Fixed points of soft-set valued and fuzzy set-valued maps with applications

Abstract

In this paper, the notion of soft set-valued mappings and E-soft fixed points are introduced. These ideas are used to establish some analogues of classical fixed point theorems in the literature. Some Examples are given to support the hypotheses of the theorems. A few graphical and tabular illustrations are also provided so as to visualize the novel concepts pictorially. Moreover, as an application of one of the presented results, an existence condition for a solution of nonlinear discrete-type delay differential equation is provided.

Keywords

soft set soft set-valued mapping delay differential equation 2010 Mathematics Subject Classification: 46S40,47H10,54H25.

1 Introduction and Preliminary

A number of problems in economics, management sciences, engineering, environmental sciences, medical sciences,etc., involve vagueness and the complexities of modeling uncertain data. Some of these difficulties are basically subjective in nature and hence based on human feelings, understanding, vision system and the like, while others are based on existing facts, yet they are firmly embedded in an imprecise environment. Conventional mathematical tools which require all notions to be exact, are not always sufficient to handle imprecisions in a wide variety of practical fields. In recent time, a large number of theories have been proposed for handling vagueness in a more effective way. Some of these are probability and fuzzy set theory [41 , 49], intuitionistic fuzzy sets [3], rough sets [36], etc. First of these theories was the fuzzy sets introduced by Zadeh [49],in 1965. Fuzzy sets provide an appropriate framework for representing and processing vague concepts by allowing partial memberships. Each of these earlier tools has its inherent drawbacks. As pointed out by Molodtsov [30], all the weakness of earlier mathematical methods are as a result of inadequate parametrization tools. Consequently, Molodtsov [30] initiated a novel concept, called soft theory. This theory is free from the difficulties affecting earlier mathematical techniques to a large extent. Presently, more than a handful of significant results have so far been investigated in theoretical realm of soft set theory [19]. The basic properties of soft sets were proposed in [20 , 32]. Following the basic operations presented by Maji et al [32], Ali et al. [2] proposed some new algebraic operations for soft sets. Thereafter, Maji et al. [31] and Majumdar and Samanta [33] extended the notion of soft sets to fuzzy soft sets. Recently, Ma et al. [29] provided a comprehensive review of some decision making methods based on soft sets and put forward several algorithms in decision making problems by combining various kinds of existing hybrid models. Meanwhile, soft set theory has gained enormous applications in several fields such as machine learning, robotic engineering, computer science, game theory, operation research, Riemann integration, etc. Some advancements in the applications of soft set theory can be found in [1 , 47].

On the other hand, fixed point theory in the framework of metric spaces is one of the most useful mathematical tools in nonlinear functional analysis. Banach contraction principle [11] is the first limelight result in this field. Over the years, the result has witnessed several improvements in different directions. Edelstein [17] first provided a version of Banach contraction theorem by replacing a complete metric with a compact metric space. Also, he established a generalization of the contraction theorem for mappings satisfying less restrictive Lipschitz inequality such as local contraction [18] and locally contractive mappings [17]. In 1969, Nadler [34] generalized the Banach contraction theorem by using the idea of multivalued mappings and contractions. Also, he initiated the idea of multi-valued locally contractive mapping, thereby extending a fixed point theorem of Edelstein [17]. Afterwards, Edelstein’s results were studied by more than a few researchers; see, for example, [5 , 25] and references therein.

In continuation of the above development, Azam eta al [4] extended two Edelstein’s results in [17, 18] on contractive mappings defined on a compact metric space to fuzzy globally and locally contractive mappings and hence obtained fuzzy fixed point of the said mappings. Along the line, the concept of fuzzy mappings was introduced by Heilpern [24] which is a fuzzy generalization of Banach contraction theorem. He also introduced the existence of fuzzy fixed point theorem which improved Nadler’s [34] fixed point theorem for multivalued mappings. Since then, several papers have come up with fixed point theory and fuzzy fixed point theorems of various definitions of contractive mappings. For example, see [23 , 45] and references therein.

In this paper, the concept of soft set-valued mappings is introduced. First, we define the main idea and provide a few examples of the novel scheme. Thereafter, by using the new notion, an analogue of Nadler’s fixed point theorem, Edelstein’s type fixed point theorem on fuzzy globally contractive mappings and fuzzy locally contractive mappings on compact connected metric space are established. In general, our concepts are soft set extensions of fixed point theorems for point-to-point and point-to-set maps presented in [14 , 34] and related results in the literature. Moreover, as an application, we employ one of our results to establish an existence condition for solutions of a delay differential equation.

Let X be an initial universe set and E be a set of parameters. Molodstov [30] defined the notion of soft set in the following manner.

Definition 1.1 [30] A pair (F, E) is called a soft set (over X) if and only if F is a mapping of E into the set of all subsets of X.

In other words, the soft set (F, E) is a parameterized family of subsets of the set X. For each e ∈ E, every set F (ɛ), from this family may be considered as the set of e-elements of (F, E), or as the set of ɛ-approximate elements of the soft set.

In what follows, we initiate the study of soft set-valued mappings. Hereafter, an element e ∈ E associated with a point x ∈ X, shall be written as a (x). In other words, a (x) is not a function of x but rather a notation representing an element of the parameter set E.

2 Soft Set-Valued Mappings

In this section, we reintroduce the notion of soft set-valued mapping, i.e., a mapping with values in the family of soft sets. Whereas, this idea was first time floated by Akbar Azam in “The 11th International Workshop on Dynamical Systems and Applications, June 26–28 (2012), Cankaya University, Ankara, Turkey” in an invited talk entitled: Coincidence Points of Fuzzy Mappings. See [9]. This concept is continued as follows.

Let X be an initial universe, I = [0, 1] and E be a set of parameters. Let A ⊂ E and P (X) be the power set of X. Recall that (F, A) is called a soft set over X, where F : A ⟶ P (X). Denote by [P (X)] ^E, the set of all soft sets over X.

Definition 2.1 A mapping T : X ⟶ [P (X)] ^E is called a soft set-valued mapping. A point x^* ∈ X is said to be an e- soft fixed point of T if x^* ∈ (Tx^*) (a (x^*)), for some a (x^*) = e ∈ DomTx^*. If DomTx^* = E and x^* ∈ (Tx^*) (a (x^*)) for all a (x) ∈ E, then we say that x^* is an E-soft fixed point of T. We write it as x^* ∈ Tx^*.

