The existence of fixed points for new nonlinear multivalued maps and their applications

Abstract

In this paper, we first establish some new fixed point theorems for $\mathcal{M}\mathcal{T}$-functions. By using these results, we can obtain some generalizations of Kannan's fixed point theorem and Chatterjea's fixed point theorem for nonlinear multivalued contractive maps in complete metric spaces. Our results generalize and improve some main results in the literature and references therein.

Mathematics Subject Classifications

47H10; 54H25

1. Introduction

Throughout this paper, we denote by ℕ and ℝ, the sets of positive integers and real numbers, respectively. Let (X, d) be a metric space. For each x ∈ X and A ⊆ X, let d(x, A) = inf_{y ∈ A}d(x, y). Denote by $\mathcal{N}\left(X\right)$ the family of all nonempty subsets of $X,\mathcal{C}\left(X\right)$ the family of all nonempty closed subsets of X and $\mathcal{C}\mathcal{B}\left(X\right)$ the class of all nonempty closed bounded subsets of X, respectively.

For any $A,B\in \mathcal{C}\mathcal{B}\left(X\right)$, define a function $\mathcal{H}:\mathcal{C}\mathcal{B}\left(X\right)\times \mathcal{C}\mathcal{B}\left(X\right)\to \left[0,\infty \right)$ by

$$\mathcal{H}\left(A,B\right)=max\left\{\underset{x\in B}{sup}d\left(x,A\right),\underset{x\in A}{sup}d\left(x,B\right)\right\},$$

then $\mathcal{H}$ is said to be the Hausdorff metric on $\mathcal{C}\mathcal{B}\left(X\right)$ induced by the metric d on X. A point x in X is a fixed point of a map T if Tx = x (when T: X → X is a single-valued map) or x ∈ Tx (when T: X → 2^Xis a multivalued map). The set of fixed points of T is denoted by $\mathcal{F}\left(T\right)$.

It is known that many metric fixed point theorems were motivated from the Banach contraction principle (see, e.g., [1]) that plays an important role in various fields of applied mathematical analysis. Later, Kannan [2, 3] and Chatterjea [4] established the following fixed point theorems.

Theorem K. (Kannan [2, 3]) Let (X,d) be a complete metric space and T: X → X be a selfmap. Suppose that there exists $\gamma \in \left[0,\frac{1}{2}\right)$ such that

$$d\left(Tx,Ty\right)\le \gamma \left(d\left(x,Tx\right)+d\left(y,Ty\right)\right)\mathsf{\text{for}}\mathsf{\text{all}}x,y\in X.$$

Then, T has a unique fixed point in X.

Theorem C. (Chatterjea [4]) Let (X,d) be a complete metric space and T: X → X be a selfmap. Suppose that there exists $\gamma \in \left[0,\frac{1}{2}\right)$ such that

$$d\left(Tx,Ty\right)\le \gamma \left(d\left(x,Ty\right)+d\left(y,Tx\right)\right)\mathsf{\text{for}}\mathsf{\text{all}}x,y\in X.$$

Then, T has a unique fixed point in X.

Let f be a real-valued function defined on ℝ. For c ∈ ℝ, we recall that

$$\underset{x\to c}{limsup}f\left(x\right)=\underset{\epsilon >0}{inf}\underset{0<\left|x-c\right|<\epsilon }{sup}f\left(x\right)$$

and

$$\underset{x\to c^{+}}{limsup}f\left(x\right)=\underset{\epsilon >0}{inf}\underset{0<x-c<\epsilon }{sup}f\left(x\right).$$

Definition 1.1. [5–10] A function φ: [0, ∞) → [0,1) is said to be an $\mathcal{M}\mathcal{T}$-function if it satisfies Mizoguchi-Takahashi's condition ( i.e., lim sup_{s → t}+ φ(s) < 1 for all t ∈ [0, ∞)).

It is obvious that if φ: [0, ∞) → [0,1) is a nondecreasing function or a nonincreasing function, then φ is an $\mathcal{M}\mathcal{T}$-function. So the set of $\mathcal{M}\mathcal{T}$-functions is a rich class. But it is worth to mention that there exist functions that are not $\mathcal{M}\mathcal{T}$-functions.

Example 1.1. [8] Let φ: [0, ∞) → [0, 1) be defined by

$$\phi \left(t\right):=\left\{\begin{array}{cc}\hfill \frac{sint}{t}\hfill & \hfill ,\mathsf{\text{if}}t\in \left(0,\frac{\pi }{2}\right]\hfill \\ \hfill 0\hfill & \hfill ,\mathsf{\text{otherwise}}\mathsf{\text{.}}\hfill \end{array}\right.$$

Since $lim\underset{s\to 0^{+}}{sup}\phi \left(s\right)=1,\phi$ is not an $\mathcal{M}\mathcal{T}$-function.

Very recently, Du [8] first proved some characterizations of $\mathcal{M}\mathcal{T}$-functions.

Theorem D. [8] Let φ: [0, ∞) → [0,1) be a function. Then, the following statements are equivalent.

• (a) φ is an $\mathcal{M}\mathcal{T}$-function.

