Generalization of Mizoguchi-Takahashi type contraction and related fixed point theorems

Abstract

In this paper, we introduce a new notion to generalize a Mizoguchi-Takahashi type contraction. Then, using this notion, we obtain a fixed point theorem for multivalued maps. Our results generalize some results by Minak and Altun, Kamran and those contained therein.

MSC:47H10, 54H25.

1 Introduction and preliminaries

The notions of α-ψ-contractive and α-admissible mappings were introduced by Samet et al. [1]. They proved some fixed point results for such mappings in complete metric spaces. These notions were generalized by Karapınar and Samet [2]. Asl et al. [3] extended these notions to multifunctions and introduced the notions of ${\alpha }_{\ast }$-ψ-contractive and ${\alpha }_{\ast }$-admissible mappings. Afterwards Ali and Kamran [4] further generalized the notion of ${\alpha }_{\ast }$-ψ-contractive mappings and obtained some fixed point theorems for multivalued mappings. Some interesting extensions of results by Samet et al. [1] are available in [5–13]. Nadler initiated a fixed point theorem for multivalued mappings. Some extensions of Nadler’s result can also be found in [14–31]. Mizoguchi and Takahashi [32] extended the Nadler fixed point theorem. Recently, Minak and Altun generalized Mizoguchi and Takahashi’s theorem by introducing a function $\alpha :X\times X\to [0,\mathrm{\infty })$. In this paper, we introduce the notion of ${\alpha }_{\ast }$-Mizoguchi-Takahashi type contraction. By using this notion, we generalize some fixed point theorems presented by Minak and Altun [7], Kamran [26] and those contained therein.

We denote by $\mathit{CL}(X)$ the class of all nonempty closed subsets of X and by $\mathit{CB}(X)$ the class of all nonempty closed and bounded subsets of X. For $A\in \mathit{CL}(X)$ or $\mathit{CB}(X)$ and $x\in X$, $d(x,A)=inf\{d(x,a):a\in A\}$, and H is a generalized Hausdorff metric induced by d. Now we recollect some basic definitions and results for the sake of completeness.

If, for $x_0\in X$, there exists a sequence $\{x_n\}$ in X such that $x_n\in T{x}_{n-1}$, then $O(T,x_0)=\{x_0,x_1,x_2,\dots \}$ is said to be an orbit of $T:X\to \mathit{CL}(X)$ at $x_0$. A mapping $h:X\to \mathbb{R}$ is said to be T-orbitally lower semicontinuous at $\xi \in X$, if $\{x_n\}$ is a sequence in $O(T,x_0)$ and $x_n\to \xi$ implies $h(\xi )\le lim\hspace{0.17em}infh(x_n)$. The following definition is due to Asl et al. [3].

Definition 1.1 [3]

Let $(X,d)$ be a metric space, $\alpha :X\times X\to [0,\mathrm{\infty })$ and $T:X\to \mathit{CL}(X)$. Then T is ${\alpha }_{\ast }$-admissible if for each $x,y\in X$ with $\alpha (x,y)\ge 1\Rightarrow {\alpha }_{\ast }(Tx,Ty)\ge 1$, where ${\alpha }_{\ast }(Tx,Ty)=inf\{\alpha (a,b):a\in Tx,b\in Ty\}$.

Minak and Altun [7] generalized Mizoguchi and Takahashi’s theorem in the following way.

Theorem 1.2 [7]

Let $(X,d)$ be a complete metric space, $T:X\to \mathit{CB}(X)$ be a mapping satisfying

$${\alpha }_{\ast }(Tx,Ty)H(Tx,Ty)\le \varphi (d(x,y))d(x,y)\mathit{\text{for each }}x,y\in X,$$

where $\varphi :[0,\mathrm{\infty })\to [0,1)$ such that ${lim\hspace{0.17em}sup}_{r\to t^{+}}\varphi (r)<1$ for each $t\in [0,\mathrm{\infty })$. Also assume that

• (i) T is ${\alpha }_{\ast }$-admissible;

• (ii) there exists $x_0\in X$ with $\alpha (x_0,x_1)\ge 1$ for some $x_1\in T{x}_0$;

• (iii)

  • (a) T is continuous,

  or

  • (b) if $\{x_n\}$ is a sequence in X with $x_n\to x$ as $n\to \mathrm{\infty }$ and $\alpha (x_n,x_{n+1})\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$, then we have $\alpha (x_n,x)\ge 1$ for each $n\in \mathbb{N}\cup \{0\}$.