A clear picture of the concept of soft set-valued mappings can be seen in Fig. 1, where X (theinitialuniverse) = {x₁, x₂, x₃, ⋯} E (setofparameters) = {e₁, e₂, e₃, ⋯} $A_{i} = {e_{i} : i \in Δ \subseteq ℕ} \subset E .$

Fig.1

Pictorial representation of soft set-valued mappings.

A soft set-valued mapping T : X ⟶ [P (X)] ^E can also be seen geometrically by a 3 dimensional(3D) graph as shown in Fig. 2.

Fig.2

3D representation of soft set-valued mappings.

Notice that in Fig. 2, at the cross of e_i and x_j, a dotted vertical line (in the direction of Tx) is the representation of the set (Tx_j) (e_i) ∈ P (X) for each i = 1, 2, 3, ⋯ and j = 1, 2, 3, ⋯.

Example 2.2 Let X = [0, 1] and E = [0, 1]. $Define T : X ⟶ [P (X)]^{E} by$ $(Tx) (a (x)) = {\begin{matrix} [0, \frac{x}{3}], & if a (x) = e_{1} \in E \\ (0, \frac{x}{6}) ⋃ (\frac{x}{4}, \frac{x}{3}], & if a (x) = e_{2} \in E \\ \emptyset, otherwise \end{matrix}$ Then T is a soft set-valued mapping.

Figure 4 is a three-dimensional(3D) graphical representation of the soft set-valued mapping in Example 2.2.

In the following example, we elaborate the concept of soft set-valued mapping and its expected role in decision making problems.

Example 2.3 Suppose that a cricket board of a country organizes a training camp to select best players for finalization of national cricket team. Firstly, a training session has been arranged by three coaches. Secondly, final selection trials are held and these instructors/coaches provide their independent opinions as selectors. After the selection of the team, each player is also asked to give his opinion about coaching and selection by indicating best coach of his choice. The final ranking of players is based on individual aggregate scores from the coaches. This is further illustrated as follows:

Let the initial universe of all players be given by X = {x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀, x₁₁, x₁₂, x₁₃, x₁₄, x₁₅} and M ⊂ E be given by M = {e₁, e₂, e₃}, where e₁, e₂, and e₃ stand for batting, bowling, and fielding, respectively. The independent opinions from the three coaches

$f, g, h : M ⟶ P (X) are given by$ f (e₁) = {x₁, x₃, x₄, x₆, x₈, x₁₀, x₁₁, x₁₃, x₁₄, x₁₅} ,

f (e₂) = {x₁, x₂, x₅, x₆, x₈, x₁₀, x₁₃, x₁₄, x₁₅} ,

f (e₃) = {x₁, x₂, x₃, x₅, x₆, x₈, x₁₀, x₁₁, x₁₂, x₁₃, x₁₄, x₁₅} ,

g (e₁) = {x₁, x₂, x₃, x₅, x₇, x₈, x₁₀, x₁₁, x₁₂, x₁₃, x₁₄} ,

g (e₂) = {x₁, x₂, x₄, x₅, x₆, x₉, x₁₀, x₁₁, x₁₂, x₁₃, x₁₄} ,

g (e₃) = {x₁, x₂, x₄, x₅, x₆, x₈, x₁₁, x₁₂, x₁₃, x₁₄, x₁₅, }

h (e₁) = {x₃, x₄, x₆, x₇, x₉, x₁₀, x₁₃, x₁₅} ,

h (e₂) = {x₁, x₂, x₃, x₇, x₈, x₁₁, x₁₃, x₁₄, x₁₅} ,

h (e₃) = {x₃, x₄, x₈, x₁₀, x₁₃, x₁₄} .

This phenomenon can be seen in Fig. 3.

Fig.3

Graphical representation of Example 2.3.

Fig.4

3D representation of the soft set-valued mappings in Example 3.2.

Key representation of Fig. 3 is shown in Table 1.

Table 1

Keys

Note that in Fig. 3, at the cross of x₁ and e₁, ⊛ means the player x₁ is fit for batting according to the opinion of two coaches f and g, but this player is not fit for batting according to the opinion of coach h. At the cross of x₁ and e₂, boxed⊛ means the player x₁ is fit for bowling according to the opinion of the three coaches. At the cross of x₁ and e₃, ⊛ means the player x₁ is fit for fielding according to the opinion of two coaches f and g, but the player is not fit for fielding in the opinion of coach h. Thus, the total score of the player x₁ is 7 according to the opinion of the three coaches. Similarly, the total scores of the rest players are calculated accordingly. At a glance from the table, one can see that the first two best players are x₁₃ and x₁₄ with respective scores 9 and 8. Also, players x₁, x₈ and x₁₀ have a tie in the second position with a score of 7.

Next, the opinions of the players regarding the best coach of their choices is represented in Table 2.

Table 2

Opinion Polls of the Players

Coaches	Players
f	x ₁	x ₃	x ₆	x ₁₀	x ₁₄	x ₁₅
g	x ₂	x ₅	x ₉	x ₁₁	x ₁₃
h	x ₄	x ₈	x ₇	x ₁₂

According to Table 2, coach f has the highest opinion polls of the players and hence is considered as the best coach for the national cricket team.

Now, define a soft set-valued mapping T : X ⟶ [P (X)] ^E as $Tx = {\begin{matrix} f, & if x \in {x_{1}, x_{3}, x_{6}, x_{10}, x_{14}, x_{15}} \\ g, & if x \in {x_{2}, x_{5}, x_{9}, x_{11}, x_{13}} \\ h, & if x \in {x_{4}, x_{7}, x_{8}, x_{12}} . \end{matrix}$ Note that x₁₃ ∈ Tx₁₃. That is, the best player is an E-soft fixed point of T; whereas the weakest player x₉ is not an E-soft fixed point of T.

For computer purpose, the soft set-valued mapping in Example 2.3 may be presented in tabular form as in Table 3.

Table 3

Tabular representation of soft set-valued mapping

Tx	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈	x ₉	x ₁₀	x ₁₁	x ₁₂	x ₁₃	x ₁₄	x ₁₅
f	1	0	1	0	0	1	0	0	0	1	0	0	0	1	1
g	0	1	0	0	1	0	0	0	1	0	1	0	1	0	0
h	0	0	0	1	0	0	1	1	0	0	0	1	0	0	0

Remark 2.4. It is known that every fuzzy set may be considered as a soft set (See [30]).

