UDC 517.911

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF BOUNDED SOLUTIONS OF A SYSTEM OF DIFFERENTIAL EQUATIONS

N. V. MEDVEDEV

In the present paper we consider questions of the existence and location of bounded solutions of a system of differential equations. In the investigation we make extensive use of the methods employed in the paper [1].

1. Consider the linear differential equation

\[ \frac{dx}{dt}=Ax+f(t), \tag{1} \]

where \(f(t)\) is a strongly continuous bounded function on the interval \((-\infty,+\infty)\) with values in the Banach space \(E\); \(x(t)\) is the unknown function with values in \(E\); \(A\) is a constant linear bounded operator acting in the space \(E\).

Let us represent a bounded solution of this equation in a form convenient for us.

Lemma 1. If the operator \(e^{-A\omega}-I\) is invertible, where \(I\) is the identity operator, and \(f(t)\) is an \(\omega\)-periodic function, then equation (1) has an \(\omega\)-periodic solution:

\[ x(t)=\int_t^{t+\omega} e^{A(t-s)}(e^{-A\omega}-I)^{-1} f(s)\,ds . \tag{2} \]

Formula (2) is verified by direct differentiation. The following lemmas hold analogously.

Lemma 2. If the real parts of the points of the spectrum of the operator \(A\) are not less than some positive number \(\sigma\) \((\operatorname{Re}\lambda\geq\sigma)\), then equation (1) has a strongly continuous bounded solution

\[ x(t)=\int_t^{t+\omega} e^{A(t-s)}F(s)\,ds, \tag{3} \]

where \(\omega\) is a positive number such that \(\|e^{-A\omega}\|<1\); \(F(t)\) is a bounded solution of the difference equation

\[ e^{-A\omega}F(t+\omega)-F(t)=f(t), \tag{4} \]

i.e.

\[ F(t)=-f(t)-e^{-A\omega}f(t+\omega)-e^{-2A\omega}f(t+2\omega)-\cdots . \tag{5} \]

Lemma 3. If for the operator \(A\) the inequality \(\operatorname{Re}\lambda\leq\sigma\) is satisfied, where \(\sigma\) is some negative number, then equation (1) has a bounded solution (3), where now \(\omega\) is a positive number such that

\[ \left\| e^{A\omega}\right\|<1,\quad F(t) \text{ is a solution of equation (4), namely} \]
\[ F(t)=e^{A\omega}f(t-\omega)+e^{2A\omega}f(t-2\omega)+\cdots . \tag{6} \]

Let now \(E\) be an \(n\)-dimensional space, and let \(A=\{a_{ij}\}\) be a real matrix satisfying the conditions
\[ a_{ij}\geq 0\quad (i,j=1,2,\ldots,n;\ i\ne j). \tag{7} \]

As is known, a vector \(x\in E\) is called positive if all its components are positive (notation: \(x>0\)), and nonnegative if all its components are nonnegative (notation: \(x\geq 0\)). If \(x-y\geq 0\), then the relation \(x\geq y\) is defined. The set of vectors of the form \(x\geq 0\) forms a cone \(K\).

Lemma 4. If the real parts of all eigenvalues of the matrix \(A\) are negative and conditions (7) hold, then the matrix \(-A^{-1}\) has a positive eigenvector \(a\), corresponding to its positive eigenvalue \(\lambda\).

Indeed, consider the equation
\[ \frac{dx}{dt}=Ax+b, \tag{8} \]
where \(b\) is some constant vector such that \(b\geq 0,\ b\ne 0\). Equation (8) has the stationary solution \(x=-A^{-1}b\). Obviously, all other solutions tend to this solution as \(t\to +\infty\). Any point lying in the cone \(K\), under motion along the trajectories of system (8) in the direction of increasing time, remains in the cone \(K\) by virtue of conditions (7), and therefore it can approach the point \(-A^{-1}b\) only under the condition \(-A^{-1}b\geq 0\). Hence it follows that the matrix \(-A^{-1}\) is positive, i.e., all its elements are greater than zero. Now the existence of the eigenvector \(a\) and the eigenvalue \(\lambda\) of the matrix \(-A^{-1}\) follows from Perron’s theorem [2].

