Abstract
Full Text
UDC 517.917
MATHEMATICS
E. L. TONKOV
ON A PERIODIC EQUATION OF SECOND ORDER
(Presented by Academician I. M. Vinogradov on 15 V 1968)
The present note continues the investigations published in ((^{10},{}^{16})).
Let (u_0, u_1) (\bigl(u_0(0)=1,\ \dot u_0(0)=0,\ u_1(0)=0,\ \dot u_1(0)=1\bigr)) be a fundamental system of solutions of the equation
[
\mathcal{L}x \equiv \ddot x+q(t)\dot x+p(t)x=0,\qquad t\in(-\infty,\infty),
\tag{1}
]
where (q,p\in L[0,\omega]) are (\omega)-periodic.
By the multipliers of equation (1) (see ((^{18})), p. 98) we shall mean the roots of the characteristic equation
[
\lambda^2-\bigl[u_0(\omega)+\dot u_1(\omega)\bigr]\lambda+\exp(-\omega I_q)=0
]
[
\left(I_q=\frac1\omega\int_0^\omega q(t)\,dt\right).
]
It is easy to see that a real number (\lambda) is not a multiplier if and only if there exists a Green’s function for the problem
[
\mathcal{L}x=0,\qquad x(\omega)=\lambda x(0),\qquad \dot x(\omega)=\lambda \dot x(0),
\tag{2}
]
i.e., if and only if (2) has the unique trivial solution (x(t)\equiv0).
Many questions in the theory of equation (1) reduce to the estimation of multipliers (see ((^{14}))). For example, this equation is asymptotically stable for (t\in[0,\infty)) if and only if (I_q>0) and, for all (\lambda), (|\lambda|\ge1), there exists a Green’s function for problem (2).
As usual, we identify two functions that differ only on a set of measure zero, and, in order to shorten formulations, omit qualifications necessary when dealing with summable functions (the inequality holds almost everywhere, etc.).
- As in ((^{10},{}^{16})), denote by (\alpha=r_0(\alpha), r_1(\alpha), r_2(\alpha),\ldots) the successive zeros of a nontrivial solution of equation (1) having its first zero at (t=\alpha) ((r_i<r_{i+1})).
Lemma 1. A Green’s function of problem (2) exists for every (\lambda<0) (there are no negative multipliers of equation (1)) if and only if, for all (\alpha\in[0,\omega]), either (\alpha+\omega<r_1(\alpha)), or, for some (n=0,1,\ldots),
[
r_{2n+1}(\alpha)<\alpha+\omega<r_{2n+3}(\alpha).
]
- Consider problem (2) for (\lambda>0). We shall say that condition (A) is satisfied if problem (2) has no eigenfunction of constant sign, and condition (B) if (2) has no sign-changing eigenfunction. Thus, for the existence of a Green’s function of problem (2), it is necessary and sufficient that conditions (A) and (B) be satisfied simultaneously.
Put
[
\rho=\frac1\omega\ln\lambda,\qquad
p^*(\rho,t)\equiv \rho^2+\rho I_q+\frac14 I_q^2+p(t)-\frac14 q^2(t)-\frac12 \dot q(t).
]
Lemma 2. Let (\lambda>0). Condition (A) is satisfied if and only if, for some natural (k), on the interval ([0,k\omega]) there exists a funct-
function (v(t)) with absolutely continuous first derivative such that
(v(k\omega)=v(0)), (\dot v(k\omega)\leq \dot v(0)) ((\geq 0)),
(\mathcal L_\rho v\equiv \mathcal L v+2\rho \dot v+(\rho^2+\rho q)v\geq 0)
((\mathcal L_\rho v\leq 0)), and either (\mathcal L_\rho v\not\equiv 0), or (\dot v(k\omega)\ne \dot v(0)).
Theorem 1. Suppose that any one of the following conditions is satisfied:
a) (p\ne 0) on ([0,\omega]), (I_q\geq -\rho), and for all (\alpha\in[0,\omega])
[
I(\alpha)\equiv \int_{\alpha}^{\alpha+\omega} dt\int_{\alpha}^{t}
\exp\left(-\int_{s}^{t} q(\tau)\,d\tau\right)\,ds\geq -\rho\omega;
]
b) (p^(\rho,t)\in L[0,\omega]), (p^(\rho,t)\ne 0) on ([0,\omega]),
[
\int_{0}^{\omega} p^*(\rho,t)\,dt\geq 0.
