Existence of three periodic solutions for a quasilinear periodic boundary value problem

1.   Introduction and main results

In this paper, we consider the quasilinear periodic boundary value problem

where $f(t, x):{{\bf R}}\times{{\bf R}}\to{{\bf R}}$ is an $L^1$-Carathéodory function which is $1$-periodic in $t$ and $\lambda$ is a positive parameter.

We need the following assumptions:

$\mathrm{(Q_1)}$ $p:{{\bf R}}\to(0, +\infty)$ is a continuous function such that there exist two positive numbers $M\geq m$ and

$\mathrm{(Q_2)}$ $\alpha(t)\in C({{\bf R}})$ is a $1$-periodic positive weight function, that is, there exist $\alpha_1\geq \alpha_0 > 0$ such that

Since the three critical points theorem was obtained by Ricceri ^[11], it has been one of the most frequently applied abstract multiplicity theorems. After that, Averna-Bonanno ^[1] and Ricceri ^[12,13] have given some general three critical points theorems due to Ricceri ^[11]. These three critical points theorems in ^[1,11,12,13] are widely used to solve differential equations (see, for example ^{[1,2,3,4,5,6,7,8,9,10,13]}). In particular, in ^[5], using the three critical points theorem of ^[11], Bonanno and Livrea have studied the existence and multiplicity of solutions for the periodic boundary value problem

where $A(t) = (a_{i, j}(t))_{n\times n}$ is positive definite matrix for all $t\in[0, T], a_{i, j}(t)\in C([0, T], {{\bf R}}), G\in C^1({{\bf R}}^n, {{\bf R}})$ and $b(t)\in L^1([0, T])\setminus\{0\}$ that is a.e. nonnegative. Noticing that when $p(x')\equiv1, T = 1$ and $f(t, x) = b(t)g(x)$, the $n$-dimensional problem (1.4) from ^[5] reduces to the one-dimensional problem (1.1) in case $n = 1$. Recently, in ^[7], using two general three critical points theorems of ^[1] and ^[12], Li et al. have studied the existence of three periodic solutions for $p$-Hamiltonian systems

where $\lambda, \mu\in[0, +\infty), p > 1, A(t) = (a_{i, j}(t))_{n\times n}$ is positive definite matrix for all $t\in[0, T], a_{i, j}(t)\in C([0, T], {{\bf R}}), F: [0, T]\times{{\bf R}}^n\to{{\bf R}}$ is a function such that $F(\cdot, x)$ is continuous in $[0, T]$ for all $x\in{{\bf R}}^n$ and $F(t, \cdot)$ is a $C^1$-function in ${{\bf R}}^n$ for a.e. $t\in[0, T]$, and $G\in [0, T]\times{{\bf R}}^n\to{{\bf R}}$ is measurable in $[0, T]$ and $C^1$ in ${{\bf R}}^n$. Noticing that when $p = 2, T = 1, n = 1$ and $\mu = 0$, problem (1.5) becomes problem (1.1) as $p(x')\equiv1$.

It is well known, the second order Hamiltonian systems satisfying periodic boundary conditions is motivated by celestial mechanics(see ^[15]). Finding periodic solutions for the system is a classic problem. The authors of ^[5] and ^[7] have proved the existence of three periodic solutions for this system. We also want to point out that, in ^[5] and ^[7], the nonlinear terms in the differential equations of the problems studied there do not depend on the derivatives of the unknown functions, i.e., $p(x')\equiv1$, in (1.1). So we are interested in problem (1.1).

On the other hand, in ^[3], using the three critical points theorem of ^[11], Afrouzi and Heidarkhani established a three solutions result for the following quasilinear two point boundary value problem

where $f:[a, b]\times{{\bf R}}\to{{\bf R}}$ is an $L^1$-Carathéodory function, $h:{{\bf R}}\to(0, +\infty)$ is a continuous function and $\lambda > 0, a, b\in{{\bf R}}$, and extend the main result of ^[8] to problem (1.6). Inspired by the ideas of ^[3,8], we discuss the existence of three periodic solutions for problem (1.1).

