 [ d ( Fx 2 k , Fx 2 k + 2 ) + d ( Fx 2 k + 1 , Fx 2 k + 1 ) ] } ) ≤ φ ( max { d ( Fx 2 k , Fx 2 k + 1 ) , d ( Fx 2 k + 1 , Fx 2 k + 2 ) } ) .

If max { d ( Fx 2 k , Fx 2 k + 1 ) , d ( Fx 2 k + 1 , Fx 2 k + 2 ) } = d ( Fx 2 k + 1 , Fx 2 k + 2 ) , then the above inequality implies that which is a contradiction as Fx_n+1 ≠ Fx _n  for n = 0, 1, 2 ⋯ ,  . It follows that  max { d ( Fx 2 k , Fx 2 k + 1 ) , d ( Fx 2 k + 1 , Fx 2 k + 2 ) } = d ( Fx 2 k , Fx 2 k + 1 ) .

It further implies that d ( Fx 2 k + 1 , Fx 2 k + 2 ) ≤ φ ( d ( Fx 2 k , Fx 2 k + 1 ) ) .

Consequently, d ( Fx n + 1 , Fx n ) ≤ φ ( d ( Fx n , Fx n - 1 ) ) ≤ φ 2 ( d ( Fx n - 1 , Fx n - 2 ) ) ≤ ⋯ ≤ φ n ( d ( Fx 1 , Fx 0 ) ) , for each n ≥ 1 . Now for each positive integer m and n  (n > m) ,  we have d ( Fx m , Fx n ) ≤ d ( Fx m , Fx m + 1 ) + d ( Fx m + 1 , Fx m + 2 ) + ⋯ + d ( Fx n - 1 , Fx n ) ≤ φ m ( d ( Fx 1 , Fx 0 ) ) + φ m + 1 ( d ( Fx 1 , Fx 0 ) ) + ⋯ + φ n - 1 ( d ( Fx 1 , Fx 0 ) ) ≤ ∑ i = m n - 1 φ i ( d ( Fx 1 , Fx 0 ) ) ≤ ∑ i = m ∞ φ i ( d ( Fx 1 , Fx 0 ) ) .

Since, ∑ i = 1 ∞ φ i ( t ) < 1 for each t > 0 .  It yields that {Fx _n } is a Cauchy sequence. As FY is complete, there exists u ∈ Y such that Fx _n  → Fu (this holds also if ⋃_x∈Y [[Φx]  _{α Φ L (x)}  ∪ [Ψx]  _{α Psi L (x)} ] is complete with lim n → ∞ Fx n ∈ ⋃ x ∈ Y [ [ Φ x ] α Φ L ( x ) ∪ [ Ψ x ] α Ψ L ( x ) ] ⊆ FY ).

