Analysis study on multi-order $ \varrho $-Hilfer fractional pantograph implicit differential equation on unbounded domains

Abstract

In this paper, we investigate a multi-order $ \varrho $-Hilfer fractional pantograph implicit differential equation on unbounded domains $ (a, \infty), a\geq 0 $. The existence and uniqueness of solution are established for a such problem by utilizing the Banach fixed point theorem in an applicable Banach space. In addition, stability of the types Ulam-Hyers ($ \mathcal UH $), Ulam-Hyers-Rassias ($ \mathcal UHR $) and semi-Ulam-Hyers-Rassias (s-$ \mathcal UHR $) are discussed by using nonlinear analysis topics. Finally, a concrete example includes some particular cases is enhanced to illustrate rightful of our results.

Keywords

• $ \varrho $-hilfer fractional derivatives
• multi-order
• pantograph implicit differential equations
• ulam-hyers-rassias stability
• fixed point theorems