Similarly, every fuzzy mapping S : X ⟶ I^X can be thought as soft set-valued mapping Ω_S : X ⟶ [P (X)] ^[0,1], defined by $Ω_{S} (x) (e) = {t \in X : (Sx) (t) \geq e} .$

Notice that X ⟼ P (X) is embedding by x ⟶ {x} and P (X) ⟼ I^X is embedding by A ⟶ χ_A, for every subset A of P (X); where χ_A is the characteristic function of the crisp set A. Similarly, I^X ⟼ [P (X)] ^[0,1] is embedding by F ⟶ Γ_F, for every F in I^X; where $Γ_{F} (e) = F_{e} = {t \in X : F (t) \geq e} .$

In short words, every fuzzy mapping is a special kind of soft set-valued mapping. Therefore, it is reasonable to study the properties of soft set-valued mapping regarding solution of functional equations in connection with fixed point theorems.

3 Fixed Point Theorems

Now, we present a few fixed point results in the setting of soft set-valued mappings. First, we give the following preliminary definitions/concepts. For a metric space X, let CB (X) and C (X) denote the set of all nonempty closed bounded and compact subsets of X, respectively. Consider two soft sets (F, A) and (G, B), (a, b) ∈ A × B. Assume that F (a) , G (b) ∈ CB (X). For ∊ > 0, define N^d (∊, F (a)), $S_{EX}^{(a, b)} (F, G)$ , $E_{(F_{a}, G_{b})}^{d}$ as follows: $N^{d} (∊, F (a)) = {x \in X : d (x, y) < ∊,$ $for some y \in F (a)},$ $\begin{matrix} E_{(F_{a}, G_{b})}^{d} = & {∊ > 0 : F (a) \subseteq N^{d} (∊, G (b)), \\ G (b) \subseteq N^{d} (∊, F (a))}, \end{matrix}$ $S_{EX}^{(a, b)} (F, G) = inf E_{(F_{a}, G_{b})}^{d},$ $p_{EX}^{a (x)} (x, F) = inf {d (x, y) : y \in F (a)} .$ Define a distance function $S_{EX}^{\infty} : [P (X)]^{E} \times [P (X)]^{E} ⟶ ℝ by$ $S_{EX}^{\infty} (F, G) = sup_{(a, b) \in \bar{A} \times \bar{B}} S_{EX}^{(a, b)} (F, G), where$ $\bar{A} \times \bar{B} = {(a, b) \in A \times B : F (a), G (b) \in CB (X)} .$ and ${\hat{S}}_{EX}^{\infty} : [P (X)]^{E} \times [P (X)]^{E} ⟶ ℝ$ by ${\hat{S}}_{EX}^{\infty} (F, G) = sup_{(a, b) \in \hat{A} \times \hat{B}} {\hat{S}}_{EX}^{(a, b)} (F, G), where$ $\hat{A} \times \hat{B} = {(a, b) \in A \times B : F (a), G (b) \in C (X)} .$

Theorem 3.1. Let (X, d) be a complete metric space, f : X ⟶ X be a surjection with $fx = \bar{x}$ , E be the parameter set and T : X ⟶ [P (X)] ^E a soft set-valued mapping. Suppose that for x ∈ X, there exists a (x) ∈ DomTx such that (Tx) (a (x)) ∈ CB (X). If there exists λ ∈ (0, 1) such that for all x, y ∈ X, $S_{EX}^{(a (x), a (y))} (Tx, Ty) \leq λ d (\bar{x}, \bar{y}),$ (3.1) then there exists $\bar{u} \in X$ such that $\bar{u} \in (Tu) (a (u))$ , for some a (u) ∈ E.

Proof. Let x₀ ∈ X, then ∅ ≠ (Tx₀) (a (x₀)) is a closed and bounded subset of X. Let $\bar{x_{1}} \in ({Tx}_{0}) (a (x_{0}))$ . It follows that ∅ ≠ (Tx₁) (a (x₁)) is closed and bounded and for $β = \sqrt{λ}$ , $\begin{matrix} S_{EX}^{(a (x_{0}), a (x_{1}))} ({Tx}_{0}, {Tx}_{1}) & \leq & λ d (\bar{x_{0}}, \bar{x_{1}}) \\ < & β d (\bar{x_{0}}, \bar{x_{1}}) = η (say) . \end{matrix}$ Then $η \in E_{(({Tx}_{0}) (a (x_{0})), ({Tx}_{1}) (a (x_{1})))}^{d} .$ This implies that $({Tx}_{0}) (a (x_{0})) \subseteq N^{d} (η, ({Tx}_{1}) (a (x_{1})))$ and $({Tx}_{1}) (a (x_{1})) \subseteq N^{d} (η, ({Tx}_{0}) (a (x_{0}))) .$ Thus, $\bar{x_{1}} \in N^{d} (η, ({Tx}_{1}) (a (x_{1})))$ . Consequently, there exists some $\bar{x_{2}} \in ({Tx}_{1}) (a (x_{1}))$ such that $d (\bar{x_{1}}, \bar{x_{2}}) < η .$ In the same way, there exists $\bar{x_{2}} \in N^{d} (β d (\bar{x_{1}}, \bar{x_{2}}), ({Tx}_{2}) (a (x_{2})))$ . Hence, we can find $\bar{x_{3}} \in ({Tx}_{2}) (a (x_{2}))$ such that $\begin{matrix} d (\bar{x_{2}}, \bar{x_{3}}) & < & β d (\bar{x_{1}}, \bar{x_{2}}) \\ < & β^{2} d (\bar{x_{0}}, \bar{x_{1}}) . \end{matrix}$ By continuing this process, we can generate a sequence ${\bar{x_{n}}}_{n \in ℕ}$ such that ${\bar{x}}_{n + 1} \in ({Tx}_{n}) (a (x_{n}))$ with $d ({\bar{x}}_{n}, {\bar{x}}_{n + 1}) \leq β^{n} d ({\bar{x}}_{0}, {\bar{x}}_{1}) .$ Moreover, for $m, n \in ℕ$ , with m > n, by triangle inequality, we have $\begin{matrix} d ({\bar{x}}_{n}, {\bar{x}}_{m}) & \leq & d ({\bar{x}}_{n}, {\bar{x}}_{n + 1}) + d ({\bar{x}}_{n + 1}, {\bar{x}}_{n + 2}) \\ + \dots + d ({\bar{x}}_{m - 1}, {\bar{x}}_{m}) \\ \leq & (β^{n} + β^{n + 1} + \dots + β^{m - 1}) d ({\bar{x}}_{0}, {\bar{x}}_{1}) \\ \leq & \frac{β^{n}}{1 - β} d ({\bar{x}}_{0}, {\bar{x}}_{1}) ⟶ 0 as ⟶ \infty . \end{matrix}$ Hence, $d ({\bar{x}}_{n}, {\bar{x}}_{m}) ⟶ 0$ as n, m⟶ ∞. This proves that ${{\bar{x}}_{n}}_{n \in ℕ}$ is a Cauchy sequence of elements of X. Therefore, the completeness of X implies that there exists $\bar{u} \in X$ such that ${\bar{x}}_{n} ⟶ \bar{u}$ as n⟶ ∞.

