On nonlinear F -contractive fuzzy mappings

1 Introduction and Preliminaries

The metric fixed point theory is a useful tool in the study of various pratical problems appearing in the framework of applied sciences. The Banach contraction principle is a famous fundamental metric fixed point result and has an important role in fixed point theory. This principle has become very popular and has applications in various fields of mathematics and sciences due to its simple forms and iterations which can be easily implemented on the computers. The Banach contraction principle has been extended and generalized in many directions by putting conditions on the spaces or on the mappings.

The idea of fuzzy set was first laid down by Zadeh [26]. Weiss [23] and Butnariu [9] introduced the notion of fixed points for fuzzy mappings. Heilpern [13] initiated the idea of fuzzy contraction mappings and proved a fixed point theorem for fuzzy contraction mappings which is a fuzzy analogue of the result of Nadler [18] for multivalued mappings. Afterward, many researchers have extended and generalized Heilpern fixed point result for fuzzy mappings satisfying different contractive type conditions in (generalized) metric spaces (see, e.g., [1, 21] and references therein).

In this paper, we establish the existence of (common) fuzzy fixed points for fuzzy mappings concerning F- contractions (the concept was introduced by Wardowski [24] in 2012) and Feng - Liu type condition [11] in compelete metric spaces with/without using the Pompeiu - Hausdorff distance between α-level sets of fuzzy mappings. Our results generalize and strengthen known results in the literature. An example and an application for the solution of a system of functional equations arising in dynamic programming are given to illustrate the usability of the obtained results.

We next recall some denifitions and results which will be needed in the sequel.

Let (X, d) be a metric space. Denote by C (X) the set of all nonempty closed subsets of X, by CB (X) the set of all nonempty closed bounded subsets of X and by K (X) the set of all nonempty compact subsets of X. The distance from a point x ∈ X to a nonempty subset A of X is defined by d ( x , A ) = inf { d ( x , y ) : y ∈ A } . For A, B ∈ CB (X), the Pompeiu - Hausdorff distance between A and B is defined by H ( A , B ) = max { sup a ∈ A d ( a , B ) , sup b ∈ B d ( b , A ) } .

A fuzzy set A in X is a function with domain X and values in I : = [0, 1] and, for each x ∈ X, the value A (x) is called the grade of membership of x in A. Denote by I ^X the collection of all fuzzy sets in X. Let A be a fuzzy set in X. If α ∈ (0, 1], then the α-level set of A is denoted by [A]  _α and it is defined as [ A ] α = { x ∈ X : A ( x ) ≥ α } .

For α = 0, we define [ A ] 0 = { x ∈ X : A ( x ) > 0 } ¯ , where C ¯ is the closure of a set C in (X, d). We say that a fuzzy set A is more accurate than a fuzzy set B, denoted by A ⊂ B, if and only if A (x) ≤ B (x) for each x ∈ X. It is easy to see that if 0 < α ≤ β ≤ 1, then [A]  _β  ⊂ [A]  _α . Let A and B be two fuzzy sets and α ∈ [0, 1] be such that [A]  _α and [B]  _α are nonempty subsets of X. We define p α ( A , B ) = inf { d ( x , y ) : x ∈ [ A ] α , y ∈ [ B ] α } . If [A]  _α , [B]  _α ≠ ∅ for each α ∈ [0, 1], then we define p ( A , B ) = sup { p α ( A , B ) : α ∈ [ 0 , 1 ] } . If [A]  _α , [B]  _α  ∈ CB (X) for each α ∈ [0, 1], then we define d ∞ ( A , B ) = sup { D α ( A , B ) : α ∈ [ 0 , 1 ] } , with D α ( A , B ) = H ( [ A ] α , [ B ] α ) .

Let Y be a subset of X. A mapping T is called a fuzzy mapping if T is a mapping from Y into I ^X . That is, T is a mapping which associates with each y ∈ Y the fuzzy set T (y) ∈ I ^X . As a fuzzy set, T (y) in X is characterized by a membership function T (y) : X → [0, 1]. So, T (y) (x) is the grade of membership of x in T (y). For convenience, we denote the α-level set of T (x) by [Tx]  _α instead of [T (x)]  _α .

Let x ∈ X and α ∈ [0, 1]. The fuzzy point x _α of X is the fuzzy set x _α  : X → [0, 1] given by x α ( y ) = { α if x = y 0 otherwise . For α = 1, we have the following indicator function of the set {x}: x 1 ( y ) = { 1 if x = y 0 otherwise and we also denote this fuzzy set x₁ : X → [0, 1] by {x}.

Definition 1.1. Let T : X → I ^X be a fuzzy mapping and α ∈ [0, 1]. A point x ∈ X is called an α- fuzzy fixed point of T if x _α  ⊂ Tx, that is, T (x) (x) ≥ α or, equivalently, x ∈ [Tx]  _α . A point x ∈ X is called a fuzzy fixed point of T if {x} ⊂ Tx.

