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The convergence theorems of Ishikawa iterative process with errors for Φhemicontractive mappings in uniformly smooth Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 206 (2012)
Abstract
Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and T:D\to D be a generalized Lipschitz Φhemicontractive mapping with q\in F(T)\ne \mathrm{\varnothing}. Let \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{d}_{n}\} be four real sequences in [0,1] and satisfy the conditions (i) {a}_{n},{b}_{n},{d}_{n}\to 0 as n\to \mathrm{\infty} and {c}_{n}=o({a}_{n}); (ii) {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}=\mathrm{\infty}. For some {x}_{0}\in D, let \{{u}_{n}\}, \{{v}_{n}\} be any bounded sequences in D, and \{{x}_{n}\} be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φquasiaccretive mapping.
MSC:47H10.
1 Introduction and preliminary
Let E be a real Banach space and {E}^{\ast} be its dual space. The normalized duality mapping J:E\to {2}^{{E}^{\ast}} is defined by
where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing. It is well known that

(i)
If E is a smooth Banach space, then the mapping J is singlevalued;

(ii)
J(\alpha x)=\alpha J(x) for all x\in E and \alpha \in \mathrm{\Re};

(iii)
If E is a uniformly smooth Banach space, then the mapping J is uniformly continuous on any bounded subset of E. We denote the singlevalued normalized duality mapping by j.
Definition 1.1 ([1])
Let D be a nonempty closed convex subset of E, T:D\to D be a mapping.

(1)
T is called strongly pseudocontractive if there is a constant k\in (0,1) such that for all x,y\in D,
\u3008TxTy,j(xy)\u3009\le k{\parallel xy\parallel}^{2}; 
(2)
T is called ϕstrongly pseudocontractive if for all x,y\in D, there exist j(xy)\in J(xy) and a strictly increasing continuous function \varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \varphi (0)=0 such that
\u3008TxTy,j(xy)\u3009\le {\parallel xy\parallel}^{2}\varphi (\parallel xy\parallel )\parallel xy\parallel ; 
(3)
T is called Φpseudocontractive if for all x,y\in D, there exist j(xy)\in J(xy) and a strictly increasing continuous function \mathrm{\Phi}:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \mathrm{\Phi}(0)=0 such that
\u3008TxTy,j(xy)\u3009\le {\parallel xy\parallel}^{2}\mathrm{\Phi}(\parallel xy\parallel ).
It is obvious that Φpseudocontractive mappings not only include ϕstrongly pseudocontractive mappings, but also strongly pseudocontractive mappings.
Definition 1.2 ([1])
Let T:D\to D be a mapping and F(T)=\{x\in D:Tx=x\}\ne \mathrm{\varnothing}.

(1)
T is called ϕstronglyhemipseudocontractive if for all x\in D, q\in F(T), there exist j(xq)\in J(xq) and a strictly increasing continuous function \varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \varphi (0)=0 such that
\u3008TxTq,j(xq)\u3009\le {\parallel xq\parallel}^{2}\varphi (\parallel xq\parallel )\parallel xq\parallel . 
(2)
T is called Φhemipseudocontractive if for all x\in D, q\in F(T), there exist j(xq)\in J(xq) and the strictly increasing continuous function \mathrm{\Phi}:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \mathrm{\Phi}(0)=0 such that
\u3008TxTq,j(xq)\u3009\le {\parallel xq\parallel}^{2}\mathrm{\Phi}(\parallel xq\parallel ).
Closely related to the class of pseudocontractivetype mappings are those of accretive type.
Definition 1.3 ([1])
Let N(T)=\{x\in E:Tx=0\}\ne \mathrm{\varnothing}. The mapping T:E\to E is called strongly quasiaccretive if for all x\in E, {x}^{\ast}\in N(T), there exist j(x{x}^{\ast})\in J(x{x}^{\ast}) and a constant k\in (0,1) such that \u3008TxT{x}^{\ast},j(x{x}^{\ast})\u3009\ge k{\parallel x{x}^{\ast}\parallel}^{2}; T is called ϕstrongly quasiaccretive if for all x\in E, {x}^{\ast}\in N(T), there exist j(x{x}^{\ast})\in J(x{x}^{\ast}) and a strictly increasing continuous function \varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \varphi (0)=0 such that \u3008TxT{x}^{\ast},j(x{x}^{\ast})\u3009\ge \varphi (\parallel x{x}^{\ast}\parallel )\parallel x{x}^{\ast}\parallel; T is called Φquasiaccretive if for all x\in E, {x}^{\ast}\in N(T), there exist j(x{x}^{\ast})\in J(x{x}^{\ast}) and a strictly increasing continuous function \mathrm{\Phi}:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \mathrm{\Phi}(0)=0 such that \u3008TxT{x}^{\ast},j(x{x}^{\ast})\u3009\ge \mathrm{\Phi}(\parallel x{x}^{\ast}\parallel ).
