In this article, we study the notion of the variational inequalities for lattice-valued fuzzy relations. In this context, a variational inequality problem has been proposed that generalizes many results in the literature. The conditions for the existence of solutions of the proposed problem have been discussed. It has been shown that the proposed variational inequality problem is equivalent to a fixed point problem. This fixed point formulation allows us to present an iterative algorithm to approximate solution of the variational inequality problem. For applications, first the existence result for the solutions of an ℒ-fuzzy Caputo-Fabrizio fractional differential inclusion initial value problem involving a projection operator has been proved. Then the solutions of an obstacle boundary value variational inequality problem in function spaces has been obtained.
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