A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator

In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the psi-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.1118710

Multiplicity of solutions for fractional-order differential equations via the k(x)-Laplacian operator and the genus theory

In this paper, we investigate the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the x-fractional space (H y,β;x k(x)) (△). Using the Genus Theory, the Concentration-Compactness Principle, and the Mountain Pass Theorem, we show that under certain suitable assumptions the considered problem has at least k pairs of non-trivial solutions.FacultyReviewe