# Convergence of Antiperiodic Boundary Value Problems for First-Order Integro-Differential Equations

Authors: Yameng Wang; Juan Zhang; Yufeng Sun
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we investigate the convergence of approximate solutions for a class of first-order integro-differential equations with antiperiodic boundary value conditions. By introducing the definitions of the coupled lower and upper solutions which are different from the former ones and establishing some new comparison principles, the results of the existence and uniqueness of solutions of the problem are given. Finally, we obtain the uniform and rapid convergence of the iterative sequences of approximate solutions via the coupled lower and upper solutions and quasilinearization method. In addition, an example is given to illustrate the feasibility of the method.


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## Body

## 1. Introduction

In recent decades, the integro-differential equations have developed rapidly because the models described by various of integro-differential equations have appeared in a number of fields such as fluid dynamics, biology, economics, and control theory, for details and examples, we can refer to references [1–7] and cited therein. Meanwhile, the qualitative theory of integral differential equations creates an branch of nonlinear analysis. Boundary value problems for various first-order integral differential equations have been studied by several researchers, and there are some results on the existence of solutions and extremal solutions, the controllability problem controllability of integral boundary value conditions, and antiperiodic boundary value conditions, such as ordinary differential equations [8–12], difference equations [13, 14], fractional differential equations [15–20], impulsive differential equations [9, 14, 21, 22], integro-differential equations, and impulsive functional differential equations [18, 23–26].However, we found that most of these known results concerned with the existence and uniformly convergence results of solutions and extremal solutions via the method of upper and lower solutions coupled with the monotone iterative technique (see [27]). It is well known that the method of quasilinearization (QSL) provides a powerful tool for obtaining convergence of approximate solutions of nonlinear problems [28, 29]. The technique of upper and lower solutions coupled with the QSL have been applied successfully to obtain monotone sequences of approximate solutions converging uniformly and quadratically to the unique solution of integro-differential equations with antiperiodic boundary value conditions [30–32]. In terms of applications, it is important to pay attention to the high-order convergence of sequences of approximate solutions. The high-order convergence results of various differential equations can be found in [33–39].In this paper, we consider the following first-order integro-differential equations with antiperiodic boundary value conditions (APBVP):(1)x′t=ft,xt,Txt,t∈J,x0=−xT,where f∈CJ×R2,R,J=0,T,Txt=∫0tkt,sxsds,k∈CD,R+,k0=supkt,s,D=t,s∈J×J:t≥s.The aim of this paper is to investigate the convergence of approximate solutions of the problem. We give the particular definitions of the coupled lower and upper related solutions which are new and establish some new comparison principles in order to discuss the existence and uniqueness of the solutions. Then, by using the method of quasilinearization, we obtain the two monotone sequences of approximate solutions converging to the unique solution of the problem with rate of convergence of orderk. Finally, we give an example to illustrate our main results.


