# Rapid Convergence for Telegraph Systems with Periodic Boundary Conditions

Authors: Peiguang Wang; Xiang Liu
Journal: Journal of Function Spaces
 (2017)

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## Abstract

The generalized quasilinearization method is applied in this paper to a telegraph system with periodic boundary conditions. We consider the case in which the forcing functionF(t,x,U) satisfies the following condition: ∂n-1F(t,x,U)/∂Un-1 exists and is quasimonotone nondecreasing or nonincreasing. We develop nonlinear iterates of order n-1 which will be different with n being even or odd. Finally, we develop two sequences which converge to the solution of the telegraph system and the convergence is of order n.


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## Body

## 1. Introduction

It is well known that the telegraph equation can be used to model many problems such as fluid mechanics, thermodynamics, and elastic mechanics; examples can be found in [1–4]. Recently, the maximum principle for the telegraph equation has been studied by many authors. Ortega and Robles-Perez [5] established a maximum principle for periodic solutions of the telegraph equation (1)utt-uxx+cut+λu=ht,x,t,x∈R2with doubly periodic boundary condition. Mawhin et al. [6] gave a maximum principle for bounded solutions of the telegraph equation which are 2π-periodic with respect to x and bounded over R with respect to t. Li [7] built a maximum principle for time-periodic solutions of the telegraph equation (2)utt-uxx+cut+at,xu=ht,x,t,x∈R2,which are bounded on R for x and 2π-periodic for t. Wang and An [8] gave the maximum principle for the linear telegraph system (3)utt-uxx+c1ut+a11t,xu+a12t,xv=f1t,x,vtt-vxx+c2vt+a21t,xu+a22t,xv=f2t,x,ut+2π,x=ut,x+2π=ut,x,vt+2π,x=vt,x+2π=vt,x,and they obtained a monotone sequence of approximate solutions converging uniformly and quadratically to a solution of the semilinear telegraph system.The convergence of the solution plays an important role in the development of the qualitative theory, and the higher-order convergence of solutions is also very important in practical applications. Quasilinearization is an efficient method for constructing approximate solutions of a variety of nonlinear problems. It was introduced by Bellman and Kalaba [9] and generalized by Lakshmikantham and Vatsala [10]. Up till now, there are some higher-order convergence results for a variety of differential equations under different assumptions. In [11], Mohapatra et al. applied the generalized quasilinearization method for first-order initial value problems and obtained the nth order of convergence on conditions that the nth derivatives ∂nf(t,x)/∂xn and ∂nϕ(t,x)/∂xn of the functions f(t,x) and ϕ(t,x) exist and are continuous and ϕ and f+ϕ are n-1-hyperconvex or n-1-hyperconcave in x. Cabada et al. [12] used the generalized quasilinearization method for first-order boundary value problems to obtain the nth order of convergence by assuming that ∂nf(t,x)/∂xn exists and is continuous. Afterwards, the results of nth-order convergence, such as second-order boundary value problems [13], impulsive differential equations [14], reaction diffusion equations [15], parabolic integrodifferential equations [16], nonlinear boundary value problems [17], and singular differential systems [18], were also obtained under weaker assumptions; namely, ∂n-1f(t,x,u)/∂un-1 exists and is nondecreasing or nonincreasing in u and one-sided Lipschitzian in u for n>2. The nonlinear iterate is of order n-1 and it is different when n is even or odd. To the best of our knowledge, no results on the rapid convergence of solutions for a nonlinear telegraph system can be found.In this paper, we attempt to extend this generalized quasilinearization method to a telegraph system by assuming that∂n-1F(t,x,U)/∂Un-1 exists and is quasimonotone nondecreasing or nonincreasing. We obtain two sequences which converge to the solution of the telegraph system and the convergence is of order n (n being even or odd) for n>2.


