Fixed point theorems for a new nonlinear mapping similar to a nonspreading mapping
 Tomonari Suzuki^{1}Email author
https://doi.org/10.1186/16871812201447
© Suzuki; licensee Springer. 2014
Received: 31 October 2013
Accepted: 24 January 2014
Published: 21 February 2014
Abstract
We introduce a new nonlinear mapping similar to the nonspreading mapping which is defined by Kohsaka and Takahashi (Arch. Math. 91:166177, 2008). We prove fixed point theorems for such a mapping without the convexity of the domain. We also prove convergence theorems.
MSC:47H10, 47H09, 47J25.
Keywords
1 Introduction
We note that the concept of nonspreading mapping is very important because of useful applications.
where α is constant belonging to $(0,1/2)$.
Motivated by the above, we introduce a new concept, named Chatterjea mapping. The condition (5) of this concept is weaker than (2). It is meaningful to study Chatterjea mapping because the concept of nonspreading mapping is very important. In this paper, we prove fixed point theorems for Chatterjea mappings without the convexity of the domain. We also prove convergence theorems to a fixed point.
2 Preliminaries
Throughout this paper we denote by ℕ the set of all positive integers and by ℝ the set of all real numbers. For $n\in \mathbb{N}\cup \{0\}$, we define n! by $0!=1$ and $(n+1)!=n!(n+1)$, that is, n! is the factorial of n. Stirling’s formula ${lim}_{n}\sqrt{2\pi n}{(n/e)}^{n}/(n!)=1$ is well known. For $n,k\in \mathbb{N}\cup \{0\}$ with $k\le n$, we define $C(n,k)=n!/(k!(nk)!)$, that is, $C(n,k)$ is the binomial coefficient of $(n;k)$.
A Banach space E is said to be smooth if the limit ${lim}_{t\to 0}(\parallel x+ty\parallel \parallel x\parallel )/t$ exists for each $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$. The normalized duality mapping J from E into ${E}^{\ast}$ is defined by $\u3008x,Jx\u3009={\parallel x\parallel}^{2}={\parallel Jx\parallel}^{2}$ for all $x\in E$.
for all $x,y\in E$ with $\parallel x\parallel \le 1$, $\parallel y\parallel \le 1$ and $xy\in \{tz:t\in [2,\epsilon ]\cup [+\epsilon ,+2]\}$. It is obvious that UCED implies strictly convexity. We know that every separable Banach space can be equivalently renormed so that it is UCED. See [3, 4] and others. We know UCED is characterized as follows.
Lemma 1 ([5])
 (i)
E is UCED.
 (ii)If $\{{u}_{n}\}$ is a bounded sequence in E, then a function g on E defined by$g(x)=\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}\parallel {u}_{n}x\parallel $(3)
for all $\lambda \in (0,1)$ and $x,y\in E$ with $x\ne y$.
holds for $y\in C$ with $y\ne z$. We remark that we may replace ‘lim inf’ by ‘lim sup’. All nonempty compact subsets have the Opial property. Also, all Hilbert spaces, ${\ell}^{p}$ ($1\le p<\mathrm{\infty}$) and finite dimensional Banach spaces have the Opial property. A Banach space with a duality mapping which is weakly sequentially continuous also has the Opial property [7]. We know that every separable Banach space can be equivalently renormed so that it has the Opial property [8].
Then $minf(C)$ exists.
and hence $f(z)=inff(C)$. □
Let C be a subset of a Banach space E and let f be a function from C into ℝ. f is said to be nonincreasing with respect to a mapping T on C if $f(Tx)\le f(x)$ for all $x\in C$. Also, from now on, in the case where C is bounded, we consider every function f to satisfy (4).
The proof of the following lemma is obvious.
 (i)
$s\le t$ if and only if $\eta (s)\le \eta (t)$.
 (ii)
If ${lim\hspace{0.17em}sup}_{n}{t}_{n}\in \mathbb{R}$, then $\eta ({lim\hspace{0.17em}sup}_{n}{t}_{n})={lim\hspace{0.17em}sup}_{n}\eta ({t}_{n})$.
