- Open Access
Existence of positive solutions of advanced differential equations
© Li et al.; licensee Springer 2013
- Received: 6 November 2012
- Accepted: 20 May 2013
- Published: 5 June 2013
In this paper, we study the advanced differential equations
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.
- advanced differential equations
- positive solutions
where , , , and , , , are ω-periodic functions, , , , and are constants.
where and , .
where , , , .
where and .
Throughout this work, we always assume that the following conditions hold:
(H1) , ;
(H2) , , and .
Definition 1.1 A function x is said to be a solution of Eq. (1.1) if , , which has the property and it satisfies Eq. (1.1) for . We say that a solution of Eq. (1.1) is oscillatory if it has arbitrarily large zeros. Otherwise, it is nonoscillatory.
Eq. (1.1) has an eventually positive solution;
There is a function , , such that ω solves the Riccati equation (1.6).
⇒ (i). Let ω be a continuously differentiable solution of Eq. (1.6) for .
is the solution of Eq. (1.1).
The proof of (ii) ⇒ (i) is complete. The proof is complete. □
- (a)There is a solution of the Riccati equation (1.6) for some such that(2.4)
- (b)There is a function for some such that(2.5)
Integrating the last inequality and using , we see that , which contradicts the assumption that is eventually positive. Therefore (2.7) must hold.
⇒ (a). Assume that there is a function satisfying Eq. (2.5) on . Differentiation of (2.5) then shows that is a solution of (1.6) for , and it satisfies (2.4). The proof of (b) ⇒ (a) is complete. □
Then there exists a continuous solution of Eq. (2.5) which satisfies the inequality .
By (3.2), we see that the functions in the image set SF are uniformly bounded on any finite interval of .
Due to (3.1) and (3.4), there exists such that for , , hence SF is equicontinuous.
for every and , where is between and .
for . Thus, uniformly on a finite interval.
We obtained that the conditions of the Schauder-Tyichonoff theorem are satisfied, hence the mapping S has at least one fixed point ν in F, and because for , ν is the continuous solution of Eq. (2.5). □
holds for t large enough. Then Eq. (1.1) has a positive solution with the property .
Proof Let be given such that the conditions of the theorem hold. We show that the conditions of Theorem 3.1 are satisfied with and for t large enough.
Therefore, by Theorem 3.1, Lemma 2.1 and Lemma 2.2, Eq. (1.1) has a positive solution, and the proof is complete. □
f is nondecreasing continuous function and , .
The following fixed point theorem will be used to prove the main results.
Lemma 3.1 (Schauder’s fixed point theorem)
Let Ω be a closed, convex and nonempty subset of a Banach space X. Let be a continuous mapping such that T Ω is a relatively compact subset of X. Then T has at least one fixed point in Ω. That is, there exists an such that .
Then Eq. (3.6) has a positive solution which tends to zero.
We can show that for any , .
Second: We prove that T is continuous.
Third: We show that TΩ is relatively compact.
The proof is similar to Theorem 2.1 of , we omitted it. □
The authors sincerely thank the anonymous referees for their valuable suggestions and comments which greatly helped improve this article. Supported by NSF of China (11071054), Natural Science Foundation of Hebei Province (A2011205012).