Then T has a fixed point.

Kamran in [26] generalized Mizoguchi and Takahashi’s theorem in the following way.

Theorem 1.3 [26]

Let $(X,d)$ be a complete metric space and $T:X\to \mathit{CL}(X)$ be a mapping satisfying

$$d(y,Ty)\le \varphi (d(x,y))d(x,y)\mathit{\text{for each }}x\in X\mathit{\text{ and }}y\in Tx,$$

where $\varphi :[0,\mathrm{\infty })\to [0,1)$ such that ${lim\hspace{0.17em}sup}_{r\to t^{+}}\varphi (r)<1$ for each $t\in [0,\mathrm{\infty })$. Then,

• (i) for each $x_0\in X$, there exists an orbit $\{x_n\}$ of T and $\xi \in X$ such that ${lim}_n{x}_n=\xi$;

• (ii) ξ is a fixed point of T if and only if the function $h(x):=d(x,Tx)$ is T-orbitally lower semicontinuous at ξ.

2 Main results

We begin this section with the following definition.

Definition 2.1 Let $(X,d)$ be a metric space, $T:X\to \mathit{CL}(X)$ is said to be an ${\alpha }_{\ast }$-Mizoguchi-Takahashi type contraction if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty })$ and $\varphi :[0,\mathrm{\infty })\to [0,1)$ satisfying ${lim\hspace{0.17em}sup}_{r\to t^{+}}\varphi (r)<1$ for every $t\in [0,\mathrm{\infty })$ such that

$${\alpha }_{\ast }(Tx,Ty)d(y,Ty)\le \varphi (d(x,y))d(x,y)\text{for each }x\in X\text{ and }y\in Tx.$$

(2.1)

Before moving toward our main results, we prove some lemmas.

Lemma 2.2 Let $(X,d)$ be a metric space, $\{A_k\}$ be a sequence in $\mathit{CL}(X)$, $\{x_k\}$ be a sequence in X such that $x_k\in A_{k-1}$. Let $\varphi :[0,\mathrm{\infty })\to [0,1)$ be a function satisfying ${lim\hspace{0.17em}sup}_{r\to t^{+}}\varphi (r)<1$ for every $t\in [0,\mathrm{\infty })$. Suppose that $\{d(x_{k-1},x_k)\}$ is a nonincreasing sequence such that

$$d(x_k,A_k)\le \varphi (d(x_{k-1},x_k))d(x_{k-1},x_k),$$

(2.2)

$$d(x_k,x_{k+1})\le d(x_k,A_k)+{\varphi }^{n_k}(d(x_{k-1},x_k)),$$

(2.3)

where $n_1<n_2<\cdots$ , $k,n_k\in \mathbb{N}$. Then $\{x_k\}$ is a Cauchy sequence in X.

Proof The proof runs on the same lines as the proof of [[18], Lemma 3.2]. We include its details for completeness. Let $d_k:=d(x_{k-1},x_k)$. Since $d_k$ is a nonincreasing sequence of nonnegative real numbers, therefore ${lim}_{k\to \mathrm{\infty }}{d}_k=c\ge 0$. By hypothesis, for $t=c$, we get ${lim\hspace{0.17em}sup}_{t\to c^{+}}\varphi (t)<1$. Therefore, there exists $k_0$ such that $k\ge k_0$ implies that $\varphi (d_k)<h$, where ${lim\hspace{0.17em}sup}_{t\to c^{+}}\varphi (t)\le h<1$. From (2.2) and (2.3), we have

