Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A SYSTEM OF NONLINEAR VARIATIONAL INCLUSIONS IN REAL BANACH SPACES

  • Bai, Chuan-Zhi (Department of Mathematics, Huaiyin Normal College) ;
  • Fang, Jin-Xuan (Department of Mathematics, Nanjing Normal University)
  • Published : 2003.08.01
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Abstract

In this paper, we introduce and study a system of nonlinear implicit variational inclusions (SNIVI) in real Banach spaces: determine elements $x^{*},\;y^{*},\;z^{*}\;\in\;E$ such that ${\theta}\;{\in}\;{\alpha}T(y^{*})\;+\;g(x^{*})\;-\;g(y^{*})\;+\;A(g(x^{*}))\;\;\;for\;{\alpha}\;>\;0,\;{\theta}\;{\in}\;{\beta}T(z^{*})\;+\;g(y^{*})\;-\;g(z^{*})\;+\;A(g(y^{*}))\;\;\;for\;{\beta}\;>\;0,\;{\theta}\;{\in}\;{\gamma}T(x^{*})\;+\;g(z^{*})\;-\;g(x^{*})\;+\;A(g(z^{*}))\;\;\;for\;{\gamma}\;>\;0,$ where T, g : $E\;{\rightarrow}\;E,\;{\theta}$ is zero element in Banach space E, and A : $E\;{\rightarrow}\;{2^E}$ be m-accretive mapping. By using resolvent operator technique for n-secretive mapping in real Banach spaces, we construct some new iterative algorithms for solving this system of nonlinear implicit variational inclusions. The convergence of iterative algorithms be proved in q-uniformly smooth Banach spaces and in real Banach spaces, respectively.

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References

  1. Nonlinear semigroups and differential equations in Banach spaces V.Barbu
  2. Nonlinear Functional Analysis K.Deimling
  3. J. Math. Anal. Appl. v.185 A perturbed algorithm for variational inclusions A.Hassouni;A.Moudafi https://doi.org/10.1006/jmaa.1994.1277
  4. Comput. Math. Appl. v.35 no.10 A new completely general class of variational inclusions with noncompact valued mappings N.J.Huang https://doi.org/10.1016/S0898-1221(98)00067-4
  5. Comput. Math. Appl. v.41 A new class of generalized set-valued implicit variational inclusions in Banach spaces with an application https://doi.org/10.1016/S0898-1221(00)00331-X
  6. J. Math. Anal. Appl. v.209 Mann and Ishikawa type perturbed iterative algorithms for generalized quasi-variational inclusions K.R.Kazmi https://doi.org/10.1006/jmaa.1997.5368
  7. J. Math. Anal. Appl. v.255 Three-step iterative algorithms for multivalued quasi variational inclusions M.Aslam Noor https://doi.org/10.1006/jmaa.2000.7298
  8. J. Funct. Anal. v.6 A characterization of strict cinvexity of Banach spaces and other uses of duality mappings W.V.Petryshyn https://doi.org/10.1016/0022-1236(70)90061-3
  9. J. Math. Anal. Appl. v.220 Strongly nonlinear generalized equations L.U.Uko https://doi.org/10.1006/jmaa.1997.5796
  10. Comput. Math. Appl. v.41 Projection methods, algorithms, and a new system of nonlinear variational inequalities R.U.Verma https://doi.org/10.1016/S0898-1221(00)00336-9
  11. Arch. Math. (Basel) v.58 Approximation of fixed points of nonexpansive mappings R.Wittmann https://doi.org/10.1007/BF01190119
  12. Nonlinear Anal. v.16 Inequalities in Banach spaces with applications H.K.Xu https://doi.org/10.1016/0362-546X(91)90200-K
  13. J. Math. Anal. Appl. v.157 Chracteristic inequalities for uniformly convex and uniformly smooth spaces Z.B.Xu;G.F.Roach https://doi.org/10.1016/0022-247X(91)90144-O
  14. Nonlinear functional analysis and its applications, Ⅱ/A Nonlinear monotone operators E.Zeidler