Existence and multiplicity of solutions to a ψ-Hilfer fractional p-Laplacian equations.

In this paper, by Symmetric Mountain Pass Lemma, we study the existence and multiplicity of solutions to the following nonlocal ψ -Hilfer fractional p -Laplasian equation: H D T α , β ; ψ (| H D 0 + α , β ; ψ ξ (x) | p − 2 H D 0 + α , β ; ψ ξ (x)) − γ (x) | ξ (x) | r − 2 ξ (x) − g (x , ξ (x)) = 0 , I 0 + β (β − 1) ; ψ ξ (0) = I T β (β − 1) ; ψ ξ (T) = 0 , where H D 0 + α , β ; ψ ξ (x) and H D T α , β ; ψ are ψ -Hilfer fractional derivatives left-sided and right-sided of order 1 p < α < 1 , 0 ≤ β ≤ 1 , r > p and I 0 + β (β − 1) ; ψ (.) and I T β (β − 1) ; ψ (.) are ψ -Riemann–Liouville fractional integrals left-sided and right-sided, γ : [ 0 , T ] → ℝ and g : [ 0. T ] × ℝ → ℝ are continuous functions. Finally, we give some examples to illustrate the main results. [ABSTRACT FROM AUTHOR]