Now, by using triangle inequality along with the definition of $S_{EX}^{(a, b)} (,)$ we get $\begin{matrix} d (\bar{u}, (Tu) (a (u))) & \leq & d (\bar{u}, {\bar{x}}_{n}) + d ({\bar{x}}_{n}, (Tu) (a (u))) \\ \leq & d (\bar{u}, {\bar{x}}_{n}) + λ d ({\bar{x}}_{n - 1}, \bar{u}) \\ \leq & 0 as n ⟶ \infty . \end{matrix}$ Consequently, $\bar{u} \in (Tu) (a (u))$ , for some a (u) ∈ E.

Next, we give an example to validate the assumptions of Theorem 3.1.

Example 3.2. Let X = [0, ∞), E = [0, 1], and $d : X \times X ⟶ ℝ be defined by$ $d (x, y) = | x - y |, for all x, y \in X with x \neq y .$ Then (X, d) is a complete metric space. Define a soft set-valued mapping T : X ⟶ [P (X)] ^E by $(Tx) (a (x)) = {\begin{matrix} [0, 3 x], & if a (x) = e_{1} \in E \\ (0, 3 x), & if a (x) = e_{2} \in E \\ \emptyset, & otherwise \end{matrix}$ where e₁ < e₂ and f : X ⟶ X is defined by $fx = 4 x = \bar{x}$ . Then for x ∈ X, there exists a (x) = e₁ ∈ E such that (Tx) (a (x)) ∈ CB (X). For two soft sets Tx, Ty : E ⟶ P (X) and $\bar{E} \times \bar{E} = {(e_{1}, e_{1}) \in E \times E}$ , consider the following two cases:

Case I: If x < y, then $E_{((Tx) (e_{1}), (Ty) (e_{1}))}^{d} = [3 y - 3 x, \infty) .$ Case II: If x > y, then $E_{((Tx) (e_{1}), (Ty) (e_{1}))}^{d} = [3 x - 3 y, \infty) .$ From the above two cases, we have $\begin{matrix} S_{EX}^{(a (x), a (y))} (Tx, Ty) & \leq & S_{EX}^{\infty} (Tx, Ty) \\ = & | 3 x - 3 y | \\ \leq & 3 | x - y | \\ \leq & \frac{3}{4} | 4 x - 4 y | \\ \leq & \frac{3}{4} | \bar{x} - \bar{y} | \\ \leq & λ d (\bar{x}, \bar{y}) . \end{matrix}$ Therefore, all the conditions of Theorem 3.1 are satisfied to obtain $\bar{0} \in (T 0) (e_{1})$ for some e₁ ∈ E.

Corollary 3.3. Let (X, d) be a complete metric space, f : X ⟶ X be a surjection with $fx = \bar{x}$ , E be the parameter set and T : X ⟶ [P (X)] ^E a soft set-valued mapping. Suppose that for x ∈ X, there exists a (x) ∈ DomTx such that (Tx) (a (x)) ∈ CB (X). If there exists λ ∈ (0, 1) such that for all x, y ∈ X, $S_{EX}^{\infty} (Tx, Ty) \leq λ d (\bar{x}, \bar{y}),$ (3.2) then there exists $\bar{u} \in X$ such that $\bar{u} \in (Tu) (a (u))$ , for some a (u) ∈ E.

Proof. Since $S_{EX}^{(a (x), a (y))} (Tx, Ty) \leq S_{EX}^{\infty} (Tx, Ty),$ then the proof follows directly by Theorem 3.1.

A well-known fixed point theorem due to Edelstein [17] states that if (X, d) is a compact metric space and T : X ⟶ X is a contractive mapping (i . e d (Tx, Ty) < d (x, y) , ∀ x, y ∈ X), then T has a unique fixed point in X. This result has been generalized and applied in different directions. A few of these extensions can be found in [4 , 25]. In the next theorem, we extend this idea to a soft contractive mapping in the setting of soft set-valued mapping.

Definition 3.4. A soft set-valued mapping T : X ⟶ [P (X)] ^E is called soft contraction (or soft globally contraction) if there exists an η ∈ (0, 1) such that ${\hat{S}}_{EX}^{\infty} (Tx, Ty) \leq η d (x, y), for all x, y \in X .$

Definition 3.5. A soft set-valued mapping T : X ⟶ [P (X)] ^E is said to be soft contractive (soft globally contractive) if for all x, y ∈ X, x ≠ y, ${\hat{S}}_{EX}^{\infty} (Tx, Ty) < d (x, y) .$ (3.3)

Theorem 3.6. Let (X, d) be a compact metric space and T : X ⟶ [P (X)] ^E be a soft contractive mapping. If for x ∈ X, there exists a (x) ∈ DomTx such that (Tx) (a (x)) is a nonempty compact subset of X, then there exists u ∈ X such that u ∈ (Tu) (a (u)), for some a (u) ∈ E.}

