Lyapunov Vector Functions, Rotation of Vector Fields, Guiding Functions, and the Existence of Poisson Bounded Solutions

Abstract

We use the Lyapunov vector function method and the guiding function method to obtain sufficient conditions for the existence of Poisson bounded and partially Poisson bounded solutions of systems of differential equations.

The application of the Lyapunov function method [1] to the study of the boundedness of solutions of differential systems is given in [2], and its application to the study of the boundedness of solutions with respect to part of variables is given in the monograph [3, pp. 223–228]. A generalization of the Lyapunov function method—the Lyapunov vector function method—is presented in the monographs [4, 5]. The book [5] shows how the Lyapunov vector function method can be applied to the derivation of conditions ensuring the boundedness of all solutions of an arbitrary nonlinear system. Independently of the methods in the paper [2], the monograph [6] developed the guiding function method based on the technique of rotation of vector fields. The method was used in [6] to obtain sufficient conditions for the existence of solutions of an arbitrary nonlinear system bounded on the entire real line.

On the other hand, the present author (see, e.g., [7]) has recently begun a study of a new form of solution boundedness—the Poisson boundedness. The concept of Poisson boundedness of a solution on the half-line is that there exists a ball in the state space and a countable system of disjoint intervals on the time half-line with the sequence of their right endpoints tending to $+\infty$ such that the solution is contained in the ball at all times belonging to these intervals. Obviously, a bounded solution is Poisson bounded; the opposite, as is easily seen, is not true. In papers by the present author, conditions have been studied under which all solutions of a differential system are Poisson bounded.

To study conditions for the existence of Poisson bounded solutions, the present paper develops a method that is a synthesis of the Lyapunov vector function method and the guiding function method. Using this method, we obtain sufficient conditions for the existence of Poisson bounded solutions (Theorem 1) as well as partially (in part of the variables) Poisson bounded solutions (Theorem 2). Let us now proceed to rigorous definitions and statements.

Consider the system of differential equations

$$ \frac {dx}{dt}=F(t,x),\quad F(t,x)=\big (F_1(t,x),\ldots ,F_n(t,x)\big )^{\mathrm {T}},\quad x\in \mathbb {R}^n,\quad t\in \mathbb {R}_{+}\equiv [0,+\infty ),\quad n\geq 2,$$

(1)

where $F\colon \mathbb {R}_{+}\times \mathbb {R}^n\to \mathbb {R}^n$ is a continuous function satisfying the local Lipschitz condition in the variable $x\in \mathbb {R}^n$. In addition, we assume that all solutions of system (1) are extendible to the time half-line $\mathbb {R}_{+}$.

In what follows, by $\|\cdot \|$ we denote the Euclidean norm on $\mathbb {R}^n$. The solution $x=x(t)$ of system (1) with the initial condition $(t_0,x_0)\in \mathbb {R}_{+}\times \mathbb {R}^n$ is denoted by $x(t,t_0,x_0)$. Any nonnegative increasing number sequence $(\tau _i)_{i\in \mathbb {N}}$ such that $\lim \limits _{i\to +\infty }\tau _i=+\infty$ will be called a $\mathcal {P}$-sequence.

Recall [2] that a solution $x=x(t,t_0,x_0)$ of system (1) is said to be bounded if there exists a number $\beta >0$ such that $\|x(t,t_0,x_0)\|\leq \beta$ for all $t\in \mathbb {R}_{+}$.

FormalPara Definition 1.

A solution $x=x(t,t_0,x_0)$ of system (1) is said to be Poisson bounded if there exists a $\mathcal {P}$ -sequence $(\tau _i)_{i\in \mathbb {N}}$ and a number $\beta >0$ such that $\|x(\tau _i,t_0,x_0)\|\leq \beta$ for all $i\in \mathbb {N}$.

Obviously, if a solution of system (1) is bounded, then it is also Poisson bounded, because in this case one can take any $\mathcal {P}$-sequence for the required $\mathcal {P}$-sequence. It is also obvious that increasing the number $\beta$ if necessary, we can assume that the inequality $\|x(t,t_0,x_0)\|\leq \beta$ for a Poisson bounded solution is satisfied for all $t$ belonging to some sequence of intervals whose right endpoints tend to $+\infty$, as was underlined above. Therefore, the above definition is equivalent to the definition of a Poisson bounded solution given in the paper [8].

