Multi-valued (ψ, φ, ε, λ)-contraction in probabilistic metric space

Abstract

In this article, we present a new definition of a class of contraction for multi-valued case. Also we prove some fixed point theorems for multivalued (ψ, φ, ε, λ)-contraction mappings in probabilistic metric space.

1 Introduction

The class of (ε, λ)-contraction as a subclass of B-contraction in probabilistic metric space was introduced by Mihet [1]. He and other researchers achieved to some interesting results about existence of fixed point in probabilistic and fuzzy metric spaces [2–4]. Mihet defined the class of (ψ, φ, ε, λ)-contraction in fuzzy metric spaces [4]. On the other hand, Hadzic et al. extended the concept of contraction to the multi valued case [5]. They introduced multi valued (ψ - C)-contraction [6] and obtained fixed point theorem for multi valued contraction [7]. Also Žikić generalized multi valued case of Hick's contraction [8]. We extended (φ - k) - B contraction which introduced by Mihet [9] to multi valued case [10]. Now, we will define the class of (ψ, φ, ε, λ)-contraction in the sense of multi valued and obtain fixed point theorem.

The structure of article is as follows: Section 2 recalls some notions and known results in probabilistic metric spaces and probabilistic contractions. In Section 3, we will prove three theorems for multi valued (ψ, φ, ε, λ)- contraction.

2 Preliminaries

We recall some concepts from the books [11–13].

Definition 2.1. A mapping T : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (a t-norm) if the following conditions are satisfied:

• (1) T (a, 1) = a for every a ∈ [0, 1];

• (2) T (a, b) = T (b, a) for every a, b ∈ [0, 1];

• (3) a ≥ b, c ≥ d ⇒ T(a, c) ≥ T(b, d) a, b, c, d ∈ [0, 1];

• (4) T(T(a, b), c) = T(a, T(b, c)), a, b, c ∈ [0, 1].

Basic examples are, T _L (a, b) = max{a + b - 1, 0}, T _P (a, b) = ab and T _M (a, b) = min{a, b}.

Definition 2.2. If T is a t-norm and $\left(x_1,x_2,...,x_n\right)\in {\left[0,1\right]}^n\left(n\ge 1\right),{\top }_{i=1}^{\mathrm{\infty }}{x}_i$ is defined recurrently by ${\top }_{i=1}^1{x}_i=x_1$ and ${\top }_{i=1}^n{x}_i=T\left({\top }_{i=1}^{n-1}{x}_i,x_n\right)$ for all n ≥ 2. T can be extended to a countable infinitary operation by defining ${\top }_{i=1}^{\mathrm{\infty }}{x}_i$ for any sequence ${\left(x_i\right)}_{i\in N*}\mathsf{\text{as}}{lim}_{n\to \mathrm{\infty }}{\top }_{i=1}^n{x}_i$.

Definition 2.3. Let Δ₊ be the class of all distribution of functions F : [0, ∞] → [0, 1] such that:

• (1) F (0) = 0,

• (2) F is a non-decreasing,

• (3) F is left continuous mapping on [0, ∞].

D₊ is the subset of Δ₊ which lim_x→∞F(x) = 1.

Definition 2.4. The ordered pair (S, F) is said to be a probabilistic metric space if S is a nonempty set and F : S × S → D₊ (F(p, q) written by F _pq for every (p, q) ∈ S × S) satisfies the following conditions:

• (1) F _uv (x) = 1 for every x > 0 ⇒ u = v (u, v ∈ S),

• (2) F _uv = F _vu for every u, v ∈ S,

• (3) F _uv (x) = 1 and F _vw (y) = 1 ⇒ F _u,w (x + y) = 1 for every u, v,w ∈ S, and every x, y ∈ R⁺.

A Menger space is a triple (S, F, T) where (S, F) is a probabilistic metric space, T is a triangular norm (abbreviated t-norm) and the following inequality holds F _uv (x + y) ≥ T (F _uw (x), F _wv (y)) for every u, v, w ∈ S, and every x, y ∈ R⁺.

