On uniqueness for a family of nonstandard problems

Abstract

In this study we investigate the uniqueness of solutions of the nonstandard problem d 2 u d t 2 = A u + F , α u ( 0 ) + u ( T ) = g , β d u d t ( 0 ) + d u d t ( T ) = h , in the general case where we do not assume the positivity of the operator A . We prove that whenever α = − β with | α | ≠ 1 we have always uniqueness of solutions. We also obtain some families of the parameters α , β where uniqueness fails. It is worth noting that the intersection of these families of parameters α , β with the families obtained in [L.E. Payne, P.W. Schaefer, Energy bounds for some nonstandard problems in partial differential equations, J. Math. Anal. Appl. 273 (2002) 75–92] of parameters α , β where the uniqueness holds is not empty. Thus the assumption of positivity of the operator A assumed in [L.E. Payne, P.W. Schaefer, Energy bounds for some nonstandard problems in partial differential equations, J. Math. Anal. Appl. 273 (2002) 75–92] plays a relevant role. We end this note by giving sufficient conditions for guaranteeing the uniqueness of solutions for two concrete problems.