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

Authors: D. K. Igobi; U. Abasiekwere
Journal: International Journal of Differential Equations
 (2019)

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## 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.


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## Body

## 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 ft→f(x,t+s) 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 convergence(1)fk→f0if∫0tfkx,τdτ→∫0tfox,τdτ,with the following properties:(i)
V⊂W is a compact set; then there exists a locally Lebesgue integrable function MV(s) such that (2)fx,s≤MVs,x∈V,and∫0s+hMV(τ)dτ is uniformly continuous in s. That is, there exists μV(ε)>0 such that ∫0s+hMV(τ)dτ<εh≤μV(ε)(ii)
V⊂W is a compact set; then there exists a locally Lebesgue integrable function KA(s) such that (3)fx,s-fy,s≤KVsx-y,x,y∈Vand∫0sKVτdτ≤NVfor alls and NA fixedThe metric convergence offk fulfilling (i) and (ii) for some NV and μV guarantees the continuity of (t,f)→ft(t,x) and the precompactness of the function space, but not completeness. That is, by equation (1), the Cauchy sequence fi implies ∫0sf(x,τ)dτ converges for all (x,s). However,(4)Asx=lim⁡∫0sfix,τdτdoes not have the integral representation(5)Asx=∫0sfx,τdτ.In 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 equation(6)x•=fx,t,with integral equivalent(7)xt-xt0=∫t0tfx,τdτ.Suppose the integral is a Riemann integral, we can define a δ-fine partitiont 0 = s 1 ≤ s 2 ≤ ⋯ ≤ s n + 1 = t on [t0,t] and a ηi∈[si-1,si],i=1,2,3,…. The differential approximation of Equation (7) in the Riemann sense is(8)∑i=1n∫sisi+1fxτiδidδ.If we defined (9)Asxt=∫t0tfx,δdδ,So, by Equation (8), we have (10)∑i=1tAsi+1-Asixηi,=SdA,x,P.Equation (10) defined the Riemann-Kurzweil sum approximation of x(t)-x(t0) if and only if x(t) is a solution of (7) and [si,si+1] is a fine partition. Thus, for any I∈Rn, (11)I=∫t0tdAsxηdefines the Kurzweil integral if there exists an ε>0 such that(12)I-SdA,x,P≤εThe 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 form(13)x•t=B0xt+B1xt+utxt0=φ,coupled with impulses(14)Δxtk=Ikxtk,k=1,2,…m,where B0n×n,B1n×1 are constant matrices, x∈G([t0,t0+δ],X), u(t):[t0,t0+δ]→X, xt:[-r,0]→X expresses the history of x on [t-r,t]by xt(ϕ)=x(t+ϕ),ϕ∈[-r,0]φ ∈ G ( [ - r , 0 ] , X ) is the initial term, and tk,k=1,2,… with t0<t1<⋯<tk⋯<tm<t0+δ are the impulses time.We will employ the fundamental matrix theorem to derive the integral equivalent of Equation (13) and define Lebesgue integrable functions L(φ(t)):G([t0-r,t0-δ],X)→X and f(t):[t0,t0+δ]→X for δ>0,r≥0 satisfying the conditions(15)(A)∫-r0K0t-s-rφsds≤∫t0tLφ,sds(B)∫0tK0t-s-rusds≤∫t0tfsds.We 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

LetX be a Banach space and L(X) a Banach space of bounded linear operators on X, with .X and .L(X) defining the topological norms in X and L(X), respectively. A partition is any finite set U=s0,s1,…,si+1 such that a=s0<s1<⋯<si+1=b. Given any finite step function A(t):[a,b]→L(X), for A(t) being a constant on (si-1,si), then varabA(t)=sup⁡∑i=1n(P)A(si)-A(si-1)X is the variation of A(t) on [a,b]. The function A(t) is of bounded variation on [a,b] if varabA(t)<∞.The functionA(t):[a,b]→L(X) is regulated on [a,b] if the one-sided limits A(t-)=lims→t-⁡A(s) and A(t+)=lims→t+⁡A(s) exist at every point of t∈[a,b]. That is A(a-)=A(a) and A(b+)=A(b) such that Δ-A(t)=A(t)-A(t-) and Δ+A(t)=A(t+)-A(t) for all t∈[a,b]. By G([a,b],L(X)) we denote the set of all regulated functions A(t):[a,b]→L(X), which is a Banach space when endowed with the usual supremum norm A∞=sup⁡A(t)X,t∈[a,b]A tagged division of a compact intervalU[a,b]⊂R is a finite collection of point-interval pairs P=(η,U), where U=s0,s1,…,si+1 and ηi∈[si-1,si] (that is (ηi,[si-1,si])). A gauge on [a,b] is any positive function δ:[a,b]→(0,∞). A tagged division P(ηi[si-1,si]) is δ-fine if for every i=1,2,…. [si-1,si]⊂(ηi-δ(ηi),ηi+δ(ηi)).Definition 1.
