Some new fixed point theorems for set-valued contractions in complete metric spaces

Abstract

In this article, we obtain some new fixed point theorems for set-valued contractions in complete metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.

MSC: 47H10, 54C60, 54H25, 55M20.

1 Introduction and preliminaries

Let (X, d) be a metric space, D a subset of X and f : D → X be a map. We say f is contractive if there exists α ∈ [0, 1) such that for all x, y ∈ D,

$$d\left(fx,fy\right)\le \alpha \cdot d\left(x,y\right).$$

The well-known Banach's fixed point theorem asserts that if D = X, f is contractive and (X, d) is complete, then f has a unique fixed point in X. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, a mapping f : X → X is called a quasi-contraction if there exists k < 1 such that

$$d\left(fx,fy\right)\le k\cdot max\left\{d\left(x,y\right),d\left(x,fx\right),d\left(y,fy\right),d\left(x,fy\right),d\left(y,fx\right)\right\}$$

for any x, y ∈ X. In 1974, C'iric' [2] introduced these maps and proved an existence and uniqueness fixed point theorem.

Throughout we denote the family of all nonempty closed and bounded subsets of X by CB(X). The existence of fixed points for various multi-valued contractive mappings had been studied by many authors under different conditions. In 1969, Nadler [3] extended the famous Banach Contraction Principle from single-valued mapping to multi-valued mapping and proved the below fixed point theorem for multi-valued contraction.

Theorem 1[3]Let (X, d) be a complete metric space and T : X → CB(X). Assume that there exists c ∈ [0, 1) such that

$$\mathcal{H}\left(Tx,Ty\right)\le cd\left(x,y\right)forallx,y\in X,$$

where$\mathcal{H}$denotes the Hausdorff metric on CB(X) induced by d, that is, H(A, B) = max{sup_x∈AD(x, B), sup_y∈BD(y, A)}, for all A, B ∈ CB(X) and D(x, B) = inf_z∈Bd(x, z). Then, T has a fixed point in X.

In 1989, Mizoguchi-Takahashi [4] proved the following fixed point theorem.

Theorem 2[4]Let (X, d) be a complete metric space and T : X → CB(X). Assume that

$$\mathcal{H}\left(Tx,Ty\right)\le \xi \left(d\left(x,y\right)\right)\cdot d\left(x,y\right)$$

for all x, y ∈ X, where ξ : [0, ∞) → [0, 1) satisfies$lim\underset{s\to t^{+}}{sup}\xi \left(s\right)<1$for all t ∈ [0, ∞). Then, T has a fixed point in X.

In the recent, Amini-Harandi [5] gave the following fixed point theorem for set-valued quasi-contraction maps in metric spaces.

Theorem 3[5]Let (X, d) be a complete metric space. Let T : X → CB(X) be a k-set-valued quasi-contraction with $k<\frac{1}{2}$, that is,

$$\mathscr{H}(Tx\mathrm{,}Ty)\le k\cdot \max \{(x\mathrm{,}y),D(x\mathrm{,}Tx),D(y\mathrm{,}Ty),D(x\mathrm{,}Ty)),D(y\mathrm{,}Tx)\}$$

for any x, y ∈ X. Then, T has a fixed point in X.

2 Fixed point theorem (I)

In this section, we assume that the function ψ : ℝ⁺⁵ → ℝ⁺ satisfies the following conditions:

(C1) ψ is a strictly increasing, continuous function in each coordinate, and

(C2) for all t ∈ ℝ⁺, ψ(t, t, t, 0, 2t) < t, ψ(t, t, t, 2t, 0) < t, ψ(0, 0, t, t, 0) < t and ψ(t, 0, 0, t, t) < t.

Definition 1 Let (X, d) be a metric space. The set-valued map T : X → X is said to be a set-valued ψ-contraction, if

$$\mathscr{H}(Tx\mathrm{,}Ty)\le \psi (d(x\mathrm{,}y),D(x\mathrm{,}Tx),D(y\mathrm{,}Ty),D(x\mathrm{,}Ty)),D(y\mathrm{,}Tx))$$

for all x, y ∈ X.

We now state the main fixed point theorem for a set-valued ψ-contraction in metric spaces, as follows:

Theorem 4 Let (X, d) be a complete metric space. Let T : X → CB(X) be a set-valued ψ-contraction. Then, T has a fixed point in X.