The aim of this paper is to establish some new criteria for problem (1.1) to have at least three periodic solutions by applying the three critical points theorem due to B. Ricceri. In addition, we give an example to illustrate the validity of our result.

Next we state our results.

Theorem 1.1. Let $g(y) = \int_0^y\left(\int_0^\tau p(\xi)d\xi\right)d\tau\; (\forall y\in{{\bf R}})$ and $F(t, x) = \int_0^xf(t, \xi)d\xi$. Assume that $f: {{\bf R}}\times{{\bf R}}\to{{\bf R}}$ is an $L^1$-Carathéodory function with $1$-periodic in $t$, $p:{{\bf R}}\to(0, +\infty)$ satisfies $\mathrm{(Q_1)}$ and $\alpha(t)\in C({{\bf R}})$ satisfies $\mathrm{(Q_2)}$. Assume that there exist three positive constants $c, d$ and $s$ with $s < 2$, $g(2d)+g(-2d)+\frac{\alpha_0d^2}{2} > c^2\min\{m, \alpha_0\}$, and a function $\gamma(t)\in L^1([0, 1])$ such that

(ⅰ) $f(t, x)\geq0$ for each $(t, x)\in[0, 1]\times [0, \frac{3d}{2}]$;

(ⅱ) $F(t, x)\leq\gamma(t)(1+|x|^s)$ for all $x\in{{\bf R}}$ and a.e. $t\in[0, 1]$;

(ⅲ)

Then, there exist an open set $\Lambda\in [0, +\infty)$ and a positive number $r_0$ such that for every $\lambda\in\Lambda$, problem (1.1) has at least three periodic solutions whose norms in $Z$ are less than $r_0$, where

When $f(t, x) = f_1(t)f_2(x)$, we have the following result by using Theorem 1.1.

Theorem 1.2. Assume that $f_1:{{\bf R}}\to{{\bf R}}^+$ is a $1$-periodic continuous function, $f_2:{{\bf R}}\to{{\bf R}}$ is a continuous function, $p:{{\bf R}}\to(0, +\infty)$ satisfies $\mathrm{(Q_1)}$ and $\alpha(t)\in C({{\bf R}})$ satisfies $\mathrm{(Q_2)}$. Assume that there exist three positive constants $c, d$ and $s$ with $s < 2$, $g(2d)+g(-2d)+\frac{\alpha_0d^2}{2} > c^2\min\{m, \alpha_0\}$, and a positive constant $\gamma$ such that

(ⅰ) $f_2(x)\geq0$ for each $x\in [0, \frac{3d}{2}]$;

(ⅱ) $\int_0^xf_2(\xi)d\xi\leq\gamma(1+|x|^s)$ for all $x\in{{\bf R}}$ and a.e. $t\in[0, 1]$;

(ⅲ)

Then, there exist an open set $\Lambda\in [0, +\infty)$ and a positive number $r_0$ such that for every $\lambda\in\Lambda$, problem

has at least three periodic solutions whose norms in $Z$ are less than $r_0$.

Remark 1.. If we take $p(x')\equiv1$ in Theorem 1.2, then we can get the corresponding results of problem (1.4) as $T = 1, n = 1$. Inspecting the conditions of Theorem 3.1 in ^[5], it is not difficult to find that its hypothesis is different from that of our Theorem 1.2, so this is a new result.

Furthermore, when $\alpha(t)$ and $f(t, x)$ don't depend on $t$, we have the following autonomous version of Theorem 1.1.

