Multiple positive solutions to a coupled systems of nonlinear fractional differential equations

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.

differential equations, researchers of various field of mathematics, engineering, physics and computer science etc, gave much attention to study fractional differential equations. Another important application of fractional calculus has been found in condensed matter physics, where the fractional quantum Hall effect is one of the most attracting phenomena. For the present or planned technologies fractional order models are used by the implementation of an optical lattice setup. For detail, see Nielsen et al. (2013), Hagerstrom et al. (2012) and the reference there in. In networking systems it has been proved that several real networks in their degree of distribution obey a power-law. The presence of highly connected nodes in a scale-free network causes well known robustness against random failures. But on the other hand suffers from vulnerability to malicious attacks at their highly connected nodes. Fractional order models provide more realistic and accurate approach as compared to classical order models to study the afore said phenonmena, see Shang (2014). In last few years, the study of existence and uniqueness of solutions to boundary value problems for fractional order differential equations got much attention from many researchers and a number of research articles are available in the literature, we refer few of them in Deren (2015), Li et al. (2010), Khan and Shah (2015) and the reference therein. The iterative solutions to boundary and initial values problems of nonlinear fractional order differential equations were also studied by some authors (see Ahmad and Nieto 2008;Cui and Zou 2014;Shah et al. 2016 and the references therein). Moreover, 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). Bai and Lü (2005), have studied the existence of multiple solutions for the following boundary value problems where D α is the standard Riemann-Liouville fractional derivative of order α, and f : [0, 1] × [0, ∞) → [0, ∞) is continuous function. By means of classical fixed point theorems sufficient conditions were obtained for multiplicity of solutions. Recently, Goodrich (2010), considered the following class of nonlinear fractional differential equations with the given boundary conditions for multiplicity of positive solutions as where n > 3 and f : [0, 1] × [0, ∞) → [0, ∞) is continuous function. Many problems in applied sciences can be modeled as coupled system of differential equations with different type of boundary conditions. The coupled system of fractional order differential equations have many application in computer networking, see Li et al. (2015a, b), Suo et al. (2013). Boundary values problems for coupled systems with ordinary derivatives are well studied, however, coupled systems with fractional derivatives have attracted the attention quite recently. Most of the biological, physical, computer network model and chemical models etc, are in the form of coupled system (see Anastassiou et al. 2011;Chasnov 2009;Lia et al. 2015). Due to these important applications and uses of coupled 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), , Su (2009 and the references therein. As Bai and Feng (2004), established sufficient conditions for existence of positive solution to a coupled system of fractional differential equations as given by the where 1 < α, β < 1, f , g : [0, 1] × R → R are nonlinear continuous functions. Wang et al. (2010), developed sufficient conditions for existence and uniqueness of solution for the coupled system with three point boundary conditions of the form where 1 < α, β < 2, 0 ≤ a, b ≤ 1, 0 < ξ < 1, and f , g : [0, 1] × [0, ∞) → [0, ∞) are nonlinear continuous functions. 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 where n − 1 < α, β ≤ n, > 0, f , g : [0, ∞) → [0, ∞) are continuous. 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 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 u(t), v(t)) = 0; t ∈ (0, 1); n − 1 < α ≤ n, D β v(t) + ψ(t, u(t), v(t)) = 0; t ∈ (0, 1); n − 1 < β ≤ n, where n > 3, α − δ ≥ 1, β − γ ≥ 1 and 0 < δ, γ ≤ n − 2, , µ ∈ (0, ∞), 0 < η, , ξ < 1, φ, ψ : [0, 1] × [0, ∞) × [0, ∞) → [0, ∞) are continuous functions and D α , D β stand for Riemann-Liouville fractional derivative of order α, β respectively. Sufficient conditions are developed for uniqueness of solution of system (1), by using Perov's fixed theorem. Moreover by means of some classical fixed point theorems of cone type, we develop necessary and sufficient conditions under which the considered system has at least one , two or more positive solutions. Also, we develop conditions for nonexistence of positive solution for system (1). We also provide some examples to illustrate our main results.