3 Chatterjea mapping
In this section, we introduce the concept of Chatterjea mapping.
for all $x,y\in C$.
From the definition, we can obtain the following propositions.
Proposition 4 Let T be a nonspreading mapping on a subset C of a Hilbert space E. Then T is Chatterjea with $t\mapsto {t}^{2}$.
Proof Obvious. □
for all $x,y\in C$. Then T is Chatterjea with η.
Adding the both inequalities, we obtain the desired result. □
Proposition 6 Let p and q be positive real numbers with $p<q$. Let T be a mapping on a subset C of a Banach space E. Assume T is Chatterjea with $t\mapsto {t}^{p}$. Then T is also Chatterjea with $t\mapsto {t}^{q}$.
which implies that T is Chatterjea with $t\mapsto {t}^{q}$. □
Then T is Chatterjea with $t\mapsto {t}^{q}$, however, T is not Chatterjea with $t\mapsto {t}^{p}$.
which implies that T is not Chatterjea with $t\mapsto {t}^{p}$. □
Remark Example 7 also informs that Chatterjea mappings are not necessarily continuous.
4 Basic properties
In this section, we prove basic properties of Chatterjea mapping.
for all $x\in C$ and $z\in F(T)$.
Proposition 8 Assume that a mapping T on C is Chatterjea and has a fixed point. Then T is a quasinonexpansive mapping.
Using this and the strict increasingness of η, we obtain (6). □
From Proposition 8, we obtain the following.
Lemma 9 Assume that a mapping T on C is Chatterjea and has a fixed point. Then $\{{T}^{n}u\}$ is bounded for all $u\in C$.
Proposition 10 Let T be a Chatterjea mapping on a closed subset C of a Banach space E. Then $F(T)$ is closed. Moreover, if E is strictly convex and C is convex, then $F(T)$ is also convex.
The following lemma plays a very important role in this paper.
Lemma 11 Put ${I}_{0}=\{(m,n):m,n\in \mathbb{N}\cup \{0\},m\le n\}$ and $I=\{(m,n):m,n\in \mathbb{N},m<n\}$. Let A be a function from ${I}_{0}$ into $[0,\mathrm{\infty})$ satisfying the following:

$A(0,n)\le 1$ for $n\in \mathbb{N}\cup \{0\}$;

$A(n,n)=0$ for $n\in \mathbb{N}$;

$A(m,n)\le (1/2)A(m1,n)+(1/2)A(m,n1)$ for $(m,n)\in I$.
 (i)
$A(m,n)\le 1$ for $(m,n)\in {I}_{0}$;
 (ii)
$A(j+2n,j+2n+1)\le \frac{1}{{2}^{2n}}{\sum}_{k=0}^{n}C(2n,k)\frac{2n2k+1}{2nk+1}A(j+k,j+2n+1k)$ for $j,n\in \mathbb{N}\cup \{0\}$;
 (iii)
$A(j+2n+1,j+2n+2)\le \frac{1}{{2}^{2n+1}}{\sum}_{k=0}^{n}C(2n+1,k)\frac{2n2k+2}{2nk+2}A(j+k,j+2n+2k)$ for $j,n\in \mathbb{N}\cup \{0\}$;
 (iv)
${lim}_{n\to \mathrm{\infty}}A(n,n+1)=0$.
Similarly we can prove ${lim}_{n}A(2n+1,2n+2)=0$. We have shown (iv). □
T is said to be asymptotically regular on C if T is asymptotically regular at all $x\in C$.
Lemma 12 Let T be a Chatterjea mapping on a subset C of a Banach space E. Assume $\{{T}^{n}x\}$ is bounded for some $x\in C$. Then T is asymptotically regular at x.
Therefore T is asymptotically regular at x. □
 (i)
$\{{T}^{n}x\}$ is bounded for all $x\in C$.