Definition 2.5. Let φ : (0, 1) → (0, 1) be a mapping, we say that the t-norm T is φ-convergent if

$$\forall \delta \in \left(0,1\right)\forall \lambda \in \left(0,1\right)\exists s=s\left(\delta ,\lambda \right)\in \mathsf{\text{N }}{\top }_{i=1}^n\left(1-{\phi }^{s+i}\left(\delta \right)\right)>1-\lambda ,\forall n\ge 1.$$

Definition 2.6. A sequence (x _n )_{n∈ N}is called a convergent sequence to x ∈ S if for every ε > 0 and λ ∈ (0, 1) there exists N = N(ε, λ) ∈ N such that $F_{x_{n}x}\left(\epsilon \right)>1-\lambda ,\forall n\ge N.$

Definition 2.7. A sequence (x _n )_{n∈ N}is called a Cauchy sequence if for every ε > 0 and λ ∈ (0, 1) there exists N = N(ε, λ)∈ N such that $F_{x_n{x}_{n+m}}\left(\epsilon \right)>1-\lambda ,\forall n\ge N\forall m\in \mathbb{N}.$

We also have

$$x_n{\to }^{F}x\iff F_{x_{n}x}\left(t\right)\to 1\forall t>0.$$

A probabilistic metric space (S, F, T) is called sequentially complete if every Cauchy sequence is convergent.

In the following, 2^S denotes the class of all nonempty subsets of the set S and C(S) is the class of all nonempty closed (in the F-topology) subsets of S.

Definition 2.8 [14]. Let F be a probabilistic distance on S and M ∈ 2^S. A mapping f: S → 2^S is called continuous if for every ε > 0 there exists δ > 0, such that

$$F_{uv}\left(\delta \right)>1-\delta \Rightarrow \forall x\in fu\exists y\in fv:F_{xy}\left(\epsilon \right)>1-\epsilon .$$

Theorem 2.1 [14]. Let (S, F, T) be a complete Menger space, sup _{0≤ t < 1}T (t, t) = 1 and f : S → C(S) be a continuous mapping. If there exist a sequence (t _n )_n∈N⊂ (0, ∞) with ${\sum }_1^{\mathrm{\infty }}{t}_n<\mathrm{\infty }$ and a sequence (x _n ) _n∈N⊂ S with the properties:

$$x_{n+1}\in f{x}_n\mathsf{\text{for}}\mathsf{\text{all}}n\mathsf{\text{and}}\underset{n\to \mathrm{\infty }}{lim}{\top }_{i=1}^{\mathrm{\infty }}{g}_{n+i-1}=1,$$

Where $g_n:=F_{x_n{x}_{n+1}}\left(t_n\right),$ then f has a fixed point.

The concept of (ψ, φ, ε, λ) - B contraction has been introduced by Mihet [15]. We will consider comparison functions from the class ϕ of all mapping φ : (0, 1) → (0, 1) with the properties:

• (1) φ is an increasing bijection;

• (2) φ (λ) < λ ∀λ ∈ (0, 1).

Since every such a comparison mapping is continuous, it is easy to see that if φ ∈ ϕ, then lim_n→∞φⁿ(λ) = 0 ∀λ ∈ (0, 1).

Definition 2.9[15]. Let (X, M, *) be a fuzzy Metric space. ψ be a map from (0, ∞) to (0, ∞) and φ be a map from (0, 1) to (0, 1). A mapping f: X → X is called (ψ, φ, ε, λ)-contraction if for any x, y ∈ X, ε > 0 and λ ∈ (0, 1).

$$M\left(x,y,\epsilon \right)>1-\lambda \Rightarrow M\left(f\left(x\right),f\left(y\right),\psi \left(\epsilon \right)\right)>1-\phi \left(\lambda \right).$$

If ψ is of the form of ψ(ε) = kε (k ∈ (0, 1)), one obtains the contractive mapping considered in [3].

3 Main results

In this section we will generalize the Definition 2.9 to multi valued case in probabilistic metric spaces.