$$\begin{array}{rcl}{d}_{k+1}& \le & \varphi (d_k)d_k+{\varphi }^{n_k}(d_k)\\ \le & \varphi (d_k)\varphi (d_{k-1})d_{k-1}+\varphi (d_k){\varphi }^{n_{k-1}}(d_{k-1})+{\varphi }^{n_k}(d_k)\\ \cdots \\ \le & \prod _{i=1}^k\varphi (d_i)d_1+\sum _{m=1}^{k-1}\prod _{i=m+1}^k\varphi (d_i){\varphi }^{n_m}(d_m)+{\varphi }^{n_k}(d_k)\\ \le & \prod _{i=1}^k\varphi (d_i)d_1+\sum _{m=1}^{k-1}\prod _{i=max\{k_0,m+1\}}^k\varphi (d_i){\varphi }^{n_m}(d_m)+{\varphi }^{n_k}(d_k).\end{array}$$

(2.4)

We have deleted some factors of ϕ from the product in (2.4) using the fact that $\varphi <1$. Let S denote the second term on the right-hand side of (2.4),

$$\begin{array}{rcl}S& \le & (k_0-1)h^{k-k_0+1}\sum _{m=1}^{k_0-1}{\varphi }^{n_m}(d_m)+\sum _{m=k_0}^{k-1}{h}^{k-m}{\varphi }^{n_m}(d_m)\\ \le & (k_0-1)h^{k-k_0+1}\sum _{m=1}^{k_0-1}{\varphi }^{n_m}(d_m)+\sum _{m=k_0}^{k-1}{h}^{k-m+n_m}\\ \le & C{h}^k+\sum _{m=k_0}^{k-1}{h}^{k-m+n_m}\\ \le & C{h}^k+h^{k+n_{k_0}-k_0}+h^{k+n_{k_0-1}-(k_0-1)}+\cdots +h^{k+n_{k-1}-(k-1)}\\ \le & C{h}^k+\sum _{m=k+n_{k_0}-k_0}^{k+n_{k-1}-(k-1)}{h}^m\\ =& C{h}^k+\frac{h^{k+n_{k_0}-k_0+1}-h^{k+n_{k-1}-k+2}}{1-h}\\ <& C{h}^k+h^k\frac{h^{n_{k_0}-k_0+1}}{1-h}\\ =& C{h}^k,\end{array}$$

where C is a generic positive constant. Now, it follows from (2.4) that

$$\begin{array}{rcl}{d}_{k+1}& \le & \prod _{i=1}^k\varphi (d_i)d_1+C{h}^k+{\varphi }^{n_k}(d_k)\\ <& h^{k-k_0+1}\prod _{i=1}^{k_0-1}\varphi (d_i)d_1+C{h}^k+h^{n_k}\\ <& C{h}^k+C{h}^k+k\\ =& C{h}^k,\end{array}$$

C again being a generic constant. Now, for $k\ge k_0$, $m\in \mathbb{N}$,

$$\begin{array}{rcl}d(x_k,x_{k+m})& \le & \sum _{i=k+1}^{k+m}{d}_i\\ <& \sum _{i=k+1}^{k+m}C{h}^{i-1}\\ =& C\frac{h^{k+1}-h^{k+m}}{1-h}\\ \le & h^k,\end{array}$$

which shows that $\{x_k\}$ is a Cauchy sequence in X. □

Lemma 2.3 Let $(X,d)$ be a metric space, $T:X\to \mathit{CL}(X)$ be an ${\alpha }_{\ast }$-Mizoguchi-Takahashi type contraction. Let $\{x_k\}$ be an orbit of T at $x_0$ such that ${\alpha }_{\ast }(T{x}_{k-1},T{x}_k)\ge 1$ and

$$d(x_k,x_{k+1})\le d(x_k,T{x}_k)+{\varphi }^{n_k}(d(x_{k-1},x_k)),$$

(2.5)

where $x_k\in T{x}_{k-1}$, $n_1<n_2<\cdots$ and $k,n_k\in \mathbb{N}$ and $\{d(x_{k-1},x_k)\}$ is a nonincreasing sequence. Then $\{x_k\}$ is a Cauchy sequence in X.