LetA:[a,b]→L(X) be a given function. A Kurzweil integral over the interval [a,b] exists if there is a unique element I∈X such that, for every ε>0 and a gauge δ on [a,b], we have (16)I-SdA,x,P<εsatisfied for all δ-fine partition P of [a,b], where S(dA,x,P)=∑i=1n(p)[A(si)-A(si-1)]x(ηi), P=(ηi,[si-1,si]), and I=∫abd[A(s)]x(η) is the Kurzweil integral. If the Kurzweil integral exists over [a,b], then ∫abd[A(s)]x(η)=-∫bad[A(s)]x(η). In the Jaroslav Kurzweil sense d[A(s)]x(η) is not defined; only ∫abd[A(s)]x(η) might exist.The Kurzweil integral is related to the Riemann integral when the spaceX is the set of real numbers such that A:[a,b]→R and the Riemann sum is defined as(17)SdA,x,P=∑i=1npAηisi-si-1,for all P=[a,b]. 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.
LetA:[a,b]→L(X) be continuous in s for each η. If ∫abd[A(s)]x(η) exists then for each s1<s2 the integral ∫s1s2d[A(s)]x(η) exists, and ∫atd[A(s)]x(η) is continuous in t∈[a,b].Proposition 3.
LetA:[a,b]→L(X), and s1<t∈[a,b] such that t-s1<δ(s1); then (18)(i)∫s1tdAsxη-At-As1xη<2ε(ii) The continuity of A(t) in t implies that ∫s1td[A(s)]x converges to zero as s1→t.The consequent of Lemma2 is the result by Schwabik [13] stated as Lemma 4.Lemma 4.
LetA:[a,b]→L(X) be a given function such that A(s) is integrable over [a,s] for s∈[a,b) and let the limit(19)lims→b⁡∫asdAsxη-As-Abxη=I∈Xexist.Then the function A(s) is integrable over [a,b] and∫abdA(s)x(η)=I.
Similarly, ifA(s) is integrable over [s,b] for s∈(a,b], let the limit(20)lims→a⁡∫sbdAsxη-Aa-Asxη=I∈Xexist.Then the function A(s) is integrable over [a,b] and ∫abd[A(s)]x(η)=I.The result of Lemma4 is a follow-up of Lemma (A.2.) in Artstein [7]Lemma 5.
Ifx(η) is piecewise continuous in [a,b] then G(x(η),t)=∫abd[A(s)]x(η) exists, where G(x(η),t)∈G([t0,t0+δ],X) is a regulated function.Definition 6.
LetA:[a,b]→L(X),g:[a,b]→X and x0∈X; the linear nonhomogeneous generalized ordinary differential equation is of the form(21)dxdt=dAtxt+gtxt0=x0Definition 7.
The linear nonhomogeneous generalized integral solution of (21) is of the form (22)xt=x0+∫t0tdAtx+gt-gt0,t∈a,b,if the Kurzweil integral ∫t0td[A(t)]x exists and x(t0)=x0 satisfies Equation (22) for each t∈[a,b]. 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 Section4.Definition 8.
A matrixKn×n(t) is a fundamental matrix of the system (13) if it satisfies the matrix equation(23)ddtKt=B0Kt+B1Kt-r,t>0,Kt=0fort<0,K0=IDefinition 9.
The solution of Equation (23) with an identity initial condition K(0)=I has a recurrent form(24)Ki+1t=eB0t-irKiir+∫irteB0t-sB1Kis-rds,where Ki(t) is defined in the interval i-1r≤t≤ir,i=0,1,…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 K(0)=I has the form(25)K0t=0-∞≤t≤-rI-r≤t≤0eB0t+u1t0≤t≤reB0t+eB0t-rur+u1tr≤t≤2reB0t-mrummr+uiti-1r≤t≤irwhere(26)upt=∑∑ij=1p∏i=p1∑k=0∞B0kiB1ijt-p-1kpkp!∏s=p-11τ1-is+1ks1-is+1ks!kv=kv+iv1+kv-1+iv-11+⋯i21+k1+i1⋯,ip=1,ij=0,1Definition 11.
