On the Existence and Stability of Bounded Solutions for Abstract Dynamic Equations on Time Scales

Duque, Cosme; Leiva, Hugo; Gallo, Rodolfo; Tridane, Abdessamad

doi:https://doi.org/10.1155/2023/8489196

International Journal of Differential Equations

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Research Article | Open Access

Volume 2023 | Article ID 8489196 | https://doi.org/10.1155/2023/8489196

On the Existence and Stability of Bounded Solutions for Abstract Dynamic Equations on Time Scales

Cosme Duque,¹Hugo Leiva,²Rodolfo Gallo,²and Abdessamad Tridane³

Academic Editor: Jaume Giné

Received21 Feb 2022

Revised02 Sept 2022

Accepted12 Sept 2022

Published29 Apr 2023

Abstract

In this article we study the existence and stability of bounded solutions for semilinear abstract dynamic equations on time scales in Banach spaces. In order to do so, we use the definition of the Riemann delta-integral to prove a result about closed operator in Banach spaces and then we just use the representation of bounded solutions as an improper delta-integral from minus infinite to . We prove the existence, uniqueness, and exponential stability of such bounded solutions. As particular cases, we study the existence of periodic and almost periodic solutions as well. Finally, we present some equations on time scales where our results can be applied.

1. Introduction

The time scales theory was introduced by Hilger (see [1, 2]) with the purpose to study difference and differential equations from a unified perspective. In recent decades, a large number of researchers have directed their efforts to study this powerful tool which has relevant applications in economics, population dynamics, quantum physics, controllability, and among others (see, for instance [3–9] and the references therein).

Particularly, this paper is devoted to study the existence and stability of bounded solutions of certain abstract dynamic equations on times scales, but before presenting the subject of this paper we recall that a time scale, denoted by , is any arbitrary nonempty closed subset of . On , the forward and backward jump operators are defined, respectively, as A point is said to be right-dense if , right-scattered if , left-dense if , left-scattered if , isolated if . The function defined by is known as graininess function. has the topology inherited from standard topology on the real numbers, and the time scale interval is defined by , with , similarly is defined open intervals and open neighborhoods. The set is given by if or if .

A function is called rd-continuous (right-dense continuous) if is continuous at every right-dense point of and its left-sided limit exists and is finite at left-dense points of . The set of all rd-continuous functions is denoted by . The function is said to be -differentiable (delta differentiable) at if there exists a vector with the following property: given , there exists a neighborhood , for some , such that for all . In this case the vector is called the delta derivative of in . When is -differentiable at it is easy to show that if is right-scattered, or if is right-dense. For more details about calculus on time scale we refer the reader to Bohner and Peterson’s book [4].

For our purposes, in the sequel, we will assume that , , , if and , and . We will also denote by .

The main concern of this paper is to study the existence and stability of bounded solutions for the following abstract dynamic equation on time scales:

Here, we are considering as a rd-continuous function which is locally Lipschitz on a Banach space , uniformly on and is a linear operator which generates a -semigroup of bounded linear operator .

The motivation to study equation (1) was given by the works presented in [10, 11] where the authors studied the existence and stability of bounded solutions of the discrete-dependent system of difference equationand the evolution equation,respectively. Therefore, this paper allows us to unify the results presented in [10, 11] and extend them.

Now, to accomplish this task, we will assume the following hypotheses for problem (1): (H1) for some constant (H2) Given there exits such that for all and

The organization of this paper is as follows: Next section is devoted to present some preliminary results about integration, generalized exponential function, and semigroup theory on time scales. In the third section, we analyze the existence and stability of bounded solutions. The fourth section is devoted to study, as particular case, the existence of periodic and almost periodic solutions. In the fifth section, we show that the obtained bounded mild solution is also, under certain conditions, a classical solution. Finally, we presented some examples as applications of our results.

2. Preliminaries

We start to present some facts about integration, the exponential function, and semigroup theory on time scales, that will be of utility in the development of this work.

A partition on (see [12]) is a finite sequence of points such that , this partition will be denoted by . It is clear that depends on the partition , so we get .

Suppose that is a bounded function on interval and let be a partition of , the sumis called the Riemann -sum of on , where is an arbitrary point on for each . If , the set denotes the set of all partitions of such that for every either or and . The set by Lemma 2.7 in [12].

Definition 1 (see [12]). The function is said to be Riemann -integrable on if there exists an element , denoted by , with the property for every , there exists such thatfor all independently offor.
The element is unique and it is called the Riemann - integral of from to .
Let . The improper -integral of is defined by (see [13])provided that this limit exists. Analogously,The function is called regressive (resp. positively regressive) if (resp. ) for . The set of all regressive (resp. positively regressive) and rd-continuous functions is denoted by (resp. ).

