Existence of solutions for nonlinear problems involving mixed fractional derivatives with \(p(x)\)-Laplacian operator (Q6589986)

Resonance problems are among the most intriguing and widely studied topics in the field of differential boundary value problems (BVPs). In the present paper, the authors get the existence of solutions for the functional BVP (1.1) involving the mixed fractional derivatives with the \(p(x)\)-Laplacian operator in the case of non-resonance and the case of resonance. Here, the \(p(x)\)-Laplacian operator is a generalization of the \(p\)-Laplacian operator. The authors establish the solvability results of the problem for the non-resonant case using the Banach fixed point theorem and demonstrate the existence of solutions in the resonance case through an extended version of Mawhin's coincidence theory. The existence of solutions for the functional BVPs for mixed fractional differential equation with the \(p(x)\)-Laplacian has not been studied and contributes to the theoretical and applied nature. Also, several examples are provided to validate the main results.