Next Article in Journal
Independent Rainbow Domination Numbers of Generalized Petersen Graphs P(n,2) and P(n,3)
Next Article in Special Issue
Fuzzy Stability Results of Finite Variable Additive Functional Equation: Direct and Fixed Point Methods
Previous Article in Journal
Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron
Previous Article in Special Issue
A Characterization of Strong Completeness in Fuzzy Metric Spaces
Open AccessArticle

Some New Fuzzy Fixed Point Results with Applications

1
Department of Mathematics, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia
2
Institute of Research and Development of Processes IIDP, Campus of Leioa, University of the Basque Country, Bizkaia 48940, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(6), 995; https://doi.org/10.3390/math8060995
Received: 7 May 2020 / Revised: 12 June 2020 / Accepted: 15 June 2020 / Published: 17 June 2020

Abstract

The aim of this article is to establish some fixed point results for fuzzy mappings and derive some corresponding multivalued mappings results of literature. For this purpose, we define some new and generalized contractions in the setting of b-metric spaces. As applications, we find solutions of integral inclusions by our obtained results.
Keywords: fuzzy mappings; multivalued mappings; integral inclusions; generalized contractions fuzzy mappings; multivalued mappings; integral inclusions; generalized contractions

1. Introduction and Preliminaries

In 1981, Heilpern [1] utilized the approach of fuzzy set to initiate a family of fuzzy mappings which are extensions of multivalued mappings and obtained a result for these mappings in metric linear space. In this paper, we shall use the following notations which have been recorded from [2,3,4,5,6,7,8,9,10,11,12].
A fuzzy set in M is a function with domain M and values in [ 0 , 1 ] , I M is the collection of all fuzzy sets in M . If Θ is a fuzzy set and μ M , then the function values Θ ( μ ) is called the grade of membership of μ in Θ . The α -level set of A is denoted by Θ α and is defined as follows:
Θ α = { μ : Θ ( μ ) α } if α ( 0 , 1 ] ,
Θ 0 = { μ : Θ ( μ ) > 0 } . ¯
Here Θ ¯ denotes the closure of the set Θ . Let F ( M ) be the collection of all fuzzy sets in a metric space M .
Czerwik [13] in 1993 extended the conception of metric space by initiating the notion of b-metric space and obtained the celebrated Banach fixed point theorem in this generalized metric space.
Definition 1.
A b-metric on a nonempty set M is a function d b : M × M R 0 + such that these assertions hold:
  • ( b 1 ) d b ( μ , ϖ ) = 0 μ = ϖ ,
  • ( b 2 ) d b ( μ , ϖ ) = d b ( ϖ , μ ) ,
  • ( b 3 ) d b ( μ , υ ) s ( d b ( μ , ϖ ) + d b ( ϖ , υ ) )
for all μ , ϖ , υ M , where s 1 .
The triple ( M , d b , s ) is said to be a b-metric space. Clearly, every metric space is a b-metric space whenever s = 1 , but the converse need not be true.
Example 1
([13]). Let M = [ 0 , ) and d b : M × M [ 0 , ) defined by
d b ( μ , ϖ ) = | μ ϖ | 2
for all μ , ϖ M . Clearly ( M , d b , 2 ) is a b-metric space, but not a metric space.
Example 2
([14]). Let p ( 0 , 1 ) and l p ( R ) = μ n R : n = 1 μ n p < endowed with the function d b : l p ( R ) × l p ( R ) R defined by
d b μ n , ϖ n = n = 1 μ n ϖ n p 1 p
for each μ n , ϖ n l p ( R ) , is a b-metric space with s = 2 1 p .
Definition 2
([13]). Let ( M , d b , s ) is a b-metric space.
(i) A sequence μ n in M converges to μ M if lim n d b μ n , μ = 0 .
(ii) A sequence μ n in M is a Cauchy sequence, if for each ϵ > 0 there exists a natural number N ( ϵ ) such that d b μ n , μ m < ϵ for each m , n N ( ϵ ) .
(iii) We say that ( M , d b , s ) is a complete if each Cauchy sequence in M converges to some point of M .
Definition 3
([15]). Let ( M , d b , s ) is a b-metric space. A subset A M is said to be open if and only if for any μ A ¯ , there exists ϵ > 0 0 such that the open ball B O ( μ , ϵ ) A . The family of all open subsets of M will be denoted by τ .