Theorem 1.3. Assume that $f:{{\bf R}}\to{{\bf R}}$ is a continuous function and $p:{{\bf R}}\to(0, +\infty)$ satisfies $\mathrm{(Q_1)}$. Assume that there exist three positive constants $c, d$ and $s$ with $s < 2$, $g(2d)+g(-2d)+\frac{\alpha_0 d^2}{2} > c^2\min\{m, \alpha_0\}$, and a positive constant $\gamma$ such that

(ⅰ) $f(x)\geq0$ for each $x\in [0, \frac{3d}{2}]$;

(ⅱ) $\int_0^xf(\xi)d\xi\leq\gamma(1+|x|^s)$ for all $x\in{{\bf R}}$;

(ⅲ)

Then, there exist an open set $\Lambda\in [0, +\infty)$ and a positive number $r_0$ such that for every $\lambda\in\Lambda$, problem

has at least three periodic solutions whose norms in $Z$ are less than $r_0$, where $\alpha > 0$.

We postpone the proofs to the next section and turn to give an example to illustrate the validity of Theorem 1.1.

Example 1. Let $p(y) = 2-\cos y$, $\alpha(t)\equiv 1$ and

Then, we have $m = 1, M = 3, \alpha_0 = \alpha_1 = 1$,

and

where $A = -15e^{16}+\frac{61}{3}$. If $d = 16$ and $c = \sqrt{\frac{4}{\min\{m, \alpha_0\}}} = 2$, then

and

This shows that (1.7) of Theorem 1.1 holds. Further, if $\gamma(t)\equiv e^{16}$ and $s = \frac{2}{3}$, then all the assumptions of Theorem 1.1 are satisfied. Hence, there exist an open interval $\Lambda\in [0, +\infty)$ and a positive number $r_0$ such that for every $\lambda\in\Lambda$, problem (1.1) has at least three periodic solutions whose norms in $Z$ are less than $r_0$.

Remark 2 If we take $p(x')\equiv1$ in Theorem 1.1, then we can get the corresponding results of problem (1.5) as $T = 1, p = 2, n = 1$ and $\mu = 0$. Let $F(t, x)$ and $\alpha(t)$ be the functions in Example 1 respectively, and then after a simple calculation, it is not difficult to verify that

satisfy the conditions of Theorem 1 of ^[7]. Moreover, Example 1 does not satisfy the assumptions of Theorem 1 of ^[7], so Theorem 1.1 represents a development of Theorem 1 of ^[7] in some sense.

2.   Variational setting and proof of Theorems

For the reader's convenience, we first recall here the three critical points theorem of ^[12] and Proposition 3.1 of ^[13].

Lemma 2.1. (^[12], Theorem 1) Let $Z$ be a separable and reflexive real Banach space, $\Phi: Z\to{{\bf R}}$ a continuously G$\hat{a}$teaux differentiable and sequentially weakly lower semi-continuous functional whose G$\hat{a}$teaux derivative admits a continuous inverse on $Z^*$ and $\Phi$ is bounded on each bounded subset of $Z; \Psi : Z\to {{\bf R}}$ a continuously G$\hat{a}$teaux differentiable functional whose G$\hat{a}$teaux derivative is compact. Assume that

for all $\lambda\in[0, +\infty)$, and that there exists $\beta\in{{\bf R}}$ such that

Then, there exist an open set $\Lambda\in [0, +\infty)$ and a positive number $r_0$ such that for every $\lambda\in\Lambda$, the equation

has at least three solutions whose norms in $Z$ are less than $r_0$.

Proposition 2.2. (^[13], Proposition 3.1) Let $Z$ be a nonempty set, and $\Phi, \Psi$ two real functions on $Z$. Assume that there are $r > 0$ and $x_0, x_1\in Z$ such that

Then, for each $\beta\in{{\bf R}}$ satisfying

one has

Remark 3 We recall that a $L^1$-Carathéodory function $f:[0, T]\times{{\bf R}}\to{{\bf R}}$ is defined by

$\mathrm{(C_1)}$ $t\to f(t, x)$ is measurable for every $x\in{{\bf R}}$;

$\mathrm{(C_2)}$ $x\to f(t, x)$ is continuous for almost every $t\in[0, T]$;

$\mathrm{(C_3)}$ for every $\rho > 0$ there exists a function $l_\rho\in L^1([0, T])$ such that

Next, we establish the variational setting for problem (1.1).