## 2. Comparison Theorems

In this section, we begin with some comparison principles that will be useful in later discussions.Lemma 1.
Assume that there exist constantsM>0, N>0, and γ≥0, such that(2)M+k0NTT≤1.If there exists a functionp∈C1J,R, such that(3)p′t≤−Mpt+γ−NTpt,t∈J,p0≤pT,then pt≤0 on J.Proof.
Suppose the conclusion is not true, and we consider the following two cases, wherep0≤0 and p0>0, respectively.Case 1.
Whenp0≤0, there exists a t∗∈0,T, such that pt∗>0. Let t∗∈0,t∗, such that pt∗=infpt=−b, b≥0. By equation (3), we have(4)p′t≤bM+k0TN.
Integrating inequality (4) from t∗ to t∗, we obtain(5)0<pt∗≤pt∗+b∫t∗t∗M+k0TNdt.
Thus,(6)b<b∫0TM+k0TNdt,which contradicts (2), therefore pt≤0.Case 2.
Whenp0>0, there are two cases: pt>0 for t∈J or there exist t¯, t¯, such that pt¯≤0 and pt¯>0 for t¯,t¯∈J.Case 3.
Whenpt>0, by equation (3), we have p′t≤0, which contradicts the condition of equation (3).Case 4.
If there existt¯ and t¯, such that pt¯≤0 and pt¯>0, we have qt˜∗=infpt=−b, where t˜∗∈0,T, b≥0. Then, equation (4) holds. Integrating inequality (4) from t˜∗ to T, we have(7)0<pt¯≤pt˜∗+b∫t˜∗TM+k0TNdt,which is also a contradiction. The proof of Lemma 1 is completed.
Next, consider the linear APBVP:(8)x′t+Mxt+γ+NTxt=0,t∈J,x0=−xT.Corollary 1.
Assume thatM>N>0 and M+k0NTT≤1, then APBVP (8) has at most one solution.Proof.
Letx1,x2 be any solution of APBVP (8), x1≥x2, and yt=x1t−x2t, then(9)y′t+Myt+NTyt=0,y0=−yT.
IfyT>0, then it follows from (9) that y0<0. By Lemma 1, we have yt≤0, that is a contradiction. On the contrary, if yT<0, we have y0>0. By the proof of Lemma 1, we have yt≤0, that is also a contradiction. Therefore, we have yT=y0=0. Furthermore, by Lemma 1, we have yt≤0, that is, yt=0 for t∈J. The proof of Corollary 1 is completed.
Similar to the proof of Lemma1, we have the following lemma.Lemma 2.
Assume that there exist integrable functionsϕit<0, i=1,2, such that(10)∫0Tϕ1t+ϕ2tk0Tdt≥−1.If there exist functionspi∈C1J,R, i=1,2, such that(11)p1′t≤ϕ1tp1t+γ1+ϕ2tTp1t,fort∈J,p10≤p2T,p2′t≤ϕ1tp2t+γ2+ϕ2tTp2t,fort∈J,p20≤p1T,where(12)γ1=p1T−p201−e∫0Tϕ1tdt,γ2=p2T−p101−e∫0Tϕ2tdt.Then,p1t≤0 and p2t≤0 on J.Proof.
We just prove that the case ofp1t≤0. Suppose that the conclusion is not true, we can consider the following two cases, where p10≤0 and p10>0, respectively.Case 5.
Whenp10≤0, by the proof of Lemma 1, we have ∫0Tϕ1t+ϕ2tk0Tdt<−1, which contradicts (10).Case 6.
Whenp10>0, there are two cases: p1t>0 for t∈J or there exist t¯ and t¯, such that p1t¯≤0 and p1t¯>0 for t¯,t¯∈J.Case 7.
Whenp1t>0, t∈J, if p20≤0, by the proof of Lemma 1, we have p2t≤0, that implies p10≤p2T≤0, which is a contradiction.
Ifp20>0, we have p2T≥p10>0. Then, there are two cases: p2t>0 for t∈J and there exist t˜ and t^, such that p2t˜≤0 and p2t^>0, respectively.Case 8.
Whenp2t>0 for all t∈J, we have p2′t<0; hence, p2t is decreasing. By pt>0 and equation (11), imply p1′t<0; then, p1t is decreasing and p20>p2T≥p10>p1T≥p20, which is a contradiction.Case 9.
For another case, we havep2t˜∗=infp2t=−b, where t˜∗∈0,T, b≥0. Equation (11) implies that(13)p2′t≤−bϕ1t+k0Tϕ2t.
Integrating inequality (13) from t˜∗ to T, we have(14)0<p2T≤p2t˜∗−b∫t˜∗Tϕ1t+k0Tϕ2tdt.
Thus,(15)b<−b∫0Tϕ1t+k0Tϕ2tdt,which is also a contradiction.
The proof of Case4 is analogous to the proof of Lemma 1, and we omit its details here. This completes the proof of Lemma 2.Remark 1.
Whenγ=0 and γi=0, i=1,2, respectively, the conclusion of Lemmas 1 and 2 is also true.


