On soft set-valued maps and integral inclusions

1 Introduction and preliminaries

One of the first most celebrated fixed point theorems in spaces with metric structure appeared explicitly in Banach’s thesis in 1922 (see [9]), where it was initially applied to obtain the existence of a solution to an integral equation. The theorem is now popularly known as Banach fixed point theorem (or the contraction mapping principle). Indeed, the contraction mapping principle is a reformulation of the successive approximation techniques originally used by some earlier mathematicians, namely Cauchy, Liouville, Picard, Lipschitz, and so on. The original idea of fixed point theorem due to Banach has been developed and applied in different directions. In some generalizations of the contraction mapping principle, the contractive inequality is weakened (e.g., see [12]) and in others, the topology of the ground space is weakened (e.g., see [19, 28]).

Throughout this article, ℝ , ℝ + and ℕ , represent the set of real numbers, positive real numbers and natural numbers, respectively.

In 2012, Wardowski [37] established a generalization of the contraction mapping principle by introducing a new type of contraction known as F-contraction as follows.

Definition 1.1. [37] Let (ℑ , σ) be a metric space. A mapping Θ :ℑ ⟶ ℑ is called an F-contraction, if there exists ξ > 0 such that σ ( Θ ς , Θ  ) > 0 ⇒ ξ + F ( σ ( Θ ς , Θ  ) ) ≤ F ( σ ( ς ,  ) )(1.1) for all ς,  ∈ ℑ, where F : ℝ + ⟶ ℝ is a mapping satisfying:

F is strictly increasing; that is, α< β  ⇒ F (α) < F (β),

We denote the family of functions satisfying (F₁) - (F₃) by Ω.

Example 1.2. [37] The functions F : ℝ + ⟶ ℝ defined by

F (α) = ln(α),

From (i) and (ii) in Example 1.2, it is easy to see that every F-contraction is a contractive mapping, that is, for all ς,  ∈ ℑ with Θς≠ Θ , we get σ ( Θ ς , Θ  ) < σ ( ς ,  ) . Shortly, Wardowski and Van Dung [38] initiated the concept of F-weak contraction and obtained a generalization of F-contraction in the following manner.

Definition 1.3. [38] Let (ℑ , σ) be a metric space. A mapping Θ :ℑ ⟶ ℑ is called an F-weak contraction, if there exist F ∈ Ω and ξ > 0 such that for all ς,  ∈ ℑ satisfying σ (Θς, Θ ) >0, ξ + F ( σ ( Θ ς , Θ  ) ) ≤ F ( M * ( ς ,  ) ) , where M * ( ς ,  ) = max { σ ( ς ,  ) , σ ( ς , Θ ς ) , σ (  , Θ  ) , σ ( ς , Θ  ) + σ (  , Θ ς ) 2 } .

Motivated by the pioneer work in [37, 38], more than a handful of modifications of F-contraction in the framework of single-valued mappings have been established. For a few of such results, we refer [2, 22].

Denote by ℑ^* and ℑ b * , the set of all nonempty compact subsets of ℑ and the set of all nonempty closed and bounded subsets of ℑ, respectively. For Δ, ℧ ∈ ℑ ^*, the mapping H : ℑ * × ℑ * ⟶ ℝ defined by H ( Δ , ℧ ) = max { sup ς ∈ ℧ σ ( ς , Δ ) , sup  ∈ Δ σ (  , ℧ ) } , where σ ( ς , Δ ) = inf  ∈ Δ σ ( ς ,  ) , is called the Hausdorff metric induced by the metric σ.

Let Θ : ℑ ⟶ ℑ b * be a multivalued mapping. Then, Θ is called a multivalued contraction (see [32]) if for all ς,  ∈ ℑ, there exists λ ∈ [0, 1) such that H ( Θ ς , Θ  ) ≤ λ σ ( ς ,  ) . In 1969, Nadler [32] proved that every multivalued contraction on a complete metric space has a fixed point. This result marked the first metric fixed point theorem for multivalued mapping. Following Wardowski [37] and inspired by multivalued contraction due to Nadler [32], Altun et al. [5] introduced the idea of multivalued F-contraction and established some fixed point theorems via these type of mappings defined on complete metric space.

Definition 1.4. Let (ℑ , σ) be a metric space and Θ : ℑ ⟶ ℑ b * be a multivalued mapping. Then, Θ is called a multivalued F-contraction, if there exist F ∈ Ω and ξ > 0 such that for all ς,  ∈ ℑ, H (Θς, Θ ) >0 implies ξ + F ( H ( Θ ς , Θ  ) ) ≤ F ( σ ( ς ,  ) ) .

By taking F (α) = ln(α) for all α > 0, every multivalued contraction in the sense of Nadler is also a multivalued F-contraction. After the result in [5], the notion of F-contraction in the setting of set-valued mappings has been extended in different directions, for example, see [1, 24] and references therein.

In 2012, Samet et al. [36] introduced the concepts of α - ϑ-contractive and α-admissible mappings and extended many existing results, in particular, the contraction mapping principle due to Banach [9].

Denote by M , the family of nondecreasing functions ϑ : ℝ + ⟶ ℝ such that ∑ n = 1 ∞ ϑ n ( t ) < ∞ , for each t > 0, where ϑ ⁿ  (t) is the n ^th -iterate of ϑ.

Definition 1.5. [36] Let (ℑ , σ) be a metric space and Θ :ℑ ⟶ ℑ be a given mapping. Then Θ is called an α - ϑ-contractive mapping if there exist two functions α : ℑ × ℑ ⟶ ℝ + and ϑ ∈ M such that for all ς,  ∈ ℑ, α ( ς ,  ) σ ( Θ ς , Θ  ) ≤ ϑ ( σ ( ς ,  ) ) .

Definition 1.6. [36] Let Θ :ℑ ⟶ ℑ and α : ℑ × ℑ ⟶ ℝ + be mappings. Then Θ is said to be α-admissible if for all ς,  ∈ ℑ, α ( ς ,  ) ≥ 1 ⇒ α ( Θ ς , Θ  ) ≥ 1 .

