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Convergence theorems of a threestep iteration method for a countable family of pseudocontractive mappings
Fixed Point Theory and Applications volume 2013, Article number: 100 (2013)
Abstract
The purpose of this paper is to construct a threestep iteration method (as follows) and obtain the convergence theorem for a countable family of Lipschitz pseudocontractive mappings in Hilbert space H. For the iteration format,
under suitable conditions, we prove that the sequence \{{x}_{n}\} generated from above converges strongly to a common fixed point of {\{{T}_{n}\}}_{n\ge 1}. The results obtained in this paper improve and extend previous results that have been proved for this class of nonlinear mappings.
MSC:47H05, 47H09, 47H10.
1 Introduction
Let C be a nonempty subset of H. A mapping T:C\to H is said to be nonexpansive, if
A mapping T:C\to H is called αstrictly pseudocontractive in the terminology of Browder and Petryshyn [1] if for all x,y\in C there exists \alpha >0 such that
Without loss of generality, we may assume that \alpha \in (0,1). If I denotes the identity operator, then (1.2) can be rewritten as
A mapping T is called pseudocontractive if
Note that inequality (1.3) can be equivalently written as
Apart from their being an important generalization of nonexpansive mappings and αstrict pseudocontractive mappings, interest in pseudocontractive mappings stems mainly from their firm connection with the important class of nonlinear monotone mappings, where a mapping A with domain D(A) and range R(A) in H is called monotone if the inequality
holds for every x,y\in D(A) and for all s>0. We observe that A is monotone if and only if T:=IA is pseudocontractive, and thus a zero of A, N(A):=\{x\in D(A):Ax=0\}, is a fixed point of T, F(T):=\{x\in D(T):Tx=x\}. It is now well known (see, e.g., [2]) that if A is monotone then the solutions of the equation Ax=0 correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts, especially within the past 20 years or so, have been devoted to iterative methods for approximating fixed points of T when T is pseudocontractive (see, for example, [3–5] and the references contained therein).
The most general iterative algorithm for nonexpansive mappings studied by many authors is the following:
where {\{{\alpha}_{n}\}}_{n\ge 0}\subset (0,1) and satisfies the following additional assumptions: (i) {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0; (ii) {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, the sequence {\{{x}_{n}\}}_{n\ge 1} generated by (1.5) is generally referred to as the Mann iteration scheme in the light of Mann [6].
The Mann iteration process does not generally converge to a fixed point of T even when the fixed point exists. If, for example, C is a nonempty, closed, convex and bounded subset of a real Hilbert space, T:C\to C is nonexpansive, and the Mann iteration process is defined by (1.5) with (i) {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0; (ii) {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}, one can only prove that the sequence is an approximate fixed point sequence, that is, \parallel {x}_{n}T{x}_{n}\parallel \to 0 as n\to \mathrm{\infty}. To get the sequence {\{{x}_{n}\}}_{n\ge 1} to converge to a fixed point of T (when such a fixed point exists), some type of compactness condition must be additionally imposed either on C (e.g., C is compact) or on T.
Later, some authors tried to prove convergence of Mann iteration scheme to a fixed point of a much more general and important class of Lipschitz pseudocontractive mappings. But, in 2001, Chidume and Mutangadura [7] gave an example of a Lipschitz pseudocontractive selfmap of a compact convex subset of a Hilbert space with a unique fixed point for which no Mann sequence converges. Consequently, for this class of maps, the Mann sequence may not converge to a fixed point of Lipschitz pseudocontractive mappings even when C is a compact convex subset of H.
In 1974, Ishikawa [8] introduced an iteration process, which in some sense is more general than that of Mann and which converges to a fixed point of a Lipschitz pseudocontractive selfmap T of C. The following theorem is proved.
Theorem IS [8]
If C is a compact convex subset of a Hilbert space H, T:C\to C is a Lipschitz pseudocontractive mapping and {x}_{0} is any point of C, then the sequence \{{x}_{n}\} converges strongly to a fixed point of T, where \{{x}_{n}\} is defined iteratively for each integer n\ge 0 by
where \{{\alpha}_{n}\}, \{{\beta}_{n}\} are sequences of positive numbers satisfying the conditions:
The iteration method of Theorem IS, which is now referred to as the Ishikawa iterative method has been studied extensively by various authors. But it is still an open question whether or not this method can be employed to approximate fixed points of Lipschitz pseudocontractive mappings without the compactness assumption on C or T (see, e.g., [4, 9, 10]).
