μ-Pseudo compact almost automorphic weak solutions for some partial functional differential inclusions

Abstract

The aim of this work is to investigate the existence and uniqueness of μ-pseudo almost periodic (resp. μ-pseudo compact almost automorphic) weak solutions for the following partial functional differential inclusion: $x^{\mathrm{\prime }}\left(t\right)+\mathcal{A}x\left(t\right)\ni f(t,x_t)\mathrm{f}\mathrm{o}\mathrm{r}t\in \mathbb{R},$where $\mathcal{A}:\mathcal{H}⟶2^{\mathcal{H}}$ is a strongly maximal monotone operator on a real Hilbert space $\mathcal{H}$, $f:\mathbb{R}\times \mathcal{C}⟶\mathcal{H}$ is a Stepanov μ-pseudo almost periodic (resp. μ-pseudo compact almost automorphic) function of class r, $\mathcal{C}=C\left(\right[-r,0],\mathcal{H})$ is the Banach space of all continuous functions from $[-r,0]$ to $\mathcal{H}$ and the history function $x_t$ is defined by $x_t\left(\theta \right)=x(t+\theta )$ for $\theta \in [-r,0]$. Two examples are given for parabolic and subdifferential systems.

Keywords:

• μ-pseudo almost peridicity
• μ-pseudo compact almost automorphy
• maximal monotone operator
• partial functional differential inclusion
• weak solution

Mathematics Subject Classifications:

• 34C12
• 34C27
• 35B15
• 35B40
• 35D30

Acknowledgements

The authors are grateful to the anonymous referee for carefully reading this manuscript and giving valuable suggestions for improvements.

Disclosure statement

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