Positive solutions for classes of multi-parameter fourth-order impulsive differential equations with one-dimensional singular p-Laplacian
© Zhang and Feng; licensee Springer. 2014
Received: 29 January 2014
Accepted: 29 April 2014
Published: 13 May 2014
The authors consider the following impulsive differential equations involving the one-dimensional singular p-Laplacian: , , , , , , , , , where and are two parameters. Several new and more general existence and multiplicity results are derived in terms of different values of and . In this case, our results cover equations without impulsive effects and are compared with some recent results.
Keywordsmulti-parameter impulsive differential equations one-dimensional singular p-Laplacian positive solution cone and partial ordering
The theory and applications of the fourth-order ordinary differential equation are emerging as an important area of investigation; it is often referred to as the beam equation. In , Sun and Wang pointed out that it is necessary and important to consider various fourth-order boundary value problems (BVPs for short) according to different forms of supporting. Owing to its importance in engineering, physics, and material mechanics, fourth-order BVPs have attracted much attention from many authors; see, for example [2–29] and the references therein.
where , . By using the upper and lower solution method, fixed point theorems, and the properties of the Green’s function and , the authors give sufficient conditions for the existence of one positive solution.
where and are two parameters, , , is a p-Laplace operator, i.e., , , , , ω is a nonnegative measurable function on , on any open subinterval in which may be singular at and/or , () (where m is fixed positive integer) are fixed points with , , where and represent the right-hand limit and left-hand limit of at , respectively. In addition, ω, f, , g, and h satisfy
(H2) with for all t and ;
(H3) with () for all t and ;
Some special cases of (1.1) have been investigated. For example, Bai and Wang  studied the existence of multiple solutions of problem (1.1) with , , and for . By using a fixed point theorem and degree theory, the authors proved the existence of one or two positive solutions of problem (1.1).
Feng  considered problem (1.1) with , , for and . By using a suitably constructed cone and fixed point theory for cones, the author proved the existence results of multiple positive solutions of problem (1.1).
Motivated by the papers mentioned above, we will extend the results of [14, 30, 31] to problem (1.1). We remark that on impulsive differential equations with a parameter only a few results have been obtained, not to mention impulsive differential equations with two parameters; see, for instance, [32–34]. However, these results only dealt with the case that and .
The rest of the paper is organized as follows: in Section 2, we state the main results of problem (1.1). In Section 3, we provide some preliminary results, and the proofs of the main results together with several technical lemmas are given in Section 4.
2 Main results
In this section, we state the main results, including existence and multiplicity of positive solutions for problem (1.1).
where δ is defined in (3.20).
- (i)If and , then there exist and such that, for any and , problem (1.1) has a positive solution , with(2.2)
- (ii)If and , then there exist and such that, for any and , problem (1.1) has a positive solution with(2.3)
- (iii)If , then there exist and such that, for any and , problem (1.1) has at least two positive solutions and with(2.4)
If and , then there exist and such that, for any and , problem (1.1) has a positive solution , with property (2.2).
If and , then there exist and such that, for any and , problem (1.1) has a positive solution , with property (2.3).
- (iii)If , then there exist and such that, for any and , problem (1.1) has at least two positive solutions and with(2.5)
where , .
A function with is called a solution of problem (1.1) if it satisfies (1.1).
Proof The proof of Lemma 3.1 is similar to that of Lemma 2.1 in . □
Write . Then from (3.6) and (3.7), we can prove that and have the following properties.
Proof The proof of Lemma 3.2 is similar to that of Lemma 2.2 in . □
From (3.14) and (3.15), we can prove that and have the following properties.
It is easy to see K is a closed convex cone of .
From (3.21), we know that is a solution of problem (1.1) if and only if y is a fixed point of operator .
Lemma 3.3 Suppose that (H1)-(H4) hold. Then and is completely continuous.
Proof The proof of Lemma 3.3 is similar to that of Lemma 2.4 in . □
To obtain positive solutions of problem (1.1), the following fixed point theorem in cones is fundamental, which can be found in [, p.94].
, and , , or
, and , ,
then A has at least one fixed point in .
with , where .
4 Proofs of the main results
where is a constant.
Applying (b) of Lemma 3.4 to (4.1) and (4.6) shows that has a fixed point with . Hence, since for we have , , it follows that (2.2) holds. This gives the proof of part (i).
where and satisfy (4.2) and (4.3), respectively.
Applying (a) of Lemma 3.4 to (4.7) and (4.8) shows that has a fixed point with . Hence, since for we have for , it follows that (2.3) holds. This gives the proof of part (ii).
Applying Lemma 3.4 to (4.9)-(4.11) shows that has two fixed points and such that and . These are the desired distinct positive solutions of problem (1.1) for and satisfying (2.4). Then the result of part (iii) follows. □
Applying (b) of Lemma 3.4 to (4.14) and (4.17) shows that has a fixed point with . Hence, since for we have , , it follows that (2.2) holds. This gives the proof of part (i).
where and satisfy (4.15) and (4.16), respectively.
Applying (a) of Lemma 3.4 to (4.20) and (4.21) shows that has a fixed point with . Hence, since for we have , , it follows that (2.3) holds. This gives the proof of part (ii).
Applying Lemma 3.4 to (4.22)-(4.24) shows that has two fixed points and such that and . These are the desired distinct positive solutions of problem (1.1) for and satisfying (2.5). Then the proof of part (iii) is complete. □
Two parameters and are considered.
, not only for .
It follows from the proof of Theorem 2.1 that the conditions of Corollary 3.2 in  are not the optimal conditions, which guarantee the existence of at least one positive solution for problem (1.1). In fact, if , or , , we can prove that problem (1.1) has at least one positive solution, respectively.
This work is sponsored by the project NSFC (11301178, 11171032) and the Fundamental Research Funds for the Central Universities (2014MS58). The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.
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