Positive solutions for classes of multi-parameter fourth-order impulsive differential equations with one-dimensional singular p-Laplacian

The authors consider the following impulsive differential equations involving the one-dimensional singular p-Laplacian: {({\varphi }_p(y^{″}(t)))}^{″}=\lambda \omega (t)f(t,y(t)), t\in J, t\ne t_k, k=1,2,\dots ,m, \mathrm{\Delta }{y}^{\prime }{|}_{t=t_k}=-\mu I_k(t_k,y(t_k)), k=1,2,\dots ,m, ay(0)-b{y}^{\prime }(0)={\int }_0^{1}h(s)y(s)ds, ay(1)+b{y}^{\prime }(1)={\int }_0^{1}h(s)y(s)ds, {\varphi }_p(y^{″}(0))={\varphi }_p(y^{″}(1))={\int }_0^{1}h(t){\varphi }_p(y^{″}(t))dt, where \lambda >0 and \mu >0 are two parameters. Several new and more general existence and multiplicity results are derived in terms of different values of \lambda >0 and \mu >0. In this case, our results cover equations without impulsive effects and are compared with some recent results.

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 [1], 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.

Very recently, Zhang and Liu [30] studied the following fourth-order four-point 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 G(t,s) and H(t,s), the authors give sufficient conditions for the existence of one positive solution.

In this paper, we investigate the existence of positive solutions of fourth-order impulsive differential equations with two parameters

where \lambda >0 and \mu >0 are two parameters, a,b>0, J=[0,1], {\varphi }_p(s) is a p-Laplace operator, i.e., {\varphi }_p(s)={|s|}^{p-2}s, p>1, {({\varphi }_p)}^{-1}={\varphi }_q, \frac{1}{p}+\frac{1}{q}=1, ω is a nonnegative measurable function on (0,1), \omega \ne 0 on any open subinterval in (0,1) which may be singular at t=0 and/or t=1, t_k (k=1,2,\dots ,m) (where m is fixed positive integer) are fixed points with , \mathrm{\Delta }{y}^{\prime }{|}_{t=t_k}=y^{\prime }(t_k^{+})-x^{\prime }(t_k^{-}), where y^{\prime }(t_k^{+}) and y^{\prime }(t_k^{-}) represent the right-hand limit and left-hand limit of y^{\prime }(t) at t=t_k, respectively. In addition, ω, f, I_k, g, and h satisfy

(H₁) \omega \in L_{\mathrm{loc}}^1(0,1);

(H₂) f\in C([0,1]\times [0,+\mathrm{\infty }),[0,+\mathrm{\infty })) with f(t,y)>0 for all t and y>0;

(H₃) I_k\in C([0,1]\times [0,+\mathrm{\infty }),[0,+\mathrm{\infty })) with I_k(t,y)>0 (k=1,2,\dots ,n) for all t and y>0;

(H₄) g,h\in L^1[0,1] are nonnegative and \xi \in [0,a), \nu \in [0,1), 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 p=2, I_k=0, k=1,2,\dots ,m and \omega \equiv 1 for t\in J. 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 \lambda =1, I_k(t_k,y(t_k))=I_k(y(t_k)), \omega \equiv 1 for t\in J and \mu =1. 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 p=2 and \mu =1.

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.

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, r_1, r_2, and R satisfying

where δ is defined in (3.20).

Theorem 2.1 Assume that (H₁)-(H₄) hold.

1. (i)

   If f^{\mathrm{\infty }}=0 and I^{\mathrm{\infty }}=0, then there exist {\lambda }_0>0 and {\mu }_0>0 such that, for any \lambda >{\lambda }_0 and \mu >{\mu }_0, problem (1.1) has a positive solution u(t), t\in J with

   \delta r\le u(t)\le \frac{1}{\delta }R,t\in J.

   (2.2)
2. (ii)

   If f^0=0 and I^0=0, then there exist {\lambda }_0>0 and {\mu }_0>0 such that, for any \lambda >{\lambda }_0 and \mu >{\mu }_0, problem (1.1) has a positive solution u(t) with

   \delta r\le u(t)\le R,t\in J.