• (b) For each t ∈ [0, ∞), there exist $r_t^{\left(1\right)}\in \left[0,1\right)$ and ${\epsilon }_t^{\left(1\right)}>0$ such that $\phi \left(s\right)\le r_t^{\left(1\right)}$ for all $s\in \left(t,t+{\epsilon }_t^{\left(1\right)}\right)$.

• (c) For each t ∈ [0, ∞), there exist $r_t^{\left(2\right)}\in \left[0,1\right)$ and ${\epsilon }_t^{\left(2\right)}>0$ such that $\phi \left(s\right)\le r_t^{\left(2\right)}$ for all $s\in \left[t,t+{\epsilon }_t^{\left(2\right)}\right]$.

• (d) For each t ∈ [0, ∞), there exist $r_t^{\left(3\right)}\in \left[0,1\right)$ and ${\epsilon }_t^{\left(3\right)}>0$ such that $\phi \left(s\right)\le r_t^{\left(3\right)}$ for all $s\in \left(t,t+{\epsilon }_t^{\left(3\right)}\right]$.

• (e) For each t ∈ [0, ∞), there exist $r_t^{\left(4\right)}\in \left[0,1\right)$ and ${\epsilon }_t^{\left(4\right)}>0$ such that $\phi \left(s\right)\le r_t^{\left(4\right)}$ for all $s\in \left[t,t+{\epsilon }_t^{\left(4\right)}\right)$.

• (f) For any nonincreasing sequence {x _n }_{n ∈ℕ}in [0, ∞), we have 0 ≤ sup_{n ∈ℕ}φ(x _n ) < 1.

• (g) φ is a function of contractive factor [10]; that is, for any strictly decreasing sequence {x _n }_{n ∈ℕ}in [0, ∞), we have 0 ≤ sup_{n ∈ℕ}φ(x _n ) < 1.

In 2007, Berinde and Berinde [11] proved the following interesting fixed point theorem.

Theorem BB. (Berinde and Berinde [11]) Let (X,d) be a complete metric space, $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a multivalued map, φ: [0, ∞) → [0,1) be an $\mathcal{M}\mathcal{T}$-function and L ≥ 0. Assume that

$$\mathcal{H}\left(Tx,Ty\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)+Ld\left(y,Tx\right)\mathsf{\text{for}}\mathsf{\text{all}}x,y\in X.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

It is quite obvious that if let L = 0 in Theorem BB, then we can obtain Mizoguchi-Takahashi's fixed point theorem [12] that is a partial answer of Problem 9 in Reich [13, 14].

Theorem MT. (Mizoguchi and Takahashi [12]) Let (X,d) be a complete metric space, $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a multivalued map and φ: [0, ∞) → [0,1) be an $\mathcal{M}\mathcal{T}$-function. Assume that

$$\mathcal{H}\left(Tx,Ty\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)\mathsf{\text{for}}\mathsf{\text{all}}x,y\in X.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

In fact, Mizoguchi-Takahashi's fixed point theorem is a generalization of Nadler's fixed point theorem, but its primitive proof is difficult. Later, Suzuki [15] give a very simple proof of Theorem MT. Recently, Du [5] established new fixed point theorems for τ⁰-metric (see Def. 2.1 below) and $\mathcal{M}\mathcal{T}$-functions to extend Berinde-Berinde's fixed point theorem. In [5], some generalizations of Kannan's fixed point theorem, Chatterjea's fixed point theorem and other new fixed point theorems for nonlinear multivalued contractive maps were given.

In this paper, we first establish some new fixed point theorems for $\mathcal{M}\mathcal{T}$-functions. By using these results, we can obtain some generalizations of Kannan's fixed point theorem and Chatterjea's fixed point theorem for nonlinear multivalued contractive maps in complete metric spaces. Our results generalize and improve some main results in [1–5, 7–9, 12–15] and references therein.

2. Preliminaries

Let (X, d) be a metric space. Recall that a function p: X × X → [0, ∞) is called a w-distance [1, 16, 17], if the following are satisfied:

(w 1) p(x, z) ≤ p(x, y) + p(y, z) for any x, y, z ∈ X;

(w 2) for any x ∈ X,p(x, ⋅) : X → [0, ∞) is l.s.c;

(w 3) for any ε > 0, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ imply d(x, y) ≤ ε.

Recently, Lin and Du introduced and studied τ-functions [5, 9, 18–22]. A function p: X × X → [0, ∞) is said to be a τ-function, if the following conditions hold:

(τ 1) p(x, z) ≤ p(x, y) + p(y, z) for all x, y, z ∈ X;

(τ 2) If x ∈ X and {y _n } in X with lim_{n →∞}y _n = y such that p(x, y _n ) ≤ M for some M = M(x) > 0, then p(x, y) ≤ M;

(τ 3) For any sequence {x _n } in X with lim_{n →∞}sup{p(x _n , x _m ): m >n} = 0, if there exists a sequence {y _n } in X such that lim_{n →∞}p(x _n , y _n ) = 0, then lim_{n →∞}d(x _n , y _n ) = 0;

(τ 4) For x, y, z ∈ X,p(x, y) = 0 and p(x, z) = 0 imply y = z.

Note that not either of the implications p(x, y) = 0 ⇔ x = y necessarily holds and p is nonsymmetric in general. It is well-known that the metric d is a w-distance and any w-distance is a τ-function, but the converse is not true; see [5, 19].

