Mizoguchi-Takahashi’s type common fixed point theorems without T-weakly commuting condition and invariant approximations

Abstract

The purpose of this paper is to introduce a concept of $T_f$-orbitally lower semi-continuous mappings which is more general than the concept of T-orbitally lower semi-continuous mappings and continuous mappings and also prove Mizoguchi-Takahashi’s type coincidence point theorems by using this concept. Moreover, we show that the existence of common fixed points for Mizoguchi-Takahashi’s type multi-valued mappings do not require the condition of T-weakly commuting mappings. Finally, some invariant approximation results are obtained as applications. Our results unify, extend, and complement several well-known results.

MSC:47H10, 54H25.

1 Introduction and preliminaries

Throughout this paper, we denote by ℕ the set of all positive integers, by ℝ the set of all real numbers, and by ${\mathbb{R}}_{+}$ the set of all nonnegative real numbers.

Let $(X,d)$ be a metric space. We denote by $2^X$ the class of all nonempty subsets of X, by $K(X)$ the class of all nonempty compact subsets of X, by $CL(X)$ the class of all nonempty closed subsets of X, by $CB(X)$ the class of all nonempty closed bounded subsets of X. A functional $H:CL(X)\times CL(X)\to {\mathbb{R}}_{+}\cup \{+\mathrm{\infty }\}$ is said to be the Pompeiu-Hausdorff generalized metric induced by d is given by

$$H(A,B)=\{\begin{array}{ll}max\{{sup}_{a\in A}d(a,B),{sup}_{b\in B}d(b,A)\},& \text{if the maximum exists};\\ +\mathrm{\infty },& \text{otherwise},\end{array}$$

for all $A,B\in CB(X)$, where $d(a,B)=inf\{d(a,b):b\in B\}$ is the distance from a to $B\subseteq X$.

Remark 1.1 The following properties of the Pompeiu-Hausdorff generalized metric induced by d are well known:

• (1) H is a metric on $CB(X)$.

• (2) If $A,B\in CB(X)$ and $q>1$, then, for all $a\in A$, there exists $b\in B$ such that $d(a,b)\le qH(A,B)$.

• (3) $(CB(X),H)$ is a complete metric space provided $(X,d)$ is a complete metric space.

Definition 1.1 Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to 2^X$ be mappings.

• (1) A point $x\in X$ is said to be a fixed point of f (resp., T) if $x=fx$ (resp., $x\in Tx$). The set of all fixed points of f (resp., T) is denoted by $F(f)$ (resp., $F(T)$).

• (2) A point $x\in X$ is said to be a coincidence point of f and T if $fx\in Tx$. The set of all coincidence points of f and T is denoted by $C(f,T)$.

• (3) A point $x\in X$ is said to be a common fixed point of f and T if $x=fx\in Tx$. The set of all common fixed points of f and T is denoted by $F(f,T)$.

Definition 1.2 Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to 2^X$ be mappings.

• (1) If, for any $x_0\in X$, there exists a sequence $\{x_n\}$ in X such that $x_n\in T{x}_{n-1}$ for all $n\in \mathbb{N}$, then $O(T,x_0):=\{x_0,x_1,x_2,\dots \}$ is said to be an orbit of T.

• (2) If, for any $x_0\in X$, there exists a sequence $\{f{x}_n\}$ in $f(X)$ such that $f{x}_n\in T{x}_{n-1}$ for all $n\in \mathbb{N}$, then $O_f(T,x_0):=\{f{x}_0,f{x}_1,f{x}_2,\dots \}$ is said to be an f-orbit of T.

In 1969, Nadler [1] extended the Banach contraction principle to multi-valued mappings as follows.

Theorem 1.1 (Nadler [1])

Let $(X,d)$ be a complete metric space and $T:X\to CB(X)$ such that

$$H(Tx,Ty)\le kd(x,y)$$

(1.1)

for all $x,y\in X$, where $k\in [0,1)$. Then T has at least one fixed point.

Since the theory of multi-valued mappings has many applications in many areas, a number of authors have focused on the topic and have published some interesting fixed point theorems in this frame. Following this trend, in 1972, Reich [2] extended Theorem 1.1 in the following way.

Theorem 1.2 (Reich [2])

Let $(X,d)$ be a complete metric space and $T:X\to K(X)$ be a mapping satisfying

$$H(Tx,Ty)\le \alpha (d(x,y))d(x,y)$$

(1.2)

for all $x,y\in X$, where $\alpha :(0,\mathrm{\infty })\to [0,1)$ is R-function, that is,

$$\underset{x\to t^{+}}{lim\hspace{0.17em}sup}\alpha (x)<1$$

for all $t\in (0,\mathrm{\infty })$. Then T has at least one fixed point.

