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Transformation technique, fixed point theorem and positive solutions for second-order impulsive differential equations with deviating arguments
Advances in Difference Equations volume 2014, Article number: 312 (2014)
Abstract
This paper investigates the boundary value problems of second-order impulsive differential equations with deviating arguments
where is a real sequence with , , ω may be singular at and/or . Several new and more general results are obtained for the existence of positive solutions for the above problem by using transformation technique and Krasnosel’skii’s fixed point theorem. We discuss our problems under two cases when the deviating arguments are delayed and advanced. The approach to deal with the impulsive term is different from earlier approaches. It is the first paper where the transformation technique and a fixed point theorem for cones are applied to second-order differential equations with impulsive effects and deviating arguments. An example is included to verify the theoretical results.
1 Introduction
Impulsive differential equations, which provide a natural description of observed evolution processes, are regarded as important mathematical tools for better understanding of several real world problems in applied sciences, such as population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc. Therefore, the study of this class of impulsive differential equations has gained prominence and it is a rapidly growing field. For the general theory of impulsive differential equations, we refer the reader to [1–3], whereas the applications of impulsive differential equations can be found in [4–14]. Nieto and O’Regan [15] pointed out that in a second-order differential equation one usually considers impulses in the position u and the velocity . However, in the motion of spacecraft one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position [16]. The impulses only on velocity occur also in impulsive mechanics [17].
Some classical tools such as bifurcation theory [18, 19], fixed point theorems in cones [20–24], the method of lower and upper solutions [25, 26], the theory of critical point theory and variational methods [7, 15, 27–30] and the technique via appropriate transformation [31–34] have been widely used to study impulsive differential equations. But it is quite difficult to apply these approaches to an impulsive differential equation with deviating arguments; therefore, there was no result in this area for a long time. Only in the recent eight years, there appeared a few articles which dealt with some impulsive differential equations with deviating arguments by using fixed point theorems in cones [35–38]. Motivated by [35–38], in this article we shall use a different approach to discuss the existence of positive solutions for a class of impulsive differential equations with deviating arguments.
Consider the second-order nonlinear impulsive differential equation of the type
where , , , (, here n is a fixed positive integer) are fixed points with , , is a real sequence with , , () represent the right-hand limit of at , is nonnegative.
Throughout this paper we assume that on . In addition, ω, f, , α and h satisfy
(H1) with and ω does not vanish on any subinterval of ;
(H2) , ;
(H3) is a real sequence with , , ;
(H4) is nonnegative with , where
Remark 1.1 Throughout this paper, we always assume that a product equals unity if the number of factors is equal to zero, and let
Remark 1.2 Combining (H3) and the definition of , we know that is a step function and bounded on J, and
Some special cases of (1.1) have been investigated. For example, Zhang and Feng [34] considered problem (1.1) under the case that and on J. By using fixed point theories in cones, the authors proved the existence of positive solutions for problem (1.1).
At the same time, a class of boundary value problems with delay has been investigated; for example, see [39–44]. It is not difficult to see that the corresponding functions f appearing on the right-hand side depend on , , where initial functions x are given on the initial set, for example, . Jankowski [45, 46] pointed out that in such cases , there are some problems with a constant delay τ. If we consider the differential problem on intervals , where , then it means that we have no delays; we have such a situation in paper [44]. If , then it is easy to solve the differential equation on , since we have the solution on the initial set . Continuing this process, we can find a solution on the whole interval by using the method of steps. In the present paper, for example, the deviating argument α can have a form with a fixed number , so the delay is a function of t. In this case, the initial set reduces to one point , and we cannot apply the step method. To the authors’ knowledge, it is the first paper when positive solutions have been investigated for a class of second-order impulsive differential equations with deviating arguments both of advanced and delayed type.
Remark 1.3 There are almost no papers, except [35–37], studying second-order impulsive differential equations with deviating arguments using fixed point theory. However, in [35–37], Jankowski only considered , not ω is singular at and/or , and dealt with the nonlinear term that is in the form of , not ; see (2.12).
Remark 1.4 Comparing with [35–37], we transform problem (1.1) into a differential system without impulse, i.e., the technique to deal with impulses is completely different from that of [35–37]. According to the authors’ knowledge, it is probably the first paper where this technique is applied to second-order impulsive boundary value problems with deviating arguments.
Remark 1.5 The technique to deal with is completely different from that of Zhang et al. [31], Zhang et al. [32] and Sun et al. [33].
Being directly inspired by [31–37], the authors will prove several new and more general results for the existence of positive solutions for problem (1.1) by using fixed point theories.