## 2. Preliminaries

For convenience, we give some notions and lemmas.LetT2 be the torus defined as (4)T2=R2πZ×R2πZ,where R is a real number set and Z is an integer set. Doubly 2π-periodic functions will be regarded as functions defined on T2. L1(T2) and C(T2) denote the Lebesgue integrable function space and continuous function space on T2. The norms can be defined as hL1=∫T2|h|dtdx and uc=max(t,x)∈T2⁡x(s).LetD′(T2) denote the space of distribution on T2 and E represent a Banach space C(T2)×C(T2).First, we consider the linear equation(5)utt-uxx+ciut-λiu=hit,x,inD′T2,where ci>0, λi∈R, and hi∈L1(T2) for i=1,2.LetŁλi be the differential operator,(6)Łλiu=utt-uxx+ciut-λiu,acting on functions on T2. From the discussion in [5, 7], we know that if λi<0,Łλi has the resolvent Rλi(7)Rλiu:L1T2⟶CT2,hi⟶ui,where ui is the unique solution of (5), and the restriction of Rλi on Lp(T2)(1<p<∞) or C(T2) is compact. In particular, Rλi:C(T2)→C(T2) is a completely continuous operator. Furthermore, for λi=-ci2/4, the Green function Gi(t,x) of the differential operator Łλi is explicitly expressed; we can refer to Lemma 5.2 in [5]. According to the definition of Gi(t,x), we have (8)Gi_≔essinf⁡Git,x=e-3ciπ/21-e-ciπ2,Gi¯≔esssup⁡Git,x=1+e-ciπ21-e-ciπ2.We assume the following conditions throughout this paper.(H2.1)
aii∈C(T2), 0≤aii(t,x)≤ci2/4 for (t,x)∈T2, and ∫T2aii(t,x)dtdx>0.(H2.2)
a12, a21∈C(T2,R-),f1, f2∈C(T2,R+).When-λi=aii(t,x)≤ci2/4, we have the following unique existence and positive estimate result.Lemma 1 (see [7]).
Assume thathi∈L1(T2) and X is the Banach space C(T2).
Then, (5) has a unique solution ui=Pihi,Pi:L1(T2)→X is a linear bounded operator with the following properties:(i)
Pi:C(T2)→C(T2) is a completely continuous operator.(ii)
Ifhi>0, a.e.(t,x)∈T2, Pihi has the positive estimate (9)Gi_hiL1≤Pihi≤Gi¯Gi_aiiL1hiL1.LetE be a Banach space ordered by the positive cone K⊂E and equipped with norm · and L^:E→E be a bounded linear operator. By maximum principle to the following system: (10)U=L^U+F,U∈E;we mean that F≥0(i.e.,F∈K) implies U≥0 since U is a solution of (10).Now, one has the following linear telegraph system:(11)utt-uxx+c1ut+a11t,xu+a12t,xv=f1t,x,vtt-vxx+c2vt+a21t,xu+a22t,xv=f2t,x,ut+2π,x=ut,x+2π=ut,x,vt+2π,x=vt,x+2π=vt,x.Here and in what follows, by a doubly periodic solution of (11), we mean that (u,v)∈L1(T2)×L1(T2) satisfies (11) in the distribution sense; that is, (12)∫T2uϕtt-ϕxx-c1ϕt+a11ϕ+∫T2a12vϕ=∫T2f1ϕ,∀ϕ∈D′T2,∫T2vφtt-φxx-c2φt+a22φ+∫T2a21uφ=∫T2f2φ,∀φ∈D′T2.We have the following result.Lemma 2 (see [5]).
Assume that conditions(H2.1) and (H2.2) hold, and (H2.3)
a12L1G1¯<G1_a11L1, a21L1G2¯<G2_a22L1.
Then, system (11) has at least one solution in E and satisfies the maximum principle.In this paper, we consider the nonlinear telegraph system with doubly periodic boundary conditions:(13)utt-uxx+c1ut=f1t,x,u,v,vtt-vxx+c2vt=f2t,x,u,v,ut+2π,x=ut,x+2π=ut,x,vt+2π,x=vt,x+2π=vt,x.Let system (13) be equivalent to (14)ŁU=Ft,x,U,U∈E,where (15)ŁU=Ł100Ł2U=utt-uxx+c1utvtt-vxx+c2vt,Ft,x,U=f1t,x,u,vf2t,x,u,v.Remark 3.
Here and in what follows, the inequalities related to upper and lower solutions are in the distribution sense.Definition 4.
