UDC 517.925+517.948.33

MATHEMATICS

Yu. G. BORISOVICH, V. F. SUBBOTIN

THE SHIFT OPERATOR ALONG TRAJECTORIES OF EVOLUTION EQUATIONS AND PERIODIC SOLUTIONS

(Presented by Academician P. S. Aleksandrov on 5 IX 1966)

1. The article considers linear and nonlinear systems of differential equations of the form

\[ x'(t)=f(t,x_t), \tag{1} \]

where \(x_t=x(t+s)\), \(-\omega\le s\le 0\). Here \(f(t,x_0)=(f_1,f_2,\ldots,f_n)\), and each component \(f_i(t,x_0(s))\) is a functional on elements \(x_0(s)=(x_{01}(s),\ldots,x_{0n}(s))\) of the space \(C^n[-\omega,0]\) of continuous vector-functions. System (1) includes, for example, systems of integro-differential equations with retarded argument of the form

\[ x'(t)=F(t,x(t),x(t-h))+\int_{-H}^{0}G(t,s,x(t),x(t+s))\,ds,\quad h,H\le \omega, \tag{1'} \]

and in essence is a convenient notation for such systems. We assume these functionals to be defined on some sphere \(T_a=(\|x\|<a)\), continuous in \(x_0\) uniformly with respect to \((t,x_0)\), and bounded. We also assume measurability of the superposition \(f(t,x_t)\), if \(x(t)\) is measurable (superpositional measurability).

We shall consider the problem of periodic solutions of system (1), assuming the right-hand side to be \(\omega\)-periodic. If the Cauchy problem \(x_{t=0}=x_0(s)\) is uniquely and nonlocally solvable for any initial function from some closed domain \(\overline{\Omega}\subset T_a\), then the shift operator is defined by

\[ \Pi_t x_0=x_t,\quad t\ge 0,\quad x_0\in\overline{\Omega}, \tag{2} \]

where \(x(t)\) is the solution of the Cauchy problem generated by the initial function \(x_0(s)\). It is well known that the fixed points of the operator \(\Pi_\omega\), and only they, generate \(\omega\)-periodic solutions of system (1) that start in the domain \(\overline{\Omega}\) (see, for example, \((^{1,2})\)). In \((^{2,3})\), topological methods were applied to the study of the operator \(\Pi_\omega\).

In the present article we shall apply methods of cone theory \((^{4,5})\).

2. Let us recall some notions. Denote by \(K\) the cone of nonnegative functions in the space \(C[-\omega,0]\). By a wedge \(W_m\) in the space \(C^n[-\omega,0]\) we shall mean the set of those vector-functions \(x_0(s)\) which satisfy the condition \(x_{0k}(s)\in K\) for all \(k=1,\ldots,m\) \((m\le n)\). In particular, \(W_n\) is the cone \(K_n\) of nonnegative vector-functions. We shall call the elements of the wedge semipositive, and the elements of the cone positive.

Let the domain \(\overline{\Omega}\) belong to \(W_m\). We shall call the operator \(\Pi_\omega\) semipositive on \(\overline{\Omega}\) if \(\Pi_\omega\overline{\Omega}\subset W_m\). In particular, when \(W_n=K_n\), the semipositivity of the operator \(\Pi_\omega\) means its positivity on \(\overline{\Omega}\).

Let, for every \(t\in[0,\omega]\) and every \(k=1,\ldots,m\),

\[ f_k(t,x_0(s))\ge 0,\quad \text{if } x_{0k}(0)=0,\ x_0(s)\in W_m. \tag{3} \]

Theorem 1. If conditions No. 1 and condition (3) are satisfied, then the operator \(\Pi_\omega\) is semipositive on \(\overline{\Omega}\) and completely continuous.

Below, the conditions of Theorem 1 are always assumed to be fulfilled.