   (2.3)
3. (iii)

   If f^0=f^{\mathrm{\infty }}=I^{\mathrm{\infty }}=I^0=0, then there exist {\lambda }_0>0 and {\mu }_0>0 such that, for any \lambda >{\lambda }_0 and \mu >{\mu }_0, problem (1.1) has at least two positive solutions u_1(t) and u_2(t) with

   (2.4)

Theorem 2.2 Assume that (H₁)-(H₄) hold.

1. (i)

   If f_{\mathrm{\infty }}=+\mathrm{\infty } and I_{\mathrm{\infty }}=+\mathrm{\infty }, then there exist {\overline{\lambda }}_0>0 and {\overline{\mu }}_0>0 such that, for any and , problem (1.1) has a positive solution u(t), t\in J with property (2.2).

2. (ii)

   If f_0=+\mathrm{\infty } and I_0=+\mathrm{\infty }, then there exist {\overline{\lambda }}_0>0 and {\overline{\mu }}_0>0 such that, for any and , problem (1.1) has a positive solution u(t), t\in J with property (2.3).

3. (iii)

   If f_0=f_{\mathrm{\infty }}=I_{\mathrm{\infty }}=I_0=+\mathrm{\infty }, then there exist {\overline{\lambda }}_0>0 and {\overline{\mu }}_0>0 such that, for any and , problem (1.1) has at least two positive solutions u_1(t) and u_2(t) with

   (2.5)

Let J^{\prime }=J\setminus \{t_1,t_2,\dots ,t_m\}, and

Then P{C}^1[0,1] is a real Banach space with norm

where {\parallel y\parallel }_{\mathrm{\infty }}={sup}_{t\in J}|y(t)|, {\parallel y^{\prime }\parallel }_{\mathrm{\infty }}={sup}_{t\in J}|y^{\prime }(t)|.

A function y\in P{C}^1[0,1]\cap C^4(J^{\prime }) with {\phi }_p(y^{″})\in C^2(0,1) 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 3.1 If (H₁), (H₂), and (H₄) hold, then problem (3.3) has a unique solution x given by

where

Proof The proof of Lemma 3.1 is similar to that of Lemma 2.1 in [31]. □

Write e(t)=t(1-t). Then from (3.6) and (3.7), we can prove that H(t,s) and G(t,s) have the following properties.

Proposition 3.1 If (H₄) holds, then we have

where

Remark 3.1 From (3.6) and (3.11), we obtain

Lemma 3.2 If (H₁), (H₃), and (H₄) hold, then problem (3.4) has a unique solution y and y can 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 H_1(t,s) and G_1(t,s) have the following properties.

Proposition 3.2 If (H₄) 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

Define a cone in P{C}^1[0,1] by

where

It is easy to see K is a closed convex cone of P{C}^1[0,1].

Define an operator T_{\lambda }^{\mu }:K\to P{C}^1[0,1] by

From (3.21), we know that y\in P{C}^1[0,1] is a solution of problem (1.1) if and only if y is a fixed point of operator T_{\lambda }^{\mu }.

Lemma 3.3 Suppose that (H₁)-(H₄) hold. Then T_{\lambda }^{\mu }(K)\subset K and T_{\lambda }^{\mu }:K\to K is 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 3.4 Let P be a cone in a real Banach space E. Assume {\mathrm{\Omega }}_1, {\mathrm{\Omega }}_2 are bounded open sets in E with 0\in {\mathrm{\Omega }}_1, {\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2. If

is completely continuous such that either

1. (a)

   \parallel Ax\parallel \le \parallel x\parallel, \mathrm{\forall }x\in P\cap \partial {\mathrm{\Omega }}_1 and \parallel Ax\parallel \ge \parallel x\parallel, \mathrm{\forall }x\in P\cap \partial {\mathrm{\Omega }}_2, or

2. (b)

   \parallel Ax\parallel \ge \parallel x\parallel, \mathrm{\forall }x\in P\cap \partial {\mathrm{\Omega }}_1 and \parallel Ax\parallel \le \parallel x\parallel, \mathrm{\forall }x\in P\cap \partial {\mathrm{\Omega }}_2,

then A has at least one fixed point in P\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1).