The functionα0∈C(T2,R2) is called a lower solution of system (14) if it satisfies (16)Łα0≤Ft,x,α0,inD′T2.
If the above inequality is reversed, the functionα0∈C(T2,R2) is called an upper solution of system (14).
Now, we will give two important lemmas which are necessary in our further discussion.Lemma 5.
Assume that conditions(H2.1)–(H2.3) hold. (H2.4)
The functionsα0,β0∈C(T2,R2) are lower and upper solutions of system (14) with α0≤β0.(H2.5)
The functionF satisfies the inequality (17)Ft,x,U-Ft,x,V≥-M0U-VforU≥V, where(18)M0=At0,x0=a11t0,x0a12t0,x0a11t0,x0a22t0,x0,t0,x0∈T2.Then, system (14) has a solution U∈E that satisfies α0≤U≤β0.Proof.
Letαn+1, βn+1 be the solutions of the following linear system: (19)Łαn+1=Ft,x,αn-M0αn+1-αn,inD′T2,Łβn+1=Ft,x,βn-M0βn+1-βn,inD′T2,which exist because of Lemma 2. According to the iterative schemes (19), we obtain the sequences {αn} and {βn} which were generated by the initial conditions α0 and β0, respectively. The regularity of solutions of the linear operator implies that αn and βn are continuous if n≥1.
We first show thatα0≤α1≤β1≤β0 on T2.
For this purpose, we setp=α1-α0. Using condition (H2.4), we obtain (20)Łp≥Ft,x,α0-M0α1-α0-Ft,x,α0=-M0p,inD′T2.By Lemma 2, we have p≥0; that is, α0≤α1 on T2. Similarly, letting p=β0-β1, we can find that (21)Łp≥Ft,x,β0-Ft,x,β0-M0β1-β0=-M0p,inD′T2.Using Lemma 2, we get p≥0, showing that β1≤β0 on T2.
To proveα1≤β1, set p=β1-α1. From condition (H2.5), we have (22)Łp=Ft,x,β0-M0β1-β0-Ft,x,α0-M0α1-α0≥-M0β0-α0-M0β1-β0+M0α1-α0=-M0p,inD′T2.As before, this implies that α1≤β1 on T2. Thus, we conclude that (23)α0≤α1≤β1≤β0.
The process can be continued successively to obtain(24)α0≤α1≤⋯≤αn≤βn≤⋯≤β1≤β0,onT2.In particular, the sequence {αn} is nondecreasing and converges pointwise to a function U that satisfies α0≤U≤β0. A passage to the limit based on the Dominated Convergence Theorem shows that U is a solution of (25)ŁU=Ft,x,U-M0U-U;that is, U is a solution of (14). Again, the regularity theory allows us to conclude that U belongs to E.


## 3. Generalized Quasilinearization Method

In this section, we apply the method of generalized quasilinearization for a telegraph system. We obtain that the convergence of the sequences of successive approximations is of ordern where n is 2k+1 or 2k.Theorem 6.
Assume that conditions(H2.1), (H2.2), (H2.4), and (H2.5) hold: A3.1
a12L1G1¯<(1/2)G1_a11L1, a21L1G2¯<(1/2)G2_a22L1.(A3.2)
The Fréchet derivatives∂iF(t,x,U)/∂Ui(i=0,1,2,…,2k) exist and are continuous satisfying 0≤∂2kF(t,x,U)/∂U2k-∂2kF(t,x,V)/∂U2k≤LU-V for α0≤V≤U≤β0, (t,x)∈T2, and L=(l) is a positive matrix.
Then, there exist monotone sequences{αn} and {βn}, which converge uniformly to the solution U of system (14) in E, and the convergence is of order 2k+1; that is, there exist positive constants M1 and N1 such that (26)U-αn+1≤M1U-αn2k+1,βn+1-U≤N1βn-U2k+1hold, where |Si|c=max|Si(t,x)|, S=|S1|c+|S2|c.Proof.
