Convergence results for the zero-finding problem and fixed points of nonexpansive semigroups and strict pseudocontractions

In this work, we establish strong convergence theorems for solving the fixed point problem of nonexpansive semigroups and strict pseudocontractions, and the zero-finding problem of maximal monotone operators in a Hilbert space. We further apply our result to the convex minimization problem and commutative semigroups.

MSC: 47H09, 47H10.

Let H be a real Hilbert space and K a nonempty, closed, and convex subset of H. Let be a nonlinear mapping. Then T is said to be nonexpansive if for all . The fixed points set of T is denoted by .

In 1953, Mann [21] introduced the following classical iteration for a nonexpansive mapping in a real Hilbert space: and

where .

In 1967, Halpern [13] introduced another classical iteration for a nonexpansive mapping in a real Hilbert space: and

where and is fixed.

Let be a contraction (i.e., for all and ). In 2000, Moudafi [25] introduced the viscosity approximation method for a nonexpansive mapping T as follows: and

where . It was proved, in a Hilbert space that the sequence generated by (1.2) strongly converges to a fixed point of T under suitable conditions.

Let A be a strongly positive bounded linear operator on H: that is, there is a constant with property

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:

where K is the fixed point set of a nonexpansive mapping T on H and b is a given point in H.

Recently, Marino-Xu [22] introduced the following general iterative method for a nonexpansive mapping T in a Hilbert space: and

where , f is a contraction and A is a strongly positive bounded linear operator.

Since then, there have been a number of modified viscosity approximation methods for nonexpansive mappings or nonexpansive semigroups (see, for example, [6,7,9,26,32,35,38,42,43]).

Recall that is called a κ-strict pseudocontraction if there exists a constant such that

for all . It is known that (1.3) is equivalent to the following:

for all .

The class of strict pseudocontractions was introduced, in 1967, by Browder-Petryshyn [3]. The existence and weak convergence theorems were proved in a real Hilbert space by using Mann iterative algorithm (1.1) with a constant sequence for all . Recently, Marino-Xu [23] and Zhou [44] extended the results of Browder-Petryshyn [3] to Mann’s iteration process (1.1). Since 1967, the study of fixed points for strict pseudocontractions has been investigated by many authors (see, e.g., [1,28]).

A set-valued mapping is called monotone if for all , , and imply . A monotone mapping M is maximal if its graph of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for , for all imply . Let , be the resolvent of M. It is well known that is single-valued and for any . For each , the Yosida approximation of M is defined by . We know that for all and .

A fundamental problem of monotone operators is that of finding an element x such that . Such a problem is called the zero-finding problem (denoted by the set of solutions) and also includes many concrete examples, such as convex programming and monotone variational inequalities. It is known that if is a proper lower semicontinuous convex function, then ∂g is maximal monotone and the equation is reduced to (see [29,30]).

Initiated by Martinet [24], Rockafellar [30] introduced the following iterative scheme: and

where and M is a maximal monotone operator on H. Such an algorithm is called the proximal point algorithm. It was proved that the sequence generated by (1.4) converges weakly to an element in if .

The convergence of the zero-finding problem of monotone operators has been studied by many authors in several setting (see, for example, [8,10,14,15,27,34]).

In this work, motivated by Lau et al.[16-20], Marino-Xu [22], and Saeidi [32], we introduce a new general iterative scheme for solving the fixed- point problem of a nonexpansive semigroup involving a strict pseudocontraction and the zero-finding problem of a maximal monotone operator in the framework of a Hilbert space. Some applications concerning the convex minimization problem and commutative semigroups are also presented.

In this section, we state some preliminaries and lemmas which will be used in the sequel.

Let S be a semigroup. We denote by the Banach space of all bounded real-valued functionals on S with supremum norm. For each , we define the left and right translation operators and on by

for each and , respectively. Let X be a subspace of containing 1. An element μ in the dual space of X is said to be a mean on X if . It is well known that μ is a mean on X if and only if

for each . We often write instead of for and .

Let X be a translation invariant subspace of (i.e., and for each ) containing 1. Then a mean μ on X is said to be left invariant (resp. right invariant) if (resp. ) for each and . A mean μ on X is said to be invariant if μ is both left and right invariant [16-18]. S is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. S is a amenable if S is left and right amenable. In this case, also has an invariant mean. It is known that is amenable when S is commutative semigroup or solvable group. However, the free group or semigroup of two generators is not left or right amenable (see [11,20]). A net of means on X is said to be left regular[11] if

for each , where is the adjoint operator of .

