Ważewski type theorem for non-autonomous systems of equations with a disconnected set of egress points

Applied Mathematics and Computation 265 (2015) 358–369

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

˙ Wazewski type theorem for non-autonomous systems of equations with a disconnected set of egress points Grzegorz Gabor a, Sebastian Ruszkowski a, Jiˇrí Vítovec b,∗ a b

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toru´ n, Poland Brno University of Technology, CEITEC – Central European Institute of Technology, Technická 3058/10, CZ-616 00 Brno, Czech Republic

a r t i c l e

i n f o

MSC: 34N05 39A10

a b s t r a c t In this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form

Keywords: Time scale Dynamic system Non-autonomous system Difference equation Asymptotic behavior of solution Retract method

y (t ) = f (t, y(t )), where f : T × Rn → Rn , and T is a time scale. For a given set  ⊂ T × Rn , we formulate conditions for function f which guarantee that at least one solution y of the above system stays in . Unlike previous papers the set  is considered in more general form, i.e., the time section t is an arbitrary closed bounded set homeomorphic to the disk (for every t ∈ T) and the boundary ∂T  does not contain only egress points. Thanks to this, we can investigate a substantially wider range of equations with various types of bounded solutions. A relevant example is considered. The results are new also for non-autonomous systems of difference equations and the systems of impulsive differential equations. © 2015 Elsevier Inc. All rights reserved.

1. Introduction In this paper we investigate an asymptotic behavior of solutions of systems of n dynamic equations on time scales of the form

y (t ) = f (t, y(t )),

(1)

where f : T × Rn → Rn , and T is a time scale. We show that, for a given set  ⊂ T × Rn and a function f satisfying some conditions, there exists at least one solution y = y(t ) of system (1) such that (t, y(t)) ∈  for every t ∈ T. One of the main situations that may be represented in the above form is a non-autonomous system of difference equations

y(n ) = f (n, y(n )).

(2)

The other motivation to work in a time scale setting is that the impulsive problems with impulses in fixed moments can be easily rewritten as dynamic equations on time scales. Unlike the previous works devoted to n-dimensional discrete or dynamic systems, the present paper assumes that the boundary ∂T  of  does not have to be constituted only by points of (strict) egress and simultaneously its shape is more general then ∗

Corresponding author. Tel.: +420 541143134. E-mail addresses: [email protected] (G. Gabor), [email protected] (S. Ruszkowski), [email protected] (J. Vítovec).

http://dx.doi.org/10.1016/j.amc.2015.05.027 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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in previous papers. More precisely, in previous papers, see, e.g., [6–8,10], devoted to n-dimensional systems, one considered the time section t (set  with fixed t ∈ T) of a simple hyper-prism shape. In the present paper t (for every t ∈ T) is an arbitrary closed bounded set homeomorphic to the disk. ˙ ˙ Wazewski method (see [18]) is a simple but excellent topological principle, frequently called the Wazewski retract method which has been used by many authors to prove the existence of a bounded solution, resp. solution which remains in a prescribed domain. It generalizes the direct method of Lyapunov and is based on examining so-called “egress” and “strict egress” points on a boundary of the set of constraints. ˙ The Wazewski retract method was firstly adapted to differential equations and inclusions, see, e.g., [11,13,17] and references therein. Then, this method was extended on difference equations, see, e.g. [4–6]. Finally and recently, it was used for dynamic equations on time scales, see e.g., [1,7–10,12]. ˙ The main aim of this paper is to extend the Wazewski retract method to a wider class of problems on time scales with attention to have essentially new results for the important class of non-autonomous difference equations. The result given here generalizes the result in [12] of the first and second author, where the planar dynamic system on time scales is studied. Moreover, it generalizes the result in [8] of the third author and J. Diblík, where the system of n dynamic equations on time scales is studied with  of a simple hyper-prism shape. The geometrical features of  ⊂ T × Rn and its parts we use in our investigations become much more complicated in a higher dimensional space and need a deep geometrical study. In particular, the linking number is applied as a topological tool. The paper is organized as follows. In Section 2, we recall some information on dynamic equations on time scales and define the so-called “local -process” on time scales. In Section 3, we introduce the set (-tube)  with further concepts necessary in the sequel. In Section 4, the announced main result is formulated and proved. There is also a formulation of the result for a system of difference equations in the form of a corollary, and the second corollary for a system of impulsive differential equations. In Section 5, one example is given to illustrate the obtained results. Some remarks about the method used and further possibilities of extending the results derived in this paper are given in Section 6. 2. Preliminaries At the beginning, let us remind a notation for problems on time scales. The theory of time scales was introduced by Stefan Hilger in his Ph.D. thesis in 1988, see [15], in order to unify the continuous and discrete calculus. Nowadays it is well-known calculus and often studied in applications. Note that a time scale T is an arbitrary nonempty closed subset of reals and that [a, b]T := [a, b] ∩ T (resp. (a, b)T := (a, b) ∩ T, (a, b]T := (a, b] ∩ T or [a, b)T := [a, b) ∩ T) stands for an arbitrary finite time scale interval. Moreover, [a, ∞ )T := [a, ∞ ) ∩ T, or (a, ∞ )T := (a, ∞ ) ∩ T, denotes an infinite time scale interval. The symbols σ , ρ , μ, f σ and f  stand for the forward jump operator, backward jump operator, graininess, f ◦ σ and -derivative of f, respectively. Further, we will use the symbols Crd (T ) for a class of rd-continuous (i.e., right-dense continuous) functions defined on a time scale T. Finally, we will work with all types of points on time scale T, i.e., with right-dense points or right-scattered points and with left-dense points or left-scattered points. See [14], which is the initiating paper of the time scale theory, and [2,3] including a great deal of information on the time scale calculus. Throughout this paper, we will assume that the time scale T is unbounded from above, and t0 ∈ T. We use the standard symbol || · || for an arbitrary vector norm in Rn . Note that (in this paper) the type of a norm is not important. Further, note that A ⊆ B means that A is an arbitrary subset of B, while A ⊂ B means that A is strictly a proper subset of B. In the next definition, we recall further aspects needed later. See [3, Definition 8.14], where the following concepts are similarly defined. Definition 2.1. Let T be a time scale. A function f : T × Rn → Rn is called (i) rd-continuous, if g defined by g(t) := f(t, y(t)) is rd-continuous for any rd-continuous function y : T → Rn , (ii) bounded on a set M ⊆ T × Rn if there exists a constant K > 0 such that

|| f (t, y )|| ≤ K for every (t, y ) ∈ M, (iii) Lipschitz continuous in time t on a set S ⊆ Rn if there exists a constant Lt > 0 such that

|| f (t, y1 ) − f (t, y2 )|| ≤ Lt ||y1 − y2 || for every y1 , y2 ∈ S.

