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Open AccessArticle

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

by Alberto Cabada 1,*,† and Om Kalthoum Wanassi 2,†
1
Departamento de Estatística, Análise Matemática e Optimización Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2
Department of Mathematics, University of Monastir, Monastir 5000, Tunisia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(2), 255; https://doi.org/10.3390/math8020255
Received: 9 January 2020 / Revised: 5 February 2020 / Accepted: 7 February 2020 / Published: 14 February 2020
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the α -Riemann-Liouville fractional derivative, with α ( 1 , 2 ] . To deduce the existence and non-existence results, we first study the linear equation, by deducing the main properties of the related Green functions. We obtain the optimal set of parameters where the Green 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. View Full-Text
Keywords: fractional equations; Green functions; integral boundary conditions; fixed-point index; existence and non-existence fractional equations; Green functions; integral boundary conditions; fixed-point index; existence and non-existence
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Cabada, A.; Wanassi, O.K. Existence Results for Nonlinear Fractional Problems with Non-Homogeneous Integral Boundary Conditions. Mathematics 2020, 8, 255.

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