© Hong Gang Li et al. 2010
Received: 30 April 2010
Accepted: 8 June 2010
Published: 28 June 2010
A general nonlinear framework for an Ishikawa-hybrid proximal point algorithm using the notion of -accretive is developed. Convergence analysis for the algorithm of solving a nonlinear set-valued inclusions problem and existence analysis of solution for the nonlinear set-valued inclusions problem are explored along with some results on the resolvent operator corresponding to -accretive mapping due to Lan-Cho-Verma in Banach space. The result that sequence generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem with the convergence rate is proved.
The set-valued inclusions problem, which was introduced and studied by Di Bella , Huang et al. , and Jeong , is a useful extension of the mathematics analysis. And the variational inclusion(inequality) is an important context in the set-valued inclusions problem. It provides us with a unified, natural, novel, innovative, and general technique to study a wide class of problems arising in different branches of mathematical and engineering sciences. Various variational inclusions have been intensively studied in recent years. Ding and Luo, Verma , Huang , Fang and Huang , Lan et al. , Fang et al. , and Zhang et al.  introduced the concepts of -subdifferential operators, maximal -monotone operators, -monotone operators, -monotone operators, -monotone operators, -accretive mappings, -monotone operators, and defined resolvent operators associated with them, respectively. Moreover, by using the resolvent operator technique, many authors constructed some approximation algorithms for some nonlinear variational inclusions in Hilbert spaces or Banach spaces. Recently, Verma has developed a hybrid version of the Eckstein and Bertsekas  proximal point algorithm, introduced the algorithm based on the -maximal monotonicity framework , and studied convergence of the algorithm.
On the other hand, in 2008, Li  studied the existence of solutions and the stability of perturbed Ishikawa iterative algorithm for nonlinear mixed quasivariational inclusions involving -accretive mappings in Banach spaces by using the resolvent operator technique in .
Inspired and motivated by recent research work in this field, in this paper, a general nonlinear framework for a Ishikawa-hybrid proximal point algorithm using the notion of -accretive is developed. Convergence analysis for the algorithm of solving a nonlinear set-valued inclusions problem and existence analysis of solution for the nonlinear set-valued inclusions problem are explored along with some results on the resolvent operator corresponding to -accretive mapping due to Lan et al. in Banach space. The result that sequence generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem as the convergence rate is proved.
In particular, is the usual normalized duality mapping, and (for all ). If is strictly convex , or is uniformly smooth Banach space, then is single valued. In what follows we always denote the single-valued generalized duality mapping by in real uniformly smooth Banach space unless otherwise stated.
A special case of problem (2.5) is the following.
(i)If is a Hilbert space, is the zero operator in , is the identity operator in , and , then problem (2.5) becomes the parametric usual variational inclusion with a -maximal monotone mapping , which was studied by Verma .
(ii)If is a real Banach space, is the identity operator in , and , then problem (2.5) becomes the parametric usual variational inclusion with a -accretive mapping, which was studied by Li .
Let us recall the following results and concepts.
Based on the literature , we can define the resolvent operator as follows.
Definition 2.7 (see ).
The -accretive mappings are more general than -monotone mappings and -accretive mappings in Banach space or Hilbert space, and the resolvent operators associated with -accretive mappings include as special cases the corresponding resolvent operators associated with -monotone operators, -accretive mappings, -monotone operators, -subdifferential operators [3–14, 16, 17].
Lemma 2.9 (see ).
In the study of characteristic inequalities in -uniformly smooth Banach spaces, Xu  proved the following result.
Lemma 2.10 (see ).
3. The Existence of Solutions
Now, we are studing the existence for solutions of problem (2.5).
4. Ishikawa-Hybrid Proximal Point Algorithm
Based on Lemma 3.1, we develop an Ishikawa-hybrid proximal point algorithm for finding an iterative sequence solving problem (2.5) as follows.
For a suitable choice of the mappings , space , and nonnegative sequences , , Algorithm 4.1 can be degenerated to a number of algorithms involving many known algorithms which are due to classes of variational inequalities and variational inclusions [12–14].
and the convergence rate is .By (4.4), if , then it follows that and . Therefor, the sequence generated hybrid proximal point Algorithm 4.1 converges linearly to a solution of problem (2.5) with convergence rate . This completes the proof.
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