Definition 1
The fractional integral of order α > 0 of a function y : (0, ∞) → R is defined by provided the integral is pointwise defined on (0, ∞).
Then A has at least one fixed point in P ∩ (� 2 \ � 1 ).
Definition 6 Jalili and Samet (2014) For a nonempty set X, the mapping d : Moreover the pairs (X, d) is called generalized metric space.
Definition 7 Jalili and Samet (2014) Let M = {M n,n ∈ R n×n + }, the system of all n × n matrices with positive element. For any matrix A the spectral radius is defined by

Main results
This section is concerned to the existence, uniqueness and multiplicity results of positive solutions for boundary value problem (1). We begin with the following lemma.

Lemma 9
Let y ∈ C[0, 1] then the boundary value problem where 0 < η < 1, ∈ (0, ∞), 0 < δ ≤ n − 2, has a unique positive solution given by where G α (t, s) is a Green's function given by Proof By applying I α and using Lemma 3, the general solution of linear boundary value problem (2) is given by With the help of boundary and initial conditions of Eq.
Let J = [ω, 1 − ω], then, we define the cone P ⊂ X × X by Now inview of Lemma 9, we can write system (1) as an equivalent coupled system of integral equations given as Then the fixed point of operator A coincides with the solution of coupled system (1).
Then A(P) ⊂ P and A : P → P is completely continuous, where A is defined in (7).
Proof To derive A(P) ⊂ P, let (u, v) ∈ P,then by Lemma 10, we have A(u, v) ∈ P. Further from property (P 4 ) and for all t ∈ J, we get Also from (P 3 ), we obtain Thus from (8) and (9), we have Similarly, one can write that Hence we have A(u, v) ∈ P ⇒ A(P) ⊂ P. Next by similar proof of Theorem 1 of  and using Arzelá-Ascoli's theorem, one can easily prove that A : P → P is completely continuous.
Theorem 12 Assume that φ and ψ are continuous + } is a matrix given by Then the system (1) has a unique positive solution (u, v) ∈ P.
Proof Let us define a generalized metric d : Obviously (X × X, d) is a generalized complete metric space. Then for any (u, v), (ū,v) ∈ X × X and using property (P 3 ) we get Similarly we can show that Then the system (1) has at least one positive solution in Also inview of Theorem 11, the operator A : → P is completely continuous. Let (u, v) ∈ �, such that (u, v) < r. Then, we have similarly, A 2 (u, v) < r 2 , thus A(u, v) < r. Therefore, thank to Lemma 4, we have A(u, v) ∈ �. Therefore A : → .
Proof As A defined in (7) is completely continuous.

Theorem 15
Assume that (C 1 ) − (C 3 ) hold. Further the following conditions are also satisfied: Then the boundary value problem (1) has at least one positive solutions. Moreover, if φ 0 = ψ 0 = 0 and φ ∞ = ψ ∞ = ∞, then the the boundary value problem (1) has at least one positive solution. Proof Proof is similar as like the proof of Theorem 14.
Proof Proof is like the proof of Theorem 16.

Examples
We conclude the paper with the following examples.
Example 21 Consider the system of non-linear fractional differential equations.
Also by simple calculation we can get that φ ∞ = 0 = ψ ∞ . Thus by Theorem 14, boundary value problem (19) has at least one positive solution.
Further for all (t, u, v Thus all the assumptions of Theorem 16 are satisfied and also a = 1. Hence by Theorem 16, the boundary value problem (21) has at least two positive solutions (u 1 , v 1 ) and (u 2 , v 2 ) which satisfy

Non-existence of positive solution
In this section, we discuss the non-existence of positive solution to the coupled system (1) of fractional order differential equations.
Proof On contrary let (u, v) be the positive solution of boundary value problem (1) . Then (u, v) ∈ P for 0 < t < 1 and which is contradiction. So boundary value problem (1) has no positive solution. Hence proof is completed. Proof Proof is just like the proof of Theorem 24, so we omit it.

Conclusion
In this article, we have developed sufficient conditions for the multiplicity results of positive solutions to a highly nonlinear coupled system of fractional order differential equations. Our paper is the generalization of the work carried out in Goodrich (2010), Jalili and Samet (2014), Rehman and Khan (2010). In Jalili and Samet (2014), the authors studied the concerned coupled system with homogenous boundary conditions involving fractional order derivative, but we extended this work with taking non-homogenous boundary condition involved fractional order derivative of Riemann-Liouville type. By using classical fixed point theorems, we have successfully developed conditions under which the considered coupled system has multiple solutions. Moreover, uniqueness and non existence results have also been established. Numerous examples have been provided which justify the results developed by us.