Proof. Note that for each A, B ∈ CB(X), a ∈ A and γ > 0 with $\mathcal{H}\left(A,B\right)<\gamma$, there exists b ∈ B such that d(a, b) < γ. Since T : X → CB(X) is a set-valued ψ-contraction, we have

$$\mathscr{H}(Tx\mathrm{,}Ty)\le \psi (d(x\mathrm{,}y),D(x\mathrm{,}Tx),D(y\mathrm{,}Ty),D(x\mathrm{,}Ty)),D(y\mathrm{,}Tx))$$

for all x, y ∈ X. Suppose that x₀ ∈ X and that x₁ ∈ X. Then, by induction and by the above observation, we can find a sequence {x _n } in X such that x_n+1∈ Tx _n and for each n ∈ ℕ,

$$\begin{array}{ll}\hfill d\left(x_{n+1},x_n\right)& \le \psi \left(d\left(x_n,x_{n-1}\right),D\left(x_n,T{x}_n\right),D\left(x_{n-1},T{x}_{n-1}\right),D\left(x_n,T{x}_{n-1}\right),D\left(x_{n-1},T{x}_n\right)\right)\\ \le \psi \left(d\left(x_n,x_{n-1}\right),d\left(x_n,x_{n+1}\right),d\left(x_{n-1},x_n\right),d\left(x_n,x_n\right),d\left(x_{n-1},x_{n+1}\right)\right)\\ \le \psi \left(d\left(x_n,x_{n-1}\right),d\left(x_n,x_{n+1}\right),d\left(x_{n-1},x_n\right),0,d\left(x_{n-1},x_n\right)+d\left(x_n,x_{n+1}\right)\right),\end{array}$$

and hence, we can deduce that for each n ∈ ℕ,

$$d\left(x_{n+1},x_n\right)\le d\left(x_n,x_{n-1}\right).$$

Let we denote c _m = d(x_m+1, x _m ). Then, c _m is a non-increasing sequence and bounded below. Thus, it must converges to some c ≥ 0. If c > 0, then by the above inequalities, we have

$$c\le c_{n+1}\le \psi \left(c_n,c_n,c_n,0,2c_n\right).$$

Passing to the limit, as n → ∞, we have

$$c\le c\le \psi \left(c,c,c,0,2c\right)<c,$$

which is a contradiction. Hence, c = 0.

We next claim that the following result holds:

for each γ > 0, there is n₀(γ) ∈ ℕ such that for all m > n > n₀(γ),

$$d\left(x_m,x_n\right)<\gamma .\left(*\right)$$

We shall prove (*) by contradiction. Suppose that (*)is false. Then, there exists some γ > 0 such that for all k ∈ ℕ, there exist m _k , n _k ∈ ℕ with m _k > n _k ≥ k satisfying:

• (1) m _k is even and n _k is odd;

• (2) $d\left(x_{m_k},x_{n_k}\right)\ge \gamma$;

• (3) m _k is the smallest even number such that the conditions (1), (2) hold.

Since c _m ↘ 0, by (2), we have $\underset{k\to \infty }{lim}d\left(x_{m_k},x_{n_k}\right)=\gamma$ and

$$\begin{array}{c}\gamma \le d(x_{m_k}\mathrm{,}{x}_{n_k})\le \mathscr{H}(T{x}_{m_k-1}\mathrm{,}T{x}_{n_k-1})\\ \le \psi (d(x_{m_k-1}\mathrm{,}{x}_{n_k-1}),d(x_{m_k-1}\mathrm{,}{x}_{m_k}),d(x_{n_k-1}\mathrm{,}{x}_{n_k}),d(x_{m_k-1}\mathrm{,}{x}_{n_k}),d(x_{n_k-1}\mathrm{,}{x}_{m_k}))\\ \le \psi (c_{m_k-1}+d(x_{m_k}\mathrm{,}{x}_{n_k})+c_{n_k-1}\mathrm{,}{c}_{m_k-1}\mathrm{,}{c}_{n_k-1}\mathrm{,}{c}_{m_k-1}+d(x_{m_k}\mathrm{,}{x}_{n_k}),d(x_{m_k}\mathrm{,}{x}_{n_k})+c_{n_k-1})).\end{array}$$

Letting k → ∞. Then, we get

$$\gamma \le \psi \left(\gamma ,0,0,\gamma ,\gamma \right)<\gamma ,$$

a contradiction. It follows from (*) that the sequence {x _n } must be a Cauchy sequence.