Let K be a nonempty, closed, and convex subset of H. A family is called a nonexpansive semigroup on K if for each , the mapping is nonexpansive and for each . We denote by the set of common fixed points of , i.e.,

Throughout this article, we denote the open ball of radius r centered at 0 by and also denote the closed and convex hull of by . For and a mapping , the set of ε-approximate fixed points of T will be denoted by , i.e..

The following lemmas are important in order to prove our main theorem.

Lemma 2.1[20,31,39]

Letfbe a function of a semigroupSinto a Banach spaceEsuch that the weak closure ofis weakly compact and letXbe a subspace ofcontaining all the functionswith. Then, for any, there exists a unique elementinEsuch that

for all. Moreover, ifμis a mean onXthen

We can writeby.

Lemma 2.2[20,31,39]

LetKbe a closed and convex subset of a Hilbert spaceH, be a nonexpansive semigroup fromKintoKsuch thatandXbe a subspace ofcontaining 1 and the mappingbe an element ofXfor eachand, andμbe a mean onX.

If we writeinstead of, then the following hold:

(i) is a nonexpansive mapping fromKintoK;

(ii) for each;

(iii) for each;

(iv) ifμis left invariant, thenis a nonexpansive retraction fromKonto.

Let K be a nonempty, closed, and convex subset of a real Hilbert space H. Then, for any , there exists a unique nearest point in K, denoted by , such that

for all . Such a projection is called the metric projection of H onto K. We also know that for and , if and only if

We know the following subdifferential inequality.

Lemma 2.3For all, there holds the inequality

Lemma 2.4[22]

LetAbe a strongly positive bounded linear operator on a Hilbert spaceHwith coefficientand. Then.

In the sequel, we need the following crucial lemmas.

Lemma 2.5[41]

Assumeis a sequence of nonnegative real numbers such that

whereis a sequence inandis a sequence insuch that

(a) ;

(b) or.

Then.

Lemma 2.6[36]

Letandbe bounded sequences in a Banach spaceEsuch that

whereis a real sequence inwith. If, then.

The following crucial results can be found in [1].

Lemma 2.7[1]

LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceHand letbe aκ-strict pseudocontraction such that, thenis demiclosed at zero, that is, for all sequencewithandit follows that.

Lemma 2.8[1]

LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceHand let () be a family of-strict pseudocontractions for some. Assumeis a positive sequence such that. Thenis aκ-strict pseudocontraction with. Moreover, ifhas a common fixed point, then.

Lemma 2.9[40]

Let the resolventbe defined by, . Then the following holds:

for alland.

In this section, we are now ready to prove our main theorem.

Theorem 3.1LetHbe a real Hilbert space anda nonexpansive semigroup onH. Letbe a maximal monotone operator andaκ-strict pseudocontraction such that. LetXbe a left invariant subspace ofsuch that, and the functionis an element ofXfor each. Letbe a left regular sequence of means onXsuch that. Letfbe anα-contraction onHandAa strongly positive bounded linear operator with coefficient. Letβandγbe real numbers such thatand. Letbe generated byand

where, andsatisfying the conditions:

(C1) and;

(C2) ;

(C3) ;

(C4) and.

Thenconverges strongly towhich also solves the following variational inequality:

Proof Since , we shall assume that and . So by Lemma 2.4, we have .

First, we show that is bounded. Let . Put for all . Then

which yields

Moreover, since is firmly nonexpansive,

From (3.3), we have

By an induction, we can show that

Therefore, is bounded. So are , , , and .

We next show that

Observe that

Indeed,

Since is bounded and , (3.4) holds.

For each , define . Then is nonexpansive, and hence

for some big enough constant .

On the other hand, since and ,

Put . Then

which implies

Substituting (3.5) and (3.6) into (3.7), we obtain

Using Lemma 2.9, (3.4), (C1), (C2), and (C4), we have

From Lemma 2.6, we derive

It also follows that

We next show that

Put

Set . Then D is a nonempty bounded closed convex set. Moreover, , , and are in D. To complete our proof, we follow the proof line as in [2] (see also [19,20,33]). Let . From [5], there exists such that

From Corollary 1.1 in [5], there exists a natural number N such that

for all and . Let . Since is left regular, there exists such that

for all and . So we have for all

Observe, by Lemma 2.2

Combining (3.10)-(3.12), we derive

for all and . Let and . Then there exists which satisfies (3.9). Observe

Since and , there exists such that

for all . Hence, . Since is arbitrary,

We next show that

Since is firmly nonexpansive and ,

which implies

Therefore,

which yields

for some . Thus, (3.15) holds by (3.8) and .

