East Asian mathematical journal

  • 제25권4호
  • /
  • Pages.527-532
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  • 2009
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  • 1226-6973(pISSN)
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  • 2287-2833(eISSN)
영남수학회 (The Youngnam Mathematical Society)
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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)
  • 투고 : 2009.08.13
  • 심사 : 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

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