Throughout the sequel, the Sobolev space $Z$ is defined by

with the norm

Clearly, $Z$ is a Hilbert space and $Z^* = Z$, where $Z^*$ is the dual space of $Z$.

Setting

we have

Proposition 2.3. Assume that $p(\cdot)$ satisfies $\mathrm{(Q_1)}$, then $g'$ is strongly monotone.

Proof. By $\mathrm{(Q_1)}$ and (2.4), we have

It follows that

for all $y, z\in{{\bf R}}$. This shows that $g'$ is strongly monotone.

Put

Proposition 2.4. Assume that $p(\cdot)$ and $\alpha(t)$ satisfy (1.2) and (1.3) respectively, then

(1) $\Phi$ is well-defined in $Z$;

(2) $\Phi$ is G$\hat{a}$teaux differentiable in $Z$;

(3) $\Phi'$ is a Lipschitzian operator;

(4) $\Phi$ is convex in $Z$.

Proof. From (1.2) and (1.3), we have

and

for every $x\in Z$, which implies that $\Phi$ is well-defined in $Z$.

Taken that $x, y\in Z$ and $\{a_n\}\in{{\bf R}}\backslash\{0\}$ with $\lim\limits_{n\to+\infty}a_n = 0$. By the mean value theorem of differential calculus, for a.e. $t\in[0, 1]$, we can see that there exist $u_n(t)$ and $v_n(t)$ such that

and

Then, by (1.2) and (1.3), if $n$ is large enough, we have

for almost every $t\in[0, 1]$. Again using the Lebesgue's theorem, from the continuity of $g'$ and the arbitrariness of $\{a_n\}$, we know that $\Phi$ is G$\hat{a}$teaux differentiable in $Z$ with

If we fix $x, y, z\in Z$ with $\|z\|\leq1$. Let $u(t)$ be such that

for every $t\in[0, 1]$. Thus, by the Hölder inequality, (1.2), (1.3) and (2.4), we have

which shows that

for every $x, y\in Z$ and $\Phi'$ is Lipschitzian.

Finally, from (1.2) and (2.4), we know that $g$ is convex. Noticing that $\alpha(t)x^2$ is convex in $x$, we have $\Phi$ is convex in $Z$.

For every $x\in Z$, put

Since $f:[0, 1]\times{{\bf R}}\to{{\bf R}}$ is an $L^1$-Carathéodory function, we know that $\Psi$ is a well-defined and G$\hat{a}$teaux differentiable functional with

for every $x, y\in Z$. Since the embeddings $Z\hookrightarrow L^q\; (q\geq1)$ and $Z\hookrightarrow L^\infty$ are compact (See R. A. Adams^[16]), we have $\Psi':Z\to Z^*$ is a continuous and compact operator.

Next, we consider the functional $I:Z\to{{\bf R}}$ defined by

for every $x\in Z$, where $\lambda > 0$. Clearly, $I$ is G$\hat{a}$teaux differentiable. If $x\in Z$ is a critical point for $I$, we have

for each $y\in Z$. This implies that $g'\circ x'$ has a weak derivative which equals $\alpha(t) x(t)-\lambda f(t, x(t))$ and is thus continuous, so $g'\circ x'$ is $C^1([0, 1])$. Since $g'$ is an invertible $C^1$-function, it follows that $x'$ is also in $C^1([0, 1])$, hence $x$ is in $C^2([0, 1])$.

Set

such that $\int_0^1e(t)dt = 0$. Let $y(t) = \int_0^te(\tau)d\tau$. Then $y(t)\in Z$ and $\int_0^1|e(t)|^2dt = 0$, that is, $e(t) = 0$ for a.e. $t\in[0, 1]$. This shows that

for all $t\in[0, 1]$. Hence we conclude that $x$ is a solution of problem (1.1) belongs to $C^2([0, 1])$.

