We consider a nonlinear equation in a reflexive Banach space X, governed by a history-dependent operator, over the time interval [0, T], with \(T > 0\) . We prove that, under appropriate assumptions, the equation admits a unique solution \(u:[0,T] \rightarrow X\) . We then provide necessary and sufficient conditions that guarantee the uniform convergence of any arbitrary sequence of continuous functions, defined on [0, T] with values in X, to the solution u. The proof relies on the fixed point structure of the problem. Our results are applicable to a wide range of nonlinear problems. As an example, we consider a mathematical model describing the contact of a viscoelastic body with a foundation. The contact is frictionless and is modeled by a unilateral condition. The weak formulation of the model takes the form of a history-dependent variational inequality for the displacement field. We apply our abstract results to this problem and, in particular, obtain a continuous dependence result.
