tailieunhanh - MAXIMAL REGULAR BOUNDARY VALUE PROBLEMS IN BANACH-VALUED WEIGHTED SPACE RAVI P. AGARWAL, MARTIN

MAXIMAL REGULAR BOUNDARY VALUE PROBLEMS IN BANACH-VALUED WEIGHTED SPACE RAVI P. AGARWAL, MARTIN BOHNER, AND VELI B. SHAKHMUROV Received 10 July 2004 This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banachvalued weighted L p spaces. These results are applied to. | MAXIMAL REGULAR BOUNDARY VALUE PROBLEMS IN BANACH-VALUED WEIGHTED SPACE RAVI P. AGARWAL MARTIN BOHNER AND VELI B. SHAKHMUROV Received 10 July 2004 This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fred-holmness estimates for the resolvent and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banachvalued weighted Lp spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties. 1. Introduction and notation Boundary value problems for differential-operator equations have been studied in detail in 4 15 22 35 40 42 . The solvability and the spectrum of boundary value problems for elliptic differential-operator equations have also been studied in 5 6 12 14 16 18 29 30 31 32 33 34 37 41 . A comprehensive introduction to differential-operator equations and historical references may be found in 22 42 . In these works Hilbert-valued function spaces have been considered. The main objective of the present paper is to discuss nonlocal boundary value problems for ordinary and partial differential-operator equations DOE in Banach-valued weighted Lp spaces. In this work the following is done. 1 The continuity compactness and qualitative properties of the embedding operators in the associated Banach-valued weighted function space are considered. 2 An ordinary differential-operator equation Lu jm á Am kuk x f x x e 0 b am 0 of arbitrary order on a domain with varying bound is investigated. Copyright 2005 Hindawi .