Now, Lemma 1 implies that d ( Fu , [ Φ u ] α Φ L ( u ) ) ≤ d ( Fu , Fx 2 n ) + d ( Fx 2 n , [ Φ u ] α Φ L ( u ) ) d ( Fu , Fx 2 n ) + H ( [ Ψ x 2 n - 1 ] α Ψ L ( x 2 n - 1 ) , [ Φ u ] α Φ L ( u ) ) d ( Fu , Fx 2 n ) + φ ( max { d ( Fu , Fx 2 n - 1 ) , d ( Fu , [ Φ u ] α Φ L ( u ) ) , d ( Fx 2 n - 1 , [ Ψ x 2 n - 1 ] α Ψ L ( x 2 n - 1 ) ) , d ( Fu , [ Ψ x 2 n - 1 ] α Ψ L ( x 2 n - 1 ) ) + d ( Fx 2 n - 1 , [ Φ u ] α Φ L ( u ) ) ] } ) d ( Fu , Fx 2 n ) + φ ( max { d ( Fu , Fx 2 n - 1 ) , d ( Fu , [ Φ u ] α Φ L ( u ) ) , d ( Fx 2 n - 1 , Fx 2 n ) , d ( Fu , Fx 2 n ) + d ( Fx 2 n - 1 , [ Φ u ] α Φ L ( u ) ) ] } ) . Letting n → ∞ ,  and using the fact that φ is continuous, we have, d ( Fu , [ Φ u ] α Φ L ( u ) ) ≤ φ ( d ( Fu , [ Φ u ] α Φ L ( u ) ) . Using the above inequality, we have F ( u ) ∈ [ Φ u ] α Φ L ( u ) , for all x ∈ Y . Similarly, by using d ( Fu , [ Ψ u ] α Ψ L ( u ) ) ≤ d ( Fu , x 2 n + 1 ) [ Ψ u ] α Ψ L ( u ) ) , we have  F ( u ) ∈ [ Ψ u ] α Ψ L ( u ) , for all x ∈ Y .  Hence u ∈ (C_(F,Φ) ⋂ C_(F,Ψ)) . Finally, we show that, if [Φx]  _{α Φ L (x)} ,   [Ψy]  _{α Ψ L (y)} are singleton subsets of X for all x, y ∈ X, then (C_(F,Φ) ⋂ C_(F,Ψ))  is singleton. For this, assume that there exists two distinct points u, v ∈ (C_(F,Φ) ⋂ C_(F,Ψ)) .   It follows that Fu ≠ Fv , Fu ∈ [ Φ u ] α Φ L ( u ) = [ Ψ u ] α Ψ L ( u ) and Fv ∈ [ Φ v ] α Φ L ( v ) = [ Ψ v ] α Ψ L ( v ) .

Using inequality (2), we obtain d ( Fu , Fv ) = H ( [ Φ u ] α Φ L ( u ) , [ Ψ v ] α Ψ L ( v ) ) ≤ φ ( max { d ( Fu , Fv ) , d ( Fu , [ Φ u ] α Φ L ( u ) ) , d ( Fv , [ Ψ v ] α Ψ L ( v ) ) , 1 2 [ d ( Fu , [ Ψ v ] α Ψ L ( v ) ) + d ( Fv , [ Φ u ] α Φ L ( u ) ) ] } ) ≤ φ ( max { d ( Fu , Fv ) , d ( Fu , Fu ) , d ( Fv , Fv ) , 1 2 [ d ( Fu , Fv ) + d ( Fv , Fu ) ] } ) ≤ φ ( d ( Fu , Fv ) ) < d ( Fu , Fv ) , which is a contradiction, hence d (Fu, Fv) =0 and so u = v .

Next we furnish an example to support our result. For convenience, we use the following notations.

Let L be a complete bounded distributive lattice. Define a generalized characteristic function Ω W α β : X → L of a crisp set W as Ω W α β ( x ) = { α if x ∈ W β if x ∉ W .

Note that if L  =   {0, 1} and α  =  1,  β  =  0 then Ω W α β = χ W (Characteristic function of W).

Example 5. Let X = [0, ∞) ,  d (μ, ν) = |μ - ν|. Let L  =   {η,  β,  γ,  δ} with η ⪯  _L β ⪯  _L δ and η ⪯  _L γ ⪯  _L δ, where β and γ are not comparable, then (L, ⪯  _L ) is a complete distributive Lattice. Define a pair of mappings P , Q : ( 0 ∞ ) → F L ( X ) as follows: P ( μ ) ( t ) = { γ if 0 ≤ t ≤ 3 μ - 9 μ 2 2 β if 3 μ - 9 μ 2 2 < 3 μ δ if 3 μ < t < 6 μ η if 6 μ ≤ t < ∞ , Q ( μ ) ( t ) = { δ if 0 ≤ t ≤ 3 μ - 9 μ 2 2 γ if 3 μ - 9 μ 2 2 < t ≤ 6 μ η if 6 μ < t < 9 μ β if 9 μ ≤ t < ∞ . Now define Φ , Ψ : X ⟶ F L ( X ) , F : X → X as follows: Φ ( μ ) ( t ) = { Ω { 0 } η δ if μ = 0 P ( μ ) if μ ∈ ( 0 , 1 3 ) Ω { 1 2 } η δ if μ > 1 3 Ψ ( μ ) ( t ) = { Ω { 0 } η δ if μ = 0 Q ( μ ) if μ ∈ ( 0 , 1 3 ] Ω { 1 2 } η δ if μ > 1 3 , F μ = 3 μ for all μ ∈ X . Define φ : [0, ∞) → [0, ∞) as follows:  φ ( t ) = { t - t 2 2 if t ∈ [ 0 , 1 ] 1 2 if t > 1 . It is obvious that φ (t) < t for t > 0, and for μ ∈ X, if α_{Φ L (μ)} = γ ∈ L - {0 _L } and α_{Ψ L (μ)} = δ ∈ L - {0 _L }, then [Φμ]  _γ , [Ψμ]  _δ are compact.