(3)

Definition 2.2. Let t0 ∈ T. By a local solution of an initial value problem (IVP for short) (1) and

y(t0 ) = y0 , we will mean a continuous -differentiable function y : (a, b)T → for all t ∈ (a, b)T .

(4) Rn

satisfying (4), where a < ρ (t0 ), b > σ (t0 ), and (1) is fulfilled

For the next study, it is important to know, whether the solution of IVP (1), (4) exists and is uniquely defined. The following theorem (in similar form) can be found in [3, Theorem 8.16]. Theorem 2.3 (Local existence and uniqueness). Let T be unbounded from above, t0 , t1 ∈ T with t1 > t0 , y0 ∈ Rn , and put

I1 = [t0 , t1 ]T and Yk = {y ∈ Rn : y − y0  ≤ k}

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with k > 0. Let, further, f ∈ C rd (I1 × Yk ) be bounded on a set I1 × Yk with K > 0 and Lipschitz continuous with Lt > 0 in every time t ∈ I1 on a set Yk . Denote

L := sup{Lt }. t∈I1

Then, the initial value problem (1), (4) has exactly one solution y on the interval [t0 , t0 + α ]T ⊆ [t0 , t1 ]T , where

  k 1−ε α = min t1 − t0 , , for some sufficiently small ε > 0. K

L

Proof. It is obvious that f is Lipschitz continuous on a set I1 × Yk with constant L ∈ R. Hence, theorem follows from [3, Theorem 8.16].  Definition 2.4. Let y1 and y2 be local solutions of IVP (1) and (4). A solution y1 is said to be an extension of a solution y2 , if Dom (y2 )⊂ Dom (y1 ) and y1 (t) ≡ y2 (t) for all t ∈ Dom (y2 ). If we cannot extend a local solution, then we call it a maximal solution. One can find the following theorem in a similar form in [3, Theorem 8.20]. Theorem 2.5 (Maximal existence and uniqueness). If for all t0 ∈ T and y0 ∈ Rn there exists a unique local solution of IVP (1) and (4), then for all t0 ∈ T and y0 ∈ Rn there exists a unique maximal (not extendable) solution y : (a, b)T → Rn of IVP (1) and (4), where μ(a ) = 0 or a = inf T, and b − ρ (b) = 0 or b = ∞. If additionally T = Z, then all the maximal solutions have domain Z. Now, we are ready to define a local -process on time scales, which is analogy of local process in a continuous case, see, e.g., [16,17]. The following two definitions are taken from [12]. Definition 2.6. Let M ⊆ Rn × T2 . A continuous mapping : M → Rn is a local -process if: (i) (∀ (t, y ) ∈ T × Rn ) (∃ α , β ∈ T; α < t < β ):

(μ(α ) = 0 ∨ α = inf T ) ∧ (β − ρ (β ) = 0 ∨ β = ∞ ) ∧ {s ∈ T; (y, t, s ) ∈ M} = (α , β )T , (ii) (∀ (t, y ) ∈ T × Rn ): (y, t, t ) = y, (iii) (∀ (y, t, s ), (y, t, r ) ∈ M ): ( (y, t, s ), s, r ) ∈ M ∧ ( (y, t, s ), s, r ) = (y, t, r ). Definition 2.7. We say that the system (1) generates a local -process , if for all y0 ∈ Rn and t0 ∈ T a function y(t ) = (y0 , t0 , · ) is a maximal and unique solution of IVP (1) and (4) and is a local -process. If T = Z then a local -process is additionally a process. The following proposition can be found in [12], and it says about a preservation (analogously as in R) of homeomorphisms along trajectories. Proposition 2.8. Let the system (1) generate a local -process and let all solutions of the problem



y (t ) = f (t, y(t )) y(t0 ) ∈ A

exist in time t1 . Then, (·, t0 , t1 )|A is a homeomorphism between A and its image. In further proposition, we give the sufficient conditions on a function f such that the system (1) generates a local -process

which preserves an orientation of Rn .

Proposition 2.9. Let a function f : T × Rn → Rn be rd-continuous and bounded on its domain. Further, let f be Lipschitz continuous in every time t ∈ T on Rn with a Lipschitz constant Lt satisfying μ(t)Lt < 1 for all t ∈ T. Then, the system (1) generates a local -process

and for all t ∈ T we have that (·, t, σ (t)) preserves an orientation of Rn . Proof. The system (1) has a global and unique solution (it follows from Theorems 2.3 and 2.5) with a continuous dependence on the initial conditions. Hence, (1) generates a local -process . Further, thanks to (y, t, σ (t )) = yσ = y + μ(t ) f (t, y ),

(y, t, σ (t )) − (0, t, σ (t )) = y + μ(t ) f (t, y ) − (0 + μ(t ) f (t, 0 )) = y + μ(t )( f (t, y ) − f (t, 0 )). Then, in view of (3) and μ(t)Lt < 1, for y = 0, we have that the scalar product

 (y, t, σ (t )) − (0, t, σ (t )), y − 0 = y + μ(t )( f (t, y ) − f (t, 0 )), y = y2 + μ(t ) f (t, y ) − f (t, 0 ), y ≥ y2 − μ(t ) f (t, y ) − f (t, 0 )y ≥ y2 − μ(t )Lt yy = y2 (1 − μ(t )Lt ) > 0, which means that the angle of vectors (y, t, σ (t )) − (0, t, σ (t )) and y is less than π /2. Hence, they are in the same half-space (determined by the vector y and hyperplane which is perpendicular to y). Thus, (·, t, σ (t)) preserves an orientation of Rn . 

