Abstract We consider a fixed point problem $\mathcal {S}u=u$ S u = u where ${\mathcal {S}:C(\mathbb{R}_{+};X)\to C(\mathbb{R}_{+};X)}$ S : C ( R + ; X ) → C ( R + ; X ) is an almost history-dependent operator. First, we recall the unique solvability of the problem. Then, we introduce the concept of Tykhonov triple, provide several relevant examples, and prove the corresponding well-posedness results for the considered fixed point problem. This allows us to deduce various consequences which illustrate the stability of the solution with respect to perturbations of the operator $\mathcal {S}$ S . Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in solid and contact mechanics. To give some examples, we consider two mathematical models which describe the equilibrium of a viscoelastic body in frictionless contact with a foundation. We state the mechanical problems, list the assumptions on the data, and derive their associated fixed point formulation. Then, we illustrate the use of our abstract results in order to deduce the continuous dependence of the solution with respect to the data and parameters. We also provide the corresponding mechanical interpretations.