The following Lemma is essentially proved in [19]. See also [5, 8, 20, 22].

Lemma 2.1. [5, 8, 19, 20, 22] Let (X,d) be a metric space and p: X × X → [0, ∞) be any function. Then, the following hold:

• (a) If p satisfies (w 2), then p satisfies (τ 2);

• (b) If p satisfies (w 1) and (w 3), then p satisfies (τ 3);

• (c) Assume that p satisfies (τ 3). If {x _n } is a sequence in X with lim_{n →∞}sup{p(x _n ,x _m ): m >n} = 0, then {x _n } is a Cauchy sequence in X.

Let (X, d) be a metric space and p: X × X → [0, ∞) a τ-function. For each x ∈ X and A ⊆ X, let

$$d\left(x,A\right)=\underset{y\in A}{inf}d\left(x,y\right).$$

Recall that a selfmap T: X → X is said to be

• (a) Kannan's type [2, 5, 16] if there exists $\gamma \in \left[0,\frac{1}{2}\right)$, such that d(Tx, Ty) ≤ γ{d(x, Tx)+d(y, Ty)} for all x, y ∈ X;

• (b) Chatterjea's type [3, 5] if there exists $\gamma \in \left[0,\frac{1}{2}\right)$, such that d(Tx, Ty) ≤ γ{d(x, Ty) + d(y, Tx)} for all x, y ∈ X.

Lemma 2.2. [5, 9, 21, 22] Let A be a closed subset of a metric space (X, d) and p: X × X → [0, ∞) be any function. Suppose that p satisfies (τ 3) and there exists u ∈ X such that p(u, u) = 0. Then, p(u, A) = 0 if and only if u ∈ A.

Recently, Du [5, 21] first has introduced the concepts of τ⁰-functions and τ⁰-metrics as follows.

Definition 2.1. [5, 9, 21, 22] Let (X, d) be a metric space. A function p: X × X → [0, ∞) is called a τ⁰-function if it is a τ-function on X with p(x, x) = 0 for all x ∈ X.

Remark 2.1. If p is a τ⁰-function then, from (τ 4), p(x, y) = 0 if and only if x = y.

Example 2.1. [5] Let X = ℝ with the metric d(x, y) = |x —y| and 0 <a <b. Define the function p: X × X → [0, ∞) by

$$p\left(x,y\right)=max\left\{a\left(y-x\right),b\left(x-y\right)\right\}.$$

Then, p is nonsymmetric, and hence, p is not a metric. It is easy to see that p is a τ⁰-function.

Definition 2.2. [5, 9, 21, 22] Let (X, d) be a metric space and p be a τ⁰-function (resp. w⁰-distance). For any $A,B\in \mathcal{C}\mathcal{B}\left(X\right)$, define a function ${\mathcal{D}}_p:\mathcal{C}\mathcal{B}\left(X\right)\times \mathcal{C}\mathcal{B}\left(X\right)\to \left[0,\infty \right)$ by

$${\mathcal{D}}_p\left(A,B\right)=max\left\{{\delta }_p\left(A,B\right),{\delta }_p\left(B,A\right)\right\},$$

where δ _p (A, B) = sup_{x ∈ A}p(x, B) and δ _p (B, A) = sup_{x ∈ B}p(x, A), then ${\mathcal{D}}_p$ is said to be the τ⁰-metric (resp. w⁰-metric) on $\mathcal{C}\mathcal{B}\left(X\right)$ induced by p.

Clearly, any Hausdorff metric is a τ⁰-metric, but the reverse is not true. It is well-known that every τ⁰-metric ${\mathcal{D}}_p$ is a metric on $\mathcal{C}\mathcal{B}\left(X\right)$; for more detail, see [5, 9, 21, 22].

Lemma 2.3. Let (X,d) be a metric space, $T:X\to \mathcal{C}\left(X\right)$ be a multivalued map and {z _n } be a sequence in X satisfying z_{n +1}∈ Tz _n , n ∈ ℕ, and {z _n } converge to v in X. Then, the following statements hold.

• (a) If T is closed (that is, GrT = {(x, y) ∈ X × X: y ∈ Tx}, the graph of T, is closed in X × X), then $\mathcal{F}\left(T\right)\ne \varnothing$.

• (b) Let p be a function satisfying (τ 3) and p(v, v) = 0. If lim_{n →∞}p(z _n , z_{n +1}) = 0 and the map f: X → [0, ∞) defined by f(x) = p(x, Tx) is l.s.c., then $\mathcal{F}\left(T\right)\ne \varnothing$.

• (c) If the map g: X → [0, ∞) defined by g(x) = d(x, Tx) is l.s.c., then $\mathcal{F}\left(T\right)\ne \varnothing$.

• (d) Let p be a function satisfying (τ 3). If lim_{n →∞}p(z _n , Tv) = 0 and lim _{n →∞}sup{p(z _n , z _m ): m >n} = 0, then $\mathcal{F}\left(T\right)\ne \varnothing$.

Proof.

• (a) Since T is closed, z_{n +1}∈ Tz _n , n ∈ ℕ and z _n → v as n → ∞, we have v ∈ Tv. So $\mathcal{F}\left(T\right)\ne \varnothing$.

