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ABSTRACT: The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically
strict pseudocontractive mapping in the intermediate sense and the set of solutions of the variational inequality problem
for a monotone, Lipschitz continuous mapping. We introduce a modified hybrid Mann iterative scheme with perturbed mapping
which is based on well-known CQ method, Mann iteration method and hybrid (or outer approximation) method. We establish a strong
convergence theorem for three sequences generated by this modified hybrid Mann iterative scheme with perturbed mapping. Utilizing
this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically
strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive
mappings.
KeywordsModified hybrid Mann iterative scheme with perturbed mapping–Variational inequality–Asymptotically strict pseudocontractive mapping in the intermediate sense–Fixed point–Monotone mapping–Strong convergence–Demiclosedness principle
Acta Applicandae Mathematicae 04/2012; 115(2):167-191. · 0.90 Impact Factor
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ABSTRACT: The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex
subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent method for this class of variational
inequalities is introduced. Strong convergence of this method is established under suitable assumptions imposed on the algorithm
parameters.
Applied Mathematics and Mechanics 04/2012; 28(8):1029-1036. · 0.56 Impact Factor
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ABSTRACT: In this paper, we introduce and study a relaxed extragradient method for finding solutions of a general system of variational
inequalities with inverse-strongly monotone mappings in a real Hilbert space. First, this system of variational inequalities
is proven to be equivalent to a fixed point problem of nonexpansive mapping. Second, by using the demi-closedness principle
for nonexpansive mappings, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient
method converges strongly to a solution of this system of variational inequalities. In addition, utilizing this result, we
provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.
Mathematical Methods of Operational Research 04/2012; 67(3):375-390. · 0.48 Impact Factor
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J. Applied Mathematics. 01/2012; 2012.
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ABSTRACT: It is well known that the convexity of functions plays a vital role in mathematical economics, engineering, management, optimization
theory, etc. This concept in linear spaces relies on the possibility of connecting any two points of the space by the line
segment between them. Since convexity is often not enjoyed the real problems, several classes of functions have been defined
and studied for the purpose of weakening the limitations of convexity. In 1981, Hanson [33] realized that the convexity requirement,
utilized to prove sufficient optimality conditions for a differentiable mathematical programming problem, can be further weakened
by substituting the linear term y − x appearing in the definition of differentiable convex, pseudoconvex and quasiconvex functions with an arbitrary vector-valued
function. In view of this idea, Hanson [33] (see also Craven [12]) introduced the concept of invexity by replacing the linear
term y − x in the definition of convex function by a vector-valued function η(y, x).
09/2011: pages 221-274;
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Applied Mathematics and Computation. 01/2011; 218:1112-1123.
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Computers & Mathematics with Applications. 01/2011; 61:2468-2479.
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Computers & Mathematics with Applications. 01/2011; 61:2447-2455.
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J. Optimization Theory and Applications. 01/2011; 151:489-512.
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ABSTRACT: The purpose of this article is to investigate the problem of finding a common element of the set of fixed points of a non-expansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping. We introduce a hybrid Mann iterative scheme with perturbed mapping which is based on the well-known Mann iteration method and hybrid (or outer approximation) method. We establish a strong convergence theorem for three sequences generated by this hybrid Mann iterative scheme with perturbed mapping. Utilizing this theorem, we also construct an iterative process for finding a common fixed point of two mappings, one of which is non-expansive and the other taken from the more general class of Lipschitz pseudocontractive mappings.
Optimization 08/2010; 59(6):929-944. · 0.50 Impact Factor
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J. Computational Applied Mathematics. 01/2010; 233:2902-2915.
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Computers & Mathematics with Applications. 01/2010; 59:1441-1452.
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Computers & Mathematics with Applications. 01/2009; 57:1682-1691.
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Applied Mathematics and Computation. 01/2009; 209:162-176.
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J. Global Optimization. 01/2009; 43:487-502.
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Appl. Math. Lett. 01/2009; 22:197-201.
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Computers & Mathematics with Applications. 01/2009; 58:605-617.
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Appl. Math. Lett. 01/2009; 22:428-434.
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ABSTRACT: In this article, we derive some equivalences of generalized mixed non-linear programs, generalized mixed least-element problems, generalized mixed complementarity problems and generalized mixed variational inequality problems under certain regularity and growth conditions. We also generalize the notion of a Z-type map for point-to-set maps. Our results improve and extend recent results in the literature.
Optimization 01/2009; 58(1):63-76. · 0.50 Impact Factor
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ABSTRACT: In this article, a general variable-step basic projection algorithm for solving strongly quasivariational inequalities is proposed. Under certain conditions, the convergence of the general variable-step basic projection algorithm is established. For the practical consideration, we also give the relaxed version of this algorithm, in which the projection onto a closed convex set is replaced by another projection at each iteration which is easy to calculate. The convergence of relaxed scheme is also obtained under certain assumptions.
Optimization 10/2008; 57(5):607-620. · 0.50 Impact Factor