Proposition 1
([15]). τ defines a topology on ( M , d b , s ) .
Proposition 2
([15,16]). Let ( M , d b , s ) be a metric type space and τ be the topology defined above. Then for any nonempty subset A M , we have
(i) A is closed if and only if for any sequence { μ n } in A which converges to μ , we have μ A .
(ii) if we define A ¯ to be the intersection of all closed subsets of M which contains A, then for any μ A ¯ and for any ϵ > 0 , we have
B O ( μ , ϵ ) A .
Let P c ( M ) denote the class of all non-empty and closed subsets of M and P c b ( M ) , the class of non-empty, closed and bounded subsets of M . Let μ M and Ω M ,
d b ( μ , Ω ) = inf d b ( μ , υ ) : υ Ω .
For Ω 1 , Ω 2 P c b ( M ) , the function H b : P c b ( M ) × P c b ( M ) [ 0 , + ) defined by
H b ( Ω 1 , Ω 2 ) = max { δ b ( Ω 1 , Ω 2 ) , δ b ( Ω 2 , Ω 1 ) }
where
δ b ( Ω 1 , Ω 2 ) = sup d b ( μ , ϖ ) : μ Ω 1 , ϖ Ω 2
is said to be Hausdorff b-metric [14] induced by the b-metric d b .
We recall the following properties from [14,17]:
Lemma 1.
Let ( M , d b , s ) be a b-metric space. For any Ω 1 , Ω 2 , Ω 3 P c b ( M ) and any μ , ϖ M , we have the following:
(i) 
d b ( μ , Ω 2 ) d b ( μ , b ) for any b Ω 2 .
(ii) 
δ b ( Ω 1 , Ω 2 ) H b ( Ω 1 , Ω 2 )
(iii) 
d b ( μ , Ω 2 ) H b ( Ω 1 , Ω 2 ) for any μ Ω 1 ,
(iv) 
H b ( Ω 1 , Ω 1 ) = 0 ,
(v) 
H b ( Ω 1 , Ω 2 ) = H b ( Ω 2 , Ω 1 )
(vi) 
H b ( Ω 1 , Ω 3 ) s [ H b ( Ω 1 , Ω 2 ) + H b ( Ω 2 , Ω 3 ) ]
(vii) 
d b ( μ , Ω 1 ) s [ d b ( μ , ϖ ) + d b ( ϖ , Ω 1 ) ] .
Furthermore, we will always assume that
(viii)) 
d b is continuous in its variables.
In 2012, Wardowski [18] initiated a new version of contractions which is named as F-contractions. Many researchers [19,20,21,22,23] established distinct fixed point results by utilizing these contractions. Cosentino et al. [24] used the wardowski’s approach in the setting of b-metric space defined as follows:
Definition 4.
Let ϝ s denotes the collection of functions F : ( 0 , + ) ( , + ) satisfying the properties:
( F 1 )
F is strictly increasing;
( F 2 )
{ μ n } ( 0 , + ) , lim n μ n = 0 lim n F ( μ n ) = ;
( F 3 )
0 < r < 1 such that lim n 0 + μ r F ( μ ) = 0 .
( F 4 )
for each sequence { μ n } R + of positive numbers such that ϱ + F ( s μ n ) F ( μ n 1 ) , n N and some ϱ > 0 , then ϱ + F ( s n μ n ) F ( s n 1 μ n 1 ) for all n N and s 1 .
Throughout this paper, we assume that the functions F ϝ s which are continuous from the right.
On the other hand, Constantin [25] initiated a new collection P of continuous functions σ : ( R + ) 5 R + satisfying these conditions:
( σ 1 )
σ ( 1 , 1 , 1 , 2 , 0 ) , σ ( 1 , 1 , 1 , 0 , 2 ) , σ ( 1 , 1 , 1 , 1 , 1 ) ( 0 , 1 ] ,
( σ 2 )
σ is sub-homogeneous, that is, for all ( μ 1 , μ 2 , μ 3 , μ 4 , μ 5 ) ( R + ) 5 and α 0 , we have σ ( α μ 1 , α μ 2 , α μ 3 , α μ 4 , α μ 5 ) α σ ( μ 1 , μ 2 , μ 3 , μ 4 , μ 5 ) ;
( σ 3 )
σ is a non-decreasing function, i.e, for μ i , ϖ i R + , μ i ϖ i , i = 1 , . . . , 5 , we get
σ ( μ 1 , μ 2 , μ 3 , μ 4 , μ 5 ) σ ( ϖ 1 , ϖ 2 , ϖ 3 , ϖ 4 , ϖ 5 )
and if μ i , ϖ i R + , i = 1 , . . . , 4 , then σ ( μ 1 , μ 2 , μ 3 , μ 4 , 0 ) σ ( ϖ 1 , ϖ 2 , ϖ 3 , ϖ 4 , 0 ) and σ ( μ 1 , μ 2 , μ 3 , 0 , μ 4 ) σ ( ϖ 1 , ϖ 2 , ϖ 3 , 0 , ϖ 4 )
and obtained a random fixed point theorem for multivalued mappings.
The following lemma of [26] is needed in the the proof of our main result.
Lemma 2.
If σ P and μ , ϖ R + are such that
μ < max σ ϖ , ϖ , μ , ϖ + μ , 0 , σ ϖ , ϖ , μ , 0 , ϖ + μ , σ ϖ , μ , ϖ , ϖ + μ , 0 , σ ϖ , μ , ϖ , 0 , ϖ + μ ,
then μ < ϖ .
The purpose of this paper is to present some common α -fuzzy fixed points for fuzzy mappings via F-contraction in complete b-metric space to extend the main result of Heilpern [1], Wardowski [18], Ahmad et al. [19], Sgroi et al. [21], Cosentino et al. [24] and Shahzad et al. [27] and some known results of literature.