Definition 1.2. Let S, T be fuzzy mappings from X into I ^X . A point x ∈ X is called a common α- fuzzy fixed point of S and T for some α ∈ [0, 1] if x ∈ [Sx]  _α  ∩ [Tu]  _α .

Lemma 1.1. LM Let (X, d) be a metric space, x ∈ X and α ∈ [0, 1]. Assume that A is a fuzzy set in X such that [A]  _α is nonempty closed. If p _α  (x, A) =0, then x _α  ⊂ A.

Proof. Since 0 = p α ( x , A ) = inf y ∈ [ A ] α d ( x , y ) = d ( x , [ A ] α ) , and [A]  _α is nonempty, closed, we have that x ∈ [A]  _α . That is, x _α  ⊂ A.

Definition 1.3. Let (X, d) be a metric space. A function f : X → ℝ is said to be lower semicontinuous in X if lim inf x → x 0 f ( x ) ≥ f ( x 0 ) , for each x₀ ∈ X.

A fuzzy set A in a metric linear space V is said to be an approximate quantity if and only if [A]  _α is compact and convex in V for each α ∈ [0, 1] and sup x ∈ V A ( x ) = 1 .

The collection of all approximate quantities in V is denoted by W (V).

2 Fixed point results

The following class of functions F was introduced in [24] by Wardowski. A function F : ( 0 , ∞ ) → ℝ belongs to the class F if it satisfies the following conditions:

F is strictly increasing;

Using the class F , Wardowski defined the concept of F-contraction and proved a result about the existence and uniqueness of fixed points for F-contraction mappings which is a real generalization of the Banach contraction principle. In this section, we first study the existence of common fuzzy fixed points for fuzzy mappings satifying generalized nonlinear F- contractive conditions. For our first result, we also consider the family T of all functions τ : [0, ∞) → (0, ∞) satisfying: lim inf s → t + τ ( s ) > 0 , for all t ≥ 0 .(2.1)

The first result of this paper is stated as follows.

Theorem 2.1. Let (X, d) be a complete metric space and T, S be fuzzy mappings from X into I ^X such that there exist functions α _T , α _S  : X → (0, 1] for which [Sx] _{α S (x)}, [Tx] _{α T (x)} are nonempty compact subsets of X for each x ∈ X. Assume that there exist L ≥ 0, τ ∈ T and F ∈ F such that

τ ( d ( x , y ) ) + F ( H ( [ Sx ] α S ( x ) , [ Ty ] α T ( y ) ) ) ≤ F ( M ( x , y ) ) + Lm ( x , y )(2.2) for all x, y ∈ X with H ([Sx] _{α S (x)}, [Ty] _{α T (y)}) > 0, where M ( x , y ) = max { d ( x , y ) , d ( x , [ Sx ] α S ( x ) ) , d ( y , [ Ty ] α T ( y ) ) , 1 2 [ d ( x , [ Ty ] α T ( y ) ) + d ( y , [ Sx ] α S ( x ) ) ] } and m ( x , y ) = min { d ( y , [ Sx ] α S ( x ) ) , d ( x , [ Ty ] α T ( y ) ) } . If F is lower semicontinuous, then there exists u in X such that u ∈ [ Su ] α S ( u ) ∩ [ Tu ] α T ( u ) .

Proof. Let x₀ be an arbitrary point in X and let x₁ ∈ [Sx₀] _{α S (x0)}. By the compactness of [Sx₀] _{α S (x0)} and [Tx₁] _{α T (x1)}, there exists x₂ ∈ [Tx₁] _{α T (x1)} such that d ( x 1 , x 2 ) ≤ H ( [ Sx 0 ] α S ( x 0 ) , [ Tx 1 ] α T ( x 1 ) ) .

For this x₂, by the compactness of [Tx₁] _{α T (x1)} and [Sx₂] _{α S (x2)}, there exists x₃ ∈ [Sx₂] _{α S (x2)} such that d ( x 2 , x 3 ) ≤ H ( [ Tx 1 ] α T ( x 1 ) , [ Sx 2 ] α S ( x 2 ) ) .