Definition 3.1. Let S be a nonempty set, φ ∈ ϕ, ψ be a map from (0, ∞) to (0, ∞) and F be a probabilistic distance on S. A mapping f : S → 2^S is called a multi-valued (ψ, φ, ε, λ)-contraction if for every x, y ∈ S, ε > 0 and for all λ ∈ (0, 1) the following implication holds:

$$F_{xy}\left(\epsilon \right)>1-\lambda \Rightarrow \forall p\in fx\exists q\in fy:F_{pq}\left(\psi \left(\epsilon \right)\right)>1-\phi \left(\lambda \right).$$

Now, we need to define some conditions on the t-norm T or on the contraction mapping in order to be able to prove fixed point theorem. These two conditions are parallel. If one of them holds, Theorem 3.1 will obtain.

Definition 3.2[11]. Let (S, F) be a probabilistic metric space, M a nonempty subset of S and f : M → 2^S - {∅}, a mapping f is weakly demicompact if for every sequence (p _n )_{n∈ N}from M such that p_n+1∈ fp _n , for every n ∈ N and lim $F_{p_{n+1},p_n}\left(\epsilon \right)=1$, for every ε > 0, there exists a convergent subsequence ${\left(p_{n_j}\right)}_{j\in \mathsf{\text{N}}}.$

The other condition is mentioned in the Theorem 3.1.

Theorem 3.1. Let (S, F, T) be a complete Menger space with sup _{0 ≤ a < 1}T (a, a) = 1, M ∈ C(S) and f : M → C(M) be a multi-valued (ψ, φ, ε, λ)-contraction, where the series Σψⁿ(ε) is convergent for every ε > 0 and φ ∈ ϕ. Let there exists x₀ ∈ M and x₁ ∈ fx₀ such that $F_{x_0{x}_1}\in D_{+}$. If f is weakly demicompact or

$$\underset{n\to \mathrm{\infty }}{lim}{\top }_{i=1}^{\mathrm{\infty }}\left(1-{\phi }^{n+i-1}\left(\epsilon \right)\right)=1\mathsf{\text{for every}}\epsilon >0$$

(1)

then there exists at least one element x ∈ M such that x ∈ fx.

Proof. Since there exists x₀ ∈ M and x₁ ∈ fx₀ such that $F_{x_0{x}_1}\in D_{+}$, hence for every λ ∈ (0, 1) there exists ε > 0 such that $F_{x_0{x}_1}>1-\lambda$. The mapping f is a (ψ, φ, ε, λ)-contraction and therefore there exists x₂ ∈ fx₁ such that

$$F_{x_2{x}_1}\left(\psi \left(\epsilon \right)\right)>1-\phi \left(\lambda \right)$$

Continuing in this way we obtain a sequence (x _n )_n∈Nfrom M such that for every n ≥ 2, x _n ∈ fx_n-1and

$$F_{x_n,x_{n-1}}\left({\psi }^{n-1}\left(\epsilon \right)\right)>1-{\phi }^{n-1}\left(\lambda \right).$$

(2)

Since the series Σψⁿ(ε) is convergent we have lim_n→∞ψⁿ(ε) = 0 and by assumption φ ∈ ϕ, so lim_n→∞φⁿ(λ) = 0. We infer for every ε₀ > 0 that

$$\underset{n\to \mathrm{\infty }}{lim}{F}_{x_n{x}_{n-1}}\left({\epsilon }_0\right)=1.$$

(3)

Indeed, if ε₀ > 0 and λ₀ ∈ (0, 1) are given, and n₀ = n₀(ε₀, λ₀) is enough large such that for every n ≥ n₀, ψⁿ(ε) ≤ ε₀ and φⁿ(λ) ≤ λ₀ then

$$F_{x_{n+1}{x}_n}\left({\epsilon }_0\right)\ge F_{x_{n+1}{x}_n}\left({\psi }^n\left(\epsilon \right)\right)>1-{\phi }^n\left(\lambda \right)>1-{\lambda }_0\mathsf{\text{ for every }}n\ge n_0.$$

If f is weakly demicompact (3) implies that there exists a convergent subsequence ${\left(x_{n_k}\right)}_{k\in N}$.

