Subscribe Now Subscribe Today
Research Article
 

Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations



Ahmad Adawi
 
Facebook Twitter Digg Reddit Linkedin StumbleUpon E-mail
ABSTRACT

There was limited information on the analysis of differential equations of fractional order. Therefore, study on fractional differential equations is essential to understand the solution behavior of many applications in sciences. In this study, initial value problems are discussed for the fractional differential equations and various criteria on existence and uniqueness are obtained.

Services
Related Articles in ASCI
Similar Articles in this Journal
Search in Google Scholar
View Citation
Report Citation

 
  How to cite this article:

Ahmad Adawi , 2010. Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations. Journal of Applied Sciences, 10: 2132-2135.

DOI: 10.3923/jas.2010.2132.2135

URL: https://scialert.net/abstract/?doi=jas.2010.2132.2135

INTRODUCTION

Fractional differential equations have been of great interest recently. In cause, in part to both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, chemistry, mechanics engineering, etc. For more details one can see from Hadid et al. (1996a), Diethelm and Ford (2002), Kilbas et al. (2006), Rabha and Momani (2007), Lin (2007), Kosmatov (2009) and Lakshmikantham and Vatsala (2008).

This study is concerned with the existence and uniqueness of solutions for initial value problems of fractional order of the form:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(1)

with conditions:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(2)

Here, zεRn, tεI⊆R, t0 εI⊆R, z0, z1, ..., zm-1εRn and h is a function from IxRnm to Rn

Upon substituting:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(3)

and

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(4)

System (1)-(2) amounts to the system:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential EquationsImage for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(5)

with the condition:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(6)

where,

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(7)

and In is the identity matrix.

System Eq. 5 and 6 has the form:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(8)

with the condition:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(9)

The following existence and uniqueness theorems for system Eq. 8 and 9 were proved by Hadid (1995) and Hadid et al. (1996b).

Theorem 1 (Hadid, 1995): If (t, x) is continuous in a closed and bounded rectangular box:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

Then there exists at least one solution x(t) of Eq. 8-9 on 0<t0<t≤t0+β for some β>0.

Theorem 2 (Hadid et al., 1996b): The initial value problem Eq. 8-9 has a unique solution defined on the interval 0<t0<t≤t0+a if the function f(t, x) is continuous and bounded in the strip:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

And satisfies in this strip the Lipschitz condition:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

for some positive constant L.

The above theorems will be used to investigate the results.

MAIN RESULTS

In this section we give existence and uniqueness results for the IVP (Eq. 1 and 2).

Theorem 3: If h(t, Y) is continuous in a closed and bounded rectangular box:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

 

where,

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

and

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

Then there exists at least one solution z(t) of (1)-(2) on 0<t0<t≤t0 +βfor some β>0

Proof: By the substitution Eq. 3 and 4, system Eq. 1 and 2 is converted into system Eq. 5 and 6 which has the form of system Eq. 8 and 9. Now using Theorem 1 with:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

Due to the continuity of the function h(t, Y) in D, the function f is continuous in D. By Theorem 1 system (5)-(6) has at least one solution Y(t) on 0<t0<t≤t0+B for some β>0 So system (1)-(2) has at least one solution z(t) on 0<t0 <t≤t0 +β for some β>0.

Theorem 4: If the function h(t,Y) is continuous and bounded in the strip:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

where,

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

and

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

and satisfies in this strip the Lipschitz condition:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

for some positive constant L. Then the initial value problem (1)-(2) has a unique solution z(t) defined on the interval 0<t0<t≤t0+a

Proof: Upon substituting Eq. 3 and 4, system Eq. 1 and 2 amounts to system Eq. 5 and 6. Common practice involves the function:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

Since h is continuous and bounded in the strip 0<t0<t≤t0+a, ||Y||<∞ the function f satisfies in this strip:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

for some positive number L. From Theorem 2, system Eq. 5 and 6 has a unique solution Y(t) on the interval 0<t0<t≤t0+a and so system Eq. 1 and 2 has a unique solution z(t) on this interval.

Theorem 5: If A0(t), A1(t), ..., Am-1 (t) are continuous and bounded nxn matrix functions on some interval 0<t0<t≤t0+a and g(t) is a continuous and bounded vector function on some interval, then the linear fractional differential equation:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential EquationsImage for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(10)

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations
(11)

has a unique solution z(t) on the interval 0<t0<t≤t0+a

Proof: By the substitutions Eq. 3 and 4, system Eq. 1 and 2 can be written in the form:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

with the condition:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

where,

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

Let

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

From the definition of the function h it seems clear that:

Image for - Existence and Uniqueness Theorems of Higher Order Fractional Differential Equations

The desired result now follows from Theorem 2.

REFERENCES
1:  Diethelm, K. and N. Ford, 2002. Analysis of fractional differential equations. J. Math. Anal. Appl., 265: 229-248.
CrossRef  |  

2:  Hadid, S.B., 1995. Local and global existence theorems on differential equations of non-integer order. J. Fract. Calc., 7: 101-105.

3:  Hadid, S.B., B. Masaedeh and S. Momani, 1996. On the existence of maximal and minimal solutions of differential equations of non integer order. J. Fract. Calc., 9: 41-44.

4:  Hadid, S.B., A. Taani and S. Momany, 1996. Some existence on differential equations of generalized order through a fixed-point theorem. J. Fract. Calc., 9: 45-49.

5:  Kilbas, A.A., H.M. Srivastava and J.J. Trujillo, 2006. . Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, ISBN-13: 978-0-444-51832-3.

6:  Kosmatov, N., 2009. Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal., 70: 2521-2529.
CrossRef  |  

7:  Lakshmikantham, V. and A.S. Vatsala, 2008. Basic theory of fractional differential equations. Nonlinear Anal., 69: 2677-2682.
CrossRef  |  

8:  Lin, W., 2007. Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl., 332: 709-726.
CrossRef  |  

9:  Rabha, W.I. and S. Momani, 2007. On the existence and uniqueness of solutions of a class of fractional differential equations. J. Math. Anal. Appl., 334: 1-10.
CrossRef  |  

©  2021 Science Alert. All Rights Reserved