Definition 2 (see [4]). Let , the generalized exponential function is defined bywhereis the cylinder transformation defined on the Hilger complex numbers and ,.
For we define the operationsIt easy to see that forms an Abelian group, for details see [4].

Theorem 1 (see [4, 14]). For , the generalized exponential function satisfy the following properties: For ,(1),,(2),(3)(4)(5)if and for all , , then for all (6)if , then for , , and .

Definition 3 (see [15]). A one parameter family is a -semigroup if it satisfies the following properties:(i)(ii) for every (iii) for each If then the semigroup is called uniformly continuous.
The linear operator defined byis the infinitesimal generator of the semigroup and is the domain of .
One class of important functions are the generalized polynomial functions (see [4]) , , which are defined recursively of the following form: and .

Lemma 1 (see [4]). Let . Then, for all .

Theorem 2. If is a bounded linear operator on , then is the generator of a uniformly continuous semigroup which is given by

Proof. Note that for , with ,which implies the convergence in the uniform operator topology for any and defines a bounded linear operator for each .
Now,(i)(ii) We shall see that Indeed, we proceed by using induction over . If , then ; if , then . Suppose that (14) holds for , then Therefore, .(iii)Estimating the power series yieldsas .
Finally, we see that is the infinitesimal generator of .

Remark 1. The converse of Theorem 2 is also true and was proved in [15] Theorem 2.7.

Definition 4 (see [16]). Let be a -semigroup. The semigroup is called exponentially stable if there exists and such thatfor all with .
It is known (see [16, 17]) that solution of system (1) satisfies the integral equationbut not conversely since a solution of (19) is not necessarily -differentiable. We shall refer to a rd-continuous solution of (19) as a mild solution of equation (1). is a classical solution of (1) on if is rd-continuously -differentiable, for and (1) is satisfied on .

3. Existence of Bounded Solutions

In this section, we study the existence and stability of bounded solutions for system (1). The natural space to study this problem iswhich we will endow with the norm

It is easy to show that the space is a Banach space, and for we define

Lemma 2. Let a exponentially stable -semigroup with infinitesimal generator , and let . Then, is a mild solution of (1) if and only if is solution of the integral equation

Proof. If is a mild solution of (1), thenSince the semigroup is exponentially stable, we have that there exist and such thatOn the other hand for , therefore we obtain the estimateand hence . Now, passing to limit in (24), it follows thatOur next step is to show the improper integral is convergent. For this end, let us consider such that and let be the Lipschitz constant of in . Then,Therefore, (23) is well defined.
Now, suppose that is solution of (23). Then, for each we have thatHence, for we get thatwhere . Therefore, is a mild solution of equation (1).

Theorem 3. Let be a exponentially stable -semigroup with infinitesimal generator . Ifwhere is the Lipschitz constant of in , then equation (1) has a unique mild solution in . Moreover, is exponentially stable.

Proof. Let us consider the operator defined byBy considering Lemma 2 we will show that the operator has a unique fixed point in .
For , we have thatSo, and therefore . Now, for it followsSince , we get is a contraction mapping. Thus, by using the Banach fixed point theorem, has a unique fixed point in , which satisfiesHence by the preceding lemma, is a bounded mild solution of equation (1).
Finally, let us prove that is exponentially stable. Consider an arbitrary solution of (1) such that , with , then . As long as remains less than , we get the following estimate:So,By using Gronwall’s inequality (see [18]), we obtain thatwhich implies thatSince andthen we getfor . Due that for , then or , but in the second case, this contradicts the estimate . Therefore, and remains for all in the ball andIf we suppose that is globally Lipschitz, then the bounded solutions given by Theorem 3 is globally.

Theorem 4. Let be a exponentially stable -semigroup with infinitesimal generator , suppose that is globally Lipschitz with constant and such that

Then, there exists a unique mild solution of (1) defined on and exponentially stable.

Proof. If , then the condition (43) implies thatThus, from the preceding theorem, we have that equation (1) has a unique mild solution in . Since the condition (44) is satisfied for large enough, then is the unique solution of (1) on .
Now, if is any other solution of (1), thenand by using Gronwall’s inequality, it is obtained thatSince , is exponentially stable for all .

Corollary 1. Suppose thatwhere and is a Lipschitz function with constant . Then, the unique bounded solution giving by Theorems 3 and 4 depends continuously on .

Proof. Let and , be the bounded solutions of (1) given by Theorems 3 and 4. Then,Hence,soand therefore,

4. Periodic and Almost Periodic Cases

This section is dedicated to study the cases when the function is either periodic or almost periodic. We begin with the periodic case.

