MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2015/849731 849731 Research Article Rapid Convergence of Solution for Hybrid System with Causal Operators http://orcid.org/0000-0001-5169-8750 Wang Peiguang 1 Li Zhifang 1 Wu Yonghong 2 Xia Weiguo 1 College of Mathematics and Information Science Hebei University Baoding 071002 China hbu.edu.cn 2 Department of Mathematics and Statistics Curtin University Perth WA 6845 Australia curtin.edu.au 2015 20102015 2015 04 08 2015 11 10 2015 20102015 2015 Copyright © 2015 Peiguang Wang 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.

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 order k2.

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  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 .

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 . 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 k2.

2. Preliminaries

In this section, we present the following definition and lemma which will help to prove our main result.

Let E=C(I,R), where I=[0,T], T>0 is an appropriate positive constant, and QC(E,E).

Definition 1 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

The operator Q 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 0st, one also has (Qx)(s)=(Qy)(s) for 0st with t<T,  T being arbitrary.

Let the points tjI 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,ttj,tj+1,  j=0,1,,p,u0=uT, where QC(I×R×R,R) is a continuous causal operator, the functions Λj:RR are increasing, and there exist constants Lj>0 such that, for any points tjR and u(tj)v(tj), the following equalities or inequalities are satisfied: (2)Λjutj-Λjvtj=Λjutj-vtj,Λjutjk=Λjuktj,Λjutj-ΛjvtjLjutj-vtj, and if m<0, then Λjm<0; that is, Λj(-a)=-Λja, in which a>0,  j=0,1,,p.

The function α(t)C1(I,R) is called a lower solution of the PBVP (1) if the following inequalities are satisfied:(3)αtQt,αt,Λjαtj,for  ttj,tj+1,  j=0,1,,p,α0αT.

Analogously, we can define an upper solution of the PBVP (1) by introducing the inequalities in (3) in opposite directions.

Let the functions α,βC1(I,R) be such that α(t)β(t). Consider the sets (4)Ω=uCI,R:αtutβt.

Similar to the proof of Theorem 3.2.1 in , we have the following lemma.

Lemma 2.

Let v,wC(I,R) be lower and upper solutions of the PBVP (1) satisfying v(t)w(t), tI. 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), tI.

3. Main Result

Consider the Banach space CI 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 n0N and λ>0 such that (5)xn+1-xλxn-xk,nn0;  λ  is a constant.

Theorem 3.

Let the following conditions hold:

The functions α(t), β(t) are lower and upper solutions to the PBVP (1) and α(t)β(t) for t[0,T].

There exist continuous functions (t,u(t),Λj(u(tj))), iQ/(Λj(uj))i(t,u(t),Λj(u(tj))), and constants Mi0 and Ni0 such that(6)iQuit,ut,Λjutj-i!Mi,iQΛjujit,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 k2.

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-1iQuit,vt,Λjvtju-vii!+kQukt,ξt,Λjξtju-vkk!+i=0k-1iQΛjutjit,vt,ΛjvtjΛjutj-Λjvtjii!+kQΛjutjkt,ξt,ΛjξtjΛjutj-Λjvtjkk!, where ξ(t)[v,u], α(t)vuβ(t), define the following function:(8)gt,ut,Λjutj;vt,Λjvtj=i=0k-1iQuit,vt,Λjvtju-vii!-Mku-vk+i=0k-1iQΛjutjit,vt,ΛjvtjΛjutj-Λjvtjii!-NkΛjutj-Λjvtjk, in which the function gC(I×R×R×R×R,R). Using (H2), (7), and (8), we get(9)gt,ut,Λjutj;vt,ΛjvtjQt,ut,Λjutj,tI,  v,uΩ.

Now, consider the following boundary value problem:(10)ut=gt,ut,Λjutj;αt,Λjαtj,tI,u0=uT.

It follows from (9) that (11)βtQt,βt,Λjβtjgt,βt,Λjβtj;αt,Λjαtj,β0βT,(12)αtQt,αt,Λjαtj=gt,αt,Λjαtj;αt,Λjαtj,α0αT. That is, α and β are lower and upper solutions of (10), respectively.

Thus, using Lemma 2, we conclude that problem (10) has the unique solution α1 and α1[α,β].

