Functional-Type Caristi–Kirk Theorem on Menger Space and Application

)e theory of probabilistic metric spaces is of fundamental importance in random functional analysis especially due to its extensive applications in random differential and random integral equations. In 1942, Menger introduced in [1] a generalization of metric space, called a statistical metric space, by using distribution functions instead of nonnegative real numbers as values of the metric. )e connection between probabilistic and geometric concepts has been established in 1956 by Špaček [2]. However, the main influence upon the development of this theory is owed to Schweizer and Sklar and their coworkers (see [3–6]). Since then, a large number of fixed point theorems for single-valued and multivalued mappings in probabilistic metric spaces have been proved by many authors (see, for examples, [7–13]). Since every metric space is a probabilistic metric space, we can use many results in probabilistic metric spaces to prove some fixed point theorems in metric spaces and Banach spaces. In this paper, first, we extend Caristi’s fixed point theorem in metric spaces to probabilistic metric spaces. Secondly, we prove, in the setting of a Menger space, some common fixed point theorems for a pair of mappings satisfying a system of Caristi-type contractions. Two examples are given to support our main results. Let R+ denote the set of positive real numbers and N denote the set of all natural numbers. For brevity, T(x) and S(x) will be denoted by Tx and Sx, respectively. Let


Introduction and Preliminaries
e theory of probabilistic metric spaces is of fundamental importance in random functional analysis especially due to its extensive applications in random differential and random integral equations. In 1942, Menger introduced in [1] a generalization of metric space, called a statistical metric space, by using distribution functions instead of nonnegative real numbers as values of the metric. e connection between probabilistic and geometric concepts has been established in 1956 byŠpaček [2]. However, the main influence upon the development of this theory is owed to Schweizer and Sklar and their coworkers (see [3][4][5][6]). Since then, a large number of fixed point theorems for single-valued and multivalued mappings in probabilistic metric spaces have been proved by many authors (see, for examples, [7][8][9][10][11][12][13]). Since every metric space is a probabilistic metric space, we can use many results in probabilistic metric spaces to prove some fixed point theorems in metric spaces and Banach spaces.
In this paper, first, we extend Caristi's fixed point theorem in metric spaces to probabilistic metric spaces. Secondly, we prove, in the setting of a Menger space, some common fixed point theorems for a pair of mappings satisfying a system of Caristi-type contractions. Two examples are given to support our main results.
Let R + denote the set of positive real numbers and N denote the set of all natural numbers. For brevity, T(x) and S(x) will be denoted by Tx and Sx, respectively. Let c : R + ⟶ R + be some function. Let, for α ∈ R + , (1) Definition 1. We say that a function c is right lower (upper) semicontinuous at α if lim inf t ⟶ α + c(t) � c(α)(lim sup t ⟶ α + c(t) � c(α)).
From the definition, it is easy to prove that the following proposition is needed in the sequel.

Definition 2.
A mapping g: R + ⟶ R + is said to be locally bounded above if it is bounded above on each [0, a], (a > 0).
In order to set the framework needed to state our main results, we recall the following notions.
Definition 3 (distribution function). A mapping F: R ⟶ R + is called a distribution function if it is nondecreasing, left continuous with infF � 0, and supF � 1.
In what follows, we always denote by D the set of all distribution functions. Example 1. A simple example of distribution function is a Heaviside function: Definition 4 (probabilistic metric space). A probabilistic metric space (PM-space, for short) is an ordered pair (X, F), where X is an abstract set of elements and F is a mapping of X × X ⟶ D (we shall denote the distribution function F(p, q) by F p,q , and F p,q (t) will represent the value of F p,q at t ∈ R). e functions F p,q , p, q ∈ X, are assumed to satisfy the following conditions: If F p,q (t 1 ) � 1 and F q,r (t 2 ) � 1, then F p,r (t 1 + t 2 ) � 1 Definition 5 (see [14]). Let (X, F) be a probabilistic metric space. For p ∈ X and t > 0, the strong t-neighborhood of p is the set e strong neighborhood system at p is the collection N p � N p (t) : t > 0 , and the strong neighborhood system for X is the union N � ∪ p∈X N p .
Definition 6 (see [14]). e (ε, λ)− topology on (X, F) is the topology introduced on X by the family of the neighbor- Definition 7 (triangular norm). A triangular norm (briefly a t-norm) is a binary operation τ on the unit interval [0, 1] which is associative, commutative, and nondecreasing in each of its variables and such that τ(x, 1) � x for every x ∈ [0, 1].