Proof Given that $\{x_k\}$ is an orbit of T at $x_0$, i.e., $x_k\in T{x}_{k-1}$ for each $k\in \mathbb{N}$, with ${\alpha }_{\ast }(T{x}_{k-1},T{x}_k)\ge 1$ for each $k\in \mathbb{N}$, as T is an ${\alpha }_{\ast }$-Mizoguchi-Takahashi type contraction. From (2.1), we have

$$\begin{array}{rcl}d(x_k,T{x}_k)& \le & {\alpha }_{\ast }(T{x}_{k-1},T{x}_k)d(x_k,T{x}_k)\\ \le & \varphi (d(x_{k-1},x_k))d(x_{k-1},x_k).\end{array}$$

From (2.5), we have

$$d(x_k,x_{k+1})\le d(x_k,T{x}_k)+{\varphi }^{n_k}(d(x_{k-1},x_k)).$$

Since all the conditions of Lemma 2.2 are satisfied, $\{x_k\}$ is a Cauchy sequence in X. □

Theorem 2.4 Let $(X,d)$ be a complete metric space, $T:X\to \mathit{CL}(X)$ be an ${\alpha }_{\ast }$-Mizoguchi-Takahashi type contraction and ${\alpha }_{\ast }$-admissible. Suppose that there exist $x_0\in X$ and $x_1\in T{x}_0$ such that $\alpha (x_0,x_1)\ge 1$. Then,

• (i) there exists an orbit $\{x_n\}$ of T and $x^{\ast }\in X$ such that $lim{x}_n=x^{\ast }$;

• (ii) $x^{\ast }$ is a fixed point of T if and only if $h(x)=d(x,Tx)$ is T-orbitally lower semicontinuous at $x^{\ast }$.

Proof By hypothesis, we have $x_0\in X$ and $x_1\in T{x}_0$ with $\alpha (x_0,x_1)\ge 1$. Thus, for $x_1\in T{x}_0$, we can choose a positive integer $n_1$ such that

$${\varphi }^{n_1}(d(x_0,x_1))\le [1-\varphi (d(x_0,x_1))]d(x_0,x_1).$$

(2.6)

There exists $x_2\in T{x}_1$ such that

$$d(x_1,x_2)\le d(x_1,T{x}_1)+{\varphi }^{n_1}(d(x_0,x_1)).$$

(2.7)

As T is ${\alpha }_{\ast }$-admissible, we have ${\alpha }_{\ast }(T{x}_0,T{x}_1)\ge 1$. From (2.6) and (2.7) it follows that

$$\begin{array}{rcl}d(x_1,x_2)& \le & d(x_1,T{x}_1)+{\varphi }^{n_1}(d(x_0,x_1))\\ \le & {\alpha }_{\ast }(T{x}_0,T{x}_1)d(x_1,T{x}_1)+{\varphi }^{n_1}(d(x_0,x_1))\\ \le & \varphi (d(x_0,x_1))d(x_0,x_1)+[1-\varphi (d(x_0,x_1))]d(x_0,x_1)\\ =& d(x_0,x_1).\end{array}$$

Now we can choose a positive integer $n_2>n_1$ such that

$${\varphi }^{n_2}(d(x_1,x_2))\le [1-\varphi (d(x_1,x_2))]d(x_1,x_2).$$

(2.8)

There exists $x_3\in T{x}_2$ such that

$$d(x_2,x_3)\le d(x_2,T{x}_2)+{\varphi }^{n_2}(d(x_1,x_2)).$$

(2.9)

As T is ${\alpha }_{\ast }$-admissible, then $\alpha (x_1,x_2)\ge {\alpha }_{\ast }(T{x}_0,T{x}_1)\ge 1$ implies ${\alpha }_{\ast }(T{x}_1,T{x}_2)\ge 1$. Using (2.8) and (2.9) we have that

$$\begin{array}{rcl}d(x_2,x_3)& \le & d(x_2,T{x}_2)+{\varphi }^{n_2}(d(x_1,x_2))\\ \le & {\alpha }_{\ast }(T{x}_1,T{x}_2)d(x_2,T{x}_2)+{\varphi }^{n_2}(d(x_1,x_2))\\ \le & \varphi (d(x_1,x_2))d(x_1,x_2)+[1-\varphi (d(x_1,x_2))]d(x_1,x_2)\\ =& d(x_1,x_2).\end{array}$$

