# Study of a class of arbitrary order differential equations by a coincidence degree method

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Boundary Value Problems

, 2017:111

First Online: 03 August 2017Received: 08 May 2017Accepted: 18 July 2017

Abstract

In this manuscript, we investigate some appropriate conditions which ensure the existence of at least one solution to a class of fractional order differential equations FDEs provided by $$\textstyle\begin{cases} -{}^{C}\mathbf{D}^{q} zt=\theta t,zt; \quad t\in \mathfrak{J}=0,1, q\in 1, 2, \\ zt\vert {t=0}=\phi z,\qquad z1=\delta{}^{C}\mathbf{D}^{p} z\eta ,\quad p, \eta \in 0,1. \end{cases}$$ The nonlinear function \\theta :\mathfrak{J}\times \mathbf{R} ightarrow \mathbf{R}\ is continuous. Further, \\delta \in 0, 1\ and \\phi \in C\mathfrak{J},\mathbf{R}\ is a non-local function. We establish some adequate conditions for the existence of at least one solution to the considered problem by using Grönwall’s inequality and a priori estimate tools called the topological degree method. We provide two examples to verify the obtained results.Keywordsfractional order differential equations Caputo derivative condensing operator Grönwall’s inequality topological degree method MSC34A08 35R11  Download fulltext PDF

Autor: Nigar Ali - Kamal Shah - Dumitru Baleanu - Muhammad Arif - Rahmat Ali Khan