Following [4, pp. 46–48], let us recall some facts needed in what follows about Lyapunov vector functions. Given a continuously differentiable vector function

$$ V:\mathbb {R}_{+}\times \mathbb {R}^n\to \mathbb {R}^k,\quad V(t,x)=\big (V_1(t,x),\ldots ,V_k(t,x)\big )^{\mathrm {T}},\quad k\geq 1,$$

the derivative according to system (1) of this vector function is defined by the relation

$$ \dot {V}(t,x)=\big (\dot {V}_1(t,x),\ldots ,\dot {V}_k(t,x)\big )^{\mathrm {T}}, $$

where $\dot {V}_i(t,x)$ is the derivative according to system (1) of the function $V_i(t,x)$, $1\leq i\leq k$. In what follows, the notation $\xi \leq \eta$ for vectors $\xi =(\xi _1,\ldots ,\xi _k)^{\mathrm {T}}$ and $\eta =(\eta _1,\ldots ,\eta _k)^{\mathrm {T}}\in \mathbb {R}^k$ means that $\xi _i\leq \eta _i$ for each $1\leq i\leq k$. We say [3, p. 235] that a continuous vector function

$$ f\colon \mathbb {R}_{+}\times \mathbb {R}^k\to \mathbb {R}^k,\quad f(t,\xi )=\big (f_1(t,\xi ),\ldots ,f_k(t,\xi )\big )^{\mathrm {T}},\quad k\geq 1, $$

satisfies the Waűewski condition if for each $1\leq s\leq k$ the function $f_s$ is nondecreasing with respect to the variables $\xi _1,\ldots ,\xi _{s-1}$, $\xi _{s+1},\ldots ,\xi _k$, i.e., if it follows from the relations $\xi _i\leq \eta _i$, $1\leq i\leq k$, $i\ne s$, and $\xi _s=\eta _s$ that $f_s(t,\xi )\leq f_s(t,\eta )$. If $f$ satisfies the Waűewski condition, then we write $f\in W$. Note that the condition $f\in W$ degenerates for $k=1$; therefore, we adopt the convention that $f\in W$ for each continuous function $f:\mathbb {R}_{+}\times \mathbb {R}\to \mathbb {R}$.

A continuously differentiable vector function $V\colon \mathbb {R}_{+}\times \mathbb {R}^n\!\to \!\mathbb {R}^k$ satisfying the condition $V(t,x)\!\geq \! 0$ for any $t\in \mathbb {R}_{+}$ and $x\in \mathbb {R}^n$, where $0$ is the zero vector in $\mathbb {R}^k$, and a system

$$ \frac {d\xi }{dt}=f(t,\xi ),\quad f\in W, $$

(2)

are called a Lyapunov vector function and a comparison system, respectively, for system (1) if the following condition is satisfied: $\dot {V}(t,x)\leq f(t,V(t,x))$. In what follows, we always assume that the right-hand side of system (2) satisfies the local Lipschitz condition with respect to $\xi$ and, in addition, the solutions of this system are extendible to the entire half-line $\mathbb {R}_{+}$. Since we have the uniqueness of solution of the Cauchy problem for system (2), it follows from the Waűewski theorem (see, e.g., [3, p. 236]) that for each point $(t_0,x_0)\in \mathbb {R}_{+}\times \mathbb {R}^n$ the solution $x(t,t_0,x_0)$ of system (1), the Lyapunov vector function $V\colon \mathbb {R}_{+}\times \mathbb {R}^n\to \mathbb {R}^k$, and the solution $\xi (t,t_0,V(t_0,x_0))$ of the comparison system (2) for system (1) are related for all $t\geq t_0$ by the inequality

$$ V\big (t,x(t,t_0,x_0)\big )\leq \xi (t,t_0,V(t_0,x_0)). $$

(3)

Now let us recall necessary notions and constructions related to the rotation of vector fields and to operators of shift along trajectories [6] (see also [9]). Let $\Omega$ be an arbitrary compact subset of $\mathbb {R}^n$ with boundary $\partial \Omega$. Following [6], we define a continuous vector field, or, for brevity, a vector field $Q$, on $\Omega$ as an arbitrary continuous mapping $Q\colon \Omega \to \mathbb {R}^n$. For a vector field $Q$ on $\Omega$, consider its restriction to $\partial \Omega$, i.e., the vector field $Q|_{\partial \Omega }\colon \partial \Omega \subset \Omega \to \mathbb {R}^n$. A vector field $Q$ on $\Omega$ is said to be nondegenerate on $\partial \Omega$ if $Q(x)\ne 0\in \mathbb {R}^n$ for all $x\in \partial \Omega$. It can readily be seen that any vector field $Q$ nondegenerate on $\partial \Omega$ defines a continuous mapping

$$ T\colon \partial \Omega \to S^{n-1}=\big \{a\in \mathbb {R}^n:\|a\|=1\big \},\quad T(x)=Q(x)\big /\big \|Q(x)\big \|,\quad x\in \partial \Omega .$$