Example 2.
e following are the three basic t-norms: where (X, F) is a PM-space and τ is a t-norm satisfying the following triangle inequality: e following result is due to Schweizer et al. [6].
is a Hausdorff space in the topology induced by the family U p (ε, λ) p∈X,ε > 0,λ > 0 of neighborhoods.
Definition 9. Let (X, F, τ) be a Menger space. A sequence (p n ) n∈N in X is said to be (i) τ-convergent to p ∈ X if for any ε > 0 and any λ > 0, there exists a positive integer N � N(ε, λ) such that F p n ,p (ε) > 1 − λ, whenever n ≥ N (ii) τ-Cauchy sequence if for any ε > 0 and any λ > 0, there exists a positive integer N � N(ε, λ) such that F p n ,p m (ε) > 1 − λ, whenever n, m ≥ N Definition 10. A Menger space (X, F, τ) is said to be τ-complete if each τ-Cauchy sequence in X is τ-convergent to some point in X.
Definition 11. Let (X, F, τ) be a Menger space such that sup t<1 τ(t, t) � 1, and let S be a self-mapping on (X, F, τ). S is said to be a τ-continuous mapping if for each sequence (p n ) n∈N in X which is τ-convergent to a point p ∈ X, the sequence (Sp n ) n∈N is τ-convergent to Sp. e following result was established by Schweizer and Sklar in [5].
e following theorem establishes a connection between metric spaces and Menger spaces.
Theorem 3 (see [13]). Let (X, d) be a metric space. Let F : X × X ⟶ D be the mapping defined by en, We say that a t-norm τ satisfies the condition (P) if for all Remark 2. It is easy to see that if a t-norm τ satisfies the condition (P), then Theorem 4 (see [15]). Let (X, F, τ) be a Menger space with τ satisfying the condition (P). Let (p n ) n∈N and (q n ) n∈N be two sequences in X such that (p n ) n∈N is τ-convergent to p ∈ X and (q n ) n∈N is τ-convergent to q ∈ X. en, Lemma 1. (see [15], Proposition 2). If (X, F, τ) is a Menger space with τ satisfying the condition (P), then it is metrizable. In addition, if (X, F, τ) is sequentially complete, then it must be net-complete.

Main Results
Theorem 5. Let (X, F, τ) be a complete Menger space with τ satisfying the condition (P). Let T, S : X ⟶ X be two mappings such that, for all p ∈ X and for all t > 0, where ϕ: X ⟶ R + is a lower semicontinuous function, c: R + ⟶ R + is a right locally bounded from above, and h is a locally bounded function from R + × R + to R + . en, there exists an element p * ∈ X such that Proof.
us, TX 0 ⊂ X 0 and SX 0 ⊂ X 0 . And since ϕ(p), Step 2: we define a partial order "⪯" on X 0 as follows: If p, q ∈ X 0 such that p ⪯ q, then ϕ(q) ≤ ϕ(p). On the contrary, the reflexivity and e symmetry of "⪯" are obvious. Let us prove the transitivity.
Proof. Consider the mapping F : X × X ⟶ D defined as follows: d(p, q)), t ∈ R, p, q ∈ X.

(26)
By eorem 3, the space (X, F, τ M ) is a complete Menger space. It is easy to see that the t-norm τ M satisfies the condition (P). erefore, all conditions of eorem 5 are satisfied. en, the conclusion holds. □ Example 3. Consider the set X � R + endowed with the metric d defined by and define the function F : X × X ⟶ D as follows: for all(p, q) ∈ X 2 , t ∈ R.

(28)
Consider the two mappings T, S : X ⟶ X such that and the lower semicontinuous function ϕ : X ⟶ [0, +∞[ defined by Let us show that, for all p ∈ X and t > 0,

Journal of Mathematics
(43) Hence, p * ⪯ Sp * , which implies that Sp * � p * and Tp * � p * . □ Example 4. Consider the space X � R + with the metric d defined by Consider the two mappings T, S : X ⟶ X such that Consider the function F : X × X ⟶ D defined by for all(p, q) ∈ X 2 , t ∈ R. (48) Let p ∈ X. Let us show that In all the cases, we have |p − Sp| ≤ ψ(p) − ψ(Tp); Since H is nondecreasing, ∀p ∈ X, ∀t > 0.

Application
Let C 1 be the space of the functions f : N * ⟶ ]0, +∞] such that the series n≥1 2 − n /f(n) is convergent and C 2 be the space of the functions f : N * ⟶ [0, +∞[ such that the series n≥1 2 − n f(n) is convergent. We consider two metrics d 1 and d 2 defined, respectively, on C 1 and C 2 by , Since the function m : (C 1 , d 1 ) ⟶ (C 2 , d 2 ) defined by m(f) � 1/f, for all f ∈ C 1 , is a homeomorphism and (C 2 , d 2 ) is complete, then also (C 1 , d 1 ) is complete.
Let F be the family of distribution functions defined on e equation f(n) � g(n)(f(n − 1)) r(n) + h(n), n ∈ N * , of unknown f, has a solution in C 1 .
Proof. Let C g,h,c be the set of elements f of C 1 such that f(1) � c and f(n) ≥ g(n)(f(n − 1)) r(n) + h(n), for all n ∈ N * . (56) C g,h,c is nonempty since the sequence f, defined by f(1) � c and f(n) � +∞ for all n ≥ 2, is an element of C g,h,c .
en, we have f ∈ C g,h,c .

Conclusion
e results in this paper (1) Extend the main results of [16] from a standard metric space to a Menger space (2) Generalize and improve the main result of [15]

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.