2. Results and Discussion

We state our main result in this way.
Theorem 1.
Let ( M , d b , s ) be a complete b-metric space with coefficient s 1 and let R 1 , R 2 : M F ( M ) and for each μ , ϖ M , α R 1 ( μ ) , α R 2 ( ϖ ) ( 0 , 1 ] such that R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) P c b ( M ) . Assume that F ϝ s , a constant ϱ > 0 and σ P such that
2 ϱ + F ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ) F σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) )
for all μ , ϖ M with H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) > 0 . Then there exists μ * M such that μ * R 1 μ * α R 1 ( μ * ) R 2 μ * α R 2 ( μ * ) .
Proof. 
Let μ 0 M , then by hypotheses ∃ α R 1 ( μ 0 ) ( 0 , 1 ] such that R 1 μ 0 α R 1 ( μ 0 ) P c b ( M ) . Let μ 1 R 1 μ 0 α R 1 ( μ 0 ) . For this μ 1 , α R 2 ( μ 1 ) ( 0 , 1 ] such that R 2 μ 1 α R 2 ( μ 1 ) P c b ( M ) .
2 ϱ + F d b μ 1 , R 2 μ 1 α R 2 ( μ 1 ) 2 ϱ + F H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) 2 ϱ + F s H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) F σ d b ( μ 0 , μ 1 ) , d b ( μ 0 , R 1 μ 0 α R 1 ( μ 0 ) ) , d b ( μ 1 , R 2 μ 1 α R 2 ( μ 1 ) ) , d b ( μ 0 , R 2 μ 1 α R 2 ( μ 1 ) ) , d b ( μ 1 , R 1 μ 0 α R 1 ( μ 0 ) ) F σ d b ( μ 0 , μ 1 ) , d b ( μ 0 , μ 1 ) , d b ( μ 1 , R 2 μ 1 α R 2 ( μ 1 ) ) , d b ( μ 0 , R 2 μ 1 α R 2 ( μ 1 ) ) , 0
and so
d b μ 1 , R 2 μ 1 α R 2 ( μ 1 ) < σ d b ( μ 0 , μ 1 ) , d b ( μ 0 , μ 1 ) , d b ( μ 1 , R 2 μ 1 α R 2 ( μ 1 ) ) , d b ( μ 0 , R 2 μ 1 α R 2 ( μ 1 ) ) , 0 .
Then Lemma 2 gives that d b μ 1 , R 2 μ 1 α R 2 ( μ 1 ) < d b ( μ 0 , μ 1 ) . Thus, we obtain
2 ϱ + F s H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) F σ d b ( μ 0 , μ 1 ) , d b ( μ 0 , μ 1 ) , d b ( μ 1 , R 2 μ 1 α R 2 ( μ 1 ) ) , d b ( μ 0 , R 2 μ 1 α R 2 ( μ 1 ) ) , 0 F σ d b ( μ 0 , μ 1 ) , d b ( μ 0 , μ 1 ) , d b ( μ 0 , μ 1 ) , 2 d b ( μ 0 , μ 1 ) , 0 F d b ( μ 0 , μ 1 ) σ 1 , 1 , 1 , 2 , 0 F d b ( μ 0 , μ 1 ) .
Since F ϝ s , so ∃ h > 1 such that
F h s H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) < F s H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) + ϱ .
Next as d b μ 1 , R 2 μ 1 α R 2 ( μ 1 ) < h H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) , we deduce that there exists μ 2 R 2 μ 1 α R 2 ( μ 1 ) (obviously, μ 2 μ 1 ) such that d b ( μ 1 , μ 2 ) h H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) . Thus, we have
F s d b ( μ 1 , μ 2 ) F h s H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) < F s H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) + ϱ
which implies by (2) that
2 ϱ + F s d b ( μ 1 , μ 2 ) 2 ϱ + F s H b R 1 μ 0 α R 1 ( μ 0 ) , R 2 μ 1 α R 2 ( μ 1 ) + ϱ F ( d b ( μ 0 , μ 1 ) ) + ϱ .
Thus we have
ϱ + F s d b ( μ 1 , μ 2 ) F ( d b ( μ 0 , μ 1 ) ) .
For this μ 2 , α R 1 ( μ 2 ) ( 0 , 1 ] such that R 1 μ 2 α R 1 ( μ 2 ) P c b ( M ) .
2 ϱ + F d b μ 2 , R 1 μ 2 α R 1 ( μ 2 ) 2 ϱ + F H b R 1 μ 2 α R 1 ( μ 2 ) , R 2 μ 1 α R 2 ( μ 1 ) 2 ϱ + F s H b R 1 μ 2 α R 1 ( μ 2 ) , R 2 μ 1 α R 2 ( μ 1 ) F σ d b ( μ 2 , μ 1 ) , d b ( μ 2 , R 1 μ 2 α R 1 ( μ 2 ) ) , d b ( μ 1 , R 2 μ 1 α R 2 ( μ 1 ) ) , d b ( μ 2 , R 2 μ 1 α R 2 ( μ 1 ) ) , d b ( μ 1 , R 1 μ 2 α R 1 ( μ 2 ) ) F σ d b ( μ 2 , μ 1 ) , d b ( μ 2 , R 1 μ 2 α R 1 ( μ 2 ) ) , d b ( μ 1 , μ 2 ) , 0 , d b ( μ 1 , R 1 μ 2 α R 1 ( μ 2 ) ) = F σ d b ( μ 1 , μ 2 ) , d b ( μ 2 , R 1 μ 2 α R 1 ( μ 2 ) ) , d b ( μ 1 , μ 2 ) , 0 , d b ( μ 1 , R 1 μ 2 α R 1 ( μ 2 ) )
and so
d b μ 2 , R 1 μ 2 α R 1 ( μ 2 ) < σ d b ( μ 2 , μ 1 ) , d b ( μ 2 , R 1 μ 2 α R 1 ( μ 2 ) ) , d b ( μ 1 , μ 2 ) , 0 , d b ( μ 1 , R 1 μ 2 α R 1 ( μ 2 ) ) .
Then Lemma 2 gives that d b μ 2 , R 1 μ 2 α R 1 ( μ 2 ) < d b ( μ 1 , μ 2 ) . Thus, we obtain
2 ϱ + F s H b R 1 μ 2 α R 1 ( μ 2 ) , R 2 μ 1 α R 2 ( μ 1 ) F σ d b ( μ 2 , μ 1 ) , d b ( μ 2 , R 1 μ 2 α R 1 ( μ 2 ) ) , d b ( μ 1 , μ 2 ) , 0 , d b ( μ 1 , R 1 μ 2 α R 1 ( μ 2 ) ) F σ d b ( μ 1 , μ 2 ) , d b ( μ 1 , μ 2 ) , d b ( μ 1 , μ 2 ) , 0 , 2 d b ( μ 1 , μ 2 ) F d b ( μ 1 , μ 2 ) σ 1 , 1 , 1 , 0 , 2 F d b ( μ 1 , μ 2 ) .
Since F ϝ s , so ∃ h > 1 such that
F h s H b R 2 μ 1 α R 2 ( μ 1 ) , R 1 μ 2 α R 1 ( μ 2 ) = F h s H b R 1 μ 2 α R 1 ( μ 2 ) , R 2 μ 1 α R 2 ( μ 1 ) < F s H b R 1 μ 2 α R 1 ( μ 2 ) , R 2 μ 1 α R 2 ( μ 1 ) + ϱ .
Next as d b μ 2 , R 1 μ 2 α R 1 ( μ 2 ) < h H b R 2 μ 1 α R 2 ( μ 1 ) , R 1 μ 2 α R 1 ( μ 2 ) , we deduce that ∃ μ 3 R 1 μ 2 α R 1 ( μ 2 ) (obviously, μ 3 μ 2 ) such that d b ( μ 2 , μ 3 ) h H b R 2 μ 1 α R 2 ( μ 1 ) , R 1 μ 2 α R 1 ( μ 2 ) . Thus, we have
F s d b ( μ 2 , μ 3 ) F h s H b R 2 μ 1 α R 2 ( μ 1 ) , R 1 μ 2 α R 1 ( μ 2 ) < F s H b R 1 μ 2 α R 1 ( μ 2 ) , R 2 μ 1 α R 2 ( μ 1 ) + ϱ .