Let fixed. The time scale is called -periodic if for each we get that , and the function is said to be -periodic on if for all . From Theorem 2.1 in [19] we get that , and . For more details about periodic functions on times scales see [4, 19].

Theorem 5. Assume that is -periodic, and is -periodic in , then the solution obtained in Theorems 3 and 4 is also -periodic.

Proof. Let be the unique solution of (19) on . We will prove that is also a solution of (19) on the ball . In fact, given we haveThus, satisfies (19). Since the operator has a unique fixed point which is defined by (32), then for .
For studying the almost periodic case, we need some definitions and results about almost periodic functions on time scales, for details of it, one can read [20–22].
A time scale is called an almost periodic time scale ifSuppose that is an almost periodic time scale. A function is called an almost periodic function in if the -translation set of is a relatively dense set in for all ; that is, for any given , there exist a constant such that each interval of length contains a such that is called the -translation number of and is called the inclusion length of .
For a given , the hull of is defined aswhere means that , when this limit exists.

Theorem 6. Suppose that where almost periodic and globally Lipschitz with constant , then the unique solution given by Theorems 3 and 4 is almost periodic.

Proof. To prove this theorem, we shall use the Theorem 3.18 in [20]. A function is almost periodic if and only if the hull is compact in the topology of uniform convergence.
Now, let and we consider the set , by Theorem 3.27 in [21] we have that is closed. If , then is also an almost periodic function. Let us consider the function given by (32).It is enough to establish that is compact in the topology of uniform convergence. In fact, if we consider the sequence in , then by using the diagonal procedure, there is a sequence in such thatDue is almost periodic, then is relatively compact, and putting , then there exists a convergent subsequence . On the other hand,Then,Thus, is a Cauchy sequence in , which implies that is relatively compact. So is an almost periodic function. Moreover, . For this reason, the only fixed point of on is in , i.e., is almost periodic.

5. Existence of Classical Solution

In this section we will show that the bounded mild solution of (1) given by equation (32), obtained in Theorems 3 and 4, is also, under certain conditions, a classical solution of (1). In order to achieve this, we need the following theorem, which is an extension of Theorem 1.3.5 of [23]:

Theorem 7. Let be a closed linear operator defined on . Let be a function in , where . If , is -continuous on and the integralsexist, then

Proof. Suppose that . Set where is sufficiently small. Since and exist, then for each , there exists such thatfor every Riemann -sum of corresponding to a partition independently of for .
If we put , then and .
Moreover,Since is a closed operator on , then it follows thatNow, making , we have the result.
If , the result follows from the fact thatand the previous result.

Remark 2. Next theorem pretends to extend a result presented in [11] to time scales; particularly the conditions (1)–(3) of Theorem 8 can be satisfied if is an analytic semigroup (see [24, 25]).

Theorem 8. Let be the bounded mild solution of equation (1), obtained in Theorems 3 and 4. If for each _T and , where , for all , such that , , and the following statements hold:(1)(2) is rd-continuous(3) existsThen, is a classical solution of (1) on , i.e.,

Proof. We know that can be expressed throughConditions (1)–(3) and the previous theorem imply thatSinceBaring in mind that A is a closed operator, we have thatNow, we shall use the fact that, for a Riemann -integrable function the following equality holds:Next, for , we consider the following equalities:Passing to the limit when , it is obtained thatA similar discussion for , produces the same result. This concludes the proof of the theorem.

6. Examples

This section is devoted to present some applications of our results.

Example 1. Consider the equationwhere and are the positive constant; and is a rd-continuous and bounded function. Notice that and . If we define , then it follows thatwhere . If then all conditions of Theorem 3 are satisfied, therefore the equation (75) has a unique bounded solution which is exponentially stable.

Example 2. Let us consider the time scale and the dynamic equationBy using the change of variable , we get that the equation (77) can be written as first order system of dynamic equations.whereIn this case and the eigenvalues of the matrix are . Then, it is easy to show thatand . If , then all hypotheses of Theorem 3 are satisfied which implies that equation (77) has a unique bounded solution which is exponentially stable.

Example 3. We consider the following dynamic equation:where . This dynamic equation is the representation on time scales of a single self-excited neuron without delayed excitation. Here represents the voltage of the neuron, is the ratio of the capacitance to the resistance, and the feedback strength. The transfer function is given by and the function represents other input to the neuron. (see [26–28])
Notice that satisfies ; therefore, if is periodic or almost periodic, and . Then, the unique solution of (81) is periodic or almost periodic.

Data Availability

Data sharing are not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Cosme Duque et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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