The organization of this paper is as follows. In Section 2, we present some definitions and lemmas which are needed throughout this paper. In particular, we transform problem (1.1) into a differential system without impulse. In Section 3, we use a fixed point theorem to obtain the existence of positive solutions for problem (1.1) with advanced argument α. Finally, in Section 4, we formulate sufficient conditions under which delayed problem (1.1) has positive solutions. In particular, our results in these sections are new when on .
2 Preliminaries
In this section, we first present some definitions and lemmas which are needed throughout this paper.
Definition 2.1 (see [47])
Let E be a real Banach space over R. A nonempty closed set is said to be a cone provided that
-
(i)
for all and all , , and
-
(ii)
implies .
Every cone induces an ordering in E given by if and only if .
Definition 2.2 The map β is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E provided that is continuous and
for all and .
Definition 2.3 A function is said to be a solution of problem (1.1) on J if:
-
(i)
is absolutely continuous on each interval and , ;
-
(ii)
for any , , exist and ;
-
(iii)
satisfies (1.1).
We shall reduce problem (1.1) to a system without impulse. To this goal, firstly by means of the transformation
we convert problem (1.1) into
The following lemmas will be used in the proof of our main results.
Lemma 2.1 Assume that (H1)-(H4) hold. Then
-
(i)
If is a solution of problem (2.2) on J, then is a solution of problem (1.1) on J;
-
(ii)
If is a solution of problem (1.1) on J, then is a solution of problem (2.2) on J.
Proof The proof is similar to that of Lemma 2.1 in [34]. □
Lemma 2.2 If (H1)-(H4) hold, then problem (2.2) has a solution y, and y can be expressed in the form
where
Proof The proof is similar to that of Lemma 2.1 in [48]. □
Lemma 2.3 Let , G and H be given as in Lemma 2.2. Then we have the following results:
where
Proof Relation (2.6) is simple to prove. Note that
for , .
Similarly, we can prove that for , . Hence, it follows from that
This gives the proof of Lemma 2.3. □
Remark 2.1 Noticing that , it follows from (2.4) and (2.5) that
and
where
Let . Then E is a real Banach space with the norm defined by
Define a cone K in E by
Also, define for r a positive number by
Note that .
Define by
Lemma 2.4 Assume that (H1)-(H4) hold. Then and is completely continuous.
Proof For , it follows from (2.6) and (2.12) that
It follows from (2.7), (2.12) and (2.13) that
Thus, .
Next, by arguments similar to those of Theorem 1 in [49], one can prove that is completely continuous. So it is omitted, and the lemma is proved. □
Remark 2.2 From (2.12), we know that is a solution of problem (2.2) if and only if y is a fixed point of the operator T.
From Lemma 2.1 and Remark 2.2, we can obtain the following results.
Lemma 2.5 Assume that (H1)-(H4) hold. Then
-
(i)
If is a solution of problem (1.1) on J, then is a fixed point of T;
-
(ii)
If is a fixed point of T, then is a solution of problem (1.1) on J.
In the rest of this section, we state a well-known fixed point theorem which we need later.
Lemma 2.6 (see [47])
Let P be a cone in a real Banach space E. Assume that , are bounded open sets in E with , . If
is completely continuous such that either
-
(i)
, and , , or
-
(ii)
, and , ,
then A has at least one fixed point in .
3 Existence of positive solutions for problem (1.1) under on J
For convenience, we introduce the following notations:
We also define as [50] = number of zeros in the set and = number of infinities in the set . Sun and Li [51] pointed out that , and there are six possible cases: (i) and ; (ii) and ; (iii) and ; (iv) and ; (v) and ; and (vi) and . By using Krasnosel’skii’s fixed point theorem in a cone, some results are obtained for the existence of at least one or two positive solutions of problem (1.1) for on J under the above six possible cases.
3.1 For the case on J under and
In this subsection, we discuss the existence of a single positive solution for problem (1.1) for on J under and .
For convenience, we introduce the following notations:
Theorem 3.1 Assume that (H1)-(H4) hold. If and , then problem (1.1) has at least one positive solution.
Proof First, we consider the case and . Since , then there exists such that
Since on J, it follows from on J that
Let . Then, for , we have for , and then
Consequently, for any and , (2.9) and (2.12) imply
which implies
Next turning to , there exists satisfying such that
Since on J, it follows from on that
Let . Then, for , we have
Hence, for , it follows from (2.9) and (2.12) that
which implies
Thus by (i) of Lemma 2.6, it follows that T has a fixed point y in with
Lemma 2.5 implies that problem (1.1) has at least one positive solution x with
This gives the proof of Theorem 3.1. □
Remark 3.1 For and , there is another case and . However, at the moment, we give no information on the existence of a positive solution for problem (1.1) if we change and into and in Theorem 3.1.
