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Strong convergence theorems for fixed points of asymptotically strict quasiϕpseudocontractions
Fixed Point Theory and Applications volume 2012, Article number: 137 (2012)
Abstract
In this paper, the fixed point problem of asymptotically strict quasiϕpseudocontractions is investigated based on hybrid projection algorithms. Strong convergence theorems of fixed points are established in a reflexive, strictly convex, and smooth Banach space such that both E and ${E}^{\ast}$ have the KadecKlee property.
MSC:47H09, 47J25.
1 Introduction
In what follows, we always assume that E is a Banach space with the dual space ${E}^{\ast}$. Let C be a nonempty, closed, and convex subset of E. We use the symbol J to stand for the normalized duality mapping from E to ${2}^{{E}^{\ast}}$ defined by
where $\u3008\cdot ,\cdot \u3009$ denotes the generalized duality pairing of elements between E and ${E}^{\ast}$.
Let ${U}_{E}=\{x\in E:\parallel x\parallel =1\}$ be the unit sphere of E. E is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in {U}_{E}$ with $x\ne y$. It is said to be uniformly convex if for any $\u03f5\in (0,2]$ there exists $\delta >0$ such that for any $x,y\in {U}_{E}$,
It is known that a uniformly convex Banach space is reflexive and strictly convex. E is said to be smooth provided ${lim}_{t\to 0}\frac{\parallel x+ty\parallel \parallel x\parallel}{t}$ exists for all $x,y\in {U}_{E}$. It is also said to be uniformly smooth if the limit is attained uniformly for all $x,y\in {U}_{E}$.
It is well known that if ${E}^{\ast}$ is strictly convex, then J is single valued; if ${E}^{\ast}$ is reflexive, and smooth, then J is single valued and demicontinuous; for more details, see [1] and the references therein.
It is also well known that if D is a nonempty, closed, and convex subset of a Hilbert space H, and ${P}_{D}:H\to D$ is the metric projection from H onto D, then ${P}_{D}$ is nonexpansive. This fact actually characterizes Hilbert spaces; and consequently, it is not available in more general Banach spaces. In this connection, Alber [2] introduced a generalized projection operator ${\mathrm{\Pi}}_{D}$ in Banach spaces which is an analogue of the metric projection in Hilbert spaces.
Let E be a smooth Banach space. Consider the functional defined by
Notice that, in a Hilbert space H, (1.1) is reduced to $\varphi (x,y)={\parallel xy\parallel}^{2}$ for all $x,y\in H$. The generalized projection ${\mathrm{\Pi}}_{C}:E\to C$ is a mapping that assigns to an arbitrary point $x\in E$, the minimum point of the functional $\varphi (x,y)$; that is, ${\mathrm{\Pi}}_{C}x=\overline{x}$, where $\overline{x}$ is the solution to the following minimization problem:
The existence and uniqueness of the operator ${\mathrm{\Pi}}_{C}$ follow from the properties of the functional $\varphi (x,y)$ and the strict monotonicity of the mapping J; see, for example, [1–4]. In Hilbert spaces, ${\mathrm{\Pi}}_{C}={P}_{C}$. It is obvious from the definition of the function ϕ that
and
Recall the following.

(1)
A point p in C is said to be an asymptotic fixed point of T if C contains a sequence $\{{x}_{n}\}$ which converges weakly to p such that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0$. The set of asymptotic fixed points of T will be denoted by $\tilde{F}(T)$.

(2)
T is said to be relatively nonexpansive if
$$\tilde{F}(T)=F(T)\ne \mathrm{\varnothing},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\varphi (p,Tx)\le \varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in C,\mathrm{\forall}p\in F(T).$$ 
(3)
T is said to be relatively asymptotically nonexpansive if
$$\tilde{F}(T)=F(T)\ne \mathrm{\varnothing},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\varphi (p,{T}^{n}x)\le (1+{\mu}_{n})\varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in C,\mathrm{\forall}p\in F(T),\mathrm{\forall}n\ge 1,$$
where $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ is a sequence such that ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$.
Remark 1.1 The class of relatively asymptotically nonexpansive mappings was first considered in Su and Qin [5]; see also [6, 7] and the references therein.