Furthermore, Reich [2] also raised the following question in his work:

Can the range of T, that is, $K(X)$ , be replaced by $CB(X)$ or $CL(X)$ ?

In 1989, Mizoguchi and Takahashi [3] gave the positive answer for the conjecture of Reich [2], when the inequality holds also for $t=0$, as follows.

Theorem 1.3 (Mizoguchi and Takahashi [3])

Let $(X,d)$ be a complete metric space and $T:X\to CB(X)$. Assume that

$$H(Tx,Ty)\le \alpha (d(x,y))d(x,y)$$

(1.3)

for all $x,y\in X$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is MT-function, that is,

$$\underset{x\to t^{+}}{lim\hspace{0.17em}sup}\alpha (x)<1$$

for all $t\in [0,\mathrm{\infty })$. Then T has at least one fixed point.

Remark 1.2 It is well known that, if $\alpha :[0,\mathrm{\infty })\to [0,1)$ is a nondecreasing function or a nonincreasing function, then α is a MT-function. Therefore, the class of MT-functions is a rich class and so this class has been investigated heavily by many authors.

In 2007, Eldred et al. [4] claimed that Theorem 1.3 is equivalent to Theorem 1.1 in the following sense:

If a mapping $T:X\to CB(X)$ satisfies (1.3), then there exists a nonempty complete subset M of X satisfying the following:

• (1) M is T-invariant, that is, $Tx\subseteq M$ for all $x\in M$.

• (2) T satisfies (1.1) for all $x,y\in M$.

In the same year, Suzuki [5] produced an example which shows that Mizoguchi-Takahashi’s fixed point theorem for multi-valued mappings is a real generalization of Nadler’s contraction principle. Since the primitive proof of Mizoguchi-Takahashi’s fixed point theorem is quite difficult, Suzuki gave a very simple proof of Mizoguchi-Takahashi’s theorem. Several authors devoted their attention to investigate its generalizations in various different directions of the Mizoguchi-Takahashi’s fixed point theorem (see [6–14] and references therein).

In 2009, Kamran [15] extended the result of Mizoguchi and Takahashi [3] for closed multi-valued mappings and proved a fixed point theorem by using the concept of T-orbitally lower semi-continuous mappings as follows:

Definition 1.3 ([16])

Let $(X,d)$ be a metric space, $T:X\to CL(X)$ be a mapping multi-valued, and let $\xi \in X$.

• (1) A mapping $g:X\to \mathbb{R}$ is said to be lower semi-continuous at ξ if, for any sequence $\{x_n\}$ in X such that $x_n\to \xi$ as $n\to \mathrm{\infty }$,

  $$g\xi \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}g{x}_n.$$

• (2) A mapping $g:X\to \mathbb{R}$ is said to be T-orbitally lower semi-continuous at ξ if, for any sequence $\{x_n\}$ in $O(T,x_0)$ such that $x_0\in X$ and $x_n\to \xi$ as $n\to \mathrm{\infty }$,

  $$g\xi \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}g{x}_n.$$

The following result is a main result of Kamran [15].

Theorem 1.4 (Kamran [15])

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

$$d(y,Ty)\le \alpha (d(x,y))d(x,y)$$

(1.4)

for all $x\in X$ and $y\in Tx$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is MT-function. Then:

(K1) For each $x_0\in X$, there exist an orbit $\{x_n\}$ of T and $\xi \in X$ such that ${lim}_{n\to \mathrm{\infty }}{x}_n=\xi$.

(K2) ξ is a fixed point of T if and only if the function $g:X\to \mathbb{R}$ defined by $g(x):=d(x,Tx)$ for all $x\in X$ is T-orbitally lower semi-continuous at ξ.

Recently, Ali [17] extended the above result to common fixed point theorem by using the concept of T-weakly commuting as follows:

Definition 1.4 ([18])

Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to CL(X)$ be mappings. The mapping f is said to be T-weakly commuting at $x\in X$ if $ffx\in Tfx$.

The following result is a main result of Ali [17].

