Forced oscillation of certain fractional differential equations

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RESEARCH

Open Access

Forced oscillation of certain fractional differential equations Da-Xue Chen1* , Pei-Xin Qu2 and Yong-Hong Lan3 *

Correspondence: [email protected] College of Science, Hunan Institute of Engineering, 88 East Fuxing Road, Xiangtan, Hunan, 411104, P.R. China Full list of author information is available at the end of the article 1

Abstract The paper deals with the forced oscillation of the fractional differential equation

(Dqa x)(t) + f1 (t, x(t)) = v(t) + f2 (t, x(t)) for t > a ≥ 0 q–k

m–q

with the initial conditions (Da x)(a) = bk (k = 1, 2, . . . , m – 1) and limt→a+ (Ia x)(t) = bm , q where Da x is the Riemann-Liouville fractional derivative of order q of x, m – 1 < q ≤ m, m–q m ≥ 1 is an integer, Ia x is the Riemann-Liouville fractional integral of order m – q of x, and bk (k = 1, 2, . . . , m) are/is constants/constant. We obtain some oscillation theorems for the equation by reducing the fractional differential equation to the equivalent Volterra fractional integral equation and by applying Young’s inequality. We also establish some new oscillation criteria for the equation when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. The results obtained here improve and extend some existing results. An example is given to illustrate our theoretical results. MSC: 34A08; 34C10 Keywords: forced oscillation; fractional derivative; fractional differential equation

1 Introduction The aim of the paper is to establish several oscillation theorems for forced fractional differential equation with initial conditions of the form q

(Da x)(t) + f (t, x(t)) = v(t) + f (t, x(t)), q–k (Da x)(a) = bk (k = , , . . . , m – ),

t > a ≥ , m–q limt→a+ (Ia x)(t) = bm ,

 (.)

q

where Da x is the Riemann-Liouville fractional derivative of order q of x, m –  < q ≤ m, m–q m ≥  is an integer, Ia x is the Riemann-Liouville fractional integral of order m – q of x, bk (k = , , . . . , m) are/is constants/constant, fi : [a, ∞) × R → R (i = , ) are continuous functions, and v : [a, ∞) → R is a continuous function. By a solution of (.), we mean a nontrivial function x ∈ C([a, ∞), R) which has the propq erty Da x ∈ C([a, ∞), R) and satisfies (.) for t ≥ a. Our attention is restricted to those solutions of (.) which exist on [a, ∞) and satisfy sup{|x(t)| : t > t∗ } >  for any t∗ ≥ a. A solution x of (.) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (.) is said to be oscillatory if all its solutions are oscillatory. © 2013 Chen et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Chen et al. Advances in Difference Equations 2013, 2013:125 http://www.advancesindifferenceequations.com/content/2013/1/125

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Differential equations of f