# Rapid Convergence of Solution for Hybrid System with Causal Operators

Authors: Peiguang Wang; Zhifang Li; Yonghong Wu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We investigated the convergence of iterative sequences of approximate solutions to a class of periodic boundary value problem of hybrid system with causal operators and established two sequences of approximate solutions that converge to the solution of the problem with rate of orderk≥2.


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## Body

## 1. Introduction

Recently, the problem of qualitative theory of dynamic systems with causal operators has attracted much attention since such systems include several types of dynamic systems, such as ordinary differential equations, integrodifferential equations, differential equations with finite or infinite delay, Volterra integral equations, and neutral equations. Therefore, the study of the theory of causal systems becomes very important. This is because a single result involving causal operators covers interesting corresponding results from many categories of dynamic systems, thus avoiding duplication and monotony of repetition. For more details, we can refer to the monographs [1–9] and the references cited therein. Since it is difficult to find the solutions of differential equations with causal operators, we need to look for the approximate solutions. Quasilinearization combined with the technique of upper and lower solutions is an effective and fruitful technique for obtaining approximate solutions to a wide variety of nonlinear problems. The main advantages of the method are the practicality of finding successive approximations of the unknown solution as well as the quadratic convergence rate. Some recent results in the development of the method and its real-world applications can be found in [10–19].Hybrid systems have also attracted much attention in recent years. Hybrid systems are dynamical systems that evolute continuously in time but have formatting changes, called modes, at a sequence of discrete times. Some recent works on hybrid systems are included in [20–26]. However, to our best knowledge, very few results have been achieved on hybrid system with causal operators; particularly methods for finding approximate solutions with rapid convergence are yet to be developed. Hence, the purpose of this paper is to develop the method of quasilinearization for the periodic boundary value problem of hybrid system with causal operators. We will prove that the problem has solutions which can be approximated via monotone sequences with rate of convergence of order k≥2.


## 2. Preliminaries

In this section, we present the following definition and lemma which will help to prove our main result.LetE=C(I,R), where I=[0,T], T>0 is an appropriate positive constant, and Q∈C(E,E).Definition 1 (see [2]).
The operatorQ is said to be a causal or nonanticipatory operator if the following property is satisfied: for each couple of elements x, y of E such that x(s)=y(s) for 0≤s≤t, one also has (Qx)(s)=(Qy)(s) for 0≤s≤t with t<T,  T being arbitrary.Let the pointstj∈I be fixed such that t0=0,  tp+1=T and tj<tj+1,  j=0,1,2,…,p.We consider the following periodic boundary value problem (PBVP) of hybrid system with causal operators:(1)u′=Qt,ut,Λjutj,t∈tj,tj+1,j=0,1,…,p,u0=uT, where Q∈C(I×R×R,R) is a continuous causal operator, the functions Λj:R→R are increasing, and there exist constants Lj>0 such that, for any points tj∈R and u(tj)≤v(tj), the following equalities or inequalities are satisfied: (2)Λjutj-Λjvtj=Λjutj-vtj,Λjutjk=Λjuktj,Λjutj-Λjvtj≤Ljutj-vtj, and if m<0, then Λjm<0; that is, Λj(-a)=-Λja, in which a>0,  j=0,1,…,p.The functionα(t)∈C1(I,R) is called a lower solution of the PBVP (1) if the following inequalities are satisfied:(3)α′t≤Qt,αt,Λjαtj,fort∈tj,tj+1,j=0,1,…,p,α0≤αT.Analogously, we can define an upper solution of the PBVP (1) by introducing the inequalities in (3) in opposite directions.Let the functionsα,β∈C1(I,R) be such that α(t)≤β(t). Consider the sets (4)Ω=u∈CI,R:αt≤ut≤βt.Similar to the proof of Theorem 3.2.1 in [2], we have the following lemma.Lemma 2.
Letv,w∈C(I,R) be lower and upper solutions of the PBVP (1) satisfying v(t)≤w(t), t∈I. Suppose that the operator Q is bounded on Ω. Then, there exists a solution x(t) of (1) in the closed set Ω, such that v(t)≤x(t)≤w(t), t∈I.


