On existence and continuation of solutions of the statedependent impulsive dynamical system with boundary constraints
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Abstract
The statedependent impulsive dynamical system with boundary constraints is a kind of special but common system in nature. But because of the complexity of the geometry or topological structures of the impulsive surface, it is hard to determine when an event or an impulsive surface is reached. Therefore, a general statedependent impulsive nonlinear dynamical system is rarely studied. This paper presents a class of statedependent impulsive dynamical systems with boundary constraints. We obtain the existence and continuation of their viable solutions and provide sufficient conditions for the existence and uniqueness of the viable solutions to the system. Finally, two examples are given to illustrate the effectiveness of the results.
Keywords
Viable solution Statedependent impulsive Existence Continuation Boundary constraints1 Introduction
The impulsive conditions are not only involved in ordinary differential equations, these conditions may be involved in fractional differential equations as well as in partial differential equations [1, 2]. Impulsive differential equations (IDEs) are basic dynamical models to describe the dynamics of kinds of evolution processes which experience a change of state suddenly, such as harvesting, vibroimpact, natural disasters. These processes are subject to abrupt changes, which are also called perturbations. Since the duration of shortterm perturbations is negligible compared to the duration of an entire evolution [3, 4, 5], such perturbations involved in the models are generally expressed in the form of impulses. Impulsive differential equations play a very important role in the model construction and analysis of impulsive problems in electrical, mechanical, population dynamics, industrial robotics, biotechnology, optimal control, pharmacokinetics, economic and social sciences, and so on [6, 7], and they have been extensively studied in the past several years [8, 9, 10, 11, 12].
Impulsive differential systems have many kinds of different characteristics of impulsive perturbations, and we usually study three kinds of impulsive differential systems: differential systems with fixedtime impulses, differential systems with variabletime impulses, and differential systems with statedependent impulses. Most of the previous papers [13, 14, 15, 16, 17] consider differential systems with fixedtime impulses or variabletime impulses and discuss their basic qualitative problems, for example, the existence and uniqueness of the solutions of systems, stability, synchronization, bifurcation, etc.
In reality, however, the statedependent impulsive systems (the impulsive moments depend on the state of the system) are more reasonable in modeling and control due to the statedependent impulsive control strategy being more economic, efficient, and practical. So far, the statedependent impulsive systems have a number of applications especially in ecological models, mathematical biology, control theory, etc. In ecological models, the control strategies (by catching, spraying pesticide, or releasing the natural enemy) are taken only when the number of species reaches a critical level, rather than the usual fixedtime impulsive control strategy [18, 19, 20, 21, 22, 23, 24]. In particular, Tang et al. [18] studied the existence and stability of positive orderk (\(k\geq 1\)) periodic solutions of statedependent impulsive models by using the properties of the Lambert W function and Poincaré map. Nie et al. [20] studied the existence and stability of positive order1 or order2 periodic solution of an SIR epidemic model with statedependent pulse vaccination. In Chap. 8 of the book Principles of Discontinuous Dynamical Systems, Akhmet [24] studies discontinuous dynamical systems (DDS). The author mainly analyzes the dynamical properties of the solution trajectory and vector field of autonomous equations with discontinuities and studies the local existence, uniqueness, and extension by using the related properties of discontinuous flows (DF). However, due to the complexity of the topological structure of the impulsive hypersurface, the discontinuous dynamical systems this work considers are almost a twodimensional system, while the study about high dimensional autonomous systems with discontinuous properties is still rare.
The paper is structured as follows. Section 2 provides the necessary notations and definitions. In Sect. 3, sufficient conditions for the existence and continuation of viable solution of statedependent impulsive autonomous differential system (1.1) with state constraints are presented and proved. In Sect. 4, in order to illustrate our results, an example is delivered to illustrate the conclusion.