By repeating this process for all $k\in \mathbb{N}$, we can choose a positive integer $n_k$ such that

$${\varphi }^{n_k}(d(x_{k-1},x_k))\le [1-\varphi (d(x_{k-1},x_k))]d(x_{k-1},x_k).$$

(2.10)

There exists $x_k\in T{x}_{k-1}$ such that

$$d(x_k,x_{k+1})\le d(x_k,T{x}_k)+{\varphi }^{n_k}(d(x_{k-1},x_k)).$$

(2.11)

Also, by ${\alpha }_{\ast }$-admissibility of T, we have ${\alpha }_{\ast }(T{x}_{k-1},T{x}_k)\ge 1$ for each $k\in \mathbb{N}$. From (2.10) and (2.11) it follows that

$$\begin{array}{rcl}d(x_k,x_{k+1})& \le & d(x_k,T{x}_k)+{\varphi }^{n_k}(d(x_{k-1},x_k))\\ \le & {\alpha }_{\ast }(T{x}_{k-1},T{x}_k)d(x_k,T{x}_k)+{\varphi }^{n_k}(d(x_{k-1},x_k))\\ \le & \varphi (d(x_{k-1},x_k))d(x_{k-1},x_k)+[1-\varphi (d(x_{k-1},x_k))]d(x_{k-1},x_k)\\ =& d(x_{k-1},x_k),\end{array}$$

which implies that $\{d(x_k,x_{k+1})\}$ is a nonincreasing sequence of nonnegative real numbers. Thus, by Lemma 2.3, $\{x_k\}$ is a Cauchy sequence in X. Since X is complete, there exists $x^{\ast }\in X$ such that $x_k\to x^{\ast }$ as $k\to \mathrm{\infty }$. Since $x_k\in T{x}_{k-1}$, it follows from (2.1) that

$$\begin{array}{rcl}d(x_k,T{x}_k)& \le & {\alpha }_{\ast }(T{x}_{k-1},T{x}_k)d(x_k,T{x}_k)\\ \le & \varphi (d(x_{k-1},x_k))d(x_{k-1},x_k)\\ <& d(x_{k-1},x_k).\end{array}$$

Letting $k\to \mathrm{\infty }$, in the above inequality, we have

$$\underset{k\to \mathrm{\infty }}{lim}d(x_k,T{x}_k)=0.$$

(2.12)

Suppose that $h(x)=d(x,Tx)$ is T-orbitally lower semicontinuous at $x^{\ast }$, then

$$d(x^{\ast },T{x}^{\ast })=h\left(x^{\ast }\right)\le \underset{k}{lim\hspace{0.17em}inf}h(x_k)=\underset{k}{lim\hspace{0.17em}inf}d(x_k,T{x}_k)=0.$$

By the closedness of T it follows that $x^{\ast }\in T{x}^{\ast }$. Conversely, suppose that $x^{\ast }$ is a fixed point of T, then $h(x^{\ast })=0\le {lim\hspace{0.17em}inf}_{k}h(x_k)$. □

Example 2.5 Let $X=\{\frac{1}{n}:n\in \mathbb{N}\}\cup \{0\}\cup (1,\mathrm{\infty })$ be endowed with the usual metric d. Define $T:X\to \mathit{CL}(X)$ by

$$Tx=\{\begin{array}{cc}\{0\}\hfill & \text{if }x=0,\hfill \\ \{\frac{1}{n+2},\frac{1}{n+3}\}\hfill & \text{if }x=\frac{1}{n}:1\le n\le 6,\hfill \\ \{\frac{1}{n},0\}\hfill & \text{if }x=\frac{1}{n}:n>6,\hfill \\ [2x,\mathrm{\infty })\hfill & \text{if }x>1,\hfill \end{array}$$

and $\alpha :X\times X\to [0,\mathrm{\infty })$ by

$$\alpha (x,y)=\{\begin{array}{cc}1\hfill & \text{if }x,y\in \{\frac{1}{n}:n\in \mathbb{N}\}\cup \{0\},\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$$