In view of condition(A3.2), we have (27)Ft,x,V≥∑i=02k∂iFt,x,U∂UiV-Uii!for α0≤U≤V≤β0, (t,x)∈T2, and (28)Ft,x,V≤∑i=02k∂iFt,x,U∂UiV-Uii!for α0≤V≤U≤β0, (t,x)∈T2, where (29)∂iFt,x,U∂UiV-Ui=∂∂U1u∗-u+∂∂U2v∗-vif1t,x,U,∂∂U1u∗-u+∂∂U2v∗-vif2t,x,UT,for U=(u,v), V=(u∗,v∗), i=0,1,2,….
Let us first consider the following systems:(30)ŁU=∑i=02k∂iFt,x,α∂UiU-αii!≡Ht,x,U,inD′T2,(31)ŁV=∑i=02k∂iFt,x,β∂UiV-βii!≡Gt,x,V,inD′T2.
Initially, we show thatα0 and β0 are lower and upper solutions of (30), respectively. Let α=α0, β=β0 in (30) and (31). Condition (H2.4) and inequality (27) lead to (32)Łα0≤Ft,x,α0=Ht,x,α0,inD′T2,Łβ0≥Ft,x,β0≥∑i=02k∂iFt,x,α0∂Uiβ0-α0ii!=Ht,x,β0,inD′T2.We show that α0 and β0 are lower and upper solutions of (30), respectively. Hence, we obtain by Lemma 5 that there exists a solution α1 of (30) such that α0≤α1≤β0 on T2. We next prove that α1 and β0 are lower and upper solutions of (31), respectively. From inequalities (27) and (28), we get (33)Łα1=∑i=02k∂iFt,x,α0∂Uiα1-α0ii!≤Ft,x,α1≤∑i=02k∂iFt,x,β0∂Uiα1-β0ii!=Gt,x,α1,inD′T2,Łβ0≥Ft,x,β0=Gt,x,β0,inD′T2,which show that α1 and β0 are lower and upper solutions of (31), respectively. As before, by Lemma 5, we can conclude that system (31) has a solution β1 that satisfies α1≤β1≤β0 on T2.
Next, we must show thatα1 and β1 are lower and upper solutions of system (14), respectively. To prove this, we can apply inequality (27) to obtain (34)Łα1=∑i=02k∂iFt,x,α0∂Uiα1-α0ii!≤Ft,x,α1,inD′T2.It proves that α1 is a lower solution of (14). Similarly, using inequality (28), we see that (35)Łβ1=∑i=02k∂iFt,x,β0∂Uiβ1-β0ii!≥Ft,x,β1,inD′T2,which implies that β1 is an upper solution of (14). It therefore follows that α0≤α1≤β1≤β0 on T2.
Assume now thatαn and βn are lower and upper solutions of (14), respectively, and αn≤βn on T2. We need to show that αn≤αn+1≤βn+1≤βn on T2, where αn+1 and βn+1 are solutions of (30) and (31), respectively, with α=αn and β=βn.
Now, we show thatαn and βn are lower and upper solutions of (30), respectively. Let α=αn and β=βn in (30) and (31). Utilizing the assumption that αn and βn are lower and upper solutions of (14), respectively, and inequality (27), we get (36)Łαn≤Ft,x,αn=Ht,x,αn,inD′T2,Łβn≥Ft,x,βn≥∑i=02k∂iFt,x,αn∂Uiβn-αnii!=Ht,x,βn,inD′T2,which show that αn and βn are lower and upper solutions of (30), respectively. Hence, we can now apply Lemma 5 to deduce that there exists a solution αn+1 of (30) such that αn≤αn+1≤βn on T2. Using the same technique, we can show that there exists a solution βn+1 of (31) such that αn+1≤βn+1≤βn on T2.
Furthermore, we need to prove thatαn+1 and βn+1 are lower and upper solutions of (14). For this purpose, by using inequality (27), we get (37)Łαn+1=∑i=02k∂iFt,x,αn∂Uiαn+1-αnii!≤Ft,x,αn+1,inD′T2.This proves that αn+1 is a lower solution of (14). Similar reasoning gives that βn+1 is an upper solution of (14). Thus, we conclude that αn≤αn+1≤βn+1≤βn on T2. Therefore, the method of mathematical induction can be applied to obtain (38)α0≤α1≤⋯≤αn≤βn≤⋯≤β1≤β0,t,x∈T2.Since αn and βn are lower and upper solutions of (14), respectively, and all the assumptions of Lemma 5 hold, we can conclude that there exists a solution of (14) such that αn≤U≤βn. Hence, we have (39)α0≤α1≤⋯≤αn≤U≤βn≤⋯≤β1≤β0,t,x∈T2.
