Existence of periodic solutions for a prescribed mean curvature Liénard p-Laplacian equation with two delays

Recently, the study of the periodic solutions of prescribed mean curvature equations has become very active; see [1–5] and the references therein. Meanwhile, the Liénard equation, Liénard system, and p-Laplacian equations are also studied by many people; see for example [6–8]. These kinds of equations have wide applications in many fields, such as physics, mechanics, and engineering.

By using the theory of coincidence degree, the author obtained some sufficient conditions for the existence and uniqueness of periodic solution in the case of $\tau (t)=0$.

where ${\phi }_p(s)={|s|}^{p-2}s$, $p\ge 2$, τ, γ are continuous functions with period T, ${\tau }^{\prime }(t)<1$, ${\gamma }^{\prime }(t)<1$, and ${\int }_0^{T}e(t)dt=0$, $f,g,h\in C^1(R,R)$.

To the best of our knowledge, there are few results on this topic. The purpose of this paper is to establish a criterion to guarantee the existence of T-periodic solution. Our methods and results are different from the corresponding ones of [9]. So our results are essentially new.

Let X and Y be real Banach spaces and $L:D(L)\subset X\to Y$ be a Fredholm operator with index zero, that is, $X=KerL\oplus X_1$ and $Y=ImL\oplus Y_1$. Furthermore, let $P:X\to KerL$ and $Q:Y\to Y_1$ be the continuous projectors. Clearly, $KerL\cap (D(L)\cap X_1)=\{0\}$, thus the restriction $L_P=L{|}_{D(L)\cap X_1}$ is invertible. Denote by K the inverse of $L_P$.

Let Ω be an open bounded subset of X with $D(L)\cap \mathrm{\Omega }\ne \mathrm{\Phi }$. A map $N:\overline{\mathrm{\Omega }}\to Y$ is called L-compact in $\overline{\mathrm{\Omega }}$ if $QN(\overline{\mathrm{\Omega }})$ is bounded and the operator $K(I-Q)N:\overline{\mathrm{\Omega }}\to X$ is compact. The following, Mawhin’s continuation theorem, is well known.

Lemma 2.1 (Gaines and Mawhin [10])

Then the equation $Lx=Nx$ has a solution in $\overline{\mathrm{\Omega }}\cap D(L)$.

Denote by $\varphi (x)=\frac{{\phi }_q(x)}{\sqrt{1-{\phi }_q^2(x)}}$, ${\phi }_q(s)={|s|}^{q-2}s$, $1/p+1/q=1$, then prescribed mean curvature p-Laplacian operator ${\phi }_p(\frac{x}{\sqrt{1+x^2}})$ is marked by ${\varphi }^{-1}(x)$. Clearly, if $x(t)={(x_1(t),x_2(t))}^{\mathrm{\top }}$ is a T-periodic solution to (2.1), then $x_1(t)$ must be a T-periodic solution to (1.3).

From (2.3) and (2.4), one can easily see that N is L-compact on Ω, where Ω is an open bounded subset of X. The following lemma is useful to estimate a priori bounds of periodic solutions of (2.1).

Lemma 2.2 (Liu et al. [11])

Lemma 2.3 ([12])

where ${\tau }_0=\frac{1}{1-{|{\tau }^{\prime }|}_0}$, $\gamma (t)$ satisfies the above inequality and ${\gamma }_0=\frac{1}{1-{|{\gamma }^{\prime }|}_0}$.

At the end of this section, we list the basic assumptions which will be used in Section 3.

(H₁) $(g(x_1)-g(x_2))(x_1-x_2)>0$, $x_1,x_2\in R$ and $x_1\ne x_2$, $(h(x_1)-h(x_2))(x_1-x_2)>0$, $x_1,x_2\in R$ and $x_1\ne x_2$.

(H₂) There is a constant $d>0$ such that $x(g(x)+h(x))>0$ if and only if $|x|>d$.

(H₄) There is a constant $\sigma >0$ such that $|f(s)|\ge \sigma$ for all $s\in R$.