Example 1.7. [36] Let ℑ = (0, ∞). Define Θ :ℑ ⟶ ℑ and α : ℑ × ℑ ⟶ ℝ + by Θ ς = ln ς , ς ∈ ℑ and for all ς,  ∈ ℑ, α ( ς ,  ) = { 2 , if ς ≥  0 , if ς <  . Then, Θ is α-admissible.

Example 1.8. [36] Let ℑ = ℝ + . Define Θ :ℑ ⟶ ℑ and α : ℑ × ℑ ⟶ ℝ + by Θ ς = ς , for all ς∈ ℑ, and α ( ς ,  ) = { exp ( ς -  ) , if ς ≥  0 , if ς <  . Then, Θ is α-admissible.

Aydi et al. [6] combined the concepts of F-contraction and α-admissibility in the following manner.

Definition 1.9. [6] Let (ℑ , σ) be a metric space. A single-valued mapping Θ :ℑ ⟶ ℑ is said to be a modified F-contraction via α-admissible, if there exist ξ > 0, F ∈ Ω and ϑ ∈ M such that for all ς,  ∈ ℑ, σ (Θς, Θ ) >0 implies ξ + F ( α ( ς ,  ) σ ( Θ ς , Θ  ) ) ≤ F ( ϑ ( σ ( ς ,  ) ) ) .(1.2) If we take F (α) = ln(α) for all α > 0, the contractive inequality (1.2) becomes α ( ς ,  ) σ ( Θ ς , Θ  ) ≤ e - ξ ϑ ( σ ( ς ,  ) ) ≤ ϑ ( σ ( ς ,  ) ) ,(1.3) for all ς,  ∈ ℑ ,  Θς ≠ Θ . In fact, here the mapping Θ in (1.3) is an α - ϑ-contraction introduced by Samet et al. [36]. For some results in the context of α - ϑ - F-contractions, see [21, 27]. Also, for a recent survey of F-contractions with related fixed point theorems, the interested reader is referred to Karapinar et al. [23]. On the other hand, the applications of mathematics witnessed tremendous developments as a result of the introduction of soft set by Molodstov [31]. The method of handling problems in conventional mathematics is in the opposite of the technique of soft set theory. In classical mathematics, to describe any system or object, we first construct its mathematical model and then attempt to obtain the exact solution. If the exact solution is too complicated, then we define the notion of approximate solution. But, in soft set theory, the initial description of an object takes an approximate nature with no restriction, and the notion of exact solution is not essential. Thus, to describe an object in soft set theory, any convenient parametrization tool which may be words, sentences, numbers, mappings, functions, to mention a few, may be used. Thereby, making the theory more easier and flexible in terms of applications in every day life. In [31], Moldstov highlighted several directions for possible applications of soft set, such as in smoothness of functions, game theory, Riemann-integration, operation research, probability and so on. Presently, the concept of soft set is receiving more than a handful of extensions in different perspectives. For example, see [11, 34] and references therein.

Let ℑ be a nonempty set and E be the universe of discourse of all parameters related to the elements in ℑ. In this case, each parameter is a word, sentence or function. Let P (ℑ) be the power set of ℑ. Molodstov [31] defined the concept of soft set in the following sense.

Definition 1.10. [31] A pair (F, Δ) is called a soft set over ℑ, where Δ ⊆ E and F is a set- valued mapping F : Δ ⟶ P (ℑ). In this way, a soft set over ℑ is a parameterized family of subsets of ℑ.

Example 1.11. Suppose the soft (F, E) describes the structures of certain number of men. Let the reference set of all men be ℑ = {ς₁, ς₂, ς₃, ς₄, ς₅} and the universe of all parameters be represented by E = { e 1 , e 2 , e 3 , e 4 } = { fat , tall , muscular , lanky } . In this case, to define a soft set means to point out fat men, tall men, muscular men, and lanky men. Thus, we may define F : E ⟶ P (ℑ) by F (e₁) = {ς₁, ς₂, ς₅}  , F (e₂) = {ς₂, ς₄, ς₅} ,  F (e₃) = {ς₅} , F (e₄) = empty. So, the soft set (F, E) is a parameterized family {F (e _i ) : i = 1, 2, 3, 4} of P (ℑ).

It is well-known that set-valued analysis has enormous applications in control theory, game theory, biomathematics, qualitative physics, viability theory, and so on. With this motivation, not long ago, Mohammed and Azam [8, 30] studied the concept of soft set-valued maps and introduced the notions of e-soft fixed point and E-soft fixed point. It is shown in [29] that every fuzzy mapping is a particular kind of soft set-valued map. Since every fuzzy mapping has its corresponding multifunction analogue (see [14], [Theorem 2.2]), hence, the idea of e-soft fixed point theorems is a generalization of the concept of fuzzy fixed point and fixed point of multivalued mappings.

In what follows, we recall some specific concepts of soft set-valued maps from [29, 30]. Let (ℑ , σ) be a metric space. Denote by [P (ℑ)]  ^E , the family of soft sets over ℑ. Then, consider two soft sets (F, Δ) and (G, ℧), (a, b)∈ Δ × ℧. Assume that F (a) , G (b) ∈ ℑ ^*. For ε > 0, define N ^σ  (ε, F (a)), S E ℑ ( a , b ) ( F , G ) and E ( F a , G b ) σ , respectively, as follows: N σ ( ε , F ( a ) ) = { ς ∈ ℑ : σ ( ς ,  ) < ε for some  ∈ F ( a ) } , E ( F a , G b ) σ = { ε > 0 : F ( a ) ⊆ N σ ( ε , G ( b ) ) , G ( b ) ⊆ N σ ( ε , F ( a ) ) } , and S E ℑ ( a , b ) ( F , G ) = inf E ( F a , G b ) σ , Define a distance function S E ℑ ∞ : [ P ( ℑ ) ] E × [ P ( ℑ ) ] E ⟶ ℝ + by S E ℑ ∞ ( F , G ) = sup ( a , b ) ∈ Δ ¯ × ℧ ¯ S E ℑ ( a , b ) ( F , G ) , where Δ ¯ × ℧ ¯ = { ( a , b ) ∈ Δ × ℧ : F ( a ) , G ( b ) ∈ ℑ * } .

