Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation

Abstract

In this work, we consider an initial value problem of a nonhomogeneous retarded functional equation coupled with the impulsive term. The fundamental matrix theorem is employed to derive the integral equivalent of the equation which is Lebesgue integrable. The integral equivalent equation with impulses satisfying the Carathéodory and Lipschitz conditions is embedded in the space of generalized ordinary differential equations (GODEs), and the correspondence between the generalized ordinary differential equation and the nonhomogeneous retarded equation coupled with impulsive term is established by the construction of a local flow by means of a topological dynamic satisfying certain technical conditions. The uniqueness of the equation solution is proved. The results obtained follow the primitive Riemann concept of integration from a simple understanding.

1. Introduction

The dynamic of an evolving system is most often subjected to abrupt changes such as shocks, harvesting, and natural disasters. When the effects of these abrupt changes are trivial the classical differential equation is most suitable for the modeling of the system. But for short-term perturbation that acts in the form of impulses, the impulsive delay differential equation becomes handy. An impulsive retarded differential equation is a delay equation coupled with a difference equation known as the impulsive term. Among the earliest research work on impulsive differential equation was the article by Milman and Myshkis [1]. Thereafter, growing research interest in the qualitative analysis of the properties of the impulsive retarded equation increases, as seen in the works of Igobi and Ndiyo [2], Isaac and Lipcsey [3], Benchohra and Ntouyas [4], Federson and Schwabik [5], Federson and Taboas [6], Argawal and Saker [7], and Ballinger [8].

The introduction of the generalized ordinary differential equation in the Banach space function by Kurzweil [9] has become a valuable mathematical tool for the investigation of the qualitative properties of continuous and discrete systems from common sense. The topological dynamic of the Kurzweil equation considers the limit point of the translate under the assumptions that the limiting equation satisfying the Lipschitz and Carathéodory conditions is not an ordinary differential equation, and the space of the ordinary equation is not complete. But if the ordinary differential equation is embedded in the Kurzweil equations we obtained a complete and compact space, such that the techniques of the topological translate can be applied.

A more relaxed Kurzweil condition was presented in the article by Artstein [10]. He considered the metric topology characterized by the convergencewith the following properties:(i) is a compact set; then there exists a locally Lebesgue integrable function such that  and is uniformly continuous in . That is, there exists such that (ii) is a compact set; then there exists a locally Lebesgue integrable function such that  for all and fixed

The metric convergence of fulfilling (i) and (ii) for some and guarantees the continuity of and the precompactness of the function space, but not completeness. That is, by equation (1), the Cauchy sequence implies converges for all . However,does not have the integral representationIn summary, the Kurzweil equation addresses functions whose limit exists but are nowhere differentiable, but, by using the primitive definition of Riemann integral, a correspondence is established.

Consider an ordinary equationwith integral equivalentSuppose the integral is a Riemann integral, we can define a fine partition

on and a . The differential approximation of Equation (7) in the Riemann sense isIf we defined So, by Equation (8), we have Equation (10) defined the Riemann-Kurzweil sum approximation of if and only if is a solution of (7) and is a fine partition. Thus, for any , defines the Kurzweil integral if there exists an such thatThe differential equation resulting from the primitive Kurzweil integral (11) is what is known as the generalized ordinary differential equation.

The correspondence between the generalized ordinary differential equation and other types of differential system is well established in the following articles: Federson and Taboas [6], Federson and Schwabik [5], Imaz and Vorel [11], Oliva and Vorel [12], and Schwabik [13]. This was made possible by embedding the ordinary differential equation in the space of the generalized ordinary differential equation and constructing a local flow by means of a topological dynamic satisfying certain technical conditions.

In this work, we consider an initial value problem of a nonhomogeneous retarded functional equation of the formcoupled with impulseswhere are constant matrices, , , expresses the history of on by  

is the initial term, and with are the impulses time.

We will employ the fundamental matrix theorem to derive the integral equivalent of Equation (13) and define Lebesgue integrable functions and for satisfying the conditionsWe embed the integral equivalent equation with impulses satisfying conditions (A) and (B) in the space of generalized ordinary differential equations (GODEs), and using similar argument as presented by Federson and Taboas [6] and Federson and Schwabik [5] to show the relationship between the solutions of the generalized ordinary differential equation and the equivalent impulsive retarded differential equation, and establish the uniqueness of the equation solution.

