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.

Background

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). Also, 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. For the aforesaid application, see Mainardi (1995), Metzler and Klafer (2000), Scher and Montroll (1975), Wang et al. (2011). Moreover, considerable attention has been given to study mathematical epidemiology in term of fractional order models, this is due to the fact that fractional order models of epidemic disease are more realistic and provide great information as compared to classical order models. In last few years fractals and Chaos are also studied by using the tools of fractional differential equations. Besides from the above discussion, fractional order differential equations have also many applications in the fields of aerodynamics, fluid dynamics, physical chemistry, economics, polymerrheology, regular variation in thermodynamics, biophysics, blood flow phenomena, etc. Due to the important applications and uses of fractional 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 problemswhere is the standard Riemann–Liouville fractional derivative of order and 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 aswhere and 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), Shah and Khan (2015), Su (2009), Shah et al. (2015) 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 thewhere 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 formwhere and 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 bywhere 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 equationswhere are continuous.

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 conditionswhere and are continuous functions and stand for Riemann–Liouville fractional derivative of order 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.

Preliminaries

To proceed further, we recall some basic definitions and well known results of functional analysis, fixed point theory and fractional calculus (see i.e. Agarwal et al. 2004; Deimling 1985; Jalili and Samet 2014; Krasnoselskii 1964; Miller and Ross 1993; Podlubny 1993; Szilárd 2003; Zeidler 1986).

Definition 1

The fractional integral of order of a function is defined byprovided the integral is pointwise defined on .

Definition 2

The Riemann–Liouville fractional derivative of order of a function is defined bywhere represents integer part of .

The following results need in the sequel,

Lemma 3

Ahmad and Nieto (2009a, b) Letthen for arbitrarywe have

Lemma 4

Agarwal et al. (2004) LetXbe a Banach space withclosed and convex. LetUbe a relatively open subset ofPwithandbe a continuous and compact(completely continuous) mapping. Then either

The mappinghas a fixed point inor

There existandwith

Lemma 5

Podlubny (1993) LetPbe a cone of real Banach spaceXand letandbe two bounded open sets inXsuch that. Let operatorbe completely continuous operator. If one of the following two conditions holds:

for all, for all

for all, for all.

Then has at least one fixed point in

Definition 6

Jalili and Samet (2014) For a nonempty set X, the mapping is called a generalized metric on X if with satisfies Moreover the pairs (X, d) is called generalized metric space.

(symmetric property)

(tetrahedral inequality).

Definition 7

Jalili and Samet (2014) Let , the system of all matrices with positive element. For any matrix A the spectral radius is defined by , where are the eigenvalues of the matrix A.

Lemma 8

Jalili and Samet (2014) Let (M, d) be a complete generalized metric space and letbe an operator such that there exists a matrixwithif, thenhas a fixed point. Further, for anythe iterative sequenceconverges tow.

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

Letthen the boundary value problemwherehas a unique positive solution given bywhereis a Green’s function given bywhere

Proof

By applying and using Lemma 3, the general solution of linear boundary value problem (2) is given byWith the help of boundary and initial conditions of Eq. (2) and , we get and using we haveand thus (4) becomeswhere is the Green’s function of linear boundary value problem (2). Similarly, we can obtain where is the Green’s function for the second equation of the system (1) and given bywhere

We recall the following lemma found in Jalili and Samet (2014) and Yang (2012). The proof is omitted because the proof is like the proof given in Goodrich (2010).

Lemma 10

Letbe the Green’s function of (1). This Green’s functionG(t, s) has some properties given bywhere

G(t, s) is continuous function on the unit square for all

for allandfor all

for each

Further we define some fundamental results which will be used throughout in this paper. Let us define endowed with the norm Further the norm for the product space we define as Obviously is a Banach space.

Let , then, we define the cone byNow inview of Lemma 9, we can write system (1) as an equivalent coupled system of integral equations given asLet be the operator defined asThen the fixed point of operator coincides with the solution of coupled system (1).

Theorem 11

Assume thatare continuous. Thenis completely continuous, whereis defined in (7).

To derive , let ,then by Lemma 10, we have . Further from property and for all , we getAlso from , we obtainThus from (8) and (9), we haveSimilarly, one can write thatHence we have Next by similar proof of Theorem 1 of Shah and Khan (2015) and using Arzel-Ascoli’s theorem, one can easily prove that is completely continuous.

Theorem 12

Assume thatandare continuous on, and there existthat satisfyThen the system (1) has a unique positive solution.

forand

for and

whereis a matrix given by

Let us define a generalized metric byObviously is a generalized complete metric space. Then for any and using property we getSimilarly we can show thatThus we havewhereAs by means of Lemma 8, system (1) has a unique positive solution.

