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Journal of Pure and Applied Mathematics

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Kamal Shah*
 
Department of Mathematics, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan, Email: kamalshah408@gmail.com
 
*Correspondence: Kamal Shah, Department of Mathematics, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan, Email: kamalshah408@gmail.com

Received: 05-Jun-2018 Accepted Date: Jun 25, 2018; Published: 29-Jun-2018, DOI: 10.37532/2752-8081.18.2.8

Citation: Shah K. Coupled systems of boundary value problems for nonlinear fractional differential equations. J Pur Appl Math. 2018;2(2):14-17.

This open-access article is distributed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC) (http://creativecommons.org/licenses/by-nc/4.0/), which permits reuse, distribution and reproduction of the article, provided that the original work is properly cited and the reuse is restricted to noncommercial purposes. For commercial reuse, contact reprints@pulsus.com

Abstract

In this article, we study coupled systems of boundary value problems for fractional order differential equations. We use the idea of the Generalized matric space to develop necessary and sufficient conditions for uniqueness of positive solutions of the system. We also obtain sufficient conditions for existence of at least one solution via nonlinear differentiation of Leray Schauder type. We include an example to illustrate our main results.

Keywords

Coupled system; Fractional differential equations; Riemann-liouville boundary conditions; Existence; Uniqueness results

Recently the theory on existence and uniqueness of solutions of fractional differential equations have attracted much attentions and a large number of research articles on the solvability of nonlinear fractional differential equations are available. We refer to [1-4] and the references therein for some of the recent development in the theory. On the other hand, coupled systems of boundary value problems for non linear fractional differential equations are not well studied and only few results can be found dealing with existence and uniqueness of solutions [5-8]. Su [5] developed sufficient conditions for existence of solutions to the following coupled systems of two point boundary value problems

Dα u(t) = f (t,v(t),Dμ v(t)), Dβ u(t) = g(t,u(t), Dν u(t)), 0 < t <1, u(0) = u(1) = v(0) = v(1) = 0,

where 1<α ,β ≤ 2,μ,ν satisfies α −μ and β −ν ≥1 and f , g :[0,1]× R× R→ R are continuous and Dis the standard Rieman- Liouville derivative. Wang et al. [8] obtained sufficient conditions for existence and uniqueness of positive solutions to the following coupled systems of nonlinear three-point boundary values problems

Dα u(t) = f (t,v(t)), Dβ v(t) = g(t,u(t)), 0 < t <1

u(0) = 0, v(0) = 0, u(1) = au(η ),v(1) = bv(η ),

where 1<α ,β ≤ 2,0 ≤ a,b ≤1and 0 <η <1 and

f , g :[0,1]×[0,∞)→[0,∞) are continuous.

Motivated by the above studies, we develop some new existence and uniqueness results for the following coupled systems of nonlinear boundary values problems

(1.1)

image

where 2 <α ,β ≤ 3 and f , g : I × R× R→ R are continuous and D, I denote Riemann-Liouville’s fractional derivative and fractional integral respectively. We use Perov fixed point theorem [9] and Leray-Schauder fixed point theorem to obtain sufficient conditions for existence and uniqueness results. We also provide an example to illustrated our results.

Preliminaries

We recall some fundamental results and definitions [10,11].

Definition 2.1

The fractional integral of order α ∈R+ of a function y : (0,∞)→ R is defined by

image

provided the integral converges.

Definition 2.2

The Riemann-Liouville fractional order derivative of a function y : (0,∞)→ R is defined by

image

where n = [α ]+1and[α ] represents the integer part of α provided that the right side is point wise defined on (0,∞).

Lemma 2.3

The following result holds for fractional derivative and integral image, for arbitraryimage.

Lemma 2.4

(7) Let X be a Banach space with image closed and convex. Let Ω be a relatively open subset of image with 0∈Ω and image be a continuous and compact(completely continuous) mapping. Then either

1. The mappingT has a fixed point in image or

2. There exist μ ∈∂Ωand k ∈(0,1) with Ω = kTu.

Definition 2.5

For a nonempty set Z, a mapping d : Z × Z → Rn is called a generalized metric on Z if the following hold

image

(M2) d(u,v) = d(v,u),∀u,v∈ X, (symmetric property);

(M3) d(x, y) ≤ d(x,v) + d(v,u) + d(u,v),∀x, y,u,v,∈ X, (tetrahedral inequality).

Note: The properties such as convergent sequence, cauchy sequence, open/ closed subset are the same for generalized metric spaces as hold for the usual metric spaces.