2 Preliminaries
Remark 2.1
According to the definition of \(\tau (\cdot )\), we may know that \(\tau (x)>0\) for \(x\notin \mathcal{M}\) and \(\tau (x)=0\) for \(x\in \mathcal{M}\). Furthermore, if \(\mathcal{M}^{+}(x)=\emptyset \), then \(\tau (x)=\infty \).
The impulsive dynamical system (1.1) is called discontinuous dynamical system (DDS) when \(\mathcal{M}^{+}(x_{0}) \neq \emptyset \) [24, 31, 32].
Definition 2.1
(Viable solution)
A solution \(\pi _{x_{0}}(t)\) of (2.1) on \(\mathbb {R}_{+}\) with initial condition \(x(0)=x_{0} \in \mathcal{K}\) is said to be viable in the viability constraints \(\mathcal{K}\subset \mathbb {R}^{n}\) on \(\mathbb {R}_{+}\) if, for every time \(t\geq 0\), \(\pi _{x_{0}}(t)\in \mathcal{K}\).
 \(\mathit{(H1)}\)
 \(\mathcal{M}\neq \emptyset \) and there exists a continuously differentiable function \(H:\partial {\mathcal{K}}\rightarrow \mathbb {R}\) such that the hypersurface \(\mathcal{M}\) is defined by$$ \mathcal{M}\triangleq \bigl\{ x\in \partial {\mathcal{K}}H(x)=0 \text{ and } \nabla H(x)\neq 0\bigr\} . $$(2.3)
 \(\mathit{(H2)}\)

\(J: \mathcal{M}\rightarrow \mathcal{N}\) is a continuous differentiable function, and \(\det [\frac{\partial J(x)}{ \partial x} ]\neq 0\) for \(x\in \mathcal{M}\).
 \(\mathit{(H3)}\)

\(\overline{\mathcal{N}}\cap \mathcal{M}=\emptyset \), where \(\overline{\mathcal{N}}\) is the closure of \(\mathcal{N}\).
 \(\mathit{(H4)}\)

The vector field \(f(x)\) satisfies the following transversality condition: \(\langle \nabla H (x),f(x)\rangle \neq 0 \) for all \(x\in \mathcal{M}\).
 \(\mathit{(H5)}\)

For \(x\in \mathcal{N}\), \(\langle \nabla \widetilde{H}(x),f(x)\rangle \neq 0\).
In order to define a solution of (1.1), we need the following definition.
Definition 2.2
 (i)
\(\widetilde{\pi }_{x_{0}}(t)\) is right continuous on \([t_{0},t_{f})\);
 (ii)
For every \(t\in [t_{0},t_{f})\), left and right limits of \(\widetilde{\pi }_{x_{0}}(t)\) exist, denoted by \(\widetilde{\pi }_{x _{0}}^{}(t)\triangleq \lim_{s\rightarrow t^{}}\widetilde{\pi }_{x _{0}}(s)\) and \(\widetilde{\pi }_{x_{0}}^{+}(t)\triangleq \lim_{s\rightarrow t^{+}}\widetilde{\pi }_{x_{0}}(s)\);
 (iii)
There exists a closed discrete subset \(\mathcal{I}_{x _{0}}\subset [t_{0},t_{f})\) called impulsive times such that (a) for \(t\notin \mathcal{I}_{x_{0}}\), \(\widetilde{\pi } _{x_{0}}(t)\) is differentiable, \(\frac{\mathrm {d}\widetilde{\pi }_{x_{0}}(t)}{\mathrm {d}t}=f(\widetilde{\pi } _{x_{0}}(t))\), and \(\widetilde{\pi }_{x_{0}}(t)\notin \mathcal{M}\); (b) for \(t\in \mathcal{I}_{x_{0}}\), \(\widetilde{\pi }_{x _{0}}^{}(t)\in \mathcal{M}\) and \(\widetilde{\pi }_{x_{0}}^{+}(t)=J( \widetilde{\pi }_{x_{0}}^{}(t))\).