The rotation $\gamma (Q,\partial \Omega )$ of a vector field $Q$ nondegenerate on $\partial \Omega$ is the degree $\mathrm {deg}\,(T)\in \mathbb {Z}$ of the mapping $T:\partial \Omega \to S^{n-1}$. In the case where the compact subset $\Omega$ in $\mathbb {R}^n$ is an $n$-dimensional smooth orientable manifold with boundary $\partial \Omega$ (see, e.g., [10]), the integer $\mathrm {deg}\,(T)$ is easy to define, for example, by using the functor $H_{n-1}(-;\mathbb {Z})$ of singular $(n-1)$-dimensional homology of topological spaces with integer coefficients [11]. Indeed, the continuous mapping $T:\partial \Omega \to S^{n-1}$ induces a homomorphism $H_{n-1}(T;\mathbb {Z})\colon H_{n-1}(\partial \Omega ;\mathbb {Z})\to H_{n-1}(S^{n-1};\mathbb {Z})$ of the singular homology groups. It is well known (see, e.g., [11]) that the groups $H_{n-1}(\partial \Omega ;\mathbb {Z})$ and $H_{n-1}(S^{n-1};\mathbb {Z})$ are isomorphic to the group $\mathbb {Z}$ and the generators of these groups are the fundamental classes $[\partial \Omega ]$ and $[S^{n-1}]$ of the manifolds $\partial \Omega$ and $S^{n-1}$, respectively. Using the group homomorphism $H_{n-1}(T;\mathbb {Z})$ and the fundamental classes $[\partial \Omega ]$ and $[S^{n-1}]$, the degree of the mapping $\mathrm {deg}\,(T)\in \mathbb {Z}$ is defined by the rule

$$ H_{n-1}(T;\mathbb {Z})\big ([\partial \Omega ]\big ) =\mathrm {deg}\,(T)[S^{n-1}]. $$

In the general case, where $\Omega$ is an arbitrary compact subset in $\mathbb {R}^n$, the definition of the degree $\mathrm {deg}\,(T)\in \mathbb {Z}$ of the mapping $T:\partial \Omega \to S^{n-1}$ is described in detail in the monograph [9, p. 9–29].

We use the following terminology. The subset

$$ \mathrm {Tr}\,(x_0)=\big \{x\in \mathbb {R}^n:x=x(t,0,x_0),\enspace t\geq 0\big \}\subset \mathbb {R}^n, $$

where $x(t,0,x_0)$ is the solution of system (1) and $x_0$ is an arbitrary point in $\mathbb {R}^n$, will be called the trajectory of system (1) issuing from the point $x_0$. For each $\tau >0$, consider the continuous mapping

$$ U(\tau )\colon \Omega \to \mathbb {R}^n,\quad U(\tau )(x_0)=x(\tau ,0,x_0),$$

where $x(t,0,x_0)$ is the solution of system (1) and $x_0$ is an arbitrary point in $\Omega$. The mapping $U(\tau )$ is called [6, pp. 11–12] the operator of shift along the trajectories of system (1) for time $0\leq t\leq \tau$. A $\tau$-nonrecurrent point of a trajectory of system (1) is a point $x_0\in \mathbb {R}^n$ such that the solution $x(t,0,x_0)$ of system (1) satisfies the condition $x(t,0,x_0)\ne x_0$ for all $0<t\leq \tau$ [6, p. 101].

Consider the vector field

$$ S_0\colon \Omega \to \mathbb {R}^n,\quad S_0(x)=-F(0,x), $$

where $F(t,x)$ is the right-hand side of system (1). The rotation $\gamma (S_0,\partial \Omega )$ of this vector field is closely related to the problem on the existence of fixed points of the operator $U(\tau )$ of shift along the trajectories of system (1). Indeed, it was shown in [6, p. 102–104] that if a vector field $S_0:\Omega \to \mathbb {R}^n$ nondegenerate on $\partial \Omega$ has rotation $\gamma (S_0,\partial \Omega )\ne 0$ and all points of the boundary $\partial \Omega$ are $\tau$-nonrecurrent points of the trajectories of system (1), then the operator $U(\tau )$ of shift along the trajectories of system (1) has at least one fixed point inside $\Omega$, i.e., a point $x\in \Omega \setminus \partial \Omega$ such that $U(\tau )(x)=x$. We will use the following terminology. The subsets

$$ \eqalign { \mathrm {Tr}^{+}(x_0,t_0)&=\big \{x\in \mathbb {R}^n:x =x(t,t_0,x_0),\enspace t>t_0\big \}\subset \mathbb {R}^n,\cr \mathrm {Tr}^{-}(x_0,t_0)&=\big \{x\in \mathbb {R}^n:x =x(t,t_0,x_0),\enspace 0\leq t\leq t_0\big \}\subset \mathbb {R}^n,}$$

where $x(t,t_0,x_0)$ is the solution of system (1) and $(t_0,x_0)$ is any point in $\mathbb {R}_{+}\times \mathbb {R}^n$, will be called the right part and the left part, respectively, of the trajectory $\mathrm {Tr}\,(x(0,t_0,x_0))$ of system (1).