We next show that

From (3.2), we have

So, we obtain

It follows that

From (C1) and (C3), we conclude that (3.16) holds. Moreover, we get that

It is easy to see that is a contraction. So, by Banach’s contraction principle, there exists a unique point p which satisfies the following variational inequality:

We next show that

To this end, we choose a subsequence of such that

Since is bounded and H is reflexive, there exists a point such that . From (3.15) and (3.17), there exists a corresponding subsequence of (resp. of ) such that (resp. ).

We next show that . Since ,

From (3.15) and , we have

Noting that , by the monotonicity of M, we have

for all . So we obtain

for all . Hence, by the maximality of M.

On the other hand, from (3.14), we get that by the demiclosedness of a nonexpansive mapping [4,12]. Applying Lemma 2.7 to (3.16), we also get that . This shows that , and hence

We finally show that as . From Lemmas 2.3 and 2.4, we have

It follows that

From (3.19) and (C1), we can apply Lemma 2.5 to conclude that as . This completes the proof. □

From Rockafellar’s theorem [29,30], we next apply our result to the convex minimization problem in a Hilbert space.

Corollary 3.2LetHbe a real Hilbert space anda nonexpansive semigroup onH. Letbe a proper lower semi-continuous convex function andaκ-strict pseudocontraction such that. LetXbe a left invariant subspace ofsuch that, and the functionis an element ofXfor each. Letbe a left regular sequence of means onXsuch that. Letfbe anα-contraction onHandAa strongly positive bounded linear operator with coefficient. Let, β, γ, andbe as in Theorem 3.1. Then the sequencegenerated byand

converges strongly towhich also solves the variational inequality (3.1).

Using Lemma 2.8, we next apply our result to a finite family of strict pseudocontractions in a Hilbert space.

Corollary 3.3LetHbe a real Hilbert space anda nonexpansive semigroup onH. Letbe a maximal monotone operator anda family of-strict pseudocontractions such that. Let. LetXbe a left invariant subspace ofsuch that, and the functionis an element ofXfor each. Letbe a left regular sequence of means onXsuch that. Letfbe anα-contraction onHandAa strongly positive bounded linear operator with coefficient. Let, β, γ, andbe as in Theorem 3.1 andwith. Then the sequencegenerated byand

converges strongly towhich also solves the variational inequality (3.1).

Using the results proved in [37] (see also [19]), we obtain the following corollaries.

Corollary 3.4LetHbe a real Hilbert space. Letandbe nonexpansive mappings onHwith. Letbe a maximal monotone operator and letbe aκ-strict pseudocontraction such that. Letfbe anα-contraction onHandAa strongly positive bounded linear operator with coefficient . Let, β, γ, , andbe as in Theorem 3.1. Then the sequencegenerated byand

converges strongly towhich also solves the variational inequality (3.1).

Corollary 3.5LetHbe a real Hilbert space. Letbe a strongly continuous nonexpansive semigroup onH. Letbe a maximal monotone operator andaκ-strict pseudocontraction such that. Letfbe anα-contraction onHandAa strongly positive bounded linear operator with coefficient . Let, β, γ, , andbe as in Theorem 3.1. Then the sequencegenerated byand

whereis an increasing sequence insuch thatand, converges strongly towhich also solves the variational inequality (3.1).

Corollary 3.6LetHbe a real Hilbert space. Letbe a strongly continuous nonexpansive semigroup onH. Letbe a maximal monotone operator andaκ-strict pseudocontraction such that. Letfbe anα-contraction onHandAa strongly positive bounded linear operator with coefficient . Let, β, γ, andbe as in Theorem 3.1. Then the sequencegenerated byand

whereis a decreasing sequence insuch that, converges strongly towhich also solves the variational inequality (3.1).

The authors declare that they have no competing interests.

The author wishes to thank Professor Anthony To-Ming Lau for the hospitality and guidance when stayed in University of Alberta during Spring/Summer 2011 and Professor Suthep Suantai for the valuable suggestion. The author was supported by the Thailand Research Fund, the Commission on Higher Education, and University of Phayao under Grant MRG5580016.

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