If \(\mathcal{M}^{+}(x_{0})=\emptyset \), then \(\widetilde{\pi }_{x_{0}}(t)= \pi _{x_{0}}(t)\), that is, the trajectory \(\widetilde{\varPi }^{+}(x_{0},t)\) does not intersect with impulse surface \(\mathcal{M}\), there is no impulsive effect. Thus, the trajectory \(\widetilde{\varPi }^{+}({x_{0}},t)\) starting at the initial point \({x_{0}}\in \mathcal{K}\) will remain in the viability constraints \(\mathcal{K}\) forever. Therefore, by the existence and uniqueness theorem for ordinary differential equation, \(\widetilde{\pi }_{x_{0}}(t)\) exists and is unique on an interval \([0,t_{f})\) as a viable solution of system (2.1).
Remark 2.2
Note that \(\widetilde{\pi }_{x_{0}}(t_{k})\in \mathcal{M}\), \(\widetilde{\pi }_{x_{0}}(t_{k}^{+})\in \mathcal{N}\). Moreover, \(\widetilde{\pi }_{x_{0}}(t_{k})\in \mathcal{M}\) for \(t_{k}>t_{0}\) and \(\widetilde{\pi }_{x_{0}}(t_{k})\in \mathcal{N}\) for \(t_{k}>t_{0}\), where \(t_{k}\in \mathcal{I}_{x_{0}}\).
Remark 2.3
If \(t_{f}<\infty \), then \(\widetilde{\pi }_{x_{0}}:[t_{0},t_{f}) \rightarrow \mathbb {R}^{n}\) is a maximal solution of (1.1), where \(\mathcal{I}_{x_{0}}\neq \emptyset \), \(\widetilde{\pi }_{x_{0}}:[ \max (\mathcal{I}_{x_{0}}),t_{f})\rightarrow \mathbb {R}^{n}\) is a maximal solution of (1.1), and when \(\mathcal{I}=\emptyset \), \(\widetilde{\pi }_{x_{0}}:[t_{0},t_{f})\rightarrow \mathbb {R}^{n}\) is a maximal solution of (2.1). If \(t_{f}=\infty \), then the solution is obviously maximal.
We shall use \(\mathcal{PC}([t_{0},t_{f}),\mathbb {R}^{n})\) to denote the class of piecewise continuous functions from \([t_{0},t_{f})\) to \(\mathbb {R}^{n}\), with discontinuities of the first kind only at \(t=t_{k}\), \(k=1,2, \ldots \) . Thus, \(\widetilde{\pi }_{x_{0}}(t)\in \mathcal{PC}^{1}([t _{0},t_{f}),\mathbb {R}^{n})\).
Now we give the Schauder fixed point theorem, the definitions of the impulsive viable solution and continuation of the solution of (1.1).
Theorem 2.1
(Schauder fixed point theorem [33])
Let\(\mathcal{C}\subseteq \mathbb {R}^{n}\)be a nonempty, convex, and closed set, let\(f:\mathcal{C}\rightarrow \mathcal{C}\)be continuous, and assume that\(f(\mathcal{C})\)is bounded. Show that there exists\(x\in \mathcal{C}\)such that\(f(x)=x\).
Definition 2.3
(Impulsive viable solution)
A solution \(\widetilde{\pi }_{x_{0}}(t) \in \mathcal{PC}^{1}([t_{0},t_{f}),\mathbb {R}^{n})\) of (1.1) on the interval \([t_{0},t_{f})\) with initial condition \(x(0^{+})=x_{0}\) is said to be viable in the viability constraints \(\mathcal{K}\subset \mathbb {R}^{n}\) on \([t_{0},t_{f})\) if, for every time \(t\in [t_{0},t_{f})\backslash \mathcal{I}_{x_{0}}\), \(\widetilde{\pi }_{x_{0}}(t)\in \mathcal{K}\).
Definition 2.4
([24])
A solution \(\widetilde{\pi }_{x_{0}}(t)\) of (1.1) is said to be continuable to a set \(U\in \mathbb {R}^{n}\) as time decreases (increases) if there exists a time \(s\in \mathbb {R}\) such that \(s\leq 0\) (\(s\geq 0\)) and \(\widetilde{\pi }_{x_{0}}(s)\in U\).