Now, suppose that α0=αα1αnβ, where αn is the unique solution of (13)ut=gt,ut,Λjutj;αn-1t,Λjαn-1tj,tI,u0=uT.

In this case, we have (14)βtQt,βt,Λjβtjgt,βt,Λjβtj;αnt,Λjαntj,β0βT,αnt=gt,αnt,Λjαntj;αn-1t,Λjαn-1tjQt,αnt,Λjαntj=gt,αnt,Λjαntj;αnt,Λjαntj,αn0=αT.

We conclude, using again Lemma 2, that there exists a unique solution αn+1[αn,β] for(15)ut=gt,ut,Λjutj;αnt,Λjαntj,tI,u0=uT.

Thus, we know that {αn} is a nondecreasing sequence and is bounded in C1(I). According to the standard arguments (see ), 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 order k. For this purpose, using (7), we have(18)ψt=i=0k-1iQuit,αnt,Λjαntjψt-αntii!+kQukt,ρnt,Λjρntjψt-αntkk!+i=0k-1iQΛjutjit,αnt,ΛjαntjΛjψtj-Λjαntjii!+kQΛjutjkt,ρnt,ΛjρntjΛjψtj-Λjαntjkk!,ψ0=ψT,where  ρnαn,ψ.

On the other hand, by (8) and (15), it is verified that, for n0, (19)αn+1t=gt,αn+1t,Λjαn+1tj;αnt,Λjαntj=i=0k-1iQuit,αnt,Λjαntjαn+1t-αntii!-Mkαn+1t-αntk+i=0k-1iQΛjutjit,αnt,ΛjαntjΛjαn+1tj-Λjαntjii!-NkΛjαn+1tj-Λjαntjk,αn+10=αn+1T.

Let en+1=ψ-αn+1 and an=αn+1-αn; then ank(t)enk(t), for all nN and tI. Thus, we have (20)en+1=i=0k-1iQuit,αnt,Λjαntjenit-aniti!+kQukt,ρnt,Λjρntjenktk!+Mkankt+i=0k-1iQΛjutjit,αnt,ΛjαntjΛjentji-Λjantjii!+kQΛjutjkt,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)kQukt,ut,Λjutjk!Ak,kQΛjutjkt,ut,Λjutjk!Bk,t,uΩ. Finally, as for all E,FR,  Ei-Fi=(E-F)j=0i-1Ei-1-jFj, we get that (22)en+1t-Pnten+1t-HntΛjen+1tjCkenkt+DkΛjenktj,tI, where Ck=Ak+Mk>0, Dk=Bk+Nk>0, and (23)Pnt=i=0k-1iQuit,αnt,Λjαntji!j=0i-1eni-1-jtanjtHnt=i=0k-1iQΛjutjit,αnt,Λjαntji!j=0i-1Λjentji-1-jΛjantjj.

Since {αn} converges uniformly to ψ in I, (21) implies that there exists n0N and P>0,H>0, such that Pn(t)-P<0, Hn(t)-H<0, for n>n0, and tI. Then, there exists a continuous function σn0 on I such that (24)en+1t+Pen+1t+HΛjen+1tj=Ckenkt+σnt+DkΛjenktj+Λjσntj,tI,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 , 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 tI and nn0, (28)0ψt-αn+1tCkPmaxtIenkt+DkHΛjenktjCkPmaxtIenkt+LjDkHenktjCkPmaxtIenkt+LjDkHenkt, where enktjenkt=max{enkt:t[0,T]}. Hence, (29)ψt-αn+1tλψ-αntk, for all nn0, 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-1iQuit,vt,Λjvtju-vii!-Mku-vk+i=0k-1iQΛjutjit,vt,ΛjvtjΛjutj-Λjvtjii!-NkΛjutj-Λjvtjk,if  k  is odd,i=0k-1iQuit,vt,Λjvtju-vii!+Aku-vk+i=0k-1iQΛjutjit,vt,ΛjvtjΛjutj-Λjvtjii!+BkΛjutj-Λjvtjk,if  k  is even,where the function hC(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,ΛjvtjQt,ut,Λjutj,tI,  u,vΩ.