3.2 For the case on J under and
In this subsection, we discuss the existence for the positive solutions of problem (1.1) under and . For convenience, we introduce the following notations:
and
Now, we shall state and prove the following main result.
Theorem 3.2 Suppose that (H1)-(H4) hold and on J. In addition, let the following two conditions hold:
(H5) There exist and such that ;
(H6) There exist and such that for , ; furthermore, .
Then problem (1.1) has at least one positive solution.
Proof Without loss of generality, we may assume that . Considering , we have for , .
Since on J, it follows from on J that
Let . Then, for , we have for , and then
Consequently, for any and , (2.9) and (2.12) imply
which implies
On the other hand, from (H6), when is fixed, there exists such that
for and . Since on J, it follows from on J that
Let . Then, for , we have
Hence, for , it follows from (2.9) and (2.12) that
which implies
Thus by (i) of Lemma 2.6, it follows that T has a fixed point y in with
Thus, it follows from Lemma 2.5 that problem (1.1) has at least one positive solution x with
This finishes the proof of Theorem 3.2. □
We remark that condition (H5) in Theorem 3.2 can be replaced by the following condition:
,
which is a special case of (H5).
Corollary 3.1 Suppose that (H1)-(H4), , (H6) hold and on J. Then problem (1.1) has at least one positive solution.
Proof We show that implies (H5). Suppose that holds. Then there exists a positive number such that
Hence, we obtain
Therefore, (H5) holds. Hence, by Theorem 3.2, problem (1.1) has at least one positive solution. □
Theorem 3.3 Suppose that (H1)-(H5) hold and on J. In addition, let the following condition hold:
(H7) .
Then problem (1.1) has at least one positive solution.
Proof The proof is similar to those of (3.2) and (3.3), respectively. □
Corollary 3.2 Suppose that (H1)-(H4), , (H7) hold and on J. Then problem (1.1) has at least one positive solution.
3.3 For the case on J under and or and
In this subsection, we discuss the existence for the positive solutions of problem (1.1) for the case on J under and or and .
Theorem 3.4 Suppose that (H1)-(H4) hold, on J and and . Then problem (1.1) has at least one positive solution.
Proof The proof is similar to that of Theorem 3.2. □
Theorem 3.5 Suppose that (H1)-(H4) hold, on J and and . Then problem (1.1) has at least one positive solution.
Proof Consider , then there exists such that for , .
Since on J, it follows from on J that
Let . Then, for , we have for , and then
Consequently, for , it follows from (2.9) and (2.12) that
which implies
Next, turn to . In fact, we can show that implies (H5).
Let . Then there exists such that for . Let
Then we have
This implies that . Hence, implies that (H5).
Similarly to the proof of (3.3), we have
where .
Thus by (ii) of Lemma 2.6, it follows that T has a fixed point y in with
This finishes the proof of Theorem 3.5. □
From Theorems 3.4 and 3.5, we have the following result.
Corollary 3.3 Suppose that and condition (H6) in Theorem 3.2 hold. Then problem (1.1) has at least one positive solution.
Theorem 3.6 Suppose that (H1)-(H4), on J, and . Then problem (1.1) has at least one positive solution.
Proof The proof is similar to that of Theorem 3.2. □
Theorem 3.7 Suppose (H1)-(H4), on J, and . Then problem (1.1) has at least one positive solution.
Proof The proof is similar to that of Theorem 3.2. □
From Theorems 3.6 and 3.7, the following corollaries are easily obtained.
Corollary 3.4 Suppose that and condition (H5) in Theorem 3.2 hold. Then problem (1.1) has at least one positive solution.
Corollary 3.5 Suppose that and condition (H5) in Theorem 3.2 hold. Then problem (1.1) has at least one positive solution.
3.4 For the case on J under and or and
In this subsection we study the existence of multiple positive solutions for problem (1.1) for the case on J under and or and .
Combining the proofs of Theorems 3.1 and 3.2, the following theorem is easily proved.
Theorem 3.8 Suppose that (H1)-(H4), on J, and and condition (H5) of Theorem 3.2 hold. Then problem (1.1) has at least two positive solutions.
Corollary 3.6 Suppose that (H1)-(H4), on J, and and condition of Corollary 3.1 hold. Then problem (1.1) has at least two positive solutions.
Remark 3.2 Noticing Remark 3.1, at the moment we give no information on the existence of a positive solution for problem (1.1) under and .
4 Positive solutions of problem (1.1) for the case of on J
Now we deal with problem (1.1) for the case of on J. Similarly as in Lemmas 2.3 and 2.4, we can prove the following results.