A completely continuous semipositive operator \(\Pi_\omega\) generates on the boundary \(\dot{\Omega}\) of the open domain \(\Omega\) (in the relative topology \(W_m\)) the vector field
\[ \Phi x_0=\Pi_\omega x_0-x_0, \]
directed into the wedge \(W_m\), and if this field has no singular points, then its rotation \(\gamma\{\Phi,\dot{\Omega},W_m\}\) relative to the wedge \(W_m\) is defined \({}^{(5)}\); the rotation of the vector field is a topological invariant equal to the algebraic number of fixed points of the operator \(\Pi_\omega\) lying in \(\Omega\).

The problem of computing the rotation \(\gamma\) is difficult. The general principle of Leray for a homotopic transformation of a vector field into a simpler one, in our situation, can be formulated as follows.

Suppose that in equation (1) one can introduce a numerical parameter \(\lambda\), \(0\le \lambda\le 1\), so that the following conditions are satisfied:

1) \(f(t,x_0,\lambda)\) satisfies condition (3) for all \(\lambda\);

2) for all \(\lambda\) the system
\[ x'(t)=f(t,x_t,\lambda) \tag{4} \]
satisfies conditions No. 1 and the operator \(\Pi_\omega(x_0,\lambda)\) is completely continuous jointly in \((x_0,\lambda)\);

3) \(f(t,x_0,1)=f(t,x_0)\);

4) \(f(t,x_0,0)=f_0(t,x_0(0))\), where \(f_0(t,c)\) is a function of finitely many variables \((t,c)\) \((c\in E^n\), finite-dimensional Euclidean space).

Under these conditions we shall say that system (1) is homotopic to the system of ordinary differential equations
\[ x'(t)=f_0(t,x(t)). \tag{5} \]

Denote by \(\Pi_\omega^0 x_0\) the shift operator for system (5) in the finite-dimensional space \(E^n\), and denote the corresponding vector field by \(\Phi^0 x_0\). Define in \(E^n\) the wedge \(W_m^n\) as the set of vectors \(x_0=(x_{01},\ldots,x_{0n})\) whose first \(m\) coordinates are nonnegative, and let \(\Omega^n\) be some open subset of the wedge \(W_m^n\).

It turns out that the rotations of the vector fields \(\Phi\) and \(\Phi^0\) on the boundaries \(\dot{\Omega}\) and \(\dot{\Omega}^n\) are related to one another for an appropriate choice of the domains \(\Omega,\Omega^n\). Connections of this kind were first noted in the works \({}^{(2,6)}\) (in \({}^{(6)}\), in a more general form for domains in \(C^n[-\omega,0]\)). We shall generalize the result of \({}^{(2)}\) for the operator \(\Pi_\omega\) acting in the wedge \(W_m\).

Theorem 2. Let the set \(\mathfrak{M}\) of semipositive \(\omega\)-periodic solutions of system (4) be bounded for all \(\lambda\), and let \(\Omega^n\) and \(\Omega\) be relative spheres in the wedges \(W_m^n\), \(W_m\), respectively, with center at the point \(\theta\) and containing the set \(\mathfrak{M}\). Then
\[ \gamma\{\Phi,\dot{\Omega},W_m\}=\gamma\{\Phi^0,\dot{\Omega}^n,W_m^n\}. \]

In the case when equation (1) is linearized at zero or at infinity, the rotation of the field \(\Phi\) on relative spheres \(\dot{\Omega}_0\) or \(\dot{\Omega}_\infty\) of sufficiently small, respectively large, radius is computed from the corresponding linear systems
\[ x'=L_0(t,x_t),\qquad x'=L_\infty(t,x_t), \]
provided that they have no semipositive \(\omega\)-periodic solutions other than zero. In this case the formula is valid
\[ \gamma\{\Phi,\dot{\Omega}_0,W_m\}=\gamma_0(\theta),\qquad \gamma\{\Phi,\dot{\Omega}_\infty,W_m\}=\gamma_\infty(\theta), \]
where \(\gamma_0(\theta)\), \(\gamma_\infty(\theta)\) are the indices of the point \(\theta\) of the shift operators of the linear systems.

