Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators

Abstract

In this paper, we study the existence of solutions for evolution problems of the form − du dr (t) ∈ A(t)u(t)+F (t, u(t))+f (t, u(t)), where, for each t, A(t) : D(A(t)) → 2 H is a maximal monotone operator in a Hilbert space H with continuous, Lipschitz or absolutely continuous variation in time. The perturbation f is separately integrable on [0, T ] and separately Lipschitz on H, while F is scalarly measurable and separately scalarly upper semicontinuous on H, with convex and weakly compact values. Several new applications are provided.

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

“…We mentioned [439,427] as classical references on this topic, but similar results can be found in [147,209,512,531,446,445,444,559,199,116,487]. Among extensions and similar studies, we also find discretization for different kinds of variational inequalities [27,23,43,388,462], set-valued perturbations [169,237], state-dependent FOSwP [39,46,26,170,293,296,376], time-delay FOSwP and SOSwP [235,236,117,450,174,177], time-dependent maximal monotone DIs [64,153], proximal algorithms, i.e., implicit or backward Euler discretization of (2.16), for optimization problems [48,57,286,412,477] (it is known that an explicit Euler method has strong drawbacks [477,Sections 3.3,6.3]). Systems with prox-regular set-valued mappings have also been studied using time-stepping algorithms [42,…”

“…(Perturbation) Classical FOSwP involves single-valued perturbations. However many articles consider set-valued perturbations f (t, x) [67,65,64,169,241,296,338,444,446,535] (initiated in [425], see also [169] for references to the Séminaire d'Analyse Convexe de Montpellier), integral perturbations are studied in [203,119].…”

“…Well-posedness of (2.16) is proved in [524] under passivity-like conditions, in both cases of AC and LBV solutions, hence allowing jumps, and relying on [368,Theorem 3]: this implies to re-interpret (2.16) as an MDI, beingẋ a measure, in a similar way as done for the FOSwP (see (2.3) and (2.4)). A more general case is studied in [64,Theorem 4.4] with F (t, x) = N S(t) (x) and f (t, x) = H(t)(x) for some maximal monotone set-valued operator H(•), dom(H(t)) ⊂ S(t) for each t, S(•) closed and AC set-valued mapping. As alluded to above, one important class of such DIs is when F (x) = ∂ϕ(x) with ϕ(•) proper, convex LSC [100].…”

Some results associated to first-order set-valued evolution problems with subdifferentials

“…For some contributions on the first-order differential inclusions governed by subdifferential operators and sweeping processes, we refer to [7]- [22] and the references therein. For the more general time-dependent maximal monotone operators case, the reader is referred to [23,24,25,26].…”

“…Several related results have been developed in [31] and [32] for the case that the sets C(t) are uniformly prox-regular and move with bounded variation. Dealing with time-dependent maximal monotone operator A(t) instead of ∂ ϕ(t, •) in the variant above, we refer to [23]. We will proceed, to state the main existence theorem, by a discretisation method.…”