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A Mixed Discontinuous Galerkin Approximation of Time Dependent Convection Diffusion Optimal Control Problem.

1. Introduction

The objective of this paper mainly focuses on developing a mixed discontinuous Galerkin scheme for the following control constrained optimal control problem governed by a transient convection diffusion equation:

[mathematical expression not reproducible]. (1)

More details will be specified later.

This kind of problem plays an important role in many fields, such as air pollution ([1]) and waste water treatment ([2]). In recent years the research of numerical method for this kind of problem forms a hot topic. Lots of literatures are devoted to developing effective numerical methods for this kind of problem. In [3-6] the stabilization method such as local projection stabilization, SUPG, and continuous interior penalty method are investigated. The discontinuous Galerkin approximation including primary discontinuous Galerkin method and local discontinuous Galerkin method is addressed in [7-12]. In [13] the authors discuss the characteristic finite element approximation of transient convection

diffusion optimal control problem. For more references, one can refer to [14].

In this paper, we investigate a mixed discontinuous Galerkin approximation of transient convection diffusion optimal control problem with control constraints. This scheme is based on the combination of a discontinuous Galerkin method for the hyperbolic part and a mixed finite element method for the elliptic part of the state equation. Variational discretization approach is utilized to approximate the control variable. The work of this paper is motivated by [8, 15] where a similar scheme was proposed for convection diffusion equations and stationary convection diffusion optimal control problems, respectively. Similar to other discontinuous Galerkin methods, this scheme is also locally conservative, which makes it much suitable for problems where conservation is important, for example, for time dependent convection diffusion problems. Moreover, when the diffusion coefficient tends to zero, this scheme reduces to the classical discontinuous Galerkin method. Thus it inherits the stabilizing features of discontinuous Galerkin methods. We derive a priori error estimates of the state, adjoint state, and control for both semidiscrete scheme and fully discrete scheme. Numerical experiment is carried out to show the performance of our scheme.

The rest of the paper is organized as follows: In Section 2, semidiscrete and fully discrete mixed discontinuous Galerkin scheme are defined for control constrained optimal control problems governed by the time dependent convection diffusion equations. In Section 3, a priori error estimates of the semidiscrete scheme are derived. In Section 4, we derive a priori error estimates of the fully discrete scheme. Finally numerical example is given to illustrate the theoretical findings.

2. Mixed Discontinuous Galerkin Scheme

Consider the following optimal control problem with control constraints:

[mathematical expression not reproducible], (2)

where [OMEGA] is a convex polygon with piecewise smooth boundary [omega][OMEGA]. Here K is the admissible set defined by

K ={q [member of] [L.sup.2] ([[OMEGA].sub.T]) : a [less than or equal to] q(x, t) [less than or equal to] b a.e. in [[OMEGA].sub.T] with a, b [member of] R, a [less than or equal to] b}. (3)

[alpha] and f are given functions, [epsilon] > 0 is a constant, and [beta] is a given vector valued function, which satisfies [alpha] - (1/2)[nabla] x [beta] [greater than or equal to] [??] > 0 with a constant [??] > 0.

To define a mixed discontinuous Galerkin scheme for (2), we introduce a new variable:

q = -[[epsilon].sup.1/2][nabla]y. (4)

Then the optimal control problem (2) can be rewritten as

[mathematical expression not reproducible]. (5)

Let [T.sup.h] be a regular triangulation of [mathematical expression not reproducible]. Let [mathematical expression not reproducible] denotes the diameter of the element [tau]. We define the following spaces:

[mathematical expression not reproducible]. (6)

For simplicity, we set

[mathematical expression not reproducible], (7)

where

[partial derivative][[tau].sub.-] = {l [member of] [partial derivative][tau], n x [beta][|.sub.l] < 0}, (8)

n is the outward norm direction on [partial derivative][tau],

[mathematical expression not reproducible], (9)

and [w] = [w.sub.+] on [partial derivative][tau] when [partial derivative][[tau].sub.-] [subset] [partial derivative][OMEGA]. Then the weak formulation for the optimal control problem (5) reads as follows: finding (q, y, u) [subset] V x W x K such that

[mathematical expression not reproducible] (10)

[mathematical expression not reproducible]. (11)

Standard arguments techniques imply that optimal control problem (10)-(11) admits a unique solution and the following first-order optimality condition holds:

[mathematical expression not reproducible]. (12)

Let [V.sub.h] x [W.sub.h] [subset] V x W denote the Raviart-Thomas element space of the lowest order associated with a triangular or rectangular mesh [T.sup.h] of [OMEGA] (see [16] for details). Then the approximation scheme of (10)-(11) is as follows: finding ([q.sub.h], [y.sub.h], [u.sub.h]) [member of] [V.sub.h] x [W.sub.h] x K such that

[mathematical expression not reproducible]. (13)

Here [[??].sub.0] is an approximation of the initial value [y.sub.0](x). Variational discretization approach is used for the control u.