1. The theorem 3 stated below is based on the computation of the indices \(\gamma_0(\theta)\), \(\gamma_\infty(\theta)\). We represent the elements of the wedge \(x_0(s)\) in the form \(x_0=(y_0,z_0)\), where \(y_0\) are the nonnegative components of the vector \(x_0\), forming the cone \(K_m\),

and \(z_0\) the remaining ones. Then for the functionals \(L_0, L_\infty\), by condition (3), we have the decomposition
\[ L_k(t,x_0)=\bigl(L_{k1}(t,y_0),\,L_{k2}(t,y_0,z_0)\bigr)\quad (k=0,\infty). \]
Consider the linear systems
\[ y'(t)=L_{k1}(t,y_t), \tag{6} \]
\[ z'(t)=L_{k2}(t,0,z_t)\quad (k=0,\infty), \tag{7} \]
and denote the fixed-point indices of the corresponding translation operators by \(\gamma_{ki}\), \(i=1,2\) (\(\gamma_{k1}\) relative to the cone \(K_m\), and \(\gamma_{k2}\) relative to the space \(C^{n-m}[-\omega,0]\)).

Theorem 3. Suppose that: 1) the functional \(f(t,x_0)\) has the asymptotic Fréchet derivative; 2) for \(k=\infty\) system (7) has no nonzero \(\omega\)-periodic solutions, and system (6) has no nonzero positive \(\omega\)-periodic solutions. Finally, let \(\gamma_{\infty1}\cdot\gamma_{\infty2}\ne 0\). Then system (1) has a semipositive \(\omega\)-periodic solution.

Suppose, moreover, that the functional \(f(t,x_0)\) has a Fréchet derivative at zero, \(f(t,\theta)\equiv 0\), and that condition 2) is satisfied for systems (7), (6) (\(k=0\)). Let \(\gamma_{01}\cdot\gamma_{02}\ne\gamma_{\infty1}\cdot\gamma_{\infty2}\). Then system (1) has, in addition to the zero solution, a nonzero \(\omega\)-periodic semipositive solution.

Below, for simplicity, we shall omit the index \(k\) in the notation.

We note that the index \(\gamma_2(\theta)\) can be computed by the formula \(\gamma_2(\theta)=(-1)^\beta\), where \(\beta\) is the sum of the orders of the eigenvalues of the translation operator for system (7) that are greater than one \((^7)\). Similarly, the index \(\gamma_1(\theta)\) can be related to the positive spectrum of the translation operator \(A_\omega\) of system (6): if \(A_\omega\) is a strongly positive operator, then it has a unique simple eigenvalue \(\lambda_1^+\) \((^4)\), and \(\gamma_1(\theta)=1\) if \(\lambda_1^+<1\), and \(\gamma_1(\theta)=0\) if \(\lambda_1^+>1\) \((^5)\).

Let us formulate a sufficient condition for the strong positivity of the operator \(A_\omega\). Let system (6) have the form
\[ y'(t)=S(t)y(t)+R(t,y_t), \tag{6'} \]
where \(S(t)\) is an \(\omega\)-periodic matrix, and \(R(t,y_0)\) is a nonnegative functional on \(K_m\). Under these conditions the following is true.

Theorem 4. If the translation operator \(D_\omega\) of the system of ordinary differential equations \(y'=S(t)y\) is strongly positive on the cone of nonnegative vectors \(y\geq\theta\), and the vector-function \(R(t)=R(t,y_t)\) is not identically zero on the interval \([0,\omega]\) for any solution \(y(t)\) of system (6) with a nonzero initial condition \(y_0(s)\geq 0,\ y_0(0)=0\), then the operator \(A_\omega\) is strongly positive.

We note that an analogous assertion also holds for nonlinear systems.

Sufficient conditions for the strong positivity of the operator \(D_\omega\) are found in work \((^8)\).

1. Let us compute the indices \(\gamma_i\) in terms of the right-hand side of systems (6), (7). Let systems (6), (7) be stationary, i.e.,
   \[ y'(t)=L_1(y_t),\qquad z'(t)=L_2(0,z_t). \tag{8} \]
   Then, to compute the indices \(\gamma_1(\theta), \gamma_2(\theta)\) of systems (8), one may use the spectrum of the infinitesimal generators \(A_t, B_t\), \(t\geq 0\) \((^{9-11})\), generated by equations (8).