• (b) Since z _n → v as n → ∞, by the lower semicontinuity of f, we obtain

  $$p\left(v,Tv\right)=f\left(v\right)\le \underset{m\to \infty }{liminf}p\left(z_n,T{z}_n\right)\le \underset{n\to \infty }{lim}p\left(z_n,z_{n+1}\right)=0,$$

which implies p(v, Tv) = 0. By Lemma 2.2, we get $v\in \mathcal{F}\left(T\right)$.

• (c) Since {z _n } is convergent in X, lim_{n →∞}d(z _n , z _n +1) = 0. Since

  $$d\left(v,Tv\right)=g\left(v\right)\le \underset{m\to \infty }{liminf}d\left(z_n,T{z}_n\right)\le \underset{n\to \infty }{lim}d\left(z_n,z_{n+1}\right)=0,$$

we have d(v,Tv) = 0 and hence $v\in \mathcal{F}\left(T\right)$.

• (d) Since lim_{n →∞}sup{p(z _n , z _m ): m >n} = 0 and lim_{n →∞}p(z _n , Tv) = 0, there exists {a _n } ⊂ {z _n } with lim_{n →∞}sup{p(a _n , a _m ): m >n} = 0 and {b _n } ⊂ Tv such that lim_{n →∞}p(a _n , b _n ) = 0. By (τ 3), lim_{n →∞}d(a _n , b _n ) = 0. Since a _n → v as n → ∞ and d(b _n ,v) ≤ d(b _n ,a _n ) + d(a _n ,v), it implies b _n → v as n → ∞. By the closedness of Tv, we have v ∈ Tv or $v\in \mathcal{F}\left(T\right)$.   □

In this paper, we first introduce the concepts of capable maps as follows.

Definition 2.3. Let (X, d) be a metric space and $T:X\to \mathcal{C}\left(X\right)$ be a multivalued map. We say that T is capable if T satisfies one of the following conditions:

(D1) T is closed;

(D2) the map f: X → [0, ∞) defined by f(x) = p(x, Tx) is l.s.c;

(D3) the map g: X → [0, ∞) defined by g(x) = d(x, Tx) is l.s.c;

(D4) for each sequence {x _n } in X with x_{n +1}∈ Tx _n , n ∈ ℕ and lim_{n →∞}x _n = v, we have lim_{n →∞}p(x _n , Tv) = 0;

(D5) inf{p(x, z) + p(x,Tx) : x ∈ X} > 0 for every $z\notin \mathcal{F}\left(T\right)$.

Remark 2.2.

• (1) Let (X, ||⋅||) be a Banach space. If $T:X\to \mathcal{C}\left(X\right)$ is u.s.c, then T is a capable map since it is closed (for more detail, see [5, 23]).

• (2) Let (X, d) be a metric space and $T:X\to \mathcal{C}\left(X\right)$ be u.s.c. Since the function f: X → [0, ∞) defined by f(x) = d(x,Tx) is l.s.c. (see, e.g., [24, Lemma 3.1] and [25, Lemma 2]), T is a capable map.

• (3) Let (X, d) be a metric space and $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a generalized multivalued (φ, L)-weak contraction [11], that is, there exists an $\mathcal{M}\mathcal{T}$-function φ and L ≥ 0 such that

  $$\mathcal{H}\left(Tx,Ty\right)\le \phi \left(d\left(x,y\right)\right)d\left(x,y\right)+Ld\left(y,Tx\right)\mathsf{\text{for}}\mathsf{\text{all}}x,y\in X.$$

Then, T is a capable map. Indeed, let {x _n } in X with x_{n +1}∈ Tx _n , n ∈ ℕ and lim_{n →∞}x _n = v.

Then

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

which means that T satisfies (D4).

• (4) Let (X, d) be a metric space and T: X → X is a single-valued map of Kannan's type, then T is a capable map since (D5) holds; for more detail, see [[16], Corollary 3].

3. Fixed point theorems of generalized Chatterjea's type and others

Below, unless otherwise specified, let (X, d) be a complete metric space, p be a τ⁰-function and ${\mathcal{D}}_p$ be a τ⁰-metric on $\mathcal{C}\mathcal{B}\left(X\right)$ induced by p.

In this section, we will establish some fixed point theorems of generalized Chatterjea's type.

Theorem 3.1. Let $T:X\to \mathcal{C}\left(X\right)$ be a capable map. Suppose that there exists an $\mathcal{M}\mathcal{T}$-function φ: [0, ∞) → [0,1) such that for each x ∈ X,

$$2p\left(y,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,Ty\right)\mathsf{\text{for all }}y\in Tx.$$

(3.1)

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Proof. Let κ: [0, ∞) → [0,1) be defined by $\kappa \left(t\right)=\frac{1+\phi \left(t\right)}{2}$. Then

$$0\le \phi \left(t\right)<\kappa \left(t\right)<1\mathsf{\text{for all }}t\in \left[0,\infty \right).$$