which implies by (5) that
2 ϱ + F s d b ( μ 2 , μ 3 ) 2 ϱ + F s H b R 1 μ 2 α R 1 ( μ 2 ) , R 2 μ 1 α R 2 ( μ 1 ) + ϱ F ( d b ( μ 2 , μ 1 ) ) + ϱ = F ( d b ( μ 1 , μ 2 ) ) + ϱ .
Consequently, we get
ϱ + F s d b ( μ 2 , μ 3 ) F ( d b ( μ 1 , μ 2 ) ) .
So, pursuing in this way, we obtain a sequence { μ n } in M such that μ 2 n + 1 R 1 μ 2 n α R 1 ( μ 2 n ) and μ 2 n + 2 R 2 μ 2 n + 1 α R 2 ( μ 2 n + 1 ) and
ϱ + F ( s d b ( μ 2 n + 1 , μ 2 n + 2 ) ) F ( d b ( μ 2 n , μ 2 n + 1 ) )
and
ϱ + F ( s d b ( μ 2 n + 2 , μ 2 n + 3 ) ) F ( d b ( μ 2 n + 1 , μ 2 n + 2 ) )
n N . By (8) and (9), we get
ϱ + F ( s d b ( μ n , μ n + 1 ) ) F ( d b ( μ n 1 , μ n ) )
n N . By (10) and ( F 4 ), we have
ϱ + F ( s n d b ( μ n , μ n + 1 ) ) F ( s n 1 d b ( μ n 1 , μ n ) ) .
Thus by (11), we obtain
F s n d b ( μ n , μ n + 1 ) F s n 1 d b ( μ n 1 , μ n ) ϱ F s n 2 d b ( μ n 2 , μ n 1 ) 2 ϱ F d b ( μ 0 , μ 1 ) n ϱ .
Taking n , we get lim n F s n d b ( μ n , μ n + 1 ) = . Along with ( F 2 ), we have
lim n s n d b ( μ n , μ n + 1 ) = 0 .
By ( F 3 ), ∃ r 0 , 1 so that
lim n [ s n d b ( μ n , μ n + 1 ) ] r F s n d b ( μ n , μ n + 1 ) = 0 .
From (12), we have
[ s n d b ( μ n , μ n + 1 ) ] r F s n d b ( μ n , μ n + 1 ) [ s n d b ( μ n , μ n + 1 ) ] r F d b ( μ 0 , μ 1 ) [ s n d b ( μ n , μ n + 1 ) ] r [ F d b ( μ 0 , μ 1 ) n ϱ ] [ s n d b ( μ n , μ n + 1 ) ] r F d b ( μ 0 , μ 1 ) n ϱ [ s n d b ( μ n , μ n + 1 ) ] r 0 .
Taking n in the above expression, we get
lim n n [ s n d b ( μ n , μ n + 1 ) ] r = 0 .
Hence lim n n 1 r s n d b ( μ n , μ n + 1 ) = 0 . Now, the last limit implies that the series n = 1 s n d b ( μ n , μ n + 1 ) is convergent. Thus { μ n } is a Cauchy sequence in M . Since ( M , d b , s ) is a complete b-metric space, so ∃ μ * M such that
lim n μ n = μ * .
Now, we prove that μ * R 2 μ * α R 2 ( μ * ) . We assume on the contrary that μ * R 2 μ * α R 2 ( μ * ) . Then by (14), ∃ n 0 N and { μ n k } of { μ n } such that d b ( μ 2 n k + 1 , R 2 μ * α R 2 ( μ * ) ) > 0 , n k n 0 . Now, using (1) with μ = μ 2 n k + 1 and ϖ = μ * , we obtain
2 ϱ + F s H b ( R 1 μ 2 n k α R 1 ( μ 2 n k ) , R 2 μ * α R 2 ( μ * ) )
F σ d b ( μ 2 n k , μ * ) , d b ( μ 2 n k , R 1 μ 2 n k α R 1 ( μ 2 n k ) ) , d b ( μ * , R 2 μ * α R 2 ( μ * ) , d b ( μ 2 n k , R 2 μ * α R 2 ( μ * ) ) , d b ( μ * , R 1 μ 2 n k α R 1 ( μ 2 n k ) )
This implies that
2 ϱ + F d b ( μ 2 n k + 1 , R 2 μ * α R 2 ( μ * ) ) 2 ϱ + F s H b ( R 1 μ 2 n k α R 1 ( μ 2 n k ) , R 2 μ * α R 2 ( μ * ) ) F σ d b ( μ 2 n k , μ * ) , d b ( μ 2 n k , R 1 μ 2 n k α R 1 ( μ 2 n k ) ) , d b ( μ * , R 2 μ * α R 2 ( μ * ) , d b ( μ 2 n k , R 2 μ * α R 2 ( μ * ) ) , d b ( μ * , R 1 μ 2 n k α R 1 ( μ 2 n k ) ) .
As ϱ > 0 , so by ( F 1 ), we obtain
d b ( μ 2 n k + 1 , R 2 μ * α R 2 ( μ * ) ) < σ d b ( μ 2 n k , μ * ) , d b ( μ 2 n k , R 1 μ 2 n k α R 1 ( μ 2 n k ) ) , d b ( μ * , R 2 μ * α R 2 ( μ * ) , d b ( μ 2 n k , R 2 μ * α R 2 ( μ * ) ) , d b ( μ * , R 1 μ 2 n k α R 1 ( μ 2 n k ) ) .
Letting n in the above expression, we have
d b ( μ * , R 2 μ * α R 2 ( μ * ) ) 0 .
Hence μ * R 2 μ * α R 2 ( μ * ) . Similarly, one can easily prove that μ * R 1 μ * α R 1 ( μ * ) . Thus μ * R 1 μ * α R 1 ( μ * ) R 2 μ * α R 2 ( μ * )  □.
For one fuzzy mapping, we deduce the following result.
Theorem 2.
Let ( M , d b , s ) be a complete b-metric space with coefficient s 1 and let R : M F ( M ) and for each μ , ϖ M , α R ( μ ) , α R ( ϖ ) ( 0 , 1 ] such that R μ α R ( μ ) , R ϖ α R ( ϖ ) P c b ( M ) . Assume that F ϝ s , a constant ϱ > 0 and σ P such that
2 ϱ + F ( s H b ( R μ α R ( μ ) , R ϖ α R ( ϖ ) ) ) F σ d b ( μ , ϖ ) , d b ( μ , R μ α R ( μ ) ) , d b ( ϖ , R ϖ α R ( ϖ ) ) , d b ( μ , R ϖ α R ( ϖ ) ) , d b ( ϖ , R μ α R ( μ ) )
for all μ , ϖ M with H b ( R μ α R ( μ ) , R ϖ α R ( ϖ ) ) > 0 . Then there exists μ * M such that μ * R μ * α R ( μ * ) .
Corollary 1.
Let ( M , d b , s ) be a complete b-metric space with coefficient s > 1 and let R 1 , R 2 : M F ( M ) and for each μ , ϖ M , α R 1 ( μ ) , α R 2 ( ϖ ) ( 0 , 1 ] such that R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) P c b ( M ) . Assume that k ( 0 , 1 ) and σ P such that
s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) k σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) )
μ , ϖ M . Then μ * M such that μ * R 1 μ * α R 1 ( μ * ) R 2 μ * α R 2 ( μ * ) .
Proof. 
Let k ( 0 , 1 ) be such that k = e 2 ϱ where ϱ > 0 and F ( θ ) = l n ( θ ) for θ > 0 . From (15), for all μ , ϖ M with H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) > 0 , we get
F ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ) 2 ϱ + F σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) )
that is
2 ϱ + F ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ) F σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) )
Thus we can apply Theorem 1 to deduce that R 1 and R 2 have a common fuzzy fixed point. ☐
Corollary 2.
Let ( M , d b , s ) be a complete b-metric space with coefficient s > 1 and let R 1 , R 2 : M F ( M ) and for each μ , ϖ M , α R 1 ( μ ) , α R 2 ( ϖ ) ( 0 , 1 ] such that R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) P c b ( M ) . Assume that k ( 0 , 1 ) and σ P such that
s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) e s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) ) k σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) )
μ , ϖ M . Then μ * M such that μ * R 1 μ * α R 1 ( μ * ) R 2 μ * α R 2 ( μ * ) .
Proof. 
Let k ( 0 , 1 ) be such that k = e 2 ϱ where ϱ > 0 and F ( θ ) = θ + l n ( θ ) for θ > 0 . From (16), for all μ , ϖ M with H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) > 0 , we get
F ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ) 2 ϱ + F σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) )
that is
2 ϱ + F ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ) F σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) ) .
Thus we can apply Theorem 1 to deduce that R 1 and R 2 have a common fuzzy fixed point. ☐
Corollary 3.
Let ( M , d b , s ) be a complete b-metric space with coefficient s > 1 and let R 1 , R 2 : M F ( M ) and for each μ , ϖ M , α R 1 ( μ ) , α R 2 ( ϖ ) ( 0 , 1 ] such that R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) P c b ( M ) . Assume that k ( 0 , 1 ) and σ P such that
s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) + 1 ) k σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) ) σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) ) + 1
for all μ , ϖ M . Then μ * M such that μ * R 1 μ * α R 1 ( μ * ) R 2 μ * α R 2 ( μ * ) .
Proof. 
Let k ( 0 , 1 ) be such that k = e 2 ϱ where ϱ > 0 and F ( θ ) = l n ( θ 2 + θ ) for θ > 0 . From (17), for all μ , ϖ M with H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) > 0 , we get
F ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ) 2 ϱ + F σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) )
that is
2 ϱ + F ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ) F σ d b ( μ , ϖ ) , d b ( μ , R 1 μ α R 1 ( μ ) ) , d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) , d b ( ϖ , R 1 μ α R 1 ( μ ) ) .
Thus we can apply Theorem 1 to deduce that R 1 and R 2 have a common fuzzy fixed point. ☐
Remark 1.
If we take s = 1 , then b-metric spaces turn into complete metric spaces and we get some new fixed point theorems for fuzzy mappings in metric spaces.
We obtain the following result from our main theorem by taking one fuzzy mapping.
Corollary 4.
Let ( M , d b , s ) be a complete b-metric space with coefficient s 1 and let R : M F ( M ) and for each μ , ϖ M , α R ( μ ) , α R ( ϖ ) ( 0 , 1 ] such that R μ α R ( μ ) , R ϖ α R ( ϖ ) P c b ( M ) . Assume that F ϝ s , a constant ϱ > 0 and σ P such that
2 ϱ + F ( s H b ( R μ α R ( μ ) , R ϖ α R ( ϖ ) ) ) F σ d b ( μ , ϖ ) , d b ( μ , R μ α R ( μ ) ) , d b ( ϖ , R ϖ α R ( ϖ ) ) , d b ( μ , R ϖ α R ( ϖ ) ) , d b ( ϖ , R μ α R ( μ ) )
for all μ , ϖ M with H b ( R μ α R ( μ ) , R 2 ϖ α R ( ϖ ) ) > 0 . Then μ * M such that μ * R μ * α R ( μ * ) .
Proof. 
Take R 1 = R 2 = R in Theorem 1. □
Corollary 5.
Let ( M , d b , s ) be a complete b-metric space with coefficient s 1 and let R 1 , R 2 : M F ( M ) and for each μ , ϖ M , α R 1 ( μ ) , α R 2 ( ϖ ) ( 0 , 1 ] such that R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) P c b ( M ) . Assume that F ϝ s , a constant ϱ > 0 and σ P such that
2 ϱ + F ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ) F d b ( μ , ϖ )
for all μ , ϖ M with H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) > 0 . Then there exists μ * M such that μ * R 1 μ * α R 1 ( μ * ) R 2 μ * α R 2 ( μ * ) .
Proof. 
Considering σ P given by σ ( μ 1 , μ 2 , μ 3 , μ 4 , μ 5 ) = μ 1 in Theorem 1. □
Remark 2.
If we take R 1 , R 2 : M F ( M ) as R 1 = R 2 = R : M F ( M ) and s = 1 in the above Corollary, we get the main result of Ahmad et al. [19]. With this, if F ( θ ) = θ + ln ( θ ) , for θ > 0 , then a result by Heilpern [1].
Corollary 6.
Let ( M , d b , s ) be a complete b-metric space with coefficient s 1 and let R 1 , R 2 : M F ( M ) and for each μ , ϖ M , α R 1 ( μ ) , α R 2 ( ϖ ) ( 0 , 1 ] such that R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) P c b ( M ) . Assume that F ϝ s , ϱ > 0 , σ P and λ i 0 ( i = 1 , 5 ) such that