## 3. Main Result

Consider the Banach spaceCI with the usual norm ·∞. For a given sequence {xn}⊂C(I), we say that {xn} converges to x with order of convergence k, if {xn} converges to x in C(I) and there exist n0∈N and λ>0 such that (5)xn+1-x∞≤λxn-x∞k,n≥n0;λis a constant.Theorem 3.
Let the following conditions hold:(H1)
The functionsα(t), β(t) are lower and upper solutions to the PBVP (1) and α(t)≤β(t) for t∈[0,T].(H2)
There exist continuous functions(t,u(t),Λj(u(tj))), ∂iQ/∂(Λj(uj))i(t,u(t),Λj(u(tj))), and constants Mi≥0 and Ni≥0 such that(6)∂iQ∂uit,ut,Λjutj≥-i!Mi,∂iQ∂Λjujit,ut,Λjutj≥-i!Ni,i=0,1,…,k. Then there exist two monotone sequences {αn} and {βn} with α0=α and β0=β, which converge uniformly to the unique solution ψ of the PBVP (1), and the convergence is of order k≥2.Proof.
Firstly, we note that the condition(H2) implies that the PBVP (1) has a unique solution ψ(t) between α(t) and β(t). To construct the sequence {αn}, for given(7)Qt,ut,Λjutj=∑i=0k-1∂iQ∂uit,vt,Λjvtju-vii!+∂kQ∂ukt,ξt,Λjξtju-vkk!+∑i=0k-1∂iQ∂Λjutjit,vt,ΛjvtjΛjutj-Λjvtjii!+∂kQ∂Λjutjkt,ξt,ΛjξtjΛjutj-Λjvtjkk!, where ξ(t)∈[v,u], α(t)≤v≤u≤β(t), define the following function:(8)gt,ut,Λjutj;vt,Λjvtj=∑i=0k-1∂iQ∂uit,vt,Λjvtju-vii!-Mku-vk+∑i=0k-1∂iQ∂Λjutjit,vt,ΛjvtjΛjutj-Λjvtjii!-NkΛjutj-Λjvtjk, in which the function g∈C(I×R×R×R×R,R). Using (H2), (7), and (8), we get(9)gt,ut,Λjutj;vt,Λjvtj≤Qt,ut,Λjutj,t∈I,v,u∈Ω.
Now, consider the following boundary value problem:(10)u′t=gt,ut,Λjutj;αt,Λjαtj,t∈I,u0=uT.
It follows from (9) that (11)β′t≥Qt,βt,Λjβtj≥gt,βt,Λjβtj;αt,Λjαtj,β0≥βT,(12)α′t≤Qt,αt,Λjαtj=gt,αt,Λjαtj;αt,Λjαtj,α0≤αT. That is, α and β are lower and upper solutions of (10), respectively.
Thus, using Lemma2, we conclude that problem (10) has the unique solution α1 and α1∈[α,β].
Now, suppose thatα0=α≤α1≤⋯αn≤β, where αn is the unique solution of (13)u′t=gt,ut,Λjutj;αn-1t,Λjαn-1tj,t∈I,u0=uT.
In this case, we have(14)β′t≥Qt,βt,Λjβtj≥gt,βt,Λjβtj;αnt,Λjαntj,β0≥βT,αn′t=gt,αnt,Λjαntj;αn-1t,Λjαn-1tj≤Qt,αnt,Λjαntj=gt,αnt,Λjαntj;αnt,Λjαntj,αn0=αT.