 \(\mathit{(H6)}\)

\(\sup \f(x)\<+\infty \) for all \(x\in \mathcal{K}\).
 \(\mathit{(H7)}\)

 \(\mathit{(a)}\)

Every solution \(\pi _{x_{0}}(t)\), \(x_{0}\in \mathcal{K}\), of (2.1) is continuable to either ∞ or \(\mathcal{M}\) as time increases.
 \(\mathit{(b)}\)

Every solution \(\pi _{x_{0}}(t)\), \(x_{0}\in \mathcal{K}\), of (2.1) is continuable to either −∞ or \(\mathcal{N}\) as time decreases.
3 Main results
In this section we prove the existence and continuation of solution of (1.1).
The following theorem gives sufficient conditions for the existence and uniqueness of solutions of (1.1).
Theorem 3.1
If hypotheses\(\mathit{(H1)}\)–\(\mathit{(H3)}\)hold, then for every\(x_{0}\in \mathcal{K}\), there exist\(r< t_{0}\)and\(s>t_{0}\)such that (1.1) has a unique viable solution\(x:[r,s]\rightarrow \mathcal{K}\)over the interval\([r,s]\).
Proof
 (C1)If \(x_{0}\notin M\cup N\), then this implies that there exists a constant \(\alpha >0\) small enough such that \(\mathcal{B}_{ \alpha }(x_{0})\cap (\mathcal{M}\cup \mathcal{N})=\emptyset \) and \(\mathcal{B}_{\alpha }(x_{0})\subseteq \mathcal{K}\). Let \(M\triangleq \sup \{\f(x)\:x\in \mathcal{B}_{\alpha }(x_{0})\}\). Further, let \(\xi ,\eta >0\) be such that \(M\xi \leq \eta \leq \varepsilon \), and letwhere \(s\triangleq t_{0}+\xi \). It is easy to see that Ω is a convex closed set and bounded. Let \(G: C[t_{0},s]\rightarrow C[t_{0},s]\) be given by$$ \varOmega \triangleq \bigl\{ x(\cdot )\in C[t_{0},s] \Vert xx_{0} \Vert \leq \alpha , x(t _{0})=x_{0}, t \in [t_{0},s]\bigr\} , $$It follows that$$ (Gx) (t)\triangleq x_{0}+ \int _{t_{0}}^{t} f\bigl(x(\upsilon )\bigr)\,\mathrm {d}\upsilon , \quad t\in [t_{0},s]. $$(3.1)where \(t\in [t_{0},s]\), \(G(\varOmega )\) is bounded by (3.2). Furthermore, because f is continuous on \(\mathcal{K}\), it follows that, for every \(\varepsilon >0\), there exists \(\delta >0\) such that \(\sup_{t_{0}\leq t\leq s}\x(t)\bar{x}(t)\<\delta \), we have$$ \begin{aligned}[b] \bigl\Vert (Gx) (t)x_{0} \bigr\Vert &= \biggl\Vert \int _{t_{0}}^{t} f\bigl(x(\upsilon )\bigr)\,\mathrm {d}\upsilon \biggr\Vert \\ &\leq \int _{t_{0}}^{t} \bigl\Vert f\bigl(x(\upsilon ) \bigr) \bigr\Vert \,\mathrm {d}\upsilon \\ &\leq M \vert tt_{0} \vert \\ &\leq M\xi \\ &\leq \eta , \end{aligned} $$(3.2)Therefore, by the Schauder fixed point theorem 2.1, we know that \(x(t)=(Gx)(t)\) is a solution of (1.1) if and only if \(x(t)\) is a fixed point of G for all \(t\in [t_{0},s]\) (for more details, see [33]). On the other hand, according to the uniqueness theorem of nonlinear dynamical system (2.1), we obtain that system (1.1) has a unique solution \(x:[t_{0},s] \rightarrow \mathbb {R}^{n}\) over the interval \([t_{0},s]\).