2 ϱ + F ( s H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) ) F λ 1 d b ( μ , ϖ ) + λ 2 d b ( μ , R 1 μ α R 1 ( μ ) ) + λ 3 d b ( ϖ , R 2 ϖ α R 2 ( ϖ ) ) + λ 4 d b ( μ , R 2 ϖ α R 2 ( ϖ ) ) + λ 5 d b ( ϖ , R 1 μ α R 1 ( μ ) )
for all μ , ϖ M with H b ( R 1 μ α R 1 ( μ ) , R 2 ϖ α R 2 ( ϖ ) ) > 0 and ( λ 1 + λ 2 ) ( s + 1 ) + s ( λ 3 + λ 4 ) ( s + 1 ) + 2 s λ 5 < 2 . Then there exists μ * M such that μ * R 1 μ * α R 1 ( μ * ) R 2 μ * α R 2 ( μ * ) .
Proof. 
Considering σ P given by σ ( μ 1 , μ 2 , μ 3 , μ 4 , μ 5 ) = λ 1 μ 1 + λ 2 μ 2 + λ 3 μ 3 + λ 4 μ 4 + λ 5 μ 5 in Theorem 1, where λ i 0 such that ( λ 1 + λ 2 ) ( s + 1 ) + s ( λ 3 + λ 4 ) ( s + 1 ) + 2 s λ 5 < 2 □.
Remark 3.
Taking F ( θ ) = θ + ln ( θ ) , for θ > 0 , we get Theorem 2.2 of Shahzad et al. [27].
Example 3.
Let M = { 0 , 1 , 2 } and define metric d b : M × M R 0 + by
d b ( μ , ϖ ) = 0 , i f μ = ϖ 1 6 , i f μ ϖ and μ , ϖ { 0 , 1 } 1 2 , if μ ϖ and μ , ϖ { 0 , 2 } 1 , i f μ ϖ and μ , ϖ { 1 , 2 } .
It is easy to see that ( M , d ) is a complete b-metric space with coefficient s = 3 2 , which the ordinary triangle inequality does not hold. Define
( R 0 ) ( t ) = ( R 1 ) ( t ) = 1 2 , i f t = 0 0 , i f t = 1 , 2
and
( R 2 ) ( t ) = 0 , i f t = 0 , 2 1 2 , i f t = 1 .
Define α : M ( 0 , 1 ] by α ( μ ) = 1 2 for all μ M . Now we obtain that
[ R μ ] 1 2 = { 0 , 1 } , i f μ = 0 , 1 { 1 } , i f μ = 2 .
For μ , ϖ M , we get
H b ( [ R 0 ] 1 2 , [ R 2 ] 1 2 ) = H b ( [ R 1 ] 1 2 , [ R 2 ] 1 2 ) = H b ( { 0 } , { 1 } ) = 1 6 .
Taking F ( θ ) = θ + ln ( θ ) , for θ > 0 and ϱ = 1 100 > 0 . Then
2 ϱ + F ( s H b ( [ R 0 ] 1 2 , [ R 2 ] 1 2 ) ) = 1 50 + 1 4 + ln ( 3 2 . 1 6 ) 1 2 + ln ( 1 2 ) = F ( d b ( 0 , 2 ) )
also
2 ϱ + F ( s H b ( [ R 1 ] 1 2 , [ R 2 ] 1 2 ) ) = 1 50 + 1 4 + ln ( 3 2 . 1 6 ) 1 + ln ( 1 ) = F ( d b ( 1 , 2 ) )
for all μ , ϖ M . As a result, all assumptions of Theorem 2 hold by considering σ P as σ ( μ 1 , μ 2 , μ 3 , μ 4 , μ 5 ) = μ 1 and there exists a point 0 M such that 0 [ R 0 ] 1 2 is an α-fuzzy fixed point of R .
Fixed point results for multivalued mappings can be deduced from fuzzy fixed point results in this way.
Theorem 3.
Let ( M , d b , s ) be a complete b-metric space with coefficient s 1 and let G 1 , G 2 : M P c b ( M ) . If F ϝ s and ϱ > 0 such that
2 ϱ + F ( s H b ( G 1 μ , G 2 ϖ ) ) F σ d b ( μ , ϖ ) , d ( μ , G 1 μ ) , d ( ϖ , G 2 ϖ ) , d ( μ , G 2 ϖ ) , d ( ϖ , G 1 μ )
for all μ , ϖ M with H b G 1 μ , G 2 ϖ > 0 . Then μ * M such that μ * G 1 μ * G 2 μ * .
Proof. 
Consider α : M ( 0 , 1 ] and R 1 , R 2 : M F ( M ) defined by
R 1 ( μ ) ( t ) = α ( μ ) , if t G 1 μ , 0 , if t G 1 μ
and
R 2 ( μ ) ( t ) = α ( μ ) , if t G 2 μ , 0 , if t G 2 μ .
Then
R 1 μ α ( μ ) = t : R 1 ( μ ) ( t ) α ( μ ) = G 1 μ and R 2 μ α ( μ ) = t : R 2 ( μ ) ( t ) α ( μ ) = G 2 μ .
Hence, Theorem 1 can be applied to get μ * M such that μ * R 1 μ * α R 1 ( μ * ) R 2 μ * α R 2 ( μ * ) = G 1 μ * G 2 μ * .
Corollary 7.
Let ( M , d b , s ) be a complete b-metric space with coefficient s 1 and let G : M P c b ( M ) . If F ϝ s and ϱ > 0 such that
2 ϱ + F ( s H b ( G μ , G ϖ ) ) F σ d b ( μ , ϖ ) , d b ( μ , G μ ) , d b ( ϖ , G ϖ ) , d b ( μ , G ϖ ) , d b ( ϖ , G μ )
for all μ , ϖ M with H b G μ , G ϖ > 0 . Then μ * M such that μ * G μ * .
Proof. 
Taking G 1 = G 2 = G in Theorem 3. □
We can get the following result of Cosentino et al. [24] in this way.
Corollary 8
([24]). Let ( M , d b , s ) be a complete b-metric space with coefficient s 1 and let G : M P c b ( M ) . If F ϝ s and ϱ > 0 such that
2 ϱ + F ( s H b ( G μ , G ϖ ) ) F d b ( μ , ϖ )
for all μ , ϖ M with H b G μ , G ϖ > 0 . Then μ * M such that μ * G μ * .
Proof. 
Considering σ P given by σ ( μ 1 , μ 2 , μ 3 , μ 4 , μ 5 ) = μ 1 in Corollary 8. □
Corollary 9.
Let ( M , d ) be a complete metric space and let G : M P c b ( M ) . If F ϝ , ϱ > 0 and λ i 0 ( i = 1 , 5 ) such that
2 ϱ + F ( H ( G μ , G ϖ ) ) F λ 1 d ( μ , ϖ ) + λ 2 d ( μ , G μ ) + λ 3 d ( ϖ , G ϖ ) + λ 4 d ( μ , G ϖ ) + λ 5 d ( ϖ , G μ )
for all μ , ϖ M with H G μ , G ϖ > 0 and λ 1 + λ 2 + λ 3 + 2 λ 5 = 1 and λ 4 1 . Then μ * M such that μ * G μ * .
Proof. 
Considering σ P given by σ ( μ 1 , μ 2 , μ 3 , μ 4 , μ 5 ) = λ 1 μ 1 + λ 2 μ 2 + λ 3 μ 3 + λ 4 μ 4 + λ 5 μ 5 , where λ i 0 ( i = 1 , 5 ) and λ 1 + λ 2 + λ 3 + 2 λ 5 = 1 and λ 4 1 and taking s = 1 in Corollary 8, we get the main result of Sgroi et al. [21]. □
Remark 4.
If we consider G : M M , s = 1 and λ 1 = 1 , λ 2 = λ 3 = λ 4 = λ 4 = 0 , we get the main result of Wardowski [18].