Let x₁ ∈ X and x₂ ∈ Tx₁. If x₁ = x₂, then $x_1\in \mathcal{F}\left(T\right)$ and we are done. Otherwise, if x₂ ≠ x₁, by Remark 2.1, we have p(x₁,x₂) > 0. If x₁ ∈ Tx₂, then it follows from (3.1) that

$$2p\left(x_2,T{x}_2\right)\le \phi \left(p\left(x_1,x_2\right)\right)p\left(x_1,T{x}_2\right)=0,$$

which implies p(x₂,Tx₂) = 0. Since p is a τ⁰-function and Tx₂ is closed in X, by Lemma 2.2, x₂ ∈ Tx₂ and $x_2\in \mathcal{F}\left(T\right)$. If x₁ ∉ Tx₂, then p(x₁,Tx₂) > 0 and, by (3.1), there exists x₃ ∈ Tx₂ such that

$$\begin{array}{ll}\hfill 2p\left(x_2,x_3\right)& <\kappa \left(p\left(x_1,x_2\right)\right)p\left(x_1,x_3\right)\\ \le \kappa \left(p\left(x_1,x_2\right)\right)\left[p\left(x_1,x_2\right)+p\left(x_2,x_3\right)\right].\end{array}$$

By induction, we can obtain a sequence {x _n } in X satisfying x_{n +1}∈ Tx _n , n ∈ ℕ, p(x _n , x_{n +1}) > 0

and

$$2p\left(x_{n+1},x_{n+2}\right)<\kappa \left(p\left(x_n,x_{n+1}\right)\right)\left[p\left(x_n,x_{n+1}\right)+p\left(x_{n+1},x_{n+2}\right)\right]$$

(3.2)

By (3.2), we get

$$p\left(x_{n+1},x_{n+2}\right)<\frac{\kappa \left(p\left(x_n,x_{n+1}\right)\right)}{2-\kappa \left(p\left(x_n,x_{n+1}\right)\right)}p\left(x_n,x_{n+1}\right)$$

(3.3)

Since 0 <κ(t) < 1 for all $t\in \left[0,\infty \right),\frac{\kappa \left(p\left(x_n,x_{n+1}\right)\right)}{2-\kappa \left(p\left(x_n,x_{n+1}\right)\right)}\in \left(0,1\right)$ for all n ∈ ℕ. So the sequence {p(x _n , x_{n +1})} is strictly decreasing in [0, ∞). Since φ is an $\mathcal{M}\mathcal{T}$-function, by applying (g) of Theorem D, we have

$$0\le \underset{n\in \mathbb{N}}{sup}\phi \left(p\left(x_n,x_{n+1}\right)\right)<1.$$

Hence, it follows that

$$0<\underset{n\in \mathbb{N}}{sup}\kappa \left(p\left(x_n,x_{n+1}\right)\right)=\frac{1}{2}\left[1+\underset{n\in \mathbb{N}}{sup}\phi \left(p\left(x_n,x_{n+1}\right)\right)\right]<1.$$

Let λ:= sup_{n ∈ℕ}κ(p(x _n , x_{n +1})) and take $c:=\frac{\lambda }{2-\lambda }$. Then λ, c ∈ (0,1). We claim that {x _n } is a Cauchy sequence in X. Indeed, by (3.3), we have

$$p\left(x_{n+1},x_{n+2}\right)<\frac{\kappa \left(p\left(x_n,x_{n+1}\right)\right)}{2-\kappa \left(p\left(x_n,x_{n+1}\right)\right)}p\left(x_n,x_{n+1}\right)\le cp\left(x_n,x_{n+1}\right).$$

(3.4)

It implies from (3.4) that

$$p\left(x_{n+1},x_{n+2}\right)<cp\left(x_n,x_{n+1}\right)<\cdots <c^{n}p\left(x_1,x_2\right)\mathsf{\text{for each }}n\in \mathbb{N}.$$

We have lim_{n →∞}sup{p(x _n ,x _m ): m >n} = 0. Indeed, let ${\alpha }_n=\frac{c^{n-1}}{1-c}p\left(x_1,x_2\right),n\in \mathbb{Z}$. For m, n ∈ ℕ with m >n, we have

$$p\left(x_n,x_m\right)\le \sum _{j=n}^{m-1}p\left(x_j,x_{j+1}\right)<{\alpha }_n.$$

(3.5)

Since c ∈ (0,1), lim_{n →∞}α_n = 0 and, by (3.5), we get

$$\underset{n\to \infty }{lim}sup\left\{p\left(x_n,x_m\right):m>n\right\}=0.$$

(3.6)

Applying (c) of Lemma 2.1, {x _n } is a Cauchy sequence in X. By the completeness of X, there exists v ∈ X such that x _n → v as n → ∞. From (τ 2) and (3.5), we have

$$p\left(x_n,v\right)\le {\alpha }_n\mathsf{\text{for all }}n\in \mathbb{N}.$$

(3.7)

Now, we verify that $v\in \mathcal{F}\left(T\right)$. Applying Lemma 2.3, we know that $v\in \mathcal{F}\left(T\right)$ if T satisfies one of the conditions (D1), (D2), (D3) and (D4).

Finally, assume (D5) holds. On the contrary, suppose that v ∉ Tv. Then, by (3.5) and (3.7), we have

$$\begin{array}{ll}\hfill 0& <\underset{x\in X}{inf}\left\{p\left(x,v\right)+p\left(x,Tx\right)\right\}\\ \le \underset{n\in \mathbb{N}}{inf}\left\{p\left(x_n,v\right)+p\left(x_n,T{x}_n\right)\right\}\\ \le \underset{n\in \mathbb{N}}{inf}\left\{p\left(x_n,v\right)+p\left(x_n,x_{n+1}\right)\right\}\\ \le \underset{n\to \infty }{lim}2{\alpha }_n\\ =0,\end{array}$$

a contradiction. Therefore $v\in \mathcal{F}\left(T\right)$. The proof is completed.   □

Here, we give a simple example illustrating Theorem 3.1.