We conclude, using again Lemma2, that there exists a unique solution αn+1∈[αn,β] for(15)u′t=gt,ut,Λjutj;αnt,Λjαntj,t∈I,u0=uT.
Thus, we know that{αn} is a nondecreasing sequence and is bounded in C1(I). According to the standard arguments (see [12]), the Ascoli-Arzela Theorem guarantees the existence of a subsequence which converges uniformly to a continuous function ψ∈[α,β].
Since(16)αnt=u0+∫0tgs,αns,Λjαntj;αn-1t,Λjαn-1tjds we have (17)ψt=u0+∫0tgs,ψs,Λjψtj;ψt,Λjψtjds=u0+∫0tQs,ψs,Λjψtjds, and ψ is the unique solution of the PBVP (1) in [α,β].
Now, we prove that the convergence is of orderk. For this purpose, using (7), we have(18)ψ′t=∑i=0k-1∂iQ∂uit,αnt,Λjαntjψt-αntii!+∂kQ∂ukt,ρnt,Λjρntjψt-αntkk!+∑i=0k-1∂iQ∂Λjutjit,αnt,ΛjαntjΛjψtj-Λjαntjii!+∂kQ∂Λjutjkt,ρnt,ΛjρntjΛjψtj-Λjαntjkk!,ψ0=ψT,whereρn∈αn,ψ.
On the other hand, by (8) and (15), it is verified that, for n≥0, (19)αn+1′t=gt,αn+1t,Λjαn+1tj;αnt,Λjαntj=∑i=0k-1∂iQ∂uit,αnt,Λjαntjαn+1t-αntii!-Mkαn+1t-αntk+∑i=0k-1∂iQ∂Λjutjit,αnt,ΛjαntjΛjαn+1tj-Λjαntjii!-NkΛjαn+1tj-Λjαntjk,αn+10=αn+1T.
Leten+1=ψ-αn+1 and an=αn+1-αn; then ank(t)≤enk(t), for all n∈N and t∈I. Thus, we have (20)en+1′=∑i=0k-1∂iQ∂uit,αnt,Λjαntjenit-aniti!+∂kQ∂ukt,ρnt,Λjρntjenktk!+Mkankt+∑i=0k-1∂iQ∂Λjutjit,αnt,ΛjαntjΛjentji-Λjantjii!+∂kQ∂Λjutjkt,ρnt,ΛjρntjΛjentjkk!+NkΛjantjk. The continuity of ∂kQ/∂uk and ∂kQ/∂(Λj(u(tj)))k in Ω implies that there exist Ak>0 and Bk>0 such that(21)∂kQ∂ukt,ut,Λjutj≤k!Ak,∂kQ∂Λjutjkt,ut,Λjutj≤k!Bk,t,u∈Ω. Finally, as for all E,F∈R,  Ei-Fi=(E-F)∑j=0i-1Ei-1-jFj, we get that (22)en+1′t-Pnten+1t-HntΛjen+1tj≤Ckenkt+DkΛjenktj,t∈I, where Ck=Ak+Mk>0, Dk=Bk+Nk>0, and (23)Pnt=∑i=0k-1∂iQ∂uit,αnt,Λjαntji!∑j=0i-1eni-1-jtanjtHnt=∑i=0k-1∂iQ∂Λjutjit,αnt,Λjαntji!∑j=0i-1Λjentji-1-jΛjantjj.
Since{αn} converges uniformly to ψ in I, (21) implies that there exists n0∈N and P>0,H>0, such that Pn(t)≤-P<0, Hn(t)≤-H<0, for n>n0, and t∈I. Then, there exists a continuous function σn≤0 on I such that (24)en+1′t+Pen+1t+HΛjen+1tj=Ckenkt+σnt+DkΛjenktj+Λjσntj,t∈I,en+10=en+1T, or equivalently (25)en+1t=∫0TGt,s,PCkenkt+σntds+∫0TGt,s,HDkΛjenktj+Λjσntjds, where G is the Green function associated