In order to obtain a strong convergence theorem for pseudocontractive mappings without the compactness assumption, Zhou [11] established the hybrid Ishikawa algorithm for Lipschitz pseudocontractive mappings as follows:
He proved that the sequence \{{x}_{n}\} defined by (1.8) converges strongly to {P}_{F(T)}{x}_{0}, where {P}_{C} is the metric projection from H into C. We observe that the iterative algorithm (1.7) generates a sequence \{{x}_{n}\} by projecting {x}_{0} onto the intersection of closed convex sets {C}_{n} and {Q}_{n} for each n\ge 0.
In 2009, Yao et al. [12] introduced the hybrid Mann algorithm as follows. Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T:C\to C be a LLipschitz pseudocontractive mapping such that F(T)\ne \mathrm{\varnothing}. Assume that the sequence \{{\alpha}_{n}\}\subset [a,b] for some a,b\in (0,\frac{1}{1+L}). Then for {C}_{1}=C and {x}_{1}={P}_{{C}_{1}}{x}_{0}, they proved that the sequence \{{x}_{n}\} defined by
converges strongly to {P}_{F(T)}{x}_{0}.
More recently, Tang et al. [13] generalized algorithm (1.8) to the hybrid Ishikawa iterative process. Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T:C\to C be a Lipschitz pseudocontractive mapping. Let \{{\alpha}_{n}\}, \{{\beta}_{n}\} be a sequence in [0,1]. Suppose that {x}_{0}\in H. For {C}_{1}=C and {x}_{1}={P}_{{C}_{1}}{x}_{0}, define a sequence \{{x}_{n}\} of C as follows:
Then they proved that the hybrid algorithm (1.9) strongly converges to a fixed point of Lipschitz pseudocontractive mappings. It is worth mentioning that the schemes in (1.7)(1.9) are not easy to compute. They involve computation of the intersection of {C}_{n} and {Q}_{n} for each n\ge 1.
Recently, Habtu Zegeye et al. [18] generalized algorithm (1.9) to Ishikawa iterative process (not hybrid) as follows. Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let {T}_{i}:C\to C, i=1,2,\dots ,N, be a finite family of Lipschitz pseudocontractive mappings with Lipschitzian constants {L}_{i}, for i=1,2,\dots ,N, respectively. Assume that the interior of F:={\bigcap}_{i=1}^{N}F({T}_{i}) is nonempty. Let \{{x}_{n}\} be a sequence generated from an arbitrary {x}_{0}\in C by
Under some conditions, \{{x}_{n}\} converges strongly to {x}^{\ast}\in F.
Our concern now is the following: Is it possible to construct a threestep iteration method and obtain a convergence theorem for a countable family of pseudocontractive mappings?
It is our purpose in this paper to construct a threestep iteration method and obtain the convergence theorem for a countable family of pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on one of the mappings or on C. The results obtained in this paper improve and extend the results of Theorem IS, Zhou [14], Yao et al. [12], Tang et al. [13] and Habtu Zegeye et al. [18].
2 Preliminaries
Let C be a nonempty subset of a real Hilbert space H. The mapping T:C\to H is called Lipschitz or Lipschitz continuous if there exists L>0 such that
If L=1, then T is called nonexpansive; and if L<1 then T is called a contraction. It is easy to see from Eq. (2.1) that every contraction mapping is nonexpansive and every nonexpansive mapping is Lipschitz.
A countable family of mapping {\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}:C\to H is called uniformly Lipschitz with Lipschitz constant {L}_{n}>0, n\ge 1, if there exists 0<L={sup}_{n\ge 1}{L}_{n} such that
A countable family of mapping {\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}:C\to H is called uniformly closed if {x}_{n}\to {x}^{\ast} and \parallel {x}_{n}{T}_{n}{x}_{n}\parallel \to 0 imply {x}^{\ast}\in {\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}).
In the sequel, we also need the following definition and lemma.