Continuing this way, we obtain a sequence {x _n } in X such that x_2n+1 ∈ [Sx_2n] _{α S (x2n)} and x_2n+2 ∈ [Tx_2n+1] _{α T (x2n+1)} satisfying d ( x 2 n + 1 , x 2 n + 2 ) ≤ H ( [ Sx 2 n ] α S ( x 2 n ) , [ Tx 2 n + 1 ] α T ( x 2 n + 1 ) ) and d ( x 2 n + 2 , x 2 n + 3 ) ≤ H ( [ Tx 2 n + 1 ] α T ( x 2 n + 1 ) , [ Sx 2 n + 2 ] α S ( x 2 n + 2 ) ) , for all n = 0, 1, 2⋯ We claim that if there is some n₀ ≥ 0 such that d (x _{n 0} , x_n0+1) =0, then d (x_n0+1, x_n0+2) =0. That is, if d (x _{n 0} , x_n0+1) =0 for some n₀ ≥ 0, then d (x _n , x_n+1) =0 for all n ≥ n₀. Without loss of generality, we may assume that n₀ = 2k for some k ≥ 0. Suppose on the contrary that d (x_2k+1, x_2k+2) >0. By (2.2) and the monotonicity of F, we have τ ( d ( x 2 k , x 2 k + 1 ) ) + F ( d ( x 2 k + 1 , x 2 k + 2 ) ) ≤ τ ( d ( x 2 k , x 2 k + 1 ) ) + F ( H ( [ Sx 2 k ] α S ( x 2 k ) , [ Tx 2 k + 1 ] α T ( x 2 k + 1 ) ) ) ≤ F ( max { d ( x 2 k , x 2 k + 1 ) , d ( x 2 k , [ Sx 2 k ] α S ( x 2 k ) ) , d ( x 2 k + 1 , [ Tx 2 k + 1 ] α T ( x 2 k + 1 ) ) , 1 2 [ d ( x 2 k , [ Tx 2 k + 1 ] α T ( x 2 k + 1 ) ) + d ( x 2 k + 1 , [ Sx 2 k ] α S ( x 2 k ) ) ] } ) + L min { d ( x 2 k + 1 , [ Sx 2 k ] α S ( x 2 k ) ) d ( x 2 k , [ Tx 2 k + 1 ] α T ( x 2 k + 1 ) ) } ≤ F ( max { d ( x 2 k , x 2 k + 1 ) , d ( x 2 k + 1 , x 2 k + 2 ) } ) = F ( d ( x 2 k + 1 , x 2 k + 2 ) ) .

This implies that τ (0) = τ (d (x_2k, x_2k+1)) ≤0 which is a contradiction. Hence, our claim is true. Now, if there is n 0 ∈ ℕ such that d (x _{n 0} , x_n0+1) =0, then x n 0 ∈ [ Sx n 0 ] α S ( x n 0 ) ∩ [ Tx n 0 ] α T ( x n 0 ) , and so the proof is complete.

Assume now that d (x _n , x_n+1) >0 for all n ≥ 0. Using (2.2) and the monotonicity of F as above, we have, for all n ≥ 0, that τ ( d ( x 2 n , x 2 n + 1 ) ) + F ( d ( x 2 n + 1 , x 2 n + 2 ) ) ≤ F ( max { d ( x 2 n , x 2 n + 1 ) , d ( x 2 n + 1 , x 2 n + 2 ) } ) .

Assume that d (x_2n+1, x_2n+2) ≥ d (x_2n, x_2n+1). Then, we obtain τ ( d ( x 2 n , x 2 n + 1 ) ) + F ( d ( x 2 n + 1 , x 2 n + 2 ) ) ≤ F ( d ( x 2 n + 1 , x 2 n + 2 ) ) which implies that τ ( d ( x 2 n , x 2 n + 1 ) ) ≤ 0 , a contradiction. Hence, d ( x 2 n + 1 , x 2 n + 2 ) < d ( x 2 n , x 2 n + 1 ) and so τ ( d ( x 2 n , x 2 n + 1 ) ) + F ( d ( x 2 n + 1 , x 2 n + 2 ) ) ≤ F ( d ( x 2 n , x 2 n + 1 ) ) .

Similarly, we can show that d ( x 2 n + 2 , x 2 n + 3 ) < d ( x 2 n + 1 , x 2 n + 2 ) and τ ( d ( x 2 n + 1 , x 2 n + 2 ) ) + F ( d ( x 2 n + 2 , x 2 n + 3 ) ) ≤ F ( d ( x 2 n + 1 , x 2 n + 2 ) ) , for all n ≥ 0. Thus, {d (x _n , x_n+1)} is a strictly increasing sequence of positive real numbers and for all n ≥ 0

τ ( d ( x n - 1 , x n ) ) + F ( d ( x n , x n + 1 ) ) ≤ F ( d ( x n - 1 , x n ) )(2.3)

Denote by d _n  = d (x_n-1, x _n ) for each n ≥ 1. Since {d _n } is decreasing, there exists t ≥ 0 such that lim n → ∞ d n = t + . By (2.1), there is a positive number γ > 0 and n₀ ≥ 1 such that τ (d _n ) > γ for all n ≥ n₀. From (2.3), we have

F ( d n ) ≤ F ( d n - 1 ) - τ ( d n - 1 ) ≤ ⋯ ≤ F ( d 1 ) - ∑ i = 1 n - 1 τ ( d i ) < F ( d 1 ) - ∑ i = n 0 n - 1 τ ( d i ) < F ( d 1 ) - ( n - n 0 ) γ , n > n 0 .(2.4)

Letting n→ ∞, we get F (d _n )→ - ∞. Then, by (F2), we obtain lim n → ∞ d n = 0 .

By (F3), there exists k ∈ (0, 1) such that lim n → ∞ d n k F ( d n ) = 0 .

