On Solvability of a Generalized Nonlinear Variational-Like Inequality
© Zeqing Liu et al. 2009
Received: 12 July 2009
Accepted: 28 September 2009
Published: 11 October 2009
A new generalized nonlinear variational-like inequality is introduced and studied. By applying the auxiliary principle technique and KKM theory, we construct a new iterative algorithm for solving the generalized nonlinear variational-like inequality. By means of the Banach fixed-point theorem, we establish the existence and uniqueness of solution for the generalized nonlinear variational-like inequality. The convergence of the sequence generated by the iterative algorithm is also discussed.
It is well known that variational inequality theory has become a very effective and powerful tool for studying a wide range of problems arising in many diverse fields. One of the most interesting and important problems in the variational inequality theory is the development of an efficient iterative algorithm to compute approximate solutions. The researchers in [1–10] suggested a lot of iterative algorithms for solving various variational inequalities and variational-like inequalities. By using the auxiliary principle technique, Ding and Yao , Ding et al. , Huang and Deng , Liu et al. [8, 9], and others studied several classes of nonlinear variational inequalities and variational-like inequalities in reflexive Banach and Hilbert spaces, respectively, suggested some iterative algorithms to compute approximate solutions for these nonlinear variational inequalities and variational-like inequalities, and proved the existence of solutions for the nonlinear variational inequalities and variational-like inequalities involving different monotone mappings.
Motivated and inspired by the research work in [1–10], we introduce and study a generalized nonlinear variational-like inequality. By using the auxiliary principle technique and KKM theorem due to Zhang and Xiang , we suggest a new iterative scheme for solving the generalized nonlinear variational-like inequality. Utilizing the Banach fixed-point theorem, we prove the existence and uniqueness of solution for the generalized nonlinear variational-like inequality. Under certain conditions, we discuss the convergence of the iterative sequence generalized by the iterative algorithm.
Throughout this paper, let be a real Hilbert space endowed with an inner product and a norm , respectively, and . Let be a nonempty closed convex subset of . Assume that is a coercive continuous bilinear form, that is, there exist positive constants such that
It is clear that for appropriate and suitable choices of the mappings and , the generalized nonlinear variational-like inequality (2.3) includes some variational inequalities and variational-like inequalities in [1–10] as special cases.
Recall the following concepts and results.
Lemma 2.4 (see ).
Lemma 2.5 (see ).
3. Auxiliary Problem and Algorithm
The proof of the below result is similar to that of Theorem 3.1 and is omitted.
Based on Theorems 3.1 and 3.2, we suggest the following iterative algorithm with errors for solving the generalized nonlinear variational-like inequality (2.3).
4. Existence and Convergence
In this section, we prove the existence of solution for the generalized nonlinear variational-like inequality (2.3) and discuss the convergence of the sequence generated by Algorithm 3.3.
As in the proof of Theorem 4.1, we have the following theorem.
The conditions of Theorems 3.1, 3.2, 4.1, and 4.2 are different from the conditions of the results in [1–10]. In particular, the mappings and with respect to the second argument in Theorems 3.2 and 4.2 are Lipschitz continuous, but other mappings in Theorems 3.2 and 4.2 are not Lipschitz continuous.
This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (2009A419) and the Korea Research Foundation(KRF) Grant funded by the Korea government (MEST) (2009-0073655).
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