If μ = ν = 0 or μ , ν > 1 3 , then [Φμ]  _γ  = [Ψμ]  _δ and H ( [ Φ μ ] γ , [ Ψ ν ] δ ) = 0 . If μ = 0 , ν ∈ ( 0 , 1 3 ] , then [Φ (0)]  _γ  ={ 0 }, [ Ψ ν ] δ = [ 0 , 3 ν - 9 ν 2 2 ] and H ( [ Φ ( 0 ) ] γ , [ Ψ ν ] δ ) = | 3 ν - 9 ν 2 2 | = φ ( 3 ν ) = φ ( | F ν - F 0 | ) . If μ , ν ∈ ( 0 , 1 3 ] , then H ( [ Φ μ ] γ , [ Ψ μ ] δ ) = | 3 μ - 9 μ 2 2 - 3 ν + 9 ν 2 2 | = | 3 ( μ - ν ) - 9 ( μ 2 - ν 2 2 ) | = | 3 ( μ - ν ) ( 1 - 3 ( μ + ν ) 2 ) | ≤ 3 | μ - ν | | 1 - 3 | μ - ν | 2 | ≤ 3 | μ - ν | - ( 3 | μ - ν | ) 2 2 = φ ( | 3 ( μ - ν ) | ) = φ ( | F μ - F ν | ) . If μ ∈ ( 0 , 1 3 ] , ν > 1 3 , then H ( [ Φ μ ] γ , [ Ψ μ ] δ ) = H ( [ 0 , 3 μ − 9 μ 2 2 ] , { 1 2 } ) = | 1 − 1 2 − ( 3 μ − 9 μ 2 2 ) | = | ( 1 − 3 μ ) − 1 2 ( 1 − 9 μ 2 ) | = | ( 1 − 3 μ ) − ( 1 − 1 2 ( 1 + 3 μ ) ) | ≤ | ( 1 − 3 μ ) | | ( 1 − 1 2 ( 1 − 3 μ ) ) | = | ( 1 − 3 μ ) | − 1 2 | 1 − 3 μ | 2 = ϕ ( | 3 ( 1 3 − μ ) | ) ≤ ϕ ( 3 | v − μ | ) = ϕ ( | F v − F μ | ) .

Note that [Φμ]  _γ , [Ψμ]  _δ are not singleton subsets of X for all μ, however, all other assumptions of theorem 4 are satisfied. Let μ₀ = 1, then F μ 1 ∈ [ Φ μ 0 ] γ = { 1 2 } and d ( F μ 0 , F μ 1 ) = d ( 3 , F μ 1 ) = d ( F μ 0 , [ Φ μ 0 ] γ ) = d ( 3 , { 1 2 } ) , imply that F μ 1 = 1 2 . Furthermore, F μ 2 ∈ [ Ψ μ 1 ] δ = [ 0 , 3 8 ] and  d ( F μ 1 , F μ 2 ) = d ( 1 2 , F μ 2 ) = d ( F μ 1 , [ Ψ μ 1 ] δ ) = d ( 1 2 , [ 0 , 3 8 ] ) , yield F μ 2 = 3 8 . This process achieves lim_n→∞Fμ _n  = 0 .