(4)
T is said to be quasiϕnonexpansive if
$$F(T)\ne \mathrm{\varnothing},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\varphi (p,Tx)\le \varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in C,\mathrm{\forall}p\in F(T).$$ 
(5)
T is said to be asymptotically quasiϕnonexpansive if there exists a sequence $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ with ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$ such that
$$F(T)\ne \mathrm{\varnothing},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\varphi (p,{T}^{n}x)\le (1+{\mu}_{n})\varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in C,\mathrm{\forall}p\in F(T),\mathrm{\forall}n\ge 1.$$
Remark 1.2 The class of quasiϕnonexpansive mappings and the class of asymptotically quasiϕnonexpansive mappings were first considered in Zhou, Gao, and Tan [8]; see also [9–12].
Remark 1.3 The class of quasiϕnonexpansive mappings and the class of asymptotically quasiϕnonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasiϕnonexpansive mappings and asymptotically quasiϕnonexpansive do not require $F(T)=\tilde{F}(T)$.
Remark 1.4 The class of quasiϕnonexpansive mappings and the class of asymptotically quasiϕnonexpansive mappings are generalizations of the class of quasinonexpansive mappings and the class of asymptotically quasinonexpansive mappings in Banach spaces, respectively.

(6)
T is said to be a strict quasiϕpseudocontraction if $F(T)\ne \mathrm{\varnothing}$, and a constant $\kappa \in [0,1)$ such that
$$\varphi (p,Tx)\le \varphi (p,x)+\kappa \varphi (x,Tx),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in C,p\in F(T).$$
Remark 1.5 The class of strict quasiϕpseudocontractions was first considered in Zhou and Gao [13]; see also Qin, Wang, and Cho [14].

(7)
T is said to be an asymptotically strict quasiϕpseudocontraction if $F(T)\ne \mathrm{\varnothing}$, and there exists a sequence $\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$ with ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$ and a constant $\kappa \in [0,1)$ such that
$$\varphi (p,{T}^{n}x)\le (1+{\mu}_{n})\varphi (p,x)+\kappa \varphi (x,{T}^{n}x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in C,p\in F(T).$$
Remark 1.6 The class of asymptotically strict quasiϕpseudocontractions was first considered in Qin, Wang, and Cho [14].
Remark 1.7 The class of strict quasiϕpseudocontractions and the class of asymptotically strict quasiϕpseudocontractions are generalizations of the class of asymptotically strict quasipseudocontractions and the class of asymptotically strict quasipseudocontractions in Banach spaces, respectively.
The following example can be found in [14].
Let $E={l}_{2}:=\{x=\{{x}_{1},{x}_{2},\dots \}:{\sum}_{n=1}^{\mathrm{\infty}}{{x}_{n}}^{2}<\mathrm{\infty}\}$ and ${B}_{E}$ be the closed unit ball in E. Define a mapping $T:{B}_{E}\to {B}_{E}$ by
where $\{{a}_{i}\}$ is a sequence of real numbers such that ${a}_{2}>0$, $0<{a}_{j}<1$, where $i\ne 2$, and ${\prod}_{i=2}^{\mathrm{\infty}}{a}_{j}=\frac{1}{2}$. Then T is an asymptotically strict quasiϕpseudocontraction.

(8)
T is said to be asymptotically regular on C if, for any bounded subset K of C,
$$\underset{n\to \mathrm{\infty}}{lim}\underset{x\in K}{sup}\left\{\parallel {T}^{n+1}x{T}^{n}x\parallel \right\}=0.$$
During the past five decades, many famous existence theorems of fixed points of nonlinear mappings were established. However, from the standpoint of real world applications, it is not only to know the existence of fixed points of nonlinear mappings, but also to be able to construct an iterative process to approximate their fixed points. The simplest and oldest iterative algorithm is the wellknown Picard iterative algorithm which generates an iterative sequence from an arbitrary initial ${x}_{0}$ in the following manner:
where T is some mapping. The Picard iterative algorithm is a beautiful tool in the study of contractions. A wellknown result is the Banach contraction principle. The class of nonexpansive mappings as a class of important nonlinear mappings finds many applications in signal processing, image reconstruction and so on. However, the Picard iterative algorithm fails to converge fixed points of nonexpansive mappings even when the fixed point set is not empty. For overcoming this, a Mann iterative algorithm has been studied extensively recently. The Mann iterative algorithm generates an iterative sequence for an arbitrary initial ${x}_{0}$ in the following manner:
where T is some mapping and $\{{\alpha}_{n}\}$ is some control sequence in $(0,1)$. The classic convergence theorem for fixed points of nonexpansive mappings based on the Mann iterative algorithm was established by Reich [15] in Banach spaces; for more details, see [15] and the reference therein.
