Convergence Analysis of a Power Penalty Approach for a Class of Nonlocal Double Phase Complementarity Systems

In the present paper, we consider a nonlinear complementarity problem (NCP, for short) with a nonlinear and nonhomogeneous partial differential operator (called double phase differential operator), a convection term (i.e., a reaction depending on the gradient), a generalized multivalued boundary condition, and two nonlocal terms which appear in the domain and on the boundary. We employ a power penalty method to NCP for introducing an approximating problem associated with NCP which is a nonlinear and nonlocal elliptic equation with mixed boundary value conditions. Denoting by S-infinity the solution set of NCP and by S-rho(epsilon) the solution set of the approximating problem corresponding to penalty parameter rho>0 and regularized parameter epsilon>0, we establish a critical convergence result in which S-infinity can be approached by the solution sets S-rho(epsilon) of approximating problems in the sense of Kuratowski when rho goes to zero, i.e., the following convergence relation holds empty set not equal w-lim sup(rho -> 0)S(rho)(epsilon)=s-lim sup(rho -> 0)S(rho)(epsilon) subset of S-infinity, where w-lim sup(rho -> 0)S(rho)(epsilon) and s-lim sup(rho -> 0)S(rho)(epsilon) are the weak and the strong Kuratowski upper limits of S-rho(epsilon), respectively.