Resolvent positive operator and two-parameter abstract Cauchy problem

Abstract

We use the semigroup theory to study the homogeneous two-parameter abstract Cauchy problem 2-RACPWhere u is a function from [0 , a 1 ] × [0 , a 2 ] to ordered Banach space E whose positive cone is normal and generating and a 1 , a 2 > 0, Mathematics Subject Classification: 34A12, 34A30, 47D03, 47D62, 47D99, 35D05Keywords: N-parameter semigroup, two-parameter resolvent positive operator, two-parameter integrated semigroup and two-parameter abstract Cauchy problem 1.IntroductionLet X be a Banach space, B(X) is the Banach space of all bounded linear operators on X and IR

Multi-dimensional almost periodic type functions and applications

“…has been analyzed by M. Janfada and A. Niknam in [59, Theorem 2.1], who proved that, if (A 1 , A 2 , • • •, A n ) is the infinitesimal generator of a strongly continuous semigroup (T (t)) t∈[0,∞) n , then (ACP) has a unique solution u(t) = T (t)x, t ∈ I for all initial values x ∈ i∈Nn D(A i ); a converse of this statement has been analyzed in [59, Theorem 2.2] (see also Theorem 2.5 in this paper, where the authors have shown a negative result about the uniqueness of solutions of the abstract Cauchy problem (ACP) as well as the paper [64] where the authors have considered the special case n = 2 by using the notion of a two-parameter integrated semigroup; the special case n = 2 has been also analyzed in [58] and [66], where the authors have introduced the notion of a two-parameter C-regularized semigroup and the notion of a two-parameter N -times integrated semigroup, respectively, where the operator C ∈ L(X) is injective and N ∈ N). A Hille-Yosida type theorem for multiparameter semigroups has been analyzed by Yu.…”