Proof. For each x ∈ X, by hypothesis, there exists a (x) ∈ DomTx such that (Tx) (a (x)) is compact. Define a real-valued function $f : X ⟶ ℝ$ by $f (x) = p_{EX}^{a (x)} (x, Tx) .$ This implies that $\begin{matrix} f (x) & = & p_{EX}^{a (x)} (x, Tx) \\ \leq d (x, y) + p_{EX}^{a (x)} (y, Tx) \\ \leq d (x, y) + p_{EX}^{a (x)} (y, Ty) \\ + {\hat{S}}_{EX}^{(a (x), a (y))} (Tx, Ty) \\ \leq d (x, y) + p_{EX}^{a (x)} (y, Ty) \\ + sup_{(a, b) \in \hat{E} \times \hat{E}} {\hat{S}}_{EX}^{(a (x), a (y))} (Tx, Ty) \\ = d (x, y) + f (y) + {\hat{S}}_{EX}^{\infty} (Tx, Ty) . \end{matrix}$ This gives $f (x) - f (y) \leq d (x, y) + {\hat{S}}_{EX}^{\infty} (Tx, Ty) .$ By symmetry, we have $| f (x) - f (y) | \leq d (x, y) + {\hat{S}}_{EX}^{\infty} (Tx, Ty) .$ (3.4) By using definition 3.5 together with (3.4), it follows that the mapping f is continuous. By compactness, f assumes all its bounds in X. Suppose f attains its minimum at u. Since (Tu) (a (u)) ∈ C (X), we can choose x₁ ∈ X such that x₁ ∈ (Tu) (a (u)) and $d (u, x_{1}) = p_{EX}^{a (x)} (u, Tu) = f (u) .$ Then u ∈ (Tu) (a (u)). If not, then by definition of ${\hat{S}}_{EX}^{\infty} (,)$ and soft contractive mapping, it implies that $\begin{matrix} f (x_{1}) & = & p_{EX}^{a (x)} (x_{1}, {Tx}_{1}) \\ \leq & {\hat{S}}_{EX}^{\infty} ({Tx}_{1}, Tu) \\ < & d (x_{1}, u) = f (u), \end{matrix}$ which is a contradiction to the fact that f attains its minimum at u. Consequently, u ∈ (Tu) (a (u)), for some a (u) ∈ E.

Another important result due to Edelstein [17] states that if (X, d) is a compact and connected metric space and T : X ⟶ X is a locally contractive mapping (that is, for each x ∈ X, there exists an open set G containing x such that for any y, z ∈ X with y ≠ z, d (Ty, Tz) < d (y, z)), then T has a unique fixed point in X. This theorem has been extended by several authors (see, for example [4, 15]). In what follows, we extend this concept to soft locally contractive mappings.

Definition 3.7. A soft set-valued mapping T : X ⟶ [P (X)] ^E is known as soft locally contractive if each x in X belongs to an open set G such that if y, z ∈ G, y ≠ z, ${\hat{S}}_{EX}^{\infty} (Ty, Tz) < d (y, z) .$ Definition 3.8. Let (X, d) be a metric space. For x, y ∈ X, an ∊-chain from x to y is a finite set of points x₀, x₁, x₂, ⋯ , x_n such that x = x₀, y = x_n and d (x_i, x_i+1) ≤ ∊ for all i = 0, 1, 2, ⋯ n - 1.

Lemma 3.9. [42] Let (X, d) be a compact and connected metric space. Then for each ∊ > 0 and x, y ∈ X, there exists an ∊-chain from x to y and a mapping $d^{∊} : X \times X ⟶ ℝ$ defined by $\begin{matrix} d^{∊} (x, y) = & inf {\sum_{i = 0}^{n - 1} d (x_{i}, x_{i + 1}) \\ : x_{0}, x_{1}, x_{2}, \dots, x_{n} is an ∊ - chain from x to y} \end{matrix}$ is a metric on X equivalent to d. Furthermore, for any x, y ∈ X and ∊ > 0, there exists an ∊-chain x = x₀, x₁, x₂, ⋯ , x_n = y such that $d^{∊} (x, y) = \sum_{i = 0}^{n - 1} d (x_{i}, x_{i - 1}) .$ Theorem 3.10. Let (X, d) be a compact connected metric space and E be the parameter set. Suppose that for x ∈ X, there exists a (x) ∈ E such that (Tx) (a (x)) is a compact subset of X. If T : X ⟶ [P (X)] ^E is a soft locally contractive mapping, then there exists u ∈ X such that u ∈ (Tu) (a (u)), for some a (u) ∈ E.}

Proof. By Definition 3.7, it follows that for each x ∈ X, there exists an open set G such that for any y ≠ z ∈ G, ${\hat{S}}_{EX}^{(a (y), a (z))} (Ty, Tz) < d (y, z) .$ Now, by Lemma 3.9, for each ∊ > 0 and p, q ∈ X, there exists an ∊-chain p = x₀, x₁, x₂, ⋯ , x_n = q from p to q. Since X is compact, we can find an η > 0 such that if x ≠ y and d (x, y) < η, then ${\hat{S}}_{EX}^{(a (x), a (y))} (Tx, Ty) < d (x, y) .$ Define $d^{*} : X \times X ⟶ ℝ$ by $d^{*} (p, q) = inf {\sum_{i = 0}^{n - 1} d (x_{i}, x_{i + 1}) : x_{0}, x_{1}, x_{2}, \dots, x_{n}}$ is an $\frac{η}{2}$ - chain from p to q. That is, $d^{*} = d^{\frac{η}{2}}$ . By Lemma 3.9, d^* is a metric on X equivalent to d and there exists an $\frac{η}{2}$ -chain p = x₀, x₁, x₂, ⋯ , x_n = q from p to q such that $d^{*} (p, q) = \sum_{i = 0}^{n - 1} d (x_{i}, x_{i + 1}) .$ (3.5)

Now, $d (x_{i}, x_{i + 1}) \leq \frac{η}{2} < η$ implies ${\hat{S}}_{EX}^{(a (x), a (y))} (Tx, Ty) < d (x_{i}, x_{i + 1}) < η .$

