Annals of Communications in Mathematics

ABSTRACT.This paper is devoted to find the form of the solution of the following rational difference equation:\[x_{n+1}=\frac{x_{n-3}x_{n-5}}{x_{n-1}\left( \pm 1 \pm x_{n-3}x_{n-5} \right)},\]where the initial conditions \( x_{-5}, x_{-4}, x_{-3}, x_{-2}, x_{-1}, x_{0} \) are arbitrary non-zero real numbers. Also, we study the behavior of the solutions.

AbstractThis manuscript concerns the fuzzy differential equation involving ψ-Hilfer type fractional derivative with nonlocal condition. By using successive approximation, we obtain the existence, uniqueness results of solution for ψ-Hilfer fuzzy differential equation. Further, nonlocal conditions are extended to the existence results. Furthermore, an application is shown to demonstrate the theoretical conclusions utility.

AbstractThe sufficient conditions are estabilished for the existence of solutions for a class of boundary value problems for pantograph equations involving the ψ-type fractional deerivative in Caputo sense.