Octavian G. Mustafa

On the global existence of solutions to a class of fractional differential equations

We present two global existence results for an initial value problem associated to a large class of fractional differential equations. Our approach differs substantially from the techniques employed in the recent literature. By introducing an easily verifiable hypothesis, we allow for immediate applications of a general comparison type result from [V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA 69 (2008), 2677-2682].

Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations

where the nonlinearity f satis9es certain monotonicity conditions speci9ed in the sequel. We always assume without mentioning that the function f(t; u; v) is continuous in the domain D= {(t; u; v): t ∈ [t0;+∞); u; v∈R}, where t0 ? 1. It is supposed also that all equations and inequalities involving t hold for t? t0 unless otherwise speci9ed, the symbols “o” and “O” have their standard meanings as t → ∞; and we use the notation R+ def =(0;+∞). Eq. (1) is often used for mathematical modelling of various physical, chemical and biological systems and attracts constant interest of researchers. A great deal of papers published during the last three decades are concerned with local and global existence of solutions of Eq. (1) and its numerous particular cases, uniqueness of solutions,

An existence result for a superlinear fractional differential equation

Abstract We establish the existence and uniqueness of solution for the boundary value problem 0 D t α ( x ′ ) + a ( t ) x λ = 0 , t > 0 , x ′ ( 0 ) = 0 , lim t → + ∞ x ( t ) = 1 , where 0 D t α designates the Riemann–Liouville derivative of order α ∈ ( 0 , 1 ) and λ > 1 . Our result might be useful for establishing a non-integer variant of the Atkinson classical theorem on the oscillation of Emden–Fowler equations.

A Note on the Degasperis-Procesi Equation

Abstract We prove that smooth solutions of the Degasperis-Procesi equation have infinite propagation speed: they loose instantly the property of having compact support.

On Lp-solutions for a class of sequential fractional differential equations

Abstract Under some simple conditions on the coefficient a ( t ), we establish that the initial value problem 0 D t α x ′ + a ( t ) x = 0 , t > 0 , lim t ↘ 0 [ t 1 - α x ( t ) ] = 0 has no solution in L p ( ( 1 , + ∞ ) , R ) , where p - 1 p > α > 1 p and 0 D t α designates the Riemann–Liouville derivative of order α . Our result might be useful for developing a non-integer variant of H. Weyl’s limit-circle/limit-point classification of differential equations.

Asymptotically Linear Solutions for Some Linear Fractional Differential Equations

We establish here that under some simple restrictions on the functional coefficient the fractional differential equation , has a solution expressible as for , where designates the Riemann-Liouville derivative of order and .

Asymptotic integration of a class of nonlinear differential equations

This work is concerned with the behavior of solutions of a class of second-order nonlinear differential equations locally near infinity. Using methods of the fixed point theory, the existence of solutions with different asymptotic representations at infinity is established. A novel technique unifies different approaches to asymptotic integration and addresses a new type of asymptotic behavior.

A Nagumo-like uniqueness theorem for fractional differential equations

We extend to fractional differential equations a recent generalization of the Nagumo uniqueness theorem for ordinary differential equations of first order.

Existence and asymptotic behavior of solutions of a boundary value problem on an infinite interval

In this paper, we shall study a boundary value problem on an infinite interval involving a semilinear second-order differential equation. Existence result extending recent researches is obtained by using a fixed-point theorem due to Furi and Pera. Asymptotic behavior of solutions and their first-order derivatives at infinity is discussed. Comparison with relevant known results in literature is also made.

On the asymptotic integration of a class of sublinear fractional differential equations

We estimate the growth in time of the solutions to a class of nonlinear fractional differential equations