Abstract
Full Text
UDC 517.91+517.948
MATHEMATICS
N. V. MEDVEDEV
PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS IN A BANACH SPACE
(Presented by Academician I. G. Petrovskii on 27 V 1968)
Consider the differential equation
[
dx / dt = f(t,x),
\tag{1}
]
where (x=x(t)) is the unknown function with values in a certain complex Banach space (E); (f(t,x)) is an operator which, for each fixed (t), acts in (E) and is periodic in (t) with period (2\pi). It is assumed that the operator (f(t,x)) maps a strongly continuous (2\pi)-periodic function (x(t)) into the likewise strongly continuous function (f(t,x(t))).
There are various methods for studying periodic solutions of equation (1), for example, ((^{1,2})). In the present work one more method is proposed for solving the problem of the existence and finding of periodic solutions of equation (1), which is close to Cesari’s method ((^2)).
Consider the operator (B_1), defined by the equality
[
B_1x(t)=a+\frac{1}{2\pi}\int_t^{t+2\pi}(s-t-2\pi)f(s,x(s))\,ds.
]
Let (X) be the set consisting of strongly continuous (2\pi)-periodic functions. Then the operator (B_1) acts in (X) for any value of the parameter (a\in E).
Lemma. Let (x(t,a)) be a fixed point of the operator (B_1). If
[
\int_0^{2\pi} f(s,x(s,a))\,ds=0,
\tag{2}
]
then the function (x(t,a)) is a (2\pi)-periodic solution of equation (1).
The operator (B_1) is convenient for studying the equation
[
dx/dt=A(t)x+f(t),
\tag{3}
]
where (A(t)) is a linear bounded operator, continuous and periodic in (t) with period (2\pi), and (f(t)) is a continuous (2\pi)-periodic function.
Introduce the following series:
[
U(t)=-\frac{1}{2\pi}\int_{t+2\pi}^{t} A(s)\,ds
+\frac{1}{2\pi}\int_{t+2\pi}^{t}\left(\int_{t+2\pi}^{s} A(s)\,ds\right)A(s)\,ds-\ldots,
]
[
g(t)=\frac{1}{2\pi}\int_{t+2\pi}^{t} f(s)\,ds
-\frac{1}{2\pi}\int_{t+2\pi}^{t}\left(\int_{t+2\pi}^{s} A(s)\,ds\right)f(s)\,ds+\ldots
]
In this case we have
[
U(t)=\frac{1}{2\pi}V(t,0)(V(2\pi,0)-I)V(0,t),
]
where (I) is the identity operator, and (V(t,\tau)) is the solution of the integral equation
[
V(t,\tau)=I+\int_{\tau}^{t}V(t,s)A(s)\,ds.
]
Theorem 1. If equation (3) has a (2\pi)-periodic solution (x(t)), then the equality (U(t)x(t)+g(t)=0) holds.
The proof is carried out using the operator (B_1).
One can give the following sufficient condition for the invertibility of the operator (U(t)).
Theorem 2. Suppose
[
\alpha=|A(0)|,\qquad \gamma=\sup_t|A(t)|,\qquad
\beta=\frac{1}{2\pi}\int_0^{2\pi}|A(s)-A(0)|\,ds.
]
Suppose the operator (I-2\pi A(0)) is invertible and
[
\left|(I-2\pi A(0))^{-1}\right|=(1+2\pi\rho)^{-1}.
]
Then, if
[
\frac{1+2\pi\beta\exp(2\pi\gamma)}
{1+2\pi\rho-4\pi^2\alpha\gamma\exp(2\pi\gamma)}<1,
]
equation (3) has one and only one (2\pi)-periodic solution.
We now introduce a more general operator (B_2), where
[
B_2y(t)=-\varphi(t)+\frac{1}{2\pi}\int_t^{t+2\pi}(s-t-\pi)f(s,a_0+\varphi(s)+y(s))\,ds,
]
[
\varphi(t)=\sum_{0<|n|\le k} a_n e^{int}.
]
The operator (B_2), for arbitrary values of the parameters (a_0,a_1,a_{-1},\ldots,a_k,a_{-k}), also acts in (X). If (y(t,a_0,\ldots,a_{-k})) is a fixed point of this operator, then (x(t)=a_0+\varphi(t)+y(t,a_0,\ldots,a_{-k})) is a (2\pi)-periodic solution of equation (1), when the parameters satisfy equation (2).