## 3. Linear APBVP

In this section, we consider the linear APBVP:(16)x′t+Mxt+NTxt=σt,t∈J,x0+xT=0.We can get the result of the existence and unique solution of equation (16).Theorem 1.
Assume thatM,N>0, M>2NT, and M+k0NTT≤1. Then, APBVP (16) possesses a unique solution.Proof.
For anyx∈CJ,R, denoting x=maxt∈Jxt. Let(17)ω0=maxt∈Jσt,ω1=21−e−MTω0M−2NT.
We define an operatorS:E⟶CJ,R as follows:(18)Sxt=e−Mt+Te−MT+1∫0Tσs−NTxseMsds+e−Mt∫0tσs−NTxseMsds,where E=x∈CJ,R:|x|≤ω1,x0−xT=0.
It is easy to see thatE is a closed, bounded, and convex set. Furthermore, for any x∈E, we have(19)Sxt≤e−Mt+Te−MT+1∫0Tσs+NTxseMsds,+e−Mt∫0Tσs+NTxseMsds≤e−Mt+Te−MT+1∫0Tω0+NTω1eMsds+e−Mt∫0tω0+NTω1eMsds≤2ω0+Nω1T1−e−MTM=ω1,which implies that Sx≤ω1, that is, SE⊂E and S is uniformly bounded. Furthermore, for any t1, t2∈J, we have(20)Sxt1−Sxt2=e−Mt1+Te−MT+1∫0Tσs−NTxseMsds−e−Mt2+Te−MT+1∫0Tσs−NTxseMsds+e−Mt1∫0t1σs−NTxseMsds−e−Mt2∫0t2σs−NTxseMsds≤e−Mt1+T−e−Mt2+Te−MT+1∫0Tσs+NTxseMsds+e−Mt1−e−Mt2∫0t1σs+NTxseMsds+e−Mt2∫mint1,t2maxt1,t2σs+NTxseMsds.
Sinceσ and x are bounded, thus S is uniformly continuous. According to Ascoli–Arzela’s theorem, there exists the subsequences Sxn converging uniformly on J to the continuous functions Sx and Sx∈E, then, we can see that S is compact. Therefore, there exists a solution of APBVP (16) by Schauder’s fixed point theorem. The uniqueness of solutions of APBVP (16) follows from Corollary 1. The proof is completed.


## 4. Nonlinear APBVP

In this section, we give the existence and uniqueness of the solutions of APBVP (1).Definition 1.
The functionsv,w∈C′J,R are said to be a pair of coupled lower and upper solutions for APBVP (1) if the following inequalities(21)v′t≤ft,vt,Tvt−Mγ1,v0≤−wT,(22)w′t≥ft,wt,Twt+Mγ2,w0≥−vT,hold, where M>0,γ1=vT+w0/1−e−MT, and γ2=−wT−v0/1−e−MT.Theorem 2.
Assume that the following conditions hold.H1v,w∈C′J,R are a pair of coupled lower and upper solutions of APBVP (1) such that v≤w on J;H2 There exist constants M>0andN>0 such that M>2NT≥0, M+k0NTT≤1, and(23)ft,u¯,Tu¯−ft,u,Tu≤Mu¯−u+NTu¯−Tu,while v≤u≤u¯≤wandTv≤Tu≤Tu¯≤Tw for t∈J.Then, APBVP (1) has a unique solution x∈v,w.Proof.
We construct iterative sequencesvn,wn⊂C′J,R as follows, v1=vandw1=w on J, and for n>1, vn and wn are the solutions of(24)vn′t=ft,vn−1t,Tvn−1t−Mvnt−vn−1t−NTvnt−Tvn−1t,vn0=−wnT,(25)wn′t=ft,wn−1t,Twn−1t−Mwnt−wn−1t−NTwnt−Twn−1t,wn0=−vnT.
The existence and uniqueness of the solution can be obtained by standard arguments for IVP (24) and (25).
We next prove thatv1≤v2≤w2≤w1.
Letp1=v1−v2andp2=w2−w1, by the condition of H2, and we have p10≤p2T, p20≤p1T, and(26)p1′t≤−Mv1t−v2t+γ1−NTv1t−Tv2t=−Mp1t+γ1−NTp1t,p2′t≤−Mw2t−w1t+γ2−NTw2t−Tw1t=−Mp2t+γ2−NTp2t,where γ1=p1T−p20/1−e−MT and γ2=p2T−p10/1−e−MT.
By Lemma2, we have p1=v1−v2≤0andp2=w2−w1≤0 on J.
Letp=v2−w2, by the condition of H2, and we have(27)p′t≤−Mpt−NTpt,p0=pT.
By using similar arguments of Lemma1, we have p=v2−w2≤0. Therefore, it is easy to see that these sequences satisfy(28)vn≤vn+1≤wn+1≤wn,n≥1.
Then, we have two monotone sequences which are bounded, and there existρ and μ, which satisfy limn⟶∞vn=ρ,limn⟶∞wn=μ, and ρ≤μ. Moreover, the convergence is uniform on J.
Setp=μ−ρ, then we obtain(29)p′t≤−Mpt−NTpt,p0=pT.
By Lemma1, we have pt≤0 for t∈J. Hence, ρ≡μ for t∈J, and we can conclude ρ≡μ≡x, in which x is the solution of APBVP (1). The proof of Theorem 2 is completed.