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3. -tube  and related concepts Let Bn (y, r) be an open ball in the space Rn with a center y ∈ Rn and radius r. Further, let Dn (y, r) := cl (Bn (y, r)) be a disk (the closure of Bn (y, r)) and Sn (y, r ) := ∂ (Bn+1 (y, r )) be a sphere (the boundary of Bn+1 (y, r )). Finally, by Dn := Dn (0, 1) and Sn := Sn (0, 1) we mean the unit n-dimensional disk and unit n-dimensional sphere centered in the origin. Definition 3.1. Let A ⊂ T × Rn . Then we define a time section At for arbitrary fixed t ∈ T as

At := {y ∈ Rn ; (t, y ) ∈ A}. Now we are ready to define a -tube . Definition 3.2. Let  : T × Dn → Rn be a continuous mapping such that t : Dn → t (Dn ) ⊂ Rn , where (for fixed t ∈ T) the image t (Dn ) := (t, Dn ) is a closed bounded set in the space Rn homeomorphic to the disk Dn . Then we define the -tube  as a set

 :=



({t } × t (Dn )).

t∈T

Further, we define a y-boundary ∂T  of  as

∂T  :=



({t } × ∂ (t )).

t∈T

Remark 3.3. Definition 3.2 of the -tube  seems to be complicated at first sight. However, this definition assures that the y-boundary ∂T  is continuous for every t ∈ T and every time section t is a closed bounded set homeomorphic to Dn . Finally note that  (in a special case) can be a constant -tube  = T × 0 , where 0 is homeomorphic to Dn . The following points and subset of  play an important role in the formulation of the main result. Definition 3.4. (Egress point, set E of egress points). Let the system (1) generate a local -process . A point (t, y ) ∈ ∂T  is said to be an egress point for the -tube  with respect to the system (1), if



(∀ s ∈ (t, ∞ )T ) :





(τ , (y, t, τ )) ⊂ .

τ ∈(t,s]T

Moreover, E will denote the set of all egress points for the -tube  with respect to the system (1). In view of the previous definition, the geometrical meaning of the egress point is the following. If (t ∗ , y(t ∗ )) ∈ ∂T  is the egress point, then for the solution y = y(t ) of (1) the point (t ∗ , y(t ∗ )) ∈ ∂T  is locally the last in . More precisely, if t∗ is rightscattered, (σ (t ∗ ), y(σ (t ∗ ))) ∈ . Further, if t∗ is right-dense, then for every s > t∗ there exists t ∈ (t ∗ , s )T such that (t, y(t )) ∈ . 4. Main result In this section we prove the main result of this paper. Theorem 4.1. Let  ⊂ T × Rn be a -tube and let (1) generates a local -process such that (·, t, σ (t)) preserves an orientation of Rn for all t ∈ T. Further, let there exist closed sets W, W 1 , W 2 ⊂ Sn−1 such that (H1) W1 and W2 are homeomorphic to Dn−1 and (H1a)

W 1 ∩ W 2 = ∅,

W 1 ∩ W = ∅

and W 2 ∩ W = ∅,

(H1b) for all right-scattered t ∈ T

t (W ) ⊂ Et ⊂ t (W 1 ∪ W 2 ), (H1c) for all right-dense t ∈ T

t (W ) = Et ⊂ t (W 1 ∪ W 2 ), (H2) for all left-scattered t ∈ T there is a connection between all connected components of t (W) inside (ρ (t) , ρ (t), t) ∩ t , i.e., there exists a connected set Xt ⊂ Rn satisfying

Xt ⊆ (ρ (t ) , ρ (t ), t ) ∩ t and Xt ∪ t (W ) is connected, (H3) for all left-scattered t ∈ T

(ρ (t ) , ρ (t ), t ) ∩ ∂T t ⊂ t (W ). Then, for all t0 ∈ T, there exists a point y0 ∈ t0 such that the solution y of IVP (1) and (4) remains in -tube  for every t ∈ [t0 , ∞ )T .

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Fig. 1. Geometrical meaning of conditions H1, H2 and H3.

Before the proof let us add some comments. First, Eq. (1) generates a local -process with the preservation of an orientation of Rn , e.g., if the function f satisfies all assumptions of Proposition 2.9. Second, let us explain the conditions (H1)–(H3). The assumption (H1) states that a time section Et of the egress set E is not connected for all t ∈ T and we can find two sets W1 and W2 whose images t (W1 ) and t (W2 ) are separated and cover the set Et . Moreover, the set W is disconnected with the property W ⊆ W1 ∪ W2 and can be understood as a closed set whose image t (W) almost covers the set Et (for μ(t ) = 0 this image has to be equal Et ). The assumption (H2) means that for t = ρ (t), there is a connection between connected components of t (W) inside t , consisted of points that were in ρ (t) before the jump from ρ (t) to t. Thanks to the assumption (H3), “we can jump” from the inside of ρ (t) to the boundary of t only through the image t (W). The geometrical meaning of these conditions is shown in Fig. 1. Third, the idea of the proof is the following. On the contrary, we suppose that every solution y(t) satisfying IVP (1) and (4) leaves the -tube  in a finite time in [t0 , ∞ )T . Then, we will be able to prove an existence of a continuous mapping from a disk Dn onto a disconnected set containing W. Hence, we will get a contradiction, because this mapping cannot be continuous. Without any special comment, we use the fact that the initial value problem has a unique solution and this solution depends continuously on its initial data (this is guaranteed by the assumption that (1) generates a local -process ). Proof. (of Theorem 4.1) Let us fix a time t0 ∈ T. Assume that every solution y satisfying (1) and (4) leaves the -tube . Before the main part of the proof we need some preparations. First, by assumption (H2), we define uniquely a set X˜t for each left-scattered t ∈ T as the connected component of a set