Definition 1.4 ([2])
For arbitrary given {x}_{0}\in D, the Ishikawa iterative process with errors {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} is defined by
where \{{u}_{n}\}, \{{v}_{n}\} are any bounded sequences in D; \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{d}_{n}\} are four real sequences in [0,1] and satisfy {a}_{n}+{c}_{n}\le 1, {b}_{n}+{d}_{n}\le 1, for all n\ge 0. If {b}_{n}={d}_{n}=0, then the sequence \{{x}_{n}\} defined by
is called the Mann iterative process with errors.
A mapping T:D\to D is called generalized Lipschitz if there exists a constant L>0 such that \parallel TxTy\parallel \le L(1+\parallel xy\parallel ), \mathrm{\forall}x,y\in D.
The aim of this paper is to prove the convergent results of the above Ishikawa and Mann iterations with errors for generalized Lipschitz Φhemicontractive mappings in uniformly smooth real Banach spaces. For this, we need the following lemmas.
Lemma 1.6 ([5])
Let E be a uniformly smooth real Banach space, and let J:E\to {2}^{{E}^{\ast}} be a normalized duality mapping. Then
for all x,y\in E.
Lemma 1.7 ([6])
Let {\{{\rho}_{n}\}}_{n=0}^{\mathrm{\infty}} be a nonnegative sequence which satisfies the following inequality:
where {\lambda}_{n}\in [0,1] with {\sum}_{n=0}^{\mathrm{\infty}}{\lambda}_{n}=\mathrm{\infty}, {\sigma}_{n}=o({\lambda}_{n}). Then {\rho}_{n}\to 0 as n\to \mathrm{\infty}.
2 Main results
Theorem 2.1 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and T:D\to D be a generalized Lipschitz Φhemicontractive mapping with q\in F(T)\ne \mathrm{\varnothing}. Let \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{d}_{n}\} be four real sequences in [0,1] and satisfy the conditions (i) {a}_{n},{b}_{n},{d}_{n}\to 0 as n\to \mathrm{\infty} and {c}_{n}=o({a}_{n}); (ii) {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}=\mathrm{\infty}. For some {x}_{0}\in D, let \{{u}_{n}\}, \{{v}_{n}\} be any bounded sequences in D, and \{{x}_{n}\} be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the unique fixed point q of T.
Proof Since T:D\to D is a generalized Lipschitz Φhemicontractive mapping, there exists a strictly increasing continuous function \mathrm{\Phi}:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) with \mathrm{\Phi}(0)=0 such that
i.e.,
and
for any x,y\in D and q\in F(T).
Step 1. There exists {x}_{0}\in D and {x}_{0}\ne T{x}_{0} such that {r}_{0}=\parallel {x}_{0}T{x}_{0}\parallel \cdot \parallel {x}_{0}q\parallel \in R(\mathrm{\Phi}) (range of Φ). Indeed, if \mathrm{\Phi}(r)\to +\mathrm{\infty} as r\to +\mathrm{\infty}, then {r}_{0}\in R(\mathrm{\Phi}); if sup\{\mathrm{\Phi}(r):r\in [0,+\mathrm{\infty})\}={r}_{1}<+\mathrm{\infty} with {r}_{0}<{r}_{1}, then for q\in D, there exists a sequence \{{w}_{n}\} in D such that {w}_{n}\to q as n\to \mathrm{\infty} with {w}_{n}\ne q. Furthermore, we obtain that \{{w}_{n}T{w}_{n}\} is bounded. Hence, there exists a natural number {n}_{0} such that \parallel {w}_{n}T{w}_{n}\parallel \cdot \parallel {w}_{n}q\parallel <\frac{{r}_{1}}{2} for n\ge {n}_{0}, then we redefine {x}_{0}={w}_{{n}_{0}} and \parallel {x}_{0}T{x}_{0}\parallel \cdot \parallel {x}_{0}q\parallel \in R(\mathrm{\Phi}).