Example 3.1. Let X = [0,1] with the metric d(x,y) = |x — y| for x,y ∈ X. Then, (X,d) is a complete metric space. Let $T:X\to \mathcal{C}\left(X\right)$ be defined by

$$T\left(x\right)=\left\{\begin{array}{c}\hfill \left\{0,1\right\},\mathsf{\text{if }}x=0,\hfill \\ \hfill \left\{\frac{1}{2}{x}^3,1\right\},\mathsf{\text{if }}x\in \left(0,\frac{1}{2}\right],\hfill \\ \hfill \left\{0,\frac{1}{2}{x}^3\right\},\mathsf{\text{if }}x\in \left(\frac{1}{2},1\right),\hfill \\ \hfill \left\{1\right\},\mathsf{\text{if }}x=1.\hfill \end{array}\right.$$

and φ: [0, ∞) → [0,1) be defined by

$$\phi \left(t\right)=\left\{\begin{array}{cc}\hfill 2t,\hfill & \hfill \mathsf{\text{if }}t\in \left[0,\frac{1}{2}\right),\hfill \\ \hfill 0,\hfill & \hfill \mathsf{\text{if }}t\in \left[\frac{1}{2},\infty \right).\hfill \end{array}\right.$$

Then, φ is an $\mathcal{M}\mathcal{T}$-function and $\mathcal{F}\left(T\right)=\left\{0,1\right\}\ne 0$.

On the other hand, one can easily see that

$$d\left(x,Tx\right)=\left\{\begin{array}{cc}\hfill x-\frac{1}{2}{x}^3,\hfill & \hfill \mathsf{\text{if }}x\in \left[0,1\right),\hfill \\ \hfill 0,\hfill & \hfill \mathsf{\text{if }}x=1.\hfill \end{array}\right.$$

So f(x): = d(x,Tx) is l.s.c., and hence, T is a capable map. Moreover, it is not hard to verify that for each x ∈ X,

$$2p\left(y,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,Ty\right)\mathsf{\text{for all }}y\in Tx.$$

Therefore, all the assumptions of Theorem 3.1 are satisfied, and we also show that $\mathcal{F}\left(T\right)\ne \varnothing$ from Theorem 3.1.

Theorem 3.2. Let $T:X\to \mathcal{C}\left(X\right)$ be a capable map and φ: [0, ∞) → [0,1) be an $\mathcal{M}\mathcal{T}$-function. Let k ∈ ℝ with k ≥ 2. Suppose that for each x ∈ X

$$kp\left(y,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,Ty\right)\mathsf{\text{for all }}y\in Tx.$$

(3.9)

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Proof. Since k ≥ 2, (3.9) implies (3.1). Therefore, the conclusion follows from Theorem 3.1.   □

The following result is immediate from the definition of ${\mathcal{D}}_p$ and Theorem 3.1.

Theorem 3.3. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exists an $\mathcal{M}\mathcal{T}$-function φ: [0, ∞) → [0,1) such that for each x ∈ X,

$$2{\mathcal{D}}_p\left(Tx,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,Ty\right)\mathsf{\text{for all }}y\in Tx.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Theorem 3.4. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exist two $\mathcal{M}\mathcal{T}$-functions φ, τ: [0, ∞) → [0,1) such that

$$2{\mathcal{D}}_p\left(Tx,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,Ty\right)+\tau \left(p\left(x,y\right)\right)p\left(y,Tx\right)\mathsf{\text{for all }}x,y\in X.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Proof. For each x ∈ X, let y ∈ Tx be arbitrary. Since p(y,Tx) = 0, we have $2{\mathcal{D}}_p\left(Tx,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,Ty\right)$. Therefore, the conclusion follows from Theorem 3.3.   □

Theorem 3.5. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exists an $\mathcal{M}\mathcal{T}$-function φ: [0, ∞) → [0,1) such that

$$2{\mathcal{D}}_p\left(Tx,Ty\right)\le \phi \left(p\left(x,y\right)\right)\left(p\left(x,Ty\right)+p\left(y,Tx\right)\right)\mathsf{\text{for all }}x,y\in X.$$

(3.10)

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Proof. Let τ = φ. Then, the conclusion follows from Theorem 3.4.   □

Theorem 3.6. Let T: X → X be a selfmap. Suppose that there exists an $\mathcal{M}\mathcal{T}$-function φ: [0, ∞) → [0,1) such that

$$2d\left(Tx,Ty\right)\le \phi \left(d\left(x,y\right)\right)\left(d\left(x,Ty\right)+d\left(y,Tx\right)\right)\mathsf{\text{for all }}x,y\in X.$$

(3.11)

Then, T has a unique fixed point in X.