Theorem 13

Letandare continuous onand there existsatisfying:

Then the system (1) has at least one positive solution in

Defined with

Also inview of Theorem 11, the operator is completely continuous. Let , such that Then, we havesimilarly, , thus . Therefore, thank to Lemma 4, we have Therefore

Let there exist and such that Then inview of assumptions and using Property of Lemma 10, we obtain for all similarly, one can show that

From which, we have which is a contradiction that as Thus by mean of Lemma 4, has at least one fixed point

Next we use the following assumptions and notations:

are continuous and uniformly with respect to t on [0, 1].

defined in Lemma 10 satisfy

Let these limit hold

Theorem 14

If the assumptionshold and one of the following conditions is also satisfied:Then boundary value problem (1) has at least one positive solution.

Moreover, and

there exist two constantswithsuch thatandare nondecreasing onfor all,

As defined in (7) is completely continuous.

Case I. Let the conditions of hold. Taking then there exists a constant such that where r1 > 0, and satisfies the conditionsSo for we haveAnalogouslyTherefore, we haveAlso for and there exists a constant say such that for where satisfies the conditions Let , then Now setting

So for any we obtainSimilarly , as , thus, we haveCase II. If assumptions in hold, then inview of definition of P for , we have Then from , we haveSimilarly one can also obtain for , we getAlso for , we get that for . Then from , one can getSimilarly, one can also obtain that . Hence, we haveNow inview of application of Lemma 5 to (10) and (11) or (12) and (13) implies that has a fixed point or such that and . From which it follows that boundary value problem (1) has at least one positive solution.

Theorem 15

Assume thathold. Further the following conditions are also satisfied:Then the boundary value problem (1) has at least one positive solutions. Moreover, ifandthen the the boundary value problem (1) has at least one positive solution.

Ifand

Proof is similar as like the proof of Theorem 14.

Theorem 16

Assume thathold. Also the following conditions are also satisfied:Then the boundary value problem (1) has at least two positive solutionssuch that

IfandMoreover, is also hold;

there existssuch that

and.

Let hold. Select such that . Now if

Then like the proof of Theorem 14, we haveNow, if

Then like the proof of Theorem 14, we haveAlso from we getSimilarly, we have as . Hence, we haveNow by the use of Lemma 5 to (15) to (17) gives that has a fixed point and a fixed point in Hence it implies that the boundary value problem (1) has at least two positive solutions such that and . Thus relation (14) holds. The proof is completed.

Theorem 17

Assume thathold and also the following conditions are satisfied:Then the boundary value problem (1) has at least two positive solutions.

andand

there existsuch thatsuch that

Proof is like the proof of Theorem 16.

Similarly for multiplicity the following theorems can be easily deduced:

Theorem 18

Lethold. If there exist 2mpositive numberswithandsuch that

forand

Then the boundary value problem (1) has at leastm-positive solutionssatisfying

Theorem 19

Suppose thatholds. If there exist 2mpositive numbers, withsuch thatThen the boundary value problem (1) has at leastm-positive solutionssatisfying

andare non-decreasing onfor all

Examples

We conclude the paper with the following examples.

Example 20

Consider the coupled system as followSince Also as and Thenwhere from which we haveBy standard calculation, we can obtain that hence by the use of Theorem 13 boundary value problem (18) has a unique positive solution.

Example 21

Consider the system of non-linear fractional differential equations.where and and are continuous.

Now , similarly Also by simple calculation we can get that Thus by Theorem 14, boundary value problem (19) has at least one positive solution.

Example 22

Consider the following boundary value problemfrom (20), we have as By simple calculation we obtain that and Thus by Theorem 15, the boundary value problem (20) has a positive solution.

Example 23

Consider the following boundary value problemwhere as Also, by simple calculation we obtain that and

Further for all we haveThus all the assumptions of Theorem 16 are satisfied and also Hence by Theorem 16, the boundary value problem (21) has at least two positive solutions and 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.

Theorem 24

Assume thathold andandfor allthen the boundary value problem (1) has no positive solution.

On contrary let (u, v) be the positive solution of boundary value problem (1) . Then for andwhich is contradiction. So boundary value problem (1) has no positive solution. Hence proof is completed.

Theorem 25

Letholds and iffor alland. Then boundary value problem (1) has no positive solution.

Proof is just like the proof of Theorem 24, so we omit it.

Example 26

In this section we provide an example which illustrate the results of Theorems 24 and 25 respectively. Consider the system of non linear fractional differential equations:Since holds and also as .where and

Case I: Now implies that and Thus by Theorem 24, boundary value problem (22) has no positive solution.

Case II: Also and

Then by Theorem 25, boundary value problem (22) has no solution.

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.