This implies that $d (x_{i}, x_{i + 1}) - {\hat{S}}_{EX}^{(a (x), a (y))} (Tx, Ty) > 0 .$ Suppose $t_{i} = d (x_{i}, x_{i + 1}) - {\hat{S}}_{EX}^{(a (x), a (y))} (Tx, Ty),$ for i = 0, 1, 2, ⋯ , n - 1. Then t_i > 0 and ${\hat{S}}_{EX}^{(a (x), a (y))} (Tx, Ty) < d (x_{i}, x_{i + 1}) - \frac{t_{i}}{2},$ (3.6) for i = 0, 1, 2, ⋯ , n - 1. Next, we show that $({Tx}_{0}) (a (x_{0})) \subset N^{d^{*}} (l, ({Tx}_{n}) (a (x_{n}))),$ (3.7) for some l > 0. Let y₀ ∈ (Tx₀) (a (x₀)). Using (3.6), we can choose y₁ ∈ (Tx₁) (a (x₁)) such that $d (y_{0}, y_{1}) < d (x_{0}, x_{1}) - \frac{t_{0}}{2} .$ On similar steps, we can find y₂ ∈ (Tx₂) (a (x₂)) such that $d (y_{1}, y_{2}) < d (x_{1}, x_{2}) - \frac{t_{1}}{2} .$ Continuing this process, we can find a set of points y_i, i = 0, 1, 2, ⋯ , n, where y_i ∈ (Tx_i) (a (x_i)) such that $d (y_{i - 1}, y_{i}) < d (x_{i - 1}, x_{i}) - \frac{t_{i - 1}}{2},$ for i = 0, 1, 2, ⋯ , n - 1. Consequently, y₀, y₁, y₂, ⋯ , y_n is an $\frac{η}{2}$ -chain from y₀ to y_n. Therefore, $d^{*} (y_{0}, y_{1}) = inf {\sum_{i = 0}^{n - 1} d (x_{i}, x_{i + 1}) : x_{0}, x_{1}, \dots, x_{n}}$ is an $\frac{η}{2}$ - chain from y₀ to y_n. This implies that $\begin{matrix} d^{*} (y_{0}, y_{1}) & \leq & \sum_{i = 0}^{n - 1} d (y_{i}, y_{i + 1}) \\ < & \sum_{i = 0}^{n - 1} (d (x_{i}, x_{i + 1}) - \frac{t_{i}}{2}) \\ = & \sum_{i = o}^{n - 1} d (x_{i}, x_{i - 1}) - \sum_{i = 0}^{n - 1} (\frac{t_{i}}{2}) . \end{matrix}$ Now, by (3.5), we have $d^{*} (y_{0}, y_{n}) < d^{*} (p, q) - \sum_{i = 0}^{n - 1} (\frac{t_{i}}{2}) .$ Let $l = d^{*} (p, q) - \sum_{i = 0}^{n - 1} (\frac{t_{i}}{2}) .$ Then l > 0 and y₀ ∈ N^{d
^*} (l, (Tx_n) (a (x_n))). Hence (3.7) holds. Next, we show that $({Tx}_{n}) (a (x_{n})) \subset N^{d^{*}} (l, ({Tx}_{0}) (a (x_{0}))) .$ (3.8) Let w_n ∈ (Tx_n) (a (x_n)) be arbitrary. Again, using (3.6), we may choose w_n-1 ∈ (Tx_n-1) (a (x_n-1)) such that $d (w_{n - 1}, w_{n}) < d (x_{0}, x_{1}) - \frac{t_{i - 1}}{2} .$ On similar steps, we obtain an $\frac{η}{2}$ -chain w₀, w₁, w₂, ⋯ , w_n from w₀ to w_n, where $d (w_{0}, w_{n}) < d^{*} (p, q) - \sum_{i = 0}^{n - 1} (\frac{t_{i}}{2}) = l .$ It follows that w_n ∈ N^{d
^*} (l, (Tx₀) (a (x₀)). Thus, (3.8) holds. From (3.7) and (3.8), we have $l \in E_{(({Tx}_{0}) (a (x_{0})), ({Tx}_{n}) (a (x_{n})))}^{d^{*}}$ . Therefore, ${\hat{S}}_{EX}^{* (a (x_{0}), a (x_{0}))} (({Tx}_{0}) (a (x_{0})), ({Tx}_{n}) (a (x_{n}))) < l .$ Consequently, $\begin{matrix} {\hat{S}}_{EX}^{* (a (p), a (q))} ((Tp) (a (p)), (Tq) (a (q))) \\ < & d^{*} (p, q) \\ - & \sum_{i = 0}^{n - 1} (\frac{t_{i}}{2}) \\ < & d^{*} (p, q) . \end{matrix}$ Hence, for all x, y ∈ X, ${\hat{S}}_{EX}^{* (a (x), a (y))} (Tx, Ty) < d^{*} (x, y) .$ (3.9) Define a real-valued function $f : X ⟶ ℝ$ by $f (x) = P_{EX}^{* a (x)} (x, Tx) .$ This implies that $f (p) \leq d^{*} (p, q) + P_{EX}^{* a (p)} (q, Tp)$ (3.10) $\begin{matrix} \leq d^{*} (p, q) + P_{EX}^{* a (q)} (q, Tq) + {\hat{S}}_{EX}^{* (a (p), a (q))} (Tp, Tq) \\ \leq d^{*} (p, q) + P_{EX}^{* a (q)} (q, Tq) + {\hat{S}}_{EX}^{\infty} (Tp, Tq) . \end{matrix}$ Now, using (3.9) and (3.10), we obtain $f (p) \leq d^{*} (p, q) + P_{EX}^{* a (q)} (q, Tq) + d^{*} (p, q) .$ Therefore, $f (p) - f (q) \leq 2 d^{*} (p, q) .$ By symmetry, we have $| f (p) - f (q) | \leq 2 d^{*} (p, q) .$ This shows that f is continuous. By compactness of X, the function f assumes all its bounds in X. Suppose that min f (x^*) = τ. Then τ = 0. Otherwise, notice that by compactness of (Tx^*) (a (x^*)), we can choose q₁ ∈ (Tx^*) (a (x^*)) such that $d^{*} (x^{*}, q_{1}) = f (x^{*}) = τ$ and $\begin{matrix} f (q_{1}) & = & P_{EX}^{* a (q_{1})} (q_{1}, {Tq}_{1}) \\ \leq & {\hat{S}}_{EX}^{* (a (x^{*}), a (q_{1}))} ({Tx}^{*}, {Tq}_{1}) \\ \leq & {\hat{S}}_{EX}^{\infty} ({Tx}^{*}, {Tq}_{1}) \\ < & d (x^{*}, q_{1}) = τ . \end{matrix}$ (3.11) Observe that the last expression is a contradiction to the definition of minimum of f at x^*. Consequently, τ = 0. This proves that x^* = u ∈ (Tu) (a (u)) , for some a (u) ∈ E.