Theorem 3. Suppose the operator (f(t,x)) satisfies the Lipschitz condition
[
|f(t,x')-f(t,x'')|\le p(t)|x'-x''|,\qquad x',x''\in E. \tag{4}
]
Suppose (H(t)), for each fixed (t), is a linear bounded operator, continuous and (2\pi)-periodic in (t), and satisfying the condition
[
|H(t)(x'-x'')-f(t,x')+f(t,x'')|
\le q(t)|x'-x''|,\qquad x',x''\in E. \tag{5}
]
Here (p(t)) and (q(t)) are certain positive (2\pi)-periodic functions. In addition, suppose that for the operator (\int_0^{2\pi}H(t)\,dt) there exists a bounded inverse, and that the inequalities
[
\mu=\sup_t\frac{1}{2\pi}\int_{-\pi}^{\pi}|s|\,p(s+t+\pi)\,ds<1,
]
[
\frac{1}{1-\mu}
\left|\left(\int_0^{2\pi}H(t)\,dt\right)^{-1}\right|
\int_0^{2\pi}\bigl(q(t)+\mu|H^*(t)|\bigr)\,dt<1,
]
hold, where
[
H^*(t)=H(t)-\frac{1}{2\pi}\int_0^{2\pi}H(t)\,dt.
]
Then equation (1) has one and only one (2\pi)-periodic solution.
Proof. The operator (B_2) with one parameter (a_0), i.e., with (\varphi(t)\equiv0), is, by (4), a contraction operator. Further, taking into account the identity
[
\int_0^{2\pi}H(t)\,dt\, a_0
=
\int_0^{2\pi}(H(t)x(t)-f(t,x(t)))\,dt
-\int_0^{2\pi}H^*(t)y(t,a_0)\,dt
+
]
[
+\int_0^{2\pi}f(t,x(t))\,dt,
]
where (x(t)=a_0+y(t,a_0)), (y(t,a_0)) is a fixed point of (B_2), the operator is constructed
[
Fa_0=
\left(\int_0^{2\pi} H(t)\,dt\right)^{-1}
\left[
\int_0^{2\pi}\bigl(H(t)x(t)-f(t,x(t))\bigr)\,dt
-
\int_0^{2\pi} H^*(t)y(t,a_0)\,dt
\right].
]
By virtue of (5), the operator (F) has a fixed point (a_0^). Then the function
(x^(t)=a_0^+y(t,a_0^)) is a periodic solution of equation (1). The theorem is proved.
By a somewhat different method one can obtain a proposition for equation (3).
Theorem 4. Let the operator (\displaystyle \int_0^{2\pi} A(s)\,ds) have a bounded inverse with norm (\eta), and
[
M=\sup_t \frac{1}{2\pi}\int_{-\pi}^{\pi}|sA(s+t+\pi)|\,ds,
\qquad
N=\int_0^{2\pi}|A(s)|\,ds.
]
Then, if (M(1+\eta N)<1), equation (3) has a (2\pi)-periodic solution.
Proof is carried out by the method of successive approximations, putting
[
y_{n+1}(t)=\frac{1}{2\pi}\int_t^{t+2\pi}(s-t-\pi)[A(s)(a_n+y_n(s))+f(s)]\,ds,
]
[
\int_0^{2\pi} A(s)(a_n+y_n(s))\,ds+\int_0^{2\pi} f(s)\,ds=0.
]
The following proposition is, in a certain sense, a generalization of Theorem 3.
Theorem 5. Let (E) be a Banach space such that for every function (x(t)\in X) the equality
[
\frac{1}{2\pi}\int_{-\pi}^{\pi}|x(t)|^2\,dt
=
\sum_{n=-\infty}^{\infty}|a_n|^2,
\qquad
a_n=\frac{1}{2\pi}\int_{-\pi}^{\pi} x(t)e^{-int}\,dt.
\tag{6}
]
holds. Let the operator (f(t,x)) satisfy condition (4), where (p(t)=p) is constant. Let there exist a constant linear operator (H) satisfying condition (5), and suppose that the eigenvalues of the operator (H) are distinct from numbers of the form (ni), (n=0,\pm1,\ldots,\pm k), where (k) is some natural number. Then, if
[
\mu_k=2p^2\sum_{n=k+1}^{\infty}\frac{1}{n^2}<1,
\tag{7}
]
[
\frac{1}{2\pi(1-\mu_k)}
\int_0^{2\pi} q^2(t)\,dt
\sum_{|n|\le k}|(niI-H)^{-1}|^2<1,
\tag{8}
]
then equation (1) has one and only one (2\pi)-periodic solution.
Proof. In the metric space of sequences of the form
({a_n}), (n=\pm(k+1),\pm(k+2),\ldots), (a_n\in E),
(\sum|a_n|<+\infty), an operator (\Phi) is constructed, defined by the right-hand sides of the equalities
[
a_n=\frac{1}{2\pi ni}\int_{-\pi}^{\pi} f(t,a_0+\varphi(t)+y(t))e^{-int}\,dt,
\qquad |n|>k,
]
where
[
y(t)=\sum_{|n|>k} a_n e^{int}.
]
The operator (\Phi), by virtue of (7), has a fixed point ({a_n^*}), the coordinates of which depend on the parameters. Next, in the parameter space an operator (\Psi) is introduced, defined by the right-hand sides of the equalities
[
a_n=\frac{1}{2\pi}(-niI+H)^{-1}\int_{-\pi}^{\pi}(Hx(t)-f(t,x(t)))e^{-int}\,dt,\qquad |n|\leq k,
]
where
[
x(t)=a_0+\varphi(t)+y^(t),\qquad
y^(t)=\sum_{|n|>k}a_n^*e^{int}.
]
By virtue of (8), the operator (\Psi) has a fixed point with coordinates (a_n^), (|n|\leq k). Then, in turn, the function (y^(t)), for (a_n=a_n^*), (|n|\leq k), is a fixed point of the operator (B_2). The theorem is proved.
We note that for the linear equation (2), when (A) is a constant matrix, the conditions of Theorem 5 are at the same time necessary. Indeed, in this case it is necessary that the eigenvalues of the matrix (A) be distinct from numbers of the form (ni), (n=0,\pm1,\pm2,\ldots). Then the role of the operator (H) is played by the matrix (A). In this case condition (8) holds for any (k), and condition (7) for sufficiently large (k). We also note that Theorem 5 can be formulated using a variable operator (H(t)).
The preceding method can be generalized by considering equations of higher order. For example, for the equation
[
d^2x/dt^2=f(t,x),
\tag{9}
]
where (f(t,x)) is the same operator as in equation (1), the following holds.
Theorem 6. Let the operator (f(t,x)) satisfy condition (4). Let there exist an operator (H(t)), satisfying condition (5), such that the eigenvalues of the operator
[
\int_{0}^{2\pi}H(t)\,dt
]
are distinct from numbers of the form (-2\pi n^2), (n=0,1,\ldots,k), where (k) is some natural number or zero. Then, if
[
\mu_k=\frac{1}{\pi}\int_{0}^{2\pi}p(t)\,dt\sum_{n=k+1}^{\infty}\frac{1}{n^2}<1,
]
[
\frac{1}{1-\mu_k}\sum_{|n|\leq k}
\left|\left(2\pi n^2 I+\int_{0}^{2\pi}H(t)\,dt\right)^{-1}\right|
\int_{0}^{2\pi}\bigl(q(t)+|H^*(t)|\bigr)\,dt<1,
]
then equation (9) has one and only one (2\pi)-periodic solution.
Proof is carried out using the operator
[
B_3y(t)=-\varphi(t)+\frac{1}{4\pi^2}\int_{t}^{t+2\pi}(s-t-\pi)
\int_{s}^{s+2\pi}(\tau-s-\pi)f(\tau,a_0+\varphi(\tau)+y(\tau))\,d\tau .
]
We note that if, for any (x(t)\in X), equality (7) holds, then under the conditions of Theorem 6 the estimates can, in a certain sense, be improved.
Remark. The method presented can without particular difficulty be extended to equations of the form ((3))
[
dx/dt=\omega(t,\tilde{x}),
]
where (\tilde{x}=x(t)), (\omega(t,\tilde{x})) is an operator with values in (E), defined for each (t) on the set (X) and periodic in (t) with period (2\pi), i.e., the identity (\omega(t+2\pi,\tilde{x})=\omega(t,\tilde{x})) holds for all (t) and any fixed (\tilde{x}\in X).
Vladimir Pedagogical Institute
Received
21 V 1968
CITED LITERATURE
- M. A. Krasnosel’skii, UMN, 21, no. 3 (1966).
- L. Cesari, Contrib. diff. eq., 1, N. Y., 1963.
- M. A. Krasnosel’skii, DAN, 152, No. 4 (1963).