We state and prove the following sufficient condition for the existence of Poisson bounded solutions of system (1) in terms of Lyapunov vector functions and rotations of vector fields.

FormalPara Proposition 1.

Assume that for system (1) there exists a $\mathcal {P}$-sequence $(\tau _i)_{i\in \mathbb {N}}$, a Lyapunov vector function $V\colon \mathbb {R}_{+}\times \mathbb {R}^n\to \mathbb {R}^k$ with a comparison system (2), and a nondecreasing function $b\colon \mathbb {R}_{+}\to \mathbb {R}_{+}$ with the property $b(r)\to +\infty$ as $r\to +\infty$ such that the inequality

$$ b(\|x\|)\leq \sum _{q=1}^kV_q(\tau _i,x) $$

(4)

holds for any $x\in \mathbb {R}^n$ and $i\in \mathbb {N}$. Moreover, let the following conditions be satisfied for the comparison system (2):

• 1. There exists a subset $\Omega \subset \mathrm {Im}\,(V\colon \{0\} \times \mathbb {R}^n\to \mathbb {R}^k)$ compact in $\mathbb {R}^k$ such that the vector field $S_0:\Omega \to \mathbb {R}^k$, $S_0(\xi )=-f(0,\xi )$, where $f(t,\xi )$ is the right-hand side of the comparison system (2), is nondegenerate on $\partial \Omega$ and $\gamma (S_0,\partial \Omega )\ne 0$.

• 2. For each $\xi _0\in \partial \Omega$, the right part $\mathrm {Tr}^{+}(\xi _0,t_0)$ of the trajectory $\mathrm {Tr}\,(\xi (0,t_0,\xi _0))$ of the comparison system (2) does not have any points in $\Omega$ in common with the left part $\mathrm {Tr}^{-}(\xi _0,t_0)$ of the same trajectory.

Then system (1) has at least one Poisson bounded solution.

Proof. For each $m\in \mathbb {N}$, consider the operator $U(m)\colon \Omega \to \mathbb {R}^k$ of shift along the trajectories of the comparison system (2) for system (1) for time $0\leq t\leq m$. Since, by assumption, the intersection $\mathrm {Tr}^{+}(t_0,\xi _0)\cap \mathrm {Tr}^{-}(t_0,\xi _0)\cap \Omega$ is empty for each $(t_0,\xi _0)\in \mathbb {R}_{+}\times \partial \Omega$, we conclude that all points of the boundary $\partial \Omega$ are $m$-nonrecurrent points of the trajectories of system (2) for each $m\in \mathbb {N}$. As was indicated above, it follows that for each $m\in \mathbb {N}$ the operator $U(m)$ has a fixed point $\vartheta _m\in \Omega \setminus \partial \Omega$.

Consider the family of solutions $\{\xi (t,0,\vartheta _m)\}_{m\in \mathbb {N}}$ of system (2). It follows from the assumptions of the theorem to be proved that $\xi (t,0,\vartheta _m)\in \Omega \setminus \partial \Omega$ for each $0\leq t\leq m$. Indeed, if the opposite were true, then for some point $\xi _0=x(t_0,0,\vartheta _m)\in \mathrm {Tr}\,(\vartheta _m)$, where $0<t_0<m$ and $\xi _0\in \partial \Omega$, we would have

$$ \mathrm {Tr}^{+}(t_0,\xi _0)\cap \mathrm {Tr}^{-}(t_0,\xi _0)\cap \Omega =\{\vartheta _m\}\ne \varnothing , $$

which contradicts the assumptions of the theorem.

