Nikolaos Yannakakis

Nonlinear boundary value problems

In this paper we consider two quasilinear boundary value problems. The first is vector valued and has periodic boundary conditions. The second is scalar valued with nonlinear boundary conditions determined by multivalued maximal monotone maps. Using the theory of maximal monotone operators for reflexive Banach spaces and the Leray-Schauder principle we establish the existence of solutions for both problems.

Minimax control of nonlinear evolution equations

In this paper we study nonlinear parametric optimal control problems. We formulate the relevant minimax problem and prove the existence of an optimal control. Also we derive necessary conditions for saddle point optimality. We also associate to the original optimization problem a dual problem and then for the pair we prove a complete duality theorem. Finally we present two examples of parabolic distributed parameter systems, which illustrate the abstract results.

Extremal solutions and strong relaxation for nonlinear periodic evolution inclusions

We study the existence of extremal periodic solutions for nonlinear evolution inclusions defined on an evolution triple of spaces and with the nonlinear operator A being time-dependent and pseudomonotone. Using techniques of multivalued analysis and a surjectivity result for L-generalized pseudomonotone operators, we prove the existence of extremal periodic solutions. Subsequently, by assuming that A(t, •) is monotone, we prove a strong relaxation theorem for the periodic problem. Two examples of nonlinear distributed parameter systems illustrate the applicability of our results.