Fixed point theorems for L -fuzzy mappings in quasi-pseudo metric spaces

1 Introduction

In 1965 Zadeh introduced the concept of fuzzy sets in his famous paper “Fuzzy sets” [21]. Then Heilpern [10] initiated the idea of fuzzy contraction mappings and proved a fixed point theorem for fuzzy contraction mappings which is a fuzzy analogue of Nadler’s [12] fixed point theorem for multivalued mappings. In 2014, Azam and Rashid, [5] proved a coincidence theorem for a pair of fuzzy mappings satisfying a Jungck type contractive condition which generalized Heilpern’s fuzzy contraction theorem. Nashine et al. [13] presented some common fixed point results for a pair of fuzzy mappings satisfying an almost generalized contractive condition in partially ordered complete metric spaces. Recently, Qiu and Li [16] presented the concepts of gradient and convexity of fuzzy mappings on the basis of an order in the quotient space of fuzzy numbers. The authors also discussed the fuzzy optimizations of differentiable fuzzy mappings and convex fuzzy mappings. Subsequently, several other authors studied the fixed points, coincidence points and the common fixed points of fuzzy contractive mappings in metric spaces (e.g see [1–4, 20]).

Afterwards in 1967 Goguen [8] generalized the idea of fuzzy sets in form of another notion of L-fuzzy sets. The concept of fuzzy sets is a special case ofL-fuzzy sets when L = [0, 1]. In 2014, Phiangsungnoen et al. [14, 15] utilized the notion of β-admissible and β _F -admissible pair to prove fuzzy fixed point theorems. Then Rashid et al. [17] introduced the concept of β _{F L} -admissible for a pair of L-fuzzy mappings and establish the existence of a common L-fuzzy fixed point theorem.

In 1963, Kelly [11] gave a definition of Cauchy sequence for a quasi-pseudo-metric space, and proved a generalization of the Baire category theorem. This definition was further generalized by Reilly et al. [18] in seven different notions. In [6] Azam et al. proved some local versions of fixed point theorems involving fuzzy contractive mappings in left K-sequentially complete quasi-pseudo-metric spaces and right K-sequentially complete quasi-pseudo-metric spaces which was the extension of the concepts given in [9, 19].

In this paper, we establish some local versions of fixed point theorems involving L- fuzzy contractive mappings in left K-sequentially and right K-sequentially complete quasi-pseudo metric spaces. These results are based on the fact that sometimes contractions are defined on the subsets except on the whole space.

2 Preliminaries

Definition 2.1. [6] A quasi-pseudo metric on a non empty set X is a non-negative real valued function d on X × X such that, for all x, y, z ∈ X, satisfying the following:

d (x, x) =0

Then (X, d) is called as quasi-pseudo metric space.

Each quasi-pseudo metric d on X induces a topology τ (d) which has as a base the famaily of all d-balls B _ɛ  (x), where B ɛ ( x ) = { y ∈ X : d ( x , y ) < ɛ } . If d is a quasi-pseudo metric on X, then the function d^-1, on X × X defined as d^-1 (x, y) = d (y, x), is also a quasi-pseudo metric on X. Here, d∧ d^-1 = min { d, d^-1 } and d∨ d^-1 = max { d, d^-1 }.

Definition 2.2. [18] Let (X, d) be a quasi-metric space.

A sequence {x _n } where n ∈ N in X is said to be left K-Cauchy if for each ɛ > 0 there is k ∈ N, such that d (x _n , x _m ) < ɛ, for all m ∈ N with m ≥ n ≥ k .

Definition 2.3. [6] Let (X, d) be a quasi-pseduo metric space. Let A and B be the two nonempty subsets of X. Then the Hausdorff distance between the subsets A and B is defined as, H ( A , B ) = max { sup a ∈ A d ( a , B ) , sup b ∈ B d ( b , A ) } , where d ( a , B ) = inf { d ( x , a ) : x ∈ B } . Clearly, H is the usual Hausdorff distance if d is a metric on X.

Definition 2.4. [8] A partially ordered set (L, ⪯  _L ) is called

a lattice, if a ∨ b ∈ L, a ∧ b ∈ L, for any a, b∈ L;

Definition 2.5. [8] Let L be a lattice with top element 1 _L and bottom element 0 _L for 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. It is denoted by.

