This paper aims to introduce a novel self-adaptive explicit iterative algorithm for solving the split common solution problem with monotone operator equations in real Hilbert spaces. This new approach does not employ ...
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This paper aims to introduce a novel self-adaptive explicit iterative algorithm for solving the split common solution problem with monotone operator equations in real Hilbert spaces. This new approach does not employ resolvent operators nor does it use the norms of the bounded linear operators (transfer mappings) from the source space to the image spaces.
Let C be a nonempty closed convex subset of a real Banach space X whose norm is uniformly Gateaux differentiable and T : C -> C be a continuous pseudo-contraction with a nonempty fixed point set F(T). For arbitrary...
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Let C be a nonempty closed convex subset of a real Banach space X whose norm is uniformly Gateaux differentiable and T : C -> C be a continuous pseudo-contraction with a nonempty fixed point set F(T). For arbitrary given element u is an element of C and for t is an element of (0, 1), let {y(t)} be the unique continuous path such that y(t) = (1 - t)Ty(t) + tu. Assume that y(t) -> p is an element of F(T) as t -> 0. Let {alpha(n)}, {beta(n)} and {gamma(n)} be three real sequences in (0, 1) satisfying the following conditions: (i) alpha(n) + beta(n) + gamma(n) = 1;(ii) lim(n ->infinity) alpha(n) = lim(n ->infinity) beta(n) = 0;(iii) lim(n ->infinity) beta(n)/1 - gamma(n) = 0;or (iii)' Sigma(infinity)(n=0) alpha(n)/1 - gamma(n) = infinity. Let {epsilon(n)} be a summable sequence of positive numbers. For arbitrary initial datum x(0) = x(0)(0) is an element of C and a fixed n >= 0, construct elements {x(n)(m)} as follows: x(n)(m + 1) = alpha(u)(n) + beta(n)x(n) + gamma(n)Tx(n)(m), m = 0, 1, 2, .... Suppose that there exists a least positive integer N(n) satisfying the following condition: parallel to Tx(n)(N(n) + 1) - Tx(n)(N(n))parallel to <= gamma(-1)(n) (1 - gamma(n))epsilon(n). Define iteratively a sequence {x(n)} in an explicit manner as follows: x(n + 1) = x(n + 1)(0) = x(n)(N(n) + 1) = alpha(n)u + beta(n)x(n) + gamma(n)Tx(n)(N(n)), n >= 0. Then {x(n)} converges strongly to a fixed point of T. For all the continuous pseudo-contractive mappings for which is possible to construct the sequence x(n), this result improves and extends a recent result of Yao et al. [Yonghong Yao, Yeong-Cheng Liou, Rudong Chen, Strong convergence of an iterativealgorithm for pseudocontractive mapping in Banach spaces, Nonlinear Anal., 67 (2007) 3311-3317]. (C) 2008 Elsevier Ltd. All rights reserved.
In this paper, we consider a triple hierarchical variational inequality defined over the common solution set of minimization and mixed equilibrium problems. Combining the hybrid steepest-descent method, viscosity appr...
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In this paper, we consider a triple hierarchical variational inequality defined over the common solution set of minimization and mixed equilibrium problems. Combining the hybrid steepest-descent method, viscosity approximation method and averaged mapping approach to the gradient-projection algorithm, we propose two iterative methods: implicit one and explicit one, to compute the approximate solutions of our problem. The convergence analysis of the sequences generated by the proposed methods is also established.
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