Existence Result to a Kirchhoff ψ-Hilfer Fractional Equations with p-Laplacian Operator Via Nehari Method

The goal of this work is to prove the existence of multiple solutions of the following Kirchhoff equation with $\psi$-Hilfer fractional derivative involving p-Laplacian operator, given by $(P) \begin{cases}\left(a+\left.\left.b \int_0^T\right|_0 D_t^{\alpha, \beta ; \psi}(u(t))\right|^p\right)^{p-1}{ }_t D_T^{\alpha, \beta ; \psi}\left(\varphi_p\left({ }_0 D_t^{\alpha, \beta ; \psi} u(t)\right)\right) \\=f(t, u(t))+\lambda g(t)|u(t)|^{q-2} u(t), t \in(0, T), \\I^{\alpha(\alpha-1), \psi} u(0)=I^{\alpha(\alpha-1), \psi} u(T)=0 .\end{cases}$ where $D^{\alpha,\beta;\psi}$ is $\psi$-Hilfer fractional derivative of order $\alpha$ and type $\beta,a$ and b are positives constants, $q-1\lt r,0\leq\beta\leq 1,\ \displaystyle \frac{1}{p}\lt \alpha\lt1,\lambda$ is a positive parameter, $I^{\alpha(\alpha-1),\psi}$ is $\psi$-Riemann-Liouville fractional integral, $g:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and $f\in C^{1}([0,T]\times\mathbb{R},\mathbb{R})$. Mathematics Subject Classification: 26A33, 34A08, 34B10, 34B15,35J2.