Employing of Some Basic Theory for Class of Fractional Differential Equations
© A. Babakhani and D. Baleanu 2011
Received: 19 September 2010
Accepted: 15 November 2010
Published: 29 December 2010
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)).
can be found in [19, page 53] and (ii) is an immediate consequence of (1.6) and (i).
2. Strict and Nonstrict Inequalities
We now first discuss a fundamental result relative to the fractional inequalities.
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.
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.
4. Global Existence
We need the following comparison result before we proceed further.
We are now in position to prove global existence result.
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