2. Generalized Ordinary Differential Equation

Let be a Banach space and a Banach space of bounded linear operators on , with and defining the topological norms in and , respectively. A partition is any finite set such that . Given any finite step function , for being a constant on , then is the variation of on . The function is of bounded variation on if .

The function is regulated on if the one-sided limits and exist at every point of . That is and such that and for all . By we denote the set of all regulated functions , which is a Banach space when endowed with the usual supremum norm

A tagged division of a compact interval is a finite collection of point-interval pairs , where and (that is ). A gauge on is any positive function . A tagged division is if for every .

Definition 1. Let be a given function. A Kurzweil integral over the interval exists if there is a unique element such that, for every and a gauge on , we have satisfied for all partition of , where , , and is the Kurzweil integral. If the Kurzweil integral exists over , then . In the Jaroslav Kurzweil sense is not defined; only might exist.

The Kurzweil integral is related to the Riemann integral when the space is the set of real numbers such that and the Riemann sum is defined asfor all . The properties of the Kurzweil integral such as the linearity, additivity, and convergence with respect to the nearby interval have been extensively discussed in Artstein [10], Schwabik [13], and Federson and Schwabik [5].

We state here some of the fundamental results of the Kurzweil integral on a subinterval as proved in Kurzweil [9] and Artstein [10] which are the basic concepts to be employed in this work.

Lemma 2. Let be continuous in for each . If exists then for each the integral exists, and is continuous in .

Proposition 3. Let , and such that ; then (ii) The continuity of in implies that converges to zero as .

The consequent of Lemma 2 is the result by Schwabik [13] stated as Lemma 4.

Lemma 4. Let be a given function such that is integrable over for and let the limitThen the function is integrable over and.
Similarly, if is integrable over for , let the limitThen the function is integrable over and .

The result of Lemma 4 is a follow-up of Lemma (A.2.) in Artstein [7]

Lemma 5. If is piecewise continuous in then exists, where is a regulated function.

Definition 6. Let and ; the linear nonhomogeneous generalized ordinary differential equation is of the form

Definition 7. The linear nonhomogeneous generalized integral solution of (21) is of the form if the Kurzweil integral exists and satisfies Equation (22) for each The literature on Equation (21) abounds in Schwabik [13], Schwabik, Tvrdy, and Vejvoda [14], and Artstein [10].

3. Preliminary Results

In this section, we present results that are fundamental to the establishment of the main results in Section 4.

Definition 8. A matrix is a fundamental matrix of the system (13) if it satisfies the matrix equation

Definition 9. The solution of Equation (23) with an identity initial condition has a recurrent formwhere is defined in the interval

Bastinec and Piddubna [15] used the recurrent form of (24) to define the fundamental matrix solution of Equation (23) as presented in Lemma 10 and Definition 11.

Lemma 10. The fundamental matrix solution of Equation (23) with an identity initial condition has the formwhere

Definition 11. The integral solution of system (13) satisfying the given initial condition iswhere the integral exists in the Lebesgue sense (Bastinec and Piddubna, [15]).

Definition 12. Let and be Lebesgue integrable functions satisfying conditions (A) and (B). Also assume is Kurzweil integrable function; then the integral solution (27) has the form

Remark 13. One of the fundamental theories of piecewise continuous functions with respect to delay differential equation is that if is piecewise continuous, then may be discontinuous at some or all . This result was proved in Hale [16].

Lemma 14 (Hale, [16]). Assume , and let for all . Then and the only possible points of discontinuity of are , where denotes a point of discontinuity.

In consequence of Lemma 14 and the pioneering work of Imaz and Vorel [11] and Oliva and Vorel [12], for each , and being Kurzweil integrable on , we define the functions and such thatSimilarly, we define a unit step function concentrated at as so that, given and , the impulsive term in Equation (28) is defined as

Remark 15. Let and such that and is Kurzweil integrable. Then we make the following Carathéodory and Lipschitz assumptions on the integral of the function (unlike the usual imposition of the conditions on the functions):(A₁)there exists a Kurzweil integrable function , such that(A₂)there exists a Kurzweil integrable function , such that(A₃)there exists a real constant such that ,(A₄)there exist positives constants such that for and all

Proposition 16 (Federson and Taboas, [6]). Equations (29), (30), and (32) satisfying assumptions (A₁ – A₄) are continuous on

Proposition 17 (Schwabik, [13]). If and , such that is a solution of then where is an identity operator on .