## 5. Quasilinearization

In this section, we apply the quasilinesrization method in order to obtain the result on convergence of the iterative sequences of approximate solutions for APBVP (1).Consider the Banach spaceCJ,R with the usual maximum norm x1=maxt∈Jxt. For any x∈CJ,R, we call that a given sequence xn converges to x with order of convergence k, if xn converges to x in CJ,R and there exist n0∈N and k>0 such that xm+1−x1≤xm−x1k for all m≥n0.Theorem 3.
Assume that the conditions ofH1−H2 hold.H3∂if/∂xi and ∂if/∂Txi exist and are continuous for i=0,1,…,k, and(30)∑i=1kiMiw−vi−1≤M,∑i=1kiNiw−vi−1≤N,where Mi and Ni are constants with(31)∂if∂xit,u,Tu≤i!Mi,∂if∂Txit,u,Tu≤i!Ni,where t,u,Tu∈Ω=t,u,Tu:v≤u≤w.Then, there exist monotone sequencesvn,wn of approximate solutions converging to the unique solution of (1) with rate of convergence of order k.Proof.
Let the function(32)ft,u,Tu=∑i=0k−1∂if∂xit,α,Tαu−αii!+∂kf∂xkt,χ,Tχu−αkk!+∑i=0k−1∂if∂Txit,α,TαTu−αii!+∂kf∂Txkt,χ,TχTu−αkk!≡gt,u,Tu;α,Tα,where v≤α≤u≤w, χ∈α,u. Consider the following linear equation:(33)u′t=gt,u,Tu;α,Tα,t∈J,u0=−uT.
Settingv0=v, by H3, we have(34)v0′t≤ft,v0t,Tv0t−Mγ1≡gt,v0t,Tv0t;v0t,Tv0t−Mγ1,v00≤−w0T.
Similarly, settingw0=w, we obtain(35)w0′t≥ft,w0t,Tw0t+Mγ2≡gt,w0t,Tw0t;w0t,Tw0t+Mγ2,w00≥−v0T.
Then,v0 and w0 are lower and upper solutions of equation (33), respectively. Furthermore, for t∈J and v≤x≤y≤w, we have(36)gt,x,Tx;α,Tα−gt,y,Ty;α,Tα=∑i=0k−1∂if∂xit,α,Tαx−αi−y−αii!+∂kf∂xkt,χ,Tχx−αk−y−αkk!+∑i=0k−1∂if∂Txit,α,TαTx−αi−Ty−αii!+∂kf∂Txkt,χ,TχTx−αk−Ty−αkk!=x−y∑i=0k−1∂if∂xit,α,Tα1i!∑j=0i−1x−αi−1−jy−αj+∂kf∂xkt,χ,Tχx−αi−1−jy−αjk!+Tx−y∑i=0k−1∂if∂Txit,α,Tα1i!∑j=0i−1Tx−αi−1−jTy−αj+∂kf∂Txkt,χ,TχTx−αi−1−jTy−αjk!≤x−y∑i=1kMi∑j=0i−1y−xi−1+Tx−y∑i=1kNi∑j=0i−1Ty−xi−1≤Mx−y+NTx−y.
Using Theorem2, we know that problem (33) has a solution in v,w.
Letv1 be a solution of the mentioned problem, with v1∈v,w, and we suppose v=v0≤v1≤⋯≤vn≤w, where vn∈vn−1,w is the solution of(37)u′t=gt,u,Tu;vn−1,Tvn−1,t∈J,u0=−uT.where vn−1 and w are lower and upper solutions, respectively, for the following problem:(38)u′t=gt,u,Tu;vn,Tvn,t∈J,u0=−uT.
Similarly, we know thatgt,u,Tu;vn,Tvn satisfies the conditions of Theorem 2, then problem (33) has a solution in vn,w. Let vn+1 be a solution of the mentioned problem, with vn+1∈vn,w. Hence, the constructed sequence vn is nondecreasing and bounded. In the same way, we can construct the sequence wn which is nonincreasing and bounded. Therefore, we obtain the two monotone sequences converge uniformly.
Letlimvn=ρand limwn=μ, and we can get ρ≡μ≡u by using Theorem 2, in which u is the solution of (1).
Now, we show that the convergence ofvn,wn to u is of order k. Let(39)pn=u−vn≥0,qn=wn−u≥0.
Firstly, we note that(40)u′t=∑i=0k−1∂if∂xit,vn,Tvnu−vnii!+∂kf∂xkt,χn,Tχnu−vnkk!+∑i=0k−1∂if∂Txit,vn,TvnTu−vnii!+∂kf∂Txkt,χn,TχnTu−vnkk!,where χn∈vn,u. In sequence,(41)pn+1′t=∑i=0k−1∂if∂xit,vn,Tvnpni−vn+1−vnii!+∂kf∂xkt,χn,Tχnpnk−vn+1−vnkk!+∑i=0k−1∂if∂Txit,vn,TvnTpni−Tvn+1−vnii!+∂kf∂Txkt,χn,TχnTpnk−Tvn+1−vnkk!=pn+1∑i=1k−1∂if∂xit,vn,Tvn1i!∑j=0i−1pni−1−jvn+1−vnj+∂kf∂xkt,χn,Tχnpnk−vn+1−vnkk!+Tpn+1∑i=1k−1∂if∂xit,vn,Tvn1i!∑j=0i−1Tpni−1−jTvn+1−vnj+∂kf∂Txkt,χn,TχnTpnk−Tvn+1−vnkk!.
In view of conditionH3, by the continuity of ∂if/∂xi and ∂if/∂Txi in Ω, we have(42)pn+1′t≤Apn+1t+BTpn+1t+Cpnkt+DTpnkt,where C=2Mk,D=2Nk, and(43)∑i=0k−1∂if∂xit,vn,Tvn1i!∑j=0i−1pni−1−jvn+1−vnj≤A,∑i=0k−1∂if∂xit,vn,Tvn1i!∑j=0i−1Tpni−1−jTvn+1−vnj≤B.
Similarly, we have(44)qn+1′t≤Aqn+1t+BTqn+1t+Cqnkt+DTqnkt.
The boundary conditions are(45)pn0=qnT,qn0=pnT.
LetRn=pn+qn, and we obtain(46)Rn+1′t≤ARn+1t+BTRn+1t+Cpnkt+qnkt+DTpnkt+qnkt≤A+k0BRn+1t+C+k0Dpnkt+qnkt,Rn+10=Rn+1T.
Using Gronwall’s inequality for (46), we have(47)Rn+1t≤eA+k0BtRn+10+∫0teA+k0Bt−sC+k0Dpnks+qnksds.
Lett=T, and we have(48)Rn+10≤eA+k0BTRn+10+TA+k0BeA+k0BTC+k0Dmaxt∈Jpnkt+qnkt,which implies(49)Rn+10≤eA+k0BTC+k0Dmaxt∈Jpnkt+qnktT1−eA+k0BT,that is,(50)maxt∈JRn+1t≤Kmaxt∈Jpnkt+maxt∈Jqnkt,where K=eA+k0BTeA+k0BTC+k0Dmaxt∈JRnkt/T1−eA+k0BT+T/A+k0BeA+k0BTC+k0D.
Sincemaxt∈Jpnt≤maxt∈JRntandmaxt∈Jqnt≤maxt∈JRnt, we get the desired convergence. The proof is completed.


