On uniqueness for a family of nonstandard problems

In this study we investigate the uniqueness of solutions of the nonstandard problem d2udt2=Au+F,αu(0)+u(T)=g,βdudt(0)+dudt(T)=h, in the general case where we do not assume the positivity of the operator AA. We prove that whenever α=−βα=−β with |α|≠1|α|≠1 we have always uniqueness of solutions. We also obtain some families of the parameters α,βα,β where uniqueness fails. It is worth noting that the intersection of these families of parameters α,βα,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,βα,β where the uniqueness holds is not empty. Thus the assumption of positivity of the operator AA 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.