Similarly, we also conclude that for each n ∈ ℕ,

$$\begin{array}{ll}\hfill d\left(x_n,x_{n+1}\right)& \le \psi \left(d\left(x_{n-1},x_n\right),D\left(x_{n-1},T{x}_{n-1}\right),D\left(x_n,T{x}_n\right),D\left(x_{n-1},T{x}_n\right),D\left(x_n,T{x}_{n-1}\right)\right)\\ \le \psi \left(d\left(x_{n-1},x_n\right),d\left(x_{n-1},x_n\right),d\left(x_n,x_{n+1}\right),d\left(x_{n-1},x_{n+1}\right),d\left(x_n,x_n\right)\right)\\ \le \psi \left(d\left(x_{n-1},x_n\right),d\left(x_n,x_{n+1}\right),d\left(x_{n-1},x_n\right),d\left(x_{n-1},x_n\right)+d\left(x_n,x_{n+1}\right),0\right),\end{array}$$

and hence, we have that for each n ∈ ℕ,

$$d\left(x_n,x_{n+1}\right)\le d\left(x_{n-1},x_n\right).$$

Let we denote b _m = d(x _m , x_m+1). Then, b _m is a non-increasing sequence and bounded below. Thus, it must converges to some b ≥ 0. If b > 0, then by the above inequalities, we have

$$b\le b_{n+1}\le \psi \left(b_n,b_n,b_n,2b_n,0\right).$$

Passing to the limit, as n → ∞, we have

$$b\le b\le \psi \left(b,b,b,2b,0\right)<b,$$

which is a contradiction. Hence, b = 0. By the above argument, we also conclude that {x _n } is a Cauchy sequence.

Since X is complete, there exists μ ∈ X such that lim_n→∞x _n = μ. Therefore,

$$\begin{array}{ll}\hfill D\left(\mu ,T\mu \right)& =\underset{n\to \infty }{lim}D\left(x_{n+1},T\mu \right)\\ \le \underset{n\to \infty }{lim}\mathcal{H}\left(T{x}_n,T\mu \right)\\ \le \underset{n\to \infty }{lim}\psi \left(d\left(x_n,\mu \right),D\left(x_n,T{x}_n\right),D\left(\mu ,T\mu \right),D\left(x_n,T\mu \right),D\left(\mu ,T{x}_n\right)\right)\\ \le \underset{n\to \infty }{lim}\psi \left(d\left(x_n,\mu \right),d\left(x_n,x_{n+1}\right),D\left(\mu ,T\mu \right),D\left(x_n,T\mu \right),d\left(\mu ,x_{n+1}\right)\right)\\ \le \psi \left(0,0,D\left(\mu ,T\mu \right),D\left(\mu ,T\mu \right),0\right)\\ <D\left(\mu ,T\mu \right),\end{array}$$

and hence, D(μ, Tμ) = 0, that is, μ ∈ Tμ, since Tμ is closed.

3 Fixed point theorem (II)

In 1972, Chatterjea [6] introduced the following definition.

Definition 2 Let (X, d) be a metric space. A mapping f : X → X is said to be a$\mathcal{C}$-contraction if there exists$\alpha \in \left(0,\frac{1}{2}\right)$such that for all x, y ∈ X, the following inequality holds:

$$d\left(fx,fy\right)\le \alpha \cdot \left(d\left(x,fy\right)+d\left(y,fx\right)\right).$$

Choudhury [7] introduced a generalization of $\mathcal{C}$-contraction, as follows:

Definition 3 Let (X, d) be a metric space. A mapping f : X → X is said to be a weakly$\mathcal{C}$-contraction if for all x, y ∈ X,

$$d\left(fx,fy\right)\le \frac{1}{2}\left(d\left(x,fy\right)+d\left(y,fx\right)-\varphi \left(d\left(x,fy\right),d\left(y,fx\right)\right)\right),$$

where ϕ : ℝ⁺² → ℝ⁺is a continuous function such that ψ(x, y) = 0 if and only if x = y = 0.

