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MizoguchiTakahashi’s type common fixed point theorems without Tweakly commuting condition and invariant approximations
Fixed Point Theory and Applicationsvolume 2014, Article number: 112 (2014)
Abstract
The purpose of this paper is to introduce a concept of ${T}_{f}$orbitally lower semicontinuous mappings which is more general than the concept of Torbitally lower semicontinuous mappings and continuous mappings and also prove MizoguchiTakahashi’s type coincidence point theorems by using this concept. Moreover, we show that the existence of common fixed points for MizoguchiTakahashi’s type multivalued mappings do not require the condition of Tweakly commuting mappings. Finally, some invariant approximation results are obtained as applications. Our results unify, extend, and complement several wellknown 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 PompeiuHausdorff 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 PompeiuHausdorff 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 ${x}_{n}\in T{x}_{n1}$ 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 $f{x}_{n}\in T{x}_{n1}$ 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 forbit of T.
In 1969, Nadler [1] extended the Banach contraction principle to multivalued 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 multivalued 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 Rfunction, 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 MTfunction, 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 MTfunction. Therefore, the class of MTfunctions 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 Tinvariant, 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 MizoguchiTakahashi’s fixed point theorem for multivalued mappings is a real generalization of Nadler’s contraction principle. Since the primitive proof of MizoguchiTakahashi’s fixed point theorem is quite difficult, Suzuki gave a very simple proof of MizoguchiTakahashi’s theorem. Several authors devoted their attention to investigate its generalizations in various different directions of the MizoguchiTakahashi’s fixed point theorem (see [6–14] and references therein).
In 2009, Kamran [15] extended the result of Mizoguchi and Takahashi [3] for closed multivalued mappings and proved a fixed point theorem by using the concept of Torbitally lower semicontinuous mappings as follows:
Definition 1.3 ([16])
Let $(X,d)$ be a metric space, $T:X\to CL(X)$ be a mapping multivalued, and let $\xi \in X$.

(1)
A mapping $g:X\to \mathbb{R}$ is said to be lower semicontinuous 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 Torbitally lower semicontinuous 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 MTfunction. 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 Torbitally lower semicontinuous at ξ.
Recently, Ali [17] extended the above result to common fixed point theorem by using the concept of Tweakly 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 Tweakly 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 MTfunction. If $(f(X),d)$ is a complete metric space, then
(A1) For any ${x}_{0}\in X$, there exists an forbit $\{f{x}_{n}\}$ 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 semicontinuous at ξ.
(A3) If $ff\xi =f\xi $ and f is Tweakly commuting at ξ, then f and T have a common fixed point.
In this paper, we introduce the concept of ${T}_{f}$orbitally lower semicontinuous mappings and, using this concept, prove MizoguchiTakahashi’s type coincidence point theorems. Also, we show that the condition of ‘Tweakly commuting of f’ can be omit to prove MizoguchiTakahashi’s type common fixed point theorems. By the same procedure, we can improve Theorem 1.5 by dropping the condition of ‘f is Tweakly commuting at ξ’ in (A3). As applications, we derive the invariant approximation results.
2 MizoguchiTakahashi’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 semicontinuous 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 semicontinuous 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 MizoguchiTakahashi’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 onetoone.
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 MTfunction. If $(f(X),d)$ is a complete metric space, then
(S1) For each ${x}_{0}\in X$, there exist an forbit $\{f{x}_{n}\}$ 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 semicontinuous at fξ, where $S\subseteq X$ and $f{}_{S}$ is onetoone.
(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 onetoone. Now, we can define a mapping $G:f(E)\to CL(X)$ by
for all $x\in E$. Since $f{}_{E}$ is onetoone, it follows that G is well defined. Since T satisfies the contractive condition (2.1), we have
for all $x\in X$ and $fy\in Tx$. By the construction of 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 Gorbitally lower semicontinuous 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
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 MizoguchiTakahashi’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 ‘Tweakly commuting at ξ’ condition.
Consequently, Theorem 2.2 extends and improves Nadler’s contraction principle [1], MizoguchiTakahashi’s theorem [3], Theorem 2.1 of Kamran [15], Theorem 2.2 of Ali [17], and several results in the literature. Moreover, for the singlevalued case, Theorem 2.2 also unifies Banach’s contraction principle [20] and many wellknown 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 MTfunction. If $(f(X),d)$ is a complete metric space, then
(S1) For each ${x}_{0}\in X$, there exist an forbit $\{f{x}_{n}\}$ 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 semicontinuous at fξ, where $S\subseteq X$ and $f{}_{S}$ is onetoone.
(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 MTfunction. If $(f(X),d)$ is a complete metric space, then:
(S1) For each ${x}_{0}\in X$, there exist an forbit $\{f{x}_{n}\}$ 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 semicontinuous at fξ, where $S\subseteq X$ and $f{}_{S}$ is onetoone.
(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:
(S1) For each ${x}_{0}\in X$, there exist an forbit $\{f{x}_{n}\}$ 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 semicontinuous at fξ, where $S\subseteq X$ and $f{}_{S}$ is onetoone.
(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
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 forbit $\{f{x}_{n}\}$ 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 semicontinuous at fξ, where $S\subseteq X$ and $f{}_{S}$ is onetoone;
(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
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 forbit $\{f{x}_{n}\}$ 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 semicontinuous at fξ, where $S\subseteq X$ and $f{}_{S}$ is onetoone.
(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 selfmappings 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 multivalued mappings.
In this section, we study invariant approximation results for nonlinear singlevalued mapping and multivalued 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
is called the set of best Mapproximants to $p\in X$ out of M, where $d(p,M)={inf}_{y\in M}\parallel yp\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 multivalued mappings such that
for each $x\in {B}_{M}(p)$ and $fy\in Tx$, where $\alpha :[0,\mathrm{\infty})\to [0,1)$ is an MTfunction. 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 onetoone.

(4)
${sup}_{y\in Tx}\parallel yp\parallel \le \parallel fxp\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 forbit $\{f{x}_{n}\}$ 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 semicontinuous 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 singlevalued mapping from ${Best}_{M}(p)$ to ${Best}_{M}(p)$. Now, we show that $T{}_{{Best}_{M}(p)}$ is a multivalued 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 $\parallel fxp\parallel =d(p,M)$.
Now, we obtain
This implies that $\parallel zp\parallel =d(p,M)$ 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 multivalued 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 multivalued mapping such that
for all $x\in {Best}_{M}(p)$ and $y\in Tx$, where $\alpha :[0,\mathrm{\infty})\to [0,1)$ is an MTfunction. 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 yp\parallel \le \parallel xp\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 Torbitally lower semicontinuous at ξ.
Proof Take f as the identity mapping from M into M in Theorem 3.1, we get the result. □
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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 (KRF2013053358).
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Keywords
 MizoguchiTakahashi’s type common fixed point theorem
 Tweakly commuting mapping
 ℛfunction
 $\mathcal{MT}$function
 invariant approximation