(ρ (t) , ρ (t), t) ∩ t which contains Xt . For a left-dense t ∈ T, we define X˜t := t . Further, for every x ∈ t0 let us define a leaving time1 tx of a solution y(t) of (1) with an initial condition y(t0 ) = x as

tx := inf{t ∈ (t0 , ∞ )T ; (x, t0 , t ) ∈ t if μ(t ) = 0 and (x, t0 , σ (t )) ∈ X˜σ (t ) if μ(t ) > 0}. It is obvious that

tx ≤ sup{t ∈ T; (∀ s ∈ [t0 , t]T ) : (t, (x, t0 , t )) ∈ } < ∞. Next, let I be the set of indices such that for each i ∈ I and fixed t ∈ T, Yi (t) is a connected component of the set



Y (t ) := cl (ρ (t ) , ρ (t ), t ) \ X˜t



and



Yi (t ) = Y (t ).

i∈I

Notice that for each i ∈ I we have that Yi (t ) ∩ X˜t ⊂ ∂T t . Due to (H3) and (H1), we get for each i ∈ I

Yi (t ) ∩ X˜t ⊂ t (W ) ⊂ t (W 1 ∪ W 2 ). For better understanding, see Fig. 2, where the whole situation is demonstrated. 1 It does not mean that the time tx indicates the last moment of the solution in . There is a possibility to have a solution starting in (t0 , x) that stays in  for each time with a property tx < ∞.

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Fig. 2. Presentation of the sets Yi .

Finally, let us suppose that for some i ∈ I

Yi (t ) ∩ X˜t ∩ t (W k ) = ∅

for k = 1, 2.

Then, there exist points P1 ∈ t (W1 ), P2 ∈ t (W2 ) and curves  1 ⊂ Yi (t), 2 ⊂ X˜t such that P1 and P2 are connected by  1 and  2 through Yi (t) and X˜t , respectively. Furthermore, based on assumption (H1), there exists a set 0 ⊂ ∂T t \ t (W1 ∪ W2 ) homeomorphic to a sphere Sn−2 such that it separates images of W1 and W2 . It is easy to see that the linking number (see [19], p. 509) of  0 and  1 ∪  2 is 1, which means that  1 ∪  2 cannot be contracted into a point without passing through  0 . However, due to the definition of  0 it means that  1 ∪  2 cannot be contracted into a point inside of (ρ (t) , ρ (t), t) which is a contradiction with (ρ (t) , ρ (t), t) being homeomorphic to a disk. Therefore, Yi (t ) ∩ X˜t ∩ t (W k ) = ∅ for only one of k = 1 or k = 2 for each i ∈ I. Now we are ready to construct the announced continuous mapping from Dn to W1 ∪ W2 . First, (for each fixed t ∈ T) we will define a mapping ht : Yi (t) → Wk for each index i ∈ I and fixed k depending on i. For this we need an auxiliary mapping ht∗ ,



ht∗ (x ) :=

−1 if x ∈ X˜σ (t ) ∩ Yi (σ (t )), σ (t ) (x ), t−1 ( (x, σ (t ), t )), if x ∈ (t (W ), t, σ (t )) ∩ Yi (σ (t )).

If we have x ∈ (t (W), t, σ (t))2 , then by assumption (H1) and definition of E we know that x is outside of σ (t) . Hence, in particular, x ∈ X˜σ (t ) . Therefore, ht∗ is well defined. It is easy to see that ht∗ is continuous. The sets W1 , W2 are homeomorphic to disks, thus there exists continuous extension ht of ht∗ to the whole domain Yi (t). Next, based on the previous step, we define a mapping r : t0 → W 1 ∪ W 2 of variable x ∈ t0 . If μ(tx ) = 0, then (tx , (x, t0 , tx )) is already in E so we can define

x −→ t−1 ( (x, t0 , tx )) x

for

μ(tx ) = 0.

If μ(tx ) > 0, then (tx , (x, t0 , tx )) ∈  and (x, t0 , σ (tx )) ∈ X˜σ (tx ) . So we can use the mapping htx for a point (x, t, σ (tx )). Therefore, we can correctly define for every x ∈ t0

 t−1 ( (x, t0 , tx )), x r (x ) := htx ( (x, t0 , σ (tx ))),

if if

μ(tx ) = 0, μ(tx ) > 0.

Let us fix x ∈ t0 . We will prove that r is continuous in x. First case. Let tx be dense. We will show a continuity of the mapping y → ty in x. We start with the lower semicontinuity. For each ε > 0 there exists τ ∈ T such that 0 < τ − tx < ε and (x, t0 , τ ) ∈ τ . Thus, by the continuity of , for each ε > 0 there exist δ > 0 and τ ∈ T such that 0 < τ − tx < ε and (Bn (x, δ ), t0 , τ ) ∩ τ = ∅. Therefore,

(∀ ε > 0 ) (∃ δ > 0 ) : (∀ y ∈ Bn (x, δ )) : ty − tx < ε . Now, the upper semicontinuity in tx . Suppose that tx is not upper semicontinuous in x, so

(∃ ε > 0 ) (∀ δ > 0 ) : (∃ y ∈ Bn (x, δ )) : tx − ty ≥ ε . By compactness of [t0 , tx ]T we have a sequence of points {yk } with a property

(yk , t0 , tyk ) → (x, t0 , tˆ),

where tyk → tˆ < tx if k → ∞.

Hence, (x, t0 , tˆ) ∈ cl (Etˆ ). If μ(tˆ) = 0, then, by assumption (H1c), we have that Etˆ is closed so we have

(x, t0 , tˆ) ∈ Etˆ which is a contradiction with tˆ < tx . If μ(tˆ) > 0, then, by assumption (H3), we have

(x, t0 , σ (tˆ)) ∈ σ (tˆ) (W ) ⊂ Eσ (tˆ) which is a contradiction with σ (tˆ) < tx . 2

There may exist some indices i ∈ I such that the sets Yi (σ (t)) do not contain any x with x ∈ (t (W), t, σ (t)) (see Y4 ) in Fig. 2).