From (2.4), we have for all n > n₀ that d n k F ( d n ) ≤ d n k F ( d 1 ) - d n k ( n - n 0 ) , or, equivalently, nd n k ≤ ( F ( d 1 ) + n 0 ) d n k - d n k F ( d n ) .

Letting n→ ∞ in the latter inequality, we have 0 ≤ lim n → ∞ nd n k ≤ lim n → ∞ [ ( F ( d 1 ) + n 0 ) d n k - d n k F ( d n ) ] = 0 .

That is, lim n → ∞ nd n k = 0 .

Thus, there exists n₁ > 1 such that nd n k < 1 for all n ≥ n₁. So, we have for all n ≥ n₁ that

d n < 1 n 1 / k .(2.5)

For all m > n ≥ n₁, we have d ( x n , x m ) ≤ d ( x n , x n + 1 ) + ⋯ + d ( x m - 1 , x m ) = d n + ⋯ + d m - 1 ≤ ∑ i = n ∞ d i < ∑ i = n ∞ 1 i 1 / k .

By the convergence of the series ∑ i = 1 ∞ 1 i 1 / k , passing to limit, we get d (x _n , x _m ) →0 as m, n→ ∞. Hence, {x _n } is a Cauchy sequence in (X, d). Since (X, d) is complete, there exists u ∈ X such that lim n → ∞ x n = u . We now prove that u ∈ [Tu] _{α T (u)}. Assume on the contrary that u ∉ [Tu] _{α T (u)}. Then there exists n₂ > 1 such that d (x_2n+1, [Tu] _{α T (u)}) >0 for all n ≥ n₂. By (2.2) and the monotonicity of F, we have, for all n ≥ n₂, that τ ( d ( x 2 n , u ) ) + F ( d ( x 2 n + 1 , [ Tu ] α T ( u ) ) ) ≤ τ ( d ( x 2 n , u ) ) + F ( H ( [ Sx 2 n ] α S ( x 2 n ) , [ Tu ] α T ( u ) ) ) ≤ F ( max { d ( x 2 n , u ) , d ( x 2 n , [ Sx 2 n ] α S ( x 2 n ) ) , d ( u , [ Tu ] α T ( u ) ) , 1 2 [ d ( x 2 n , [ Tu ] α T ( u ) ) + d ( u , [ Sx 2 n ] α S ( x 2 n ) ) ] } ) + L min { d ( u , [ Sx 2 n ] α S ( x 2 n ) ) , d ( x 2 n , [ Tu ] α T ( u ) ) } ≤ F ( max { d ( x 2 n , u ) , d ( x 2 n , x 2 n + 1 ) , d ( u , [ Tu ] α T ( u ) ) , 1 2 [ d ( x 2 n , [ Tu ] α T ( u ) ) + d ( u , x 2 n + 1 ) ] } ) + L min { d ( u , x 2 n + 1 ) , d ( x 2 n , [ Tu ] α T ( u ) ) } .

Since x _n  → u as n→ ∞ and d (u, [Tu] _{α T (u)}) >0, there exists an integer n₃ ≥ n₂ such that max { d ( x 2 n , u ) , d ( x 2 n , x 2 n + 1 ) , 1 2 [ d ( x 2 n , [ Tu ] α T ( u ) ) + d ( u , x 2 n + 1 ) ] } < d ( u , [ Tu ] α T ( u ) ) and d ( u , x 2 n + 1 ) < d ( x 2 n , [ Tu ] α T ( u ) ) for all n ≥ n₃. Hence, one has, for all n ≥ n₃, that τ ( d ( x 2 n , u ) ) + F ( d ( x 2 n + 1 , [ Tu ] α T ( u ) ) ) < F ( d ( u , [ Tu ] α T ( u ) ) ) + Ld ( u , x 2 n + 1 ) .

Then by the lower semicontinuity of F we have lim inf n → ∞ τ ( d ( x 2 n , u ) ) + F ( d ( u , [ Tu ] α T ( u ) ) ) ≤ lim inf n → ∞ τ ( d ( x 2 n , u ) ) + lim inf n → ∞ F ( d ( x 2 n + 1 , [ Tu ] α T ( u ) ) ) ≤ lim inf n → ∞ [ τ ( d ( x 2 n , u ) ) + F ( d ( x 2 n + 1 , [ Tu ] α T ( u ) ) ) ] ≤ F ( d ( u , [ Tu ] α T ( u ) ) ) .

This implies that lim inf n → ∞ τ ( d ( x 2 n , u ) ) ≤ 0 which is a contradiction. Thus, u ∈ [Tu] _{α T (u)}. Similarly, one can show that u ∈ [Su] _{α S (u)}. Therefore, u ∈ [Su] _{α S (u)} ∩ [Tu] _{α T (u)}. The proof is complete.

The following are some consequence of Theorem 2.1.