The integral solution of system (13) satisfying the given initial condition is(27)xt=K0tφ-r+∫-rt0K0t-s-rφsds+∫t0tK0t-s-rusds+∑k=1nIkxtkwhere the integral exists in the Lebesgue sense (Bastinec and Piddubna, [15]).Definition 12.
LetL:G([t0-r,t0+δ],L(X))×[t0,t0+δ]→X and f(t):[t0,t0+δ]→X be Lebesgue integrable functions satisfying conditions (A) and (B). Also assume t→L(xt,t) is Kurzweil integrable function; then the integral solution (27) has the form(28)xt=φ+∫t0tLxs,sds+∫t0tfsds+∑k=1nIkxtk.Remark 13.
One of the fundamental theories of piecewise continuous functions with respect to delay differential equation is that ifx∈PC([t0-r,t0+δ],X) is piecewise continuous, then xt may be discontinuous at some or all t∈[t0,t0+δ]. This result was proved in Hale [16].Lemma 14 (Hale, [16]).
Assumex∈PC([t0-r,t0+δ],X), and let g(t)=xtr for all t∈[t0,t0+δ]. Then g(t)∈PC([t0-r,t0+δ],R+) and the only possible points of discontinuity of g are t∗ort∗+r, where t∗denotes a point of discontinuity.In consequence of Lemma14 and the pioneering work of Imaz and Vorel [11] and Oliva and Vorel [12], for each x∈G([t0-r,t0+δ],X), and t→L(xt,t) being Kurzweil integrable on [t0,t0+δ], we define the functions F(x,t):G([t0-r,t0+δ],X)→C([t0-r,t0+δ],X) and g(t)∈C([t0-r,t0+δ],X) such that(29)(i)Fx,tν=0,t0-r≤ν≤t0∫t0νLxs,sdst0≤ν≤t≤t0+δ∫t0tLxs,sdst0≤t≤ν≤t0+δ(30)(ii)gtν=0,t0-r≤ν≤t0∫t0νfsdst0≤ν≤t≤t0+δ∫t0tfsdst0≤t≤ν≤t0+δSimilarly, we define a unit step function concentrated at tkc∈[t0,∞) as (31)∫kct=0,t0<t<tkc1,t≥tkc,so that, given ν∈[t0,t0+δ] and x∈G([t0-r,t0+δ],X), the impulsive term in Equation (28) is defined as (32)(iii)Jx,tν=∑k=1n∫kct∫kcνIkxtk.Remark 15.
LetL:G([t0-r,t0+δ],L(X))×[t0,t0+δ]→X and f(t):[t0,t0+δ]→X such that t→L(t,xt) and t→f(t) 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 functionM0:[t0,t0+δ]→R, such that(33)∫s0s1Ls,xsds≤∫s0s1M0ds,fors1,s2∈t0,t0+δ,x∈Gt0,t0+δ,Rn,(A2)
there exists a Kurzweil integrable functionM1:[t0,t0+δ]→R, such that(34)∫s0s1Lxs-ys,sds≤∫s0s1M1sxs-ysds,fors1,s2∈t0,t0+δ,x,y∈Gt0,t0+δ,Rn,(A3)
there exists a real constantM>0 such that ∫s0s1f(s)ds≤∫s0s1Mds,s1,s2∈[t0,t0+δ],(A4)
there exist positives constantsK1,K2 such that for k=1,2,…,n and all x,y∈Rn(35)Ikx≤K1xandIkx-Iky≤K2x-y.Proposition 16 (Federson and Taboas, [6]).
Equations (29), (30), and (32) satisfying assumptions (A1 – A4) are continuous on C([t0-r,t0+δ],X)Proposition 17 (Schwabik, [13]).
IfA:([t0,t0+δ],L(X))→L(X) and g∈G([t0,t0+δ],X),v∈[t0,t0+δ], such that x:[t0,t0+δ]→X is a solution of (36)x=x0+∫vtdAsxs+gt-gv,t∈t0,t0+δ,then (37)xt=limr→t+⁡xr=I+Δ+Atxt+Δ+gtandxt=limr→t-⁡xr=I+Δ-Atxt+Δ-gtwhere I is an identity operator on X.Proof.