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Together, we get that

(∀ ε > 0 ) (∃ δ > 0 ) : (∀ y ∈ Bn (x, δ )) : |ty − tx | < ε . Hence, for ε → 0, we have ty → tx and thus

lim r (y ) = r (x ). y→x

Second case. Let tx be left-scattered and right-dense. We know that ht∗ is continuous, hence ht is continuous as well. Then, by the definition of the mapping r for y ∈ Bn (x, δ ) with ty ≥ tx , we get (by the same reasoning as previously)

lim r (y ) = r (x ).

y→x ty ≥tx

So it is enough to prove

lim r (y ) = r (x ). y→x

σ (ty )=tx

Notice that μ(ty ) > 0, so r (y ) = hty ( (y, t0 , σ (ty ))). Next, due to the continuity of ht as well as and using the fact that

(x, t0 , tx ) ∈ X˜tx ∩ Yi (tx ), we get

lim r (y ) = lim hty ( (y, t0 , σ (ty ))) = hρ (tx ) ( (x, t0 , tx )) = t−1 ( (x, t0 , tx )) = r (x ). x y→x

σ (ty )=tx

y→x

σ (ty )=tx

Third case. Let tx be left-dense and right-scattered. If (x, t0 , tx ) ∈ int(tx ), then there exists a neighbourhood of x consisted of points with the same leaving time as x. Thus, from the continuity of the mapping htx we get a continuity of r. Now, consider (x, t0 , tx ) ∈ tx (W ). Then, by (H1b),

(Bn (x, δ ), t0 , σ (tx )) ∩ σ (tx ) = ∅ for some δ > 0. Hence, for y ∈ Bn (x, δ ) with ty < tx we can use an analogical reasoning as in the first case, and for y ∈ Bn (x, δ ) with ty = tx we get needed property from a continuity of the mapping htx . This leads us to the continuity of r in x. Finally, consider

(x, t0 , tx ) ∈ tx (Sn−1 \ W ).

(5)

We will show that it is possible only if x is on the boundary of t0 . If x ∈ int(t0 ) then there exists the first time t∗ in which the solution starting from (t0 , x) is in ∂T t ∗ . If ρ (t∗ ) < t∗ then (x, t0 , t ∗ ) ∈ t ∗ (W ) which would imply that t ∗ = tx but, by (5), (x, t0 , tx ) ∈ tx (W ). If ρ (t ∗ ) = t ∗ (by properties of a -process) in each neighbourhood of x there exists a point x∗ with

(x∗ , t0 , t ∗ ) ∈ t ∗ , which implies that there exist times s(x∗ )t∗ with (s(x∗ ), (x∗ , t0 , s(x∗ ))) ∈ E. Thus

(t ∗ , (x, t0 , t ∗ )) = lim ( s ( x∗ ) , ∗ x →x

(x∗ , t0 , s(x∗ ))) ∈ cl E.

(6)

If t∗ is right-dense, then cl Et ∗ = Et ∗ = t ∗ (W ). The first equality implies that t∗ is an exit time, so t ∗ = tx , while the second one gives (x, t0 , tx ) ∈ tx (W ); a contradiction. Suppose that t∗ is right-scattered and t∗ < tx . Then, by (H3) and (H1b),

(x, t0 , σ (t ∗ )) ∈ σ (t ∗ ) (W ) ⊂ Eσ (t ∗ ) , so σ (t ∗ ) = tx . Then (x, t0 , tx ) ∈ tx (W ); a contradiction. At last, suppose that t ∗ = tx so, in particular, it is right scattered. By (6),

(x, t0 , tx ) = lim

(x∗ , t0 , s(x∗ )) ∗ x →x

with

(x∗ , t0 , s(x∗ ))

∈ Es(x∗ ) . We prove that

sup d (z, tx (W )) → 0, for every sequence t j  tx .

z∈Et j

Indeed, assuming that this is not true we could find ε > 0 and the sequences tj tx and y j ∈ Et j with d (y j , tx (W )) ≥ ε . Since t j (W ) ⊂ Et j , by the continuity of  we can assume, without any loss of generality, that y j ∈ ∂ Et j . Moreover, by the local compactness of the tube, we can assume that yj → y for some y ∈ Rn . Every subsequence of (tj ) contains a subsequence consisted of right-dense points or right-scattered points. If tj are right-dense (we use the same indices for simplicity), then y j ∈ t j (W ), and we obtain y ∈ tx (W ), since the graph of φ : t → t (W) is closed. If tj are right-scattered, then, by (H3),

(y j , t j , σ (t j )) ⊂ σ (t j ) (W ). But is continuous, so (yj , tj , σ (tj )) → y. We use the closeness of the graph of φ to get y ∈ tx (W ). Now, by (6), d ( (x∗ , t0 , s(x∗ )), tx (W )) → 0 and, consequently, (x, t0 , tx ) ∈ tx (W ). This again leads us to a contradiction with (5). We conclude that x ∈ ∂T t0 . If we perform the same reasoning for points of this trajectory in any other starting time t1 ∈ [t0 , tx ] we would get (x, t0 , t1 ) ∈ ∂T t1 , which means that the trajectory starting from x stays on ∂T  until it leaves it in the time tx .