In this section we state the main results and give its proof.

Theorem 3.1 Suppose the assumptions (H₁)-(H₄) hold. Then (1.3) has at least one T-periodic solution provided $\frac{T}{\pi }(k_1\sqrt{{\tau }_0}+k_2\sqrt{{\gamma }_0})<\sigma$.

In fact, let $\xi -\tau (\xi )=t_1$, $\xi -\gamma (\xi )=t_2$, then from (3.2), $g(x_1(t_1))+h(x_1(t_2))=0$. If $x_1(t_1)=x_1(t_2)$ holds, (3.3) is clearly true. Now assume $x_1(t_1)<x_1(t_2)$.

namely $F(x_1(t_1))<0<F(x_1(t_2))$, then there must exist a $t_0$ such that $F(x_1(t_0))=0$. That is, $g(x_1(t_0))+h(x_1(t_0))=0$. According to (H₂), (3.3) holds. If $x_1(t_1)>x_1(t_2)$ holds, by a similar method, we can show that (3.3) is also true.

Since $\frac{T}{\pi }(k_1\sqrt{{\tau }_0}+k_2\sqrt{{\gamma }_0})<\sigma$, there is a constant $M_0>0$ independent of λ such that ${|x_1^{\prime }|}_2\le M_0$, i.e., ${|x_1|}_0\le d+\sqrt{T}{|x_1^{\prime }|}_2\le d+\sqrt{T}{M}_0:=M_1$.

where ${|f|}_0={max}_{|x_1|\le M_1}|f(x_1)|$, $g_{M_1}={max}_{|x_1|\le M_1}|g(x_1)|$, $h_{M_1}={max}_{|x_1|\le M_1}|h(x_1)|$. By (3.10), (3.11), ${|x_2|}_0\le (g_{M_1}+h_{M_1}+{|e|}_{\mathrm{\infty }})T+{|f|}_0{M}_0\sqrt{T}+d=:M_2$.

Let $\mathrm{\Omega }=\{x:x=(x_1,x_2)\in X,{|x_1|}_0<M_1+1,{|x_2|}_0<M_2+1\}$. Then $\mathrm{\Omega }\supset ({\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2)$ and Ω is a bounded open set of X. So (1) and (2) of Lemma 2.1 are satisfied.

Condition (3) of Lemma 2.1 holds. By Lemma 2.1, the equation $Lx=Nx$ has a solution. This completes the proof of Theorem 3.1. □

We can use a similar method to conclude the following result, the details are omitted.

Theorem 3.2 Suppose (H₃), (H₄), and the following assumptions hold:

(${\mathrm{H}}_1^{\prime }$) $(g(x_1)-g(x_2))(x_1-x_2)<0$, $x_1,x_2\in R$, and $x_1\ne x_2$, $(h(x_1)-h(x_2))(x_1-x_2)<0$, $x_1,x_2\in R$, and $x_1\ne x_2$.

(${\mathrm{H}}_2^{\prime }$) There is a constant $d>0$ such that $x(g(x)+h(x))<0$ if and only if $|x|>d$.

Then (1.3) has at least one T-periodic solution if $\frac{T}{\pi }(k_1\sqrt{{\tau }_0}+k_2\sqrt{{\gamma }_0})<\sigma$.

It is difficult to establish the uniqueness of the T-periodic solution of (1.3), but in the special case we obtain the uniqueness of (1.3).

Theorem 4.1 Assume (${\mathrm{H}}_1^{\prime }$) holds, and $\tau (t)={\epsilon }_1$, $\gamma (t)={\epsilon }_2$ (${\epsilon }_1$, ${\epsilon }_2$ are sufficiently small constants). Then (1.3) has at most one T-periodic solution.

We argue by contradiction, in view of $v\in C^2[0,T]$ and $v(t+T)=v(t)$, $\mathrm{\forall }t\in R$, we obtain ${max}_{t\in R}v(t)>0$.

By using a similar argument, we can also show that $y_2(t)-y_1(t)\le 0$.