It is known that the Mann iterative algorithm only has weak convergence even for nonexpansive mappings in infinitedimensional Hilbert spaces; for more details, see [16] and the reference therein. To obtain the weak convergence of the Mann iterative algorithm, socalled hybrid projection algorithms have been considered; for more details, see [17–32] and the references therein.
In [24], Marino and Xu established a strong convergence theorem for fixed points of strict pseudocontraction based on hybrid projection algorithms in Hilbert spaces. Recently, Zhou and Gao [13] studied a new projection algorithm for strict quasiϕpseudocontractions and obtained a strong convergence theorem; for more details, see [13] and the reference therein. Quite recently, Qin, Wang, and Cho [14] proved a strong convergence theorem for fixed points of an asymptotically strict quasiϕpseudocontraction in a uniformly convex and uniformly smooth Banach space based on the results announced in Zhou and Gao [13]; for more details, see [14] and the reference therein.
In this paper, motivated by the results announced in Zhou and Gao [13] and Qin, Wang, and Cho [14], we consider asymptotically strict quasiϕpseudocontractions. We establish a strong convergence theorem in a reflexive, strictly convex, and smooth Banach space such that both E and ${E}^{\ast}$ have the KadecKlee property to relax the restriction imposed on the space in Qin, Wang, and Cho’s results. The results presented in this paper mainly improve the corresponding results announced in Zhou and Gao [13] and Qin, Wang, and Cho [14].
To prove our convergence theorem, we need the following lemmas:
Lemma 1.1[4]
Let C be a nonempty closed convex subset of a smooth Banach space E and$x\in E$. Then${x}_{0}={\mathrm{\Pi}}_{C}x$if and only if
Lemma 1.2[4]
Let E be a reflexive, strictly convex, and smooth Banach space, C a nonempty closed convex subset of E, and$x\in E$. Then
Lemma 1.3[21]
Let E be a reflexive, strictly convex, and smooth Banach space. Then we have the following:
2 Main results
Theorem 2.1 Let E be a reflexive, strictly convex, and smooth Banach space such that both E and${E}^{\ast}$have the KadecKlee property. Let C be a nonempty, closed, and convex subset of E. Let$T:C\to C$be a closed and asymptotically strict quasiϕpseudocontraction with a sequence$\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$such that${\mu}_{n}\to 0$as$n\to \mathrm{\infty}$. Assume that T is asymptotically regular on C and$F(T)$is nonempty and bounded. Let$\{{x}_{n}\}$be a sequence generated in the following manner:
where${M}_{n}=sup\{\varphi (p,{x}_{n}):p\in F(T)\}$. Then the sequence$\{{x}_{n}\}$converges strongly to$\overline{x}={\mathrm{\Pi}}_{F(T)}{x}_{0}$.
Proof First, we show that $F(T)$ is closed and convex. The closedness of $F(T)$ follows from the closedness of T. Next, we show that $F(T)$ is convex. Let ${p}_{1},{p}_{2}\in F(T)$, and ${p}_{t}=t{p}_{1}+(1t){p}_{2}$, where $t\in (0,1)$. We see that ${p}_{t}=T{p}_{t}$. Indeed, we see from the definition of T that
and
In view of (1.3), we obtain that
and
It follows from (2.1), (2.2), (2.3), and (2.4) that
and
Multiplying t and $(1t)$ on both sides of (2.5) and (2.6) respectively yields that
It follows that
In light of (1.2), we arrive at
It follows that
Since ${E}^{\ast}$ is reflexive, we may, without loss of generality, assume that $J({T}^{n}{p}_{t})\rightharpoonup {e}^{\ast}\in {E}^{\ast}$. In view of the reflexivity of E, we have $J(E)={E}^{\ast}$. This shows that there exists an element $e\in E$ such that $Je={e}^{\ast}$. It follows that
Taking ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}$ on both sides of the equality above, we obtain that
This implies from Lemma 1.3 that ${p}_{t}=e$, that is, $J{p}_{t}={e}^{\ast}$. It follows that $J({T}^{n}{p}_{t})\rightharpoonup J{p}_{t}\in {E}^{\ast}$. In view of the KadecKlee property of ${E}^{\ast}$, we obtain from (2.8) that
Since ${J}^{1}:{E}^{\ast}\to E$ is demicontinuous, we see that ${T}^{n}{p}_{t}\rightharpoonup {p}_{t}$. By virtue of the KadecKlee property of E, we see from (2.7) that ${T}^{n}{p}_{t}\to {p}_{t}$ as $n\to \mathrm{\infty}$. Since T is asymptotically regular, we see that
as $n\to \mathrm{\infty}$. In view of the closedness of T, we can obtain that ${p}_{t}\in F(T)$. This shows that $F(T)$ is convex. This completes the proof that $F(T)$ is closed and convex.