Definition 2.6. [8] An L-fuzzy set A on a nonempty set X is a function A : X → L, where L is complete distributive lattice with 1 _L and 0 _L .

Remark 2.7. [8] The class of L-fuzzy sets is larger than the class of fuzzy set. Also an L-fuzzy set is a fuzzy set if L = [0, 1].

The α _L -level set of L-fuzzy set A is denoted and defined as, A α L = { x : α L ⪯ L A ( x ) } if α L ∈ L ∖ { 0 L } A 0 L = cl ( { x : 0 L ⪯ L A ( x ) } ) Here, cl (B) denotes the closure of the set B.

On the basis of classes defined in [9] we are defining the following classes for L-fuzzy sets.

Definition 2.8. Let (X, d) be a quasi-pseduo metric space and (V, d _V ) be a metric linear space. The families W L ∗ ( X ) and W L ′ ( X ) of a L-fuzzy sets on (X, d) and W _L  (V) on (V, d _V ) are defined as: ( i ) W L ∗ ( X ) = { A ∈ L X : A 1 L is a nonempty d - closed and d - 1 compact } , ( ii ) W L ′ ( X ) = { A ∈ L X : A 1 L is a nonempty d - closed and d compact } , ( iii ) W L ( V ) = { A ∈ L V : A α L is compact and convex in V for each α L ∈ L and sup x ∈ V A ( x ) = 1 L } .

Definition 2.9. Let (X, d) be a quasi-pseduo metric space and let A , B ∈ W L ∗ ( X ), or W L ′ ( X ) and α _L  ∈ L. Then, we define D α L ( A , B ) = H ( A α L , B α L ) where the Hausdorff metric H is deduced from the quasi-pseudo metric d on X, and D ( A , B ) = sup { D α L ( A , B ) : α L ∈ L ∖ { 0 } } .

Definition 2.10. Let X be any nonempty set and Y be a quasi-pseudo metric space. T is said to be L-fuzzy mapping if T is a mapping from X into W L ∗ ( Y ) or ( W L ′ ( Y ) ).

Definition 2.11. We say that x is a fixed point of a L-fuzzy mapping F _L  : X → L ^X if {x} ⊂ F _L  (x).

In the following we are modifying the lemmas, given in [6], for L-fuzzy mapping.

Lemma 2.12. Let (X, d) be a quasi-pseudo metric space and let x ∈ X and A ∈ W L ∗ ( X ) or W L ′ ( X ). Then {x} ⊆ A if and only if d ( x , [ A ] 1 L ) = 0 or d ( [ A ] 1 L , x ) = 0

Lemma 2.13. Let (X, d) be a quasi-pseudo metric space and let A ∈ W L ∗ ( X ) or W L ′ ( X ). Then, d ( x , [ A ] α L ) ≤ d ( x , y ) + d ( y , [ A ] α L ) or d ( [ A ] α , x ) ≤ d ( [ A ] α L , y ) + d ( y , x ) for any x, y ∈ X and α _L  ∈ L ∖ {0 _L }.

Lemma 2.14. Let (X, d) be a quasi-pseudo metric space and let {x₀} ⊆ A. Then, d ( x 0 , [ B ] α L ) ≤ D α L ( A , B ) or d ( [ B ] α L , x 0 ) ≤ D α L ( B , A ) for each A , B ∈ W L ∗ ( X ) or W L ′ ( X ) and α _L  ∈ L ∖ {0 _L }.

Lemma 2.15. [9] Suppose K ≠φ is compact in the quasi-pseudo metric space (X, d^-1) or (X, d). If z ∈ X, then there exists k₀ ∈ K such that, d ( z , K ) = d ( z , k 0 ) or d ( K , z ) = d ( k 0 , z ) .

3 Fixed point theorems for L-fuzzy contractive maps

In the present section, we prove the local versions of fixed point results for L-fuzzy contraction mappings in a left (right) K-sequentially complete quasi-pseudo metric space.