Consider the sequence of points $(\vartheta _m)_{m\in \mathbb {N}}$ in $\Omega \setminus \partial \Omega$. Using the fact that the set $\Omega$ is compact, from the sequence $(\vartheta _m)_{m\in \mathbb {N}}$ we select a subsequence $(\vartheta _{m_i})_{i\in \mathbb {N}}$ converging to some point $\mu \in \Omega$. Let us show that the solution $\xi (t,0,\mu )$ of system (2) satisfies the condition $\xi (t,0,\mu )\in \Omega$ for all $t\geq 0$. Assume the contrary: there exists a number $\eta \geq 0$ such that $\xi (\eta ,0,\mu )\not \in \Omega$. Since system (2) satisfies the assumptions of the theorem on the continuous dependence on the initial conditions (see, e.g., [12]), it follows that $\xi (\eta ,0,\vartheta _{m_i})\not \in \Omega$, where $\eta \leq m_i$, for sufficiently large $i$. We have arrived at a contradiction with the inclusion

$$ \xi (t,0,\vartheta _{m_i})\in \Omega \setminus \partial \Omega \subset \Omega \quad \text {for all}\quad 0\leq t\leq m_i.$$

Thus, we have shown that $\xi (t,0,\mu )\in \Omega$ for any $t\geq 0$. Since the set $\Omega$ is compact in $\mathbb {R}^k$, we conclude that in $\mathbb {R}^k$ there exists a ball of radius $\alpha >0$ centered at the origin such that $\Omega$ is contained in this ball, and hence $\|\xi (t,0,\mu )\|\leq \alpha$ for all $t\geq 0$.

Now let us show that system (1) has a Poisson bounded solution $x(t,0,x_0)$ for some $x_0\in \mathbb {R}^n$. Since, by assumption, $\Omega \subset \mathrm {Im}\,(V:\{0\}\times \mathbb {R}^n\to \mathbb {R}^k)$, it follows that there exists a point $(0,x_0)\in \{0\}\times \mathbb {R}^n$ such that $V(0,x_0)=\mu$. We use condition (4) and inequality (3) to obtain the inequalities

$$ b\Big (\big \|x(\tau _i,0,x_0)\big \|\Big )\leq \sum _{q=1}^k V_q\big (\tau _i,x(\tau _i,0,x_0)\big ) \leq \sum _{q=1}^k\xi _q\big (\tau _i,0,V(0,x_0)\big ), $$

which hold for all $i\in \mathbb {N}$, for the solution $x(t,0,x_0)$ of system (1) and the solution $\xi (t,0,V(0,x_0))$ of the comparison system (2). Moreover, we have the obvious inequalities

$$ \sum _{i=1}^k\xi _i\big (t,0,V(0,x_0)\big )\leq \sum _{i=1}^k\Big |\xi _i\big (t,0,V(0,x_0)\big )\Big |\leq k\Big \|\xi \big (t,0,V(0,x_0)\big )\Big \|$$

for each $t\geq 0$. Since $V(0,x_0)=\mu$, we have $\|\xi (t,0,V(0,x_0))\|\leq \alpha$ for all $t\geq 0$. This fact, as well as the above-indicated inequalities, implies that $b(\|x(\tau _i,0,x_0)\|)\leq k\alpha$ for all $i\in \mathbb {N}$. Using the condition $b(r)\to +\infty$ as $r\to +\infty$ and the fact that the number $k\alpha$ is fixed, we select a number $\beta >0$ such that $k\alpha \leq b(\beta )$. It follows that $b(\|x(\tau _i,0,x_0)\|)\leq b(\beta )$ for all $i\in \mathbb {N}$. Since the function $b\colon \mathbb {R}_{+}\to \mathbb {R}_{+}$ is nondecreasing, it follows from the last inequality that $\|x(\tau _i,0,x_0)\|\leq \beta$ for all $i\in \mathbb {N}$. Thus, we have shown that the solution $x(t,0,x_0)$ of system (1) is Poisson bounded. The proof of the proposition is complete.

Given arbitrary positive integers $n\geq 2$ and $1\leq m<n$, for each fixed element $(t_0,x_0)\in \mathbb {R}_{+}\times \mathbb {R}^n$ and the corresponding solution $x(t,t_0,x_0)=(x_1(t,t_0,x_0),\ldots , x_n(t,t_0,x_0))^{\mathrm {T}}$ of system (1) consider the mapping

$$ y\colon \mathbb {R}_{+}\times \big \{(t_0,x_0)\big \}\to \mathbb {R}^m,\quad y(t,t_0,x_0)=\big (x_1(t,t_0,x_0), \ldots ,x_m(t,t_0,x_0)\big )^{\mathrm {T}}.$$

Recall [3] that a solution $x(t,t_0,x_0)$ of system (1) is said to be $y$-bounded if there exists a number $\beta >0$ such that $\|y(t,t_0,x_0)\|\leq \beta$ for all $t\in \mathbb {R}_{+}$.

FormalPara Definition 2.

