Convergence of approximating paths to solutions of variational inequalities involving non-Lipschitzian mappings
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Let X be a real reflexive Banach space with a uniformly Gateaux differentiable norm, C a nonempty closed convex subset of X, T : C → X a continuous pseudocontractive mapping, and A : C → C a continuous strongly pseudocontractive mapping. We show the existence of a path {xt} satisfying xt = tAxt+(1-t)Txt, t ∈ (0, 1) and prove that {xt} converges strongly to a fixed point of T, which solves the variational inequality involving the mapping A. As an application, we give strong convergence of the path {xt} defined by xt = tAxt +(1-t)(2I-T)xt to a fixed point of firmly pseudocontractive mapping T.
Abstract
Let X be a real reflexive Banach space with a uniformly Gateaux differentiable norm, C a nonempty closed convex subset of X, T : C → X a continuous pseudocontractive mapping, and A : C → C a continuous strongly pseudocontractive mapping. We show the existence of a path {xt} satisfying xt = tAxt+(1-t)Txt, t ∈ (0, 1) and prove that {xt} converges strongly to a fixed point of T, which solves the variational inequality involving the mapping A. As an application, we give strong convergence of the path {xt} defined by xt = tAxt +(1-t)(2I-T)xt to a fixed point of firmly pseudocontractive mapping T.
참고문헌 (22)
- V. Barbu, "Convexity and Optimization in Banach spaces" (1978), Editura Academiei R. S. R.
- F. E. Browder, "Nonlinear mappings of nonexpansive and accretive type in Banach spaces", Bulletin of the American Mathematical Society, 73, pp.875~882 (1967)
- L. Cioranescu, "Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems" (1990), Kluwer Aacademic Publishers
- K. Deimling, "Zeros of accretive operators", manuscripta mathematica, 13, pp.365~374 (1974)
- K. Deimling, "Nonlinear Functional Analysis" (1985), Spring-Verlag
- K. Goebel, "Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings" (1984), Marcel Dekker, Inc.
- K. Goebel, "Topics in Metric Fixed Point Theory" (1990), Cambridge University Press
- K. S. Ha, "Strong convergence theorems for accretive operators in Banach space", Journal of Mathematical Analysis and Applications, 147(2), pp.330~339 (1990)
- J. S. Jung, "Strong convergence theorems for nonexpansive nonselfmappings in Banach space", NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 33(3), pp.321~329 (1998)
- T. Kato, "Nonlinear semigroups and evolution equations", Journal of the Mathematical Society of Japan, 19, pp.508~520 (1967)
- R. H. Martin, "Differential equations on closed subsets of a Banach space", Transactions of the American Mathematical Society, 179, pp.339~414 (1973)
- C. H. Morales, "On the fixed point theory for local k-pseudocontractions", Proceedings of the American Mathematical Society, 통권 : 81, pp.71~74 (1981)
- C. H. Morales, "Sreong convergence theorems for pseudo-contractive mapping in Banach spaces", Houston Journal of Mathematics, 16(4), pp.549~557 (1990)
- C. H. Morales, "Convergence of paths for pseudo-contractive mappings in Banach spaces", Proc. Amer. Math. Soc., pp.3411~3419 (2000)
- A. Moudafi, "Viscosity approximation methods for fixed points problems", Journal of Mathematical Analysis and Applications, 241(1), pp.46~55 (2000)
- J. G. OHara, "Iterative approaches to convex feasibility problems in Banach spaces", NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 64, pp.2022~2042 (2006)
- Z. Opial, "Weak convergence of successive approximations for nonexpansive mappings", Bulletin of the American Mathematical Society, 73, pp.591~597 (1967)
- S. Reich, "Strong convergence theorems for resolvents of accretive operators in Banach spaces", Journal of Mathematical Analysis and Applications, 75, pp.287~292 (1980)
- J. Schu, "Approximating fixed points of Lipschitzian pseudocontractive mappings", Houston Journal of Mathematics, 19(1), pp.107~115 (1993)
- B. K. Sharma, "Firmly pseudo-contractive mappings and fixed points", COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE, 38(1), pp.101~108 (1997)
- H. K. Xu, "Approximating curves of nonexpansive nonself-mappings in Banach spaces", Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 325, pp.151~156 (1997)
- H. K. Xu, "Viscosity approximation methods for nonexpansive mappings", Journal of Mathematical Analysis and Applications, 298(1), pp.240~256 (2004)