Theorem 3.1. Let (X, d) be a left K-sequentially complete quasi-pseudo metric space, x₀ ∈ X, r > 0 and T : X → W L ∗ ( X ) be a L-fuzzy mapping. If there exists k ∈ (0, 1) such that, D ( T ( x ) , T ( y ) ) ≤ k ( d - 1 ∧ d ) ( x , y ) for each x , y ∈ B ¯ d ( x 0 , r ), and d ( x 0 , [ Tx 0 ] 1 L ) < ( 1 - k ) r where 1 _L  ∈ L, then there exists x ∗ ∈ B ¯ d ( x 0 , r ) such that {x^∗} ⊂ Tx^∗.

Proof. We apply Lemma 2.15 to the nonempty d^-1 compact set K = [Tx₀] _{1 L} and x₀ to find x₁ ∈ [Tx₀] _{1 L} such that, d ( x 0 , x 1 ) = d ( x 0 , [ Tx 0 ] 1 L ) < ( 1 - k ) r It also implies that x 1 ∈ B ¯ d ( x 0 , r ).

By Lemma 2.15 choose x₂ ∈ [Tx₁] _{1 L} such that, d ( x 1 , x 2 ) = d ( x 1 , [ Tx 1 ] 1 L ) ≤ D 1 L ( T ( x 0 ) , T ( x 1 ) ) ≤ D ( T ( x 0 ) , T ( x 1 ) ) ≤ k ( d - 1 ∧ d ) ( x 0 , x 1 ) ≤ kd ( x 0 , x 1 ) < k ( 1 - k ) r We can show that, x 2 ∈ B ¯ d ( x 0 , r ) since d ( x 0 , x 2 ) ≤ d ( x 0 , x 1 ) + d ( x 1 , x 2 ) < ( 1 - k ) r + k ( 1 - k ) r < ( 1 - k ) ( 1 + k ) r By Lemma 2.15, choose x₃ ∈ [Tx₂] _{1 L} such that, d ( x 2 , x 3 ) = d ( x 2 , [ Tx 2 ] 1 L ) ≤ D 1 L ( T ( x 1 ) , T ( x 2 ) ) ≤ D ( T ( x 1 ) , T ( x 2 ) ) ≤ k ( d - 1 ∧ d ) ( x 1 , x 2 ) ≤ kd ( x 1 , x 2 ) ≤ k 2 d ( x 0 , x 1 ) < k 2 ( 1 - k ) r We can show that, x 3 ∈ B ¯ d ( x 0 , r ) since d ( x 0 , x 3 ) ≤ d ( x 0 , x 1 ) + d ( x 1 , x 2 ) + d ( x 2 , x 3 ) < ( 1 - k ) r + k ( 1 - k ) r + k 2 ( 1 - k ) r < ( 1 - k ) ( 1 + k + k 2 ) r We follow the same procedure to obtain, {x _n } ⊂ Tx_n-1 such that d ( x n - 1 , x n ) < k n - 1 d ( x 0 , x 1 ) < k n - 1 ( 1 - k ) r for n = 3 , 4 , 5 , . . . . . . Now, we verify that {x _n } is a left K-Cauchy sequence, for n < m, we have d ( x n , x m ) ≤ ∑ i = 0 m - n - 1 d ( x n + i , x n + i + 1 ) < ∑ i = n m - 1 k i d ( x 0 , x 1 ) < ( k n 1 - k ) d ( x 0 , x 1 ) As, k ∈ (0, 1) and (X, d) is a left K-sequentially complete quasi-pseudo metric space, this implies that {x _n } is a left K-Cauchy sequence in X. Therefore, there exists x ∗ ∈ B ¯ d ( x 0 , r ) such that lim n → ∞ x n = x ∗.

Now, from Lemma 2.13 and Lemma 2.14, we get d ( x ∗ , [ Tx ∗ ] 1 L ) ≤ d ( x ∗ , x n ) + d ( x n , [ Tx ∗ ] 1 L ) ≤ d ( x ∗ , x n ) + D 1 L ( T ( x n - 1 ) , T ( x ∗ ) ) ≤ d ( x ∗ , x n ) + D ( T ( x n - 1 ) , T ( x ∗ ) ) ≤ d ( x ∗ , x n ) + k ( d - 1 ∧ d ) ( x n - 1 , x ∗ ) ≤ d ( x ∗ , x n ) + kd - 1 ( x n - 1 , x ∗ ) ≤ d ( x ∗ , x n ) + kd ( x ∗ , x n - 1 ) . Since d ( x ∗ , x n ) and d ( x ∗ , x n - 1 ) → 0 } n → ∞ , therefore we have d ( x ∗ , [ Tx ∗ ] 1 L ) = 0 . So, by Lemma 2.12 we get, { x ∗ } ⊂ Tx ∗ .