A solution $x=x(t,t_0,x_0)$ of system (1) is said to be $y$-bounded in the sense of Poisson if there exists a $\mathcal {P}$ -sequence $(\tau _i)_{i\in \mathbb {N}}$ and a number $\beta >0$ such that $\|y(\tau _i,t_0,x_0)\|\leq \beta$ for all $i\in \mathbb {N}$.

It is obvious that if a solution of system (1) is $y$-bounded, then it is also $y$-bounded in the sense of Poisson, because in this case we can take any $\mathcal {P}$ -sequence for the required $\mathcal {P}$ -sequence. It can readily be seen that the above definition is equivalent to the definition of solution $y$-bounded in the sense of Poisson in the paper [8].

The following assertion is a sufficient condition for system (1) to have solutions $y$-bounded in the sense of Poisson.

FormalPara Proposition 2.

Let all the assumptions in Proposition 1 with inequality (4) replaced by the inequality

$$ b(\|y\|)\leq \sum _{q=1}^kV_q(\tau _i,x) $$

(5)

be satisfied. Then system (1) has at least one solution that is $y$-bounded in the sense of Poisson.

Proof. By reproducing the argument in the proof of Proposition 1 word for word and by replacing inequality (4) with inequality (5), we obtain the inequality

$$ b(\|y(\tau _i,0,x_0)\|)\leq \sum _{q=1}^k V_q(\tau _i,x(\tau _i,0,x_0))\leq \sum _{q=1}^k \xi _q(\tau _i,0,V(0,x_0)),\quad i\in \mathbb {N}.$$

Now, by reproducing the argument in the proof of Proposition 1 word for word with $x(t,0,x_0)$ replaced by $y(t,0,x_0)$, we obtain the desired inequality $\|y(\tau _i,0,x_0)\|\leq \beta$ for all $i\in \mathbb {N}$. Thus, we have shown that the solution $x(t,0,x_0)$ of system (1) is $y$-bounded in the sense of Poisson. The proof of the proposition is complete.

Now let us recall necessary facts about guiding functions and their indices [6]. A continuously differentiable function $G\colon \mathbb {R}^n\to \mathbb {R}$ is called a guiding function, or, more precisely, an $r_0$-guiding function, for system (1) if the following condition is satisfied:

$$ \big (\mathrm {grad}\, G(x),F(t,x)\big )>0,\quad t\geq 0,\quad \|x\|\geq r_0.$$

(6)

Consider the vector field $\mathrm {grad}\,G\colon B^n(r_0)\to \mathbb {R}^n$, where $B^n(r_0)=\{x\in \mathbb {R}^n:\|x\|\leq r_0\}$. It can be seen from condition (6) that the vector field $\mathrm {grad}\,G\colon B^n(r_0)\to \mathbb {R}^n$ is nondegenerate on $\partial B^n(r_0)$, and consequently, the rotation $\gamma (\mathrm {grad}\,G,\partial B^n(r_0))$ of this vector field is well defined. It was shown in the monograph [6, p. 90–91] that if for each $r>r_0$ we consider the corresponding vector field $\mathrm {grad}\,G\colon B^n(r)\to \mathbb {R}^n$, which is obviously nondegenerate on $\partial B^n(r)$, then one has the equality of rotations $\gamma (\mathrm {grad}\,G,\partial B^n(r))=\gamma (\mathrm {grad}\,G,\partial B^n(r_0))$. The index of an $r_0$-guiding function $G$ for system (1) is the integer $\mathrm {ind}\,(G)$ defined by the formula

$$ \mathrm {ind}\,(G)=\gamma \big (\mathrm {grad}\,G,\partial B^n(r_0)\big )=\gamma \big (\mathrm {grad}\,G,\partial B^n(r)\big ),\quad r>r_0.$$

It was shown in [6, p. 110–111] that if system (1) has an $r_0$-guiding function $G$, then for each $r\geq r_0$ the rotation of the vector field $S_0\colon B(r)\to \mathbb {R}^n$, $S_0(x)=-F(0,x)$, where $F(t,x)$ is the right-hand side of system (1), and the index of the $r_0$-guiding function $G$ are related by the formula

$$ \gamma \big (S_0,\partial B^n(r)\big )=(-1)^n\,\mathrm {ind}\,G.$$

An unbounded $r_0$-guiding function for system (1) is any $r_0$-guiding function $G$ for this system satisfying the condition $G(x)\to +\infty$ as $\|x\|\to +\infty$. It was shown in [6, p. 111–113] that $\mathrm {ind}\,(G)=1$ for each unbounded $r_0$-guiding function $G$ for system (1). It follows that if system (1) has an unbounded $r_0$-guiding function, then the rotation of the above-indicated vector field $S_0\colon B^n(r)\to \mathbb {R}^n$ is calculated by the formula $\gamma (S_0,\partial B^n(r))=(-1)^n$.