Remark 1.12.

Note that in terms of the Hausdorff metric H, the distance function S E ℑ ( a , b ) ( F , G ) reduces to: S E ℑ ( a , b ) ( F , G ) = H ( F ( a ) , G ( b ) ) = max { sup ς ∈ F ( a ) σ ( ς , G ( b ) ) , sup  ∈ G ( b ) σ (  , F ( a ) ) } .

Definition 1.13. [29] A mapping Θ : ℑ ⟶ [P (ℑ)]  ^E is called a soft set-valued map. A point u∈ ℑ is called an e-soft fixed point of Θ if u ∈ (Θu) (e), for some e ∈ E. If DomΘς = E and u ∈ (Θu) (e) for all e ∈ E, then u is said to be an E-soft fixed point of Θ. We denote the set of all E-soft fixed point of a soft set-valued map Θ by E_Fix(Θ). Here, the domain of Θ, denoted as DomΘ, is given by Dom Θ = { ς ∈ ℑ : ( Θ ς ) ( e ) ⊆ ℑ , e ∈ E } . Analogously, we define the image of Θ, imΘ as imΘ = {  |  ∃ ς ∈ ℑ :  ∈ (Θς) (e) ,  e ∈ E}.

Notice that if Θ : ℑ ⟶ [P (ℑ)]  ^E is a soft set-valued map, then (Θς, E) is a soft set over ℑ, for all ς∈ ℑ. Throughout this paper, if Θ : ℑ ⟶ [P (ℑ)]  ^E is a soft set-valued map, then the set (Θς) (e) shall also be written as (Θ _e ς) (or Θς, for short).

Several examples of soft set-valued maps have been provided in [29, 30]. We give an additional example as follows.

Example 1.14. Let ℑ = {1, 2, 3} and E = {1, 2}. Define Θ : ℑ ⟶ [P (ℑ)]  ^E as follows: ( Θ e ς ) = { { 1 , 3 } , if e = 1 { 2 , 3 } , if e = 2 . Then, Θ is a soft set-valued map. Notice that 1 ∈ (Θ _e 1) for e = 1 and 2 ∈ (Θ _e 2) for e = 2; hence, 1 and 2 are e-soft fixed point of Θ. But, 2 ∉ (Θ _e 2) and 1 ∉ (Θ _e 1) for e = 1 and e = 2, respectively. If follows that 1 and 2 are not E-soft fixed point of Θ. On the other hand, 3 ∈ (Θ _e 3) for all e ∈ E; thus, the set of all E-soft fixed point of Θ is given by E_Fix(Θ) = {3}. The map Θ can be represented as in Fig. 1. Notice that in Fig. 1, the dots represent other subsets of ℑ.

OPEN IN VIEWER

Fig. 1

Graphical representation of the soft set-valued map in Example 1.14.

By combining the ideas of F-contraction and α - ϑ-contraction due to Wardowski [37] and Samet et al. [36], respectively as well as the concept of soft set-valued maps recently initiated by Mohammed and Azam [29], first in this paper, the notions of e-continuity and E-continuity of soft set-valued maps are defined. Thereafter, we introduce the ideas of α _e -admissibility and α _e  - (ϑ, F)-contraction for a pair of soft set-valued maps and prove the existence of common e-soft fixed point of such maps defined on a complete metric space. A few consequences within the neighborhoods of our main results which include fixed point theorems of fuzzy and multivalued mappings are noted and discussed. Moreover, examples and applications to a system of integral inclusions of Fredholm type are considered to illustrate the usability of the obtained results herein.

Recall that continuity of a set-valued mapping is usually defined in terms of lower and upper semi-continuity via the notion of Hausdorff separation. We complement this idea and start this section by introducing the concepts of e-continuity and E-continuity of soft set-valued map as follows.

Definition 2.1. Let (ℑ , σ) be a metric space. A soft set-valued map Θ : ℑ ⟶ [P (ℑ)]  ^E is said to be e-continuous at u∈ ℑ, if for any sequence { ς n } n ∈ ℕ in ℑ, there exists e ∈ E such that lim n ⟶ ∞ σ ( ς n , u ) = 0 ⇒ lim n ⟶ ∞ S E ℑ ( e ( ς ) , e ( u ) ) ( Θ ς , Θ u ) = 0 . We say that Θ is E-continuous if it is e-continuous at every point of ℑ for each e ∈ E.

Definition 2.1 can be reformulated as follows.

Definition 2.2. A soft set-valued map Θ : ℑ ⟶ [P (ℑ)]  ^E is e-continuous at u∈ ℑ, if for every ε > 0, there exist a δ > 0 and e ∈ E such that σ ( ς , u ) < δ ⇒ S E ℑ ( e ( ς ) , e ( u ) ) ( Θ ς , Θ u ) < ε .

Example 2.3. Let ℑ = ℝ , E = [0, ∞) and σ (ς, ) = |ς -  | for all ς,  ∈ ℑ. Define a soft set-valued map Θ : ℑ ⟶ [P (ℑ)]  ^E by ( Θ e ς ) : = Θ ς = { [ 0 , ς + 6 ] if 0 ≤ e < 10 { 1 2 } , if 10 ≤ e < ∞ . Then, for each ς∈ ℑ, there exists e ∈ [0, 10) such that Θς = [0, ς + 6]. For ε > 0, take δ = ε 5 , then, for all ς,  ∈ ℑ, σ (ς, ) < δ implies S E ℑ ( e ( ς ) , e (  ) ) ( Θ ς , Θ  ) = | ς -  | < ε . Similarly, for all ς,  ∈ ℑ, there exists e ∈ [10, ∞) such that for any δ > 0, σ (ς, ) < δ implies S E ℑ ( e ( ς ) , e (  ) ) ( Θ ς , Θ  ) = 0 < ε . Thus, Θ is E-continuous on ℑ.

