Nonconvex evolution inclusions governed by the difference of two subdifferentials

ABSTRACT

The aim of the present paper is to prove the existence result for a class of differential inclusions governed by the difference of two subdifferentials of nonconvex functions in Hilbert spaces. More precisely, by using the Moreau-Yosida regularization, the existence of local solutions for the following differential inclusion $\dot{u}\left(t\right)+\mathrm{\partial }{\mathrm{\Phi }}_1\left(u\right(t\left)\right)-\mathrm{\partial }{\mathrm{\Phi }}_2\left(u\right(t\left)\right)\ni f\left(t\right)\text{ for almost every }t\in [0,T_0],u\left(0\right)=u_0$ is proved, where both functions ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ are assumed to be primal lower nice uniformly with respect to subgradients.

Keywords:

• Subdifferential
• primal lower nice functions
• Moreau-Yosida regularization
• differential inclusions
• Moreau envelope
• proximal mapping
• set-valued map

2010 Mathematics Subject Classifications:

• 49A52
• 49J53
• 34A60
• 34G25
• 34G20

Disclosure statement

No potential conflict of interest was reported by the author(s).