Applications

3.1 Fixed points of fuzzy and multivalued mappings

As an application of the E-soft fixed point result (Theorem 3.1), in this subsection, we obtain fixed points of fuzzy mappings and multi-valued mappings of Heilpern [24, Th. 3.1] and Nadler [34, Th. 5].

Definition 3.11. [24] A fuzzy set A in X is called an approximate quantity if and only if its α-level set is a compact convex subset(nonfuzzy)subset of X for each α ∈ [0, 1] and $sup_{x \in X} A (x) = 1$ . The set of all approximate quantities in X is denoted by W (X).

Theorem 3.12. (Heilpern [24]) Let X be a complete linear metric space and F : X ⟶ W (X) be a fuzzy mapping. If there exists λ ∈ (0, 1) such that $d_{\infty} (Fx, Fy) \leq λ d (x, y), for each x, y \in X,$ then there exists u ∈ X such that χ_{u} ⊂ F (u).}

Proof. Consider a soft set-valued mapping $Ω_{F} : X ⟶ [P (X)]^{E},$ defined by $Ω_{F} x (e) = {t \in X : (Fx) (t) \geq e} = [Fx]_{e} .$ Then $\begin{matrix} S_{EX}^{(1, 1)} (Ω_{F} x, Ω_{F} y) & = & inf E_{((Fx) (1), (Fy) (1))}^{d} \\ = & H ([Fx]_{1}, [Fy]_{1}) \\ \leq & d_{\infty} (Fx, Fy) \\ \leq & λ d (x, y) . \end{matrix}$ Thus, Theorem 3.1 can be applied with f = I_X, the identity mapping on X, to obtain u ∈ X such that u ∈ (Ω_Fu) (1) = [Fu] ₁. It follows that χ_{u} ⊂ F (u).

Theorem 3.13. (Nadler [34]) Let (X, d) be a complete metric space and $M : X ⟶ CB (X)$ be a multi-valued mapping satisfying the following conditions: there exists λ ∈ (0, 1) such that $H (M_{x}, M_{y}) \leq λ d (x, y) .$ Then there exists u ∈ X such that u ∈ Mu.}

Proof. Let E = {e₁, e₂} and consider a soft set-valued mapping $Ω : X ⟶ [P (X)]^{E},$ defined by $Ω x (e) = {\begin{matrix} X, & if e = e_{1} \\ Mx, & if e = e_{2} . \end{matrix}$ Then $\begin{matrix} S_{EX}^{(e_{2} (x), e_{2} (y))} (Ω x, Ω y) & = & inf E_{((Ω x) (e_{2} (x)), (Ω y) (e_{2} (y)))}^{d} \\ = & inf E_{(Mx, My)}^{d} \\ = & H (Mx, My) \\ \leq & λ d (x, y) . \end{matrix}$ Therefore, Theorem 3.1 can be applied with f = I_X, the identity mapping on X, to find u ∈ X such that $u \in Ω u (e_{2} (u)) = Mu .$

Remark 3.14. Notice that Theorem 3.1 cannot be obtained as corollary from Theorem 3.13 of Nadler [34] and Theorem 3.12 due to Heilpern [24]. This can be seen in Example 3.2.

3.2 Application to delay differential equations

A delay differential equation is a differential equation where the differential coefficient at a given current time depends on the solution at previous time. Such equations are also called differential equations with retarded argument. Strictly speaking. It is a specific example of a functional differential equation in which the functional part of the differential equation is the evaluation of a functional on the past of the process.

If x (t) is a function of time, then the delay η in the argument of x (t - η) is called a discrete delay and delay differential equations involving only discrete delay are said to be of discrete delay type. For some introductory terminologies on delay differential equations, the interested reader may consult [46]. In this subsection, using Theorem 3.1, we establish an existence result for a solution of nonlinear discrete-type delay differential equation: $x^{'} (t) = f (t, x (t), x (t - η)), t \in [t_{0}, b],$ (3.12) with initial data $x (t) = φ (t), t \in [t_{0} - η, t_{0}] .$ (3.13)Theorem 3.15. Consider Eq.(3.12). Suppose

$f : [t_{0}, b] \times ℝ^{2} ⟶ ℝ$ and $φ : [t_{0} - η, b] ⟶ ℝ$ are continuous;

for all $x, y \in C ([t_{0} - η, b], ℝ)$ , there exists $k \in (0, \frac{1}{2 (b - t_{0})})$ such that $| f (t, p_{1}, p_{2}) - f (t, q_{1}, q_{2}) | \leq k \sum_{i = 1}^{2} | p_{i} - q_{i} |,$ $\forall p_{i}, q_{i} \in ℝ, i = 1, 2, t \in [t_{0}, b] .$

Then problem (3.12) has a solution in $C ([t_{0} - η, b], ℝ) \cap C^{'} ([t_{0}, b], ℝ)$ .