In proving Lemma 4 we have at the same time established that a bounded solution of equation (8) lies in the cone \(K\). It turns out that a more general assertion also holds.

Theorem 1. Suppose the conditions of Lemma 4 are fulfilled. Let \(f(t)\) be a continuous bounded positive vector for all \(t\in(-\infty,+\infty)\). Then the bounded solution of equation (1) lies in the cone \(K\). In particular, if \(\lambda\geq 1\), then there exists a positive vector \(b\) such that from the condition \(f(t)-b\geq 0\) it follows that \(\varphi(t)-b\geq 0\).

Proof. First consider a periodic solution of equation (1), assuming that \(f(t)\) is a periodic function with some period. In this case the differential equation
\[ \frac{dy}{dt}=-Ay-f(-t) \]
by virtue of Lemma 2 has a periodic solution
\[ y(t)=\int_t^{t+\omega} e^{A(s-t)}F(s)\,ds, \]
where \(\omega\) is a sufficiently large positive number and
\[ F(s)=f(-s)+e^{A\omega}f(-s+\omega)+e^{2A\omega}f(-s+2\omega)+\cdots . \]
Consequently,
\[ y(0)=\int_0^\omega e^{As}F(s)\,ds=\int_0^{+\infty} e^{As}f(-s)\,ds. \]

The matrix \(e^{As}\) for \(s \ge 0\) is positive [3], and therefore \(u(0)>0\). Then the first part of the lemma follows from the fact that \(u(0)\) is at the same time the initial point of the periodic solution \(\varphi(t)\) of equation (1), under motion along whose trajectories, in view of (7), no point can leave the cone \(K\). Further, without loss of generality, suppose

\[ \inf_{-\infty<t<+\infty}\varphi(t)=\varphi(0). \tag{9} \]

Then, for sufficiently small \(\mu>0\), we have

\[ \varphi(t)-\lambda\mu a \ge \int_{0}^{+\infty} e^{As}(f(-s)-\mu a)\,ds >0, \]

where \(\lambda\) is a positive eigenvalue of the matrix \(-A^{-1}\), and \(a\) is its positive eigenvector. Consequently, \(b=\mu a\). Now it is clear that, in order to prove the theorem in the general case, it suffices to replace condition (9) by the condition

\[ \inf_{-\infty<t<+\infty}\varphi(t)=\varphi(0)-\varepsilon, \]

where \(\varepsilon\) is a sufficiently small positive vector, and pass to the limit in the relation \(\varphi(t)-\lambda\mu a+\varepsilon \ge 0\) as \(\varepsilon\to0\). The theorem is proved.

Theorem 2. Let the real parts of all eigenvalues of the matrix \(A\) be greater than zero, and let conditions (7) be satisfied. Let \(f(t)\) be a continuous bounded vector and \(f(t)\gg0,\ f(t)\ne0\) for all \(t\in(-\infty,+\infty)\). Then a bounded solution of equation (1) is located outside the cone \(K\).

Suppose that a bounded solution \(\varphi(t)\) falls into the cone \(K\). Then at some moment \(t=t_0\) the point \(\varphi(t_0)\) will be a point of the cone \(K\). Consider the solution of equation (1)

\[ x(t)=e^{A(t-t_0)}\lambda+\int_{t}^{t+\infty} e^{A(t-s)}F(s)\,ds, \tag{10} \]

where the second term is the bounded solution \(\varphi(t)\). Here the matrix \(e^{A(t-t_0)}\) is positive for \(t\ge t_0\). Therefore, for any vector \(a>0\) and any \(-\lambda\ge0,\ \lambda\ne0\), one can indicate a moment \(t_1>t_0\) such that the vector \(e^{A(t-t_0)}-a\) is outside the cone \(K\). Consequently, the solution (10), for sufficiently large \(t\), will be outside the cone \(K\). Further, since \(f(t)\ne0\), one may suppose that \(\varphi(t_0)\ne0\). Then one can indicate such a vector \(-\lambda\ge0,\ \lambda\ne0\), that \(\varphi(t_0)+\lambda\in K\). Since \(x(t_0)=\varphi(t_0)+\lambda\), it follows that \(x(t)\), in view of conditions (7), is in the cone \(K\) for every \(t\ge t_0\). The contradiction obtained completes the proof of the theorem.