To prove the uniform convergence, we can see easily that the sequence{αn} is uniformly bounded and equicontinuous on E. Hence, employing Ascoli-Arzela’s theorem, we have that there exists a subsequence {αnk} which converges uniformly to U=(U1,U2). Moreover, {αn} is a nondecreasing sequence; it follows that {αn} converges to U. By the Dominated Convergence Theorem, it is easy to verify that U is a solution of (14) on T2 in distribution sense. Similarly, convergence holds for {βn}.
Finally, we have to show that the convergence of{αn} and {βn} is of order 2k+1. For that, we consider (40)Pn+1=U-αn+1≥0,Qn+1=βn+1-U≥0.Now, using the mean value theorem together with condition (A3.2), we obtain (41)ŁPn+1=Ft,x,U-∑i=02k∂iFt,x,αn∂Uiαn+1-αnii!=Ft,x,U-Ft,x,αn+1-∫01∂2kFt,x,σαn+1+1-σαn∂U2kdσαn+1-αn2k2k!+∂2kFt,x,αn∂U2kαn+1-αn2k2k!≤∫01Fut,x,σU+1-σαn+1dσU-αn+1+∂2kFt,x,αn+1∂U2k-∂2kFt,x,αn∂U2kαn+1-αn2k2k!≤-At,xPn+1+L12k!Pn2k+1,where L1=(l,l)T. An application of Lemma 2 yields that Pn+1≤W on T2, and W=(w1,w2)T is the solution of (42)ŁW+At,xW=L12k!Pn2k+1.By applying Lemma 1, we obtain that (43)w1c≤G1¯G1_a11L1a12L1w2c+C1Pn2k+1≤12w2c+D1Pn2k+1.On the other hand, we obtain (44)w2c≤G2¯G2_a22L1a21L1w1c+C2Pn2k+1≤12w1c+D2Pn2k+1.From the above discussion, we arrive at (45)Pn+1≤W≤M1Pn2k+1,where M1 is a positive constant.
Similarly, we can get(46)ŁQn+1=Łβn+1-U=∑i=02k∂iFt,x,βn∂Uiβn+1-βnii!-Ft,x,U=Ft,x,βn+1-∫01∂2kFt,x,σβn+1+1-σβn∂U2kdσβn+1-βn2k2k!+∂2kFt,x,βn∂U2kβn+1-βn2k2k!-Ft,x,U≤∫01FUt,x,σβn+1+1-σUdσβn+1-U+∂2kFt,x,βn∂U2k-∂2kFt,x,βn+1∂U2kβn+1-βn2k2k!≤-At,xQn+1+L12k!Qn2k+1.Lemma 2 shows that Qn+1≤W on T2, where W=(w1,w2)T is the solution of (47)ŁW+At,xW=L12k!Qn2k+1.We get from Lemma 1 that (48)w1c≤G1¯G1_a11L1a12L1w2c+K1Qn2k+1≤12w2c+L1Qn2k+1.Furthermore, we obtain (49)w2c≤G2¯G2_a22L1a21L1w1c+K2Qn2k+1≤12w1c+L2Qn2k+1.It follows that (50)Qn+1≤W≤N1Qn2k+1,where N1 is a positive constant. The proof is complete.Theorem 7.
Assume that conditions(H2.1), (H2.2), (H2.4), (H2.5), and (A3.1) hold:(A3.3)
The functionF∈C(T2×R2,R2), the Fréchet derivatives ∂iF(t,x,U)/∂Ui(i=0,1,2,…,2k-1) exist and are continuous satisfying 0≤∂2k-1F(t,x,U)/∂U2k-1-∂2k-1F(t,x,V)/∂U2k-1≤LU-V for α0≤V≤U≤β0, (t,x)∈T2, and L=(l) is a positive matrix.