Proof. Let p ≡ d. Then, (3.11) and (3.10) are identical. We prove that T is a capable map. In fact, it suffices to show that (D5) holds. Assume that there exists w ∈ X with w ≠ Tw and inf {d(x,w) + d(x,Tx): x ∈ X} = 0. Then, there exists a sequence {x _n } in X such that lim_{n →∞}(d(x _n , w) + d(x _n ,Tx _n )) = 0. It follows that d(x _n ,w) → 0 and d(x _n ,Tx _n ) → 0 and hence d(w,Tx _n ) → 0 or Tx _n → w as n → ∞. By hypothesis, we have

$$2d(T{x}_n,Tw)\le \phi (d(x_n,w))((d(x_n,Tw)+d(w,T{x}_n))$$

(3.12)

for all n ∈ ℕ. Letting n → ∞ in (3.12), since φ is an $\mathcal{M}\mathcal{T}$-function and d(x _n ,w) → 0, we have d(w,Tw) <d(w,Tw), which is a contradiction. So (D5) holds and hence T is a capable map. Applying Theorem 3.5, $\mathcal{F}\left(T\right)\ne \varnothing$. Suppose that there exists $u,v\in \mathcal{F}\left(T\right)$ with u ≠ v. Then, by (3.11), we have

$$2d(u,v)=2d(Tu,Tv)\le \phi (d(u,v))((d(u,Tv)+d(v,Tu))<2d(u,v),$$

a contradiction. Hence, $\mathcal{F}\left(T\right)$ is a singleton set.   □

Applying Theorem 3.6, we obtain the following primitive Chatterjea's fixed point theorem [3].

Corollary 3.1. [3] Let T: X → X be a selfmap. Suppose that there exists $\gamma \in \left[0,\frac{1}{2}\right)$ such that

$$d\left(Tx,Ty\right)\le \gamma \left(d\left(x,Ty\right)+d\left(y,Tx\right)\right)\mathsf{\text{for all }}x,y\in X.$$

(3.13)

Then, T has a unique fixed point in X.

Proof. Define φ: [0, ∞) → [0,1) by φ(t) = 2γ. Then, φ is an $\mathcal{M}\mathcal{T}$-function. So (3.13) implies (3.11), and the conclusion is immediate from Theorem 3.6.   □

Corollary 3.2. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exist $\alpha ,\beta \in \left[0,\frac{1}{2}\right)$ such that

$${\mathcal{D}}_p\left(Tx,Ty\right)\le \alpha p\left(x,Ty\right)+\beta p\left(y,Tx\right)\mathsf{\text{ for all }}x,y\in X.$$

(3.14)

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Proof. Let φ, τ: [0, ∞) → [0,1) be defined by φ(t) = 2α and τ(t) = 2β for all t ∈ [0, ∞). Then, φ and τ are $\mathcal{M}\mathcal{T}$-functions, and the conclusion follows from Theorem 3.4.   □

The following conclusion is immediate from Corollary 3.2 with α = β = γ.

Corollary 3.3. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exists $\gamma \in \left[0,\frac{1}{2}\right)$ such that

$${\mathcal{D}}_p\left(Tx,Ty\right)\le \gamma \left(p\left(x,Ty\right)+p\left(y,Tx\right)\right)\mathsf{\text{for all }}x,y\in X.$$

(3.15)

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Remark 3.1.

• (a) Corollary 3.2 and Corollary 3.3 are equivalent. Indeed, it suffices to prove that Corollary 3.2 implies Corollary 3.3. Suppose all assumptions of Corollary 3.2 are satisfied. Let γ:= max {α, β}. Then $\gamma \in \left[0,\frac{1}{2}\right)$ and (3.14) implies (3.15), and the conclusion of Corollary 3.3 follows from Corollary 3.2.

• (b) Theorems 3.1-3.4 and Corollaries 3.1 and 3.2 all generalize and improve [5, Theorem 3.4] and the primitive Chatterjea's fixed point theorem [3].

4. Fixed point theorems of generalized Kannan's type and others

The following result is given essentially in [5, Theorem 2.1].

Theorem 4.1. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exists an $\mathcal{M}\mathcal{T}$-function φ: [0, ∞) → [0,1) such that for each x ∈ X,

$$p\left(y,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,y\right)\mathsf{\text{ for all }}y\in Tx.$$

(4.1)

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Applying Theorem 4.1, we establish the following new fixed point theorem.

Theorem 4.2. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exist two $\mathcal{M}\mathcal{T}$-functions φ, τ: [0, ∞) → [0,1) such that for each x ∈ X,

$$2{\mathcal{D}}_p\left(Tx,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,Tx\right)+\tau \left(p\left(x,y\right)\right)p\left(y,Ty\right)\mathsf{\text{ for all }}y\in Tx,$$

(4.2)

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Proof. Notice that for each x ∈ X, if y ∈ Tx, then (4.2) implies

$$2p\left(y,Ty\right)\le 2{\mathcal{D}}_p\left(Tx,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,Tx\right)+\tau \left(p\left(x,y\right)\right)p\left(y,Ty\right)$$

and hence

$$p\left(y,Ty\right)\le \frac{\phi \left(p\left(x,y\right)\right)}{2-\tau \left(p\left(x,y\right)\right)}p\left(x,Tx\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,y\right).$$

Applying Theorem 4.1, we can get the thesis.   □

The following conclusion is immediate from Theorem 4.2.

