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# Application of Fixed-Point Theory for a Nonlinear Fractional Three-Point Boundary-Value Problem

by Ehsan Pourhadi 1,2, Reza Saadati 2 and Sotiris K. Ntouyas 3,4,* 1
International Center for Mathematical Modelling in Physics and Cognitive Sciences, Department of Mathematics, Linnaeus University, SE-351 95 Växjö, Sweden
2
Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran
3
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
4
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(6), 526; https://doi.org/10.3390/math7060526
Received: 10 February 2019 / Revised: 28 May 2019 / Accepted: 6 June 2019 / Published: 10 June 2019
Throughout this paper, via the Schauder fixed-point theorem, a generalization of Krasnoselskii’s fixed-point theorem in a cone, as well as some inequalities relevant to Green’s function, we study the existence of positive solutions of a nonlinear, fractional three-point boundary-value problem with a term of the first order derivative $( a C D α x ) ( t ) = f ( t , x ( t ) , x ′ ( t ) ) , a < t < b , 1 < α < 2 ,$ $x ( a ) = 0 , x ( b ) = μ x ( η ) , a < η < b , μ > λ ,$ where $λ = b − a η − a$ and $a C D α$ denotes the Caputo’s fractional derivative, and $f : [ a , b ] × R × R → R$ is a continuous function satisfying the certain conditions. View Full-Text
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Pourhadi, E.; Saadati, R.; Ntouyas, S.K. Application of Fixed-Point Theory for a Nonlinear Fractional Three-Point Boundary-Value Problem. Mathematics 2019, 7, 526.