Theorem 1.5 (Ali [17])

Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to CL(X)$ be two mappings such that $T(X)\subseteq f(X)$ and

$$d(fy,Ty)\le \alpha (d(fx,fy))d(fx,fy)$$

(1.5)

for all $x\in X$ and $fy\in Tx$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is MT-function. If $(f(X),d)$ is a complete metric space, then

(A1) For any $x_0\in X$, there exists an f-orbit $\{f{x}_n\}$ of T and $f\xi \in f(X)$ such that ${lim}_{n\to \mathrm{\infty }}f{x}_n=f\xi$.

(A2) ξ is a coincidence point of f and T if and only if the function $h:X\to \mathbb{R}$ defined by $h(x):=d(fx,Tx)$ for all $x\in X$ is lower semi-continuous at ξ.

(A3) If $ff\xi =f\xi$ and f is T-weakly commuting at ξ, then f and T have a common fixed point.

In this paper, we introduce the concept of $T_f$-orbitally lower semi-continuous mappings and, using this concept, prove Mizoguchi-Takahashi’s type coincidence point theorems. Also, we show that the condition of ‘T-weakly commuting of f’ can be omit to prove Mizoguchi-Takahashi’s type common fixed point theorems. By the same procedure, we can improve Theorem 1.5 by dropping the condition of ‘f is T-weakly commuting at ξ’ in (A3). As applications, we derive the invariant approximation results.

2 Mizoguchi-Takahashi’s type coincidence and common fixed point theorems

In this section, we start with the following concept.

Definition 2.1 Let $(X,d)$ be a metric space, $f:X\to X$, $T:X\to CL(X)$ be mappings, and let $x_0,\xi \in X$.

• (1) A mapping $h:f(X)\to \mathbb{R}$ is said to be lower semi-continuous at fξ if, for any sequence $\{f{x}_n\}$ in $f(X)$ such that $f{x}_n\to f\xi$ as $n\to \mathrm{\infty }$,

  $$h(f\xi )\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}h(f{x}_n).$$

• (2) A mapping $h:f(X)\to \mathbb{R}$ is said to be $T_f$ -orbitally lower semi-continuous at fξ if, for any sequence $\{f{x}_n\}$ in $O_f(T,x_0)$ such that $f{x}_n\to f\xi$ as $n\to \mathrm{\infty }$,

  $$h(f\xi )\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}h(f{x}_n).$$

Next, we apply the following useful lemma due to Haghi et al. [19] and Theorem 1.4 to obtain new Mizoguchi-Takahashi’s type common fixed point theorem.

Lemma 2.1 ([19])

Let X be a nonempty set and $f:X\to X$ be a mapping. Then there exists a subset E of X such that $f(E)=f(X)$ and $f{|}_E:E\to X$ is one-to-one.

The following result is a main result in this paper.

Theorem 2.2 Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to CL(X)$ be two mappings such that $T(X)\subseteq f(X)$ and

$$d(fy,Ty)\le \alpha (d(fx,fy))d(fx,fy)$$

(2.1)

for all $x\in X$ and $fy\in Tx$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is MT-function. If $(f(X),d)$ is a complete metric space, then

(S1) For each $x_0\in X$, there exist an f-orbit $\{f{x}_n\}$ of T and $f\xi \in f(X)$ such that ${lim}_{n\to \mathrm{\infty }}f{x}_n=f\xi$.

(S2) ξ is a coincidence point of f and T if and only if the function $h:f(S)\to \mathbb{R}$ defined by $h(fx):=d(fx,Tx)$ for all $fx\in f(S)$ is $T_f$-orbitally lower semi-continuous at fξ, where $S\subseteq X$ and $f{|}_S$ is one-to-one.

(S3) If ξ is a coincidence point of f and T such that $ff\xi =f\xi$, then f and T have a common fixed point.

Proof Let $f:X\to X$ be a mapping. Using Lemma 2.1, there exists $E\subseteq X$ such that $f(E)=f(X)$ and $f{|}_E$ is one-to-one. Now, we can define a mapping $G:f(E)\to CL(X)$ by

$$G(fx)=Tx$$

(2.2)

for all $x\in E$. Since $f{|}_E$ is one-to-one, it follows that G is well defined. Since T satisfies the contractive condition (2.1), we have

$$d(fy,Ty)\le \alpha (d(fx,fy))d(fx,fy)$$

(2.3)

for all $x\in X$ and $fy\in Tx$. By the construction of G, we get

$$d(fy,G(fy))\le \alpha (d(fx,fy))d(fx,fy)$$

(2.4)

for all $fx\in f(E)$ and $fy\in G(fx)$. This implies that G is satisfies the contractive condition (1.4). From (S1), it follows that, for each $x_0\in X$, there exist an orbit $\{f{x}_n\}$ of G and $f\xi \in f(E)$ such that ${lim}_{n\to \mathrm{\infty }}f{x}_n=f\xi$. This implies that (K1) in Theorem 1.4 holds.