Lett∈[t0,t0+δ]; then(38)xt-=limr→t-⁡xr=x0+limr→t-⁡∫vrdAsxs+limr→t-1gr-gv=x0+∫vtdAsxs-limr→t-⁡∫trdAsxs+limr→t-1⁡gr-gt+gt-gv=xt-limr→t-⁡∫trdAsxs+limr→t-1⁡gr-gt=xt-At--Atxt+gt--gt=I-Δ-Atxt+Δ-gt


## 4. Main Results

Consider for eacht∈[t0,t0+δ], A(t)∈G([t0-r,t0+δ],L(X)) given by A(t)y=F(y,t)+J(y,t), such that the generalized nonhomogeneous linear ordinary differential Equation (21) holds. Then the integral equation(39)y=y0t0+∫t0vdFy,t+∫t0vdJy,t+gt-gt0satisfying the initial condition(40)yt0v=φv-t0,v∈t0-r,t0φt0t0,v∈t0,t0+δis the solution of the generalized ordinary differential equation(41)dydt=dAt+gt,yt0=y0.The 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.
Assumey is a solution of Equation (39) satisfying the initial condition (40), then for all τ∈[t0,t0+δ] we have(42)ytτ=yττ,τ∈t0-r,t0+δand(43)ytτ=ytt,τ∈t0-r,t0+δTheorem 19.
Letφ∈G([-r,0],X) and L:G([-r,0],L(X))×[t0-r,t0+δ]→X be linear functions in the first variables such that t→L(xt,t) is Lebesgue integrable on [t0-r,t0+δ] and the conditions (A1), (A2), (A3), and (A4) are satisfied. Assume y(t) is a solution of Equation (41) on [t0-r,t0+δ] satisfying the initial condition (40); then v∈[t0-r,t0+δ] such that (44)xv=yt0v,v∈t0-r,t0yvv,v∈t0,t0+δ,is a solution of Equation (28) on [t0-r,t0+δ] if, for any ε>0, (45)SdA,x,P-∫dAx<εholds.Proof.
Using the result of Lemma18 and Equation (22), we have (46)xv-xt0=yv-yt0v=∫t0vdAy+gv-gt0≤∫t0vdAy-SdA,y,P+SdA,y,P+gv-gt0We make the choice of the gauge function (47)δη<min⁡tk-tk-12:k=1,…,n,η∈t0,t0+δ0<δη<min⁡η-tk,η-tk-1,forη∈tk,tk-1,k=1…nThese ensure that each subinterval of δ-fine partition contains at most one of the points tk, k=1,…,n, corresponding to a tag of the interval. Hence, by Equation (46), we have(48)xsi-xsi-1≤∑i=1nAsi-Asi-1yηiv+gv-gt0=Fyηisi-Fyηi-1si-1v+∑∫tksi-∫tksi-1yηi+gsi-gsi-1v=0,t0-r≤v≤t0∫si-1vLys,sds,si-1≤v≤si≤t0+δ∫si-1siLys,sds,si-1≤si≤v≤t0+δ+0,t0-r≤v≤t0∫si-1vfsds,si-1≤v≤si≤t0+δ∫si-1sifsds,si-1≤si≤v≤t0+δ+∑∫tksi-∫tksi-1∫tkvItyηi
This implies that(49)xsi-xsi+1≤supv∈t0-r,t0+δ⁡∫si-1vLysηi,sds,si-1≤v≤si≤t0+δ∫si-1siLysηi,sds,si-1≤si≤v≤t0+δ+supv∈t0-r,t0+δ⁡∫si-1vfsds,si-1≤v≤si≤t0+δ∫si-1sifsds,si-1≤si≤v≤t0+δ+Ikytk∫tkv=supv∈si-1,si-1⁡∫sivLysηi,sds+supv∈si-1,si-1⁡∫sivfsds+Ikytk∫tkvAlso by Lemma 14 and Equation (44), for t∈[t0,v], we have(50)ytt0,φ=yt=sup-r≤θ≤0⁡yt+θ≤supt0-r≤s≤v⁡ys=supt0-r≤s≤v⁡xvs=supt0-r≤s≤+∞⁡xvs=xv<εHence, by Remark 15, we have(51)xsi-xsi-1≤supv∈si-1,si-1⁡∫sivLxsηi,sds+supv∈si-1,si-1⁡∫sivfsds+Ikxtk∫tkvvart0tx≤∫t0tM0sdsxX+Mt-t0+K0xX=Mt-t0+K0+∫t0tM0sdsxXwhich implies that the function x∈G([t0,t0+δ],X)→X is of bounded variation, and hence the existence of solution of Equation (28).Theorem 20.
Letx(t)∈G([t0,t0+δ],X) satisfy equation (28) on [t0-r,t0+δ] and let t→L(xt,t) be Lebesgue integrable. Then, there exists a y(t) satisfying the initial condition (44) such that for any v∈[t0-r,t0+δ](52)(i)ytv=xv,v∈t0-r,t0xt,v∈t0,t0+δ(ii) for tc∈[t0,t0+δ], there exists a k=k(tc)<t0+δ,0<k<1 and Δtc>0 such that (53)vartc-Δtc,tc∩t0,t0+δ⁡At<k.Then Equation (39) has a unique solution.Proof.
