We consider an abstract minimization problem in reflexive Banach spaces together with a specific family of approximating sets, constructed by perturbing the cost functional and the set of constraints. For this problem, we state and prove various well-posedness results in the sense of Tykhonov, under different assumptions on the data. The proofs are based on arguments of lower semicontinuity, compactness and Mosco convergence of sets. Our results are useful in the study of various mathematical models in contact mechanics. To provide examples, we introduce 2 models, which describe the equilibrium of an elastic body in contact with a rigid body covered by a rigid-plastic and an elastic material, respectively. The weak formulation of each model is in the form of a minimization problem for the displacement field. We use our abstract well-posedness results in the study of these minimization problems. In this way, we obtain existence, uniqueness and convergence results, and moreover, we provide their mechanical interpretations.

我们考虑自反Banach空间中的抽象最小化问题，以及通过扰动成本函数和约束集而构造的一组特定的近似集。对于这个问题，我们根据数据的不同假设陈述并证明了Tykhonov方面的各种适定性结果。证明基于较低半连续性，紧性和集合的Mosco收敛性的论据。我们的结果可用于研究接触力学中的各种数学模型。为了提供示例，我们介绍了两个模型，分别描述了弹性体与被刚性塑料和弹性材料覆盖的刚性体接触的平衡。每个模型的弱公式是位移场的最小化问题。我们在研究这些最小化问题时使用了抽象的适定性结果。这样，我们获得了存在性，唯一性和收敛性结果，此外，我们提供了它们的机械解释。