On Controlled Fractional Variational Inequalities Involving Convex Functionals

This paper explores weak sharp solutions for a controlled variational inequality governed by a convex fractional curvilinear integral functional with path-independent properties. The fractional framework involves derivatives or integrals of non-integer order, which introduce non-local behavior by incorporating the influence of past states—commonly interpreted as memory-dependent effects in dynamic systems. This modeling approach captures phenomena with long-range temporal or spatial interactions more accurately than classical methods. Although the integral is taken along a curve, path-independence ensures that the functional depends only on the curve’s endpoints, while still being sensitive to geometric data such as orientation and boundary conditions. This structure provides a flexible and rigorous framework for representing system dynamics. The paper analyzes the properties of weak sharp solutions using a combination of variational techniques and a dual disparity functional. Under suitable conditions, it establishes an equivalence between the sufficiency of the minimum principle and the weak sharpness of the solution set for the controlled fractional variational inequality.