Again, by the construction of G, it follows that (S2) is equivalent to the following condition:

‘fξ is a fixed point of G, that is, $f\xi \in G(f\xi )$ if and only if the function $g:f(E)\to \mathbb{R}$ defined by $g(fx)=d(fx,G(fx))$ for all $fx\in f(E)$ is G-orbitally lower semi-continuous at fξ.’

Thus (S2) holds. Let ξ is a coincidence point of f and T, that is, $f\xi \in T\xi$. Next, we suppose that $ff\xi =f\xi$. Let $z:=f\xi$ and so $z=f\xi =ff\xi =fz\in T\xi$. Since $fz\in T\xi$, it follows from the contractive condition (2.1) that

$$\begin{array}{rcl}d(fz,Tz)& \le & \alpha (d(f\xi ,fz))d(f\xi ,fz)\\ =& \alpha (0)\cdot 0\\ =& 0,\end{array}$$

which shows that $fz\in Tz$. Therefore, $z=fz\in Tz$, that is, z is a common fixed point of f and T. This completes the proof. □

Remark 2.1 Theorem 2.2 generalizes many results in the following sense:

• (1) The inequality (2.1) is weaker than some kinds of the contractive conditions such as Mizoguchi-Takahashi’s contractive condition [3], Nadler’s contractive condition [1], Kamran’s contractive condition [15], etc.

• (2) The range of T is $CL(X)$ which is more general than $CB(X)$.

• (3) For the existence of coincidence point, we merely require that the condition in (S2), whereas other result demands stronger than this condition.

• (4) For the existence of common fixed point, we only requires the condition $ff\xi =f\xi$, whereas Theorem 1.5 requires both of this condition and the ‘T-weakly commuting at ξ’ condition.

Consequently, Theorem 2.2 extends and improves Nadler’s contraction principle [1], Mizoguchi-Takahashi’s theorem [3], Theorem 2.1 of Kamran [15], Theorem 2.2 of Ali [17], and several results in the literature. Moreover, for the single-valued case, Theorem 2.2 also unifies Banach’s contraction principle [20] and many well-known results.

Corollary 2.3 Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to CL(X)$ be two mappings such that $T(X)\subseteq f(X)$ and

$$H(Tx,Ty)\le \alpha (d(fx,fy))d(fx,fy)$$

(2.5)

for all $x\in X$ and $fy\in Tx$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is an MT-function. If $(f(X),d)$ is a complete metric space, then

(S1) For each $x_0\in X$, there exist an f-orbit $\{f{x}_n\}$ of T and $f\xi \in f(X)$ such that ${lim}_{n\to \mathrm{\infty }}f{x}_n=f\xi$.

(S2) ξ is a coincidence point of f and T if and only if the function $h:f(S)\to \mathbb{R}$ defined by $h(fx):=d(fx,Tx)$ for all $fx\in f(S)$ is $T_f$-orbitally lower semi-continuous at fξ, where $S\subseteq X$ and $f{|}_S$ is one-to-one.

(S3) If ξ is a coincidence point of f and T such that $ff\xi =f\xi$, then f and T have a common fixed point.

Proof Since $d(fy,Ty)\le H(Tx,Ty)$ for all $fy\in Tx$, it follows from the contractive condition (2.5) that the inequality (2.1) holds. Therefore, we get the result. □

Corollary 2.4 Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to CL(X)$ be two mappings such that $T(X)\subseteq f(X)$ and

$$H(Tx,Ty)\le \alpha (d(fx,fy))d(fx,fy)$$

(2.6)

for all $x,y\in X$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is an MT-function. If $(f(X),d)$ is a complete metric space, then:

(S1) For each $x_0\in X$, there exist an f-orbit $\{f{x}_n\}$ of T and $f\xi \in f(X)$ such that ${lim}_{n\to \mathrm{\infty }}f{x}_n=f\xi$.

(S2) ξ is a coincidence point of f and T if and only if the function $h:f(S)\to \mathbb{R}$ defined by $h(fx):=d(fx,Tx)$ for all $fx\in f(S)$ is $T_f$-orbitally lower semi-continuous at fξ, where $S\subseteq X$ and $f{|}_S$ is one-to-one.

