Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions

Version 1 : Received: 9 January 2020 / Approved: 11 January 2020 / Online: 11 January 2020 (10:23:08 CET)

A peer-reviewed article of this Preprint also exists.

Cabada, A.; Wanassi, O.K. Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions. Mathematics 2020, 8, 255. Cabada, A.; Wanassi, O.K. Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions. Mathematics 2020, 8, 255.

Journal reference: Mathematics 2020, 8, 255
DOI: 10.3390/math8020255

## Abstract

This paper deals with the study of the existence and non existence of solutions of a three parameter's family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the $\alpha$-Riemann-Liouville fractional derivative, with $\alpha \in (1,2]$. In order to deduce the existence and non existence results, we first study the linear equation, by deducing the main properties of the related Green's functions. We obtain the optimal set of parameters where the Green's function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.

## Keywords

Fractional Equations; Green's Functions; Integral Boundary Conditions; Fixed Point Index; Existence and Non-existence

## Subject

MATHEMATICS & COMPUTER SCIENCE, Analysis

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