Consider the semigroup \(A_t\). Its infinitesimal generator \(A y_0(s)=dy_0/ds\) has as its domain continuously differentiable functions satisfying the boundary condition \(y_0'=L_1(y_0)\). The complex spectrum of the operator \(A\) is determined by the characteristic equation (for example, see \((^{11})\))
\[ \left|\widetilde L_1(e^{\mu s})-\mu I\right|=0. \tag{9} \]

Here \(\widetilde L_1(e^{\mu s})\) is a matrix of order \(m\), whose \(j\)-th column has the form \(L_1(y_0^j(s))\), where \(y_0^j(s)\) is a vector-function with zero components except for the \(j\)-th, equal to \(e^{\mu s}\).

Theorem 5. Let \(A_\omega\) be a strongly positive operator. Then there exists a unique real root \(\mu_0\) of equation (9) such that the system of algebraic equations \((\widetilde L_1(e^{\mu_0 s})-\mu_0 I)c=0\) has a unique normalized strictly positive solution \(c_+\). Moreover, \(\gamma_1(\theta)=1\) if \(\mu_0<0\), and \(\gamma_1(\theta)=0\) if \(\mu_0>0\). (By a strictly positive solution we mean a solution all of whose components are positive.)

Let us note that \(\mu_0>\operatorname{Re}\mu\), where \(\mu\) is any root of equation (9).

As an example, consider the second-order system

\[ \binom{x(t)}{y(t)}' = A\binom{x(t)}{y(t)} + B\binom{x(t-h)}{y(t-h)} + \int_{-h}^{0}\beta(s)\binom{x(t+s)}{y(t+s)}\,ds . \]

Here \(A=(a_{ij})\), \(B=(b_{ij})\) are constant matrices, and the matrix \(\beta(s)=(\beta_{ij}(s))\) is summable and nonnegative for all \(s\). Let \(a_{12}, a_{21}>0\), \(b_{ij}\ge 0\), and suppose that at least one of the matrices \(B\), \(C=\int_{-h}^{0}\beta(s)\,ds\) has at least one strictly positive row.

Denote

\[ \Delta=(a_{11}+b_{11}+c_{11})(a_{22}+b_{22}+c_{22})-(a_{12}+b_{12}+c_{12})(a_{21}+b_{21}+c_{21}), \]

and denote by \(\gamma(\theta)\) the index of the point \(\theta\) of the shift operator of our system, computed with respect to the cone of nonnegative functions \((x,y)\).

Theorem 6. Let the conditions listed above be satisfied, and let

\[ a_{11}<0,\qquad a_{22}<0,\qquad |a_{11}|>b_{11}+c_{11},\qquad |a_{22}|>b_{22}+c_{22}. \tag{10} \]

Then: 1) \(\gamma(\theta)=1\), if \(\Delta>0\); 2) \(\gamma(\theta)=0\), if at least one of the conditions (10) is not fulfilled, or \(\Delta<0\); 3) \(\gamma(\theta)\) is undefined if \(\Delta=0\).

To compute the index \(\gamma_2(\theta)\), note that the root subspaces of the infinitesimal operator \(B\) of the semigroup \(B_t\) are normally split off in the sense of (12), and the operator \(B-\mu I\) is normally solvable and has finite defect numbers. Let the eigenvalues \(\mu\) of the operator \(B\) be distinct from \(2\pi k\omega^{-1}i\), \(k=0,\pm1,\ldots\). Using the relation between the spectra and root subspaces of the operators \(B\) and \(\bar B_\omega\), we obtain \(\gamma_2(\theta)=(-1)^\beta\), where \(\beta\) is the sum of the orders of the positive eigenvalues \(\mu\) of the infinitesimal operator \(B\).

In conclusion, the authors take this opportunity to express their gratitude to M. A. Krasnosel’skii and P. P. Zabreiko for a number of suggestions.

Voronezh State University

Received
23 VIII 1966

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