(S3) If ξ is a coincidence point of f and T such that $ff\xi =f\xi$, then f and T have a common fixed point.

Proof Since the condition (2.6) implies the condition (2.5), we get the result. □

If we take $\alpha (t)=k$ for all $t\in [0,\mathrm{\infty })$, where k is constant number with $k\in [0,1)$, then we get the following result.

Corollary 2.5 Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to CL(X)$ be two mappings such that $T(X)\subseteq f(X)$ and satisfying

$$d(fy,Ty)\le kd(fx,fy),$$

(2.7)

for each $x\in X$ and $fy\in Tx$, where $k\in [0,1)$. If $(f(X),d)$ is a complete metric space, then:

(S1) For each $x_0\in X$, there exist an f-orbit $\{f{x}_n\}$ of T and $f\xi \in f(X)$ such that ${lim}_{n\to \mathrm{\infty }}f{x}_n=f\xi$.

(S2) ξ is a coincidence point of f and T if and only if the function $h:f(S)\to \mathbb{R}$ defined by $h(fx):=d(fx,Tx)$ for all $fx\in f(S)$ is $T_f$-orbitally lower semi-continuous at fξ, where $S\subseteq X$ and $f{|}_S$ is one-to-one.

(S3) If ξ is a coincidence point of f and T such that $ff\xi =f\xi$, then f and T have a common fixed point.

Corollary 2.6 Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to CL(X)$ be two mappings such that $T(X)\subseteq f(X)$ and

$$H(Tx,Ty)\le kd(fx,fy)$$

(2.8)

for all $x\in X$ and $fy\in Tx$, where $k\in [0,1)$. If $(f(X),d)$ is a complete metric space, then:

(S1) For each $x_0\in X$, there exist an f-orbit $\{f{x}_n\}$ of T and $f\xi \in f(X)$ such that ${lim}_{n\to \mathrm{\infty }}f{x}_n=f\xi$;

(S2) ξ is a coincidence point of f and T if and only if the function $h:f(S)\to \mathbb{R}$ defined by $h(fx):=d(fx,Tx)$ for all $fx\in f(S)$ is $T_f$-orbitally lower semi-continuous at fξ, where $S\subseteq X$ and $f{|}_S$ is one-to-one;

(S3) If ξ is a coincidence point of f and T such that $ff\xi =f\xi$, then f and T have a common fixed point.

Corollary 2.7 Let $(X,d)$ be a metric space, $f:X\to X$ and $T:X\to CL(X)$ be two mappings such that $T(X)\subseteq f(X)$ and

$$H(Tx,Ty)\le kd(fx,fy)$$

(2.9)

for all $x,y\in X$, where $k\in [0,1)$. If $(f(X),d)$ is a complete metric space, then:

(S1) For each $x_0\in X$, there exist an f-orbit $\{f{x}_n\}$ of T and $f\xi \in f(X)$ such that ${lim}_{n\to \mathrm{\infty }}f{x}_n=f\xi$.

(S2) ξ is a coincidence point of f and T if and only if the function $h:f(S)\to \mathbb{R}$ defined by $h(fx):=d(fx,Tx)$ for all $fx\in f(S)$ is $T_f$-orbitally lower semi-continuous at fξ, where $S\subseteq X$ and $f{|}_S$ is one-to-one.

(S3) If ξ is a coincidence point of f and T such that $ff\xi =f\xi$, then f and T have a common fixed point.

3 Invariant approximation results

Several problems concerning invariant approximations for self-mappings were obtained as applications of fixed point, coincidence point, and common fixed point results (see [21–27] and references therein). Also, Kamran [18], Latif and Bano [28], and O’Regan and Shahzad [29, 30] obtained invariant approximation results for multi-valued mappings.

In this section, we study invariant approximation results for nonlinear single-valued mapping and multi-valued mapping as applications of main results in Section 2.

Let M be a subset of a normed space E and $p\in E$. The set

$${Best}_M(p):=\{x\in M:\parallel x-p\parallel =d(p,M)\}$$

is called the set of best M-approximants to $p\in X$ out of M, where $d(p,M)={inf}_{y\in M}\parallel y-p\parallel$.

Here, we derive some invariant approximation results.