$$ \begin{aligned} \bigl\Vert (Gx) (t)(G\bar{x}) (t) \bigr\Vert &= \bigg\ \int _{t_{0}}^{t} [f\bigl(x(\upsilon )\bigr)f\bigl( \bar{x}(\upsilon )\bigr)\,\mathrm {d}\upsilon \bigg\ \\ &\leq \int _{t_{0}}^{t} \bigl\Vert f\bigl(x(\upsilon ) \bigr)f\bigl(\bar{x}(\upsilon )\bigr) \bigr\Vert \,\mathrm {d}\upsilon \\ &\leq \varepsilon (tt_{0}) \\ &\leq \varepsilon \xi . \end{aligned} $$
 (C2)
If \(x_{0}\in \mathcal{M}\), then for all \(t\geq t_{0}\), we have \(x_{0}^{+}=J(x_{0})\in \mathcal{N}\in \mathcal{K}\). It follows from hypothesis \(\mathit{(H3)}\) that there exists a constant \(\alpha >0\) such that \(\mathcal{B}_{\alpha }(x_{0}^{+})\cap \mathcal{M}\neq \emptyset \) and \(\mathcal{B}_{\alpha }(x_{0}^{+})\subseteq \mathcal{K}\). Hence, \(x(t)\) can be continued at the right. Similar to (C1), (1.1) has a unique solution \(x:[t_{0},s]\rightarrow \mathbb {R}^{n}\) over the interval \([t_{0},s]\). Let us consider \(t\leq t_{0}\) now. It is easy to see that \(x_{0}^{}=J^{1}(x_{0}^{+})\in \mathcal{M}\). Hence, there exists a constant \(\alpha >0\) such that \(B_{\alpha }(x_{0}^{+})\cap \mathcal{N}\neq \emptyset \) by hypothesis \(\mathit{(H3)}\), and \(x(t)\) can be proceeded at the left. This means that there exists a constant ξ such that \(r\triangleq t_{0}\xi \). Similar to (C1), (1.1) has a unique solution \(x:[r,t_{0}]\rightarrow \mathbb {R}^{n}\) over the interval \([r,t_{0}]\). Therefore, there exists a unique solution \(x(t)\) of (1.1) on an interval \((r,s)\).
 (C3)
If \(x_{0}\in \mathcal{N}\), similar to cases (C1) or (C2).
Remark 3.1
If hypotheses \(\mathit{(H1)}\)–\(\mathit{(H5)}\) hold, then every solution of (1.1) continuously depends on the initial value \(x_{0}\) [24, 31].
Next, we discuss the continuation of the solution for (1.1). The following theorems prove that every solution of (1.1) is a continuation to \(\mathbb {R}\).
Theorem 3.2
If hypotheses\(\mathit{(H4)}\), \(\mathit{(H6)}\), and\(\mathit{(H7)}\)hold, then every solution\(\pi _{x_{0}}(t)\), \(x_{0}\in \mathcal{K}\)of (2.1) is continuable to\(\mathbb {R}\).
Proof
 (i)
If \(\widetilde{\pi }_{x_{0}}(t)\) does not intersect the impulse set \(\mathcal{M}\), then the solution \(\widetilde{\pi }_{x_{0}}(t)\) of system (2.1) starting at the initial point \(x_{0}\in \mathcal{K}\) is free from the impulsive effects and remains in the set \(\mathcal{K}\) forever. It means that \(\widetilde{\pi }_{x_{0}}(t)\) is a nonlinear dynamical system (2.1). According to hypotheses \(\mathit{(H7)}\), the solutions of (1.1) on the maximal interval \([t_{0},t_{f})\) of existence are continuable to \(\mathbb {R}\).
