Monotonicities and variational inequality models of second-order with simple integral functionals

Abstract This paper is concentrated on the study of second order invex monotonicities together with second order variational-like problems governed by $$C^2$$ C 2 -class simple integral functionals. In this process, we replace the classical concepts of gradient and Hessian by suitable integral operators (Gr and He). We present the concepts of (strictly) second order invex, second order pseodoinvex, and second order quasiinvex simple integral functionals and explore their correlations with their equivalent second order invex monotonicity of the Gr functional. We formulate the second order variational inequality problem ( SVIP ) associated with the considered variational problem ( VP ) and we study the existence and uniqueness conditions of its solution. Next, we prove the relationship between the solution of ( SVIP ) and the optimal solution of ( VP ). The theory is verified by a real world problem involving minimization of transport costs of goods, but our optimization problem could also be applied in control theory, engineering, economics, or medicine.