1. Let us now consider, in a finite-dimensional space \(E\), the nonlinear differential equation

\[ \frac{dx}{dt}=Ax+f(t,x), \tag{11} \]

where \(A\) is a real constant matrix; \(f(t,x)\) is a vector-function, continuous and bounded in the domain \(\{-\infty<t<+\infty,\ \|x\|\le M\}\), \(\|f(t,x)\|\le N\) (here \(M\) and \(N\) are some positive numbers). In addition, we additionally suppose that at all points of the domain \(\{-T\le t\le T,\ \|x\|\le M\}\), where \(T\) is an arbitrary positive number, the theorem on continuous dependence of solutions on initial conditions holds.

Bounded solutions of this equation, by virtue of (3), are solutions of the nonlinear integral equation

\[ x(t)=\int_t^{t+\omega} e^{A(t-s)}F(s,x(s))\,ds, \tag{12} \]

where \(F(s,x(s))\) is a solution of the difference equation

\[ e^{-A\omega}F(s+\omega,x(s+\omega))-F(s,x(s))=f(s,x(s)). \]

Let us note some cases of the existence of a bounded solution of equation (12), and consequently of equation (11).

Theorem 3. Let the real parts of all eigenvalues of the matrix \(A\) be negative. If \(\mu<M\) for some \(\omega>0\) such that \(\|e^{A\omega}\|<1\), where

\[ \mu=\frac{N\|e^{A\omega}\|}{1-\|e^{A\omega}\|} \sup_{-\infty<t<+\infty}\int_t^{t+\omega}\|e^{A(t-s)}\|\,ds, \]

then there exists at least one bounded solution on \((-\infty,+\infty)\) of equation (11).

Proof. Since

\[ F(s,x(s))=e^{A\omega}f(s-\omega,x(s-\omega)) +e^{2A\omega}f(s-2\omega,x(s-2\omega))+\cdots, \]

then, for \(\|x(s)\|\le M\), the inequality

\[ \|F(s,x(s))\|\le \frac{N\|e^{A\omega}\|}{1-\|e^{A\omega}\|} \]

holds.

Consequently, the operator \(B\), defined by the right-hand side of equality (12), for \(\mu<M\) maps continuous functions \(x(t)\), \(\|x(t)\|\le M\), into continuous functions \(y(t)=Bx(t)\), \(\|y(t)\|\le M\).

Consider a solution of equation (11) on some interval \([-T,T]\). Suppose that \(f(t,x(t))=f(-T,x(-T))\) for \(t<-T\), and \(f(t,x(t))=f(T,x(T))\) for \(t>T\). Then, obviously, the operator \(B\) maps continuous functions \(x(t)\), \(\|x(t)\|\le M\), on the interval \([-T,T]\) one-to-one into uniformly continuous functions \(y(t)\), \(\|y(t)\|\le\mu\), on the same interval. Consequently, by the well-known Schauder principle, the operator \(B\) has a fixed point \(x(t,T)\), depending, generally speaking, on \(T\); moreover, \(x(t,T)\) is a solution of equation (11) on the given interval. Now form a monotone sequence \(T_1,T_2,\ldots,T_n,\ldots\to+\infty\) and the sequence of initial points

\[ x(0,T_1),\ x(0,T_2),\ldots,\ x(0,T_n),\ldots . \tag{13} \]

Since \(\|x(0,T_n)\|\le\mu\), from the sequence (13) one can extract a subsequence converging to some limit \(x_0\), \(\|x_0\|\le\mu\). Equation (11) has a solution \(x(t)\) with the initial condition \(x(0)=x_0\) in some neighborhood of the point \(t=0\). By the theorem on continuous dependence on initial conditions, under further continuation in both directions the solution \(x(t)\) will remain in the domain \(\|x\|\le M\). The theorem is proved.

The following is also true.

Theorem 4. Let the real parts of all eigenvalues of the matrix \(A\) be positive. If \(\mu<M\) for some \(\omega>0\) such that \(\|e^{-A\omega}\|<1\), where

\[ \mu=\frac{N}{1-\|e^{-A\omega}\|} \sup_{-\infty<t<+\infty}\int_t^{t+\omega}\|e^{A(t-s)}\|\,ds, \]

then there exists at least one bounded solution of equation (11).

We shall now clarify some conditions for the existence of bounded solutions of equation (11) in the cone \(K\).

Theorem 5. Suppose that the conditions of Lemma 4 and Theorem 3 are satisfied. If there exists a positive vector \(b\) such that from the condition \(x(t)-b \geq 0\) there follows the relation \(f(t,x(t))-b \geq 0\) for all \(t\), and \(\lambda \gg 1\), then equation (11) has a bounded solution lying entirely in the cone \(K\). (Here \(\lambda\) is the eigenvalue of the matrix \(-A^{-1}\) corresponding to its positive eigenvector \(a\).)

Proof. For sufficiently small \(\mu>0\) we have \(b-\mu a \geq 0\). Then, by Theorem 1, the relation
\[ Bx(t)-\mu a \geq 0 \]
is satisfied. Consequently, the fixed point of the operator \(B\) lies in the cone \(K\). The theorem is proved.

Let us note that some other conditions for the existence of positive periodic solutions of equation (11) are indicated in the work of M. A. Krasnosel'skii [4].

By referring to Theorem 2 one can verify the validity of the following proposition.

Theorem 6. Suppose that conditions (7) and the conditions of Theorem 4 are satisfied. If from the nonnegativity of the vector \(x(t)\) there follows the nonnegativity of the vector \(f(t,x(t))\) for all \(t\), then equation (11) has a bounded solution located outside the cone \(K\).

1. Some results of the preceding section remain valid for a nonlinear differential equation in the infinite-dimensional space \(E\). The principal difficulty in this case consists in establishing the compactness of the set of initial points analogous to the set (13).

Consider the equation
\[ \frac{dx}{dt}=Ax+f(t,x), \tag{14} \]
where \(A\) is a constant linear operator; \(f(t,x)\) is an operator mapping all continuous and bounded on the interval \((-\infty,+\infty)\) vectors \(x(t)\), \(\|x(t)\|\leq M\), into the collection of continuous and bounded on the interval vectors \(f(t,x(t))\), \(\|f(t,x(t))\|\leq N\).

Let \(A\) be a bounded operator, the real parts of the points of whose spectrum are not greater than some negative number. Then, assuming \(f(t,x)\) to be prescribed on the interval \((-\infty,0)\) in a definite way, as is done in [1] for equation (11), so that the set of initial points analogous to the set (13) consists of only one point, one can obtain a condition for the existence of a bounded solution of equation (14) on the interval \([0,+\infty)\).

Under additional restrictions imposed on the operator \(f(t,x)\), the following is valid.

Theorem 7. Let \(A\) be a bounded operator, the real parts of the points of whose spectrum are not greater than some negative number. Suppose that the operator \(f(t,x)\) satisfies the Lipschitz condition
\[ \|f(t,x_1(t))-f(t,x_2(t))\|\leq K \sup_{-\infty<t<+\infty}\|x_1(t)-x_2(t)\|\quad (x_1,x_2\in X) \]
with a sufficiently small constant \(K\), where \(X\) is the set of continuous bounded functions, \(\|x(t)\|\leq M\). If \(\mu\leq M\) for some \(\omega>0\) such that \(\|e^{A\omega}\|<1\), where
\[ \mu=\frac{N\|e^{A\omega}\|}{1-\|e^{A\omega}\|}\sup_{-\infty<t<+\infty}\int_t^{t+\omega}\|e^{A(t-s)}\|\,ds, \]

then there exists at least one solution of equation (14) that is bounded on \((-\infty,+\infty)\).