Corollary 2.1. Let (X, d) be a complete metric space and T be fuzzy mapping from X into I ^X such that there exists a function α : X → (0, 1] for which [Tx] _α(x) is a nonempty compact subset of X for each x ∈ X. Assume that there exist L ≥ 0, τ ∈ T and F ∈ F such that τ ( d ( x , y ) ) + F ( H ( [ Tx ] α ( x ) , [ Ty ] α ( y ) ) ) ≤ F ( K ( x , y ) ) + Lk ( x , y ) for all x, y ∈ X with H ([Tx] _α(x), [Ty] _α(y)) > 0, where K ( x , y ) = max { d ( x , y ) , d ( x , [ Tx ] α ( x ) ) , d ( y , [ Ty ] α ( y ) ) , 1 2 [ d ( x , [ Ty ] α ( y ) ) + d ( y , [ Tx ] α ( x ) ) ] } and k ( x , y ) = min { d ( y , [ Tx ] α ( x ) ) , d ( x , [ Ty ] α ( y ) ) } . If F is lower semicontinuous, then there exists u in X such that u ∈ [Tu] _α(u).

Corollary 2.2. Let (X, d) be a complete metric space and R, Q : X → K (X) be multivalued mappings. Assume that there exist L ≥ 0, τ ∈ T and F ∈ F such that τ ( d ( x , y ) ) + F ( H ( Rx , Qy ) ) ≤ F ( N ( x , y ) ) + Ln ( x , y ) for all x, y ∈ X with H (Rx, Qy) > 0, where N ( x , y ) = max { d ( x , y ) , d ( x , Rx ) , d ( y , Qy ) , 1 2 [ d ( x , Qy ) + d ( y , Rx ) ] } and n ( x , y ) = min { d ( y , Rx ) , d ( x , Qy ) } . If F is lower semicontinuous, then there exists u in X such that u ∈ Ru ∩ Qu.

Proof. Let α : X → (0, 1] be a mapping and S, T : X → I ^X be defined by S ( x ) ( y ) = { α ( x ) if y ∈ Rx , 0 if y ∉ Rx and T ( x ) ( y ) = { α ( x ) if y ∈ Qx , 0 if y ∉ Qx .

Then, we have, for all x ∈ X, that [ Sx ] α ( x ) = { y : S ( x ) ( y ) ≥ α ( x ) } = Rx and [ Tx ] α ( x ) = { y : T ( x ) ( y ) ≥ α ( x ) } = Qx .

Applying Theorem 2.1 we obtain u ∈ X such that u ∈ [ Su ] α ( u ) ∩ [ Tu ] α ( u ) = Ru ∩ Qu .

The following corollary is a generalization of the main result in [16].

Corollary 2.3. Let (X, d) be a complete metric linear space and T, S : X → W (X) be fuzzy mappings. Assume that there exist L ≥ 0, τ ∈ T and F ∈ F such that τ ( p ( x , y ) ) + F ( d ∞ ( Sx , Ty ) ) ≤ F ( B ( x , y ) ) + L β ( x , y ) for all x, y ∈ X with d_∞ (Sx, Ty) >0, where B ( x , y ) = max { p ( x , y ) , p ( x , Sx ) , p ( y , Ty ) , 1 2 [ p ( x , Ty ) + p ( y , Sx ) ] } , and β ( x , y ) = min { p ( y , Sx ) , p ( x , Ty ) } . If F is lower semicontinuous, then there exists u in X such that {u} ⊂ Su and {u} ⊂ Tu.

Proof. By the definition of d_∞- metric for fuzzy sets, one has H ( [ Sx ] 1 , [ Ty ] 1 ) ≤ d ∞ ( Sx , Ty ) , ∀ x , y ∈ X . We have [Sx] ₁ ⊂ [Sx]  _α for each α ∈ (0, 1] and for all x ∈ X. Hence, d (x, [Sx]  _α ) ≤ d (x, [Sx] ₁) for each α ∈ (0, 1] and for all x ∈ X. This implies that p (x, Sx) ≤ d (x, [Sx] ₁) for all x ∈ X. Similarly, we have p (x, Tx) ≤ d (x, [Tx] ₁), p (x, Ty) ≤ d (x, [Ty] ₁), p (x, Sy) ≤ d (x, [Sy] ₁) for all x, y ∈ X. Thus, for all x, y ∈ X, one has τ ( d ( x , y ) ) + F ( H ( [ Sx ] 1 , [ Ty ] 1 ) ) ≤ τ ( p ( x , y ) ) + F ( d ∞ ( Sx , Ty ) ) ≤ F ( max { p ( x , y ) , p ( x , Sx ) , p ( y , Ty ) , 1 2 [ p ( x , Ty ) + p ( y , Sx ) ] } ) + L min { p ( y , Sx ) , p ( x , Ty ) } ≤ F ( max { d ( x , y ) , d ( x , [ Sx ] 1 ) , d ( y , [ Ty ] 1 ) , 1 2 [ d ( x , [ Ty ] 1 ) + d ( y , [ Sx ] 1 ) ] } ) + L min { d ( y , [ Sx ] 1 , d ( x , [ Ty ] 1 ) } . By Theorem 2.1, there exists u ∈ X such that u ∈ [Su] ₁ ∩ [Tu] ₁. Therefore, {u} ⊂ Tu and {u} ⊂ Su. The proof is complete.