with the following linear boundary value problem:(26)u′+Pu+HΛjutj=σt+Λjσtj,u0=uT. From [4], we have that G is positive on I×I, since the solution of problem (26) is given by (27)ut=∫0TGt,s,Pσsds+∫0TGt,s,HΛjσsjds, where ∫0TG(t,s,P)ds=1/P, ∫0TG(t,s,H)ds=1/H. We can thus conclude that, for any t∈I and n≥n0, (28)0≤ψt-αn+1t≤CkPmaxt∈I⁡enkt+DkHΛjenktj≤CkPmaxt∈I⁡enkt+LjDkHenktj≤CkPmaxt∈I⁡enkt+LjDkHenkt, where enktj≤enkt=max⁡{enkt:t∈[0,T]}. Hence, (29)ψt-αn+1t∞≤λψ-αnt∞k, for all n≥n0, and λ=max⁡{Ck/P,LjDk/H}>0.
Similarly, to construct the sequence{βn}, define the following function:(30)ht,ut,Λjutj;vt,Λjvtj=∑i=0k-1∂iQ∂uit,vt,Λjvtju-vii!-Mku-vk+∑i=0k-1∂iQΛjutjit,vt,ΛjvtjΛjutj-Λjvtjii!-NkΛjutj-Λjvtjk,ifkis odd,∑i=0k-1∂iQ∂uit,vt,Λjvtju-vii!+Aku-vk+∑i=0k-1∂iQΛjutjit,vt,ΛjvtjΛjutj-Λjvtjii!+BkΛjutj-Λjvtjk,ifkis even,where the function h∈C(I×R×R×R×R,R), and Mk,Nk,Ak, and Bk are nonnegative constants given by (6) and (21), respectively. Similar to the discussion of g(t,u(t),Λj(u(tj));v(t),Λj(v(tj))) above, we have (31)ht,ut,Λjutj;vt,Λjvtj≥Qt,ut,Λjutj,t∈I,u,v∈Ω.
Now, letβ0=β; for n≥1, we define βn by induction, as the unique solution of the following boundary value problem:(32)u′t=ht,ut,Λjutj;βn-1t,Λjβn-1tj,t∈I,u0=uT. We can obtain α≤βn≤βn-1≤⋯≤β2≤β1≤β0≤β. Similar to the discussion of {αn}, {βn} is a nonincreasing sequence and is bounded in C1(I). Then {βn} converges uniformly in C(I) to the continuous function ψ∈[α,β]. Since (33)βnt=u0+∫0tht,βns,Λjβntj;βn-1s,Λjβn-1tjds,we have (34)ψt=u0+∫0ths,ψs,Λjψtj;ψs,Λjψtjds=u0+∫0tQs,ψs,Λjψtjds.
Therefore,ψ is the unique solution of the PBVP (1) in [α,β]. Furthermore, we prove that the convergence is of order k. For this purpose, using (7), we have (35)ψ′t=∑i=0k-1∂iQ∂uit,βnt,Λjβntjψt-βntii!+∂kQ∂ukt,ρnt,Λjρntjψt-βntkk!+∑i=0k-1∂iQ∂Λjutjit,βnt,ΛjβntjΛjψtj-Λjβntjii!+∂kQ∂Λjutjkt,ρnt,ΛjρntjΛjψtj-Λjβntjkk!,ψ0=ψT,ρn∈ψ,βn.