## 6. An Example

In this section, we will provide an example which demonstrates the application of Theorem3.Example 1.
Consider the following APBVP:(51)x′t=120t1+x2t−110x−120∫0txscossds,t∈0,1,x0=−x1.
It is easy to check thatv0=−1 and w0=1 are lower and upper solutions of (51), respectively, which satisfies condition H1 of Theorem 3. And we can show that(52)fx≤15,fy≤120,fxx≤110,fyy=0,∂if∂xi=0,∂if∂Txi=0,i=3,4,…,k.
SettingM1=1/5, M2=1/10, N1=1/20, N2=0, Mi=0, Ni=0, i=3,4,…,k, M=3/5, and N=1/20, satisfy conditions H2 and H3 of Theorem 3. Then, convergence of the iterative sequences of approximate solutions for APBVP (51) are of order k≥2.


## 7. Conclusion

In this paper, we discussed the problem of rapid convergence for the first-order integro-differential equations with antiperiodic boundary value conditions. By using the particular definitions of the coupled lower and upper related solutions, which are new and some new comparison principles, we obtained the existence and uniqueness of solution of the problems. Meanwhile, by using the method of quasilinearization, we obtained the monotone sequences of approximate solutions, converging to the unique solution of such problems with the rate of convergence of orderk. Finally, we give an example to illustrate our main results.



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