Then, there exist monotone sequences{αn} and {βn}, which converge uniformly to the solution U of system (14) in E, and the convergence is of order 2k; that is, there exist positive constants M2 and N2 such that (51)U-αn+1≤M2U-αn2k,βn+1-U≤N2βn-U2k-1U-αn+βn-Uhold, where |Si|c=max|Si(t,x)|, S=|S1|c+|S2|c.Proof.
It can be noted from condition(A3.3) that (52)Ft,x,V≥∑i=02k-1∂iFt,x,U∂UiV-Uii!for α0≤U≤V≤β0, (t,x)∈T2, and (53)Ft,x,V≤∑i=02k-2∂iFt,x,U∂UiV-Uii!+∂2k-1Ft,x,V∂U2k-1V-U2k-12k-1!for α0≤V≤U≤β0, (t,x)∈T2.
Consider the following systems:(54)ŁU=∑i=02k-1∂iFt,x,α∂UiU-αii!≡Ht,x,U,inD′T2,(55)ŁV=∑i=02k-2∂iFt,x,β∂UiV-βii!+∂2k-1Ft,x,α∂U2k-1V-β2k-12k-1!≡Gt,x,V,inD′T2.
We can prove as in Theorem6 that there exists a solution α1 of (54) such that α0≤α1≤β0 on T2 with α=α0.
Now, let us show thatα1 and β0 are lower and upper solutions of (55), respectively. By setting α=α0 and β=β0 in (55), it follows from inequalities (52) and (53) that (56)Łα1=∑i=02k-1∂iFt,x,α0∂Uiα1-α0ii!≤Ft,x,α1≤∑i=02k-2∂iFt,x,β0∂Uiα1-β0ii!+∂2k-1Ft,x,α1∂U2k-1α1-β02k-12k-1!≤∑i=02k-2∂iFt,x,β0∂Uiα1-β0ii!+∂2k-1Ft,x,α0∂U2k-1α1-β02k-12k-1!=Gt,x,α1,inD′T2,Łβ0≥Ft,x,β0=Gt,x,β0,inD′T2.This shows that α1 and β0 are lower and upper solutions of (55), respectively. It follows from Lemma 5 that there exists a solution β1 of (55) such that α1≤β1≤β0 on T2.
Now, we prove thatα1 and β1 are lower and upper solutions of system (14). Utilizing inequality (52), we find that (57)Łα1=∑i=02k∂iFt,x,α0∂Uiα1-α0ii!≤Ft,x,α1,inD′T2.Hence, α1 is a lower solution of (14). In a similar way, condition (A3.3) together with inequality (53) gives (58)Łβ1=∑i=02k-2∂iFt,x,β0∂Uiβ1-β0ii!+∂2k-1Ft,x,α0∂U2k-1β1-β02k-12k-1!≥∑i=02k-2∂iFt,x,β0∂Uiβ1-β0ii!+∂2k-1Ft,x,β1∂U2k-1β1-β02k-12k-1!≥Ft,x,β1,inD′T2.This proves that β1 is an upper solution of (14). As a result, it follows that α0≤α1≤β1≤β0 on T2.
Thus, using mathematical induction, one can obtain(59)α0≤α1≤⋯≤αn≤βn≤⋯≤β1≤β0,t,x∈T2,where αn and βn are solutions of (54) and (55), respectively, with α=αn-1 and β=βn-1.
Employing the Ascoli-Arzela theorem, both sequences{αn} and {βn} converge uniformly to the solution of system (14).
At last, we show that the convergence of the sequences of{αn} and {βn} is 2k. For this purpose, we consider (60)Pn+1=U-αn+1≥0,Qn+1=βn+1-U≥0.
For the convergence of{αn}, we can prove Theorem 6 to obtain (61)Pn+1≤W≤M2Pn2k-1Pn+Qn,where M2 is a positive constant.