Example 2.4. Let ℑ = ℝ , E = [0, 1] and σ (ς, ) = |ς -  | for all ς,  ∈ ℑ. Define a soft set-valued map Θ : ℑ ⟶ [P (ℑ)]  ^E by ( Θ e ς ) : = Θ ς = { [ - 5 , 20 ] if 0 ≤ e < 1 7 ( 30 , ∞ ) , if 1 7 ≤ e ≤ 1 . Then, Θ is e-continuous for e ∈ [ 0 , 1 7 ) . But, Θ is not e-continuous for all e ∈ [ 1 7 , 1 ] . Hence, Θ is not E-continuous on ℑ.

A slight modification of the next two examples due to Aubin (see [7, Page 57]), shows that the concept of E-continuity of soft set-valued maps is more general than its corresponding crisp set-valued maps.

Example 2.5. Let ℑ = ℝ , E = [ 0 , 1 ] and define a soft set-valued map Θ : ℑ ⟶ [P (ℑ)]  ^E by ( Θ e ς ) = { [ - 1 , 1 ] , if ς ≠ 0 and 0 ≤ e < 1 2 { 0 } , if ς = 0 and 1 2 ≤ e ≤ 1 . Then, Θ is lower semicontinuous at zero but not upper semicontinuous. However, Θ is E-continuous on ℑ.

Example 2.6. Let ℑ = ℝ , E = [ 0 , 1 ] and define a soft set-valued map Θ : ℑ ⟶ [P (ℑ)]  ^E by ( Θ e ς ) = { { 0 } , if ς ≠ 0 and 0 ≤ e < 1 2 [ - 1 , 1 ] , if ς = 0 and 1 2 ≤ e ≤ 1 . Then, Θ is upper semicontinuous at zero but not lower semicontinuous. Obviously, Θ is E-continuous on ℑ.

Let Ψ denotes the family of functions ϑ : ℝ ⟶ ℝ satisfying:

lim n ⟶ ∞ ϑ n ( t ) n < 0 for all t > 0 and n > 0;

Example 2.7. The function ϑ : ℝ ⟶ ℝ defined by ϑ ( t ) = { t 5 - 3 , if t < 3 t - 3 , if t > 3 . belongs to Ψ.

Clearly, any function ϑ satisfying (ϑ₁) also posses the property that lim n ⟶ ∞ ϑ n ( t ) = - ∞ for all t ∈ (0, ∞).

Definition 2.8. Let ℑ be a nonempty set. We say that the pair (G, Θ) of soft set-valued maps G, Θ : ℑ ⟶ [P (ℑ)]  ^E is α _e -admissible, if there exist a function α : ℑ × ℑ ⟶ ℝ + and e ∈ E such that

for each ς∈ ℑ and any  ∈ (G _e ς) with α (ς, ) ≥1, we have α ( , w) ≥1 for all w ∈ (Θ _e  );

Recall that a function α : ℑ × ℑ ⟶ ℝ + is called symmetric if α (ς, ) ≥1 implies α ( , ς) ≥1 for all ς,  ∈ ℑ. Similarly, the pair (G, Θ) of soft set-valued maps G, Θ : ℑ ⟶ [P (ℑ)]  ^E is said to be symmetric α _e -admissible if there exists a symmetric function α : ℑ × ℑ ⟶ ℝ + and some e ∈ E such that (G, Θ) is α _e -admissible.

Definition 2.9. Let (ℑ , σ) be a metric space. A pair (G, Θ) of soft set-valued maps G, Θ : ℑ ⟶ [P (ℑ)]  ^E is said to be an α _e  - (ϑ, F)-contraction, if there exist α : ℑ × ℑ ⟶ ℝ + , ϑ ∈ Ψ , e ∈ E and F ∈ Ω such that for all ς,  ∈ ℑ, F ( S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ) ≤ ϑ ( F ( ∐ ( ς ,  ) ) )(2.4) for some a (ς) , a () ∈ E with α (ς, ) ≥1 and S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , where ∐ ( ς ,  ) = max { σ ( ς ,  ) , σ ( ς , G ς ) , σ (  , Θ  ) , σ ( ς , Θ  ) + σ (  , G ς ) 1 + σ ( ς ,  ) } .(2.5)

Now, we present our first main result as follows.

Theorem 2.10. Let (ℑ , σ) be a complete metric space and G, Θ : ℑ ⟶ [P (ℑ)]  ^E be two soft set-valued maps such that the pair (G, Θ) is an α _e  - (ϑ, F)-contraction. Assume that the following conditions are satisfied:

for each ς∈ ℑ, there exists a (ς) = e ∈ E such that (G _e ς) , (Θ _e ς) ∈ ℑ ^*;

Proof. Let ς₀∈ ℑ and ς₁ ∈ Gς₀ be such that α (ς₀, ς₁) ≥1. Then, we consider the following cases:

Case 1: If ∐ (ς₀, ς₁) =0, then, from (2.5), it is easy to see that ς₀ = ς₁ is a common e-soft fixed point of G and Θ. So, we presume that ∐ (ς₀, ς₁) >0. Then, ∐ ( ς 0 , ς 1 ) = max { σ ( ς 0 , ς 1 ) , ( ς 0 , G ς 0 ) , σ ( ς 1 , Θ ς 1 ) , σ ( ς 0 , Θ ς 1 ) + σ ( ς 1 , G ς 0 ) 1 + σ ( ς 0 , ς 1 ) } = max { σ ( ς 0 , ς 1 ) , σ ( ς 1 , Θ ς 1 ) , σ ( ς 0 , Θ ς 1 ) 1 + σ ( ς 0 , ς 1 ) } ≤ max { σ ( ς 0 , ς 1 ) , σ ( ς 1 , Θ ς 1 ) } .(2.6) Consider the following subcases:

