East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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ON THE EXISTENCE OF SOLUTIONS OF EXTENDED GENERALIZED VARIATIONAL INEQUALITIES IN BANACH SPACES

  • He, Xin-Feng (COLLEGE OF MATHEMATICS AND COMPUTER HEBEI UNIVERSITY) ;
  • Wang, Xian (COLLEGE OF MATHEMATICS AND COMPUTER HEBEI UNIVERSITY) ;
  • He, Zhen (COLLEGE OF MATHEMATICS AND COMPUTER HEBEI UNIVERSITY)
  • Received : 2009.08.13
  • Accepted : 2009.10.24
  • Published : 2009.12.31
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Abstract

In this paper, we study the following extended generalized variational inequality problem, introduced by Noor (for short, EGVI) : Given a closed convex subset K in q-uniformly smooth Banach space B, three nonlinear mappings T : $K\;{\rightarrow}\;B^*$, g : $K\;{\rightarrow}\;K$, h : $K\;{\rightarrow}\;K$ and a vector ${\xi}\;{\in}\;B^*$, find $x\;{\in}\;K$, $h(x)\;{\in}\;K$ such that $\xi$, g(y)-h(x)> $\geq$ 0, for all $y\;{\in}\;K$, $g(y)\;{\in}\;K$. [see [2]: M. Aslam Noor, Extended general variational inequalities, Appl. Math. Lett. 22 (2009) 182-186.] By using sunny nonexpansive retraction $Q_K$ and the well-known Banach's fixed point principle, we prove existence results for solutions of (EGVI). Our results extend some recent results from the literature.

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References

  1. Ya. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: A. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Monotonic and Accretive Type, Marcel Dekker, New York, 1996, 15-50.
  2. R. E. Bruck, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math. 47 (1973), 341-355. https://doi.org/10.2140/pjm.1973.47.341
  3. J. Fan, A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces, J. Math. Anal. Appl. 337 (2008), 1041-1047. https://doi.org/10.1016/j.jmaa.2007.04.025
  4. K. Goebel and S. Reich, Uniform convexity, in: Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
  5. J. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl. 295 (2004) 115-126. https://doi.org/10.1016/j.jmaa.2004.03.010
  6. M. Aslam Noor, Extended general variational inequalities, Appl. Math. Lett. 22 (2009) 182-186. https://doi.org/10.1016/j.aml.2008.03.007
  7. M. Aslam Noor, General variational inequalities, Appl. Math. Lett. 1 (1988), 119-121. https://doi.org/10.1016/0893-9659(88)90054-7
  8. H. K. Xu, Inequalities in Banach spaces with applications, Nonl. Anal. 16 (12) (1991), 1127-1138. https://doi.org/10.1016/0362-546X(91)90200-K