Proof. Let ; then

4. Main Results

Consider for each , given by , such that the generalized nonhomogeneous linear ordinary differential Equation (21) holds. Then the integral equationsatisfying the initial conditionis the solution of the generalized ordinary differential equationThe relationship between Equations (39) and (28) is established as in the articles Federson and Schwabik [5] and Federson and Taboas [6], though the technical manipulation of the solution in this work satisfies the Carathéodory and Lipschitz conditions in Remark 15. Sequel to this, we state a very useful assumption as stated and proved in Federson and Schwabik [5]

Lemma 18. Assume is a solution of Equation (39) satisfying the initial condition (40), then for all we haveand

Theorem 19. Let and be linear functions in the first variables such that is Lebesgue integrable on and the conditions (A₁), (A₂), (A₃), and (A₄) are satisfied. Assume is a solution of Equation (41) on satisfying the initial condition (40); then such that is a solution of Equation (28) on if, for any , holds.

Proof. Using the result of Lemma 18 and Equation (22), we have We make the choice of the gauge function These ensure that each subinterval of δ-fine partition contains at most one of the points , , corresponding to a tag of the interval. Hence, by Equation (46), we haveThis implies thatAlso by Lemma 14 and Equation (44), for , we haveHence, by Remark 15, we havewhich implies that the function is of bounded variation, and hence the existence of solution of Equation (28).

Theorem 20. Let satisfy equation (28) on and let be Lebesgue integrable. Then, there exists a satisfying the initial condition (44) such that for any (ii) for , there exists a and such that Then Equation (39) has a unique solution.

Proof. By Equations (28) and (52)Let be a Lebesgue integrable, a regulated function on , and ; then for a partition , and using the relation , we haveUsing Proposition 17 and for being a solution of equation (39) we obtainedAlso, the results in Schwabik [1] show that for a regulated function there exists such that the set is finite. This implies that the set of discontinuity points of is at most countable, and there is a finite set such that for . the operator is invertible and exists.
Therefore,andwhere is nonnegative, nondecreasing, and left continuous function, is nonnegative and bounded function.
Using preposition 2.1 (Schwabik, [1]), we defined a bounded linear operator on so that by Equation (53)If for then where satisfied the hypothesis of the theorem as stated and the operator is a contraction, and, by Banach contraction principle, it has a unique fixed point. Hence the theorem is proved.

Example 21. We consider the model of a circulating fuel reactor originally studied in [17] and modified in [18] by the inclusion of constants impulsive terms. We further modify the model equation by including an input function (a forcing term) which is Lebesgue integrable. The system equation is of the formwhere and are locally Lebesgue integrable functions such that and for all is increasing function such that with . Let there exists a function locally Lebesgue integrable such that for all , and for , maps to .

Defining , Afonso [18] proved that conditions (A₁), and (A₂) of Remark 15 are satisfied.

Also, by the hypothesis of the problem, we have thatHence condition (A₃) is satisfied.

Considering the impulsive term , we have and condition (A₄) is satisfied. Hence, in consequence of Remark 15 the integral solutionexists and is unique.

5. Conclusion

An initial value problem of a nonhomogeneous retarded functional equation coupled with the impulsive term was considered. The integral equivalent of the equation which is Lebesgue integrable was obtained using the fundamental matrix theorem. The integral equivalent equation with impulses satisfying the Carathéodory and Lipschitz conditions was embedded in the space of generalized ordinary differential equations (GODEs) and, using similar argument as presented in Federson and Taboas [6] and Federson and Schwabik [5], we showed the relationship between the generalized ordinary differential equation and the nonhomogeneous retarded functional equation coupled with impulsive term by the construction of a local flow using topological dynamic satisfying certain technical conditions. The uniqueness of the equation solution was proved.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.