Now, let β0=β; for n1, we define βn by induction, as the unique solution of the following boundary value problem:(32)ut=ht,ut,Λjutj;βn-1t,Λjβn-1tj,tI,u0=uT. We can obtain αβnβn-1β2β1β0β. 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)ψt=i=0k-1iQuit,βnt,Λjβntjψt-βntii!+kQukt,ρnt,Λjρntjψt-βntkk!+i=0k-1iQΛjutjit,βnt,ΛjβntjΛjψtj-Λjβntjii!+kQΛjutjkt,ρnt,ΛjρntjΛjψtj-Λjβntjkk!,ψ0=ψT,ρnψ,βn.

On the other hand, by (30) and (32), it is verified that, for n0, if k is odd, then (36)βn+1t=ht,βn+1t,Λjβn+1tj;βnt,Λjβntj=i=0k-1iQuit,βnt,Λjβntjβn+1t-βntii!-Mkβn+1t-βntk+i=0k-1iQΛjutjit,βnt,ΛjβntjΛjβn+1tj-Λjβntjii!-NkΛjβn+1tj-Λjβntjk,while if k is even, then (37)βn+1t=i=0k-1iQuit,βnt,Λjβntjβn+1t-βntii!+Akβn+1t-βntk+i=0k-1iQΛjutjit,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-1iQuit,βnt,Λjβntjbnit-fniti!-kQukt,ρnt,Λjρntjfnktk!-Mkbnkt+i=0k-1iQΛjutjit,βnt,ΛjβntjΛjbntji-Λjfntjii!-kQΛjutjkt,ρnt,ΛjρntjΛjfntjkk!-NkΛjbntjk, while if k is even, then (39)-fn+1=βn+1-ψ=i=0k-1iQuit,βnt,Λjβntjbnit-fniti!-kQukt,ρnt,Λjρntjfnktk!+Akbnkt+i=0k-1iQΛjutjit,βnt,ΛjβntjΛjbntji-Λjfntjii!-kQΛjutjkt,ρnt,ΛjρntjΛjfntjkk!+BkΛjbntjk. Furthermore (40)-1kbnkt-1kfnkt,if  k  odd,bnktfnkt,if  k  even, for all nN and tI. We can write that if k is odd, then (41)-fn+1t-Pnt-fn+1t-Hnt-Λjfn+1tj2Mk-fnkt+2Nk-Λjfnktj, while if k is even, then (42)-fn+1t-Pnt-fn+1t-Hnt-Λjfn+1tjCkfnkt+DkΛjfnktj, where tI,  Ck=Ak+Mk>0,  Dk=Bk+Nk>0, and (43)Pnt=i=0k-1iQuit,βnt,Λjβntji!j=0i-1bni-1-jtfnjt,Hnt=i=0k-1iQΛjutjit,βnt,Λjβntji!j=0i-1Λjbntji-1-jΛjfntjj. Since {βn} converges uniformly to ψ in I, (21) implies that there exist n0N and P>0,H>0, such that Pn(t)-P<0, Hn(t)-H<0, for n>n0 and tI. Thus, there exists a continuous function σn0 on I such that if k is odd, (44)-fn+1t-P-fn+1t-H-Λjfn+1tj=2Mk-fnkt+σnt+2Nk-Λjfnktj+Λjσntj,fn+10=fn+1T; if k is even, (45)-fn+1t-P-fn+1t-H-Λjfn+1tj=Ckfnkt+σnt+DkΛjbnktj+Λjσntj,(46)fn+10=fn+1T. Or equivalently, if k is odd, then (47)-fn+1t=0TGt,s,P2Mk-fnkt+σntds+0TGt,s,H2Nk-Λjfnktj+Λjσntjds, while if k is even, then (48)-fn+1t=0TGt,s,PCkfnkt+σntds+0TGt,s,HDkΛjfnktj+Λjσntjds, where G is the same with the above.

We conclude that, for every tI and nn0, if k is odd, then (49)0βn+1t-ψt2MkPmax-fnkt+2NkLjH-fnktj2MkPmax-fnkt+2NkLjH-fnkt,ψt-βn+1tλ1ψt-βntk, for all nn0 and λ1=max{2Mk/P,2NkLj/H}, while if k is even, then (50)0βn+1t-ψtCkPmaxfnkt+DkLjHfnktjCkPmaxfnkt+DkLjHfnkt, and hence (51)ψt-βn+1tλ2ψt-βntk for all nn0 and λ2=max{Ck/P,DkLj/H}.

The proof is complete.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).

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