Lemma 4.1 Let , G and H be given as in Lemma 2.2. Then we have the following results:
where
Proof Note that
for , .
Similarly, we can prove that for , . Hence, it follows from that
This gives the proof of Lemma 4.1. □
Let E be as defined in Section 2. We define a cone in E by
It is easy to see that is a closed convex cone of E.
Define by
It is clear that is a positive solution of problem (1.1) if and only of y is a fixed point of .
By analogous methods, we have the following results. Here we only give the proof of Lemma 4.1.
Lemma 4.2 Assume that (H1)-(H4) hold. Then and is completely continuous.
Lemma 4.3 Assume that (H1)-(H4) hold. Then
-
(i)
If is a solution of problem (1.1) on J, then is a fixed point of ;
-
(ii)
If is a fixed point of , then is a solution of problem (1.1) on J.
Similar to the proof of that in Section 3, we have the following results.
4.1 For the case on J under and
For convenience, we introduce the following notation:
Theorem 4.1 Assume that (H1)-(H4) hold. If and , then problem (1.1) has at least one positive solution.
Proof First, we consider the case and . Since , then there exists such that
Since on J, it follows from on J that
Let . Then, for , we have for , and then
Consequently, for any and , (2.9) and (2.12) imply
which implies
Next we consider , there exists satisfying such that
Since on J, it follows from on that
Let . Then, for , we have
Hence, for , it follows from (2.9) and (2.12) that
which implies
Thus by (i) of Lemma 2.6, it follows that has a fixed point y in with
This finishes the proof of Theorem 4.1. □
Remark 4.1 For and , there is another case and . However, at the moment, we give no information on the existence of a positive solution for problem (1.1) if we change and into and in Theorem 4.1.
4.2 For the case on J under and
In this subsection, we discuss the existence for the positive solutions of problem (1.1) under and . For convenience, we introduce the following notation:
Now, we shall state and prove the following main result.
Theorem 4.2 Suppose that (H1)-(H5) hold and on J. In addition, let the following condition hold:
() There exist and such that for , ; furthermore, .
Then problem (1.1) has at least one positive solution.
Corollary 4.1 Suppose that (H1)-(H4), , () hold and on J. Then problem (1.1) has at least one positive solution.
Theorem 4.3 Suppose that (H1)-(H5) hold and on J. In addition, let the following condition hold:
() .
Then problem (1.1) has at least one positive solution.
Corollary 4.2 Suppose that (H1)-(H4), , () hold and on J. Then problem (1.1) has at least one positive solution.
4.3 For the case on J under and or and
Theorem 4.4 Suppose that (H1)-(H4) hold, on J and and . Then problem (1.1) has at least one positive solution.
Theorem 4.5 Suppose that (H1)-(H4) hold, on J and and . Then problem (1.1) has at least one positive solution.
4.4 For the case on J under and or and
Combining the proofs of Theorems 4.1 and 4.2, the following theorem is easily proved.
Theorem 4.6 Suppose that (H1)-(H4), on J, and and condition (H5) of Theorem 4.2 hold. Then problem (1.1) has at least two positive solutions.
Corollary 4.3 Suppose that (H1)-(H4), on J, and and condition of Corollary 4.1 hold. Then problem (1.1) has at least two positive solutions.
5 An example
To illustrate how our main results can be used in practice, we present an example.
Example 5.1 Consider the following boundary value problem:
where , on J and
here is a positive integral number.
This means that problem (5.1) involves the advanced argument α. For example, we can take . It is clear that ω is singular at and f is both nonnegative and continuous.
Problem (5.1) can be regarded as a problem of the form (1.1), where , , , and
Hence
and
It follows from the definition of ω, f, c and h that (H1)-(H4) hold, and
Hence, by Theorem 3.1, the conclusion follows, and the proof is complete.
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Acknowledgements
This work is sponsored by the project NSFC (11301178, 11171032), the Fundamental Research Funds for the Central Universities (2014ZZD10, 2014MS58) and the improving project of graduate education of Beijing Information Science and Technology University (YJT201416). The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.
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XZ completed the main study and carried out the results of this article. MF checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
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Zhang, X., Feng, M. Transformation technique, fixed point theorem and positive solutions for second-order impulsive differential equations with deviating arguments. Adv Differ Equ 2014, 312 (2014). https://doi.org/10.1186/1687-1847-2014-312
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DOI: https://doi.org/10.1186/1687-1847-2014-312
Keywords
- advanced and delayed arguments
- impulsive differential equations
- transformation technique
- fixed point theorem
- positive solutions