Let H be a real Hilbert space. The function \varphi :H\times H\to R defined by
is studied by Alber [15], Kamimula and Takahashi [16] and Riech [17].
It is obvious from the definition of the function ϕ that
The function ϕ also has the following property:
Lemma 2.1 Let H be a real Hilbert space. Then for all x,y\in H and \alpha \in [0,1] the following inequality holds:
Remark 2.2 We now give an example of a countable family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with the interior of the common fixed points nonempty. Suppose that X:=R and C:=[1,1]\subset R. Let {\{{T}_{n}\}}_{n\ge 1}:C\to C be defined by
Then we observe that F:={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})=[1,0], and hence the interior of the common fixed points is nonempty.
Now, we show that {\{{T}_{n}\}}_{n\ge 1} is a countable family of pseudocontractive mappings. Suppose that {C}_{1}=[1,0) and {C}_{2}=[0,1]. If x,y\in {C}_{1}, we have that
If x,y\in {C}_{2}, we have that
If x\in {C}_{1}, y\in {C}_{2}, considering (\frac{1}{{2}^{n}}\frac{1}{2})y(xy)>0, we have that
Therefore, from (2.5), (2.6) and (2.7) we obtain that {\{{T}_{n}\}}_{n\ge 1} is a countable family of pseudocontractive mappings.
Next, we show that {\{{T}_{n}\}}_{n\ge 1} is uniformly Lipschitz with Lipschitz constant L={sup}_{n\ge 1}{L}_{n}=2. If x,y\in {C}_{1}, we have that
If x,y\in {C}_{2}, we have that
If x\in {C}_{1}, y\in {C}_{2}, we have that
Therefore, from (2.8), (2.9) and (2.10) we obtain that {\{{T}_{n}\}}_{n\ge 1} is uniformly Lipschitz.
Finally, we show that {\{{T}_{n}\}}_{n\ge 1} is uniformly closed.
If there exists \{{x}_{n}\}\subset {C}_{1} such that {x}_{n}\to {x}^{\ast}\in [1,0], and {x}_{n}{T}_{n}{x}_{n}=0, we observe that [1,0]\subset F;
If there exists \{{x}_{n}\}\subset {C}_{2} such that {x}_{n}\to {x}^{\ast}\in [0,1], if and only if {x}^{\ast}=0, we have that {x}_{n}{T}_{n}{x}_{n}=0, it is obvious that 0\in F;
If there exists \{{x}_{n}\}\subset C:

(i)
∃N, as n>N, {x}_{n}\in {C}_{2}. The proof is the same as the proof of the second situation;

(ii)
∃N, as n>N, {x}_{n}\in {C}_{1}. The proof is the same as the proof of the first situation;

(iii)
\{{x}_{{n}_{k}}\}\subset {C}_{1}, \{{x}_{{n}_{j}}\}\subset {C}_{2}. If there exists {x}_{n}\to {x}^{\ast}, then we have that {x}^{\ast}=0. The proof is the same as the proof of the second situation. So, we can obtain that {\{{T}_{n}\}}_{n\ge 1} is uniformly closed.
3 Main results
Theorem 3.1 Let C be a nonempty, closed and convex subset of a real Hilbert space H, let {\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}:C\to C be a countable family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with Lipschitzian constants {L}_{n}, let L:={sup}_{n\ge 1}{L}_{n}. Assume that the interior of F:={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}) is nonempty. Let \{{x}_{n}\} be a sequence generated from an arbitrary {x}_{0}\in C by the following algorithm:
where \{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1) satisfying the following conditions: (i) {\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}, \mathrm{\forall}n\ge 0; (ii) {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}=\alpha >0; (iii) {sup}_{n\ge 1}{\gamma}_{n}\le \gamma with {\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1. Then \{{x}_{n}\} converges strongly to {x}^{\ast}\in F.