]
Then for (\lambda=\exp \rho\omega) condition (A) is satisfied.
Lemma 3. In order that, for every (\lambda>0), condition (B) be satisfied, it is necessary and sufficient that, for all (\alpha\in[0,\omega]) and some (n=0,1,\ldots), the inequalities
[
r_{2n}(\alpha)<\alpha+\omega<r_{2n+2}(\alpha).
\tag{3}
]
hold.
Corollary 1. Suppose that for all (\alpha\in[0,\omega]), (\alpha+\omega<r_2^{*}(\alpha)). Suppose, moreover, that one of the following conditions is satisfied:
a) (p\ne 0) on ([0,\omega]), (I_q\geq 0), and for all (\alpha\in[0,\omega]), (I(\alpha)\geq 0);
b) (p^(0,t)\in L[0,\omega]), (p^(0,t)\ne 0) on ([0,\omega]), (I_q\geq 0),
[
\int_{0}^{\omega} p^*(0,t)\,dt\geq 0.
]
Then the Green’s function of problem (2) exists for all (\lambda\geq 1).
Remark. If inequalities (3) are satisfied for (n\geq 1), then for any (\lambda>0) the Green’s function of problem (2) exists.
- To construct effective conditions guaranteeing the inequality (\beta<r_1(\alpha)), it is convenient to use the well-known Wintner–Levinson nonoscillation criterion ((^{21})) (see also Corollary 2 of ((^{1}))), namely that (\beta<r_1(\alpha)) if and only if on ([\alpha,\beta]) there exists a function (v(t)\geq 0) such that (\mathcal L v\leq 0), with (\mathcal L v\not\equiv 0) on ([\alpha,\beta]). From Remark 2 to Theorem 1 of ((^{3})) we obtain the following oscillation criterion.
Lemma 4. (r_1(\alpha)<\beta) if and only if on ([\alpha,\beta]) there exists such a (v(t)), possessing an absolutely continuous first derivative, that
(\dot v\geq 0), (v(\alpha)=v(\beta)=0), (\mathcal L v\geq 0), with (\mathcal L v\not\equiv 0) on ([\alpha,\beta]).
Theorem 2. Suppose that (q(t)\geq -2m_1\leq 0) for (t\in[\alpha,\alpha+T_1]),
(q(t)\leq 2m_2\geq 0) for (t\in[\alpha+T_1,\alpha+T_1+T_2]),
(p(t)\geq k>m_i^2) ((i=1,2)), where
[
T_i=\frac{1}{\sqrt{k-m_i^2}}
\left(\pi-\operatorname{arcctg}\frac{m_i}{\sqrt{k-m_i^2}}\right),
\qquad i=1,2.
]
If (\beta-\alpha>T_1+T_2), then (r_1(\alpha)<\beta).
For the proof of Theorem 2 it suffices to use Lemma 4, taking as the function (v(t)) the solution of the equation
(\ddot x+M(t)\dot x+kx=0), where (M(t)=-2m_1) for (t\in[\alpha,\alpha+T_1]) and (M(t)=2m_2) for (t\in[\alpha+T_1,\alpha+T_1+T_2]).
For (q\equiv 0), Theorem 2 gives the well-known oscillation criterion of N. E. Zhukovsky ((^{5})) (see also ((^{18}))): if (p(t)\geq \pi^2/(\beta-\alpha)^2) ((\not\equiv)), then (r_1(\alpha)<\beta).
Analogously, the following assertion is proved, close in content to Corollary 3 of ((^{9})) (see also ((^{4,12,13,19}))).
Theorem 3. Suppose that (q(t)\leq 2m_1\geq 0) for (t\in[\alpha,\alpha+T_1']),
(q(t)\geq -2m_2\leq 0) for (t\in[\alpha+T_1',\alpha+T_1'+T_2']),
(p(t)\leq l>m_i^2) ((i=1,2)),
where
[
T_i'=\frac{1}{\sqrt{\,l-m_i^2\,}}\operatorname{arcctg}\frac{m_i}{\sqrt{\,l-m_i^2\,}},\qquad i=1,2.