Case 1(i) σ (ς₁, Θς₁) =0, that is, ς₁ ∈ Θς₁. Since the pair (G, Θ) is symmetric α _e -admissible, ς₁ ∈ Gς₁ and α (ς₀, ς₁) ≥1, by (ax₁), we get α (ς₁, ς₁) ≥1. Now, suppose σ (ς₁, Gς₁) >0, then since (G, Θ) is an α _e  - (ϑ, F)-contraction, then F ( σ ( ς 1 , G ς 1 ) ) ≤ F ( S E ℑ ( a ( ς 1 ) , a ( ς 1 ) ) ( Θ ς 1 , G ς 1 ) ) ≤ ϑ ( F ( ∐ ( ς 1 , ς 1 ) ) ) < F ( ∐ ( ς 1 , ς 1 ) ) = F ( σ ( ς 1 , G ς 1 ) ) gives a contradiction. It follows that ς₁ ∈ Gς₁, and hence ς₁ ∈ (G _e ς₁) ∩ (Θ _e ς₁), for some e ∈ E.

Case 1(ii) σ (ς₁, Θς₁) >0. Since α (ς₀, ς₁) ≥1 and (G, Θ) is an α _e  - (ϑ, F)-contraction, we get F ( σ ( ς 1 , Θ ς 1 ) )(2.7) ≤ F ( S E ℑ ( a ( ς 0 ) , a ( ς 1 ) ) ( G ς 0 , Θ ς 1 ) ) ≤ ϑ ( F ( ∐ ( ς 0 , ς 1 ) ) ) = ϑ ( F ( max { σ ( ς 0 , ς 1 ) , σ ( ς 1 , Θ ς 1 ) } ) ) .(2.8)

If max {σ (ς₀, ς₁) , σ (ς₁, Θς₁)} = σ (ς₁, Θς₁), then F ( σ ( ς 1 , Θ ς 1 ) ) ≤ ϑ ( F ( σ ( ς 1 , Θ ς 1 ) ) ) < F ( σ ( ς 1 , Θ ς 1 ) ) , is a contradiction. Therefore, F ( σ ( ς 1 , Θ ς 1 ) ) ≤ ϑ ( F ( σ ( ς 0 , ς 1 ) ) ) .(2.9) Since Θς₁ ∈ ℑ ^*, there exists ς₂ ∈ Θς₁ such that σ ( ς 1 , ς 2 ) = σ ( ς 1 , Θ ς 1 ) .(2.10) Putting (2.10) into (2.9), we have F ( σ ( ς 1 , ς 2 ) ) ≤ ϑ ( F ( σ ( ς 0 , ς 1 ) ) ) .(2.11)

Case 2: If ∐ (ς₁, ς₂) =0, then ς₁ = ς₂ is a common e-soft fixed point of G and Θ. Assume that ∐ (ς₁, ς₂) >0. Then, ∐ ( ς 1 , ς 2 ) = max { σ ( ς 1 , ς 2 ) , σ ( ς 2 , G ς 2 ) , σ ( ς 1 , Θ ς 1 ) , σ ( ς 1 , G ς 2 ) + σ ( ς 2 , Θ ς 1 ) 1 + σ ( ς 1 , ς 2 ) } ≤ max { σ ( ς 1 , ς 2 ) , σ ( ς 2 , G ς 2 ) } . We further consider the following subcases:

Case 2 (i) σ (ς₂, Gς₂) =0, that is, ς₂ ∈ Gς₂. Since (G, Θ) is a symmetric α _e -admissible pair, ς₂ ∈ Θς₁, α (ς₁, ς₂) ≥1 and by (ax₂), we get α (ς₂, ς₂) ≥1. Suppose that σ (ς₂, Θς₂) >0. Then, given that the pair (G, Θ) is an α _e  - (ϑ, F)-contraction, we have F ( σ ( ς 2 , Θ ς 2 ) ) ≤ F ( S E ℑ ( a ( ς 2 ) , a ( ς 2 ) ) ( G ς 2 , Θ ς 2 ) ) ≤ ϑ ( F ( ∐ ( ς 2 , ς 2 ) ) ) < F ( σ ( ς 2 , Θ ς 2 ) ) , a contradiction. Thus, ς₂ ∈ Θς₂. It follows that ς₂ ∈ (G _e ς₂) ∩ (Θ _e ς₂), for some e ∈ E.

Case 2(ii) σ (ς, Gς₂) >0. Since α (ς₁, ς₂) ≥1 and the pair (G, Θ) is an α _e  - (ϑ, F)-contraction, we have F ( σ ( ς 2 , G ς 2 ) ) ≤ F ( S E ℑ ( a ( ς 2 ) , a ( ς 1 ) ) ( G ς 2 , Θ ς 1 ) ) ≤ ϑ ( F ( ∐ ( ς 2 , ς 1 ) ) )(2.12) = ϑ ( F ( max { σ ( ς 1 , ς 2 ) , σ ( ς 2 , G ς 2 ) } ) ) .