In [6, 7], the authors proved some fixed point results for the $\mathcal{C}$-contractions. In this section, we present some fixed point results for the weakly ψ-$\mathcal{C}$-contraction in complete metric spaces.

Definition 4 Let (X, d) be a metric space. The set-valued map T : X → X is said to be a set-valued weakly ψ-$\mathcal{C}$-contraction, if for all x, y ∈ X

$$\mathcal{H}\left(Tx,Ty\right)\le \psi \left(\left[D\left(x,Ty\right)+D\left(y,Tx\right)-\varphi \left(D\left(x,Ty\right),D\left(y,Tx\right)\right)\right]\right),$$

where

(1) ψ : ℝ⁺ → ℝ⁺is a strictly increasing, continuous function with$\psi \left(t\right)\le \frac{1}{2}t$for all t > 0 and ψ(0) = 0;

(2) ϕ : ℝ⁺² → ℝ⁺is a strictly decreasing, continuous function in each coordinate, such that ϕ(x, y) = 0 if and only if x = y = 0 and ϕ(x, y) ≤ x + y for all x, y ∈ ℝ⁺.

Theorem 5 Let (X, d) be a complete metric space. Let T : X → CB(X) be a set-valued weakly$\mathcal{C}$-contraction. Then, T has a fixed point in X.

Proof. Note that for each A, B ∈ CB(X), a ∈ A and γ > 0 with $\mathcal{H}\left(A,B\right)<\gamma$, there exists b ∈ B such that d(a, b) < γ. Since T : X → CB(X) be a set-valued weakly ψ-$\mathcal{C}$-contraction, we have that

$$\mathcal{H}\left(Tx,Ty\right)\le \psi \left(\left[D\left(x,Ty\right)+D\left(y,Tx\right)-\varphi \left(D\left(x,Ty\right),D\left(y,Tx\right)\right)\right]\right)$$

for all x, y ∈ X. Suppose that x₀ ∈ X and that x₁ ∈ X. Then, by induction and by the above observation, we can find a sequence {x _n } in X such that x_n+1∈ Tx _n and for each n ∈ ℕ,

$$\begin{array}{ll}\hfill d\left(x_{n+1},x_n\right)& \le \mathcal{H}\left(T{x}_n,T{x}_{n-1}\right)\\ \le \psi \left(\left[D\left(x_n,T{x}_{n-1}\right)+D\left(x_{n-1},T{x}_n\right)-\varphi \left(D\left(x_n,T{x}_{n-1}\right),D\left(x_{n-1},T{x}_n\right)\right)\right]\right)\\ \le \psi \left(\left[d\left(x_n,x_n\right)+d\left(x_{n-1},x_{n+1}\right)-\varphi \left(d\left(x_n,x_n\right),d\left(x_{n-1},x_{n+1}\right)\right)\right]\right)\\ =\psi \left(\left[0+d\left(x_{n-1},x_{n+1}\right)-\varphi \left(0,d\left(x_{n-1},x_{n+1}\right)\right)\right]\right)\\ \le \psi \left(\left[d\left(x_{n-1},x_n\right)+d\left(x_n,x_{n+1}\right)\right]\right)\\ \le \frac{1}{2}\left[d\left(x_{n-1},x_n\right)+d\left(x_n,x_{n+1}\right)\right],\end{array}$$

and hence, we deduce that for each n ∈ ℕ,

$$d\left(x_{n+1},x_n\right)\le d\left(x_n,x_{n-1}\right).$$

Thus, {d(x_n+1, x _n )} is non-increasing sequence and bounded below and hence it is convergent. Let lim_n→∞d(x_n+1, x _n ) = ξ. Letting n → ∞ in (**), we have

$$\begin{array}{ll}\hfill \xi =\underset{n\to \infty }{lim}d\left(x_{n+1},x_n\right)& \le \underset{n\to \infty }{lim}\psi \left(\left[d\left(x_{n-1},x_{n+1}\right)\right]\right)\\ \le \underset{n\to \infty }{lim}\frac{1}{2}\left[d\left(x_{n-1},x_{n+1}\right)\right]\\ \le \underset{n\to \infty }{lim}\frac{1}{2}\left[d\left(x_{n-1},x_n\right)+d\left(x_n,x_{n+1}\right)\right]\\ \le \frac{1}{2}\left[\xi +\xi \right]=\xi ,\end{array}$$

that is,

$$\underset{n\to \infty }{lim}d\left(x_{n-1},x_{n+1}\right)=2\xi .$$

By the continuity of ψ and ϕ, letting n → ∞ in (**), we have

$$\xi \le \psi \left(2\xi -\varphi \left(0,2\xi \right)\right)\le \xi -\frac{1}{2}\cdot \varphi \left(0,2\xi \right)\le \xi .$$