Remark 3.2 To make the reader clear what {\overline{\mathrm{\Omega }}}_2, \partial {\mathrm{\Omega }}_2, \partial {\mathrm{\Omega }}_1, and {\mathrm{\Omega }}_2\setminus {\overline{\mathrm{\Omega }}}_1 mean, we give typical examples of {\mathrm{\Omega }}_1 and {\mathrm{\Omega }}_2, e.g.,

with , where {\parallel x\parallel }_{\mathrm{\infty }}={sup}_{t\in [a,b]}|x(t)|.

For convenience we introduce the following notation:

and

where r>0 is a constant.

Proof of Theorem 2.1 Part (i). Noticing that f(t,y)>0, I_k(t,y)>0 (k=1,2,\dots ,m) for all t and y>0, we can define

where r>0, and

Let

Then, for u\in K\cap \partial {\mathrm{\Omega }}_r and \lambda >{\lambda }_0, \mu >{\mu }_0, we have

which implies that

If f^{\mathrm{\infty }}=0, I^{\mathrm{\infty }}=0, then there exist l_1>0, l_2>0, and R>r>0 such that

where l_1 satisfies

l_2 satisfies

Let \alpha =\frac{R}{\delta }. Thus, when y\in K\cap \partial {\mathrm{\Omega }}_{\alpha } we have

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 T_{\lambda }^{\mu } has a fixed point y\in K\cap ({\overline{\mathrm{\Omega }}}_{\alpha }\setminus {\mathrm{\Omega }}_r) with r\le {\parallel y\parallel }_{P{C}^1}\le \alpha =\frac{1}{\delta }R. Hence, since for y\in K we have y(t)\ge \delta {\parallel y\parallel }_{P{C}^1}, t\in J, it follows that (2.2) holds. This gives the proof of part (i).

Part (ii). Noticing that f(t,y)>0, I_k(t,y)>0 (k=1,2,\dots ,m) for all t and y>0, we can define

where R>0, and

Let

Then, for y\in K\cap \partial {\mathrm{\Omega }}_R and \lambda >{\lambda }_0, \mu >{\mu }_0, we have

which implies that

If f^0=0, I^0=0, then there exist l_1>0, l_2>0, and such that

where l_1 and l_2 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 T_{\lambda }^{\mu } has a fixed point y\in K\cap ({\overline{\mathrm{\Omega }}}_R\setminus {\mathrm{\Omega }}_r) with r\le {\parallel y\parallel }_{P{C}^1}\le R. Hence, since for y\in K we have y(t)\ge \delta {\parallel y\parallel }_{P{C}^1} for t\in J, it follows that (2.3) holds. This gives the proof of part (ii).

Consider part (iii). Choose two numbers r_1 and r_2 satisfying (2.1). By part (i) and part (ii), there exist {\lambda }_0>0 and {\mu }_0>0 such that

Since f^0=f^{\mathrm{\infty }}=I^{\mathrm{\infty }}=I^0=0, from the proof of part (i) and part (ii), it follows that

and

Applying Lemma 3.4 to (4.9)-(4.11) shows that T_{\lambda }^{\mu } has two fixed points y_1 and y_2 such that y_1\in K\cap ({\overline{\mathrm{\Omega }}}_{r_1}\setminus {\mathrm{\Omega }}_r) and y_2\in K\cap ({\overline{\mathrm{\Omega }}}_R\setminus {\mathrm{\Omega }}_{r_2}). These are the desired distinct positive solutions of problem (1.1) for {\lambda }_0>0 and {\mu }_0>0 satisfying (2.4). Then the result of part (iii) follows. □

Proof of Theorem 2.2 Part (i). Noticing that f(t,y)>0, I_k(t,y)>0 (k=1,2,\dots ,m) for all t and y>0, we can define