3. Applications

Consider the integral inclusion of Fredholm
μ ( t ) g ( t ) + a b K ( t , s , μ ( s ) ) d s , t [ a , b ]
where K : [ a , b ] × [ a , b ] × R K c v ( R ) a given multivalued operator, where K c v represents the class of non-empty compact and convex subsets of R . Here g C [ a , b ] are given and μ C [ a , b ] unknown functions.
Now, for p 1 , define b-metric d b on C [ a , b ] by
d b ( μ , ϖ ) = ( max t [ a , b ] | μ ( t ) ϖ ( t ) | ) p = max t [ a , b ] | μ ( t ) ϖ ( t ) | p
for all μ , ϖ C [ a , b ] . Then ( C [ a , b ] , d b , 2 p 1 ) is a complete b-metric space.
We will assume the following:
(a) ∀ μ C [ a , b ] , the operator K : [ a , b ] × [ a , b ] × R K c v ( R ) is such that K ( t , s , μ ( s ) ) is lower semicontinuous in [ a , b ] × [ a , b ] ,
(b) O : [ a , b ] × [ a , b ] [ 0 , + ) which is continuous such that
H b ( K ( t , s , μ ) K ( t , s , ϖ ) O ( t , s ) | μ ( s ) ϖ ( s ) |
t , s [ a , b ] , μ , ϖ C [ a , b ] .
(c) ∃ ϱ > 0 such that
sup t [ a , b ] a b O ( t , s ) d s p e ϱ 2 p 1 .
Theorem 4.
Under the conditions (a)–(c), the integral inclusion (20) has a solution in C [ a , b ] .
Proof. 
Let M = C [ a , b ] . Define the fuzzy mapping R : M P c b ( M ) by
R μ α R ( μ ) = ϖ M : ϖ ( t ) g ( t ) + a b K ( t , s , μ ( s ) ) d s , t [ a , b ] .
Let μ M be arbitrary. For the multivalued operator K μ ( t , s ) : [ a , b ] × [ a , b ] K c v ( R ) , it follows from the Michael’s selection theorem that there exists a continuous operator k μ ( t , s ) : [ a , b ] × [ a , b ] R such that k μ ( t , s ) K μ ( t , s ) for each t , s [ a , b ] . It follows that g ( t ) + a b k μ ( t , s ) d s R μ . Hence R μ . It is an easy matter to show that R μ is closed, and so details are omitted (see also [28]). Furthermore, as g is continuous on [ a , b ] and K μ ( t , s ) is continuous on [ a , b ] × [ a , b ] , so their ranges are bounded. It follows that R μ is also bounded. Thus R μ P c b ( M ) .
We will check that the contractive condition (19) holds for R in M with some ϱ > 0 and F ϝ s , i.e.,
2 ϱ + F ( s δ b ( R μ 1 , R μ 2 ) ) F ( d b ( μ 1 , μ 2 ) )
for μ 1 , μ 2 M . For this, let μ 1 , μ 2 M , then there exist R μ 1 , R μ 2 such that R μ 1 , R μ 2 P c b ( M ) . Let ϖ 1 R μ 1 be arbitrary such that
ϖ 1 ( t ) g ( t ) + a b K ( t , s , μ 1 ( s ) ) d s
for t [ a , b ] holds. It means that ∀ t , s [ a , b ] , k μ 1 ( t , s ) K μ 1 ( t , s ) = K ( t , s , μ 1 ( s ) ) such that
ϖ 1 ( t ) = g ( t ) + a b k μ 1 ( t , s ) d s
for t [ a , b ] . For all μ 1 , μ 2 M , it follows from (b) that
H b ( K ( t , s , μ 1 ) K ( t , s , μ 2 ) O ( t , s ) | μ 1 ( s ) μ 2 ( s ) | .
It means that ∃ z ( t , s ) K μ 2 ( t , s ) such that
k μ 1 ( t , s ) z ( t , s ) p O ( t , s ) | μ 1 ( s ) μ 2 ( s ) | .
for all t , s [ a , b ] .
Now, we can consider the multivalued operator U defined by
U ( t , s ) = K μ 2 ( t , s ) { u R : k μ 1 ( t , s ) u O ( t , s ) | μ 1 ( s ) μ 2 ( s ) | } .
Hence, by (a), U is lower semicontinuous, it follows that there exists a continuous operator k μ 2 ( t , s ) : [ a , b ] × [ a , b ] R such that k μ 2 ( t , s ) U ( t , s ) for t , s [ a , b ] . Then ϖ 2 ( t ) = g ( t ) + a b k μ 1 ( t , s ) d s satisfies that
ϖ 2 ( t ) g ( t ) + a b K ( t , s , μ 2 ( s ) ) d s , t [ a , b ] .
t [ a , b ] . That is ϖ 2 R μ 2 and
ϖ 1 ( t ) ϖ 2 ( t ) p a b k μ 1 ( t , s ) k μ 2 ( t , s ) d s p a b O ( t , s ) | μ 1 ( s ) μ 2 ( s ) | d s p max t [ a , b ] a b O ( t , s ) d s p | μ ( t ) ϖ ( t ) | p e ϱ 2 p 1 d b ( μ 1 , μ 2 )
t , s [ a , b ] . Hence, we get
2 p 1 d b ( ϖ 1 , ϖ 2 ) e ϱ d b ( μ 1 , μ 2 ) .
Interchanging the roles of μ 1 and μ 2 , we obtain that
s H b ( R μ 1 , R μ 2 ) e ϱ d b ( μ 1 , μ 2 ) .
By passing to logarithms, we write
ln ( s H b ( R μ 1 , R μ 2 ) ) ln ( e ϱ d b ( μ 1 , μ 2 ) ) .
Taking the function F ϝ s defined by F ( θ ) = l n ( θ ) for θ > 0 , we get that the assumption (18) is fulfilled. Using the result (9), we achieve that the integral inclusion (20) has a solution.

4. Conclusions

In this article, we have established some generalized common fixed point resultss for α -fuzzy mappings in a connection with F- contraction and a family P of continuous functions σ : ( R + ) 5 R + in the setting of complete b-metric spaces. The obtained results extended and improved various well-known results in literature including Heilpern [1], Wardowski [18], Ahmad et al. [19], Sgroi et al. [21], Cosentino et al. [24] and Shahzad et al. [27]. As applications, we analyzed the existence of approximate solutions for Fredholm integral inclusions. Our results are new and significantly contribute to the existing literature in fixed point theory. Similar generalizations of these contractions for the L-fuzzy mappings R 1 , R 2 : M F L ( M ) would be a distinctive subject for future study. One can apply our results in the solution of fractional differential inclusions as a future work.