Hence, we have ϕ(0, 2ξ) = 0, that is, ξ = 0. Thus, lim_n→∞d(x_n+1, x _n ) = 0.

We next claim that the following result holds:

for each γ > 0, there is n₀(γ) ∈ ℕ such that for all m > n > n₀(γ),

$$d\left(x_m,x_n\right)<\gamma .\left(***\right)$$

We shall prove (***) by contradiction. Suppose that (***) is false. Then, there exists some γ > 0 such that for all k ∈ ℕ, there exist m _k , n _k ∈ ℕ with m _k > n _k ≥ k satisfying:

• (1) m _k is even and n _k is odd;

• (2) $d\left(x_{m_k},x_{n_k}\right)\ge \gamma$;

• (3) m _k is the smallest even number such that the conditions (1), (2) hold.

Since d(x_n+1, x _n ) ↘ 0, by (2), we have $\underset{k\to \infty }{lim}d\left(x_{m_k},x_{n_k}\right)=\gamma$ and

$$\begin{array}{ll}\hfill \gamma & \le d\left(x_{m_k},x_{n_k}\right)\le \mathcal{H}\left(T{x}_{m_k-1},T{x}_{n_k-1}\right)\\ \le \psi \left(\left[D\left(x_{m_k-1},T{x}_{n_k-1}\right)+D\left(x_{n_k-1},T{x}_{m_k-1}\right)-\varphi \left(D\left(x_{m_k-1},T{x}_{n_k-1}\right),D\left(x_{n_k-1},T{x}_{m_k-1}\right)\right)\right]\right)\\ \le \psi \left(\left[d\left(x_{m_k-1},x_{n_k}\right)+d\left(x_{n_k-1},x_{m_k}\right)-\varphi \left(d\left(x_{m_k-1},x_{n_k}\right),d\left(x_{n_k-1},T{x}_{m_k}\right)\right)\right]\right).\end{array}$$

Since

$$d\left(x_{m_k-1},x_{n_k}\right)+d\left(x_{n_k-1},x_{m_k}\right)\le d\left(x_{m_k-1},x_{m_k}\right)+d\left(x_{m_k},x_{n_k}\right)+d\left(x_{n_k},x_{m_k}\right)+d\left(x_{n_k-1},x_{n_k}\right),$$

letting k → ∞, then we get

$$\gamma \le \psi \left(2\gamma -\varphi \left(\gamma ,\gamma \right)\right)\le \gamma ,$$

and hence, ϕ(γ, γ)) = 0. By the definition of ϕ, we get γ = 0, a contradiction. This proves that the sequence {x _n } must be a Cauchy sequence.

Since X is complete, there exists z ∈ X such that lim_n→∞x _n = z. Therefore,

$$\begin{array}{ll}\hfill D\left(z,Tz\right)& =\underset{n\to \infty }{lim}D\left(x_{n+1},Tz\right)\\ \le \underset{n\to \infty }{lim}\mathcal{H}\left(T{x}_n,Tz\right)\\ \le \underset{n\to \infty }{lim}\psi \left(\left[D\left(x_n,Tz\right)+D\left(z,T{x}_n\right)-\varphi \left(D\left(x_n,Tz\right),D\left(z,T{x}_n\right)\right)\right]\right)\\ \le \underset{n\to \infty }{lim}\psi \left(\left[D\left(x_n,Tz\right)+d\left(z,x_{n+1}\right)-\varphi \left(D\left(x_n,Tz\right),d\left(z,x_{n+1}\right)\right)\right]\right)\\ \le \frac{1}{2}D\left(z,Tz\right)\end{array}$$

and hence, D(z, Tz) = 0, that is, z ∈ Tz, since Tz is closed.

