Evolution problems involving state-dependent maximal monotone operators

Abstract

The main objective of the present paper is to prove the existence of absolutely continuous solutions for an evolution problem in a Hilbert space H, of the form $-\left(\mathrm{d}u/\mathrm{d}t\right)\left(t\right)\in A(t,u(t\left)\right)u\left(t\right)$ where, for each $(t,x)\in [0,T]\times H$, the maximal monotone operator $A(t,x):H\to 2^H$ is of absolutely continuous variation in time and Lipschitz continuous in state, in the sense of the pseudo-distance introduced by A. A. Vladimirov. The perturbed problem is also considered.

COMMUNICATED BY:

• B. Mordukhovich

Keywords:

• Absolutely continuous variation
• maximal monotone operator
• pseudo-distance
• set-valued perturbation

AMS Subject Classifications 2010:

• 34H05
• 34K35
• 60H10 28A25
• 28C20

Acknowledgments

The authors are very grateful to professor Charles Castaing for the interesting discussions and remarks related to this work, and a referee for the helpful remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

M. D. P. M. M. was partially supported by Foundation for Science and Technology (FCT) [grant number UID/MAT/04561/2019].