tailieunhanh - Báo cáo hóa học: " Research Article Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 49125 13 pages doi 2007 49125 Research Article Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay Hassane Bouzahir Received 6 November 2006 Revised 16 January 2007 Accepted 19 January 2007 Recommended by Simeon Reich We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the nonlinear semigroup. Copyright 2007 Hassane Bouzahir. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Most of the existing results about functional differential equations with finite delay have been recently under verification in the case of infinite delay. Our objective in this paper is to study the solution semigroup generated by the following partial functional differential equation with infinite delay dtx t ÁTx t F xt t 0 X0 ộ E where ÁT is a nondensely defined linear operator on a Banach space E I I . The phase space can be the space Cy y being a positive real constant of all continuous functions ộ - 0 . E such that limọ . ey0ộ ỹ exists in E endowed with the norm llộHy supo 0 ey0 Iộ 0 I ộ E Cy. For every t 0 the function xt E is defined by xt 0 x t Ỡ for 0 E -TO 0 . 2 Journal of Inequalities and Applications We assume the following. H1 F E is globally Lipschitz continuous that is there exists a positive constant L such that F 1 - F 2 1 L ty1 - 2ll for all 1 ty2 G . A typical example that can be transformed into is the following d

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