Example 3.2. Let X = R ∪ {ℓ}, where ℓ ∉ R. Define d : X × X → [0, ∞) by d (x, y) = |x - y|, for all x, y ∈ R, d (ℓ , ℓ) =0, d ( x , ℓ ) = { - x + 1 2 , x < - 1 2 1 , - 1 2 ≤ x ≤ 1 2 x + 1 , x > 1 2 } and d ( ℓ , x ) = { - x - 1 2 , x < - 1 2 0 , - 1 2 ≤ x ≤ 1 2 x - 1 2 , x > 1 2 } then (X, d) is a left K-sequentially complete quasi-pseudo metric space. Now T : X → W L ∗ ( X ) defined as T ( x ) = { χ L { − 1 2 − x } , x < − 1 2 χ L { − x 2 } , − 1 2 ≤ x ≤ 1 2 χ L { x − 1 2 } , x > 1 2 χ L { 0 } , x = ℓ } is a L- fuzzy mapping. For α _L  ∈ L ∖ {0 _L }, [ T x ] α L = { t ∈ X : [ T x ] ( t ) ≥ α L } = { t ∈ X : χ L { − 1 2 − x } ( t ) ≥ L α L , x < − 1 2 t ∈ X : χ L { − x 2 } ( t ) ≥ L α L , − 1 2 ≤ x ≤ 1 2 t ∈ X : χ L { x − 1 2 } ( t ) ≥ L α L , x > 1 2 t ∈ X : χ L { 0 } ( t ) ≥ L α L , x = ℓ } = { − 1 2 − x , x < − 1 2 − x 2 , − 1 2 ≤ x ≤ 1 2 x − 1 2 , x > 1 2 0 , x = ℓ % } . Define D L : W L ∗ ( X ) × W L ∗ ( X ) → [ 0 , ∞ ) by D L ( T ( x ) , T ( y ) ) = sup α L ∈ L ∖ { 0 L } H ( [ Tx ] α L , [ Ty ] α L ) for each x , y ∈ B ¯ d ( 0 , 1 2 ) where H ( [ Tx ] α L , [ Ty ] α L ) = max { sup a ∈ [ Tx ] α L d ( a , [ Ty ] α L ) , sup b ∈ [ Ty ] α L d ( b , [ Tx ] α L ) } . Now, for k = 3 4, D L ( T ( x ) , T ( y ) ) ≤ k ( d - 1 ∧ d ) ( x , y ) for each x , y ∈ B ¯ d ( 0 , 1 2 ) and d ( 0 , [ T 0 ] 1 L ) < ( 1 - k ) . Then, 0 ∈ B ¯ d ( 0 , 1 2 ) such that {0} ⊂ T0.

Note that the L-fuzzy mapping defined in the above example is not contractive on the whole space. For example, when x and y are not in the interval [ - 1 2 , 1 2 ] .

Theorem 3.3. Let (X, d) be a right K-sequentially complete quasi-pseudo metric space, x₀ ∈ X, r > 0 and T : X → W L ′ ( X ) be a L-fuzzy mapping. If there exists k ∈ (0, 1) such that, D ( T ( x ) , T ( y ) ) ≤ k ( d - 1 ∧ d ) ( x , y ) for each x , y ∈ B ¯ d ( x 0 , r ) and d ( T [ x 0 ] 1 L , x 0 ) < ( 1 - k ) r where 1 _L  ∈ L, then there exists x ∗ ∈ B ¯ d ( x 0 , r ) such that {x^∗} ⊂ Tx^∗.

Proof. The proof is similar to the proof of Theorem 3.1, therefore omitted.

The following corollary is the generalization of the result given by Gregori and Pastor in [9].

Corollary 3.4. Let (X, d) be a left K-sequentially complete quasi-pseudo metric space and T : X → W L ∗ ( X ) be a L- fuzzy mapping. If there exists k ∈ (0, 1) such that, D ( T ( x ) , T ( y ) ) ≤ k ( d ∧ d - 1 ) ( x , y ) for each x, y ∈ X, then there exists x^∗ ∈ X such that {x^∗} ⊂ Tx^∗.

