Generalized Strongly Nonlinear Implicit Quasivariational Inequalities
© Salahuddin and M. K. Ahmad. 2009
Received: 11 February 2009
Accepted: 17 June 2009
Published: 29 July 2009
Variational inequality was initially studied by Stampacchia  in 1964. Since then, it has been extensively studied because of its crucial role in the study of mechanics, physics, economics, transportation and engineering sciences, and optimization and control. Thanks to its wide applications, the classical variational inequality has been well studied and generalized in various directions. For details, readers are referred to [2–5] and the references therein.
It is known that one of the most important and difficult problems in variational inequality theory is the development of an efficient and implementable approximation schemes for solving various classes of variational inequalities and variational inclusions. Recently, Huang [6–8] and Cho et al.  constructed some new perturbed iterative algorithms for approximation of solutions of some generalized nonlinear implicit quasi-variational inclusions (inequalities), which include many iterative algorithms for variational and quasi-variational inclusions (inequalities) as special cases. Inspired and motivated by recent research works [1, 9–19], we prove an existence theorem for solution of generalized strongly nonlinear implicit quasi-variational inequality problems and convergence of iterative sequences with errors, involving Lipschitzian, generalized pseudocontractivity and generalized -pseudocontractive mappings in Hilbert spaces.
Let be a real Hilbert space with norm and inner product . For a nonempty closed convex subset , let be the projection of onto . Let be a set valued mapping with nonempty closed convex values, and be the mappings. We consider the following problem.
The problem (2.1) is called the generalized strongly nonlinear implicit quasi-variational inequality problem.
For our main results, we need the following lemmas.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
3. Main Results
In this section, we establish an existence theorem for solution of generalized strongly nonlinear implicit quasi-variational inequality problems and convergence of the iterative sequences generated by (2.18). First, we give some definitions.
It is easy to check that (3.1) is equivalent to
For in (3.1), we get the usual concept of pseudo-contractive of , introduced by Browder and Petryshyn , that is,
In a similar way, we can define Lipschitz continuity of N with respect to the second and third arguments.
Now, we give the main result of this paper.
Let be a real Hilbert space and a set-valued mapping with nonempty closed convex values. Let be the Lipschitz continuous mappings with positive constants and respectively. Let be the mapping such that and are Lipschitz continuous with positive constants and respectively. A trimapping is generalized pseudo-contractive with respect to in the first argument of with constant and generalized -pseudo-contractive with respect to in the second argument of with constant , Lipschitz continuous with respect to the first, second, and third arguments with positive constants respectively. Suppose that is Lipschitz continuous with constant . Let , and be the three bounded sequences in and , , , , , , , and are sequences in satisfying the following conditions:
If the following conditions hold:
It follows from (3.13)–(3.16) that
From (3.17) and (3.19), it follows that
Similarly, we have
Similarly, we deduce from (3.21) the following:
From the above inequalities, we get
We now deduce Theorem 3.6 in the direction of Ishikawa iteration.
Let be a real Hilbert space and a set-valued mapping with the nonempty closed convex values. Let and be the same as in Theorem 3.6. Suppose that is Lipschitz continuous with constant . Let and be the two bounded sequences in and , , , , and be six sequences in satisfying the following conditions:
If the following conditions holds:
We can also deduce Theorem 3.6 in the direction of (2.16).
If the conditions of (3.10) hold, then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and as , where is the Mann iterative process with errors defined as follows:
Our results can be further improved in the direction of (2.25).
Now, we deduce Theorem 3.6 for three step iterative process in terms of (2.10).
Next, we state the results in terms of iterations (2.10) and (2.25).
Theorem 3.13 can also be deduce for Ishikawa and Mann iterative process.
The authors thank the editor Professor R. U. Verma and anonymous referees for their valuable useful suggestions that improved the paper.
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