Abstract
In this article, we study existence, uniqueness and nonexistence of positive solution to a highly nonlinear coupled system of fractional order differential equations. Necessary and sufficient conditions for the existence and uniqueness of positive solution are developed by using Perov’s fixed point theorem for the considered problem. Further, we also established sufficient conditions for existence of multiplicity results for positive solutions. Also, we developed some conditions under which the considered coupled system of fractional order differential equations has no positive solution. Appropriate examples are also provided which demonstrate our results.
Highlights
In recent years it is mainly proved that fractional differential equations are the best tools in the mathematical modeling of many phenomena in various field of physics, electrochemistry, viscoelasticity, control theory, image and signal processing etc
Fractional differential equations are used in the modeling of various phenomena such as nonlinear oscillation of earthquake, Nutting’s law of charge transport in amorphous semi conductors, fluid dynamics, traffic model and non-Markorian diffusion process with memory
The aim of this paper is to study the existence, uniqueness as well as non-existence conditions for positive solution to the following system of non-linear fractional order differential equations with four point boundary conditions
Summary
In recent years it is mainly proved that fractional differential equations are the best tools in the mathematical modeling of many phenomena in various field of physics, electrochemistry, viscoelasticity, control theory, image and signal processing etc, (see Hilfer 2000; Kilbas et al 2006; Lakshmikantham et al 2009; Podlubny 1999; Rossikhin and Shitikova 1997). Existence and multiplicity of positive solutions to nonlinear boundary values problem of fractional order differential equations have been studied by many authors by using classical fixed point theorems, for example see Ahmad and Nieto (2009b), Bai (2008), Cui et al (2012), Xu et al (2009). Systems of fractional order differential equations, considerable attention was given to study coupled system for the existence, uniqueness and multiplicity of positive solutions, for detail we refer Miller and Ross (1993), Shah and Khan (2015), Su (2009), Shah et al (2015) and the references therein. Rehman and Khan (2010), established sufficient conditions for multiplicity results for positive solutions to the following coupled system of nonlinear boundary value problem of fractional differential equations as given by. Jalili and Samet (2014), studied existence and uniqueness as well as multiplicity of positive solutions to the following coupled system of boundary value problems of fractional differential equations. ≤ f1(s)Gα(1, s)ds u − u + g1(s)Gα(1, s)ds v − v
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