Author Contributions

Conceptualization, S.A.A.-M.; Formal analysis, S.A.A.-M. and J.A.; Funding acquisition, S.A.A.-M. and M.D.L.S.; Investigation, S.A.A.-M. and J.A.; Methodology, S.A.A.-M. and J.A.; Project administration, M.D.L.S.; Supervision, M.D.L.S. All authors read and approved the final paper. All authors have read and agreed to the published version of the manuscript.

Funding

Deanship of Scientific Research (DSR), University of Jeddah, Jeddah. Grant No. UJ-02-007-ICGR.

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia, under grant No. UJ-02-007-ICGR. The first and second authors, therefore, acknowledge with thanks the University technical and financial support.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Heilpern, S. Fuzzy mappings and fixed point theorem. J. Math. Anal. Appl. 1981, 83, 566–569. [Google Scholar] [CrossRef]
  2. Ahmad, J.; Azam, A.; Romaguera, S. On locally contractive fuzzy set-valued mappings. J. Inequal. Appl. 2014, 2014, 74. [Google Scholar] [CrossRef]
  3. Ahmad, J.; Al-Mazrooei, A.E.; Altun, I. Generalized Θ-contractive fuzzy mappings. J. Intell. Fuzzy Syst. 2018, 35, 1935–1942. [Google Scholar] [CrossRef]
  4. Al-Mazrooei, A.E.; Ahmad, J. Fixed Point Theorems for Fuzzy Mappings with Applications. J. Intell. Fuzzy Syst. 2019, 36, 3903–3909. [Google Scholar] [CrossRef]
  5. Al-Mazrooei, A.E.; Ahmad, J. Fuzzy fixed point results of generalized almost F-contraction. J. Math. Comput. Sci. 2018, 18, 206–215. [Google Scholar] [CrossRef]
  6. Azam, A.; Beg, I. Common fixed points of fuzzy maps. Math. Comput. Model. 2009, 49, 1331–1336. [Google Scholar] [CrossRef]
  7. Azam, A.; Arshad, M.; Vetro, P. On a pair of fuzzy φ-contractive mappings. Math. Comput. Model. 2010, 52, 207–214. [Google Scholar] [CrossRef]
  8. Azam, A. Fuzzy Fixed Points of Fuzzy Mappings via a Rational Inequality. Hacettepe J. Math. Stat. 2011, 40, 421–431. [Google Scholar]
  9. Bose, R.K.; Sahani, D. Fuzzy mappings and fixed point theorems. Fuzzy Sets Syst. 1987, 21, 53–58. [Google Scholar] [CrossRef]
  10. Chang, S.S.; Cho, Y.J.; Lee, B.S.; Jung, J.S.; Kang, S.M. Coincidence point and minimization theorems in fuzzy metric spaces. Fuzzy Sets Syst. 1997, 88, 119–128. [Google Scholar] [CrossRef]
  11. Saleh, H.N.; Khan, I.A.; Imdad, M.; Alfaqih, W.M. New fuzzy φ-fixed point results employing a new class of fuzzy contractive mappings. J. Intell. Fuzzy Syst. 2019, 37, 5391–5402. [Google Scholar] [CrossRef]
  12. Sayed, A.F.; Ahmad, A. Some fixed point theorems for fuzzy soft contractive mappings in fuzzy soft metric spaces. Ital. J. Pure Appl. Math. 2018, 40, 200–214. [Google Scholar]
  13. Czerwik, S. Contraction mappings in b-metric spaces. Acta Math. Inf. Univ. Ostraviensis. 1993, 1, 5–11. [Google Scholar]
  14. Czerwik, S. Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Fis. Univ. Modena 1998, 46, 263–276. [Google Scholar]
  15. Khamsi, M.A.; Hussain, N. KKM mappings in metric type spaces. Nonlinear Anal. 2010, 73, 3123–3129. [Google Scholar] [CrossRef]
  16. Hussain, N.; Saadati, R.; Agarwal, R.P. On the topology and wt-distance on metric type spaces. Fixed Point Theory Appl. 2014, 2014, 88. [Google Scholar] [CrossRef]
  17. Czerwik, S.; Dlutek, K.; Singh, S.L. Round-off stability of iteration procedures for operators in b-metric spaces. J. Nat. Phys. Sci. 1997, 11, 87–94. [Google Scholar]
  18. Wardowski, D. Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 2012, 94. [Google Scholar] [CrossRef]
  19. Ahmad, J.; Aydi, H.; Mlaiki, N. Fuzzy fixed points of fuzzy mappings via F-contractions and an applications. J. Intell. Fuzzy Syst. 2019, 38, 1–7. [Google Scholar] [CrossRef]
  20. Imdad, M.; Khan, A.R.; Saleh, H.N.; Alfaqih, W.M. Some φ-Fixed Point Results for (F,φ,α-ψ)-Contractive Type Mappings with Applications. Mathematics 2019, 7, 122. [Google Scholar] [CrossRef]
  21. Sgroi, M.; Vetro, C. Multivalued F-contractions and the solution of certain functional and integral equations. Filomat 2013, 27, 1259–1268. [Google Scholar] [CrossRef]
  22. Hussain, N.; Ahmad, J.; Azam, A. On Suzuki-Wardowski type fixed point theorems. J. Nonlinear Sci. Appl. 2015, 8, 1095–1111. [Google Scholar] [CrossRef]
  23. Hussain, N.; Ahmad, J.; Azam, A. Generalized fixed point theorems for multi-valued α-ψ contractive mappings. J. Inequal. Appl. 2014, 2014, 348. [Google Scholar] [CrossRef]
  24. Cosentino, M.; Jleli, M.; Samet, B.; Vetro, C. Solvability of integrodifferential problems via fixed point theory in b-metric spaces. Fixed Point Theory Appl. 2015, 2015, 70. [Google Scholar] [CrossRef]
  25. Constantin, A. A random fixed point theorem for multifunctions. Stochastic Anal. Appl. 1994, 12, 65–73. [Google Scholar] [CrossRef]
  26. Isik, H. Fractional Differential Inclusions with a New Class of Set-Valued Contractions. Available online: https://arxiv.org/abs/1807.05427v1 (accessed on 12 July 2018).
  27. Shahzad, A.; Shoaib, A.; Mehmood, Q. Common fixed point theorems for fuzzy mappings in b-metric space. Ital. J. Pure Appl. Math. 2017, 38, 419–427. [Google Scholar]
  28. Sîntamarian, A. Integral inclusions of Fredholm type relative to multivalued φ-contractions. Semin. Fixed Point Theory Cluj Napoca 2002, 3, 361–368. [Google Scholar]
Back to TopTop