Theorem 3.5. Let (X, d) be a left K-sequentially complete quasi-pseudo metric space, x₀ ∈ X, r > 0 and T : X → W L ∗ ( X ) be a L-fuzzy mapping. If there exists k ∈ ( 0 , 1 2 ) such that D 1 L ( T ( x ) , T ( y ) ) ≤ k max { ( d - 1 ∧ d ) ( x , y ) , d ( x , [ Tx ] 1 L ) + d ( y , [ Ty ] 1 L ) } for each x , y ∈ B ¯ d ( x 0 , r ) and d ( x 0 , [ Tx 0 ] 1 L ) < ( 1 - k ) r where 1 _L  ∈ L,then there exists x ∗ ∈ B ¯ d ( x 0 , r ) such that {x^∗} ⊂ Tx^∗.

Proof. We apply Lemma 2.15 to the nonempty d^-1 compact set K = [Tx₀] _{1 L} and x₀ to find x₁ ∈ [Tx₀] _{1 L} such that, d ( x 0 , x 1 ) = d ( x 0 , [ Tx 0 ] 1 L ) < ( 1 - k ) r .(1) It also implies that x 1 ∈ B ¯ d ( x 0 , r ).

By Lemma 2.15 choose x₂ ∈ [Tx₁] _{1 L} such that, d ( x 1 , x 2 ) = d ( x 1 , [ Tx 1 ] 1 L ) ≤ D 1 L ( T ( x 0 ) , T ( x 1 ) ) ≤ k max { ( d - 1 ∧ d ) ( x 0 , x 1 ) , d ( x 0 , [ Tx 0 ] 1 L ) + d ( x 1 , [ Tx 1 ] 1 L ) } ≤ k max { d ( x 0 , x 1 ) , d ( x 0 , x 1 ) + d ( x 1 , x 2 ) } .(2) Now, we consider the following cases:

Case 1: If we take d (x₀, x₁) as a maximum in above inequality (2) and use the inequality (1), we get d ( x 1 , x 2 ) ≤ kd ( x 0 , x 1 ) < k ( 1 - k ) r . Case 2: If we take d (x₀, x₁) + d (x₁, x₂) as a maximum in above inequality (2), we have d ( x 1 , x 2 ) ≤ k { d ( x 0 , x 1 ) + d ( x 1 , x 2 ) } ≤ ( k 1 - k ) d ( x 0 , x 1 ) . Since, ( k 1 - k ) < k, then by using the inequality (1), we have d ( x 1 , x 2 ) ≤ kd ( x 0 , x 1 ) < k ( 1 - k ) r . It follows from above two cases that, d ( x 1 , x 2 ) < k ( 1 - k ) r . Now, we can show that x 2 ∈ B ¯ d ( x 0 , r ), since d ( x 0 , x 2 ) ≤ d ( x 0 , x 1 ) + d ( x 1 , x 2 ) < ( 1 - k ) r + k ( 1 - k ) r < ( 1 - k ) ( 1 + k ) r . Following the same way, we get {x _n } ⊂ Tx_n-1, such that d ( x n - 1 , x n ) < k n - 1 d ( x 0 , x 1 ) < k n - 1 ( 1 - k ) r for n = 3 , 4 , 5 , . . . . . . Now, we verify that {x _n } is a left K-Cauchy sequence, for n < m, we have d ( x n , x m ) ≤ ∑ i = 0 m - n - 1 d ( x n + i , x n + i + 1 ) < ∑ i = n m - 1 k i d ( x 0 , x 1 ) < ( k n 1 - k ) d ( x 0 , x 1 ) . As, k ∈ ( 0 , 1 2 ) and (X, d) is a left K-sequentially complete quasi-pseudo metric space, this implies that {x _n } is a left K-Cauchy sequence in X. Therefore, there exists x ∗ ∈ B ¯ d ( x 0 , r ) such that lim n → ∞ x n = x ∗.

