 Research
 Open Access
 Published:
Common fixedpoint results in uniformly convex Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 171 (2012)
Abstract
We introduce a condition on mappings, namely condition $(K)$. In a uniformly convex Banach space, the condition is weaker than quasinonexpansiveness and weaker than asymptotic nonexpansiveness. We also present the existence theorem of common fixed points for a commuting pair consisting of a mapping satisfying condition $(K)$ and a multivalued mapping satisfying conditions $(E)$ and $({C}_{\lambda})$ for some $\lambda \in (0,1)$.
1 Introduction
In 1978, Itoh and Takahashi [1] established the existence of common fixed points of a quasinonexpansive mapping and a multivalued nonexpansive mapping by an elementary constructive method in a Hilbert space. In 2006, Dhompongsa et al. [2] obtained a common fixed point result for a commuting pair of singlevalued and multivalued nonexpansive mappings in uniformly convex Banach spaces. The analogy result in CAT(0) spaces was also proved by Dhompongsa et al. [3]. Since then, Shahzad and Markin [4] studied an invariant approximation problem and provided sufficient conditions for the existence of $z\in E\subseteq X$ such that $d(z,y)=dist(y,E)$ and $z=t(z)\in T(z)$, where $y\in X$, t and T are commuting nonexpansive mappings on E. In 2009, Shahzad [5] also obtained a common fixed point and invariant approximation result in a CAT(0) space in which t and T are weakly commuting.
Motivated by Suzuki’s result [6], GarciaFalset et al. [7] introduced two kinds of generalizations for condition $(C)$, namely conditions $(E)$ and $({C}_{\lambda})$ and studied both the existence of fixed points and their asymptotic behavior. Recently, Abkar and Eslamian [8] proved that if E is a nonempty closed convex and bounded subset of a complete CAT(0) space X, $t:E\to E$ is a singlevalued quasinonexpansive mapping, and $T:E\to KC(E)$ is a multivalued mapping satisfying conditions $(E)$ and $({C}_{\lambda})$ for some $\lambda \in (0,1)$ such that t and T are weakly commuting, then there exists a point $z\in E$ such that $z=t(z)\in T(z)$. This result was extended to the general setting of uniformly convex metric spaces by Laowang and Panyanak [9].
In this paper, we first introduce the following condition.
Definition 1.1 Let t be a mapping on a subset E of a Banach space X. Then t is said to satisfy condition $(K)$ if

1.
the fixed point set $Fix(t)$ is nonempty closed and convex, and

2.
for every $x\in Fix(t)$, a closed convex subset A with $t(A)\subseteq A$, and $y\in A$ such that $\parallel xy\parallel =dist(x,A)$, we have $y\in Fix(t)$.
We show that, in a uniformly convex Banach space, condition $(K)$ is weaker than quasinonexpansiveness and weaker than asymptotic nonexpansiveness. We also present the existence theorem of common fixed points for a commuting pair consisting of a mapping satisfying condition $(K)$ and a multivalued mapping satisfying conditions $(E)$ and $({C}_{\lambda})$ for some $\lambda \in (0,1)$. Consequently, such a theorem extends many results in the literature.
2 Preliminaries
In this section, we give some preliminaries.
A mapping t on a subset E of a Banach space X is called an asymptotically nonexpansive mapping if for each $n\in \mathbb{N}$, there exists a positive constant ${k}_{n}\ge 1$ with ${lim}_{n\to \mathrm{\infty}}{k}_{n}=1$ such that
If ${k}_{n}\equiv 1$ for all $n\in \mathbb{N}$, then t is called a nonexpansive mapping. We denote by $Fix(t)$ the set of fixed points of t, i.e., $Fix(t)=\{x\in E:x=t(x)\}$.
We shall denote by $FB(E)$ the family of nonempty bounded closed subsets of E and by $KC(E)$ the family of nonempty compact convex subsets of E. Let $H(\cdot ,\cdot )$ be the Hausdorff distance on $FB(X)$, i.e.,
where $dist(a,B)=inf\{\parallel ab\parallel :b\in B\}$ is the distance from the point a to the subset B.
A multivalued mapping $T:E\to FB(X)$ is said to be nonexpansive if
Definition 2.1 A multivalued mapping $T:X\to FB(X)$ is said to satisfy condition $({E}_{\mu})$ provided that
We say that T satisfies condition $(E)$ whenever T satisfies $({E}_{\mu})$ for some $\mu \ge 1$.
