Equivalence of solutions to fractional $p$-Laplace type equations

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $$\mathrm{P}\mathrm{.}\mathrm{V}\mathrm{.}{\int }_{{\mathbb{R}}^n}\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^{p-2}(u(x)-u(y))}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+sp}}dy=0.$$ Solutions are defined via integration by parts with test functions, as viscosity solutions or via comparison. Our main result states that for bounded solutions, the three different notions coincide.