Proof Suppose that p\in F. Then from (3.1) and Lemma 2.1, we have that
In addition, using (3.1), we have that
Substituting (3.4) and (3.5) into (3.3), we obtain that
Since
Then, substituting (3.8) into (3.7), we obtain that
Substituting (3.6) and (3.9) into (3.2), we obtain that
Since from condition (iii), we have that
Again from condition (i), we have that {\alpha}_{n}{\beta}_{n}\le 0 and 2{\beta}_{n}{\gamma}_{n}1\le 0. So, inequality (3.10) implies that
Then
It is obviously that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel exists, then \{\parallel {x}_{n}p\parallel \} is bounded. This implies that \{{x}_{n}\}, \{{T}_{n}{x}_{n}\}, \{{z}_{n}\}, \{{T}_{n}{z}_{n}\}, \{{y}_{n}\} and \{{T}_{n}{y}_{n}\} are also bounded.
Furthermore, from (2.3), we have that
This implies that
Moreover, since the interior of F is nonempty, there exists {p}^{\ast}\in F and r>0 such that {p}^{\ast}+rh\in F whenever \parallel h\parallel \le 1. Thus, from the fact that \varphi (x,y)={\parallel xy\parallel}^{2}, and (3.12) and (3.13), we get that
Then from (3.13) and (3.14), we obtain that
and hence
Since h with \parallel h\parallel \le 1 is arbitrary, we have
So, if n>m, then we get that
But we know that \{\varphi ({p}^{\ast},{x}_{n})\} converges. Therefore, we obtain that \{{x}_{n}\} is a Cauchy sequence. Since C is closed subset of H, there exists {x}^{\ast}\in C such that
Furthermore, from (3.11) and conditions (i), (ii) and (iii), we get that
from which it follows that
Since {\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}} are uniformly closed, then from (3.15) and (3.16), we obtain that {x}^{\ast}\in {\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})=F. The proof is complete. □
Theorem 3.2 Let C be a nonempty, closed and convex subset of a real Hilbert space H, let {T}_{n}:C\to C be a finite family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with Lipschitzian constants {L}_{n}, n=1,2,\dots ,N. Assume that the interior of F:={\bigcap}_{n=1}^{N}F({T}_{n}) is nonempty. Let \{{x}_{n}\} be a sequence generated from an arbitrary {x}_{0}\in C by the following algorithm:
where {T}_{n}:={T}_{n(modN)} and \{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1) satisfying the following conditions: (i) {\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}, \mathrm{\forall}n\ge 0; (ii) {lim\hspace{0.17em}inf}_{n\mathrm{\infty}}{\alpha}_{n}=\alpha >0; (iii) {sup}_{n\ge 1}{\gamma}_{n}\le \gamma with {\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1 for L:=max\{{L}_{n}:n=1,2,\dots ,N\}. Then \{{x}_{n}\} converges strongly to {x}^{\ast}\in F.
If in Theorem 3.1, we consider a single Lipschitz pseudocontractive mapping, then we may change the conditions of Theorem 3.1.
Theorem 3.3 Let C be a nonempty, closed and convex subset of a real Hilbert space H, let T:C\to C be a Lipschitz pseudocontractive mappings with Lipschitzian constants L. Assume that the interior of F(T) is nonempty. Let \{{x}_{n}\} be a sequence generated from an arbitrary {x}_{0}\in C by the following algorithm:
where \{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1) satisfying the following conditions: (i) {\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}, \mathrm{\forall}n\ge 0; (ii) \sum {\alpha}_{n}{\beta}_{n}{\gamma}_{n}=\mathrm{\infty}; (iii) {sup}_{n\ge 1}{\gamma}_{n}\le \gamma with {\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1. Then \{{x}_{n}\} converges strongly to {x}^{\ast}\in F(T).
Proof Following the method of proof of Theorem 3.1, we obtain that {x}_{n}\to {x}^{\ast}\in C.