Definition 2.6

For an n× n matrix A, the spectral radius is defined by image, whereimage are the eigenvalues of the matrix A.

Lemma 2.7

(11), Let (Z, d) be a complete generalized metric space and let T : Z →Z be an operator such that there exist a matrix A∈M with d(Tu,Tv) ≤ Ad(u,v), for all u,v∈Z. If ρ (A) <1, then T has a fixed point Z* ∈Z, further for any Z0 the iterative sequence Zn +1 = TZn converges to Z0.

Lemma 2.8

An equivalent Fredholm integral representation of the system of boundary value problems (1.1) is given by (2.1)

image, whereimage are Green’s functions given by

(2.2) image

(2.3) image

Proof

Applying the operator Iα on the first equation of (1.1) and using lemma (2.3), we have

(2.4) image.

The boundary conditions image and image.

Hence, (2.4) takes the form

(2.5)

image

Similarly, by the same process with the second equation of the system, we obtain the second part of (2.1).

Lemma 2.9

(6) The Green’s function image of the system (2.1) has the following properties

image is continuous function on the unit square for all

(t, s)∈[0,1]×[0,1];

image for all (t, s)∈[0,1] and image for all (t, s)∈(0,1);

image;

(P4) there exist a constant γ ∈(0,1) such that

image for θ ∈(0,1), s∈[0,1] where

image.

Existence of positive solutions

Define U = {u(t) | u(t)∈C[0,1]} endowed with the Chebychev normimage. Further, define the normsimage.

Then, the product spaces image are Banach spaces. Define the conesimage by

image and

image, where J = [θ ,1−θ ], θ ∈(0,1).

Lemma 3.1

Assume that f , g :[0,1]× R× R→ R are continuous. Then (u,v)∈U ×V is a solution of (2.1), if and only if (u,v)∈U ×V is a solution of system of Fredholm integral equations (1.1).

Proof

The proof of lemma (3.1) is similar to proof of lemma (3.1) in [6].

Define T :U ×V →U ×V by

(3.1)

image

By lemma (3.1) the problem of existence of solutions of the integral equations (2.1) coincide with the problem of existence of fixed points of T.

Lemma 3.2

Assume that f , g :[0,1]×[0,∞)×[0,∞)→[0,∞) are continuous.

Then image and image, where T is defined by (3.1).

Proof

The relation image easily follow from the properties (P1) and (P2) of lemma (2.9) and all we need to show thatimage holds. Forimage, we haveimage and in view of property (P4) of lemma (2.9), for all t∈J , we obtain

(3.2)

image.

Hence, it follows that

image.

Similarly, we obtain

image.

It follows that

image,

which implies that image.

Lemma 3.3

Assume that f , g :[0,1]× R× R→ R are continuous then image is completely continuous.

Proof.

We omit the proof, because it is similar to the proof of lemma (3.2) in [6].

Lemma 3.4

Assume that f and g are continuous on [0,1] ×R× R→ R and there exist image such that the following holdimagefor image;

image for image;

image, Where the matriximage is defined by

image

Then the system (2.1) has a unique positive solution image.

Proof

Define a generalized metric d :U ×V ×U ×V → R2 by

image

Obviously (U ×V, d) is a generalized complete metric space. For any image using the property (P3) and (H3), we obtain

image

image

Similarly, we obtain

image.

Hence, it follows that

image, Where

image

image. Hence by lemma (2.7), the system (2.1) has a unique positive solutions.

Lemma 3.5

Let f and g are continuous on [0,1]× R× R→ R and there exist

image satisfying

image

image.

Then the system (2.1) has at least one positive solution (u,v) in

image.

Proof

Choose image and defineimage.

By lemma (3.3), the Operator image is completely continuous. Chooseimage and (u,v)∈∂Ωsuch that image.

Then, by properties (P1), (P3) and (H4), we obtain for all t∈[0,1]

image

Similarly, we obtain image, henceimage, which shows that (u,v)∉∂Ω.Thus by Schauder fixed point theorem,T has a fixed point in image.

Examples

Example 4.1

Consider the following coupled systems of boundary value problems

image Here

image.

Moreover

image.

Here, ρ (A) = 4.61×10−2 <1, hence by lemma (3.4) the BVP(4.1) has a unique solution. For f and g, we have

image and by simple calculation, we obtain

image

image. Hence by using lemma (3.5), BVP (4.1) has at least one positive solution.

Conclusion

With the help of Banach theorem and nonlinear Leray Schauder type, we have developed an existence theory to a coupled system of nonlinear FDEs. The concerned results have been successfully obtained and demonstrated by suitable example.

REFERENCES

 
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