Step 2. For any n\ge 0, \{{x}_{n}\} is bounded. Set R={\mathrm{\Phi}}^{1}({r}_{0}), then from Definition 1.2(2), we obtain that \parallel {x}_{0}q\parallel \le R. Denote {B}_{1}=\{x\in D:\parallel xq\parallel \le R\}, {B}_{2}=\{x\in D:\parallel xq\parallel \le 2R\}. Since T is generalized Lipschitz, so T is bounded. We may define M={sup}_{x\in {B}_{2}}\{\parallel Txq\parallel +1\}+{sup}_{n}\{\parallel {u}_{n}q\parallel \}+{sup}_{n}\{\parallel {v}_{n}q\parallel \}. Next, we want to prove that {x}_{n}\in {B}_{1}. If n=0, then {x}_{0}\in {B}_{1}. Now, assume that it holds for some n, i.e., {x}_{n}\in {B}_{1}. We prove that {x}_{n+1}\in {B}_{1}. Suppose it is not the case, then \parallel {x}_{n+1}q\parallel >R. Since J is uniformly continuous on a bounded subset of E, then for {\u03f5}_{0}=\frac{\mathrm{\Phi}(\frac{R}{4})}{24L(1+2R)}, there exists \delta >0 such that \parallel JxJy\parallel <\u03f5 when \parallel xy\parallel <\delta, \mathrm{\forall}x,y\in {B}_{2}. Now, denote
Owing to {a}_{n},{b}_{n},{c}_{n},{d}_{n}\to 0 as n\to \mathrm{\infty}, without loss of generality, assume that 0\le {a}_{n},{b}_{n},{c}_{n},{d}_{n}\le {\tau}_{0} for any n\ge 0. Since {c}_{n}=o({a}_{n}), denote {c}_{n}<{a}_{n}{\tau}_{0}. So, we have
and
Therefore,
Using Lemma 1.6 and the above formulas, we obtain
and
Substitute (2.11) into (2.10)
this is a contradiction. Thus, {x}_{n+1}\in {B}_{1}, i.e., \{{x}_{n}\} is a bounded sequence. So, \{{y}_{n}\}, \{T{y}_{n}\}, \{T{x}_{n}\} are all bounded sequences.
Step 3. We want to prove \parallel {x}_{n}q\parallel \to 0 as n\to \mathrm{\infty}. Set {M}_{1}=max\{{sup}_{n}\parallel {x}_{n}q\parallel ,{sup}_{n}\parallel {y}_{n}q\parallel ,{sup}_{n}\parallel T{x}_{n}q\parallel ,{sup}_{n}\parallel T{y}_{n}q\parallel ,{sup}_{n}\parallel {u}_{n}q\parallel ,{sup}_{n}\parallel {v}_{n}q\parallel \}.
By (2.10), (2.11), we have
and
where {A}_{n}=\parallel J({x}_{n+1}q)J({x}_{n}q)\parallel, {B}_{n}=\parallel J({x}_{n}q)J({y}_{n}q)\parallel and {A}_{n},{B}_{n}\to 0 as n\to \mathrm{\infty}.
Taking (2.14) into (2.13),
where {C}_{n}=\frac{{a}_{n}}{2}{M}_{1}^{2}+{M}_{1}{A}_{n}+3{M}_{1}{B}_{n}+2{b}_{n}{M}_{1}^{2}+2{d}_{n}{M}_{1}^{2}+\frac{{c}_{n}{M}_{1}^{2}}{{a}_{n}}\to 0 as n\to \mathrm{\infty}.