We next present a result for fuzzy mappings satifying inequalities of Feng - Liu type. This result generalizes Theorem 9 in [1] and some other results in the literature (see, e.g., [4, 20]). For our aim, let us denote by K the set of all functions κ : [0, ∞) ² → (0, ∞) satisfying lim inf t 1 , t 2 → s + κ ( t 1 , t 2 ) > 0 , ∀ s ∈ [ 0 , ∞ ) .(2.6)

Our next main result is stated as follows.

Theorem 2.2. Let (X, d) be a complete metric space and α ∈ [0, 1]. Let T : X → I ^X be a fuzzy mapping such that [Tx]  _α is a nonempty closed subset of X for all x ∈ X. Assume that there exist κ ∈ K , σ ∈ ℝ , a nondecreasing function F 1 : ( 0 , ∞ ) → ℝ and a function F 2 : ( 0 , ∞ ) → ℝ satisfying conditions (F2), (F3), such that for any x ∈ X with p _α  (x, Tx) >0, there exists y ∈ X satisfying κ ( p α ( x , Tx ) , p α ( y , Ty ) ) + G 1 ( p α ( y , Ty ) ) ≤ F 1 ( p α ( x , Tx ) ) ,(2.7) and G 2 ( d ( x , y ) ) ≤ F 1 ( p α ( x , Tx ) ) + σ ,(2.8) where G i : [ 0 , ∞ ) → ℝ ∪ { - ∞ } , i = 1, 2, are defined by G i ( t ) = { F i ( t ) if t > 0 - ∞ if t = 0

Then, there exists x ∈ X such that x _α  ⊂ Tx provided that the function x → p _α  (x, Tx) is lower semicontinuous.

Proof. Assume that p _α  (x, Tx) >0 for all x ∈ X. Let x₀ ∈ X, by assumptions, we can construct a sequence {x _n } in X such that: κ ( p α ( x n , Tx n ) , p α ( x n + 1 , Tx n + 1 ) )(2.9) + G 1 ( p α ( x n + 1 , Tx n + 1 ) ) ≤ F 1 ( p α ( x n , Tx n ) ) and G 2 ( d ( x n , x n + 1 ) ≤ F 1 ( p α ( x n , Tx n ) ) + σ(2.10) and p _α  (x _n , Tx _n ) >0 for all n.

Since κ (t₁, t₂) >0 for all t₁, t₂ ∈ [0, ∞), it follows from (2.41) that x _n  ≠ x_n+1 for all n. Hence, we can rewrite (2.41) and (2.42) as folows: κ ( p α ( x n , Tx n ) , p α ( x n + 1 , Tx n + 1 ) )(2.11) + F 1 ( p α ( x n + 1 , Tx n + 1 ) ) ≤ F 1 ( p α ( x n , Tx n ) ) and F 2 ( d ( x n , x n + 1 ) ≤ F 1 ( p α ( x n , Tx n ) ) + σ .(2.12)

By the monotonicity of F₁ and the positivity of κ, it follows from (2.43) that the sequence {p _α  (x _n , Tx _n )} is nondecreasing. Hence, there exists γ ≥ 0 such that lim n → ∞ p α ( x n , Tx n ) = γ and p _α  (x _n , Tx _n ) ≥ γ for all n. Let δ = lim inf n → ∞ κ ( p α ( x n , Tx n ) , p α ( x n + 1 , Tx n + 1 ) ) > 0 , and λ ∈ (0, δ). There exists n₀ > 1 such that κ ( p α ( x n , Tx n ) , p α ( x n + 1 , Tx n + 1 ) ) > λ , for all n ≥ n₀. By (2.43) λ + F 1 ( p α ( x n + 1 , Tx n + 1 ) ) ≤ F 1 ( p α ( x n , Tx n ) ) , for all n ≥ n₀. Hence, for all n ≥ n₀, one has F 1 ( p α ( x n , Tx n ) ≤ F 1 ( p α ( x n - 1 Tx n - 1 ) ) - λ ≤ ⋯(2.13) ≤ F 1 ( p α ( x n 0 , Tx n 0 ) ) - ( n - n 0 ) λ

Assume that γ > 0, then by the monotonicity of F₁, we have for all n ≥ n₀ that F 1 ( γ ) ≤ F 1 ( p α ( x n , Tx n ) ≤ F 1 ( p α ( x n 0 , Tx n 0 ) ) - ( n - n 0 ) λ .

Letting n→ ∞, we obtain lim n → ∞ F 1 ( γ ) ≤ - ∞ .

This is a contradiction. Thus, γ = 0, that is lim n → ∞ p α ( x n , Tx n ) = 0 .