If max {σ (ς₁, ς₂) , σ (ς₂, Gς₂)} = σ (ς₂, Gς₂), then F ( σ ( ς 2 , G ς 2 ) ) ≤ ϑ ( F ( σ ( ς 2 , G ς 2 ) ) ) < F ( σ ( ς 2 , G ς 2 ) ) yields a contradiction. Therefore, F ( σ ( ς 2 , G ς 2 ) ) ≤ ϑ ( F ( σ ( ς 1 , ς 2 ) ) ) .(2.13) Moreover, since Gς₂ ∈ ℑ ^*, there exists ς₃ ∈ Gς₂ such that σ ( ς 2 , ς 3 ) = σ ( ς 2 , G ς 2 ) .(2.14) Substituting (2.14) in (2.13), gives F ( σ ( ς 2 , ς 3 ) ) ≤ ϑ ( F ( ς 1 , ς 2 ) ) .(2.15) Combining (2.11) and (2.15), we have F (σ (ς₂, ς₃)) ≤ ϑ² (F (σ (ς₀, ς₁))). Proceeding recursively, we generate a sequence { ς n } n ∈ ℕ such that ς_2n+1 ∈ Gς_2n,  ς_2n+2 ∈ Θς_2n+1,  σ (ς _n , ς_n+1) >0,  α (ς _n , ς_n+1) ≥1 for all n ∈ ℕ and F ( σ ( ς n , ς n + 1 ) ) ≤ ϑ n ( F ( σ ( ς 0 , ς 1 ) ) ) , n ∈ ℕ .(2.16) Take δ _n  = σ (ς _n , ς_n+1). Then, from (2.16), we get F ( δ n ) ≤ ϑ n ( F ( δ 0 ) ) ⟶ - ∞ as n ⟶ ∞ .(2.17) Therefore, by (F₂), lim n ⟶ ∞ δ n = 0 . From (2.17), for all n ∈ ℕ , there exists η ∈ (0, 1) such that δ n η ( F ( δ n ) ) ≤ δ n η ϑ n ( F ( δ 0 ) ) .(2.18) As n⟶ ∞ in (2.18), we have δ n η ϑ n ( F ( δ n ) ) = 0 . Moreover, from (ϑ₁), there exists λ > 0 such that λ < | ϑ n ( F ( δ 0 ) ) n | ; from which we have n δ n η λ ≤ n δ n η | ϑ n ( F ( δ 0 ) ) n | = | δ n η ϑ n ( F ( δ 0 ) ) | .(2.19) As n⟶ ∞ in (2.19), we get lim n ⟶ ∞ n δ n η λ = 0 ; that is, lim n ⟶ ∞ n δ n η = 0 . It follows that there exists n 0 ∈ ℕ such that δ n ≤ 1 n 1 η , for all n ≥ n₀.

Now, for m , n ∈ ℕ with n < m, we obtain σ ( ς n , ς m ) ≤ ∑ i = n m - 1 δ i ≤ ∑ i = n m - 1 1 i 1 η ≤ ∑ i = n ∞ 1 i 1 η . By Cauchy root test, it is verifiable that the series ∑ i = n ∞ 1 i 1 η is convergent; and hence, { ς n } n ∈ ℕ is a Cauchy sequence in ℑ. The completeness of ℑ implies that there exists u∈ ℑ such that ς _n  ⟶ u as n⟶ ∞. Since G and Θ are E-continuous, σ ( u , Gu ) = lim n ⟶ ∞ σ ( ς 2 n + 1 , Gu ) ≤ lim n ⟶ ∞ S E ℑ ( a ( ς 2 n ) , a ( u ) ) ( G ς 2 n , Gu ) = 0 . It follows that u ∈ Gu. Similarly, we can show that σ (u, ℑ u) =0. Consequently, u is the expected e-soft fixed point of G and Θ.□

Corollary 2.11. Let (ℑ , σ) be a complete metric space and G, Θ : ℑ ⟶ [P (ℑ)]  ^E be two soft set-valued maps satisfying: ξ + F ( S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ) ≤ F ( ∐ ( ς ,  ) ) for all ς,  ∈ ℑ with S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , where a (ς) , a () ∈ E, F ∈ Ω and ∐ (ς, ) is given by (2.5). Assume that the following conditions hold:

there exist ς₀∈ ℑ and ς₁ ∈ Gς₀ such that α (ς₀, ς₁) ≥1;

Proof. Put ϑ (t) = t - ξ,  ξ > 0 in Theorem 2.10.□

Corollary 2.12. Let (ℑ , σ) be a complete metric space and G, Θ : ℑ ⟶ [P (ℑ)]  ^E be two soft set-valued maps satisfying: S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ∐ ( ς ,  ) exp ( S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) - ∐ ( ς ,  ) ) ≤ exp ( - ξ ) for all ς,  ∈ ℑ with S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , where a (ς) , a () ∈ E, ϑ ∈ Ψ, F ∈ Ω and ∐ (ς, ) is given by (2.5). Assume that the following conditions hold:

there exist ς₀∈ ℑ and ς₁ ∈ Gς₀ such that α (ς₀, ς₁) ≥1;

Proof. Set F (t) = ln t + t,  t > 0 in Corollary 2.11.□

Example 2.13. Let ℑ = {0, 2, 5, 8, 10}, E = [0, ∞) and σ (ς, ) = |ς -  |, for all ς,  ∈ ℑ. Then, (ℑ , σ) is a complete metric space. Define α : ℑ × ℑ ⟶ ℝ + by α ( ς ,  ) = { 1 , if ς ,  ∈ { 2 , 5 , 10 } 0 , otherwise . For each ς∈ ℑ and a (ς) = e ∈ E, consider two soft set-valued maps G, Θ : ℑ ⟶ [P (ℑ)]  ^E defined by ( G e ς ) = { { 0 } , if ς = 0 , e ∈ [ 0 , 10 ) { 5 } , if ς = 2 , e ∈ [ 10 , 30 ) { 8 } , if ς ∈ { 5 , 8 } , e ∈ [ 30 , 70 ) { 5 , 8 } , if ς = 10 , e ∈ [ 70 , ∞ ) ; and ( Θ e ς ) = { { 0 } , if ς = 0 , e ∈ [ 0 , 10 ) { 2 } , if ς = 2 , e ∈ [ 10 , 30 ) { 5 } , if ς , ∈ { 5 , 8 } , e ∈ [ 30 , 70 ) { 10 } , if ς = 10 , e ∈ [ 70 , ∞ ) . Then, for all ς,  ∈ ℑ with α (ς, ) ≥1 and S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , consider the following cases:

[Case 1:] For ς = 2 and =10, we have S E ℑ ( a ( 2 ) , a ( 10 ) ) ( G 2 , Θ 10 ) = inf E ( G 2 , Θ 10 ) σ = 5 and ∐ ( 2 , 10 ) = max { σ ( 2 , 10 ) , σ ( 2 , G 2 ) , σ ( 10 , Θ 10 ) , σ ( 2 , Θ 10 ) + σ ( 10 , G 2 ) 1 + σ ( 2 , 10 ) } = 8 . Therefore, S EX ( a ( 2 ) , a ( 10 ) ) ( G 2 , Θ 10 ) ∐ ( 2 , 10 ) × exp ( S EX ( a ( 2 ) , a ( 10 ) ) ( G 2 , Θ 10 ) - ∐ ( 2 , 10 ) ) = 5 8 exp ( - 3 ) < exp ( - 3 ) .

