Let E be a real Banach space, and let be the dual of E. For some real number q (), the generalized duality mapping is defined by
(1.1)
where denotes the duality pairing between elements of E and those of . In particular, is called the normalized duality mapping and for . If E is a real Hilbert space, then , where I is the identity mapping. It is well known that if E is smooth, then is single-valued, which is denoted by .
Let C be a nonempty closed convex subset of E, and let be a nonlinear map. Then the variational inequality problem with respect to C and G is to find a point such that
(1.2)
We denote by the set of solutions of this variational inequality problem.
If , a real Hilbert space, the variational inequality problem reduces to the following: Find a point such that
(1.3)
A mapping is said to be a contraction if, for some ,
(1.4)
The map T is said to be nonexpansive if
(1.5)
The map T is said to be L-Lipschitzian if there exists such that
(1.6)
A point is called a fixed point of the map T if . We denote by the set of all fixed points of the mapping T, that is,
We assume that in the sequel. It is well known that above is closed and convex (see, e.g., Goebel and Kirk [1]).
An operator is said to be accretive if , there exists such that
(1.7)
For some positive real numbers η, λ, the mapping F is said to be η-strongly accretive if for any , there exists such that
(1.8)
and it is called λ-strictly pseudocontractive if
(1.9)
It is clear that (1.9) is equivalent to the following:
(1.10)
where I denotes the identity operator.
In Hilbert spaces, accretive operators are called monotone where inequality (1.7) holds with replaced by the identity map of H.
A bounded linear operator A on H is called strongly positive with coefficient γ if there is a constant with the property
Let K be a nonempty closed convex and bounded subset of a Banach space E, and let the diameter of K be defined by . For each , let , and let denote the Chebyshev radius of K relative to itself. The normal structure coefficient of E (see, e.g., [2]) is defined by . A space E, such that , is said to have a uniform normal structure.
It is known that all uniformly convex and uniformly smooth Banach spaces have a uniform normal structure (see, e.g., [3, 4]).
Let μ be a continuous linear functional on and . We write instead of . We call μ a Banach limit if μ satisfies and for all . If μ is a Banach limit, then
for all (see, e.g., [3, 5]).
Let denote the unit sphere of a real Banach space E.
The space E is said to have a Gâteaux differentiable norm if the limit
(1.11)
exists for each . In this case, E is called smooth. E is said to be uniformly smooth if the limit (1.11) exists and is attained uniformly in . E is said to have a uniformly Gâteaux differentiable norm if, for any , the limit (1.11) exists uniformly for all .
The modulus of smoothness of E, with , is a function defined by
A Banach space E is said to be uniformly smooth if , and for , E is said to be q-uniformly smooth if there exists a fixed constant such that , .
It is well known (see, e.g., [6]) that Hilbert spaces, (or ) spaces () and Sobolev spaces, () are all uniformly smooth. More precisely, Hilbert spaces are 2-uniformly smooth, while
Also, it is well known (see, e.g., [7]) that q-uniformly smooth Banach spaces have a uniformly Gâteaux differentiable norm.
The variational inequality problem was initially introduced and studied by Stampacchia [8] in 1964. In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics and optimization. Thus, the problem of solving a variational inequality of the form (1.2) has been intensively studied by numerous authors. Iterative methods for approximating fixed points of nonexpansive mappings and their generalizations, which solve some variational inequality problems, have been studied by a number of authors (see, for example, [9–13] and the references therein).
Let H be a real Hilbert space. In 2001, Yamada [13] proposed a hybrid steepest descent method for solving variational inequality as follows: Let be chosen arbitrarily and define a sequence by
(1.12)
where T is a nonexpansive mapping on H, F is L-Lipschitzian and η-strongly monotone with , , . If is a sequence in satisfying the following conditions:
-
(C1)
,
-
(C2)
,
-
(C3)
either or ,
then he proved that the sequence converges strongly to the unique solution of the variational inequality
Besides, he also proposed the cyclic algorithm
where ; he also proved strong convergence theorems for the cyclic algorithm.
In 2006, Marino and Xu [10] considered the following general iterative method: Starting with an arbitrary initial point , define a sequence by
(1.13)
where T is a nonexpansive mapping of H, f is a contraction, A is a linear bounded strongly positive operator, and is a sequence in satisfying the following conditions:
-
(M1)
;
-
(M2)
;
-
(M3)
or .
They proved that the sequence converges strongly to a fixed point of T, which solves the variational inequality
In 2010, Tian [11] combined the iterative method (1.13) with Yamada’s iterative method (1.12) and considered the following general iterative method:
(1.14)
where T is a nonexpansive mapping on H, f is a contraction, F is k-Lipschitzian and η-strongly monotone with , , . He proved that if the sequence of parameters satisfies conditions (M1)-(M3), then the sequence generated by (1.14) converges strongly to a fixed point of T, which solves the variational inequality
Very recently, in 2011, Tian and Di [12] studied two algorithms, based on Tian’s [11] general iterative algorithm, and proved the following theorems.
Theorem 1.1 (Synchronal algorithm)
Let H be a real Hilbert space, and let be a -strictly pseudocontraction for some () such that , and f be a contraction with coefficient and be a positive constant such that . Let be an η-strongly monotone and L-Lipschitzian operator with , . Assume that , . Let be chosen arbitrarily, and let and be sequences in satisfying the following conditions:
-
(N1)
, ;
-
(N2)
, ;
-
(N3)
, .
Let
be a sequence defined by the composite process
Then converges strongly to a common fixed point of , which solves the variational inequality
(1.15)
Theorem 1.2 (Cyclic algorithm)
Let H be a real Hilbert space, and let be a -strictly pseudocontraction for some () such that , and f be a contraction with coefficient . Let be an η-strongly monotone and L-Lipschitzian operator with , . Assume that , . Let be chosen arbitrarily, and let and be sequences in satisfying the following conditions:
-
(N1′) , ;
-
(N2′) or ;
-
(N3′) , , where .
Let
be a sequence defined by the composite process
where , with , , namely is one of cyclically. Then converges strongly to a common fixed point of , which solves the variational inequality (1.15).
In this paper, we study the synchronal and cyclic algorithms for finding a common fixed point of finite strictly pseudocontractive mappings, which solves the variational inequality
(1.16)
where f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, is a positive integer, are arbitrary fixed constants, and are N-strict pseudocontractions defined on a closed convex subset C of a real q-uniformly smooth Banach space E whose norm is uniformly Gâteaux differentiable.
Let T be defined by
where such that . We will show that a sequence generated by the following synchronal algorithm:
(1.17)
converges strongly to a solution of problem (1.16).
Another approach to problem (1.16) is the cyclic algorithm. For each , let , where the constant satisfies . Beginning with , define a sequence cyclically by
Indeed, the algorithm can be written in a compact form as follows:
(1.18)
where , with , , namely is one of cyclically. We will show that (1.18) is also strongly convergent to a solution of problem (1.16) if the sequences and of parameters are appropriately chosen.
Motivated by the results of Tian and Di [12], in this paper we aim to continue the study of fixed point problems and prove new theorems for the solution of variational inequality problems in the framework of a real Banach space, which is much more general than that of Hilbert.
Throughout this research work, we will use the following notations:
-
1.
⇀ for weak convergence and → for strong convergence.
-
2.
denotes the weak ω-limit set of .