It follows from (2.44) and (2.45) that

F 2 ( d ( x n , x n + 1 ) ) ≤ F 1 ( p α ( x n 0 Tx n 0 ) ) + σ - ( n - n 0 ) λ ,(2.14) for all n ≥ n₀. Letting n→ ∞ we have lim n → ∞ F 2 ( d ( x n , x n + 1 ) ) = - ∞ .

Since F₂ satisfies (F2), we obtain lim n → ∞ d ( x n , x n + 1 ) = 0 .(2.15)

Since F₂ satisfies (F3), there exists k ∈ (0, 1) such that lim n → ∞ [ d ( x n , x n + 1 ) ] k F 2 ( d ( x n , x n + 1 ) ) = 0 .(2.16)

Using (2.46), for all n ≥ n₀, [ d ( x n , x n + 1 ) ] k F 2 ( d ( x n , x n + 1 ) ) - [ d ( x n , x n + 1 ) ] k [ F 1 ( p α ( x n 0 Tx n 0 ) ) + σ + n 0 λ ] ≤ - n λ [ d ( x n , x n + 1 ) ] k ≤ 0

This, together with (2.47) and (2.48), implies that lim n → ∞ n [ d ( x n , x n + 1 ) ] k = 0 .

Therefor, there exists N > 1 such that n [ d ( x n , x n + 1 ) ] k < 1 ∀ n ≥ N , or d ( x n , x n + 1 ) < 1 n 1 / k ∀ n ≥ N .

As in the proof of Theorem 2.1, we obtain that {x _n } is a Cauchy sequence and so, by the completeness of (X, d), x _n  → z as n→ ∞ for some z ∈ X. By the lower semicontinuity of p _α  (x, Tx), we get 0 ≤ p α ( z , Tz ) ≤ lim inf n → ∞ p α ( x n , Tx n ) = 0 , and so p _α  (z, Tz) =0 which is a contradiction. Therefore, there exists x ∈ X such that p _α  (x, Tx) =0, i.e., x _α  ⊂ Tx (by Lemma 1.1). The proof is complete.

We next give an example to illustrate the previous theorem.

Example 2.1. Let X = [0, ∞) and d (x, y) = |x - y| for all x, y ∈ X. Then, (X, d) is a complete metric space. We define the fuzzy mapping T : X → I ^X as follows. ( T 0 ) ( y ) = { 1 2 if y = 0 , 0 if y ≠ 0 , and for x > 0 ( Tx ) ( y ) = { 1 4 if 0 ≤ y < 4 x , 1 if 4 x ≤ y ≤ 5 x , 0 if 5 x < y .

Let α ∈ (1/4, 1/2]. We have [T0]  _α  = {0} and [Tx]  _α  = [4x, 5x] for x > 0. Thus, [ Tx ] α = [ 4 x , 5 x ] , ∀ x ∈ X .

We see that p α ( x , Tx ) = inf y ∈ [ Tx ] α d ( x , y ) = 3 x for all x ∈ X. Hence, x → p _α  (x, Tx) is continuous. Moreover, let F₁ (t) = F₂ (t) = ln(t) for all t > 0 and κ (t₁, t₂) = τ > 0 for t₁, t₂ ≥ 0 and σ > 0. Then, for each x ∈ X with p _α  (x, Tx) >0, i.e., x > 0, there exists y = 0 such that conditions (2.7) and (2.8) in Theorem 2.2 are satified. Indeed, since p _α  (y, Ty) =0, the inequality (2.7) is automatically satisfied. Now, with x > 0, the inequality (2.8) is rewritten as F 2 ( x ) ≤ F 1 ( 3 x ) + σ , or, ln ( x ) ≤ ln ( 3 x ) + σ .

Thus, (2.8) is satified. Therefore, all conditions of Theorem 2.2 are satified. Applying Theorem 2.2, T has a fuzzy fixed point. In fact, 0 is the fuzzy fixed point of the fuzzy mapping T.

Notice that, we cannot apply Theorem 9 in [1] to this example.

In Theorem 2.2, if κ is a suitable constant function and F₁ (t) = F₂ (t) = ln t, then we obtain the following corollary.

Corollary 2.4. Let (X, d) be a complete metric space and α ∈ [0, 1]. Let T : X → I ^X be a fuzzy mapping such that [Tx]  _α is a nonempty closed subset of X for all x ∈ X. Assume that there exist r > 1, q > 0 such that for any x ∈ X there exists y ∈ X satisfying rp α ( y , Ty ) ≤ p α ( x , Tx ) , and d ( x , y ) ≤ qp α ( x , Tx ) .

Then, there exists x ∈ X such that x _α  ⊂ Tx provided that the function x → p _α  (x, Tx) is lower semicontinuous.

The following corollary is a generalization of Theorem 2.1 in [20] and so it generalizes results in [4].

