Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis

• Published in: Nonlinear analysis 2011-12, Vol.74 (17), p.5975-5986
• Main Authors: Kou, Chunhai, Zhou, Huacheng, Yan, Ye
• Format: Article
• Language: English
• Subjects: Banach space; Derivatives; Differential equations; Exact sciences and technology; Fractional differential equation; Global existence; Half-axis; Initial value problem; Initial value problems; Mathematical analysis; Mathematics; Nonlinearity; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.05.074

The main aim of this paper is to study the global existence of solutions of initial value problems for nonlinear fractional differential equations(FDEs) on the half-axis, which is fundamental in the basic theory of FDEs and important in stability analysis of this kind of equations. In this paper, we are concerned with the nonlinear FDE D 0 + α x ( t ) = f ( t , x ) , t ∈ ( 0 , + ∞ ) , 0 < α ≤ 1 , where D 0 + α is the standard Riemann–Liouville fractional derivative, subject to the initial value condition lim t → 0 + t 1 − α x ( t ) = u 0 . By constructing a special Banach space and employing fixed-point theorems, some sufficient conditions are obtained to guarantee the global existence of solutions on the interval [ 0 , + ∞ ) . Moreover, in the case α = 1 , existence results of solutions of initial value problems for ordinary differential equations on the half-axis are also obtained. An interesting example is also included. ► We study the IVPs for nonlinear fractional differential equations. ► We construct a special Banach space. ► Some global existence results of solutions on the half-axis are obtained. ► Existence results of solutions of IVPs for ODEs on the half-axis are also included.