Proof. Consider the operator

\[ Bx(t)=\int_t^{t+\omega} e^{A(t-s)}F(s,x(s))\,ds, \]

\[ F(s,x(s))=e^{As}f(s-\omega,x(s-\omega))+e^{2A\omega}f(s-2\omega,x(s-2\omega))+\cdots . \]

The operator \(B\) maps the closed set \(X\) into itself. Moreover, the operator \(F(s,x(s))\), just like the operator \(f(t,x(t))\), satisfies the Lipschitz condition with a certain sufficiently small constant, generally speaking, depending on \(K\). Then the operator \(B\) also satisfies the Lipschitz condition

\[ \|Bx_1-Bx_2\|_X\le q\|x_1-x_2\|_X\quad (x_1,x_2\in X) \]

with some constant \(q<1\). Consequently, by the contraction mapping principle the operator \(B\) has a fixed point. The theorem is proved.

Let us note that the application of Schauder’s principle makes it possible to prove the existence of a periodic solution of equation (14) without any additional restrictions, if \(f(t,x)\) is a periodic operator.

The proofs of some of the propositions obtained can be carried over to equation (14) with an unbounded operator. We shall show, for example, the validity of the following theorem.

Suppose the operator \(A\) has an everywhere dense domain of definition \(D\). Suppose that for \(\lambda\) with \(\operatorname{Re}\lambda\ge 0\) the operator \(A-\lambda I\) has a bounded inverse and

\[ \|(A-\lambda I)^{-1}\|\le C(|\lambda|+1)^{-1}, \]

where \(C\) is some constant. In this case (see, for example, [5]) the operator \(A\) is the infinitesimal generator of the strongly continuous semigroup \(e^{At}\) \((t\ge 0)\), and the estimate

\[ \|e^{At}\|\le Ce^{-\delta t}\quad (\delta>0) \]

holds.

Theorem 8. Let \(f(t,x)\) be an \(\omega\)-periodic operator with respect to \(t\). If \(CN\le \delta M\), then equation (14) has at least one \(\omega\)-periodic solution for all \(t\in[0,+\infty)\).

Proof. Indeed, the operator \(I-e^{A\omega}\) is invertible. Thus, the periodic solutions of equation (14) are at the same time periodic solutions of the integral equation

\[ x(t)=\int_{t-\omega}^{t} e^{A(t-s)}(I-e^{A\omega})^{-1}f(s,x(s))\,ds, \tag{15} \]

as is easily verified by direct differentiation. (The differentiability of expression (15) is established, for example, in [5].) Consider the operator \(B\) defined by the right-hand side of equality (15). We have

\[ \|Bx(t)\|\le \|(I-e^{A\omega})^{-1}\|N\int_{t-\omega}^{t}\|e^{A(t-s)}\|\,ds\le \frac{CN}{\delta}\le M. \]

Consequently, by Schauder’s principle the operator \(B\) has a fixed point. The theorem is proved.

1. Finally, let us note that the results obtained in the preceding sections can, without particular difficulty, be generalized by considering differential equation (14) with a variable operator \(A(t)\). In this case bounded ...

solutions of equation (14) are bounded solutions of the integral equation

\[ x(t)=\int_t^{t+\omega} U(t)U^{-1}(s)F(s,x(s))\,ds, \]

where \(U(t)\) is the solution of the Cauchy problem

\[ \frac{dU}{dt}=A(t)U,\quad U(0)=I; \]

\(\omega\) is a positive number such that either \(\|U(t)U^{-1}(t+\omega)\|\le q\), or \(\|U(t+\omega)U^{-1}(t)\|\le q,\ 0<q<1\), for all \(t\in(-\infty,+\infty)\); \(F(s,x(s))\) is a solution of the difference equation

\[ U(s)U^{-1}(s+\omega)F(s+\omega,x(s+\omega))-F(s,x(s))=f(s,x(s)). \]

References

1. Medvedev N. V. Differential Equations, 1, No. 12, 1592—1596, 1965.
2. Gantmakher F. R. Theory of Matrices. Fizmatgiz, 1953.
3. Bekkenbach E., Bellman R. Inequalities. Translated from the English. Moscow, Mir Publishers, 1965.
4. Krasnosel’skii M. A. Positive Solutions of Operator Equations. Fizmatgiz, 1962.
5. Sobolevskii P. E. Transactions of the Moscow Mathematical Society. Fizmatgiz, 10, 1961, pp. 297—350.

Received by the editorial office
February 8, 1966

Vladimir Pedagogical
Institute