Furthermore, from (3.10) and conditions (i) and (ii), we obtain (3.11). From (3.11) and conditions (iii) and (iv), we obtain that
from which it follows that
and hence there exists a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that
Thus, {x}_{{n}_{k}}\to {x}^{\ast} and the continuity of T imply that {x}^{\ast}=T{x}^{\ast}, and hence {x}^{\ast}\in F(T). □
4 Applications
Theorem 4.1 Let H be a real Hilbert space, let {\{{A}_{n}\}}_{n=1}^{\mathrm{\infty}}:H\to H be a countable family of uniformly Lipschitz monotone mappings with Lipschitzian constants {L}_{n}, let L:={sup}_{n\ge 1}{L}_{n}. And if \parallel {A}_{n}{x}_{n}\parallel \to 0, {x}_{n}\to x, then x\in {\bigcap}_{n=1}^{\mathrm{\infty}}N({A}_{n}). Assume that the interior of F:={\bigcap}_{n=1}^{\mathrm{\infty}}N({A}_{n}) is nonempty. Let \{{x}_{n}\} be a sequence generated from an arbitrary {x}_{0}\in C by the following algorithm:
where \{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1) satisfying the following conditions: (i) {\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}, \mathrm{\forall}n\ge 0; (ii) {lim\hspace{0.17em}inf}_{n\mathrm{\infty}}{\alpha}_{n}=\alpha >0; (iii) {sup}_{n\ge 1}{\gamma}_{n}\le \gamma with {\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1. Then \{{x}_{n}\} converges strongly to {x}^{\ast}\in F.
Proof Suppose that {T}_{n}x:=(I{A}_{n})x for n\ge 1. Then we get that {\{{T}_{n}\}}_{n\ge 1} is a countable family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with {\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})={\bigcap}_{n=1}^{\mathrm{\infty}}N({A}_{n})\ne \mathrm{\varnothing}. Moreover, when {A}_{n} is replaced by I{T}_{n}, Scheme (4.1) reduces to Scheme (3.1) and hence the conclusion follows from Theorem 3.1. □
Corollary 4.2 Let H be a real Hilbert space, let {A}_{n}:H\to H be a finite family of uniformly Lipschitz monotone mappings with Lipschitzian constants {L}_{n}, n=1,2,\dots ,N. And if \parallel {A}_{n}{x}_{n}\parallel \to 0, {x}_{n}\to x, then x\in {\bigcap}_{n=1}^{N}N({A}_{n}). Assume that the interior of F:={\bigcap}_{n=1}^{N}N({A}_{n}) is nonempty. Let \{{x}_{n}\} be a sequence generated from an arbitrary {x}_{0}\in C by the following algorithm:
where {A}_{n}:={A}_{n(modN)} and \{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1) satisfying the following conditions: (i) {\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}, \mathrm{\forall}n\ge 0; (ii) {lim\hspace{0.17em}inf}_{n\mathrm{\infty}}{\alpha}_{n}=\alpha >0; (iii) {sup}_{n\ge 1}{\gamma}_{n}\le \gamma with {\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1 for L:=max\{{L}_{n}:n=1,2,\dots ,N\}. Then \{{x}_{n}\} converges strongly to {x}^{\ast}\in F.
Corollary 4.3 Let H be a real Hilbert space, let A:H\to H be a Lipschitz monotone mappings with Lipschitzian constants L. Assume that the interior of N(A) is nonempty. Let \{{x}_{n}\} be a sequence generated from an arbitrary {x}_{0}\in C by the following algorithm:
where \{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subset (0,1) satisfying the following conditions: (i) {\alpha}_{n}\le {\beta}_{n}\le {\gamma}_{n}, \mathrm{\forall}n\ge 0; (ii) \sum {\alpha}_{n}{\beta}_{n}{\gamma}_{n}=\mathrm{\infty}; (iii) {sup}_{n\ge 1}{\gamma}_{n}\le \gamma with {\gamma}^{3}{L}^{4}+2{\gamma}^{2}{L}^{3}+{\gamma}^{2}{L}^{2}+\gamma {L}^{2}+2\gamma <1. Then \{{x}_{n}\} converges strongly to {x}^{\ast}\in N(A).
Remark 4.4 In the paper [18], Scheme (2.28) of Theorem 2.5 and Scheme (2.29) of Corollary 2.6 are not correct, they are replaced by the following iterative algorithms, respectively.
and
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This project is supported by the National Natural Science Foundation of China under grant (11071279).
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Cheng, Q., Su, Y. & Zhang, J. Convergence theorems of a threestep iteration method for a countable family of pseudocontractive mappings. Fixed Point Theory Appl 2013, 100 (2013). https://doi.org/10.1186/168718122013100
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DOI: https://doi.org/10.1186/168718122013100
Keywords
 Lipschitz pseudocontractive mapping
 uniformly closed
 monotone mapping
 strong convergence
 common fixed point