To prove the convergence of{βn}, using the mean value theorem and condition (A3.3), we arrive at (62)ŁQn+1=Łβn+1-U=∑i=02k-2∂iFt,x,βn∂Uiβn+1-βnii!+∂2k-1Ft,x,αn∂U2k-1βn+1-βn2k-12k-1!-Ft,x,U=Ft,x,βn+1-∫01∂2k-1Ft,x,σβn+1+1-σβn∂U2k-1dσβn+1-βn2k-12k-1!+∂2k-1Ft,x,αn∂U2k-1βn+1-βn2k-12k-1!-Ft,x,U≤∫01FUt,x,σβn+1+1-σUdσβn+1-U+∂2k-1Ft,x,βn∂U2k-1-∂2k-1Ft,x,αn∂U2k-1βn-βn+12k-12k-1!≤-At,xQn+1+L12k-1!Qn2k-1Pn+Qn,where L1=(l,l)T. Lemma 2 shows that Qn+1≤W on T2, where W=(w1,w2)T is the solution of (63)ŁW+At,xW=L12k-1!Qn2k-1Pn+Qn.Employing Lemma 1, it follows that (64)w1c≤G1¯G1_a11L1a12L1w2c+E1Qn2k-1Pn+Qn≤12w2c+F1Qn2k-1Pn+Qn,w2c≤G2¯G2_a22L1a21L1w1c+E2Qn2k-1Pn+Qn≤12w1c+F2Qn2k-1Pn+Qn.Therefore, we conclude that (65)Qn+1≤W≤N2Qn2k-1Pn+Qn,where N2 is a positive constant. The proof is complete.Theorem 8.
Assume that conditions(H2.1), (H2.2), (H2.4), (H2.5), and (A3.1) hold:(A3.4)
The functionF∈C(T2×R2,R2), the Fréchet derivatives ∂iFt,x,U/∂Ui(i=0,1,2,…,2k) exist and are continuous satisfying -LU-V≤∂2kF(t,x,U)/∂U2k-∂2kF(t,x,V)/∂U2k≤0 for α0≤V≤U≤β0, (t,x)∈T2, and L=(l) is a positive matrix.
Then, there exist monotone sequences{αn} and {βn}, which converge uniformly to the solution U of system (14) in E, and the convergence is of order 2k+1; that is, there exist positive constants M3 and N3 such that (66)U-αn+1≤M3U-αn2kU-αn+βn-U,βn+1-U≤N3βn-U2kU-αn+βn-Uhold, where Sic=max⁡|Si(t,x)|, S=S1c+S2c.Proof.
In order to construct monotone sequences{αn} and {βn} which converge uniformly to the solution of (14), we need to consider the following systems: (67)Łαn+1=∑i=02k-1∂iFt,x,αn∂Uiαn+1-αnii!+∂2kFt,x,βn∂U2kαn+1-αn2k2k!≡Ht,x,αn+1,inD′T2,Łβn+1=∑i=02k-1∂iFt,x,βn∂Uiβn+1-βnii!+∂2kFt,x,αn∂U2kβn+1-βn2k-12k!≡Gt,x,βn+1,inD′T2.Since the proof follows on the same line as in Theorem 6, we omit the proof.Theorem 9.
Assume that conditions(H2.1), (H2.2), (H2.4), (H2.5), and (A3.1) hold:(A3.5)
The functionF∈C(T2×R2,R2), the Fréchet derivatives ∂iF(t,x,U)/∂Ui(i=0,1,2,…,2k-1) exist and are continuous satisfying -LU-V≤∂2k-1F(t,x,U)/∂U2k-1-∂2k-1F(t,x,V)/∂U2k-1≤0 for α0≤V≤U≤β0, (t,x)∈T2, and L=(l) is a positive matrix.
Then, there exist monotone sequences{αn} and {βn}, which converge uniformly to the solution U of system (14) in E, and the convergence is of order 2k; that is, there exist positive constants M4 and N4 such that (68)U-αn+1≤M4U-αn2k-1U-αn+βn-U,βn+1-U≤N4βn-U2khold, where |Si|c=max|Si(t,x)|, S=|S1|c+|S2|c.Proof.