Suppose that (1) holds and prove that (x _n )_n∈Nis a Cauchy sequence. This means that for every ε₁ > 0 and every λ₁ ∈ (0, 1) there exists n₁(ε₁, λ₁) ∈ N such that

$$F_{x_{n+p}{x}_n}\left({\epsilon }_1\right)>1-{\lambda }_1$$

(4)

for every n₁ ≥ n₁(ε₁, λ₁) and every p ∈ N.

Let n₂(ε₁) ∈ N such that ${\sum }_{n\ge n_2\left({\epsilon }_1\right)}{\psi }^n\left(\epsilon \right)<{\epsilon }_1.$ Since ${\sum }_{n=1}^{\mathrm{\infty }}{\psi }^n\left(\epsilon \right)$ is convergent series such a natural number n₂(ε₁) exists. Hence for every p ∈ N and every n ≥ n₂(ε₁) we have that

$$F_{x_{n+p+1},x_n}\left({\epsilon }_1\right)\ge {\top }_{i=1}^{p+1}{F}_{x_{n+i},x_{n+i-1}}\left({\psi }^{n+i-1}\left(\epsilon \right)\right),$$

and (2) implies that

$$F_{x_{n+p+1},x_n}\left({\epsilon }_1\right)\ge {\top }_{i=1}^{p+1}\left(1-{\phi }^{n+i-1}\left(\lambda \right)\right)$$

for every n ≥ n₂(ε₁) and every p ∈ N.

For every p ∈ N and n ≥ n₂(ε₁)

$${\top }_{i=1}^{p+1}\left(1-{\phi }^{n+i-1}\left(\lambda \right)\right)\ge {\top }_{i=1}^{\mathrm{\infty }}\left(1-{\phi }^{n+i-1}\left(\lambda \right)\right)$$

and therefore for every p ∈ N and n ≥ n₂(ε₁),

$$F_{x_{n+p+1},x_n}\left({\epsilon }_1\right)\ge {\top }_{i=1}^{\mathrm{\infty }}\left(1-{\phi }^{n+i-1}\left(\lambda \right)\right).$$

(5)

From (1) it follows that there exists n₃(λ₁) ∈ N such that

$${\top }_{i=1}^{\mathrm{\infty }}\left(1-{\phi }^{n+i-1}\left(\lambda \right)\right)>1-{\lambda }_1$$

(6)

for every n ≥ n₃(λ₁). The conditions (5) and (6) imply that (4) holds for n₁(ε₁, λ₁) = max(n₂(ε₁), n₃(λ₁)) and every p ∈ N. This means that (x _n )_n∈Nis a Cauchy sequence and since S is complete there exists lim_n→∞x _n . Hence in both cases there exists ${\left(x_{n_k}\right)}_{k\in N}$ such that

$$\underset{k\to \mathrm{\infty }}{lim}{x}_{n_k}=x.$$

It remains to prove that x ∈ fx. Since $fx=\overline{fx}$ it is enough to prove that $x\in \overline{fx}$ i.e., for every ε₂ > 0 and λ₂ ∈ (0, 1) there exists $b_{{\epsilon }_2,{\lambda }_2}\in fx$ such that

$$F_{x,b_{{\epsilon }_2,{\lambda }_2}}\left({\epsilon }_2\right)>1-{\lambda }_2.$$

(7)

Since sup_{x< 1}T(x, x) = 1 for λ₂ ∈ (0, 1) there exists δ(λ₂) ∈ (0, 1) such that T(1 - δ(λ₂), 1 - δ(λ₂)) > 1 - λ₂.

If δ'(λ₂) is such that

$$T\left(1-{\delta }^{\prime }\left({\lambda }_2\right),1-{\delta }^{\prime }\left({\lambda }_2\right)\right)>1-\delta \left({\lambda }_2\right)$$

and δ''(λ₂) = min(δ(λ₂), δ'(λ₂)) we have that

$$\begin{array}{cc}\hfill T(1-{\delta }^{″}({\lambda }_2),T((1-{\delta }^{″}\left({\lambda }_2\right),1-{\delta }^{″}\left({\lambda }_2\right)\left)\right)& \ge T(1-\delta ({\lambda }_2),T((1-{\delta }^{\prime }({\lambda }_2),1-\delta ({\lambda }_2\left)\right))\hfill \\ \ge T(1-\delta ({\lambda }_2),1-\delta ({\lambda }_2\left)\right)\hfill \\ >1-{\lambda }_2.\hfill \end{array}$$