Theorem 6. Let Y be a nonempty set,  (X, d)  be a metric space and Φ, Ψ : Y → K (X) ,  F : Y → X such that Φ ˆ x , Ψ ˆ x ∈ C ( 2 X ) for all x ∈ Y,   ⋃ x ∈ Y [ Φ ˆ x ∪ Ψ ˆ x ] ⊆ FY . Suppose that for all x, y ∈ Y and φ ∈ Δ^* H ( Φ ˆ x , Ψ ˆ y ) ≤ φ ( max { d ( Fx , Fy ) , d ( Fx , Φ ˆ x ) , d ( Fy , Ψ ˆ y ) , 1 2 [ d ( Fx , Ψ ˆ y ) + d ( Fy , Φ ˆ x ) ] } ) . If F is injective and either FY or ⋃ x ∈ Y [ Φ ˆ x ∪ Ψ ˆ x ] is complete. Then C ( F , Φ ) ⋂ C ( F , Ψ ) ≠ φ . Moreover, if Φ ˆ x , Ψ ˆ x are singleton subsets of X for all x, then C _(F,Φ)  ⋂ C _(F,Ψ)  is singleton.

Proof. By replacing, [Φx]  _{α Φ L (x)} with Φ ˆ x and [Ψx]  _{α Ψ L (x)} with Ψ ˆ y in previous theorem, we get the required results.□

Theorem 7. Let Y be a nonempty set,  (X, d)  be a metric space and Φ , Ψ : Y → C L ( X ) , F : Y → X . Let  ⋃ x ∈ Y ( [ Φ x ] 1 L ∪ [ Ψ x ] 1 L ) ⊆ FY and for all x, y ∈ Y d L ∞ ( Φ ( x ) , Ψ ( y ) ) ≤ φ ( max { d ( Fx , Fy ) , p ( Fx , Φ ( x ) ) , p ( Fy , Ψ ( y ) ) , 1 2 [ p ( Fx , Ψ ( y ) ) + p ( Fy , Φ ( x ) ) ] } ) . If F is injective and either FY or  ⋃ _x∈Y ([Φx] _{1 L}  ∪ [Ψx] _{1 L} ) is complete. Then there exists a point u  ∈ X such that χ_{Fu} ⊂ Φu,  χ_{Fu} ⊂ Ψu .

Proof. Pick x ∈ X,  then by assumptions [Φx] _{1 L} , [Ψx] _{1 L}  are nonempty compact subsets of X .  Now, as Φ ˆ x = [ Φ x ] 1 L and Ψ ˆ x = [ Ψ x ] 1 L for all x, y ∈ X,  it follows that H ( Φ ˆ x , Ψ ˆ y ) = D 1 L ( Φ ( x ) , Ψ ( y ) ) ≤ d L ∞ ( Φ ( x ) , Ψ ( y ) ) .

Thus H ( Φ ˆ x , Ψ ˆ y ) ≤ φ ( max { d ( Fx , Fy ) , p ( Fx , Φ ( x ) ) , p ( Fy , Ψ ( y ) ) , 1 2 [ p ( Fx , Ψ ( y ) ) + p ( Fy , Φ ( x ) ) ] } ) . By defination of α _L  - cuts, for α _L  ≤ β _L we have [Φx]  _{β L}  ⊆ [Φx]  _{α L}  . It follows that Φ ˆ x ⊆ [ Φ x ] α L for each α ∈ [0, 1] ,  therefore, d ( Fx , [ Φ x ] α L ) ≤ d ( Fx , Φ ˆ x ) for each α _L  ∈ L - {0 _L }  and it implies that p ( Fx , Φ ( x ) ) ≤ d ( Fx , Φ ˆ x ) . This further implies that H ( Φ ˆ x , Ψ ˆ y ) ≤ φ ( max { d ( Fx , Fy ) , d ( Fx , Φ ˆ x ) , d ( Fy , Ψ ˆ y ) , 1 2 [ d ( Fx , Ψ ˆ y ) + d ( Fy , Φ ˆ x ) ] } ) . Now, by Theorem 6 there exists u ∈ X such that Fu ∈ Φ ˆ u ∩ Ψ ˆ u . Hence, Fu ∈ [Φu] _{1 L}  ∩ [Ψu] _{1 L} , which implies that χ_{Fu} ⊂ Φu , χ_{Fu} ⊂ Ψu .