Definition 2.2 A multivalued mapping $T:X\to FB(X)$ is said to satisfy condition $({C}_{\lambda})$ for some $\lambda \in (0,1)$ provided that
A point x is called a fixed point for a multivalued mapping T if $x\in T(x)$. A single valued mapping $t:E\to E$ and a multivalued mapping $T:E\to FB(E)$ are said to be commute if $t(T(x))\subseteq T(t(x))$ for all $x\in E$.
A Banach space X is said to be strictly convex if
for all $x,y\in X$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. We recall that a Banach space X is called uniformly convex (Clarkson [10]) if, for each $\epsilon >0$, there is a $\delta >0$ such that if $\parallel x\parallel =\parallel y\parallel =1$ then
It is obvious that uniform convexity implies strict convexity.
In 1991, Xu [11] proved the characterization of uniform convexity as follows.
Theorem 2.3 [11]
A Banach space X is uniformly convex if and only if for each fixed number $r>0$, there exists a continuous function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$, $\phi (s)=0\iff s=0$, such that
for all $\lambda \in [0,1]$ and all $x,y\in X$ such that $\parallel x\parallel \le r$ and $\parallel y\parallel \le r$.
Let E be a nonempty closed and convex subset of a Banach space X and $\{{x}_{n}\}$ be a bounded sequence in X. For $x\in X$, define the asymptotic radius of $\{{x}_{n}\}$ at x as the number
Let
and
The number r and the set A are, respectively, called the asymptotic radius and asymptotic center of $\{{x}_{n}\}$ relative to E. It is known that $A(E,\{{x}_{n}\})$ is as nonempty, weakly compact and convex as E is [12]. The sequence $\{{x}_{n}\}$ is called regular relative to E if $r(E,\{{x}_{n}\})=r(E,\{{x}_{{n}_{k}}\})$ for each subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$.
Goebel [13] and Lim [14] proved the following lemma.
Lemma 2.4 Let $\{{x}_{n}\}$ be a bounded sequence in X, and let E be a nonempty closed convex subset of X. Then $\{{x}_{n}\}$ has a subsequence which is regular relative to E.
The following result was proved by Goebel and Kirk [15].
Lemma 2.5 Let $\{{z}_{n}\}$ and $\{{w}_{n}\}$ be bounded sequences in a Banach space X, and let $0<\lambda <1$. If, for every natural number n, we have ${z}_{n+1}=\lambda {w}_{n}+(1\lambda ){z}_{n}$ and $\parallel {w}_{n+1}{w}_{n}\parallel \le \parallel {z}_{n+1}{z}_{n}\parallel $, then ${lim}_{n\to \mathrm{\infty}}\parallel {w}_{n}{z}_{n}\parallel =0$.
3 Main results
We recall that a mapping t on a subset E of a Banach space X is called quasinonexpansive [16] if $\parallel xt(y)\parallel \le \parallel xy\parallel $ for all $y\in E$ and $x\in Fix(t)$. From the definition, we can see that nonexpansive mappings with a fixed point are quasinonexpansive.
We first show that a quasinonexpansive mapping defined on a strictly convex Banach space X satisfies condition $(K)$.
Proposition 3.1 Let E be a strictly convex subset of a Banach space. If $t:E\to E$ is a quasinonexpansive mapping, then t satisfies condition $(K)$.
Proof It is known that $Fix(t)$ is nonempty closed and convex [[17], Theorem 1]. Let $x\in Fix(t)$ and A be a closed convex subset with $t(A)\subseteq A$. Let $y\in A$ be such that $\parallel xy\parallel =dist(x,A)$. Quasinonexpansiveness of t implies that
Since E is strictly convex and $t(A)\subseteq A$, it must be the case that $t(y)=y$. Therefore, t satisfies condition $(K)$. □
An asymptotically nonexpansive mapping defined on a uniformly convex Banach space also satisfies condition $(K)$.
Proposition 3.2 Let X be a uniformly convex Banach space and E be a nonempty subset of X. If $t:E\to E$ is an asymptotically nonexpansive mapping with $Fix(t)\ne \mathrm{\varnothing}$, then t satisfies condition $(K)$.
