Abstract
We consider an abstract inclusion in a real Hilbert space, governed by an almost history-dependent operator and a time-dependent multimapping with prox-regular values. We establish the unique solvability of the inclusion under appropriate assumptions on the data. The proof is based on the arguments of monotonicity, fixed point, and prox-regularity. We then use our result in order to deduce some direct consequences, including an existence and uniqueness result for a class of sweeping processes associated with prox-regular sets. Finally, we provide an example in a finite dimensional case inspired by a rheological model in solid mechanics.
Highlights
1 Introduction A large variety of boundary valued problems arising in mechanics, physics, and engineering sciences lead, in a weak formulation, to nonlinear inclusions
There, existence and uniqueness results have been provided by using a fixed point theorem for almost historydependent operators
We provide an existence and uniqueness result and, to this end, we use various properties of the prox-regular sets, including the properties of the projection operators and proximal normal cone
Summary
A large variety of boundary valued problems arising in mechanics, physics, and engineering sciences lead, in a weak formulation, to nonlinear inclusions. Removing the convexity in the study of the corresponding inclusions leads to important mathematical difficulties and gives rise to new and challenging mathematical problems This can be achieved through the class of prox-regular sets ( known as positively reached, weakly convex, O(2)-convex, φ-convex, proximally smooth (see, e.g., [11] and the references therein)). We provide an existence and uniqueness result and, to this end, we use various properties of the prox-regular sets, including the properties of the projection operators and proximal normal cone. Motivated by the above mechanical problem, in this paper we shall study inclusions of the form (6) in the abstract framework of real Hilbert spaces, under the assumption that C(t) represents a family of prox-regular sets and S is an almost history-dependent operator.
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