Since ${lim}_{k\to \mathrm{\infty }}{x}_{n_k}=x$ there exists k₁ ∈ N such that $F_{x,x_{n_k}}\left(\frac{\epsilon }{3}\right)>1-{\delta }^{″}\left({\lambda }_2\right)$ for every k ≥ k₁. Let k₂ ∈ N such that

$$F_{x_{n_k},x_{n_k+1}}\left(\frac{{\epsilon }_2}{3}\right)>1-{\delta }^{″}\left({\lambda }_2\right)\mathsf{\text{for every}}k\ge k_2.$$

The existence of such a k₂ follows by (3). Let ε ∈ R₊ be such that $\psi \left(\epsilon \right)<\frac{{\epsilon }_2}{3}$ and k₃ ∈ N such that $F_{x_{n_k},x}\left(\epsilon \right)>1-\delta \prime \prime \left({\lambda }_2\right)$ for every k ≥ k₃. Since f is a (ψ, φ, ε, λ)-contraction there exists $b_{{\epsilon }_2,{\lambda }_2,k}\in fx$ such that

$$F_{x_{n_{k+1}},b_{{\epsilon }_2,{\lambda }_2,k}}\left(\psi \left(\epsilon \right)\right)>1-\phi \left({\delta }^{″}\left({\lambda }_2\right)\right)\mathsf{\text{for every}}k\ge k_3.$$

Therefore for every k ≥ k₃

$$\begin{array}{cc}\hfill F_{x_{n_{k+1,}}{b}_{{\epsilon }_2,{\lambda }_2,k}}\left(\frac{{\epsilon }_2}{2}\right)& \ge F_{x_{n_k+1},b_{{\epsilon }_2,{\lambda }_2,k}}\left(\psi \left(\epsilon \right)\right)\hfill \\ >1-\phi \left({\delta }^{″}\left({\lambda }_2\right)\right)\hfill \\ >1-{\delta }^{″}\left({\lambda }_2\right)\hfill \end{array}$$

If k ≥ max(k₁, k₂, k₃) we have

$$\begin{array}{c}{F}_{x,b_{{\epsilon }_2,{\lambda }_2,k}}\left({\epsilon }_2\right)\ge T\left(F_{x,x_{n_k}}\left(\frac{{\epsilon }_2}{3}\right),T\left(F_{x_{n_k,}{x}_{n_k+1}}\left(\frac{{\epsilon }_2}{3}\right),F_{x_{n_k+1},b_{{\epsilon }_2,{\lambda }_2,k}}\left(\frac{{\epsilon }_2}{3}\right)\right)\right)\\ T\left(1-{\delta }^{″}\left({\lambda }_2\right),T\left(1-{\delta }^{″}\left({\lambda }_2\right),1-{\delta }^{″}\left({\lambda }_2\right)\right)\right)\\ >1-{\lambda }_2\end{array}$$

and (7) is proved for $b_{{\epsilon }_2,{\lambda }_2}=b_{{\epsilon }_2,{\lambda }_2,k},k\ge max\left(k_1,k_2,k_3\right).$ Hence $x\in \overline{fx}=fx,$ which means x is a fixed point of the mapping f.

Now, suppose that instead of Σψⁿ(ε) be convergent series, ψ is increasing bijection.

Theorem 3.2. Let (S, F, T) be a complete Menger space with sup _{0 ≤ a < 1}T (a, a) = 1 and f : S → C(S) be a multi-valued (ψ, φ, ε, λ)- contraction.

If there exist p ∈ S and q ∈ fp such that F _pq ∈ D₊, ψ is increasing bijection and ${lim}_{n\to \mathrm{\infty }}{\top }_{i=1}^{\mathrm{\infty }}\left(1-{\phi }^{n+i-1}\left(\lambda \right)\right)=1$, for every λ ∈ (0, 1), then, f has a fixed point.