Corollary 2.5. Let (X, d) be a complete metric space and R : X → C (X). Assume that there exist κ ∈ K , σ ∈ ℝ , a nondecreasing function F 1 : ( 0 , ∞ ) → ℝ and a function F 2 : ( 0 , ∞ ) → ℝ satisfying conditions (F2), (F3), such that for any x ∈ X with d (x, Rx) >0, there exists y ∈ X satisfying κ ( d ( x , Rx ) , d ( y , Ry ) ) + G 1 ( d ( y , Ry ) ) ≤ F 1 ( d ( x , Rx ) ) , and G 2 ( d ( x , y ) ) ≤ F 1 ( d ( x , Rx ) ) + σ , where G i : [ 0 , ∞ ) → ℝ ∪ { - ∞ } , i = 1, 2, are defined by G i ( t ) = { F i ( t ) if t > 0 - ∞ if t = 0

Then, there exists x ∈ X such that x ∈ Rx provided that the function d (x, Rx) is lower semicontinuous.

Proof. Let α ∈ (0, 1]. Consider the fuzzy mapping T : X → I ^X defined by ( Tx ) ( y ) = { α if y ∈ Rx 0 if y ∉ Rx . Then, we have [ Tx ] α = { y ∈ X : ( Tx ) ( y ) ≥ α } = Rx . Thus, p α ( x , Tx ) = inf y ∈ [ Tx ] α d ( x , y ) = inf y ∈ Rx d ( x , y ) = d ( x , Rx ) . Applying Theorem 2.2, there exists x ∈ X such that (Tx) (x) ≥ α, i.e., x ∈ Rx.

Remark 2.1. If we take κ (t₁, t₂) = τ > 0 for all t₁, t₂ ≥ 0 in Corollary 2.5, then we obtain Theorem 2.1 in [20].

It is evident that fixed point theory has become one of the most important tools in functional analysis because of its great application in proving the existence of solutions to operator equations including differential equations, difference equations, integral equations, matrix equations, functional equations, etc., (see, e.g., [1, 25] and references therein). This section is devoted to the existence for solutions of a system of functional equations which arises in the multistage process related to dynamic programming.

Generally, a dynamic process consist of a state space (the set of the initial state, actions and transition model of the process) and a decision space (the set of possible actions that are allowed for the process). In this section, let U and V are Banach spaces, W ⊂ U is a state space and D ⊂ V is a decision space. We consider a multistage process reduced to the following system of functional equations: q i ( x ) = sup y ∈ D { g ( x , y ) + G i ( x , y , q i ( φ ( x , y ) ) ) } , x ∈ W(3.17) where φ : W × D → W, g : W × D → ℝ , G i : W × D × ℝ → ℝ , i = 1, 2, are given mappings. We refer the reader to, e.g., [6, 22] for a more detailed explanation of the background of the problem.

Our aim is to prove the existence of solutions for a system of functional equations (3.17) under certain conditions. In order to obtain our result, we need the following assumptions:

The mappings g, G₁ and G₂ are bounded.

Remark 3.1. Assumption (A2) is inspired by an assumption due to D. Wardowski in [25] where he studied the existence of solutions of an integral equation of Volterra type.

Let B (W) denote the set of all bounded real-valued functions on W. For x ∈ B (W), we define a norm in B (W) as: | | h | | = sup x ∈ W | h ( x ) | . Then, (B (W) , || · ||) is a Banach space. Indeed, the convergence in B (W) with respect to || · || is uniform and so, if {h _n } is a Cauchy sequence in B (W), then it converges uniformly to a function, say h^*, such that h^* ∈ B (W).

Our main result of this section is stated as follows.

Theorem 3.1. If (A1) and (A2) are satisfied, then the system of functional equations (3.17) has a bounded solution.

Proof. We consider operators T _i  : B (W) → B (W), i = 1, 2, defined by: for h ∈ B (W), x ∈ W T i h ( x ) = sup y ∈ D { g ( x , y ) + G i ( x , y , h ( φ ( x , y ) ) ) } .

These mappings are well-defined since g, G₁ and G₂ are bounded.

Now, fix n ≥ 2 and take h₁, h₂ ∈ B (W) such that γ_n-1 ≤ ||h₁ - h₂|| < γ _n . Then, we have

1 + | | h 1 - h 2 | | ( γ n - | | h 1 - h 2 | | ) < 1 + γ n ( γ n - γ n - 1 ) .(3.19)

For any ɛ > 0 and x ∈ W, there exist y₁, y₂ ∈ D such that T 1 h 1 ( x ) < g ( x , y 1 ) + G 1 ( x , y 1 , h 1 ( φ ( x , y 1 ) ) ) + ɛ ,(3.20) T 2 h 2 x < g ( x , y 2 ) + G 2 ( x , y 2 , h 2 ( φ ( x , y 2 ) ) ) + ɛ ,(3.21) T 1 h 1 x ≥ g ( x , y 2 ) + G 1 ( x , y 2 , h 1 ( φ ( x , y 2 ) ) ) ,(3.22) and T 2 h 2 ( x ) ≥ g ( x , y 1 ) + G 2 ( x , y 1 , h 2 ( φ ( x , y 1 ) ) ) .(3.23)