G. Gabor et al. / Applied Mathematics and Computation 265 (2015) 358–369

365

Now, if [t0 , tx ] ⊂ T, by the assumption (H3) we get the existence of a time t ≤ tx such that (x, t0 , t) ∈ t (W), so the inclusion (5) cannot happen. Hence, [t0 , tx ] ⊂ T. This implies that if in any neighbourhood of x we were able to find a point x∗ ∈ t0 with tx∗ < tx then (x∗ , t0 , tx∗ ) ∈ tx∗ (W ) and, by a compactness of the interval [t0 , tx ], we would be able to find a sequence (x∗k )∞ with x∗k → x and tx∗ → t ∗ , so (x∗k , t0 , tx∗ ) → (x, t0 , t ∗ ) where (x∗k , t0 , tx∗ ) ∈ tx∗ (W ). By the closeness of W we have k=1 k

k

k

k

(x, t0 , t ∗ ) ∈ t ∗ (W ) which contradicts (5). Hence there exists δ 1 > 0 such that for all y ∈ B(x, δ1 ) ∩ t0 we have ty ≥ tx . However, X˜tx is closed and (x, t0 , σ (tx )) ∈ X˜tx , so there exists δ < δ 1 such that (y, t0 , σ (tx )) ∈ X˜tx for all y ∈ B(x, δ ) which means that

tx = ty . Now from the continuity of htx we get the needed property. Fourth case. Let tx be an isolated point. This part can be done similarly as in the second and third case. Hence r is continuous for all points in t0 . Finally, we define a mapping R := r ◦ t0 : Dn → W 1 ∪ W 2 , that is,

R(y ) = r (t0 (y )). By the continuity of r and t0 , we get that R is continuous. R maps Dn into a disconnected set W1 ∪ W2 and the image of R contains W. This implies (by the assumption (H1a)) that R is the mapping from a connected set onto a disconnected set. Therefore, this mapping cannot be continuous. Hence, our assumption is false and there exists y0 ∈ t0 such that the solution y(t) of IVP (1) and (4) remains in -tube  for every t ∈ [t0 , ∞ )T .  As an immediate application we can write a result for the case T = Z so for the system of difference equations. For this special case T = Z, let us define -tube easily as  := k∈Z ({k} × k ), where k is a closed bounded set homeomorphic to the disk Dn . Corollary 4.2. Let  ⊂ Z × Rn be a -tube and let (2) generate a -process such that (·, k, k + 1 ) preserves an orientation of Rn for all k ∈ N. Further, let there exist closed sets Wk1 , Wk2 , Wk ⊂ ∂ k for all k ∈ N such that (H1) for all k ∈ N, Wk1 and Wk2 are homeomorphic to Dn−1 and (H1a)

Wk1 ∩ Wk2 = ∅,

Wk1 ∩ Wk = ∅

and Wk2 ∩ Wk = ∅,

(H1b)

Wk ⊂ Ek ⊂ Wk1 ∪ Wk2 , (H2) for all k ∈ N there is a connection between all connected components of Wk inside (k−1 , k − 1, k ) ∩ k , i.e., there exists a connected set Xk ⊂ Rn satisfying

Xk ⊆ (k−1 , k − 1, k ) ∩ k

and Xk ∪ Wk is connected,

(H3)

(k−1 , k − 1, k ) ∩ ∂ k ⊂ Wk . Then there exists a point y0 ∈ 0 such that the solution y of IVP (2) and y(0 ) = y0 remains in the -tube  for every k ∈ N. One can easily modify the formulation to obtain the result for a constant tube Z × 0 . We can also interpret a system of differential equations with impulses in fixed times as a system on a suitable time scale. without accumulation points. Let  ⊂ [t0 , t∞ ) × Rn be a union of k Corollary 4.3. Let us fix an increasing sequence (tk )∞ k=0 k  tubes  on intervals [tk , tk+1 ] for k = 0, . . . , ∞ and let x (t ) = f (t, x(t )) generate a local process . Moreover, let the mappings Ik be homeomorphisms of Rn such that they preserve an orientation of Rn for all k = 1, . . . , ∞. Assume that there exist closed sets W, W 1 , W 2 ⊂ Sn−1 such that (H1) W1 and W2 are homeomorphic to Dn−1 and (H1a)

W 1 ∩ W 2 = ∅,

W 1 ∩ W = ∅

and

W 2 ∩ W = ∅,

(H1b) for all k = 1, . . . , ∞

  tk−1 (W ) ⊂ Ik−1 tkk ∩ ∂ tk−1 ⊂ tk−1 (W 1 ∪ W 2 ), k k k

(H1c) for all k ≥ 1 and t ∈ (tk , tk+1 )

tk (W ) = Et ⊂ tk (W 1 ∪ W 2 ), (H2) for all k = 1, . . . , ∞ there is a connection between all connected components of tk (W ) inside Ik (tk−1 ) ∩ tk , i.e., there exists k k k a connected set Xk ⊂ Rn satisfying





Xk ⊆ Ik tk−1 ∩ tkk k

and Xk ∪ tkk (W ) is connected,

(H3) for all k = 1, . . . , ∞

Ik (tk−1 ) ∩ ∂ tkk ⊂ tkk (W ). k

366

G. Gabor et al. / Applied Mathematics and Computation 265 (2015) 358–369

Then there exists a point x0 ∈ t0 such that the solution x of

⎧  ∞ ⎨x (t ) = f (t, x(t )) for t ∈ [t0 , ∞ ) \ (tk )k=1 + x(t ) = Ik (x(tk )) for k = 1, . . . , ∞ ⎩ k x(t0 ) = x0

remains in the set  for every t ∈ [t0 , ∞). Proof. If we stretch each time of jump to interval of length one we get a time scale

T=

∞ 

[tk + k, tk+1 + k].

k=0

Define

ˆ := cl 





 {t + k} × 

k t

,

k∈N t∈[tk ,tk+1 ] k (t ) := t−k (t ) for t ∈ [tk + k, tk+1 + k],

and fˆ : T → Rn



fˆ(t, x(t )) = f (t − k, x(t − k ))

for t ∈ [tk + k, tk+1 + k] \ (tk )∞ k=1

fˆ(tk , x(tk )) = Ik (x(tk )) − x(tk ) for k = 1, . . . , ∞. Then the system of -differential equations



x (t ) = fˆ(t, x(t )) x(t0 ) = x0

ˆ and from that we conclude the claim of the generates a local -process satisfying assumptions of Theorem 4.1 for the -tube  corollary.  5. Example Let 0 be a solid of revolution defined by