Now, from Lemma 2.13 and Lemma 2.14, we get d ( x ∗ , [ Tx ∗ ] 1 L ) ≤ d ( x ∗ , x n ) + d ( x n , T [ x ∗ ] 1 L ) ≤ d ( x ∗ , x n ) + D 1 L ( T ( x n - 1 ) , T ( x ∗ ) ) ≤ d ( x ∗ , x n ) + k max { ( d - 1 ∧ d ) ( x n - 1 , x ∗ ) , d ( x n - 1 , [ Tx n - 1 ] 1 L ) + d ( x ∗ , [ Tx ∗ ] 1 L ) } ≤ d ( x ∗ , x n ) + k max { d - 1 ( x n - 1 , x ∗ ) , d ( x n - 1 , x n - 2 ) + d ( x ∗ , [ Tx ∗ ] 1 L ) } ≤ d ( x ∗ , x n ) + k max { d ( x ∗ , x n - 1 ) , d ( x n - 1 , x n - 2 ) + d ( x ∗ , [ Tx ∗ ] 1 L ) } . Since d ( x ∗ , x n ) , d ( x ∗ , x n - 1 ) and d ( x n - 1 , x n - 2 ) → 0 as n → ∞ so we have d ( x ∗ , [ Tx ∗ ] 1 L ) ≤ k max { d ( x ∗ , [ Tx ∗ ] 1 L ) } ≤ kd ( x ∗ , [ Tx ∗ ] 1 L ) which implies d ( x ∗ , [ Tx ∗ ] 1 L ) = 0 . So, by lemma 2.12 we get, { x ∗ } ⊂ Tx ∗ .

Example 3.6. Let (X, d) be the left K-sequentially complete quasi-pseudo metric space of Example 3.2. Now, T : X → W L ∗ ( X ) defined as T ( x ) = { χ L { − 1 2 − x } , x < − 1 2 χ L { − x 3 } , − 1 2 ≤ x ≤ 1 2 χ L { x − 1 2 } , x > 1 2 χ L { 0 } , x = ℓ } is a L- fuzzy mapping. For α _L  ∈ L ∖ {0 _L }, [ T x ] α L = { t ∈ X : [ T x ] ( t ) ≥ α L } = { t ∈ X : χ L { − 1 2 − x } ( t ) ≥ L α L , x < − 1 2 t ∈ X : χ L { − x 3 } ( t ) ≥ L α L , − 1 2 ≤ x ≤ 1 2 t ∈ X : χ L { x − 1 2 } ( t ) ≥ L α L , x > 1 2 t ∈ X : χ L { 0 } ( t ) ≥ L α L , x = ℓ } = { − 1 2 − x , x < − 1 2 − x 3 , − 1 2 ≤ x ≤ 1 2 x − 1 2 , x > 1 2 0 , x = ℓ % } . Define D 1 L : W L ∗ ( X ) × W L ∗ ( X ) → [ 0 , ∞ ) by D 1 L ( T ( x ) , T ( y ) ) = H ( [ T x ] 1 L , [ T y ] 1 L ) foreach x , y ∈ B ¯ d ( 0 , 1 2 ) where H ( [ Tx ] α L , [ Ty ] α L ) = max { sup a ∈ [ Tx ] α L d ( a , [ Ty ] α L ) , sup b ∈ [ Ty ] α L d ( [ Tx ] α L , b ) Now, for k = 1 3, D 1 L ( T ( x ) , T ( y ) ) ≤ k max { ( d - 1 ∧ d ) ( x , y ) , d ( x , [ Tx ] 1 L ) + d ( y , [ Ty ] 1 L ) } for each x , y ∈ B ¯ d ( 0 , 1 2 ) and d ( 0 , [ T 0 ] 1 L ) < ( 1 - k ) . Then, 0 ∈ B ¯ d ( 0 , 1 2 ) such that {0} ⊂ T0.

Theorem 3.7. Let (X, d) be a right K-sequentially complete quasi-pseudo metric space, x₀ ∈ X, r > 0 and T : X → W L ′ ( X ) be a L-fuzzy mapping. If there exists k ∈ ( 0 , 1 2 ) such that D 1 L ( T ( x ) , T ( y ) ) ≤ k max { ( d - 1 ∧ d ) ( x , y ) , d ( [ Tx ] 1 L , x ) + d ( [ Ty ] 1 L , y ) } for each x , y ∈ B ¯ d ( x 0 , r ) and d ( [ Tx 0 ] 1 L , x 0 ) < ( 1 - k ) r where 1 _L  ∈ L, then there exists x ∗ ∈ B ¯ d ( x 0 , r ) such that {x^∗} ⊂ Tx^∗.

