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 psi-Hilfer fractional p-Laplasian equation: {D-H(T)alpha,beta;psi(vertical bar D-H(0+)alpha,beta;psi xi(x)vertical bar(p-2) (H) D-0+(alpha,beta;psi)xi(x)) - gamma(x)vertical bar xi(x)vertical bar(r-2)xi(x) -g(x,xi(x)) = 0, I-0+(beta(beta-1);psi)xi(0) = I-T(beta(beta-1);psi) xi(T) = 0, where H D-0+(alpha,beta;psi) xi(x) and (H) D-T(alpha,beta;psi) are psi-Hilfer fractional derivatives left-sided and right- sided of order 1/p < alpha < 1, 0 <= beta <= 1, r > p and I-0+(beta(beta-1);psi) (.) and I-T(beta(beta-1);psi) (.) are psi-Riemann-Liouville fractional integrals left-sided and right-sided, gamma: [0, T] -> R and g : [0.T] x R -> R are continuous functions. Finally, we give some examples to illustrate the main results.