Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications(

where I ⊂R is an open interval, f(x) is a given function, is a nonlinear di erential operator, and W ( ) ∈ L(I) is a piecewise Gâteaux di erentiable function of = (u); Ua is a closed convex subspace of a re exive Banach space U. This general nonconvex, nonsmooth variational problem appears in many nonlinear systems. For example, in the nonlinear equilibrium problem of Ericksen’s bar subjected to axial extension [17], or the post-buckling analysis of extended nonlinear beam subjected to a compressed load [26], the nite strain = (u)= 2u 2 ; x − is a quadratic operator,