4 Fixed point theorem (III)

In this section, we recall the notion of the Meir-Keeler type function (see [8]). A function φ : ℝ⁺ → ℝ⁺ is said to be a Meir-Keeler type function, if for each η > 0, there exists δ > 0 such that for t ∈ ℝ⁺ with η ≤ t < η + δ, we have φ(t) < η. We now introduce the new notions of the weaker Meir-Keeler type function φ : ℝ⁺ → ℝ⁺ in a metric space and the φ-function using the weaker Meir-Keeler type function, as follow:

Definition 5 Let (X, d) be a metric space. We call φ : ℝ⁺ → ℝ⁺a weaker Meir-Keeler type function, if for each η > 0, there exists δ > 0 such that for x, y ∈ X with η ≤ d(x, y) < δ + η, there exists n₀ ∈ ℕ such that${\phi }^{n_0}\left(d\left(x,y\right)\right)<{\gamma }_{\eta }$.

Definition 6 Let (X, d) be a metric space. A weaker Meir-Keeler type function φ ; ℝ⁺ → ℝ⁺is called a φ-function, if the following conditions hold:

(φ₁) φ(0) = 0, 0 < φ(t) < t for all t > 0;

(φ ₂ ) φ is a strictly increasing function;

(φ₃) for each t ∈ ℝ⁺, {φⁿ (t)}_n∈ℕis decreasing;

(φ₄) for each$t_n\in {ℝ}^{+}\backslash \left\{0\right\}$, if lim_n→∞t _n = γ > 0, then lim_n→∞φ(t _n ) < γ;

(φ₅) for each t _n ∈ ℝ⁺, if lim_n→∞t _n = 0, then lim_n→∞φ(t _n ) = 0.

Definition 7 Let (X, d) be a metric space. The set-valued map T : X → X is said to be a set-valued weaker Meir-Keeler type φ-contraction, if

$$\mathcal{H}\left(Tx,Ty\right)\le \phi \left(\frac{1}{2}\left[D\left(x,Ty\right)+D\left(y,Tx\right)\right]\right)$$

for all x, y ∈ X.

We now state the main fixed point theorem for a set-valued weaker Meir-Keeler type ψ-contraction in metric spaces, as follows:

Theorem 6 Let (X, d) be a complete metric space. Let T : CB(X) be a set-valued weaker Meir-Keeler type ψ-contraction. Then, T has a fixed point in X.

Proof. Note that for each A, B ∈ CB(X), a ∈ A and γ > 0 with $\mathcal{H}\left(A,B\right)<\gamma$, there exists b ∈ B such that d(a, b) < γ. Since T : X → CB(X) be a set-valued ψ-contraction, we have that

$$\mathcal{H}\left(Tx,Ty\right)\le \phi \left(\frac{1}{2}\left[D\left(x,Ty\right)+D\left(y,Tx\right)\right]\right)$$

for all x, y ∈ X. Suppose that x₀ ∈ X and that x₁ ∈ X. Then, by induction and by the above observation, we can find a sequence {x _n } in X such that x_n+1∈ Tx _n and for each n ∈ ℕ,

$$\begin{array}{ll}\hfill d\left(x_{n+1},x_n\right)& \le \phi \left(\frac{1}{2}\left[D\left(x_n,T{x}_{n-1}\right)+D\left(x_{n-1},T{x}_n\right)\right]\right)\\ \le \phi \left(\frac{1}{2}\left[d\left(x_n,x_n\right)+d\left(x_{n-1},x_{n+1}\right)\right]\right)\\ \le \phi \left(\frac{1}{2}\left[d\left(x_{n-1},x_n\right)+d\left(x_n,x_{n+1}\right)\right]\right),\end{array}$$

and by the conditions (φ₁) and (φ₂), we can deduce that for each n ∈ ℕ,

$$d\left(x_{n+1},x_n\right)\le \phi \left(d\left(x_n,x_{n-1}\right)\right)<d\left(x_n,x_{n-1}\right)$$

and

$$d\left(x_{n+1},x_n\right)\le \phi \left(d\left(x_n,x_{n-1}\right)\right)\le \cdots \le {\phi }^n\left(d\left(x_1,x_0\right)\right).$$

By the condition (φ₃), {φⁿ (d(x₀, x₁))}_n∈ℕis decreasing, it must converges to some η ≥ 0. We claim that η = 0. On the contrary, assume that η > 0. Then, by the definition of the weaker Meir-Keeler type function, there exists δ > 0 such that for x₀, x₁ ∈ X with η ≤ d(x₀, x₁) < δ + η, there exists n₀ ∈ ℕ such that ${\phi }^{n_0}\left(d\left(x_0,x_1\right)\right)<\eta$. Since lim_n→∞φⁿ(d(x₀, x₁)) = η, there exists m₀ ∈ ℕ such that η ≤ φ^m (d(x₀, x₁)) < δ + η, for all m ≥ m₀. Thus, we conclude that ${\phi }^{m_0+n_0}\left(d\left(x_0,x_1\right)\right)<\eta$. Hence, we get a contradiction. Hence, lim_n→∞φⁿ(d(x₀, x₁)) = 0, and hence, lim_n→∞d(x _n , x_n+1) = 0.

Next, we let c _m = d(x _m , x_m+1), and we claim that the following result holds:

for each ε > 0, there is n₀(ε) ∈ ℕ such that for all m , n ≥ n₀(ε),

$$d\left(x_m,x_{m+1}\right)<\epsilon .\left(****\right)$$

We shall prove (****) by contradiction. Suppose that (****) is false. Then, there exists some ε > 0 such that for all p ∈ N, there are m _p , n _p ∈ ℕ with m _p > n _p ≥ p satisfying:

• (i) m _p is even and n _p is odd,

• (ii) $d\left(x_{m_p},x_{n_p}\right)\ge \epsilon$, and

• (iii) m _p is the smallest even number such that the conditions (i), (ii) hold.

Since c _m ↘ 0, by (ii), we have $\underset{p\to \infty }{lim}d\left(x_{m_p},x_{n_p}\right)=\epsilon$, and

$$\begin{array}{ll}\hfill \epsilon & \le d\left(x_{m_p},x_{n_p}\right)\\ \le \mathcal{H}\left(T{x}_{m_p-1},T{x}_{n_p-1}\right)\\ \le \phi \left(\frac{1}{2}\left[D\left(x_{m_p-1},T{x}_{n_p-1}\right)+D\left(x_{n_p-1},T{x}_{m_p-1}\right)\right]\right)\\ \le \phi \left(\frac{1}{2}\left[d\left(x_{m_p-1},x_{n_p}\right)+d\left(x_{n_p-1},x_{m_p}\right)\right]\right)\\ \le \phi \left(\frac{1}{2}\left[d\left(x_{m_p-1},x_{m_p}\right)+2d\left(x_{n_p},x_{m_p}\right)+d\left(x_{n_p-1},x_{n_p}\right)\right]\right).\end{array}$$

Letting p → ∞. By the condition (φ₄), we have

$$\epsilon \le \underset{p\to \infty }{lim}\phi \left(\frac{1}{2}\left[d\left(x_{m_p-1},x_{m_p}\right)+2d\left(x_{n_p},x_{m_p}\right)+d\left(x_{n_p-1},x_{n_p}\right)\right]\right)<\epsilon ,$$

a contradiction. Hence, {x _n } is a Cauchy sequence. Since (X, d) is a complete metric space, there exists μ ∈ X such that limn→∞x_n+1= μ. Therefore,

$$\begin{array}{ll}\hfill D\left(\mu ,T\mu \right)& =\underset{n\to \infty }{lim}D\left(x_{n+1},T\mu \right)\\ \le \underset{n\to \infty }{lim}\mathcal{H}\left(T{x}_n,T\mu \right)\\ \le \underset{n\to \infty }{lim}\phi \left(\frac{1}{2}\left[D\left(x_n,T\mu \right)\right)+D\left(\mu ,T{x}_n\right)\right)\\ \le \underset{n\to \infty }{lim}\phi \left(\frac{1}{2}\left[D\left(x_n,T\mu \right)\right)+d\left(\mu ,x_{n+1}\right)\right)\\ \le \frac{1}{2}D\left(\mu ,T\mu \right),\end{array}$$

and hence, D(μ, Tμ) = 0, that is, μ ∈ Tμ, since Tμ is closed.