Proof. From (3.12), we have $\begin{matrix} x (t) & = & x (t_{0}) + \int_{t_{0}}^{t} f (s, x (s), x (s - η)) ds \\ = & φ (t_{0}) + \int_{t_{0}}^{t} f (s, x (s), x (s - η)) ds . \end{matrix}$ Therefore, an equivalent reformulation of (3.12) is given by $\begin{matrix} x (t) = \\ {\begin{matrix} φ (t), & if t \in [t_{0} - η, t_{0}] \\ φ (t_{0}) + \int_{t_{0}}^{t} f (s, x (s), x (s - η)) ds, & if t \in [t_{0}, b] . \end{matrix} \end{matrix}$ Let E = [0, ∞) and $X = (C [t_{0} - η, b], ℝ)$ be endowed with the Tehebyshev metric $d : X \times X ⟶ ℝ$ defined by $d (x, y) = sup_{t_{0} \leq t \leq b} | x (t) - y (t) |, \forall x, y \in X .$ Then (X, d) is a compact and connected metric space. Suppose for x ∈ X, we have $w_{x} (t) = {\begin{matrix} φ (t_{0}) + \int_{t_{0}}^{t} f (s, x (s), x (s - η)) ds, \\ if t \in [t_{0}, b] . \\ φ (t), \\ if t \in [t_{0} - η, t_{0}] . \end{matrix}$ Define T : X ⟶ [P (X)] ^E by $\begin{matrix} (Tx) (a (x)) = \\ {\begin{matrix} {x \in X : x (t) < w_{x} (t)}, & if a (x) < 1 \\ {x \in X : x (t) = w_{x} (t)}, & if a (x) = 1 \\ {x \in X : x (t) > w_{x} (t)}, & if a (x) > 1 . \end{matrix} \end{matrix}$ Then for x ∈ X, there exists a (x) =1 ∈ E such that $(Tx) (a (x)) = {w_{x} (t)} .$ Now, for x, y ∈ X with x ≠ y, we have $\begin{matrix} S_{EX}^{\infty} (Tx, Ty) = & sup_{(a, b) \in E \times E} S_{EX}^{(a (x), a (y))} (Tx, Ty) \\ = & sup {| w_{x} (t) - w_{y} (t) | : t \in [t_{0}, b]} \\ \leq & \int_{t_{0}}^{t} sup_{s \in [t_{0}, b]} | f (s, x (s), x (s - η)) \\ - f (s, y (s), y (s - η)) | ds \\ \leq & \int_{t_{o}}^{t} k sup_{s \in [t_{0}, b]} (| x (s) - y (s) | \end{matrix}$ $\begin{matrix} + | x (s - η) - y (s - η) |) ds \\ \leq & \int_{t_{0}}^{t} k (sup_{s \in [t_{0}, b]} | x (s) - y (s) | \\ + sup_{s \in [t_{0}, b]} | x (s - η) - y (s - η) |) ds \\ \leq & \int_{t_{0}}^{t} k (d (x, y) + d (x, y)) ds \\ \leq & 2 k (t - t_{0}) d (x, y) \\ \leq & d (x, y) . \end{matrix}$ Hence, all the conditions of Theorem 3.1 are satisfied to obtain a continuous function x (t) ∈ X such that x (t) ∈ (Tx) (a (x)) for some a (x) ∈ E. Consequently, the e-soft fixed point x (t) will be a solution of (3.12).

Example 3.16. Consider the delay differential equation $x^{'} (t) = 2 t^{5} + \frac{x^{7} (t)}{12} + \frac{x^{7} (t - \frac{1}{5})}{12}, t \in [1, 2],$ (3.14) with initial data $x (t) = t + 2, t \in [\frac{4}{5}, 1],$ where $η = \frac{1}{5}$ represents a single delay, φ (t) = t + 2 and $f (t, x (t), x (t - η)) = 2 t^{5} + \frac{x^{7} (t)}{12} + \frac{x^{7} (t - \frac{1}{5})}{12} .$ The integral equivalent reformulation of (3.14) is given by: $x (t) = {\begin{matrix} t + 2, \\ if t \in [\frac{4}{5}, 1], \\ φ (1) + \int_{1}^{t} (2 s^{5} + \frac{x^{7} (s)}{12} + \frac{x^{7} (s - \frac{1}{5})}{12}) ds, \\ if t \in [1, 2] . \end{matrix}$ Let $p_{i}, q_{i} \in ℝ$ , i = 1, 2 and t ∈ [1, 2]. Then $\begin{matrix} | f (t, p_{1}, p_{2}) - f (t, q_{1}, q_{2}) | = & | \frac{1}{12} (p_{1}^{7} - q_{1}^{7}) \\ + \frac{1}{12} (p_{2}^{7} - q_{2}^{7}) | \\ \leq & | \frac{1}{12} (p_{1}^{7} - q_{1}^{7}) | \end{matrix}$ $\begin{matrix} + | \frac{1}{12} (p_{2}^{7} - q_{2}^{7}) | \\ \leq & \frac{1}{12} | p_{1}^{7} - q_{1}^{7} | \\ + \frac{1}{12} | p_{2}^{7} - q_{2}^{7} | \\ \leq & \frac{1}{12} \sum_{i = 1}^{2} | p_{i}^{7} - q_{i}^{7} |, \end{matrix}$ where $k = \frac{1}{12}$ . Notice that $2 k = \frac{1}{6} < 1 = \frac{1}{b - t_{0}} .$ Consequently, all the conditions of Theorem 3.15 are satisfied to guarantee the existence of a solution of (3.14).

Conclusion

In this paper, some fundamental concepts of soft set theory and fixed point theorems are unified. Motivated by the ideas of fuzzy set-valued and multivalued mappings of Heilpen and Nadler, respectively, we considered mappings whose range set is a soft set-the soft set-valued mappings. On the basis that the soft set theory can be applied to a wide range of problems in economics, engineering, physics, game theory, and so on, we provided an example (Example 2.3) as a simple application of the notion of soft set-valued mappings to highlight its prospect in decision making problems. In the process, the idea of e-soft fixed point is established. The study of fixed points of fuzzy set-valued mappings hinges largely on d_∞ metric which is useful in computing Hausdorff dimensions. It has been established that these dimensions help to understand the ɛ^∞-space with enormous applications in high energy physics and geometric problems. In this direction, we defined a distance function $S_{EX}^{\infty} : [P (X)]^{E} \times [P (X)]^{E} ⟶ ℝ$ which is a soft set extension of the concept of d_∞ metric. Along side, an analogue of Hausdorff distance function is given. Availing the above definitions, we proved a few fixed point theorems of Nadler and Edelstein type in a generalized sense. Consequently, we deduced that every fuzzy mapping is a special kind of soft set-valued mapping. To authenticate the hypotheses of our results, examples are provided with some graphical and tabular illustrations.

Moreover, existence and uniqueness of solutions of delay differential equations(DDEs) is obtained usually by method of steps, appealing to classical ODE results. However, more general delay differential equations require specialized techniques of nonlinear functional analysis. This includes some peculiar notion endemic to the problem and identification of the appropriate solution space. To this effect, Theorem 3.1 is applied to establish an existence condition for solution of a nonlinear DDE. In a nutshell, our results are soft set generalizations of fixed pint theorems for point -to-point and point-to-set maps. Knowing that mapping is a fundamental concept in mathematics and related areas of sciences and engineering, the authors are hopeful that this paper will contribute not only to the field of soft set theory but to the entire realm of science and engineering where mappings have applications.

Competing interests

The authors declare that they have no competing interests.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous referees for their valuable comments and kind suggestions to improve this paper. The first author gratefully acknowledges with thanks the World Academy of Science (TWAS), Italy, and COMSATS University, Islamabad, Pakistan, for providing him with full-time postgraduate fellowship award (FR: 3240293231).

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