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.
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. □
3 Main results
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).
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