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

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.

where $0<\xi ,\eta <1$, $0\le a<b<1$. 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.

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 $0=t_0<t_1<t_2<\cdots <t_k<\cdots <t_m<t_{m+1}=1$, $\mathrm{\Delta }{y}^{\prime }{|}_{t=t_k}=y^{\prime }(t_k^{+})-x^{\prime }(t_k^{-})$, 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 $y>0$;

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).

where δ is defined in (3.20).

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).

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.

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.

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

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].

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

with $0<r<R$, where ${\parallel x\parallel }_{\mathrm{\infty }}={sup}_{t\in [a,b]}|x(t)|$.

where $r>0$ is a constant.

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).