Let us state and prove a sufficient condition in terms of Lyapunov vector functions and guiding functions for system (1) to have Poisson bounded solutions.

FormalPara Theorem 1.

Assume that for system (1) there exists a $\mathcal {P}$-sequence $(\tau _i)_{i\in \mathbb {N}}$, a nondecreasing function $b\colon \mathbb {R}_{+}\to \mathbb {R}_{+}$ for which $b(r)\to +\infty$ as $r\to +\infty$, and a Lyapunov vector function $V\colon \mathbb {R}_{+}\times \mathbb {R}^n\to \mathbb {R}^k$ with the comparison system (2) such that inequality (4) is satisfied for any $x\in \mathbb {R}^n$ and $i\in \mathbb {N}$. Moreover, assume that there exist numbers $r_1>r_0$ and an unbounded $r_0$-guiding function $G$ for the system

$$ \frac {d\varrho }{dt}=g(t,\varrho ),\quad (t,\varrho )\in \mathbb {R}^{+}\times \mathbb {R}^k,\quad g(t,\varrho )=f(t,\varrho +\overline {r}_1),\quad \overline {r}_1=(r_1,\ldots ,r_1)^{\mathrm {T}}\in \mathbb {R}^k, $$

(7)

where $f(t,\xi )$ is the right-hand side of system (2), with the number $r_1$ satisfying the conditions

• 1. $G(\varrho )\geq M_0$ for all $\varrho \in \mathbb {R}^k$, $\|\varrho \|=r_1$, where $M_0=\max \limits _{\|\varrho \|\leq r_0}G(\varrho )$.

• 2. $B_{\overline {r}_1}^k(r_1) =\{\xi \in \mathbb {R}^k|\|\xi -\overline {r}_1\|\leq r_1\}\subset \mathrm {Im}\,(V\colon \{0\}\times \mathbb {R}^n\to \mathbb {R}^k)$.

Then system (1) has at least one Poisson bounded solution.

Proof. Consider the vector field $L_0\colon B^k(r_1)\to \mathbb {R}^k$ defined by the formula $L_0(\varrho )=-g(0,\varrho )$, where $g(t,\varrho )$ is the right-hand side of system (7). By the preceding, $\gamma (L_0,\partial B^k(r_1))=(-1)^k$. Thus, $\gamma (L_0,\partial B^k(r_1))\ne 0$.

Let us show that whatever the point $\varrho _0\in \partial B^k(r_1)$ is, the right part $\mathrm {Tr}^{+}(\varrho _0,t_0)$ of the trajectory $\mathrm {Tr}\,(\varrho (0,t_0,\varrho _0))$ of system (7) does not have points in $B^k(r_1)$ in common with the left part $\mathrm {Tr}^{-}(\varrho _0,t_0)$ of the same trajectory. Consider the function $\varphi (t)=G(\varrho (t,t_0,\varrho _0))$, $t\geq 0$, and its derivative

$$ \varphi ^{\prime }(t)=\frac {d\Big (G\big (\varrho (t,t_0,\varrho _0)\big )\Big )}{dt}= \Big (\mathrm {grad}\,G\big (\varrho (t,t_0,\varrho _0)\big ),g(t,\varrho \big (t,t_0,\varrho _0)\big )\Big ),\quad t\geq 0.$$

Since $G$ is an $r_0$-guiding function for system (7), it follows that $\varphi ^{\prime }(t)>0$ for $t\geq 0$ such that $\|\varrho (t,t_0,\varrho _0)\|\geq r_0$. It is obvious that $\varphi (t_0)=G(\varrho _0)\geq M_0$ and $\varphi (t)\leq M_0$ for $t\geq 0$ such that $\|\varrho (t,t_0,\varrho _0)\|\leq r_0$. Moreover, it is obvious that $\varphi (t)$ is an increasing function for $t\geq 0$ such that $\|\varrho (t,t_0,\varrho _0)\|\geq r_0$. It follows that $\varphi (t)\leq \varphi (t_0)$ for each point $\varrho (t,t_0,\varrho _0)\in \mathrm {Tr}^{-}(\varrho _0,t_0)$.

