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J. Garcia-Falset, E. Llorens-Fuster and S. Prus in [2] studied the existence of fixed point of J-type mappings in Banach spaces. In this paper, we extend these mappings in Menger spaces and prove the fixed point theorems of these mappings in complete Menger spaces. In this paper, we also prove theorems for the new class of mappings which is called altering J-type.

Stability of switching systems with an infinite number of subsystems is important in some structure of systems, like fuzzy systems, neural networks, and so forth. Because of the relationship between stability of a set of matrices and switching systems, this paper first studies the stability of a set of matrices, then and the results are applied for stability of switching systems. Some new conditions for globally uniformly asymptotically stability (GUAS) of discrete-time switched linear systems with an infinite number of subsystems are proposed. The paper considers some examples and simulation results.

Let A be a Banach algebra and I be a closed ideal of A. We say that A is amenable relative to I, if A/I is an amenable Banach algebra. We study the relative amenability of Banach algebras and investigate the relative amenability of triangular Banach algebras and Banach algebras associated to locally compact groups. We generalize some of the previous known results by applying the concept of relative amenability of Banach algebras, especially, we present a generalization of Johnson?s theorem in the concept of relative amenability.