Define $\varphi :[0,\mathrm{\infty })\to [0,1)$ by

$$\varphi (t)=\{\begin{array}{cc}\frac{4}{5}\hfill & \text{if }0\le t\le \frac{1}{6},\hfill \\ \frac{1}{2}\hfill & \text{if }t>\frac{1}{6}.\hfill \end{array}$$

One can check that for each $x\in X$ and $y\in Tx$, we have

$${\alpha }_{\ast }(Tx,Ty)d(y,Ty)\le \varphi (d(x,y))d(x,y).$$

Also, T is ${\alpha }_{\ast }$-admissible and for $x_0=1$ we have $x_1=\frac{1}{3}\in T{x}_0$ with $\alpha (x_0,x_1)=1$. Moreover, all the other conditions of Theorem 2.4 are satisfied. Therefore T has a fixed point. Note that Theorem 5 of Minak and Altun [7] is not applicable here; see, for example, $x=\frac{1}{7}$ and $y=\frac{1}{8}$. Further Theorem 2.1 of Kamran [26] is also not applicable; see, for example, $x=2$ and $y=4\in Tx$.

The proofs of the following theorems run on the same lines as the proof of Theorem 2.4.

Theorem 2.6 Let $(X,d)$ be a complete metric space, $T:X\to \mathit{CL}(X)$ be an ${\alpha }_{\ast }$-admissible mapping such that

$${\alpha }_{\ast }(y,Ty)d(y,Ty)\le \varphi (d(x,y))d(x,y)\mathit{\text{for each }}x\in X\mathit{\text{ and }}y\in Tx,$$

(2.13)

where $\varphi :[0,\mathrm{\infty })\to [0,1)$ satisfying ${lim\hspace{0.17em}sup}_{r\to t^{+}}\varphi (r)<1$ for every $t\in [0,\mathrm{\infty })$. Suppose that there exist $x_0\in X$ and $x_1\in T{x}_0$ such that $\alpha (x_0,x_1)\ge 1$. Then,

• (i) there exists an orbit $\{x_n\}$ of T and $x^{\ast }\in X$ such that $lim{x}_n=x^{\ast }$;

• (ii) $x^{\ast }$ is a fixed point of T if and only if $h(x)=d(x,Tx)$ is T-orbitally lower semicontinuous at $x^{\ast }$.

Theorem 2.7 Let $(X,d)$ be a complete metric space, $T:X\to \mathit{CL}(X)$ be an ${\alpha }_{\ast }$-admissible mapping such that

$$\alpha (x,y)d(y,Ty)\le \varphi (d(x,y))d(x,y)\mathit{\text{for each }}x\in X\mathit{\text{ and }}y\in Tx,$$

(2.14)

where $\varphi :[0,\mathrm{\infty })\to [0,1)$ satisfying ${lim\hspace{0.17em}sup}_{r\to t^{+}}\varphi (r)<1$ for every $t\in [0,\mathrm{\infty })$. Suppose that there exist $x_0\in X$ and $x_1\in T{x}_0$ such that $\alpha (x_0,x_1)\ge 1$. Then,

• (i) there exists an orbit $\{x_n\}$ of T and $x^{\ast }\in X$ such that $lim{x}_n=x^{\ast }$;

• (ii) $x^{\ast }$ is a fixed point of T if and only if $h(x)=d(x,Tx)$ is T-orbitally lower semicontinuous at $x^{\ast }$.