Proof of Theorem 1.1 Consider the functional $I(x) = \Phi(x)-\lambda\Psi(x)$ for every $x\in Z$ and $\lambda > 0$. From Proposition 2.4, we know that $\Phi$ is well-defined, G$\hat{a}$teaux differentiable and convex functional in $Z$, and $\Phi'$ is a Lipschitzian operator, which implies that $\Phi$ is a sequentially weakly lower semicontinuous via Theorem 1.2 of ^[15]. Further, we claim that $\Phi$ admits a continuous inverse on $Z^*$. In fact, by (1.2), (1.3), Proposition 2.3 and (2.6), we have

for all $x, y\in Z$, which shows that $\Phi'$ is uniformly monotone in $Z$. Put $y = 0$, then we have

which shows that $\Phi'$ is coercive in $Z$. Since $\Phi'$ is a Lipschitzian operator, $\Phi'$ is hemicontinuous in $Z$. By Theorem 26. A of ^[14] we can see that $\Phi$ admits a continuous inverse on $Z^*$. From the estimation formula in the proof of Proposition 2.4 $\Phi(x)\leq\frac{1}{2}\max\{M, \alpha_1\}\|x\|^2,$ we see that $\Phi$ is bounded on each bounded subset of $Z$.

On the other hand, as we saw in above, $\Psi : Z\to {{\bf R}}$ a continuously G$\hat{a}$teaux differentiable functional whose G$\hat{a}$teaux derivative is compact. Based on the previous discussion of $I$, we know that the critical points of $I = \Phi-\lambda\Psi$ in $Z$ are the solutions of problem $(1.1)$. Therefore, we only need to verify that both assumptions of Lemma 2.1 are valid.

From (1.2) and (1.3), it follows that

for all $x\in Z$. By assumption (ⅱ), we see that

and so (2.1) of Lemma 2.1 holds.

Next, We want to prove the validity of (2.2) in Lemma 2.1 by using Proposition 2.2. For $x\in Z$, taking into account

and

we have

Thus, for each $r > 0$, we can obtain

which shows that

where $c = \sqrt{\frac{4r}{\min\{m, \alpha_0\}}}$.

Put $x_1 = 0$ and

Then we have $\Phi(x_1) = \Psi(x_1) = 0$, $x_0\in Z$,

and

From $\min\limits_{t\in[\frac{1}{4}, \frac{3}{4}]}\{x_0(t)\} = d$, $\max\limits_{t\in[\frac{1}{4}, \frac{3}{4}]}\{x_0(t)\} = \frac{3d}{2}$ and the assumptions (ⅰ), we have

Moreover, choose $r = \frac{c^2\min\{m, \alpha_0\}}{4}$, and recall from the assumptions of Theorem 1.1 that

From (1.7), (2.8), (2.9), (2.10) and (2.11) we obtain that

and

By Proposition 2.2, we know that (2.2) of Lemma 2.1 holds. So, the proof is complete.

Proof of Theorem 1.2 Let $f(t, x) = f_1(t)f_2(x)$. Noting that

it is easy to verify that the assumptions of Theorem 1.1 hold. So, the proof is complete.

Proof of Theorem 1.3 Let $f(t, x) = f(x)$. Noting that

it is easy to see that the assumptions of Theorem 1.1 hold. So, the proof is complete.

3.   Conclusions

Periodic solutions of Hamiltonian systems are important in applications. For second order Hamiltonian systems or $p$-Hamiltonian systems subject to periodic boundary conditions, there are many works reported on the existence of three periodic solutions. But the results on the multiplicity of periodic solutions of quasilinear periodic boundary value problem are very rare. In this paper, we study a quasilinear second order differential equation involving periodic boundary condition. Using a three critical points theorem obtained by B. Ricceri, we establish some new existence theorem of at least three periodic solutions for the quasilinear periodic boundary value problem (1.1) under appropriate hypotheses. In addition, we give an example to illustrate the validity of our results.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 11901248) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No.18KJB110006). Moreover, the authors express their sincere gratitude to the referee for reading this paper very carefully and specially for valuable suggestions concerning improvement of the manuscript.

Conflict of interest

All authors declare no conflicts of interest in this paper.