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Existence of positive solutions of advanced differential equations
Advances in Difference Equations volume 2013, Article number: 158 (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.
In 2008, Luo et al.  investigated the existence of positive periodic solutions of the following two kinds of neutral functional differential equations:
where , , , and , , , are ω-periodic functions, , , , and are constants.
Péics et al.  obtained the existence of positive solutions of half-linear delay differential equations
where and , .
Zhang et al.  obtained the existence of nonoscillatory solutions of the first-order linear neutral delay differential equation
where , , , .
In this paper, we consider the advanced differential equation
where and .
Throughout this work, we always assume that the following conditions hold:
(H1) , ;
(H2) , , and .
For convenience, we introduce the notation
It is convenient to rewrite (1.1) in the form
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.
One of the most important methods of the study of nonoscillation is the method of generalized characteristic equation . The method was applied to second-order half-linear equations without delay, for example, in [8, 9]. Concerning cases with advanced, let us apply the Riccati-transformation
By (1.4), we have
From (1.3), we obtain
it is convenient to rewrite (1.5) in the form
Lemma 2.1 Suppose that (H1) and (H2) hold. Then the following statements are equivalent:
Eq. (1.1) has an eventually positive solution;
There is a function , , such that ω solves the Riccati equation (1.6).
Proof (i) ⇒ (ii). Let x be an eventually positive solution of Eq. (1.1) such that for . The function ω defined by
We will show that it is a solution of (1.6) on . By (1.2) and observing that
it follows that
Dividing both sides of (1.1) by gives that
From the definition of ω, we obtain
By substituting (2.2), (2.3) into (2.1), we get
We obtain (1.6), and the proof of (i) ⇒ (ii) is complete.
⇒ (i). Let ω be a continuously differentiable solution of Eq. (1.6) for .
We show that a function x defined by
is the solution of Eq. (1.1).
By (1.6), we obtain
The proof of (ii) ⇒ (i) is complete. The proof is complete. □
Lemma 2.2 Suppose that (H1) and (H2) hold. The following statements are equivalent:
There is a solution of the Riccati equation (1.6) for some such that(2.4)
There is a function for some such that(2.5)
Proof (a) ⇒ (b). Let be a solution of Eq. (1.6) for and with the property (2.4). Let be fixed arbitrarily and integrate (1.6) over :
We claim that
Assuming the contrary, if , then in view of (2.6) there is such that
for , or equivalently,
Then we have
From , it follows that , . Dividing both sides of (2.8) by gives that
Integrating the above inequality over then yields
Combining with (2.8), we have
Integrating the last inequality and using , we see that , which contradicts the assumption that is eventually positive. Therefore (2.7) must hold.
Let in (2.6). Using (2.4) and (2.7), we get . So,
⇒ (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
Theorem 3.1 Assume that there exist and functions such that ,
for every function , where
Then there exists a continuous solution of Eq. (2.5) which satisfies the inequality .
Proof Let and be real numbers such that . Then is an arbitrary compact subinterval of and set
It follows from (3.1) and (3.2), that the operator S is defined for and satisfies
By (3.2), we see that the functions in the image set SF are uniformly bounded on any finite interval of .
To prove that the functions in SF are equicontinuous on any finite interval of , we choose the finite interval as before, and let and be two arbitrary numbers from . Since is continuous on , , , such that for , we have
Due to (3.1) and (3.4), there exists such that for , , hence SF is equicontinuous.
Let the sequence tend to uniformly on any finite interval (). In particular, the convergence is uniform on the interval . Using the mean value theorem, we have
where is between and , and similarly
for every and , where is between and .
Since for , we obtain
The uniform convergence on any finite interval of implies that if n is sufficiently large,
where , and hence we obtain
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). □
Theorem 3.2 Assume that (H1), (H2) hold and there exists a positive function for such that
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.
Let be a continuous function such that . It follows from (3.5) that
Therefore, by Theorem 3.1, Lemma 2.1 and Lemma 2.2, Eq. (1.1) has a positive solution, and the proof is complete. □
Next, we consider neutral differential equations of the form
We assume that:
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 .
Theorem 3.3 Suppose that
and there exist , such that
Then Eq. (3.6) has a positive solution which tends to zero.
Proof First: Choose ,
Let be the set of all continuous functions with the norm
Then is a Banach space. We define a closed, bounded convex subset Ω of as follows:
Define the map :
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. □
Corollary 3.1 Suppose that , (3.7) holds and
Then Eq. (3.6) has a solution
Example 3.1 Consider the advanced differential equations
where and . Choose ,
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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).
The authors declare that they have no competing interest.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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Li, Q., Liu, X., Cui, F. et al. Existence of positive solutions of advanced differential equations. Adv Differ Equ 2013, 158 (2013). https://doi.org/10.1186/1687-1847-2013-158
- advanced differential equations
- positive solutions