Proof. The proof is similar to the proof of Theorem 3.5, therefore omitted.

Theorem 3.8. Let (X, d) be a left K-sequentially complete quasi-pseudo metric space, x₀ ∈ X, r > 0 and T : X → W L ∗ ( X ) be a L-fuzzy mapping. If there exists k ∈ ( 0 , 1 2 ) such that D 1 L ( T ( x ) , T ( y ) ) ≤ k max { ( d - 1 ∧ d ) ( x , y ) , d ( x , [ Ty ] 1 L ) + d ( y , [ Tx ] 1 L ) } for each x , y ∈ B ¯ d ( x 0 , r ) and d ( x 0 , [ Tx 0 ] 1 L ) < ( 1 - k ) r where 1 _L  ∈ L, then there exists x ∗ ∈ B ¯ d ( x 0 , r ) such that {x^∗} ⊂ Tx^∗.

Proof. We apply Lemma 2.15 to the nonempty d^-1 compact set K = [Tx₀] _{1 L} and x₀ to find x₁ ∈ [Tx₀] _{1 L} such that, d ( x 0 , x 1 ) = d ( x 0 , [ Tx 0 ] 1 L ) < ( 1 - k ) r .(3) It also implies that x 1 ∈ B ¯ d ( x 0 , r ).

By Lemma 2.15 choose x₂ ∈ [Tx₁] _{1 L} such that d ( x 1 , x 2 ) = d ( x 1 , [ Tx 1 ] 1 L ) ≤ D 1 L ( T ( x 0 ) , T ( x 1 ) ) ≤ k max { ( d - 1 ∧ d ) ( x 0 , x 1 ) , d ( x 0 , [ Tx 1 ] 1 L ) + d ( x 1 , [ Tx 0 ] 1 L ) } ≤ k max { d ( x 0 , x 1 ) , d ( x 0 , x 1 ) + d ( x 1 , x 2 ) } ≤ k max { d ( x 0 , x 1 ) , d ( x 0 , x 1 ) + d ( x 1 , x 2 ) } .(4) Now, we consider the following cases:

Case 1: If we take d (x₀, x₁) as a maximum in above inequality (4) and use the inequality (3), we get d ( x 1 , x 2 ) ≤ kd ( x 0 , x 1 ) < k ( 1 - k ) r .

Case 2: If we take d (x₀, x₁) + d (x₁, x₂) as a maximum in above inequality (4), we have d ( x 1 , x 2 ) ≤ k { d ( x 0 , x 1 ) + d ( x 1 , x 2 ) } ≤ ( k 1 - k ) d ( x 0 , x 1 ) . Since ( k 1 - k ) < k, then by using the inequality (3), we have d ( x 1 , x 2 ) ≤ kd ( x 0 , x 1 ) < k ( 1 - k ) r . It follows from above two cases that, d ( x 1 , x 2 ) < k ( 1 - k ) r . Now, we can show that x 2 ∈ B ¯ d ( x 0 , r ). Since d ( x 0 , x 2 ) ≤ d ( x 0 , x 1 ) + d ( x 1 , x 2 ) < ( 1 - k ) r + k ( 1 - k ) r < ( 1 - k ) ( 1 + k ) r . Following the same way, we get {x _n } ⊂ Tx_n-1, such that d ( x n - 1 , x n ) < k n - 1 d ( x 0 , x 1 ) < k n - 1 ( 1 - k ) r for n = 3 , 4 , 5 , . . . . . . Now, we verify that {x _n } is a left K-Cauchy sequence, for n < m, we have d ( x n , x m ) ≤ ∑ i = 0 m - n - 1 d ( x n + i , x n + i + 1 ) < ∑ i = n m - 1 k i d ( x 0 , x 1 ) < ( k n 1 - k ) d ( x 0 , x 1 ) . As, k ∈ ( 0 , 1 2 ) and (X, d) is a left K-sequentially complete quasi-pseudo metric space, this implies that {x _n } is a left K-Cauchy sequence in X. Therefore, there exists x ∗ ∈ B ¯ d ( x 0 , r ) such that lim n → ∞ x n = x ∗.