Theorem 4.3. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exist two $\mathcal{M}\mathcal{T}$-functions φ, τ: [0, ∞) → [0,1) such that

$$2{\mathcal{D}}_p\left(Tx,Ty\right)\le \phi \left(p\left(x,y\right)\right)p\left(x,Tx\right)+\tau \left(p\left(x,y\right)\right)p\left(y,Ty\right)\mathsf{\text{for all }}x,y\in X.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Theorem 4.4. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exists an $\mathcal{M}\mathcal{T}$-function φ: [0, ∞) → [0,1) such that for each x ∈ X,

$$2{\mathcal{D}}_p\left(Tx,Ty\right)\le \phi \left(p\left(x,y\right)\right)\left(p\left(x,Tx\right)+p\left(y,Ty\right)\right)\mathsf{\text{for all }}y\in Tx.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Theorem 4.5. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exists an $\mathcal{M}\mathcal{T}$-function φ: [0, ∞) → [0,1) such that

$$2{\mathcal{D}}_p\left(Tx,Ty\right)\le \phi \left(p\left(x,y\right)\right)\left(p\left(x,Tx\right)+p\left(y,Ty\right)\right)\mathsf{\text{for all }}x,y\in X.$$

(4.3)

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Theorem 4.6. Let T: X → X be a selfmap. Suppose that there exists an $\mathcal{M}\mathcal{T}$-function φ: [0, ∞) → [0,1) such that

$$2d\left(Tx,Ty\right)\le \phi \left(d\left(x,y\right)\right)\left(d\left(x,Tx\right)+d\left(y,Ty\right)\right)\mathsf{\text{for all }}x,y\in X.$$

(4.4)

Then, T has a unique fixed point in X.

Proof. Let p ≡ d. Then, (4.3) and (4.4) are identical. We prove that T is a capable map. In fact, it suffices to show that (D5) holds. Assume that there exists w ∈ X with w ≠ Tw and inf {d(x, w) + d(x,Tx): x ∈ X} = 0. Then, there exists a sequence {x _n } in X such that lim_n→∞ (d(x _n , w) + d(x _n ,Tx _n )) = 0. It follows that d(x _n ,w) → 0 and d(x _n ,Tx _n ) → 0 and hence d(w,Tx _n ) → 0 or Tx _n → w as n → ∞. By hypothesis, we have

$$2d(T{x}_n,Tw)\le \phi (d(x_n,w))((d(x_n,T{x}_n)+d(w,Tw))$$

(4.5)

for all n ∈ ℕ. Since d(x _n ,w) → 0 as n → ∞ and φ is an $\mathcal{M}\mathcal{T}$-function, lim_{n →∞}φ(d(x _n ,w)) < 1. Letting n → ∞ in (4.5), since Tx _n → w and d(x _n ,Tx _n ) → 0 as n → ∞, we have 2d(w,Tw) <d(w, Tw), which is a contradiction. So (D5) holds and hence T is a capable map. Applying Theorem 4.5, $\mathcal{F}\left(T\right)\ne \varnothing$. Suppose that there exists $u,v\in \mathcal{F}\left(T\right)$ with u ≠ v. Then, by (4.4), we have

$$0<2d(u,v)=2d(Tu,Tv)\le \phi (d(u,v))((d(u,Tu)+d(v,Tv))=0,$$

a contradiction. Hence, $\mathcal{F}\left(T\right)$ is a singleton set.   □

Applying Theorem 4.6, we obtain the primitive Kannan's fixed point theorem [2].

Corollary 4.1. Let T: X → X be a selfmap. Suppose that there exists $\gamma \in \left[0,\frac{1}{2}\right)$ such that

$$d\left(Tx,Ty\right)\le \gamma \left(d\left(x,Tx\right)+d\left(y,Ty\right)\right)\mathsf{\text{for all }}x,y\in X.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Corollary 4.2. Let T: X → X be a selfmap. Suppose that there exist $\alpha ,\beta \in \left[0,\frac{1}{2}\right)$ such that

$$d\left(Tx,Ty\right)\le \alpha d\left(x,Tx\right)+\beta d\left(y,Ty\right)\mathsf{\text{ for all }}x,y\in X.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Remark 4.1. Corollary 4.1 and Corollary 4.2 are indeed equivalent.

Corollary 4.3. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exist $\alpha ,\beta \in \left[0,\frac{1}{2}\right)$ such that

$${\mathcal{D}}_p\left(Tx,Ty\right)\le \alpha p\left(x,Tx\right)+\beta p\left(y,Ty\right)\mathsf{\text{for all }}x,y\in X.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Corollary 4.4. Let $T:X\to \mathcal{C}\mathcal{B}\left(X\right)$ be a capable map. Suppose that there exists $\gamma \in \left[0,\frac{1}{2}\right)$ such that

$${\mathcal{D}}_p\left(Tx,Ty\right)\le \gamma \left(p\left(x,Tx\right)+p\left(y,Ty\right)\right)\mathsf{\text{for all }}x,y\in X.$$

Then $\mathcal{F}\left(T\right)\ne \varnothing$.

Remark 4.2.

• (a) Corollary 4.3 and Corollary 4.4 are indeed equivalent.

• (b) Theorems 4.1-4.6 and Corollaries 4.1-4.4 all generalize and improve [5, Theorem 2.6] and the primitive Kannan's fixed point theorem [2].