CONVERGENCE OF VISCOSITY APPROXIMATIONS TO FIXED POINTS OF NONEXPANSIVE NONSELF-MAPPINGS IN BANACH SPACES
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Let E be a uniformly convex Banach space with a uniformly Gˆateaux differentiable norm, C a nonempty closed convex subset of E, and T : C -> E a nonexpansive mapping satisfying the weak inwardness condition. Assume that every weakly compact convex subset of E has the fixed point property. For f : C -> C a contraction and t in (0; 1), let x_t be a unique fixed point of a contraction T_t : C -> E, defined by T_t x = t f(x) + (1 - t)Tx, x in C. It is proved that if { x_t } is bounded, then x_t converges to a fixed point of T, which is the unique solution of certain variational inequality. Moreover, the strong convergence of other implicit and explicit iterative schemes involving the sunny nonexpansive retraction is also given in a reflexive and strictly convex Banach space with a uniformly Gˆateaux differentiable norm.
Abstract
Let E be a uniformly convex Banach space with a uniformly Gˆateaux differentiable norm, C a nonempty closed convex subset of E, and T : C -> E a nonexpansive mapping satisfying the weak inwardness condition. Assume that every weakly compact convex subset of E has the fixed point property. For f : C -> C a contraction and t in (0; 1), let x_t be a unique fixed point of a contraction T_t : C -> E, defined by T_t x = t f(x) + (1 - t)Tx, x in C. It is proved that if { x_t } is bounded, then x_t converges to a fixed point of T, which is the unique solution of certain variational inequality. Moreover, the strong convergence of other implicit and explicit iterative schemes involving the sunny nonexpansive retraction is also given in a reflexive and strictly convex Banach space with a uniformly Gˆateaux differentiable norm.
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