- Agarwal RP, Grace SR, O’Reagan D: Oscillation Theory of Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht; 2002.View ArticleGoogle Scholar
- Culáková I, Hanus̆tiaková L, Olach R: Existence for positive solutions of second-order neutral nonlinear differential equations. Appl. Math. Lett. 2009, 22: 1007-1010. 10.1016/j.aml.2009.01.009MathSciNetView ArticleGoogle Scholar
- Dz̆urina J, Stavroulakis IP: Oscillation criteria for second order delay differential equations. Appl. Math. Comput. 2003, 140: 445-453. 10.1016/S0096-3003(02)00243-6MathSciNetView ArticleGoogle Scholar
- Dos̆lý O, Řehák P North-Holland Math. Stud. 202. In Half-Linear Differential Equations. Elsevier, Amsterdam; 2005.Google Scholar
- Feng MX, Xie DX: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J. Comput. Appl. Math. 2009, 223: 438-448. 10.1016/j.cam.2008.01.024MathSciNetView ArticleGoogle Scholar
- Györi I, Ladas G: Oscillation Theory of Delay Differential Equations with Applications. Clarendon, Oxford; 1991.Google Scholar
- Kusano T, Lalli BS: On oscillation of half-linear functional differential equations with deviating arguments. Hiroshima Math. J. 1994, 24: 549-563.MathSciNetGoogle Scholar
- Kusano T, Naito Y, Ogata A: Strong oscillation and nonoscillation for quasi linear differential equations of second order. Differ. Equ. Dyn. Syst. 1994, 2: 1-10.MathSciNetGoogle Scholar
- Kusano T, Yoshida N: Nonoscillation theorems for a class of quasilinear differential equations of second order. J. Math. Anal. Appl. 1995, 189: 115-127. 10.1006/jmaa.1995.1007MathSciNetView ArticleGoogle Scholar
- Kulenovic MRS, Hadziomerspahic S: Existence of nonoscillatory solution of second order linear neutral delay equation. J. Math. Anal. Appl. 1998, 228: 436-448. 10.1006/jmaa.1997.6156MathSciNetView ArticleGoogle Scholar
- Luo Y, Wang WB, Shen JH: Existence of positive periodic solutions for two kinds of neutral functional differential equations. Appl. Math. Lett. 2008, 21: 581-587. 10.1016/j.aml.2007.07.009MathSciNetView ArticleGoogle Scholar
- Li HJ, Yeh CC: Nonoscillation criteria for second-order half-linear functional differential equations. Appl. Math. Lett. 1995, 8: 63-70.MathSciNetView ArticleGoogle Scholar
- Lin XN, Jiang DQ: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 2006, 321: 501-514. 10.1016/j.jmaa.2005.07.076MathSciNetView ArticleGoogle Scholar
- Ladas G, Sficas YG, Stavroulakis IP: Nonoscillatory functional differential equations. Pac. J. Math. 1984, 115: 391-398. 10.2140/pjm.1984.115.391MathSciNetView ArticleGoogle Scholar
- Péics H, Karsai J: Existence of positive solutions of halflinear delay differential equations. J. Math. Anal. Appl. 2006, 323: 1201-1212. 10.1016/j.jmaa.2005.11.033MathSciNetView ArticleGoogle Scholar
- Shen JH, Stavroulakis IP, Tang XH: Hille type oscillation and nonoscillation criteria for neutral equations with positive and negative coefficients. Stud. Univ. Zilina Math. Ser. 2001, 14: 45-59.MathSciNetGoogle Scholar
- Tian Y, Ji DH, Ge WG: Existence and nonexistence results of impulsive first order problem with integral boundary condition. Nonlinear Anal. 2009, 71: 1250-1262. 10.1016/j.na.2008.11.090MathSciNetView ArticleGoogle Scholar
- Yu YH, Wang HZ: Nonoscillatory solutions of second-order nonlinear neutral delay equations. J. Math. Anal. Appl. 2005, 311: 445-456. 10.1016/j.jmaa.2005.02.055MathSciNetView ArticleGoogle Scholar
- Zhang WP, Feng W, Yan JR, Song JS: Existence of nonoscillatory solutions of first-order linear neutral delay differential equations. Comput. Math. Appl. 2005, 49: 1021-1027. 10.1016/j.camwa.2004.12.006MathSciNetView ArticleGoogle Scholar
- Zhou Y: Existence for nonoscillatory solutions of second-order nonlinear differential equations. J. Math. Anal. Appl. 2007, 331: 91-96. 10.1016/j.jmaa.2006.08.048MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.