By Equations (28) and (52)(54)ytv=xtv=xt0v+∫t0vLxs,sds+∫t0vfsds+∑k=1nIkxtk=xt0v+Fx,t+Jx,t+∫t0vfsds=xt0v+Atx+∫t0vfsdsLet t→L(xt,t) be a Lebesgue integrable, A:G([t0,t0+δ],L(X)→L(X) a regulated function on [t0-r,t0+δ], and x:[t0,t0+δ]→X; then for a δ-fine partition si-1,si⊂ηi-δηi,ηi+δηi,i=1,2,…, and using the relation x(s)=x(ηi)=y(s),s∈[ηi,ηi-1], we have(55)ytv=xt0v+∑i=1nAsi-Asi-1xηiv+∫t0vfsds=yt0v+∫t0vdAsyη+gv-gt0Using Proposition 17 and for y(t) being a solution of equation (39) we obtained(56)I-Δ-Atyt=yt0+∫t0vdAy+gv-gt0Also, the results in Schwabik [1] show that for A∈G([t0,t0+δ],X) a regulated function there exists ε>0 such that the set t∈[t0,t0+δ],Δ-A(t)≥ε is finite. This implies that the set of discontinuity points of A is at most countable, and there is a finite set t1,t2,…,tn such that for t∈t0,t0+δ,t=tk,k=1,2,…. the operator I-Δ-A(t)∈L(X) is invertible and [I-Δ-A(t)]-1 exists.
Therefore,(57)yt=I-Δ-At-1yt0+gv-gt0+∫t0vdAysand(58)yt≤I-Δ-At-1yt0+gv+gt0+∫t0vdAys≤I-Δ-At-1N0exp⁡I-Δ-At-1vart0vA,v∈t0-r,t0+δwhere A(t) is nonnegative, nondecreasing, and left continuous function, y(t) is nonnegative and bounded function.
Using preposition 2.1 (Schwabik, [1]), we defined a bounded linear operator T:G([t0,t0+δ],X)→G([t0,t0+δ],X) on G([t0,t0+δ],X) so that by Equation (53)(59)Tyt=yt0+∫t0vdAys+gv-gt0.If y,z∈G([tc-Δtc,tc]∩[t0,t0+δ],X)→X for y(tc)=z(tc)=y(t0) then (60)Ty-Tzt≤limε→0-⁡∫tc-εtdAsys-zs≤limε→0-⁡vart,tc-ε⁡Ay-ztc-Δtc,tc∩t0,t0+δ≤vartc-Δtc,tc∩t0,t0+δ⁡Ay-ztc-Δtc,tc∩t0,t0+δ≤ky-zwhere k satisfied the hypothesis of the theorem as stated and the operator T∈G([tc-Δtc,t],X) 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) w(t):[t0,t0+δ]→X which is Lebesgue integrable. The system equation is of the form(61)x•t=∫t-rtpt-sgxsds+wt,t≠τkxtkΔxt=Ikxtk,t≠τkxtk,k=1,2,…xt0=φwhere r>0,φ∈G([-r,0],X),p:R→R+ and w(t):[t0,t0+δ]→X are locally Lebesgue integrable functions such that w(t)≤M,M>0 and p(u)≤B for all u∈R,g:R→R is increasing function such that g(0)=0,g(x)≤Kx-y,x,y∈R with K≥0. Let there exists a function m:R→R locally Lebesgue integrable such that (62)∫s1s2gxsds≤∫s1s2msds,for all s1,s2∈R, and for k=1,2…, maps R to (0,∞).DefiningL(xt,t)=∫t-rtp(t-s)g(x(s))ds, Afonso [18] proved that conditions (A1), and (A2) of Remark 15 are satisfied.Also, by the hypothesis of the problem, we have that(63)∫u1u2fsds≤∫u1u2wsds≤∫u1u2wsds≤∫u1u2Mds,Hence condition (A3) is satisfied.Considering the impulsive termΔx(t)=Ik(x(tk)), we have (64)Ikxs-Ikys≤sup⁡∑k=1mIkxs-Ikys≤K2x-y,Ikxs2≤K1x,forK1,K2>0,and condition (A4) is satisfied. Hence, in consequence of Remark 15 the integral solution(65)xt=φ+∫u1u2-∫s-rsps-ugxududs+∫u1u2wsds+∑k=1mIkxtkexists 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.



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*Source: 2523615-2019-03-20.xml*