Proof. Let ε > 0 be given and δ ∈ (0, 1) be such that δ < min{ε, ψ^-1(ε)} or ψ(δ) < ε since ψ is increasing bijection. If F _uv (δ) > 1-δ then, due to (ψ, φ, ε, λ)- contraction for each x ∈ fu we can find y ∈ fv such that F _xy (ψ(δ)) > 1 - φ(δ), from where we obtain that F _xy (ε) > F _xy (ψ(δ)) > 1 - φ(δ) > 1 - δ > 1 - ε. So f is continuous. Next, let p₀ = p and p₁ = q be in fp₀. Since F _pq ∈ D₊, hence for every λ ∈ (0, 1) there exist ε > 0 such that F _pq (ε) > 1 - λ, namely $F_{p_0{p}_1}\left(\epsilon \right)>1-\lambda$.

Using the contraction relation we can find p₂ ∈ fp₁ such that $F_{p_1{p}_2}\left(\psi \left(\epsilon \right)\right)>1-\phi \left(\lambda \right)$, and by induction, p _n such that p _n ∈ fp_n-1and $F_{p_{n-1}{p}_n}\left({\psi }^{n-1}\left(\epsilon \right)\right)>1-{\phi }^{n-1}\left(\lambda \right)$ for all n ≥ 1. Defining t _n = ψⁿ(ε), we have $g_j=F_{p_j{p}_{j+1}}\left(t_j\right)\ge 1-{\phi }^j\left(\lambda \right)$, ∀j, so ${lim}_{n\to \mathrm{\infty }}{\top }_{i=1}^{\mathrm{\infty }}{g}_{n+i-1}\ge {lim}_{n\to \mathrm{\infty }}{\top }_{i=1}^{\mathrm{\infty }}\left(1-{\phi }^{n+i-1}\left(\lambda \right)\right)=1.$

On the other hand the sequence (p _n ) is a Cauchy sequense, that is:

$$\forall \epsilon >0\exists n_0=n0\left(\epsilon \right)\in N:F_{p_n{p}_{n+m}}\left(\epsilon \right)>1-\in ,\forall n\ge n_0,\forall m\in N.$$

Suppose that ε > 0, then:

$$\underset{n\to \mathrm{\infty }}{lim}{\top }_{i=1}^{\mathrm{\infty }}{g}_{n+i+1}=1\Rightarrow \exists n_1=n_{\mathsf{\text{1}}}\left(\epsilon \right)\in N:{\top }_{i=1}^m{g}_{n+i-1}>1-\epsilon ,\forall n\ge n_1\mathsf{\text{,}}\forall m\in N.$$

Since the series ${\sum }_{n=1}^{\mathrm{\infty }}{t}_n$ is convergent, there exists n₂(= n₂(ε)) such that ${\sum }_{n=n_2}^{\mathrm{\infty }}{t}_n<\epsilon$.

Let n₀ = max{n₁, n₂}, then for all n ≥ n₀ and m ∈ N we have:

$$\begin{array}{c}{F}_{p_n{p}_{n+m}}\left(\epsilon \right)\ge F_{p_n{p}_{n+m}}\left(\sum _{i=n}^{n+m-1}{t}_i\right)\ge {\top }_{i=1}^m{F}_{p_{n+i-1}{p}_{n+1}}\left(t_{n+i-1}\right)\\ \mathsf{\text{ = }}{\top }_{i=1}^m{g}_{n+i-1}>1-\epsilon ,\end{array}$$

as desired.

Now we can apply Theorem 2.1 to find a fixed point of f. The theorem is proved. □

When ψ is increasing bijection and lim_n→∞ψⁿ(λ) be zero, by using demicompact contraction we have another theorem.

Theorem 3.3. Let (S, F, T) be a complete Menger space, T a t-norm such that sup _{0 ≤ a < 1}T (a, a) = 1, M a non-empty and closed subset of S, f : M → C(M) be a multi-valued (ψ, φ, ε, λ)- contraction and also weakly demicompact. If there exist x₀ ∈ M and x₁ ∈ fx₀ such that $F_{x_0{x}_1}\in D_{+},\psi$ is increasing bijection and lim_n→∞ψ (λ) = 0 then, f has a fixed point.

