Existence of positive solutions of advanced differential equations
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RESEARCH
Open Access
Existence of positive solutions of advanced differential equations Qiaoluan Li* , Xiaojing Liu, Feifei Cui and Weina Li *
Correspondence: [email protected] College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, P.R. China
Abstract In this paper, we study the advanced differential equations n α –1 α –1 r(t)x (t) x (t) + pi (t)x(t + τi (t)) x(t + τi (t)) = 0 i=1
and n r(t)(y(t) – P(t)y(t – τ )) + pi (t)f (y(t + σ )) = 0. i=1
By using the generalized Riccati transformation and the Schauder-Tyichonoff theorem, we establish the conditions for the existence of positive solutions of the above equations. MSC: 34K11; 39A10 Keywords: advanced differential equations; positive solutions; existence
1 Introduction In the last years, oscillation and nonoscillation of differential equations attracted a considerable attention. Many results have been obtained, and we refer the reader to the papers [–]. In , Luo et al. [] investigated the existence of positive periodic solutions of the following two kinds of neutral functional differential equations:
x(t) – cx t – τ (t) = –a(t)x(t) + f t, x t – τ (t)
and x(t) – c
Q(r)x(t + r) dr –∞
= –a(t)x(t) + b(t)
Q(r)f t, x(t + r) dr,
–∞
where a, b ∈ C(R, (, ∞)), τ ∈ C(R, R), f ∈ C(R × R, R), and a(t), b(t), τ (t), f (t, x) are ω-periodic functions, ω > , Q(r) ∈ C((–∞, ], [, ∞)), –∞ Q(r) dr = , and ω, |c| < are constants. © 2013 Li 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.
Li et al. Advances in Difference Equations 2013, 2013:158 http://www.advancesindifferenceequations.com/content/2013/1/158
Page 2 of 13
Péics et al. [] obtained the existence of positive solutions of half-linear delay differential equations n α– α– x (t) x (t) + pi (t)x t – τi (t) x t – τi (t) = , i=
where t ≥ t and α > , τi (t) ≤ t. Zhang et al. [] obtained the existence of nonoscillatory solutions of the first-order linear neutral delay differential equation
x(t) + P(t)x(t – τ ) + Q (t)x(t – σ ) – Q (t)x(t – σ ) = ,
where P ∈ C([t , ∞), R), τ ∈ (, ∞), σ , σ ∈ [, ∞), Q , Q > . In this paper, we consider the advanced differential equation
n α– α– r(t)x (t) x (t) + pi (t)x t + τi (t) x t + τi (t) = ,
(.)
i=
where t ≥ t and α > . Throughout this work, we always assume that the following conditions hold: (H ) pi ∈ C([t , ∞), R), i = , , , . . . , n; (H ) τi ∈ C([t , ∞), R+ ), i = , , , . . . , n, and < r(t) ≤ k. For convenience, we introduce the notation *
ηα = |η|α– η,
α > .
(.)
It is convenient to rewrite (.) in the form
n α* α* + pi (t)x t + τi (t) = . r(t)x (t)
(.)
i=
Definition .
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