Next, we show that ${C}_{n}$ is closed and convex for all $n\ge 1$. It is not hard to see that ${C}_{n}$ is closed for each $n\ge 1$. Therefore, we only show that ${C}_{n}$ is convex for each $n\ge 1$. It is obvious that ${C}_{1}=C$ is convex. Suppose that ${C}_{h}$ is convex for some $h\in \mathbb{N}$. Next, we show that ${C}_{h+1}$ is also convex for the same h. Let $a,b\in {C}_{h+1}$ and $c=ta+(1t)b$, where $t\in (0,1)$. It follows that
and
where $a,b\in {C}_{h}$. From the above two inequalities, we can get that
where $c\in {C}_{h}$. It follows that ${C}_{h+1}$ is closed and convex. This completes the proof that ${C}_{n}$ is closed and convex.
Next, we show that $F(T)\subset {C}_{n}$. It is obvious that $F(T)\subset C={C}_{1}$. Suppose that $F(T)\subset {C}_{h}$ for some $h\in \mathbb{N}$. For any $z\in F(T)\subset {C}_{h}$, we see that
On the other hand, we obtain from (1.3) that
Combining (2.9) with (2.10), we arrive at
which implies that $z\in {C}_{h+1}$. This shows that $F(T)\subset {C}_{h+1}$. This completes the proof that $F(T)\subset {C}_{n}$.
Next, we show that $\{{x}_{n}\}$ is a convergent sequence which strongly converges to $\overline{x}$, where $\overline{x}\in F(T)$. Since ${x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{0}$, we see that
It follows from $F(T)\subset {C}_{n}$ that
By virtue of Lemma 1.2, we obtain that
This implies that the sequence $\{\varphi ({x}_{n},{x}_{0})\}$ is bounded. It follows from (1.2) that the sequence $\{{x}_{n}\}$ is also bounded. Since the space is reflexive, we may assume that ${x}_{n}\rightharpoonup \overline{x}$. Since ${C}_{n}$ is closed, and convex, we see that $\overline{x}\in {C}_{n}$. On the other hand, we see from the weakly lower semicontinuity of the norm that
which implies that $\varphi ({x}_{n},{x}_{0})\to \varphi (\overline{x},{x}_{0})$ as $n\to \mathrm{\infty}$. Hence, $\parallel {x}_{n}\parallel \to \parallel \overline{x}\parallel $ as $n\to \mathrm{\infty}$. In view of the KadecKlee property of E, we see that ${x}_{n}\to \overline{x}$ as $n\to \mathrm{\infty}$. Notice that ${x}_{n+1}={\mathrm{\Pi}}_{{\bigcap}_{i\in \mathrm{\Delta}}F({T}_{i})}{x}_{0}\in {C}_{n+1}\subset {C}_{n}$. It follows that
Since ${x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{0}$, and ${x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{0}\in {C}_{n+1}\subset {C}_{n}$, we arrive at $\varphi ({x}_{n},{x}_{0})\le \varphi ({x}_{n+1},{x}_{0})$, $\mathrm{\forall}n\ge 1$. This shows that $\{\varphi ({x}_{n},{x}_{0})\}$ is nondecreasing. It follows from the boundedness that ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{,}{x}_{0})$ exists. It follows that
By virtue of ${x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{0}\in {C}_{n+1}$, we find that
It follows that
In view of (1.2), we see that
Since ${x}_{n}\to \overline{x}$, we find that
It follows that
This implies that $\{J({T}^{n}{x}_{n})\}$ is bounded. Note that both E and ${E}^{\ast}$ are reflexive. We may assume that $J({T}^{n}{x}_{n})\rightharpoonup {y}^{\ast}\in {E}^{\ast}$. In view of the reflexivity of E, we see that there exists an element $y\in E$ such that $Jy={y}^{\ast}$. It follows that
Taking ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}$ on both sides of the equality above yields that
That is, $\overline{x}=y$, which in turn implies that ${y}^{\ast}=J\overline{x}$. It follows that $J({T}^{n}{x}_{n})\rightharpoonup J\overline{x}\in {E}^{\ast}$. Since ${E}^{\ast}$ enjoys the KadecKlee property, we obtain from (2.15) that ${lim}_{n\to \mathrm{\infty}}J({T}^{n}{x}_{n})=J\overline{x}$. Since ${J}^{1}:{E}^{\ast}\to E$ is demicontinuous, we find that ${T}^{n}{x}_{n}\rightharpoonup \overline{x}$. This implies, from (2.14) and the KadecKlee property of E, that ${lim}_{n\to \mathrm{\infty}}{T}^{n}{x}_{n}=\overline{x}$. Notice that
It follows from the asymptotic regularity of T that
that is, $T{T}^{n}{x}_{n}\overline{x}\to 0$ as $n\to \mathrm{\infty}$. It follows from the closedness of T that $T\overline{x}=\overline{x}$.