Similar to continuous case we derive the following semidiscrete optimality conditions:

[mathematical expression not reproducible]. (14)

To define a fully discrete scheme we introduce a time partition. Let 0 = [t.sub.0] < [t.sub.1] < ... < [t.sub.N-1] < [t.sub.N] = T be a time grid with k = [t.sub.n] - [t.sub.n-1], n= 1, 2, ..., N. Let [I.sub.n] = ([t.sub.n-1], [t.sub.n]] be a half-open interval. We write [[phi].sub.n] := [phi]([t.sub.n]) for a smooth [phi].

We use backward Euler scheme for time discretization. Then a fully discrete scheme of (10)-(11) is characterized as follows: finding ([q.sup.n.sub.h], [y.sup.n.sub.h], [u.sup.n.sub.h]) [member of] [V.sub.h] x [W.sub.h] x K such that

[mathematical expression not reproducible]. (15)

To obtain the fully discrete first-order optimality condition we define a Lagrange functional as follows:

[mathematical expression not reproducible]. (16)

Here [mathematical expression not reproducible]. Then we derive the discrete first-order optimality condition:

[mathematical expression not reproducible]. (17)

3. Semidiscrete Error Estimate

The goal of this section is to prove the semidiscrete error estimates for the state, the adjoint state, and the control. We firstly decompose y - [y.sub.h] and z - [z.sub.h] as

y - [y.sub.h] = y(u) - [y.sub.h](u) + [y.sub.h](u) - [y.sub.h], z - [z.sub.h] = z(u) - [z.sub.h](u) + [z.sub.h](u) - [z.sub.h], (18)

where [y.sub.h](u) and [z.sub.h](u) satisfy the following auxiliary problems:

[mathematical expression not reproducible]. (19)

For the following analysis we introduce a new norm:

[mathematical expression not reproducible]. (20)

Following [8] we have D(y, y) [greater than or equal to] [[parallel]y[[parallel].sup.2.sub.*]

Let [mathematical expression not reproducible] be the projections defined by

[mathematical expression not reproducible], (21)

[mathematical expression not reproducible], (22)

According to [8] we have the following.

Lemma 1. If [mathematical expression not reproducible] satisfy (21)-(22), respectively, then we have

[mathematical expression not reproducible]. (23)

Lemma 2. If [mathematical expression not reproducible] satisfy (21)-(22), respectively, then the following estimates hold:

[mathematical expression not reproducible]. (24)

Proof. Differentiating (21) about t, we have

[mathematical expression not reproducible]. (25)

Set [P.sub.h] :W [right arrow] [W.sub.h] and [[PI].sub.h] : V [right arrow] [V.sub.h] to be the interpolation operators for the standard RT element space (see, e.g., [17]) such that

([psi] - [P.sub.h][psi], [w.sub.h]) = 0, [for all][w.sub.h] [member of] [W.sub.h], (div([phi] - [[PI].sub.h][phi]), [w.sub.h]) = 0, [for all][w.sub.h] [member of] [W.sub.h]. (26)

Then we have the following approximation properties:

[mathematical expression not reproducible]. (27)

Let

[mathematical expression not reproducible]. (28)

Then we derive

[mathematical expression not reproducible]. (29)

By the definition of interpolation operators [P.sub.h] and [[PI].sub.h] we obtain

B([[eta].sub.q,t], [w.sub.h]) = 0, [for all][w.sub.h] [member of] [W.sub.h], B ([v.sub.h], [[eta].sub.y,t]) = 0 [for all][v.sub.h] [member of] [V.sub.h]. (30)

Setting [w.sub.h] = [[xi].sub.y,t] and [v.sub.h] = [[xi].sub.q,t] leads to

A([[xi].sub.q,t], [[xi].sub.q,t]) + D([[xi].sub.y,t], [[xi].sub.y,t]) = A ([[eta].sub.q,t], [[xi].sub.q,t]) + D ([[eta].sub.y,t], -[[xi].sub.y,t]), (31)

namely,

[mathematical expression not reproducible]. (32)

Since the function in [W.sub.h] is piecewise constant, using the definition of D(*, *), we have for all w [member of] W and [theta] [member of] [W.sub.h]

[mathematical expression not reproducible]. (33)

Therefore, we get the estimate as follows:

[mathematical expression not reproducible]. (34)

Note that

[mathematical expression not reproducible], (35)

and according to [18] we have

[mathematical expression not reproducible]. (36)

Then we derive

[mathematical expression not reproducible]. (37)

By setting [delta] small enough we obtain

[mathematical expression not reproducible]. (38)

Using triangle inequality we have

[mathematical expression not reproducible], (39)

which implies the first theorem result. Similar to the above proof we have

[mathematical expression not reproducible]. (40)

Using the above estimates we can derive the following.

Theorem 3. Let (y, q, z, p) and ([y.sub.h](u), [q.sub.h](u), [z.sub.h](u), [p.sub.h](u)) be the solutions of (12) and (19), respectively. Then we have

[[absolute value of ([parallel](y - [y.sub.h](u), q - [q.sub.h] (u))[parallel])].sub.+] [less than or equal to] [Ch.sup.1/2]. (41)

Here

[mathematical expression not reproducible] (42)

and the constant C depends on the norms of y and q.