]
If (\beta-\alpha\beta).
From Theorems 2 and 3 we obtain
Corollary 2. Let (|q(t)|\le 2m,\ m^2<k\le p(t)\le l), and for some integers (n_1,n_2) ((0\le n_1<n_2))
[
\frac{2n_1}{\sqrt{k-m^2}}\left(\pi-\operatorname{arcctg}\frac{m}{\sqrt{k-m}}\right)
\le \beta-\alpha \le
\frac{2n_2}{\sqrt{l-m^2}}\operatorname{arcctg}\frac{m}{\sqrt{l-m^2}} .
]
Then
[
r_{n_1}(\alpha)<\beta<r_{n_2}(\alpha).
\tag{4}
]
For other effective sufficient conditions guaranteeing inequalities (4), see ((^{9,11,20})).
- The linear assertions formulated above prove useful in the study of questions of existence, uniqueness, and stability of periodic solutions of the nonlinear equation
[
\mathcal L x\equiv \ddot x+q(t)\dot x+p(t)x=f(t,x,\dot x),\qquad t\in(-\infty,\infty).
\tag{5}
]
We shall assume that (f(t,x,y)) satisfies the Carathéodory smoothness conditions ((^6)), is (\omega)-periodic in (t), and is nonnegative for (x\ge0,\ y\in(-\infty,\infty)).
The following assertion supplements Theorem 1 of ((^7)).
Theorem 4. Let the Green’s function of problem (2) for (\lambda=1) exist and be positive. Suppose, further, that for (t\in[0,\omega]), (x\ge0), (y\in(-\infty,\infty)), the inequality
[
f(t,x,y)\le a(t)x+b(t)|y|^\gamma+c,\qquad 0\le\gamma<1
]
is satisfied.
If the Green’s function of the problem (\mathcal L x=a(t)x,\ x(\omega)=x(0),\ \dot x(\omega)=\dot x(0)) exists and is positive, then equation (5) has at least one positive (\omega)-periodic solution.
Remark. Theorem 4 remains valid if (f(t,x,y)) is discontinuous in (x,y), satisfies the conditions of ((^2)), and the solution is understood in the sense of ((^2)).
The following assertion supplements the results of ((^{10})) on conditions for positivity of the Green’s function of problem (2) for (\lambda=1).
Theorem 5. Let the equation (\mathcal L x=0) be nonoscillatory on ([a,a+\omega]), i.e., for every (a\in[0,\omega)), (a+\omega<r_1(a)). Suppose, further, that (p\ne0) on ([0,\omega]) and at least one of the following conditions is fulfilled:
a) for all (a\in[0,\omega)), (I(a)\ge0);
b) for all (a\in[0,\omega)),
[
\int_a^{a+\omega} p(t)\exp\left(\int_a^t q(s)\,ds\right)dt\ge0;
]
c) (p^*(0,t)\in L[0,\omega]),
[
\int_0^\omega p^*(0,t)\,dt\ge0.
]
Then the Green’s function (G(t,s)) of problem (2) for (\lambda=1) exists and (G(t,s)>0) in the square (t,s\in[0,\omega]).
For other conditions guaranteeing positivity of the Green’s function of problem (2) for (\lambda=1), see ((^{8,15,17})).
For the case when the right-hand side of equation (5) does not depend on (\dot x), i.e., for the equation
[
\mathcal L x\equiv \ddot x+q(t)\dot x+p(t)x=f(t,x),\qquad t\in(-\infty,\infty),
\tag{6}
]
the following assertion is valid.
Theorem 6 (cf. (7)). Let the Green’s function of problem (2) for (\lambda = 1) exist and be positive. Suppose, further, that for (t \in [0,\omega]) and (x_1 \geq x_2 \geq 0) the inequalities
[
b(t)(x_1-x_2) \leq f(t,x_1)-f(t,x_2) \leq a(t)(x_1-x_2)
]
hold. If the Green’s function of the problem (\mathcal L x=a(t)x,\ x(\omega)=x(0),\ \dot x(\omega)=\dot x(0)) exists and is positive, and for the equation (\mathcal L x=b(t)x) one has (\alpha+\omega