 (ii)
If the solution of (2.1) intersects the impulse surface \(\mathcal{M}\) at the time \(t_{k}\) (i.e., \(\widetilde{\pi } _{x_{0}}(t_{k})\in \mathcal{M}\)) only finitely many times, where the impulse time sequence \(\{t_{k} \}\in \mathbb {R}\) satisfies \(\infty < t_{1}< t_{2}<\cdots <t_{k}<+\infty \). Denote by \(t_{\min }\) and \(t_{\max }\) the minimal and maximal elements of the sequence \({\tau _{i}}\), respectively. For \(t\geq t_{\max }\), the solution \(\widetilde{\pi }_{x_{0}}(t_{\max })\) of system (2.1) is subjected by impulsive effect to jump to \(x(t_{\max }^{+})=J(x(t_{ \max }^{}))\in \mathcal{N}\), and the solution \(\widetilde{\pi }_{x _{0}}(t)=\pi _{x(t_{\max }^{+})}(t)\) of system (2.1), where \(\pi _{x(t_{\max }^{+})}(t)\) is a solution of (2.1). By hypothesis \(\mathit{(H7)(a)}\), \([t_{0},t_{f})\) is continuable to \([t_{0},\infty )\) for \(t\leq t_{\min }\). Similarly, by hypothesis \(\mathit{(H7)(b)}\), \([t_{0},t_{f})\) is also continuable to \((\infty ,t_{0}]\).
 (iii)The solution \(\widetilde{\pi }_{x_{0}}(t)\) of system (1.1) intersects the impulse surface \(\mathcal{M}\) infinitely many times. It is clear that the existence of \(\tau _{\min }\) and \(\tau _{\max }\) has the following three cases:
 (a)The impulse time sequence \(\{t_{k} \}\) has a maximal element \(t_{\max }\in \mathbb {R}\), but \(t_{\min }\) does not exist. According to the proof of case (ii), we know \(\widetilde{\pi }_{x_{0}}(t)\) is continuable to +∞ as t increases. Consider t to be decreasing. Integrating both sides of the ordinary differential equation of system (2.1) that belongs to the interval \([t_{k},t_{k+1})\), we haveFrom \(\mathit{(H4)}\) and \(\mathit{(H6)}\), we denote \(Q\triangleq \sup_{\mathcal{K}}\f(x)\\) and \(\rho \triangleq \mathrm{d}( \mathcal{M},\mathcal{N})>0\). Thus, (3.3) implies that$$ \widetilde{\pi }_{x_{0}}\bigl(t_{k}^{+} \bigr)=\widetilde{\pi }_{x_{0}}\bigl(t_{k+1} ^{}\bigr)+ \int _{t_{k+1}}^{t_{k}}f\bigl(\widetilde{\pi }_{x_{0}}(\theta )\bigr)\,\mathrm {d}\theta . $$(3.3)Therefore,$$ \frac{\rho }{Q}\leq (t_{k+1}t_{k}). $$where \(k^{*}\) is fixed, \(k< k^{*}\), and \(k,k^{*}\) is the index of the impulse time sequence \(\{t_{k} \}\) and is fixed. From (3.4), this implies that \(t_{k}\rightarrow \infty \) as \(k\rightarrow \infty \). According to hypothesis \(\mathit{(H7)(b)}\), \([t_{0},t_{f})\) is continuable to \((\infty , t_{\max })\). Thus, \(\widetilde{\pi }_{x_{0}}(t)\) is continuable to −∞ as t decreases.$$ \frac{\rho }{Q}\bigl(kk^{*}\bigr)\geq t_{k}t_{k^{*}}, $$(3.4)
 (b)The sequence \(\{t_{k}\}\) has a minimal element \(t_{\min } \in \mathbb {R}\), but does not have a maximal one. Then by the arguments of (ii) \(x(t)\) is continuable to −∞. It follows now that we consider the continuation of \(x(t)\) with the increasing of time t. We haveSimilarly, we have$$ \widetilde{\pi }_{x_{0}}\bigl(t_{k+1}^{}\bigr)= \widetilde{\pi }_{x_{0}}\bigl(t_{k} ^{+}\bigr)+ \int _{t_{i}}^{t_{k+1}}f\bigl(\widetilde{\pi }_{x_{0}}(\theta )\bigr)\,\mathrm {d}\theta . $$or$$ \frac{\rho }{Q}\leq t_{k+1}\tau _{k}, $$where \(k^{*}\) is fixed, and \(k>k^{*}\). From (3.5), we get \(t_{k}\rightarrow +\infty \) as \(k\rightarrow +\infty \). According to hypothesis \(\mathit{(H7)(a)}\), \([t_{0},t_{f})\) is continuable to \([t_{\min },\infty )\).$$ \frac{\rho }{Q}\bigl(kk^{*}\bigr)\leq t_{k}t_{k^{*}}, $$(3.5)
 (c)
The sequence \(\{t_{k}\}\) has neither a minimal nor a maximal element. The proof of this case is similar to that of (a) and (b). We obtain that \(\widetilde{\pi }_{x_{0}}(t)\) is continuable to \(\mathbb {R}\).
