Abstract
The authors consider the following impulsive differential equations involving the onedimensional singular pLaplacian: , , , , , , , , , 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.
Keywords:
multiparameter; impulsive differential equations; onedimensional singular pLaplacian; positive solution; cone and partial ordering1 Introduction
The theory and applications of the fourthorder ordinary differential equation are emerging as an important area of investigation; it is often referred to as the beam equation. In [1], Sun and Wang pointed out that it is necessary and important to consider various fourthorder boundary value problems (BVPs for short) according to different forms of supporting. Owing to its importance in engineering, physics, and material mechanics, fourthorder BVPs have attracted much attention from many authors; see, for example [229] and the references therein.
Very recently, Zhang and Liu [30] studied the following fourthorder fourpoint boundary value problem without impulsive effect:
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.
In this paper, we investigate the existence of positive solutions of fourthorder impulsive differential equations with two parameters
where and are two parameters, , , is a pLaplace 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 righthand limit and lefthand limit of at , respectively. In addition, ω, f, , g, and h satisfy
(H_{3}) with () for all t and ;
(H_{4}) are nonnegative and , , where
Some special cases of (1.1) have been investigated. For example, Bai and Wang [14] 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 [31] 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, [3234]. 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).
We begin by introducing the notation
We also choose four numbers r, , , and R satisfying
where δ is defined in (3.20).
Theorem 2.1Assume that (H_{1})(H_{4}) hold.
(i) Ifand, then there existandsuch that, for anyand, problem (1.1) has a positive solution, with
(ii) Ifand, then there existandsuch that, for anyand, problem (1.1) has a positive solutionwith
(iii) If, then there existandsuch that, for anyand, problem (1.1) has at least two positive solutionsandwith
Theorem 2.2Assume that (H_{1})(H_{4}) hold.
(i) Ifand, then there existandsuch that, for anyand, problem (1.1) has a positive solution, with property (2.2).
(ii) Ifand, then there existandsuch that, for anyand, problem (1.1) has a positive solution, with property (2.3).
(iii) If, then there existandsuch that, for anyand, problem (1.1) has at least two positive solutionsandwith
3 Preliminaries
Then is a real Banach space with norm
A function with is called a solution of problem (1.1) if it satisfies (1.1).
We shall reduce problem (1.1) to an integral equation. To this goal, firstly by means of the transformation
we convert problem (1.1) into
and
Lemma 3.1If (H_{1}), (H_{2}), and (H_{4}) hold, then problem (3.3) has a unique solutionxgiven by
where
Proof The proof of Lemma 3.1 is similar to that of Lemma 2.1 in [31]. □
Write . Then from (3.6) and (3.7), we can prove that and have the following properties.
Proposition 3.1If (H_{4}) holds, then we have
where
Remark 3.1 From (3.6) and (3.11), we obtain
Lemma 3.2If (H_{1}), (H_{3}), and (H_{4}) hold, then problem (3.4) has a unique solutionyandycan be expressed in the form
where
Proof The proof of Lemma 3.2 is similar to that of Lemma 2.2 in [31]. □
From (3.14) and (3.15), we can prove that and have the following properties.
Proposition 3.2If (H_{4}) holds, then we have
where
Suppose that y is a solution of problem (1.1). Then from Lemma 3.1 and Lemma 3.2, we have
where
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 3.3Suppose that (H_{1})(H_{4}) hold. Thenandis completely continuous.
Proof The proof of Lemma 3.3 is similar to that of Lemma 2.4 in [31]. □
To obtain positive solutions of problem (1.1), the following fixed point theorem in cones is fundamental, which can be found in [[35], p.94].
Lemma 3.4LetPbe a cone in a real Banach spaceE. Assume, are bounded open sets inEwith, . If
is completely continuous such that either
thenAhas at least one fixed point in.
Remark 3.2 To make the reader clear what , , , and mean, we give typical examples of and , e.g.,
4 Proofs of the main results
For convenience we introduce the following notation:
and
Proof of Theorem 2.1 Part (i). Noticing that , () for all t and , we can define
Let
which implies that
If , , then there exist , , and such that
and then we get
where
and
It follows from (4.4) and (4.5) that
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).
Part (ii). Noticing that , () for all t and , we can define
Let
which implies that
If , , then there exist , , and such that
where and satisfy (4.2) and (4.3), respectively.
Similar to the proof of (4.6), we can prove that
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).
Consider part (iii). Choose two numbers and satisfying (2.1). By part (i) and part (ii), there exist and such that
Since , from the proof of part (i) and part (ii), it follows that
and
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. □
Proof of Theorem 2.2 Part (i). Noticing that , () for all t and , we can define
Let
Similar to the proof of (4.5), we can prove
It follows from (4.12) and (4.13) that
If , , then there exist , , and such that
and then we get
This yields
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).
Part (ii). Noticing that , () for all t and , we can define
Let
Similar to the proof of (4.5), we can prove
It follows from (4.18) and (4.19) that
If , , then there exist , , and such that
where and satisfy (4.15) and (4.16), respectively.
This yields
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).
Consider part (iii). Choose two numbers and satisfying (2.1). By part (i) and part (ii), there exist and such that
Since , from the proof of part (i) and part (ii), it follows that
and
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. □
Remark 4.1 Comparing with Feng [31], the main features of this paper are as follows.
(i) Two parameters and are considered.
(iii) It follows from the proof of Theorem 2.1 that the conditions of Corollary 3.2 in [31] 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Acknowledgements
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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