Proof The set $Fix(t)$ is closed and convex [[18], Theorem 2]. Let $x\in Fix(t)$, A be a closed convex subset of E with $t(A)\subseteq A$, and $y\in A$ be such that $\parallel xy\parallel =dist(x,A)$. By Theorem 2.3, there exists a continuous function ψ such that for all integers $l,m\ge 1$,
Since $\parallel xy\parallel =dist(x,A)$ and A is convex, we have
Thus,
Since t is asymptotically nonexpansive, the righthand side of the inequality tends to zero as l, m tend to infinity. Hence, $\{{t}^{i}(y)\}$ is a Cauchy sequence. Let ${lim}_{i\to \mathrm{\infty}}{t}^{i}(y)=z\in A$. We have
By letting $i\to \mathrm{\infty}$, we can conclude that $\parallel t(z)z\parallel \le 0$, that is $z\in Fix(t)$. Now, letting $l,m\to \mathrm{\infty}$ in (3.1) yields
Since $z\in A$ and $\parallel xy\parallel =dist(x,A)$, thus $y=z$. Therefore, $y\in Fix(t)$. □
The following example shows that the class of mappings satisfying condition $(K)$ is strictly wider than the class of quasinonexpansive mappings and asymptotically nonexpansive mappings.
Example 3.3 Let f be a function on $[0,1]$ defined by
Then f is neither quasinonexpansive nor asymptotically nonexpansive. However, f satisfies condition $(K)$.
We are now in a position to state our main theorem.
Theorem 3.4 Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let $t:E\to E$ be a mapping satisfying condition $(K)$, and let $T:E\to KC(E)$ be a multivalued mapping satisfying conditions $(E)$ and $({C}_{\lambda})$ for some $\lambda \in (0,1)$. If t and T are commute, then they have a common fixed point, that is, there exists a point $z\in E$ such that $z=t(z)\in T(z)$.
Proof Commutative property of t and T implies that $t(T(x))\subseteq T(x)$ for all $x\in Fix(t)$. Then we have $Fix(t)\cap T(x)\ne \mathrm{\varnothing}$ for all $x\in Fix(t)$ since t satisfies condition $(K)$.
Now, we find an approximate fixed point sequence in $Fix(t)$ for T. Take ${x}_{0}\in Fix(t)$. Since $Fix(t)\cap T({x}_{0})\ne \mathrm{\varnothing}$, we choose ${y}_{0}\in Fix(t)\cap T({x}_{0})$. Define
Since $Fix(t)$ is convex, we have ${x}_{1}\in Fix(t)$. Let ${y}_{1}\in T({x}_{1})$ such that $\parallel {y}_{0}{y}_{1}\parallel =dist({y}_{0},T({x}_{1}))$. We get ${y}_{1}\in Fix(t)$ since t satisfies condition $(K)$. Put
Again, choose ${y}_{2}\in T({x}_{2})$ such that $\parallel {y}_{1}{y}_{2}\parallel =dist({y}_{1},T({x}_{2}))$. Similarly, we get ${y}_{2}\in Fix(t)$. We have a sequence $\{{x}_{n}\}\subseteq Fix(t)$ such that
where ${y}_{n}\in T({x}_{n})\cap Fix(t)$ and $\parallel {y}_{n1}{y}_{n}\parallel =dist({y}_{n1},T({x}_{n}))$. For every natural number $n\ge 1$, we have
It follows that
Since T satisfies condition (${C}_{\lambda}$), we have
Hence, for each $n\ge 1$, we have
We now apply Lemma 2.5 to conclude that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0$. That is, as $n\to \mathrm{\infty}$,
Passing through a subsequence, if necessary, we can assume that $\{{x}_{n}\}$ is regular. Let $A(Fix(t),\{{x}_{n}\})=\{z\}$. For each $n\ge 1$, we choose ${z}_{n}\in T(z)$ such that $\parallel {x}_{n}{z}_{n}\parallel =dist({x}_{n},T(z))$. Since t satisfies condition $(K)$, we have ${z}_{n}\in Fix(t)$. The compactness of $T(z)$ implies that the sequence $\{{z}_{n}\}$ has a convergent subsequence $\{{z}_{{n}_{k}}\}$ with the limit point $w\in T(z)$. We also obtain $w\in Fix(t)$ since $Fix(t)$ is closed. By the condition $(E)$ of T, we have for some $\mu \ge 1$,
Note that
These entail
Since $\{{x}_{n}\}$ is regular, and an asymptotic center of a bounded sequence in a uniformly convex Banach space is a singleton set, these show that $z=w\in T(z)$. Hence, $z=t(z)\in T(z)$. □
As a consequence of Proposition 3.1, Proposition 3.2, and Theorem 3.4, we obtain the following corollaries.