 (a)
According to the above discussion, we obtain that every solution \(\widetilde{\pi }_{x_{0}}(t)\) of (1.1) is continuable to \(\mathbb {R}\). This completes the proof. □
The main results claim that every viable solution of (1.1) is continuable to +∞ and −∞. In other words, \(\mathbb {R}\) is a maximal interval of existence of each solution \(\widetilde{\pi }_{x_{0}}(t)\), \(x_{0}\in \mathcal{K}\) of (1.1). That is, \(\widetilde{\pi }_{x_{0}}(t)\in \mathcal{PC}(\mathbb {R})\).
4 Numerical examples
In this section, the validity of the results will be illustrated by two numerical examples.
Example 4.1
To further illustrate the significance of the study, we consider a specific biological model.
Example 4.2
It is easy to see that assumptions \(\mathit{(H1)}\)–\(\mathit{(H6)}\) are true. Let \(a_{1}=a_{2}=3\), \(\epsilon =0.7\), \(\rho _{1}=2\), \(\rho _{2}=1\). Figure 4(a) shows that the solution of system (4.4) starting from the initial value \((x_{10},x_{20})\in \mathcal{K}\) will leave the viability constraints \(\mathcal{K}\). Figure 4(b) shows that the solution of system (4.4) will eventually stay in the viability constraints \(\mathcal{K}\) and tend to a periodic solution.
5 Conclusion
The statedependent impulsive autonomous differential system (1.1) with boundary constraints has been considered in this paper. The main purpose is to investigate the existence and uniqueness of viable solutions of system (1.1). From Theorem 3.1, some sufficient conditions on the existence of viable solutions of system (1.1) are provided. Furthermore, we obtain sufficient conditions for the continuation of a viable solution of system (1.1) by Theorem 3.2. Finally, two examples are given to illustrate the existence and continuation of viable solutions of (1.1).
Notes
Acknowledgements
The authors are grateful to the anonymous referees for their careful reading of the manuscript and for their invaluable comments and suggestions, which largely helped to improve this paper.
Authors’ contributions
All authors read and approved the final manuscript.
Funding
This work is supported by the Natural Science Foundation of China (grant no: 61873213, 61503307, 61633011, 61702066, 11474233, 11747125), Chongqing Research Program of Basic Research and Frontier Technological Science Foundation (cstc2016jcyjA0261, cstc2017jcyjAX0256). It is also supported by Fundamental Research Funds for the Central Universities (no: XDJK2019B009) and China Postdoctoral Science Foundation. The research reported here was supported by the Natural Science Foundation Project of Chongqing CSTC (Grant no. cstc2018jcyjAX0810) and the Foundation of CQUE (18GZKP02, KY201702A). It was partially supported by Research Foundation of Key Laboratory of Machine Perception and Children’s Intelligence Development funded by CQUE (16xjpt07), China.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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