Multiplicity of positive solutions to second-order singular differential equations with a parameter

where p is a strictly positive absolutely continuous function. Such equations, even in the case $p\equiv 1$, where they are referred to as being of Schrödinger or Klein-Gordon type, appear in many scientific areas including quantum field theory, gas dynamics, fluid mechanics and chemistry.

Electrostatic or gravitational forces are the most important examples of singular interactions. During the last few decades, the study of the existence of positive solutions for singular differential equations has deserved the attention of many researchers [1–7]. Some strong force conditions introduced by Gordon [8] are standard in the related earlier works [6, 7, 9]. Compared with the case of strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity is more recent, but has also attracted many researchers [2, 3, 10, 11]. In particular, the degree theory [6, 7], the method of upper and lower solutions [11, 12], Schauder’s fixed point theorem [2, 10], some fixed point theorems in cones for completely continuous operators [13–16] and a nonlinear Leray-Schauder alternative principle [17–19] are the most relevant tools.

However, singular differential equation (1.1), in which the nonlinearity is dependent on the derivative and does not require f to be nonnegative, has not attracted much attention in the literature. There are not so many existence results for (1.1) even when the nonlinearity is independent of the derivative. In this paper, we try to fill this gap and establish the existence of positive T-periodic solutions of (1.1) using the Guo-Krasnosel’skii fixed point theorem on compression and expansion of cones, which has been used to study positive solutions for systems of ordinary, functional differential equations [14–16]. We remark that it is sufficient to prove that $T:K\cap ({\tilde{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1)\to K$ is continuous and completely continuous in Lemma 3.2 (Section 3). This point is essential and advantageous.

and prove its positiveness. The fact is very crucial to our arguments. Moreover, the one-signed property of the Green function implies that a maximum principle and an anti-maximum principle hold for the corresponding linear differential equations subject to various boundary conditions, which is an important topic in differential equations (see [20, 21]). In Section 3, by employing the Guo-Krasnosel’skii fixed point theorem, we prove the existence of twin positive solutions for (1.1) under the positiveness of the Green function associated with (1.2)-(1.3). To illustrate the new results, some applications are also given.

which implies that the series ${\sum }_{n=0}^{\mathrm{\infty }}{x}_n(t)$ converges uniformly with respect to $t\in [0,T]$. Obviously, $x(t)={\sum }_{n=0}^{\mathrm{\infty }}{x}_n(t)$ is a continuous solution of (2.2). Moreover, inequality (2.3) holds because $\phi (t)$ and $K(s,\tau )$ are nonnegative functions.

Next we prove the uniqueness. To do so, we first show that the solution of (2.2) is unique on $[t_1,t_2]\subset [0,T]$ with $t_2<\sqrt{2/{\beta }_1}$. Then the uniqueness of the solution on $[0,T]$ is direct using the continuation property.

Hence it follows that $x_1(t)=x_2(t)$ for all $t\in [t_1,t_2]$. □

To obtain (2.4), we only need to integrate the above equality from 0 to t and notice $u(0)=1$. In a similar way, we can prove (2.5). □

is the Green function, the number D is defined by $D=u(T)+v^{\prime }(T)-2$.

After not very complicated calculations, we can get (2.7) and (2.8). □

Lemma 2.5 Assume that (H) holds. Then the Green function $G(t,s)$ associated with (2.1)-(1.3) is positive for all $s,t\in [0,T]$.

Using Lemma 2.1, we get from (2.12) that $E_1(t,s)>0$ for all $t\in (s,T]$.

Again using Lemma 2.1, we get from (2.13) that $E_2(t,s)>0$ if $(t,s)\ne (T,0)$, and the proof is completed. □

In this section, we state and prove the new existence results for (1.1). The proof is based on the following well-known fixed point theorem on compression and expansion of cones, which we state here for the convenience of the reader, after introducing the definition of a cone.

We also recall that a compact operator means an operator which transforms every bounded set into a relatively compact set. Let us define the function $\omega (t)=\lambda {\int }_0^{T}G(t,s)ds$ and use ${\parallel \cdot \parallel }_1$ to denote the usual $L^1$-norm over $(0,T)$, by $\parallel \cdot \parallel$ we denote the supremum norm of $\mathbb{C}[0,T]$.

Lemma 3.2 [22]

Then $\mathcal{A}$ has a fixed point in $K\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1)$.

where σ is as in (2.14).

for $x\in X$ and $t\in [0,T]$.

Lemma 3.3 $\mathcal{A}:X\to K$ is well defined.

This implies that $\mathcal{A}(X)\subset K$ and the proof is completed. □

It is easy to prove.

Lemma 3.4 $\mathcal{A}$ is continuous and completely continuous.

Now we present our main result.

Theorem 3.5 Suppose that (1.1) satisfies (H). Furthermore, assume that

(H₂) ${lim}_{x\to 0^{+}}f(t,x,y)=+\mathrm{\infty }$ and ${lim}_{x\to +\mathrm{\infty }}f(t,x,y)/x=+\mathrm{\infty }$ uniformly $(t,y)\in {\mathbb{R}}^2$.

Then (1.1) has at least two positive T-periodic solutions for sufficiently small λ.

where $-{\omega }^{″}(t)+a(t){\omega }^{\prime }(t)+b(t)\omega (t)=\lambda$ is used.

For $r>0$, let ${\mathrm{\Omega }}_r=\{x\in K:\parallel x\parallel <r\}$ and note that $\partial {\mathrm{\Omega }}_r=\{x\in K:\parallel x\parallel =r\}$.

This implies $\parallel \mathcal{A}x\parallel \le \parallel x\parallel$.

So, equation (1.1) has a positive solution $x_1(t)={\tilde{x}}_1(t)-M\omega (t)\ge \sigma r_1-\frac{\sigma MT}{2}\ge \frac{\sigma MT}{2}$.

where γ satisfies $\lambda \gamma \sigma AT>1$.

we can conclude that $x_1$ and $x_2$ are the desired distinct positive periodic solutions of (1.1) for $\lambda <min\{{\lambda }^{\ast },{\lambda }^{\ast \ast }\}$. □

where $\alpha >0$, $\beta >1$, $\gamma \ge 0$, $c(t),d(t),e(x)\in \mathbb{C}[0,T]$. It is clear that $f(t,x,y)$ satisfies conditions (H₁), (H₂). Then (1.1) has at least two positive T-periodic solutions for sufficiently small λ.