Now let us show that if $\varrho (t,t_0,\varrho _0)\in \mathrm {Tr}^{+}(\varrho _0,t_0)$, then

$$ \big \|\varrho (t,t_0,\varrho _0)\big \|>r_0.$$

Assume the contrary: $\|\varrho (t_1,t_0,\varrho _0)\|\leq r_0$ for some $t_1>t_0$ and hence $\varphi (t_1)\leq M_0$. Since $\varphi (t_0)\geq M_0$ and $\varphi ^{\prime }(t_0)>0$, it follows that there exists a $t_0<t^{\prime }<t_1$ such that $\varphi (t^{\prime })>M_0$ and hence $\|\varrho (t^{\prime },t_0,\varrho _0)\|>r_0$. Since the function $\|\varrho (t,t_0,\varrho _0)\|$ is continuous, it follows from the preceding that there exists a $t^{\prime }<t_2\leq t_1$ such that $\|\varrho (t_2,t_0,\varrho _0)\|=r_0$ and $\|\varrho (t,t_0,\varrho _0)\|\geq r_0$ for $t^{\prime }<t\leq t_2$. It is obvious that $\varphi (t_2)\leq M_0$, because $\|\varrho (t_2,t_0,\varrho _0)\|=r_0$. Since $\|\varrho (t,t_0,\varrho _0)\|\geq r_0$ for $t^{\prime }\leq t\leq t_2$, we conclude that $\varphi ^{\prime }(t)>0$ for $t^{\prime }\leq t\leq t_2$ and hence $\varphi (t^{\prime })<\varphi (t_2)$. Based on this, we obtain $\varphi (t_2)>M_0$; this contradicts the above-indicated inequality $\varphi (t_2)\leq M_0$. Thus, $\|\varrho (t,t_0,\varrho _0)\|>r_0$ for each point $\varrho (t,t_0,\varrho _0)\in \mathrm {Tr}^{+}(\varrho _0,t_0)$.

Based on the preceding, we conclude that $\varphi ^{\prime }(t)>0$ for $t>t_0$ and hence $\varphi (t)>\varphi (t_0)$ for $t>t_0$. Thus, $\varphi (t)\leq \varphi (t_0)$ for $0\leq t\leq t_0$ and $\varphi (t)>\varphi (t_0)$ for $t>t_0$. It follows that the right part $\mathrm {Tr}^{+}(\varrho _0,t_0)$ of the trajectory $\mathrm {Tr}\,(\varrho (0,t_0,\varrho _0))$ of system (7) does not have points in $B^k(r_1)$ in common with the left part $\mathrm {Tr}^{-}(\varrho _0,t_0)$ of the same trajectory. Arguing by analogy with the proof of Theorem 1, we obtain a solution $\varrho (t,0,\mu )$ (where $\mu \in B^k(r_1)$) of system (7) for which the inclusion $\varrho (t,0,\mu )\in B^k(r_1)$ is satisfied for all $t\geq 0$.

Since system (7) has been obtained from system (2) by the change of variables $\xi =\varrho +\overline {r}_1$, we see that the solution $\xi (t,0,\mu +\overline {r}_1)=\varrho (t,0,\mu )+\overline {r}_1$ of system (2) satisfies $\xi (t,0,\mu +\overline {r}_1)\in B_{\overline {r}_1}^k(r_1)$ for all $t\geq 0$. One obviously has the inclusion $\mu +\overline {r}_1\in B_{\overline {r}_1}^k(r_1)$. It follows from condition 2 in the theorem that there exists a point $(0,x_0)\in \{0\}\times \mathbb {R}^n$ such that $V(0,x_0)=\mu +\overline {r}_1$. Arguing by analogy with the proof of Proposition 1, we obtain the Poisson boundedness of the solution $x(t,0,x_0)$ of system (1). The proof of the theorem is complete.

The next assertion is a sufficient condition for system (1) to have solutions $y$-bounded in the sense of Poisson.

FormalPara Theorem 2.

Let all assumptions of Theorem 1 with inequality (4) replaced by (5) be satisfied. Then system (1) has at least one solution $y$-bounded in the sense of Poisson.

Proof. Let us repeat word for word the argument in the proof of Theorem 1 up to the point of considering the solutions $\xi (t,0,\mu +\overline {r}_1)=\varrho (t,0,\mu )+\overline {r}_1$ of system (2) for which the inclusion $\xi (t,0,\mu +\overline {r}_1)\in B_{\overline {r}_1}^k(r_1)$ holds for all $t\geq 0$, where $V(0,x_0)=\mu +\overline {r}_1\in B_{\overline {r}_1}^k(r_1)$. After this, arguing by analogy with the proof of Proposition 2, we conclude that the solution $x(t,0,x_0)$ of system (1) is $y$-bounded in the sense of Poisson. The proof of the theorem is complete.