 (ii)
T is asymptotically regular on C.
which implies $f(Tx)\le f(x)$. Thus, f is nonincreasing with respect to T. Hence $f({T}^{n}x)\le f(x)$ for $n\in \mathbb{N}$. This implies that $\{{T}^{n}x\}$ is bounded. We have shown (i). By Lemma 12, we obtain (ii). □
5 Convergence theorems
In this section, we prove convergence theorems under the assumption that the domain C has the Opial property.
Proposition 14 Let T be a Chatterjea mapping on a subset C of a Banach space E. Assume C has the Opial property. If $\{{x}_{n}\}$ converges weakly to $z\in C$ and ${lim}_{n}\parallel T{x}_{n}{x}_{n}\parallel =0$, then $Tz=z$. That is, $IT$ is demiclosed at zero.
Since C has the Opial property, we obtain $Tz=z$. □
Remark A function $y\mapsto {lim\hspace{0.17em}sup}_{n}\eta (\parallel {x}_{n}y\parallel )$ from C into $[0,\mathrm{\infty})$ is also nonincreasing with respect to T.
Theorem 15 Let T be a Chatterjea mapping on a subset C of a Banach space E. Assume $\{{T}^{n}u\}$ is bounded for some $u\in C$; and C is boundedly weakly compact and has the Opial property. Then $\{{T}^{n}x\}$ converges weakly to a fixed point of T for all $x\in C$.
Remark We do not need the convexity of C.
This is a contradiction. Therefore $\{{T}^{n}x\}$ converges weakly to z. □
As direct consequences of Theorem 15, we obtain the following.
Corollary 16 Let T be a Chatterjea mapping on a weakly compact subset C of a Banach space E. Assume C has the Opial property. Then $\{{T}^{n}x\}$ converges weakly to a fixed point of T for all $x\in C$.
Corollary 17 Let T be a Chatterjea mapping on a compact subset C of a Banach space E. Then $\{{T}^{n}x\}$ converges strongly to a fixed point of T for all $x\in C$.
6 Existence theorems
In this section, we prove the existence of fixed points of Chatterjea mappings. By Lemma 9 and Theorem 15, we obtain the following.
 (i)
$\{{T}^{n}u\}$ is bounded for some $u\in C$.
 (ii)
T has a fixed point.
As direct consequences of Theorem 18, we obtain the following.
Corollary 19 Let T be a Chatterjea mapping on a subset C of a Banach space E. Assume that either of the following holds:

C is compact;

C is weakly compact and has the Opial property.
Then T has a fixed point.
Remark It is obvious that Corollary 19 also can be proved by Corollaries 16 and 17.
 (i)
$\{{T}^{n}u\}$ is bounded for some $u\in C$.
 (ii)
T has a fixed point.
Remark Corollary 20 is a generalization of Corollary 4.4 in [1] because we do not assume the convexity of C.
We shall prove fixed point theorems in UCED Banach spaces.
Lemma 21 Let C be a boundedly weakly compact and convex subset of a Banach space E. Let T be a mapping on a subset C. Assume that there exists a lower semicontinuous, strictly quasiconvex function f from C into ℝ such that f is nonincreasing with respect to T and f satisfies (4). Then T has a fixed point.
This is a contradiction. Hence $Tz=z$. □
Lemma 22 Let C be a boundedly weakly compact and convex subset of a Banach space E. Let ${T}_{0},{T}_{1},{T}_{2},\dots ,{T}_{\ell}$ be commuting mappings on C. Assume that for every $j=0,1,2,\dots ,\ell $, there exists a lower semicontinuous, strictly quasiconvex function ${f}_{j}$ from C into ℝ such that ${f}_{j}$ is nonincreasing with respect to ${T}_{j}$ and ${f}_{j}$ satisfies (4). Assume also that $F({T}_{j})$ is closed and convex for $j=1,2,\dots ,\ell $. Then ${\bigcap}_{j=0}^{\ell}F({T}_{j})$ is nonempty.