To construct monotone sequences{αn} and {βn} which converge uniformly to the solution of (14), we consider the following systems: (69)Łαn+1=∑i=02k-2∂iFt,x,αn∂Uiαn+1-αnii!+∂2k-1Ft,x,βn∂U2k-1αn+1-αn2k-12k-1!≡Ht,x,αn+1,inD′T2,Łβn+1=∑i=02k-1∂iFt,x,βn∂Uiβn+1-βnii!≡Gt,x,βn+1,inD′T2.We omit the proof, since it follows the same way as in Theorem 6.


## 4. Example

Now, we give an example to illustrate the result in the previous section.Example 1.
Consider the following telegraph system with periodic boundary conditions:(70)utt=2cos2⁡t-2u,0=v2-u,ut+2π,x=ut,x+2π=ut,x,vt+2π,x=vt,x+2π=vt,x,where(71)Ft,x,U=f1t,x,u,vf2t,x,u,v=2cos2⁡t-2uv2-u.One can see easily thatsin2⁡t,-sin⁡tT is a solution of the telegraph system (70). Taking α0(t)=0,-1T, β0(t)=2,1T, it is easy to verify that α0(t) and β0(t) are lower and upper solutions of the telegraph system (70), respectively, and ∂f1(t,x,u,v)/∂u=-2, ∂f1(t,x,u,v)/∂v=0, ∂f2(t,x,u,v)/∂u=-1, and ∂f2(t,x,u,v)/∂v=2v; that is, the conditions of Theorem 7 are satisfied. According to Theorem 7, we can construct two monotone sequences {αn} and {βn}.The successive approximationsαn+1(t) and βn+1(t) are solutions of the telegraph systems (72)ŁU=∑i=01∂iFt,x,αn∂UiU-αnii!,inD′T2,(73)ŁV=∑i=01∂iFt,x,βn∂UiV-βnii!,inD′T2.Firstly, we construct an increasing sequence of lower solutions which converges uniformly to the solutionsin2⁡t,-sin⁡tT. In view of the fact that α0(t)=0,-1T, the telegraph system (72) reduces to (74)utt=2cos2⁡t-2u,0=-2v-u-1,ut+2π,x=ut,x+2π=ut,x,vt+2π,x=vt,x+2π=vt,x.Then, the telegraph system (74) has a solution α1(t)=sin2⁡t,-sin2⁡t/2-1/2T.Using the solutionα1(t)=sin2⁡t,-sin2⁡t/2-1/2T, the telegraph system (72) reduces to the following: (75)utt=2cos2⁡t-2u,0=-u-2vsin2⁡t2+12-sin2⁡t2+122,ut+2π,x=ut,x+2π=ut,x,vt+2π,x=vt,x+2π=vt,x.The telegraph system (75) has an exact solution α2(t)=sin2⁡t,-sin2⁡t/sin2⁡t+1-(1/4)sin2⁡t-1/4T.Similarly, we can find all successive approximationsαn(t), which converge uniformly to the solution sin2⁡t,-sin⁡tT of the telegraph system (70).Next, we shall construct a decreasing sequence of upper solutions which will converge uniformly to the exact solution. Sinceβ0(t)=2,1T, then the telegraph system (73) can be written as (76)utt=2cos2⁡t-2u,0=2v-u-1,ut+2π,x=ut,x+2π=ut,x,vt+2π,x=vt,x+2π=vt,x.The function β1(t)=sin2⁡t,sin2⁡t/2+1/2T is a solution of the telegraph system (76).According to the solutionβ1(t)=sin2⁡t,sin2⁡t/2+1/2T, the telegraph system (73) reduces to the following: (77)utt=2cos2⁡t-2u,0=-u+2vsin2⁡t2+12-sin2⁡t2+122,ut+2π,x=ut,x+2π=ut,x,vt+2π,x=vt,x+2π=vt,x.The telegraph system (77) has a solution β2(t)=(sin2⁡t,sin2⁡t/(sin2⁡t+1)+(1/4)sin2⁡t+1/4)T.Proceeding as before, we can find all successive approximationsβn(t), which converge uniformly to the solution sin2⁡t,-sin⁡tT of the telegraph system (70).This illustrates the application of the procedure proven above for approximately obtaining the solution.



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*Source: 1982568-2017-11-13.xml*