On the other hand, by (30) and (32), it is verified that, for n≥0, if k is odd, then (36)βn+1′t=ht,βn+1t,Λjβn+1tj;βnt,Λjβntj=∑i=0k-1∂iQ∂uit,βnt,Λjβntjβn+1t-βntii!-Mkβn+1t-βntk+∑i=0k-1∂iQ∂Λjutjit,βnt,ΛjβntjΛjβn+1tj-Λjβntjii!-NkΛjβn+1tj-Λjβntjk,while if k is even, then (37)βn+1′t=∑i=0k-1∂iQ∂uit,βnt,Λjβntjβn+1t-βntii!+Akβn+1t-βntk+∑i=0k-1∂iQ∂Λjutjit,βnt,ΛjβntjΛjβn+1tj-Λjβntjii!+BkΛjβn+1tj-Λjβntjk. Let fn=ψ-βn, and bn=βn+1-βn. Then, we have that if k is odd, then (38)-fn+1′=βn+1′-ψ′=∑i=0k-1∂iQ∂uit,βnt,Λjβntjbnit-fniti!-∂kQ∂ukt,ρnt,Λjρntjfnktk!-Mkbnkt+∑i=0k-1∂iQ∂Λjutjit,βnt,ΛjβntjΛjbntji-Λjfntjii!-∂kQ∂Λjutjkt,ρnt,ΛjρntjΛjfntjkk!-NkΛjbntjk, while if k is even, then (39)-fn+1′=βn+1′-ψ′=∑i=0k-1∂iQ∂uit,βnt,Λjβntjbnit-fniti!-∂kQ∂ukt,ρnt,Λjρntjfnktk!+Akbnkt+∑i=0k-1∂iQ∂Λjutjit,βnt,ΛjβntjΛjbntji-Λjfntjii!-∂kQ∂Λjutjkt,ρnt,ΛjρntjΛjfntjkk!+BkΛjbntjk. Furthermore (40)-1kbnkt≤-1kfnkt,ifkodd,bnkt≤fnkt,ifkeven, for all n∈N and t∈I. We can write that if k is odd, then (41)-fn+1′t-Pnt-fn+1t-Hnt-Λjfn+1tj≤2Mk-fnkt+2Nk-Λjfnktj, while if k is even, then (42)-fn+1′t-Pnt-fn+1t-Hnt-Λjfn+1tj≤Ckfnkt+DkΛjfnktj, where t∈I,  Ck=Ak+Mk>0,  Dk=Bk+Nk>0, and (43)Pnt=∑i=0k-1∂iQ∂uit,βnt,Λjβntji!∑j=0i-1bni-1-jtfnjt,Hnt=∑i=0k-1∂iQ∂Λjutjit,βnt,Λjβntji!∑j=0i-1Λjbntji-1-jΛjfntjj. Since {βn} converges uniformly to ψ in I, (21) implies that there exist n0∈N and P>0,H>0, such that Pn(t)≤-P<0, Hn(t)≤-H<0, for n>n0 and t∈I. Thus, there exists a continuous function σn≤0 on I such that if k is odd, (44)-fn+1′t-P-fn+1t-H-Λjfn+1tj=2Mk-fnkt+σnt+2Nk-Λjfnktj+Λjσntj,fn+10=fn+1T; if k is even, (45)-fn+1′t-P-fn+1t-H-Λjfn+1tj=Ckfnkt+σnt+DkΛjbnktj+Λjσntj,(46)fn+10=fn+1T. Or equivalently, if k is odd, then (47)-fn+1′t=∫0TGt,s,P2Mk-fnkt+σntds+∫0TGt,s,H2Nk-Λjfnktj+Λjσntjds, while if k is even, then (48)-fn+1′t=∫0TGt,s,PCkfnkt+σntds+∫0TGt,s,HDkΛjfnktj+Λjσntjds, where G is the same with the above.
We conclude that, for everyt∈I and n≥n0, if k is odd, then (49)0≤βn+1t-ψt≤2MkPmax⁡-fnkt+2NkLjH-fnktj≤2MkPmax⁡-fnkt+2NkLjH-fnkt,ψt-βn+1t∞≤λ1ψt-βnt∞k, for all n≥n0 and λ1=max{2Mk/P,2NkLj/H}, while if k is even, then (50)0≤βn+1t-ψt≤CkPmax⁡fnkt+DkLjHfnktj≤CkPmax⁡fnkt+DkLjHfnkt, and hence (51)ψt-βn+1t∞≤λ2ψt-βnt∞k for all n≥n0 and λ2=max⁡{Ck/P,DkLj/H}.
The proof is complete.



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