Proof. We can construct a sequence (p _n )_{n∈ N}from M, such that p₁ = x₁ ∈ fx₀, p_n+1∈ fp _n . Given t > 0 and λ ∈ (0, 1), we will show that

$$\underset{n\to \mathrm{\infty }}{lim}{F}_{p_{n+1}{p}_n}\left(t\right)=1.$$

(11)

Indeed, since $F_{x_0{x}_1}\in D_{+}$, hence for every ξ > 0 there exist η > 0 such that $F_{x_0{x}_1}\left(\eta \right)>1-\xi$, and by induction $F_{p_{n-1}{p}_n}\left({\psi }^n\left(\eta \right)\right)>1-{\phi }^n\left(\xi \right)$ for all n ∈ N. By choosing n such that ψⁿ(η) < t and φⁿ(ξ) < λ, we obtain

$$F_{p_{n+1}{p}_n}\left(t\right)>1-\lambda .$$

Since t and λ are arbitrary, the proof of (1) is complete.

By Definition 3.2, there exists a subsequence ${\left(p_{n_j}\right)}_{j\in \mathsf{\text{N}}}$ such that ${lim}_{j\to \mathrm{\infty }}{p}_{n_j}$ exists. We shall prove that $x=\underset{j\to \mathrm{\infty }}{lim}{p}_{n_j}$ is a fixed point of f. Since fx is closed, $fx=\overline{fx}$, and therefore, it remains to prove that $x=\overline{fx}$, i.e., for every ε > 0 and λ ∈ (0, 1), there exist b(ε, λ) ∈ fx, such that F_x,b(ε,λ)(ε) > 1 - λ. From the condition sup _{0 ≤ a < 1}T (a, a) = 1 it follows that there exists η(λ) ∈ (0, 1) such that

$$u>1-\eta \left(\lambda \right)\Rightarrow T\left(u,u\right)>1-\lambda .$$

Let j₁(ε, λ) ∈ N be such that

$$F_{p_{n_j},x}\left({\psi }^{-1}\left(\frac{\epsilon }{2}\right)\right)>1-\frac{\eta \left(\lambda \right)}{2}\mathsf{\text{for}}\mathsf{\text{every }}j\ge j_1\left(\epsilon ,\lambda \right).$$

Since $x={lim}_{j\to \mathrm{\infty }}{p}_{n_j}$, such a number j₁(ε, λ) exists. Since f is (ψ, φ, ε, λ)-contraction and ψ is increasing bijection, for $p_{n_j+1}\in f{p}_{n_j}$ there exists b _j (ε)∈ fx such that

$$F_{p_{n_j+1},b_{j\left(\epsilon \right)}}\left(\frac{\epsilon }{2}\right)>1-\phi \left(\frac{\eta \left(\lambda \right)}{2}\right)>1-\frac{\eta \left(\lambda \right)}{2}\mathsf{\text{for}}\mathsf{\text{every}}j\ge j_1\left(\epsilon ,\lambda \right).$$

From (1) it follows that ${lim}_{j\to \mathrm{\infty }}{p}_{n_j+1}=x$ and therefore, there exists j₂(ε, λ) ∈ N such that $F_{x,p_{n_j+1}}\left(\frac{\epsilon }{2}\right)>1-\frac{\eta \left(\lambda \right)}{2}$ for every j ≥ j₂(ε, λ). Let j₃(ε, λ) = max{j₁(ε, λ), j₂(ε, λ)}. Then, for every j ≥ j₃(ε, λ) we have $F_{x,b_j\left(\epsilon \right)}\left(\epsilon \right)\ge T\left(F_{x,p_{n_j+1}}\left(\frac{\epsilon }{2}\right),F_{p_{n_j+1},b_{j\left(\epsilon \right)}}\left(\frac{\epsilon }{2}\right)\right)>1-\lambda$. Hence, if j > j₃(ε, λ), then, we can choose b(ε, λ) = b _j (ε)∈ fx. The proof is complete. □