Case 2 For ς = 10 and =2, we have S E ℑ ( a ( 10 ) a ( 2 ) ) ( G 10 , Θ 2 ) = inf E ( G 10 , Θ 2 ) σ = 6 and ∐ ( 10 , 2 ) = max { σ ( 10 , 2 ) , σ ( 10 , G 10 ) , σ ( 2 , Θ 2 ) , σ ( 10 , Θ 2 ) + σ ( 2 , G 10 ) 1 + σ ( 10 , 2 ) } = 8 . Hence, S E ℑ ( a ( 10 ) , a ( 2 ) ) ( G 10 , Θ 2 ) ∐ ( 10 , 2 ) × exp ( S EX ( a ( 10 ) , a ( 2 ) ) ( G 10 , Θ 2 ) - ∐ ( 10 , 2 ) ) = 3 4 exp ( - 2 ) < exp ( - 2 ) .

Case 3 For ς = 5 and =10, we have S EX ( a ( 5 ) , a ( 10 ) ) ( G 5 , Θ 10 ) = inf E ( G 5 , Θ 10 ) σ = 2 and ∐ ( 5 , 10 ) = max { σ ( 5 , 10 ) , σ ( 5 , G 5 ) , σ ( 10 , Θ 10 ) , σ ( 5 , Θ 10 ) + σ ( 10 , G 5 ) 1 + σ ( 5 , 10 ) } = 5 . Therefore, S E ℑ ( a ( 5 ) , a ( 10 ) ) ( G 5 , Θ 10 ) ∐ ( 5 , 10 ) × exp ( S E ℑ ( a ( 5 ) , a ( 10 ) ) ( G 5 , Θ 10 ) - ∐ ( 5 , 10 ) ) = 2 5 exp ( - 3 ) < exp ( - 3 ) .

Case 4 For ς = 10 and =5, we get S E ℑ ( a ( 10 ) , a ( 5 ) ) ( G 10 , Θ 5 ) = inf E ( G 10 , Θ 5 ) σ = 3 and ∐ ( 10 , 5 ) = max { σ ( 10 , 5 ) , σ ( 10 , G 10 ) , σ ( 5 , Θ 5 ) , σ ( 10 , Θ 5 ) + σ ( 5 , G 10 ) 1 + σ ( 10 , 5 ) } = 5 . Thus, S E ℑ ( a ( 10 ) , a ( 5 ) ) ( G 10 , Θ 5 ) ∐ ( 10 , 5 ) × exp ( S E ℑ ( a ( 10 ) , a ( 5 ) ) ( G 10 , Θ 5 ) - ∐ ( 10 , 5 ) ) = 3 5 exp ( - 2 ) < exp ( - 2 ) .

Therefore, for any ξ ∈ {2, 3}, the contractive condition of Corollary 2.12 is satisfied. Moreover, it is clear that the pair (G, Θ) is α _e -admissible and E-continuous. And, if we take ς₀ = 10 and ς₁ = 5, then ς₁ ∈ Gς₀ and α (10, 5) = α (ς₀, ς₁) ≥1. Finally, it is easy to see that for each ς∈ ℑ, there exists a (ς) : = e ∈ E such that Gς and Θς are nonempty compact subsets of ℑ. Consequently, all the hypotheses of Corollary 2.12 are satisfied. Notice that E_Fix(G) = {0, 8} and E_Fix(Θ) = {0, 2, 5, 10}. Hence, 0 is the common e-soft fixed point of G and Θ.

By considering more variants of functions F ∈ Ω, we obtain further consequences of our main result as follows.

Corollary 2.14. Let (ℑ , σ) be a complete metric space and G, Θ : ℑ ⟶ [P (ℑ)]  ^E be two soft set-valued maps satisfying: S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ≤ 1 ( 1 + ξ ( ∐ ( ς ,  ) ) 1 2 ) 2 ∐ ( ς ,  ) , for all ς,  ∈ ℑ with S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , where a (ς) , a () ∈ E, ϑ ∈ Ψ, and ∐ (ς, ) is given by (2.5). Assume that conditions (i)–(iv) of Corollary 2.11 hold. Then, G and Θ have a common e-soft fixed point in ℑ.

Proof. Put F ( t ) = - 1 t , t > 0 in Corollary 2.11.□

Corollary 2.15. Let (ℑ , σ) be a complete metric space and G, Θ : ℑ ⟶ [P (ℑ)]  ^E be two soft set-valued maps satisfying: S E ℑ ( a ( ς ) , a (  ) ) ( G ς , G  ) ( 1 + S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ) ∐ ( ς ,  ) ( 1 + ∐ ( ς ,  ) ) ≤ exp ( - ξ ) , for all ς,  ∈ ℑ with S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , where a (ς) , a () ∈ E, ϑ ∈ Ψ, F ∈ Ω and ∐ (ς, ) is given by (2.5). Assume that conditions (i)–(iv) of Corollary 2.11 hold. Then, G and Θ have a common e-soft fixed point in ℑ.

