Abstract
It is well known that many popular variational, quasi-variational, hemivariational inequalities and variational inclusions involving constraints in a Banach space to convert a fixed point problems for finding the solution of such problems. This paper is to infuse a sequence of penalized problems without constraints and we show under the few reasonable assumptions to the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems is nonempty. As an application, we explore two complicated partial differential systems of elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints.