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MATHEMATICS
Yu. G. BORISOVICH
ON THE POINCARÉ–ANDRONOV METHOD IN THE PROBLEM OF PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH DELAY
(Presented by Academician A. Yu. Ishlinskii on 6 IV 1953)
The paper considers a system of differential equations (in vector notation)
\[ x'(t)=f(t,T_t x), \tag{1} \]
in which the derivative \(x'(t)\) depends on the “prehistory” \(x(\tau)\), \(t-h\leq \tau \leq t\), and the right-hand side is \(\omega\)-periodic.
The method of the Poincaré–Andronov point transformation is extended to equations (1) for arbitrary \(h\) and \(\omega\). A method is justified for continuing a periodic solution with respect to the delay parameter \(h\) from an ordinary system of differential equations.
- Let us describe equation (1) in greater detail. Let \(0<h<\infty\), and let \(M[-h,0]\) be the set of piecewise continuous (i.e., on a finite interval having no more than a finite number of points of discontinuity, at which finite left and right limits exist) vector functions \(x(s)=(x_1(s),\ldots,x_n(s))\), \(-h\leq s\leq 0\); the set \(M[-h,\omega]\) is defined similarly. Define the operator \(T_t x=x(t+s)\), \(-h\leq s\leq 0\), acting from any \(M[-h,t]\), \(t\geq 0\), into \(M[-h,0]\). Let
\[ f(t,x)=(f_1(t,x),\ldots,f_n(t,x)), \]
where \(f_i(t,x)\), for almost every \(t\), \(-\infty<t<\infty\), is a functional on \(M[-h,0]\), and for every function \(x\in M[-h,\omega]\) the function \(f(t)=f(t,T_t x)\) is absolutely summable on \([0,\omega]\). Assume also that \(f(t,x)\) is \(\omega\)-periodic in \(t\), i.e., for almost all \(t\) and all \(x\), \(f(t+\omega,x)\equiv f(t,x)\).
We pose the Cauchy problem for equation (1):
\[ T_t x\big|_{t=0}=x^0,\quad \text{where } x^0\in M[-h,0]. \tag{2} \]
By a solution of problem (1)—(2) we shall mean a function \(x\in M[-h,l]\), \(l>0\), absolutely continuous and satisfying equation (1) for \(t\geq 0\) and condition (2).
Equations of the form (1) were considered by N. N. Krasovskii (¹). As an example we give the equation
\[ x'(t)=Ax(t)+Bx(t-h(t))+\int_{t-h(t)}^{t}\Phi(t,s;x(t),x(s))\,ds, \tag{3} \]
where \(h(t)\) is an \(\omega\)-periodic continuous function.
- The point-transformation operator. Equation (1) with delay parameter \(h\) may be regarded as an equation with parameter \(\bar h\geq h\). In general, we shall call two equations of type (1) equivalent if their solutions for \(t\geq 0\) coincide when the initial conditions coincide on the common part of their domains of definition. Therefore
one may assume that \(h=k\omega\) (\(k\)—an integer or \(\infty\)); we shall call such a parameter normalized.
Denote by \(C(-\omega,0]\) the space of uniformly continuous functions \(x^0(s)\) on the interval \(-\omega<s\leq 0\), with the usual norm
\(\|x^0\|=\sup |x^0(s)|\). By \(C^k(-\omega,0]\) we denote the set of initial functions for problem (1)—(2) of the form
\[ x(\tau)=x^0(\tau+i\omega)\quad \text{for }-(i+1)\omega<\tau\leq -i\omega,\qquad (i=0,1,\ldots,k-1). \]
Suppose that the solution \(x(t,\tilde x)\) with initial condition \(\tilde x\) is unique and is defined up to \(t=\omega\). Then in the space \(C(-\omega,0]\) the operator
\[ \Pi_\omega x^0=x(\omega+s,\tilde x),\qquad -\omega<s\leq 0, \tag{4} \]
is defined; its significance for periodic solutions is asserted by the following theorem:
Theorem 1. In order that the solution \(x(t)\) with initial condition \(\tilde x\) be \(\omega\)-periodic, it is necessary and sufficient that the function \(x^0(s)\) be a fixed point of the operator \(\Pi_\omega\): \(\Pi_\omega x^0=x^0\).
Let us show that the operator (4) is completely continuous. Indeed, let the conditions
\[ \left|\int_0^t f(\tau,T_\tau x)\,d\tau\right|\leq M(r),\qquad 0\leq t\leq \omega; \tag{5} \]
\[ \int_0^t f(\tau,T_\tau x)\,d\tau \quad \text{are equicontinuous},\qquad 0\leq t\leq \omega, \tag{6} \]
be fulfilled for any bounded set of functions \(\|x\|\leq r,\ x\in M[-k\omega,\omega]\),
\[ \int_0^t f(\tau,T_\tau x^m)\,d\tau \to \int_0^t f(\tau,T_\tau x)\,d\tau,\qquad 0\leq t\leq \omega, \tag{7} \]
if \(\|x^m-x\|\to 0,\ x^m,x\in M[-k\omega,\omega]\).
Theorem 2. Suppose that problem (1)—(2) has a unique solution on the interval \([-k\omega,\omega]\) for any choice of the initial condition (2), and that conditions (5)—(7) are fulfilled; suppose that \(\Pi_\omega\) is a bounded operator (i.e., transforms every bounded set into a bounded set).
Then the operator \(\Pi_\omega\) acts completely continuously in the space \(C(-\omega,0]\).
Let us note that in the case \(h\leq \omega\) the complete continuity of the operator \(\Pi_\omega\) was established by A. Halanay [2], who studied it in connection with periodic solutions. Below, for the study of fixed points of the operator \(\Pi_\omega\), we shall apply the rotation of the vector field [3].
3. Continuation of periodic solutions with respect to the lag parameter \(h\). Suppose that \(h=h(\lambda)\) depends on the numerical parameter \(\lambda\), \(h(1)=k\omega\), and for all \(\lambda\), \(0\leq \lambda\leq 1\), \(h(\lambda)\leq k\omega\).
The functional \(f(\lambda,t,x)\) in this case depends on \(\lambda\); its lag parameter may not be normalized for \(\lambda\ne 1\). Introduce another lag parameter \(\bar h=k\omega\). If the functional \(f(\lambda)\) is regarded as a functional \(\bar f(\lambda)\) with lag parameter (already normalized for all \(\lambda\)) \(\bar h\), then equations (1) and
\[ x'=\bar f(\lambda,t,T_t x) \tag{1'} \]
are equivalent, and it suffices to study periodic solutions of equation (1′).
Thus, the problem of continuation of a periodic solution with respect to the parameter \(h\) is equivalently reduced to the study of the dependence of a periodic solution of equation \((1')\) on the parameter \(\lambda\).
Under certain conditions \(((5)—(7)\), etc.) with respect to equation \((1')\), the operator (4), depending on \(\lambda\), \(\Pi_\omega(\lambda)\), is quite continuous in the aggregate \((\lambda, x^0)\). Therefore, for the study of the fixed points of \(\Pi_\omega(\lambda)\) one may apply the following general scheme \((^3,^4)\): for all \(\lambda\), \(0 \leqslant \lambda \leqslant 1\), an a priori estimate is known for the fixed points of the operator \(\Pi_\omega(\lambda)\); for \(\lambda=0\) it is possible to compute the rotation of the field \(x^0-\Pi_\omega(0)x^0\) on a sufficiently large sphere in \(C(-\omega,0]\); let it be different from zero; then for all \(0 \leqslant \lambda \leqslant 1\) there exists a fixed point.
For \(\lambda=0\) the rotation of the field may turn out to be simply computable. For concrete equations this will be the case, for example, when for \(\lambda=0\) the function \(h(t)\) is equal to zero or has the property \(h(0)=0\). Then the rotation of the field \(x^0-\Pi_\omega(0)x^0\) is computed through its finite-dimensional component. For example, replacing in (3) the function \(h(t)\) by the function \(\lambda h(t)\), \(0 \leqslant \lambda \leqslant 1\), we obtain, for \(\lambda=0\), an equation equivalent to the ordinary equation \(x'=(A+B)x\). We note that concrete equations with delay parameter \(h(t)\), under the condition \(t-h(t)\geqslant 0\), have been studied by a number of authors (see the survey \((^5)\)).
- We shall illustrate the method described above by one theorem on the existence of a second periodic solution. Consider equation (1), depending on the parameters \(\mu,\lambda\):
\[ x' = f(\mu,\lambda,t,T_t x), \qquad -P \leqslant \mu,\lambda \leqslant P, \tag{8} \]
with delay parameter \(h=k\omega\). Suppose that conditions (5), (6) are satisfied uniformly for all \((\mu,\lambda)\), and the condition
\[ \int_0^t f(\mu^m,\lambda^m,\tau,T_\tau x^m)\,d\tau \to \int_0^t f(\mu,\lambda,\tau,T_\tau x)\,d\tau, \qquad 0 \leqslant t \leqslant \omega, \tag{7'} \]
when \(\mu^m \to \mu,\ \lambda^m \to \lambda,\ \|x^m-x\|\to 0,\ x^m,x\in M[-k\omega,\omega]\). Let the inequality
\[ |f(\mu,\lambda,t,x)-f(\mu,\lambda,t,y)| \leqslant M(t)\|x-y\|,\qquad 0 \leqslant t \leqslant \omega, \tag{9} \]
be satisfied, where \(x,y\in M[-k\omega,0]\) and the norm is taken in this space, \(M(t)\geqslant 0\) is summable on \([0,\omega]\). Suppose that for \(\lambda=0\) the equalities
\[ f(\mu;0,t,x)=A_i(\mu,t,x)+\varphi_i(\mu,t,x) \qquad (i=1,2), \qquad -p \leqslant \mu \leqslant p, \tag{10} \]
hold, where \(p\leqslant P\), and \(A_i\) are \(\omega\)-periodic in \(t\), linear in \(x\) functionals satisfying the inequalities
\[ |A_i(\mu,t,x)|\leqslant M_i(t)\|x\| \qquad (i=1,2). \tag{11} \]
Here \(M_i\geqslant 0\) and are summable on \([0,\omega]\); let \(A_1\) and \(A_2\) satisfy conditions (5), (6) uniformly in \(\mu\), and a condition of type \((7')\) with \(\lambda=0\). Let the equations
\[ x'=A_i(\mu,t,T_t x) \qquad (i=1,2) \tag{12} \]
for all \(\mu,\ -p\leqslant \mu \leqslant p\), have only the zero \(\omega\)-periodic solution and, for \(\mu=0\), be equivalent to the \(\omega\)-periodic linear ordinary differential equations
\[ x'=A_i^*(t)x \qquad (i=1,2). \tag{13} \]
With respect to the latter, assume that they are reducible, by means of a real \(\omega\)-periodic nonsingular transformation, to equations with constant matrices.
We impose conditions on the functionals \(\varphi_1\) and \(\varphi_2\). Let
\[ |\varphi_i(t,x)| \leq L(t) O_i(\|x\|), \qquad 0 \leq t \leq \omega \quad (i=1,2), \tag{14} \]
where \(L(t) \geq 0\) is summable on \([0,\omega]\), \(O_i(u)\) are monotonically increasing functions of \(u\), \(0 \leq u < \infty\), and \(\frac1u O_1(u) \to 0\) as \(u \to 0\), \(\frac1u O_2(u) \to 0\) as \(u \to \infty\). We note that for \(\varphi_1\) it is sufficient that (14) hold for small \(x\).
Theorem 3. Suppose the preceding conditions are satisfied. Suppose all characteristic exponents \(\beta\) of equations (13) are nonzero, and in the case \(\beta=\pm ir\) (purely imaginary) the condition \(r<2\pi\omega^{-1}\) is satisfied. Suppose the numbers of positive characteristic exponents (counting their multiplicities) for each of equations (13) have different parity.
Then there exist two numbers \(\delta_1>0\), \(\delta_2>0\) such that, for all \(\mu,\lambda\), \(|\mu|\leq p+\delta_1\), \(|\lambda|\leq\delta_2\), equation (8) has at least two \(\omega\)-periodic solutions (\(\delta_1=0\) in the case \(p=P\)).
As an example, consider the equation
\[ x'=\varphi[t,x(t),x(t-h(t))]+\lambda\int_{t-h(t)}^{t} K(t,s-t)\psi[t,x(t),x(s)]\,ds, \tag{15} \]
where \(\varphi(t,x,u)\), \(\psi(t,x,u)\) are continuous \(n\)-dimensional vector functions, \(-\infty<t<\infty\), \(x,u\) belong to \(n\)-dimensional space, \(\omega\)-periodic in \(t\) and satisfying a Lipschitz condition in \((x,u)\), with
\[ |\varphi(t,x,u)-A_1x-B_1u| \leq a(|x|+|u|)^{1-\varepsilon}+b,\qquad a,b>0, \]
\[ |\varphi(t,x,u)-A_2x-B_2u| \leq c(|x|+|u|)^{1+\gamma},\qquad |x|+|u|<\delta. \]
Here \(A_i,B_i\) are constant matrices, with \(A_i+B_i\) nonsingular, and their numbers of positive eigenvalues \(\beta\) have different parity; in the purely imaginary case \(\beta=\pm ir\) suppose \(r<2\pi\omega^{-1}\). The square matrix kernel \(K(t,s)\) is continuous in \((t,s)\) and \(\omega\)-periodic in \(t\); \(h(t)\) is continuous and \(\omega\)-periodic. If the equations
\[ x'=A_i x(t)+B_i x[t-\mu h(t)] \qquad (i=1,2) \]
for all \(\mu\), \(0\leq\mu\leq 1\), have only the zero \(\omega\)-periodic solution, then equation (15), for small \(\lambda\), has at least two \(\omega\)-periodic solutions.
In conclusion we mention a number of works \({}^{(6-8)}\) devoted to the method of the point transformation. The problem of the topological study of this transformation for arbitrary \(h,\omega\) was posed at M. A. Krasnosel’skii’s seminar. He reported that results close to ours were obtained by the method of integral equations.
Received
23 III 1963
REFERENCES
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