Periodic Solutions For Parabolic Fractional $p$-Laplacian Problems via Topological Degree

Abstract

In this work, we consider the nonlinear parabolic initial boundary value problem involving the fractional $p$-Laplacian operator. We employ topological degree methods to establish the existence of a time periodic nontrivial weak solutions in the appropriate space $\mathcal{X}:=L^{p}(0,T ; W^{s,p}(\Omega))$. Our approach to proving the main result is based on transforming this nonlinear parabolic problem into an operator equation of the shape $\mathcal{K}u+\mathcal{B}u=h$, where $\mathcal{B}$ is of type $(S_+)$ relative in the domain of densely defined linear maximal monotone operator $\mathcal{K}$.