Proof. Put F (t) = ln(t² + t) ,  t > 0 in Corollary 2.11.□

Corollary 2.16. Let (ℑ , σ) be a complete metric space and G, Θ : ℑ ⟶ [P (ℑ)]  ^E be two soft set-valued maps satisfying: F ( S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ) ≤ ϑ ( F ( ∐ ( ς ,  ) ) ) for all ς,  ∈ ℑ with S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , where a (ς) , a () ∈ E, ϑ ∈ Ψ, F ∈ Ω and ∐ (ς, ) is given by (2.5). Assume that the following conditions hold:

for each ς∈ ℑ, there exists a (ς) : = e ∈ E such that (G _e ς) : = Gς and (Θ _e ς) : = Θς are nonempty compact subsets of ℑ;

Proof. For all ς,  ∈ ℑ, put α (ς, ) =1 in Theorem 2.10.□

Corollary 2.17. Let (ℑ , σ) be a complete metric space and G, Θ : ℑ ⟶ [P (ℑ)]  ^E be two soft set-valued maps satisfying: ξ + F ( S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ) ≤ F ( ∐ ( ς ,  ) ) for all ς,  ∈ ℑ with S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , where a (ς) , a () ∈ E, F ∈ Ω and ∐ (ς, ) is given by (2.5). Assume that the following conditions hold:

for each ς∈ ℑ, there exists a (ς) : = e ∈ E such that (G _e ς) and (Θ _e ς) are nonempty compact subsets of ℑ;

Proof. Put ϑ (t) = t - ξ,  ξ > 0 in Corollary 2.16.□

Corollary 2.18. Let (ℑ , σ) be a complete metric space and G, Θ : ℑ ⟶ [P (ℑ)]  ^E be two soft set-valued maps satisfying: ξ + F ( S E ℑ ∞ ( G ς , Θ  ) ) ≤ F ( ∐ ( ς ,  ) ) for all ς,  ∈ ℑ with S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , where a (ς) , a () ∈ E, F ∈ Ω and ∐ (ς, ) is given by (2.5). Assume that the following conditions hold:

for each ς∈ ℑ, there exists a (ς) : = e ∈ E such that (G _e ς) and (Θ _e ς) are nonempty compact subsets of ℑ;

Proof. Since the function F is increasing and S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ≤ S E ℑ ∞ ( G ς , Θ  ) for all ς,  ∈ ℑ, we have ξ + F ( S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ) ≤ ξ + F ( S E ℑ ∞ ( G ς , Θ  ) ) ≤ F ( ∐ ( ς ,  ) ) . Consequently, Corollary 2.17 can be applied to find a common e-soft fixed point of G and Θ.□

Corollary 2.19. Let (ℑ , σ) be a complete metric space and G, Θ : ℑ ⟶ [P (ℑ)]  ^E be two soft set-valued maps satisfying: S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) ∐ ( ς ,  ) exp ( S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) - ∐ ( ς ,  ) ) ≤ exp ( - ξ ) for all ς,  ∈ ℑ with S E ℑ ( a ( ς ) , a (  ) ) ( G ς , Θ  ) > 0 , where a (ς) , a () ∈ E, ϑ ∈ Ψ, and ∐ (ς, ) is given by (2.5). Assume that the following conditions hold:

for each ς∈ ℑ, there exists a (ς) : = e ∈ E such that (G _e ς) and (Θ _e ς) are nonempty compact subsets of ℑ;

Proof. Set F (t) = ln t + t,  t > 0 in Corollary 2.17.□

Corollary 2.20. Let (ℑ , σ) be a complete metric space and G : ℑ ⟶ [P (ℑ)]  ^E be an E-continuous soft set-valued map. Assume that for ς∈ ℑ, there exist η ∈ (0, 1), a (ς) : = e ∈ E such that (Θ _e ς) : = Θς is a nonempty compact subset of ℑ, and S E ℑ ( a ( ς ) , a (  ) ) ( G ς , G  ) ≤ η σ ( ς ,  ) ,(2.20) for all ς,  ∈ ℑ with S E ℑ ( a ( ς ) , a (  ) ) ( G ς , G  ) > 0 . Then, G has an e-soft fixed point in ℑ.

Proof. Taking ln on both sides of (2.20), gives - ln η + ln ( S E ℑ ( a ( ς ) , a (  ) ) ( G ς , G  ) ) ≤ ln ( σ ( ς ,  ) )(2.21) for all ς,  ∈ ℑ with σ (ς, ) >0. Setting ξ = - ln η and ln t = F (t) ,  t > 0 in (2.21), yields the condition S E ℑ ( a ( ς ) , a (  ) ) ( G ς , G  ) > 0 implies ξ + F ( S E ℑ ( a ( ς ) , a (  ) ) ( G ς , G  ) ) ≤ F ( σ ( ς ,  ) ) . Consequently, all the hypotheses of Corollary 2.17 are satisfied with G = Θ. Hence, G has an e-soft fixed point in ℑ.□

Example 2.21. Let ℑ = { 1 n 2 : n ∈ ℕ } ∪ { 0 } , E = [0, 1] and σ (ς, ) = |ς -  |, for all ς,  ∈ ℑ. Then, (ℑ , σ) is a complete metric space. Define a soft set-valued map G : ℑ ⟶ [P (ℑ)]  ^E by ( G e ς ) = { { 1 ρ ( n + 1 ) 2 } , if ς = 1 n 2 , 0 ≤ e < 1 10 , ρ ≥ 2 { 0 } , if ς = 0 , 1 10 ≤ e ≤ 1 . For ς,  ∈ ℑ with ς≠ , without loss of generality, take ς = 1 n 2 and  = 1 m 2 with n , m ∈ ℕ ( n ≠ m ) . Then, S E ℑ ( a ( ς ) , a (  ) ) ( G ς , G  ) = inf E ( G ς , G  ) σ = | 1 ρ ( n + 1 ) 2 - 1 ρ ( m + 1 ) 2 | ≤ 1 ρ | 1 n 2 - 1 m 2 | = η σ ( ς ,  ) . Further, it is clear that G is E-continuous and there exists e ∈ E such that (G _e ς) is a compact subset of ℑ. Hence, all the conditions of Corollary 2.20 are satisfied and 0 is the e-soft fixed point of G. However, G is not a contraction, since sup ς ,  ∈ ℑ , ς ≠  S E ℑ ( a ( ς ) , a (  ) ) ( G ς , G  ) σ ( ς ,  ) = 1 . Therefore, the result of Nadler [32, Theorem 5] is not applicable to this example.

Remark 2.22. When we consider in Definition 2.9, various types of the mapping F, then more corollaries can be derived by using the contractive inequality (2.4). For further reading, one may refer to Wardowski [37].