  π 0 := (x, y, z ) ∈ R3 : x2 + y2 + |z| ≤ π and |z| ≤ . 2

Further, let us consider a three-dimensional dynamic system of type (1) in the form

x (t ) = f1 (t, x(t ), y(t ), z(t )) := x(t ) sin(z(t )) + sin( y (t ) = f2 (t, x(t ), y(t ), z(t )) := y(t ) sin(z(t )) + 

z (t ) = f3 (t, x(t ), y(t ), z(t )) := e

−t 2





x2 (t ) + y2 (t ) + |z(t )| ),

x2 (t ) + y2 (t ) + |z(t )| − π ,

− z(t ) sin(z(t ))

defined on an arbitrary time scale T with μ(t) ≤ 1/8 for each t ∈ T. Using Theorem 4.1, we will show that for every t0 ∈ T there exists at least one solution

u∗ (t ) = (x∗ (t ), y∗ (t ), z∗ (t )) with u∗ (t0 ) ∈ 0 of system (7) which stays in a constant -tube  = T × 0 for every t ∈ [t0 , ∞ )T . At first, we consider the following auxiliary system:

x (t ) = fˆ1 (t, x(t ), y(t ), z(t )) := f1 (t, xˆ(t ), yˆ(t ), zˆ(t )), y (t ) = fˆ2 (t, x(t ), y(t ), z(t )) := f2 (t, xˆ(t ), yˆ(t ), zˆ(t )), z (t ) = fˆ3 (t, x(t ), y(t ), z(t )) := f3 (t, xˆ(t ), yˆ(t ), zˆ(t )), where

zˆ =

⎧ ⎨π /2 z

⎩

z > π /2,

|z| ≤ π /2,

−π /2 z < π /2

(7)

G. Gabor et al. / Applied Mathematics and Computation 265 (2015) 358–369

and

 ˆ yˆ ) = (x,

(x, y )

367

(x, y, zˆ) ∈ ,

(x,y ) (x,y ) (π

− zˆ )

(x, y, zˆ) ∈ .

This system is equal to system (7) on the set  in the sense that all solutions of (7) that stay in  are the same as solutions of the auxiliary system that stay in . Moreover, we have

 fˆ(t, x1 , y1 , z1 ) − fˆ(t, x2 , y2 , z2 )2   



+ yˆ1 sin zˆ1 − yˆ2 sin zˆ2 +





xˆ21 + yˆ21 + |zˆ1 | − sin

= xˆ1 sin zˆ1 − xˆ2 sin zˆ2 + sin

xˆ21 + yˆ21 −





xˆ21 + yˆ21 +



+ xˆ21 + yˆ21 −









 2

xˆ22 + yˆ22 + |zˆ1 | + |zˆ2 | /2

2

xˆ22 + yˆ22 + |zˆ1 | − |zˆ2 |

≤ (|xˆ1 | + 1 )|z1 − z2 | + |zˆ2 ||x1 − x2 | +

2

xˆ21 + yˆ21 −

+ [ˆz2 sin zˆ2 − zˆ1 sin zˆ1 ]2





 

xˆ22 + yˆ22 + |zˆ1 | − |zˆ2 | /2

+ yˆ1 (sin zˆ1 − sin zˆ2 ) + (yˆ1 − yˆ2 ) sin zˆ2



+ zˆ1 (sin zˆ1 − sin zˆ2 ) + (zˆ1 − zˆ2 ) sin zˆ2





2

xˆ22 + yˆ22 + |zˆ2 |

xˆ22 + yˆ22 + |zˆ1 | − |zˆ2 |

= xˆ1 (sin zˆ1 − sin zˆ2 ) + (xˆ1 − xˆ2 ) sin zˆ2 + 2 sin · cos



( x 1 − x2 ) 2 + ( y 1 − y 2 ) 2

2

2

2 (x1 − x2 )2 + (y1 − y2 )2 + [(|zˆ1 | + 1 )|z1 − z2 |]2   ≤ 3(|xˆ1 | + 1 )2 (z1 − z2 )2 + 3ˆz22 (x1 − x2 )2 + 3 (x1 − x2 )2 + (y1 − y2 )2 2    + 3(|yˆ1 | + 1 )2 (z1 − z2 )2 + 3ˆz22 (y1 − y2 )2 + 3 (x1 − x2 )2 + (y1 − y2 )2 + (|zˆ1 | + 1 )|z1 − z2 |     ≤ (3ˆz22 + 6 ) (x1 − x2 )2 + (y1 − y2 )2 + 3(xˆ21 + yˆ21 ) + zˆ12 + 6(|xˆ1 | + |yˆ1 | ) + 2|zˆ1 | + 7 (z1 − z2 )2 √   ≤ (3(π /2 )2 + 6 ) (x1 − x2 )2 + (y1 − y2 )2 + (3π 2 + 6 2π + 7 )(z1 − z2 )2 √   ≤ (3π 2 + 6 2π + 7 ) (x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 , + (|yˆ1 | + 1 )|z1 − z2 | + |zˆ2 ||y1 − y2 | +



which means that fˆ is a Lipschitz continuous function with the Lipschitz constant

L=



√ 3π 2 + 6 2π + 7 < 8,

hence μ(t)L < 1. It is easy to see that fˆ is rd-continuous for every t ∈ T, therefore by Proposition 2.9 fˆ = ( fˆ1 , fˆ2 , fˆ3 ) generates a local -process with the preservation of orientation of R3 for all t ∈ T. Note that the function f itself does not fulfil assumptions of Proposition 2.9. Now we try to check if the condition (H1) from Theorem 4.1 is fulfilled. We consider a constant tube T × 0 , where the set 0 is homeomorphic to a ball, so to simplify the notation we will use the identity mapping t , defined for every (x, y, z) ∈ 0 as t (x, y, z ) = (x, y, z ). To find a set W which would satisfy (H1c) we will find the set of egress points on boundary ∂T  for right-dense times t. We will use the fact that for an egress point P = (t, x, y, z ) the scalar product of vectors (f1 (t, x, y, z), f2 (t, x, y, z), f3 (t, x, y, z)) and (u, v, w), which is the vector perpendicular to ∂T t in point P oriented outside of ∂T t , is grater then zero. For “a top of the set 0 ”, i.e., for the set