Finally, we show that $\overline{x}={\mathrm{\Pi}}_{F(T)}{x}_{0}$. Letting $n\to \mathrm{\infty}$ in (2.11), we arrive at
It follows from Lemma 1.1 that $\overline{x}={\mathrm{\Pi}}_{F(T)}{x}_{0}$. The proof of Theorem 2.1 is completed. □
Remark 2.2 Comparing with the results in Zhou and Gao [13], the mapping was generalized from strict quasiϕpseudocontractions to asymptotically strict quasiϕpseudocontractions.
Remark 2.3 Comparing with the results in Qin, Wang, and Cho [14], the restriction imposed on the space was relaxed from uniform convexness to strict convexness.
Since the class of asymptotically strict quasiϕpseudocontractions includes the class asymptotically quasiϕnonexpansive mappings as a special case, we find the following subresults from Theorem 2.1.
Corollary 2.4 Let E be a reflexive, strictly convex, and smooth Banach space such that both E and${E}^{\ast}$have the KadecKlee property. Let C be a nonempty, closed, and convex subset of E. Let$T:C\to C$be a closed and asymptotically quasiϕnonexpansive mapping with a sequence$\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$such that${\mu}_{n}\to 0$as$n\to \mathrm{\infty}$. Assume that T is asymptotically regular on C, and$F(T)$is nonempty and bounded. Let$\{{x}_{n}\}$be a sequence generated in the following manner:
where${M}_{n}=sup\{\varphi (p,{x}_{n}):p\in F(T)\}$. Then the sequence$\{{x}_{n}\}$converges strongly to$\overline{x}={\mathrm{\Pi}}_{F(T)}{x}_{0}$.
In Hilbert spaces, asymptotically strict quasiϕpseudocontractions are reduced to asymptotically strict quasipseudocontractions. The following results are not hard to derive.
Corollary 2.5 Let C be a nonempty, closed, and convex subset of a Hilbert space E. Let$T:C\to C$be a closed and asymptotically strict quasipseudocontraction with a sequence$\{{\mu}_{n}\}\subset [0,\mathrm{\infty})$such that${\mu}_{n}\to 0$as$n\to \mathrm{\infty}$. Assume that T is asymptotically regular on C and$F(T)$is nonempty and bounded. Let$\{{x}_{n}\}$be a sequence generated in the following manner:
where${M}_{n}=sup\{{\parallel p{x}_{n}\parallel}^{2}:p\in F(T)\}$. Then the sequence$\{{x}_{n}\}$converges strongly to$\overline{x}={P}_{F(T)}{x}_{0}$.
For strict quasipseudocontractions, we have the following.
Corollary 2.6 Let C be a nonempty, closed, and convex subset of a Hilbert space E. Let$T:C\to C$be a closed and strict quasipseudocontraction with a nonempty fixed point set. Let$\{{x}_{n}\}$be a sequence generated in the following manner:
Then the sequence$\{{x}_{n}\}$converges strongly to$\overline{x}={P}_{F(T)}{x}_{0}$.
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Zhang, Q. Strong convergence theorems for fixed points of asymptotically strict quasiϕpseudocontractions. Fixed Point Theory Appl 2012, 137 (2012). https://doi.org/10.1186/168718122012137
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Keywords
 asymptotically strict quasiϕpseudocontraction
 fixed point
 quasiϕnonexpansive mapping
 relatively asymptotically nonexpansive mapping
 relatively nonexpansive mapping