Now, from Lemma 2.13 and Lemma 2.14, we get d ( x ∗ , [ Tx ∗ ] 1 L ) ≤ d ( x ∗ , x n ) + d ( x n , [ Tx ∗ ] 1 L ) ≤ d ( x ∗ , x n ) + D 1 L ( T ( x n - 1 ) , T ( x ∗ ) ) ≤ d ( x ∗ , x n ) + k max { ( d - 1 ∧ d ) ( x n - 1 , x ∗ ) , d ( x n - 1 , [ Tx ∗ ] 1 L ) + d ( x ∗ , [ Tx n - 1 ] 1 L ) } ≤ d ( x ∗ , x n ) + k max { d - 1 ( x n - 1 , x ∗ ) , d ( x n - 1 , [ Tx ∗ ] 1 L ) + d ( x ∗ , x n - 2 ) } ≤ d ( x ∗ , x n ) + k max { d ( x ∗ , x n - 1 ) , d ( x n - 1 , [ Tx ∗ ] 1 L ) + d ( x ∗ , x n - 2 ) } . Since, d ( x ∗ , x n ) , d ( x ∗ , x n - 1 ) and d ( x ∗ , x n - 2 ) → 0 as n → ∞ so we have d ( x ∗ , [ Tx ∗ ] 1 L ) ≤ k max { d ( x ∗ , T [ x ∗ ] 1 L ) } ≤ kd ( x ∗ , T [ x ∗ ] 1 L ) which implies d ( x ∗ , [ Tx ∗ ] 1 L ) = 0 . So, by Lemma 2.12 we get, { x ∗ } ⊂ Tx ∗ .

Example 3.9. Let (X, d) be the left K-sequentially complete quasi-pseudo metric space of Example 3.2. Now, T : X → W L ∗ ( X ) defined as T ( x ) = { χ L { − 1 2 − x } , x < − 1 2 χ L { − x 4 } , − 1 2 ≤ x ≤ 1 2 χ L { x − 1 2 } , x > 1 2 χ L { 0 } , x = ℓ } is a L- fuzzy mapping. For α _L  ∈ L ∖ {0 _L }, [ T x ] α L = { t ∈ X : [ T x ] ( t ) ≥ α L } = { t ∈ X : χ L { − 1 2 − x } ( t ) ≥ L α L , x < − 1 2 t ∈ X : χ L { − x 4 } ( t ) ≥ L α L , − 1 2 ≤ x ≤ 1 2 t ∈ X : χ L { x − 1 2 } ( t ) ≥ L α L , x > 1 2 t ∈ X : χ L { 0 } ( t ) ≥ L α L , x = ℓ } = { − 1 2 − x , x < − 1 2 − x 4 , − 1 2 ≤ x ≤ 1 2 x − 1 2 , x > 1 2 0 , x = ℓ % } .

Now, for k = 1 4, D 1 L ( T ( x ) , T ( y ) ) ≤ k max { ( d - 1 ∧ d ) ( x , y ) , d ( x , [ Ty ] 1 L ) + d ( y , [ Tx ] 1 L ) for each x , y ∈ B ¯ d ( 0 , 1 2 ) and d ( 0 , [ T 0 ] 1 L ) < ( 1 - k ) . Then, 0 ∈ B ¯ d ( 0 , 1 ) such that {0} ⊂ T0.

Theorem 3.10. Let (X, d) be a right K-sequentially complete quasi-pseudo metric space, x₀ ∈ X, r > 0 and T : X → W L ′ ( X ) be a L-fuzzy mapping. If there exists k ∈ ( 0 , 1 2 ) such that D 1 L ( T ( x ) , T ( y ) ) ≤ k max { ( d - 1 ∧ d ) ( x , y ) , d ( [ Ty ] 1 L , x ) + d ( [ Tx ] 1 L , y ) } for each x , y ∈ B ¯ d ( x 0 , r ) and d ( [ Tx 0 ] 1 L , x 0 ) < ( 1 - k ) r where 1 _L  ∈ L, then there exists x ∗ ∈ B ¯ d ( x 0 , r ) such that {x^∗} ⊂ Tx^∗.