(x, y, z ) : x2 + y2 <

Tc = we get



π2 4

∧ z=

π



2

  π π π  2 π π π  π 2 2 + x + y2 − , e−t − sin < 0. (0, 0, 1 ), x sin + sin , y sin = e−t − x 2 + y2 + 2

2

2

2

2

Thus, Tc ∩ Et = ∅. For “an upper lateral surface of the set 0 ”, i.e., for the set



Uc = (x, y, z ) : 0 ≤ z ≤ we get

 x, y,



π 2

x 2 + y2 + z = π





x2 + y2 , (x sin z + sin π , y sin z + π − π , e−t − z sin z )

= x2 sin z + y2 sin z + e−t Thus, Uc ⊂ Et .



∧

2

2







x2 + y2 − z sin z x2 + y2 ≥ e−t

2



x2 + y2 > 0.

2

2

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G. Gabor et al. / Applied Mathematics and Computation 265 (2015) 358–369

Fig. 3. Set 0 and its parts.

For “a lower lateral surface of the set 0 ”, i.e., for the set



Lc = (x, y, z ) : − we get





π 2




x2 + y2 − z = π



x, y, − x2 + y2 , (x sin z + sin π , y sin z + π − π , e−t − z sin z ) 2

= x2 sin z + y2 sin z − e−t

2







Bc =



(x, y, z ) : x2 + y2 ≤

π2 4



x2 + y2 + z sin z x2 + y2 ≤ −e−t

Thus, Lc ∩ Et = ∅. For “a bottom of the set 0 ”, i.e., for the set

we get



∧ z=−

π

2



x2 + y2 < 0.



2

   π π π  π 2 2 > 0. (0, 0, −1 ), −x + sin x 2 + y2 + , −y + x2 + y2 − , e−t − = −e−t + 2

2

2

2

Thus, Bc ⊂ Et . The whole situation one can see in Fig. 3. Based on that let us define:

W = Uc ∪ Bc , W 1 = Uc ∪ Tc ∪ {(x, y, z ) ∈ Lc : −1 ≤ z ≤ 0},   11 π . W 2 = Bc ∪ (x, y, z ) ∈ Lc : − ≤ z ≤ − 2 10 The set  is convex so for right-scattered times t we additionally need to check if Et ⊂ W1 ∪ W2 . Notice that 1 · sin (1) > 8/10, thus for z(t ) < −1 we have





z(σ (t )) = z(t ) + μ(t ) e−t − z(t ) sin(z(t )) ≤ z(t ) + 2



1 8 1− 8 10



< 0.

Furthermore, (11/10) · sin (11/10) < 1, thus for −11/10 < z(t ) < 0 we have





z(σ (t )) = z(t ) + μ(t ) e−t − z(t ) sin(z(t )) ≥ z(t ) − 2

1 1 11 π z(t ) sin(z(t )) ≥ − − >− . 8 10 8 2

For (x(t), y(t), z(t)) from the set Lc the vector (x (t), y (t)) points directly at the point (0, 0), which corresponds to the axis of symmetry of the set 0 , and μ(t)(x , y )(t) ≤ π /8 < π /2. Therefore there cannot be egress points outside of W1 and W2 , which leads to a conclusion that sets W, W1 and W2 satisfy (H1). Moreover, looking at directions of the flow on the boundary (we computed them using scalar products) and above estimations, we get also that

(W 1 , ρ (t ), t ) ∩ ∂T t ⊂ Uc and (W 2 , ρ (t ), t ) ∩ ∂T t ⊂ Bc , hence (H3) is fulfilled. 2 2 Let us notice that z (t ) = e−t for z(t ) = 0, and z(σ (t )) = μ(t )e−t ≤ μ(t ) < 1/8, hence

([0, π ] × {0} × {0}, ρ (t ), t ) ∩ ∂T t ⊂ Uc . Moreover, if x(t ) = y(t ) = 0 then |x (t)|μ(t) ≤ 1/8, |y (t)|μ(t) ≤ π /8, and (1/8 )2 + (π /8 )2 < (π /2 )2 , thus

  π  

{0} × {0} × − , 0 , ρ (t ), t ∩ ∂T t ⊂ Bc . 2

G. Gabor et al. / Applied Mathematics and Computation 265 (2015) 358–369

It implies that there exists the set



Xt := ([0, π ] × {0} × {0}, ρ (t ), t ) ∪

369

 π   {0} × {0} × − , 0 , ρ (t ), t , 2

  π   Xt ⊂ {0} × {0} × − , 0 ∪ [0, π ] × {0} × {0}, ρ (t ), t ∩ t 2

(W 1 )

connecting t = and t (W 2 ) = W 2 . Hence, (H2) also holds. We have verified that all assumptions of Theorem 4.1 are fulfilled, hence there exists a solution u∗ (t) of system (7) such that u∗ (t) ∈  for every t ∈ [t0 , ∞ )T . If we multiply the right-hand side of (7) by 1/8 then we will have analogical calculations with the same results for greater graininess, namely not bigger than 1, therefore the assumptions will be fulfilled for T = Z, hence for a system of difference equations. W1

6. Concluding remarks • In general, assumptions (H2) and (H3) may be quite hard to check. Are there any sufficiently general and more easily verifiable conditions (see [12] for n = 2)? • The main result of our paper allows to consider systems with disconnected sets of egress points that differ in points with non-zero graininess. It may be important for applications with time scale equal to the sum of intervals. • The open problem is how to prove analogous results for connected sets of egress points that are not retracts of . Acknowledgments The first and second author was supported by the Polish NCN grant no. 2013/09/B/ST1/01963. The third author was supported by the project CZ.1.07/2.3.00/30.0039 of Brno University of Technology. The third author was also supported by the project FEKT-S-14-2200 of Brno University of Technology. The work of third author was partially realized in CEITEC – Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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