In this paper, we consider a differential inclusion governed by a set-valued nonexpansive mapping and study the asymptotic behavior (weak and strong convergence) of its solutions with various assumptions on this mapping. Then for a set-valued nonexpansive mapping, we define the corresponding resolvent (proximal) operator as a set-valued mapping and study some of its elementary properties. Subsequently, we apply the resolvent operator to state the implicit discretization of the differential inclusion and study the asymptotic behavior of its solutions which yields similar convergence results as in the continuous case. This provides an algorithm for approximating a fixed point of a set-valued nonexpansive mapping which extends the classical proximal point algorithm. An application to variational inequalities and a numerical comparison with another iterative method for approximating a fixed point of set-valued nonexpansive mappings are also presented.

在本文中，我们考虑了由集值非膨胀映射控制的微分包含，并在此映射的各种假设下研究了其解的渐近行为（弱收敛和强收敛）。然后，对于集值非膨胀映射，我们将相应的解析（近端）运算符定义为集值映射，并研究其一些基本属性。随后，我们使用可分辨算子来说明微分包含的隐式离散化，并研究其解的渐近行为，这将产生与连续情况相似的收敛结果。这提供了一种用于近似集值非膨胀映射的固定点的算法，该算法扩展了经典的近端点算法。