Corollary 2.8 [26]

Let $(X,d)$ be a complete metric space and $T:X\to \mathit{CL}(X)$ be a mapping satisfying

$$d(y,Ty)\le \varphi (d(x,y))d(x,y)\mathit{\text{for each }}x\in X\mathit{\text{ and }}y\in Tx,$$

where $\varphi :[0,\mathrm{\infty })\to [0,1)$ such that ${lim\hspace{0.17em}sup}_{r\to t^{+}}\varphi (r)<1$ for each $t\in [0,\mathrm{\infty })$. Then,

• (i) for each $x_0\in X$, there exists an orbit $\{x_n\}$ of T and $\xi \in X$ such that ${lim}_n{x}_n=\xi$;

• (ii) ξ is a fixed point of T if and only if the function $h(x):=d(x,Tx)$ is T-orbitally lower semicontinuous at ξ.

Proof Define $\alpha :X\times X\to [0,\mathrm{\infty })$ by $\alpha (x,y)=1$ for each $x,y\in X$. Then the proof follows from Theorem 2.4 as well as from Theorem 2.6, and from Theorem 2.7. □

3 Application

From Definition 2.1, we get the following definition by considering only those $x\in X$ and $y\in Tx$ for which we have ${\alpha }_{\ast }(Tx,Ty)\ge 1$.

Definition 3.1 Let $(X,d)$ be a metric space, $T:X\to \mathit{CL}(X)$ is said to be a modified ${\alpha }_{\ast }$-Mizoguchi-Takahashi type contraction if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty })$ and $\varphi :[0,\mathrm{\infty })\to [0,1)$ satisfying ${lim\hspace{0.17em}sup}_{r\to t^{+}}\varphi (r)<1$ for every $t\in [0,\mathrm{\infty })$ such that for each $x\in X$ and $y\in Tx$,

$${\alpha }_{\ast }(Tx,Ty)\ge 1\Rightarrow d(y,Ty)\le \varphi (d(x,y))d(x,y).$$

(3.1)

Lemma 3.2 Let $(X,d)$ be a metric space, $T:X\to \mathit{CL}(X)$ be a modified ${\alpha }_{\ast }$-Mizoguchi-Takahashi contraction. Let $\{x_k\}$ be an orbit of T at $x_0$ such that ${\alpha }_{\ast }(T{x}_{k-1},T{x}_k)\ge 1$ and

$$d(x_k,x_{k+1})\le d(x_k,T{x}_k)+{\varphi }^{n_k}(d(x_{k-1},x_k)),$$

(3.2)

where $x_k\in T{x}_{k-1}$, $n_1<n_2<\cdots$ and $k,n_k\in \mathbb{N}$ and $\{d(x_{k-1},x_k)\}$ is a nonincreasing sequence. Then $\{x_k\}$ is a Cauchy sequence in X.

Proof Given that $\{x_k\}$ is an orbit of T at $x_0$, i.e., $x_k\in T{x}_{k-1}$ for each $k\in \mathbb{N}$, with ${\alpha }_{\ast }(T{x}_{k-1},T{x}_k)\ge 1$ for each $k\in \mathbb{N}$, as T is a modified ${\alpha }_{\ast }$-Mizoguchi-Takahashi contraction. From (3.1), we have

$$d(x_k,T{x}_k)\le \varphi (d(x_{k-1},x_k))d(x_{k-1},x_k).$$

From (3.2), we have

$$d(x_k,x_{k+1})\le d(x_k,T{x}_k)+{\varphi }^{n_k}(d(x_{k-1},x_k)).$$

Since all the conditions of Lemma 2.2 are satisfied, $\{x_k\}$ is a Cauchy sequence in X. □

Working on the same lines as the proof of Theorem 2.4 is done, one may obtain the proof of the following result.

Theorem 3.3 Let $(X,d)$ be a complete metric space, $T:X\to \mathit{CL}(X)$ be a modified ${\alpha }_{\ast }$-Mizoguchi-Takahashi contraction and ${\alpha }_{\ast }$-admissible. Suppose that there exist $x_0\in X$ and $x_1\in T{x}_0$ such that $\alpha (x_0,x_1)\ge 1$. Then,

• (i) there exists an orbit $\{x_n\}$ of T and $x^{\ast }\in X$ such that $lim{x}_n=x^{\ast }$;

• (ii) $x^{\ast }$ is a fixed point of T if and only if $h(x)=d(x,Tx)$ is T-orbitally lower semicontinuous at $x^{\ast }$.