Theorem 3.1 Let M be subset of normed space $(E,\parallel \cdot \parallel )$, $p\in E$, $f:M\to M$ be a mapping and $T:M\to CL(M)$ be a multi-valued mappings such that

$$d(fy,Ty)\le \alpha (\parallel fx-fy\parallel )\parallel fx-fy\parallel$$

(3.1)

for each $x\in B_M(p)$ and $fy\in Tx$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is an MT-function. Suppose that the following conditions hold:

• (1) $T({Best}_M(p))\subseteq f({Best}_M(p))={Best}_M(p)$.

• (2) $f({Best}_M(p))$ is a complete subspace of M.

• (3) $f{|}_{{Best}_M(p)}$ is one-to-one.

• (4) ${sup}_{y\in Tx}\parallel y-p\parallel \le \parallel fx-p\parallel$ for all $x\in {Best}_M(p)$.

Then we have the following:

(S1) For each $x_0\in {Best}_M(p)$, there exists an f-orbit $\{f{x}_n\}$ of T and $f\xi \in f({Best}_M(p))$ such that ${lim}_{n\to \mathrm{\infty }}f{x}_n=f\xi$.

(S2) $\xi \in C(f,T)\cap {Best}_M(p)$ if and only if the function $h:f({Best}_M(p))\to \mathbb{R}$ defined by $h(fx):=d(fx,Tx)$ for all $fx\in f({Best}_M(p))$ is $T_f$-orbitally lower semi-continuous at fξ.

(S3) If $\xi \in C(f,T)\cap {Best}_M(p)$ such that $ff\xi =f\xi$, then $f\xi \in F(f,T)\cap {Best}_M(p)$.

Proof From the assumption, it follows that $f{|}_{{Best}_M(p)}$ is a single-valued mapping from ${Best}_M(p)$ to ${Best}_M(p)$. Now, we show that $T{|}_{{Best}_M(p)}$ is a multi-valued mapping from ${Best}_M(p)$ to $CL({Best}_M(p))$. First, we claim that $Tx\subseteq {Best}_M(p)$ for all $x\in {Best}_M(p)$. Let $x\in {Best}_M(p)$ and $z\in Tx$. Since $f({Best}_M(p))={Best}_M(p)$, we have $fx\in {Best}_M(p)$ and hence $\parallel fx-p\parallel =d(p,M)$.

Now, we obtain

$$d(p,M)\le \parallel z-p\parallel \le \underset{y\in Tx}{sup}\parallel y-p\parallel \le \parallel fx-p\parallel =d(p,M).$$

This implies that $\parallel z-p\parallel =d(p,M)$ and thus $z\in {Best}_M(p)$. Therefore, $Tx\subseteq {Best}_M(p)$ for all $x\in {Best}_M(p)$. Since Tx is closed for all $x\in M$, it follows that Tx is closed for all $x\in {Best}_M(p)$. Hence $T{|}_{{Best}_M(p)}$ is a multi-valued mapping from ${Best}_M(p)$ to $CL({Best}_M(p))$. It is easy to obtain that

$$F(f{|}_{{Best}_M(p)},T{|}_{{Best}_M(p)})=F(f,T)\cap {Best}_M(p).$$

Thus the result follows from Theorem 2.2 with $X={Best}_M(p)$. This completes the proof. □

Theorem 3.2 Let M be subset of normed space $(E,\parallel \cdot \parallel )$, $p\in E$, and $T:M\to CL(M)$ be a multi-valued mapping such that

$$d(y,Ty)\le \alpha (\parallel x-y\parallel )\parallel x-y\parallel$$

(3.2)

for all $x\in {Best}_M(p)$ and $y\in Tx$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is an MT-function. Suppose that the following conditions hold:

• (1) $T({Best}_M(p))\subseteq {Best}_M(p)$;

• (2) ${Best}_M(p)$ is complete subspace of M;

• (3) ${sup}_{y\in Tx}\parallel y-p\parallel \le \parallel x-p\parallel$ for all $x\in {Best}_M(p)$.

Then we have the following:

(S1) For each $x_0\in {Best}_M(p)$, there exists an orbit $\{x_n\}$ of T and $\xi \in {Best}_M(p)$ such that ${lim}_{n\to \mathrm{\infty }}{x}_n=\xi$;

(S2) $\xi \in F(T)\cap {Best}_M(p)$ if and only if the function $g:{Best}_M(p)\to \mathbb{R}$, defined by $g(x):=d(x,Tx)$ for all $x\in f({Best}_M(p))$, is T-orbitally lower semi-continuous at ξ.

Proof Take f as the identity mapping from M into M in Theorem 3.1, we get the result. □