In the following we present some known results as corollaries of the above theorem.

Corollary 8. Let (V, d _v )  be a complete metric linear space, Γ , Λ : V ⟶ W L ( V ) be an L-fuzzy mappings and there exists q ∈ [ 0 , 1 2 ) such that for all x, y ∈ V d L ∞ ( Γ ( x ) , Λ ( y ) ) ≤ q max { d v ( x , y ) , p ( x , Γ ( x ) ) , p ( y , Λ ( y ) ) , p ( x , Λ ( y ) ) , p ( y , Γ ( x ) ) } .

Then there exists a point u  ∈ X such that χ_{u} ⊂ Γu χ_{u} ⊂ Λu .

Proof. q ∈ [ 0 , 1 2 ) implies that q = k 2 where k < 1 . Then d L ∞ ( Γ ( x ) , Λ ( y ) ) ≤ k 2 max { d v ( x , y ) , p ( x , Γ ( x ) ) , p ( y , Λ ( y ) ) , p ( x , Λ ( y ) ) , p ( y , Γ ( x ) ) } ≤ k 2 max { d v ( x , y ) , p ( x , Γ ( x ) ) , p ( y , Λ ( y ) ) , p ( x , Λ ( y ) ) + p ( y , Γ ( x ) ) } ≤ k max { d v ( x , y ) , p ( x , Γ ( x ) ) , p ( y , Λ ( y ) ) , p ( x , Λ ( y ) ) + p ( y , Γ ( x ) ) 2 }

It follows that φ ( max { d v ( x , y ) , p ( x , Γ ( x ) ) , p ( y , Λ ( y ) ) , p ( x , Λ ( y ) ) + p ( y , Γ ( x ) ) 2 } ) , where φ (r) = kr .  For F = I,  Y = X, Theorem 7 finds u  ∈ V such that χ_{u} ⊂ Γu χ_{u} ⊂ Λu . Corollary 9. Let (V, d _v )  be a complete metric linear space, Φ , Ψ : V ⟶ W L ( V ) and there exists k ∈ [0, 1) such that for all x, y ∈ V, d L ∞ ( Φ ( x ) , Ψ ( y ) ) ≤ k max { d v ( x , y ) , p ( x , Φ ( x ) ) , p ( y , Ψ ( y ) ) , 1 2 [ p ( x , Ψ ( y ) ) + p ( y , Φ ( x ) ) ] } Then there exists a point u  ∈ X such that χ_{u} ⊂ Φu χ_{u} ⊂ Ψu .

Corollary 10. Let (V, d _v )  be a complete metric linear space, T : V ⟶ W L ( V ) be an L-fuzzy mapping such that for all x, y ∈ V, d L ∞ ( T ( x ) , T ( y ) ) ≤ λ d v ( x , y ) where 0 ≤ λ < 1, then there exists u ∈ X such that χ_{u} ⊂ Tu .

Most of the physical problems of applied sciences and engineering are usually formulated in the form of functional equations. The solutions of some scalar or operational equations which cannot be solved exactly can be found by formulating in a fixed point problem. Many researchers use the iteration schemes to approximate the fixed point by considering the solution of differential equations, fractional differential equations, nonlinear integral equation, moment problems and many others. A lot of initial and boundary value problems can be easily solved by converting them into Fredholm and Volterra integral equations.

In this section, a common coincidence point result for a single valued mapping and a pair of set-valued mappings has been obtained. Further, we apply it to establish some hypothesis which guarantee the existence of unique solutions of a class of nonlinear integral equations.

Theorem 11. Let Y be a nonempty set,  (X, d)  be a metric space and Γ, Λ : Y → C (2 ^X ) ,  F : Y → X. Let  ⋃ x ∈ Y [ Γ x ∪ Λ x ] ⊆ FY . and for all x, y ∈ Y