Linear and nonlinear nonlocal boundary value problems for differential-operator equations

Abstract

This study focuses on nonlocal boundary value problems (BVPs) for linear and nonlinear elliptic differential-operator equations (DOEs) that are defined in Banach-valued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal L_p -regularity and Fredholmness of linear BVPs, uniformly with respect to this parameter, are presented. This fact implies that the appropriate differential operator is a generator of an analytic semigroup. Then, by using these results, the existence, uniqueness and maximal smoothness of solutions of nonlocal BVPs for nonlinear DOEs are shown. These results are applied to nonlocal BVPs for regular elliptic partial differential equations, finite and infinite systems of differential equations on cylindrical domains, in order to obtain the algebraic conditions that guarantee the same properties.

Keywords:

• Boundary value problems
• Differential-operator equations
• Maximal L_p -regularity
• Banach-valued function spaces
• Operator-valued multipliers
• Interpolation of Banach spaces
• Nonlinear DOE
• Existence
• 1991 Mathematics Subject Classifications: 34G10
• 35J25
• 35J70