In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the S -iteration process recently introduced by Sahu in  [14] converges strongly to a unique fixed point of a mapping T , where T is κ -strongly pseudocontractive mapping from a nonempty, closed and convex subset C of a smooth Banach space into itself. It is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of λ i -strictly pseudocontractive mappings from C into itself. Our results extend and improve some very recent theorems in fixed point theory and variational inequality problems. Particularly, the results presented here extend some theorems of Reich (1980)  [1] and Yamada (2001)  [15] to a general class of λ -strictly pseudocontractive mappings in uniformly smooth Banach spaces.

Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces

On Mann implicit iterations for strongly accretive and strongly pseudo-contractive mappings

In this paper, we consider last level cache (LLC) sharing problems in large-scale cloud networks with a fair payoff function. We formulate the problem as a strategic decision-making problem (i.e., a game). We examine the resource-sharing game with finite and infinite number of players. Exploiting the aggregate structure of the payoff functions, we show that the resource-sharing game has a Nash equilibrium in a wide range of return index. We show that the Nash equilibrium is not an evolutionarily stable strategy in the finite regime. Then, we introduce a myopic mean-field response where each player implements a mean-field-taking strategy. We show that such a mean-field-taking strategy is an evolutionarily stable strategy in both finite and infinite regime. We provide closed-form expression of the optimal pricing that gives an efficient resource-sharing policy. As the number of active players grows without bound, we show that the equilibrium strategy converges to a mean-field equilibrium, and the optimal prices for resources converge to the optimal price of the mean-field game. Then, we address the demand satisfaction problem for which a necessary and sufficient condition for satisfactory solutions is provided. In addition, a very fast mean-field learning algorithm is provided.

Mean-field games for resource sharing in cloud-based networks

In this paper, we study the strong convergence of SP iterative scheme with mixed errors for the accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces. We show that this iterative scheme is almost stable for both types of operators. Moreover, with the help of computer programs in C++, comparison between SP, Ishikawa and Noor iterative schemes with errors is also shown for both types of operators through examples.

Convergence of SP iterative scheme with mixed errors for accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach spaces

In this paper, we study the convergence theorems of the Ishikawa
iterative sequences with mixed errors for the accretive Lipschitzian mappings
in Banach spaces.

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Convergence of ishikawa iterative sequences for accretive lipschitzian mappings in banach spaces

We show that any fixed point of a Lipschitzian, strictly pseudocontractive mapping T on a closed, convex subset K of a Banach space X is necessarily unique, and may be norm approximated by an iterative procedure. Our argument provides a convergence rate estimate and removes the boundedness assumption on K, generalizing theorems of Liu. Let (X, ‖ · ‖) be a Banach space. Let K be a non-empty closed, convex subset of X and T : K → K. We will assume that T is Lipschitzian, i.e. there exists L > 0 such that ‖T (x)− T (y)‖ ≤ L‖x− y‖, for all x, y ∈ K. Of course, we are most interested in the case where L ≥ 1. We also assume that T is strictly pseudocontractive. Following Liu [1] this may be stated as: there exists k ∈ (0, 1) for which ‖x− y‖ ≤ ‖x− y + r[(I − T − kI)x− (I − T − kI)y]‖, for all r > 0 and all x, y ∈ K. Throughout, N will denote the set of positive integers. The following results generalize Liu [1, Theorems 1 and 2], because we remove the assumption that K is bounded and we provide a general convergence rate estimate. We note in passing, however, that the proof of Theorem 2 of Liu [1] does not use the stated boundedness assumption. Our results still extend this enhanced version of Liu [1, Theorem 2], by improving the convergence rate estimate. Theorem 1. Let (X, ‖ · ‖),K, T, L and k be as described above. Let q ∈ K be a fixed point of T . Suppose that (αn)n∈N is a sequence in (0, 1] such that for some η ∈ (0, k), for all n ∈ N, αn ≤ k − η (L+ 1)(L+ 2− k) ; while ∞ ∑

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K. P. R. Sastry

Papers

Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space