- Research
- Open Access
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
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
- Mizoguchi-Takahashi’s type common fixed point theorem
- T-weakly commuting mapping
- ℛ-function
- $\mathcal{MT}$-function
- invariant 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.
for all $A,B\in CB(X)$, where $d(a,B)=inf\{d(a,b):b\in B\}$ is the distance from a to $B\subseteq X$.
- (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.
- (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 $F(f,T)$.
- (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])
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])
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])
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:
- (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])
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])
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.
- (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.
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.
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.’
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. □
- (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.
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. □
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.
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.
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.
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.
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.
- (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 $\parallel fx-p\parallel =d(p,M)$.
Thus the result follows from Theorem 2.2 with $X={Best}_{M}(p)$. This completes the proof. □
- (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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