Set {inf}_{n\ge 0}\frac{\mathrm{\Phi}(\parallel {y}_{n}q\parallel )}{1+{\parallel {x}_{n+1}q\parallel}^{2}}=\lambda, then \lambda =0. If it is not the case, we assume that \lambda >0. Let 0<\gamma <min\{1,\lambda \}, then \frac{\mathrm{\Phi}(\parallel {y}_{n}q\parallel )}{1+{\parallel {x}_{n+1}q\parallel}^{2}}\ge \gamma, i.e., \mathrm{\Phi}(\parallel {y}_{n}q\parallel )\ge \gamma +\gamma {\parallel {x}_{n+1}q\parallel}^{2}\ge \gamma {\parallel {x}_{n+1}q\parallel}^{2}. Thus, from (2.15) it follows that
This implies that
Let {\rho}_{n}={\parallel {x}_{n}q\parallel}^{2}, {\lambda}_{n}=\frac{2{a}_{n}\gamma}{1+2{a}_{n}\gamma}, {\sigma}_{n}=\frac{2{a}_{n}{C}_{n}}{1+2{a}_{n}\gamma}. Then we get that
Applying Lemma 1.7, we get that {\rho}_{n}\to 0 as n\to \mathrm{\infty}. This is a contradiction and so \lambda =0. Therefore, there exists an infinite subsequence such that \frac{\mathrm{\Phi}(\parallel {y}_{{n}_{i}}q\parallel )}{1+{\parallel {x}_{{n}_{i}+1}q\parallel}^{2}}\to 0 as i\to \mathrm{\infty}. Since 0\le \frac{\mathrm{\Phi}(\parallel {y}_{{n}_{i}}q\parallel )}{1+{M}_{1}^{2}}\le \frac{\mathrm{\Phi}(\parallel {y}_{{n}_{i}}q\parallel )}{1+{\parallel {x}_{{n}_{i}+1}q\parallel}^{2}}, then \mathrm{\Phi}(\parallel {y}_{{n}_{i}}q\parallel )\to 0 as i\to \mathrm{\infty}. In view of the strictly increasing continuity of Φ, we have \parallel {y}_{{n}_{i}}q\parallel \to 0 as i\to \mathrm{\infty}. Hence, \parallel {x}_{{n}_{i}}q\parallel \to 0 as i\to \mathrm{\infty}. Next, we want to prove \parallel {x}_{n}q\parallel \to 0 as n\to \mathrm{\infty}. Let \mathrm{\forall}\epsilon \in (0,1), there exists {n}_{{i}_{0}} such that \parallel {x}_{{n}_{i}}q\parallel <\u03f5, {a}_{n},{c}_{n}<\frac{\u03f5}{8{M}_{1}}, {b}_{n},{d}_{n}<\frac{\u03f5}{16{M}_{1}}, {C}_{n}<\frac{\mathrm{\Phi}(\u03f5)}{2}, for any {n}_{i},n\ge {n}_{{i}_{0}}. First, we want to prove \parallel {x}_{{n}_{i}+1}q\parallel <\u03f5. Suppose it is not the case, then \parallel {x}_{{n}_{i}+1}q\parallel \ge \u03f5. Using (1.1), we may get the following estimates:
Since Φ is strictly increasing, then (2.19) leads to \mathrm{\Phi}(\parallel {y}_{{n}_{i}}q\parallel )\ge \mathrm{\Phi}(\frac{\u03f5}{4}). From (2.15), we have
which is a contradiction. Hence, \parallel {x}_{{n}_{i}+1}q\parallel <\u03f5. Suppose that \parallel {x}_{{n}_{i}+m}q\parallel <\u03f5 holds. Repeating the above course, we can easily prove that \parallel {x}_{{n}_{i}+m+1}q\parallel <\u03f5 holds. Therefore, for any m, we obtain that \parallel {x}_{{n}_{i}+m}q\parallel <\u03f5, which means \parallel {x}_{n}q\parallel \to 0 as n\to \mathrm{\infty}. This completes the proof. □
Theorem 2.2 Let E be an arbitrary uniformly smooth real Banach space, and T:E\to E be a generalized Lipschitz Φquasiaccretive mapping with N(T)\ne \mathrm{\varnothing}. Let \{{a}_{n}\}, \{{b}_{n}\}, \{{c}_{n}\}, \{{d}_{n}\} be four real sequences in [0,1] and satisfy the conditions (i) {a}_{n},{b}_{n},{d}_{n}\to 0 as n\to \mathrm{\infty} and {c}_{n}=o({a}_{n}); (ii) {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}=\mathrm{\infty}. For some {x}_{0}\in D, let \{{u}_{n}\}, \{{v}_{n}\} be any bounded sequences in E, and \{{x}_{n}\} be an Ishikawa iterative sequence with errors defined by
where S:E\to E is defined by Sx=xTx for any x\in E. Then \{{x}_{n}\} converges strongly to the unique solution of the equation Tx=0 (or the unique fixed point of S).
Proof Since T is a generalized Lipschitz and Φquasiaccretive mapping, it follows that
i.e.,
i.e.,
for all x,y\in E, q\in N(T). The rest of the proof is the same as that of Theorem 2.1. □
Corollary 2.3 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and T:D\to D be a generalized Lipschitz Φhemicontractive mapping with q\in F(T)\ne \mathrm{\varnothing}. Let \{{a}_{n}\}, \{{c}_{n}\} be two real sequences in [0,1] and satisfy the conditions (i) {a}_{n}\to 0 as n\to \mathrm{\infty} and {c}_{n}=o({a}_{n}); (ii) {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}=\mathrm{\infty}. For some {x}_{0}\in D, let \{{u}_{n}\} be any bounded sequence in D, and \{{x}_{n}\} be the Mann iterative sequence with errors defined by (1.2). Then (1.2) converges strongly to the unique fixed point q of T.