Corollary 3.5 Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let $t:E\to E$ be a quasinonexpansive mapping, and let $T:E\to KC(E)$ be a multivalued mapping satisfying conditions $(E)$ and $({C}_{\lambda})$ for some $\lambda \in (0,1)$. If t and T are commute, then they have a common fixed point, that is, there exists a point $z\in E$ such that $z=t(z)\in T(z)$.
Corollary 3.6 Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let $t:E\to E$ be an asymptotically nonexpansive mapping, and let $T:E\to KC(E)$ be a multivalued mapping satisfying conditions $(E)$ and $({C}_{\lambda})$ for some $\lambda \in (0,1)$. If t and T are commute, then they have a common fixed point, that is, there exists a point $z\in E$ such that $z=t(z)\in T(z)$.
References
 1.
Itoh S, Takahashi W: The common fixed point theory of singlevalued mappings and multivalued mappings. Pac. J. Math. 1978, 79: 493–508. 10.2140/pjm.1978.79.493
 2.
Dhompongsa S, Kaewcharoen A, Kaewkhao A: The DominguezLorenzo condition and multivalued nonexpansive mappings. Nonlinear Anal. 2006, 64: 958–970. 10.1016/j.na.2005.05.051
 3.
Dhompongsa S, Kaewkhao A, Panyanak B: Lim’s theorems for multivalued mappings in CAT(0) spaces. J. Math. Anal. Appl. 2005, 312: 478–487. doi:10.1016/j.jmaa.2005.03.055 10.1016/j.jmaa.2005.03.055
 4.
Shahzad N, Markin J: Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces. J. Math. Anal. Appl. 2008, 337: 1457–1464. doi:10.1016/j.jmaa.2007.04.041 10.1016/j.jmaa.2007.04.041
 5.
Shahzad N: Fixed point results for multimaps in CAT(0) spaces. Topol. Appl. 2009, 156: 997–1001. doi:10.1016/j.topol.2008.11.016 10.1016/j.topol.2008.11.016
 6.
Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mapping. J. Math. Anal. Appl. 2008, 340: 1088–1095. doi:10.1016/j.jmaa.2007.09.023 10.1016/j.jmaa.2007.09.023
 7.
GarciaFalset J, LorensFuster E, Suzuki T: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 2011, 375: 185–195. doi:10.1016/j.jmaa.2010.08.069 10.1016/j.jmaa.2010.08.069
 8.
Abkar A, Eslamian M: Common fixed point results in CAT(0) spaces. Nonlinear Anal. TMA 2011, 74: 1835–1840. doi:10.1016/j.na.2010.10.056 10.1016/j.na.2010.10.056
 9.
Laowang W, Panyanak P: Common fixed points for some generalized multivalued nonexpansive mappings in uniformly convex metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 20
 10.
Clarkson JA: Uniformly convex spaces. Trans. Am. Math. Soc. 1936, 40: 396–414. 10.1090/S00029947193615018804
 11.
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. TMA 1991, 16: 1127–1138. 10.1016/0362546X(91)90200K
 12.
Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.
 13.
Goebel K: On a fixed point theorem for multivalued nonexpansive mappings. Ann. Univ. Mariae CurieSkłodowska, Sect. A 1972, 29: 70–72.
 14.
Lim TC: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space. Bull. Am. Math. Soc. 1974, 80: 1123–1126. 10.1090/S000299041974136402
 15.
Goebel K, Kirk WA: Iteration processes for nonexpansive mapping in a Banach space. Contemp. Math. 1983, 21: 115–123.
 16.
Diaz JB, Metcalf FT: On the structure of the set of subsequential limit points of successive approximations. Bull. Am. Math. Soc. 1967, 73: 516–519. 10.1090/S000299041967117257
 17.
Dotson WG: Fixed points of quasinonexpansive mappings. J. Aust. Math. Soc. 1972, 13: 167–170. 10.1017/S144678870001123X
 18.
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S00029939197202985003
Acknowledgements
The first author would like to thank the Office of the Higher Education Commission, Thailand, for supporting grant fund under the program of Strategic Scholarships for Frontier Research Network for the PhD Program, Thai Doctoral degree, for this research.
Author information
Rights and permissions
About this article
Cite this article
Akkasriworn, N., Kaewkhao, A., Keawkhao, A. et al. Common fixedpoint results in uniformly convex Banach spaces. Fixed Point Theory Appl 2012, 171 (2012). https://doi.org/10.1186/168718122012171
Received:
Accepted:
Published:
Keywords
 common fixed point
 quasinonexpansive mapping
 generalized nonexpansive mapping
 uniformly convex Banach space