Proof By Lemma 21, $F({T}_{1})$ is nonempty. Since $F({T}_{1})$ is closed and convex, $F({T}_{1})$ is weakly closed. Thus $F({T}_{1})$ is boundedly weakly compact. We assume that ${A}_{k1}:={\bigcap}_{j=1}^{k1}F({T}_{j})$ is nonempty, boundedly weakly compact and convex for some $k\in \mathbb{N}$ with $1<k\le \ell $. For $x\in {A}_{k1}$ and $j\in \mathbb{N}$ with $1\le j<k$, since ${T}_{k}\circ {T}_{j}={T}_{j}\circ {T}_{k}$, we have ${T}_{k}x={T}_{k}\circ {T}_{j}x={T}_{j}\circ {T}_{k}x$, thus ${T}_{k}x$ is a fixed point of ${T}_{j}$. Therefore ${T}_{k}({A}_{k1})\subset {A}_{k1}$. By Lemma 21 again, ${T}_{k}$ has a fixed point in ${A}_{k1}$, thus, ${A}_{k}:={\bigcap}_{j=1}^{k}F({T}_{j})\ne \mathrm{\varnothing}$. Since ${A}_{k}$ is closed and convex, ${A}_{k}$ is nonempty, boundedly weakly compact and convex. By induction, ${A}_{\ell}$ is nonempty, boundedly weakly compact and convex. By Lemma 21, ${T}_{0}$ has a fixed point in ${A}_{\ell}$. This completes the proof. □
Lemma 23 Let C be a weakly compact and convex subset of a Banach space E. Let $S=\{{T}_{0}\}\cup {S}^{\prime}$ be a family of commuting mappings on C. Assume that for every $T\in S$, there exists a lower semicontinuous, strictly quasiconvex function ${f}_{T}$ from C into ℝ such that ${f}_{T}$ is nonincreasing with respect to T. Assume also that $F(T)$ is closed and convex for $T\in {S}^{\prime}$. Then S has a common fixed point.
Proof By Lemma 22, $\{F(T):T\in {S}^{\prime}\}$ has the finite intersection property. Since C is weakly compact and $F(T)$ is weakly closed for every $T\in {S}^{\prime}$, we have $A:={\bigcap}_{T\in {S}^{\prime}}F(T)\ne \mathrm{\varnothing}$. Since A is weakly compact and ${T}_{0}(A)\subset A$, ${T}_{0}$ has a fixed point in A. Thus S has a common fixed point. □
Lemma 24 Let C be a convex subset of a UCED Banach space E. Let T be a Chatterjea mapping on C. Assume that $\{{T}^{n}u\}$ is bounded for some $u\in C$. Define a function f from C into $[0,\mathrm{\infty})$ by (9). Then f is a continuous, strictly quasiconvex function such that f is nonincreasing with respect to T and f satisfies (4).
Proof We note that a function g defined by (3) is continuous and strictly quasiconvex; and g satisfies (4). So f is also continuous and strictly quasiconvex; and f satisfies (4). We have shown that f is nonincreasing with respect to T in the proof of Proposition 13. □
Using Lemmas 2124, we obtain the following.
 (i)
$\{{T}^{n}u\}$ is bounded for some $u\in C$.
 (ii)
T has a fixed point.
Theorem 26 Let C be a boundedly weakly compact and convex subset of a UCED Banach space E. Let ${T}_{1},{T}_{2},\dots ,{T}_{\ell}$ be commuting Chatterjea mappings on C. Assume that $\{{{T}_{j}}^{n}u\}$ is bounded for all $u\in C$ and j. Then ${\bigcap}_{j=1}^{\ell}F({T}_{j})$ is nonempty.
Theorem 27 Let C be a weakly compact and convex subset of a UCED Banach space E. Let S be a family of commuting Chatterjea mappings on C. Then S has a common fixed point.
Declarations
Acknowledgements
The author is grateful to the referees for their careful reading. The author is supported in part by a GrantinAid for Scientific Research from the Japan Society for the Promotion of Science.
Authors’ Affiliations
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