Corollary 2.4 Let E be an arbitrary uniformly smooth real Banach space, and T:E\to E be a generalized Lipschitz Φquasiaccretive mapping with N(T)\ne \mathrm{\varnothing}. Let \{{a}_{n}\}, \{{d}_{n}\} be two real sequences in [0,1] and satisfy the conditions (i) {a}_{n}\to 0 as n\to \mathrm{\infty} and {c}_{n}=o({a}_{n}); (ii) {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}=\mathrm{\infty}. For some {x}_{0}\in D, let \{{u}_{n}\} be any bounded sequence in E, and \{{x}_{n}\} be the Mann iterative sequence with errors defined by
where S:E\to E is defined by Sx=xTx for any x\in E. Then \{{x}_{n}\} converges strongly to the unique solution of the equation Tx=0 (or the unique fixed point of S).
Remark 2.5 It is mentioned that in 2006, Chidume and Chidume [1] proved the approximative theorem for zeros of generalized Lipschitz generalized Φquasiaccretive operators. This result provided significant improvements of some recent important results. Their result is as follows.
Theorem CC ([[1], Theorem 3.1])
Let E be a uniformly smooth real Banach space and A:E\to E be a mapping with N(A)\ne \mathrm{\varnothing}. Suppose A is a generalized Lipschitz Φquasiaccretive mapping. Let \{{a}_{n}\}, \{{b}_{n}\} and \{{c}_{n}\} be real sequences in [0,1] satisfying the following conditions: (i) {a}_{n}+{b}_{n}+{c}_{n}=1; (ii) {\sum}_{n=0}^{\mathrm{\infty}}({b}_{n}+{c}_{n})=\mathrm{\infty}; (iii) {\sum}_{n=0}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}; (iv) {lim}_{n\to \mathrm{\infty}}{b}_{n}=0. Let \{{x}_{n}\} be generated iteratively from arbitrary {x}_{0}\in E by
where S:E\to E is defined by Sx:=f+xAx, \mathrm{\forall}x\in E and \{{u}_{n}\} is an arbitrary bounded sequence in E. Then, there exists {\gamma}_{0}\in \mathrm{\Re} such that if {b}_{n}+{c}_{n}\le {\gamma}_{0}, \mathrm{\forall}n\ge 0, the sequence \{{x}_{n}\} converges strongly to the unique solution of the equation Au=0.
However, there exists a gap in the proof process of above Theorem CC. Here, {c}_{n}=min\{\frac{\u03f5}{4\beta},\frac{1}{2\sigma}\mathrm{\Phi}(\frac{\u03f5}{2}){\alpha}_{n}\} ({\alpha}_{n}={b}_{n}+{c}_{n}) does not hold in line 14 of Claim 2 on page 248, i.e., {c}_{n}\le \frac{1}{2\sigma}\mathrm{\Phi}(\frac{\u03f5}{2}){\alpha}_{n} is a wrong case. For instance, set the iteration parameters: {a}_{n}=1{b}_{n}{c}_{n}, where \{{b}_{n}\}:{b}_{1}=\frac{1}{4}, {b}_{n}=\frac{1}{n}, n\ge 2; \{{c}_{n}\}:\frac{1}{4},\frac{1}{{2}^{2}},\frac{1}{{3}^{2}},\frac{1}{4},\frac{1}{{5}^{2}},\dots ,\frac{1}{{8}^{2}},\frac{1}{9},\frac{1}{{10}^{2}},\dots ,\frac{1}{{15}^{2}},\frac{1}{16},\frac{1}{{17}^{2}},\dots ,\frac{1}{{24}^{2}},\frac{1}{25},\frac{1}{{26}^{2}},\dots ,\frac{1}{{35}^{2}},\frac{1}{36},\frac{1}{{37}^{2}},\dots . Then {\sum}_{n=0}^{\mathrm{\infty}}{c}_{n}<+\mathrm{\infty}, but {c}_{n}\ne o({b}_{n}+{c}_{n}). Therefore, the proof of above Theorem CC is not reasonable. Up to now, we do not know the validity of Theorem CC. This will be an open question left for the readers!
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Xue, Z., Lv, G. & Rhoades, B. The convergence theorems of Ishikawa iterative process with errors for Φhemicontractive mappings in uniformly smooth Banach spaces. Fixed Point Theory Appl 2012, 206 (2012). https://doi.org/10.1186/168718122012206
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DOI: https://doi.org/10.1186/168718122012206
Keywords
 generalized Lipschitz mapping
 Φhemicontractive mapping
 Ishikawa iterative sequence with errors
 uniformly smooth real Banach space