- Research Article
- Open Access

# Employing of Some Basic Theory for Class of Fractional Differential Equations

- Azizollah Babakhani
^{1}Email author and - Dumitru Baleanu
^{2, 3}Email author

**2011**:296353

https://doi.org/10.1155/2011/296353

© A. Babakhani and D. Baleanu 2011

**Received:**19 September 2010**Accepted:**15 November 2010**Published:**29 December 2010

## Abstract

Basic theory on a class of initial value problem of some fractional differential equation involving Riemann-Liouville differential operators is discussed by employing the classical approach from the work of Lakshmikantham and A. S. Vatsala (2008). The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered. Our work employed recent literature from the work of (Lakshmikantham and A. S. Vatsala, (2008)).

## Keywords

- Fractional Order
- Fractional Derivative
- Global Existence
- Fractional Differential Equation
- Initial Value Problem

## 1. Introduction

where and . Recall that is the Banach space of continuous functions from the interval into endowed with uniform norm.

We begin in this section with the recall of some definitions and results for fractional calculus which are used throughout this paper [4, 19, 20].

where . We denote by and by . Also and refer to and , respectively.

Proposition 1.1.

Let and . Then

- (i)
,

- (ii)
,

- (i)
can be found in [19, page 53] and (ii) is an immediate consequence of (1.6) and (i).

## 2. Strict and Nonstrict Inequalities

Note.

We now first discuss a fundamental result relative to the fractional inequalities.

Theorem 2.1.

Let , , and

- (i)
,

- (ii)
,

one of the foregoing inequalities being strict. Moreover, if is nondecreasing in for each and , then one has .

Proof.

which is a contradiction in view of (2.3). Hence the conclusion of this theorem holds and the proof is complete.

The next result is for nonstrict inequalities, which require a one-sided Lipschitz type condition.

Theorem 2.2.

Proof.

in view of the condition . We now apply Theorem 2.1 to the inequalities (i), (2.9), and (2.10) to get , . Since is arbitrary, we conclude that (2.6) is true, and we are done.

## 3. Local Existence and Extremal Conditions

In this section, we will consider the local existence and the existence of extremal solutions for the IVP (1.2). We now first discuss Peano's type existence result.

Theorem 3.1.

so that and will be observing in the proof of this theorem.

Proof.

This proves that is a solution of the IVP (1.2), and the proof is complete.

Theorem 3.2.

Under the hypothesis of Theorem 3.1, there exists extremal solution for the IVP (1.2) on the interval , provided is nondecreasing in for each , where will be observed in the proof of this theorem.

Proof.

exists on . Clearly . The uniform continuity of gives argument as before (as in Theorem 3.1), that is a solution of IVP (1.2).

where , , and .

Using Theorem 2.1, we get on as . Therefore, the proof is complete.

## 4. Global Existence

We need the following comparison result before we proceed further.

Theorem 4.1.

Proof.

where . Then applying Theorem 2.1, we get immediately (4.1) and since uniformly on each , the proof is complete.

We are now in position to prove global existence result.

Theorem 4.2.

with exist in .

Proof.

By the assumed local existence, we find that can be continued beyond , contradicting our assumption. Hence, every solution of (4.9) exists on , and the proof is complete.

## Authors’ Affiliations

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