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

- Wutiphol Sintunavarat
^{1}Email author, - Dong Min Lee
^{2}and - Yeol Je Cho
^{3, 4}Email author

**2014**:112

https://doi.org/10.1186/1687-1812-2014-112

© Sintunavarat et al.; licensee Springer. 2014

**Received: **9 February 2014

**Accepted: **23 April 2014

**Published: **8 May 2014

## 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.

## Keywords

*T*-weakly commuting mappingℛ-function $\mathcal{MT}$-functioninvariant approximation

## 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.

*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

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*

*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*

*for all*$x,y\in X$,

*where*$\alpha :(0,\mathrm{\infty})\to [0,1)$

*is*

*R*-

*function*,

*that is*,

*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*

*for all*$x,y\in X$,

*where*$\alpha :[0,\mathrm{\infty})\to [0,1)$

*is*

*MT*-

*function*,

*that is*,

*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:

*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])

- (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*

*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*

*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*

*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*is well defined. Since

*T*satisfies the contractive condition (2.1), we have

*G*, we get

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*.’

*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

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*

*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*

*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*

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

*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 $.

*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*.

*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*

*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*;

*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*

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

*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 $.

*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*.

*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.

*M*be a subset of a normed space

*E*and $p\in E$. The set

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*

*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 $\parallel fx-p\parallel =d(p,M)$.

*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

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*

*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. □

## Declarations

### Acknowledgements

The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (KRF-2013053358).

## Authors’ Affiliations

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