Proof. Using (12) and (19) we derive the following error equations:

[mathematical expression not reproducible]. (43)

We split the errors y - [y.sub.h](u) and q - [q.sub.h](u) into

[mathematical expression not reproducible], (44)

where [mathematical expression not reproducible] is defined in (21). Then we have

[mathematical expression not reproducible]. (45)

Setting [chi] = [??] - [y.sub.h](u) and [sigma] = [??] - [q.sub.h](u) gives

[mathematical expression not reproducible]. (46)

Choosing [w.sub.h] = [chi] and [v.sub.h] = [sigma] in the above equation leads to

([[chi].sub.t] [chi]) + D ([chi], [chi]) + A ([sigma], [sigma]) = -([y.sub.t] - [[??].sub.t], [chi]). (47)

Since D([chi], [chi]) [greater than or equal to] [[parallel][chi][parallel].sup.2.sub.*], then we deduce by Young inequality that

[mathematical expression not reproducible]. (48)

By setting [delta] small enough we derive

[mathematical expression not reproducible]. (49)

Note that [chi](0) = 0. Integrating (49) from 0 [right arrow] t leads to

[mathematical expression not reproducible]. (50)

Similarly, integrating (49) from 0 [right arrow] T yields

[mathematical expression not reproducible], (51)

which implies

[mathematical expression not reproducible]. (52)

Combining (50) and (52) we deduce that

[mathematical expression not reproducible]. (53)

By Lemma 2 we further derive

[mathematical expression not reproducible]. (54)

Then by triangle inequality and Lemma 1 we obtain

[mathematical expression not reproducible]. (55)

This implies the theorem result.

Theorem 4. Let (y, q, z, p) and ([y.sub.h](u), [q.sub.h](u), [z.sub.h](u), [p.sub.h](u)) be the solutions of (12) and (19), respectively. Then we have

[[absolute value of ([parallel](z - [z.sub.h] (u), p - [p.sub.h] (u))[parallel])].sub.+] [less than or equal to] [Ch.sup.1/2]. (56)

Here the constant C depends on the norms of y, z and q, p.

Proof. Using (12) and (19) we derive the following error equations:

[mathematical expression not reproducible]. (57)

We rewrite the errors z - [z.sub.h](u) and p - [p.sub.h](u) as

[mathematical expression not reproducible], (58)

where [mathematical expression not reproducible] is defined in (22). Then we have

[mathematical expression not reproducible]. (59)

Setting [mathematical expression not reproducible] we have

-([[xi].sub.t], [[psi].sub.h] - B([rho], [[psi].sub.h] + D([[psi].sub.h], [xi]) + A([rho], [[phi].sub.h]) + B([[phi].sub.h], [xi]) = ([z.sub.t] - [[??].sub.t], [[psi].sub.h]) + (y - [y.sub.h](u), [[psi].sub.h]). (60)

Taking [[psi].sub.h] = [xi] and [[phi].sub.h] = [rho] yields

-([[xi].sub.t], [xi]) + D([xi], [xi]) + A([rho], [rho]) = ([z.sub.t] - [[??].sub.t], [xi]) + (y - [y.sub.h](u), [xi]). (61)

Since D([xi], [xi]) [greater than or equal to] [[parallel][xi][parallel].sup.2.sub.*], then

[mathematical expression not reproducible]. (62)

Using Holder inequality and Young inequality we derive

[mathematical expression not reproducible]. (63)

Choosing [delta] small enough we obtain

[mathematical expression not reproducible]. (64)

Integrating (64) with respect to t from t [right arrow] T yields

[mathematical expression not reproducible], (65)

where [xi](T) = 0 was used. In an analogue way, integrating (64) from 0 [right arrow] T results in

[mathematical expression not reproducible]. (66)

We further have

[mathematical expression not reproducible]. (67)

Combining (65) and (67) we deduce that

[mathematical expression not reproducible]. (68)

Then we can derive the theorem result by using Lemmas 1 and 2 and Theorem 8 and triangle inequality.

Lemma 5. Let ([y.sub.h], [q.sub.h], [z.sub.h], [p.sub.h]) and ([y.sub.h](u), [q.sub.h](u), [z.sub.h](u), [p.sub.h](u)) be the solutions of (14) and (19), respectively. Then we have

[mathematical expression not reproducible]. (69)

Proof. Using (19) along with (14) leads to

[mathematical expression not reproducible]. (70)

Setting [Y.sub.h] = [y.sub.h] - [y.sub.h](u) and [Q.sub.h] = [q.sub.h] - [q.sub.h](u) and choosing [w.sub.h] = [Y.sub.h] and [v.sub.h] = [Q.sub.h] lead to

([Y.sub.h,t], [Y.sub.h]) + A ([Q.sub.h], [Q.sub.h]) + D([Y.sub.h], [Y.sub.h]) = ([u.sub.h] - u, [Y.sub.h]). (71)

By Young inequality we have

[mathematical expression not reproducible]. (72)

Similar to the proof of Lemmas 1 and 2 we have

[mathematical expression not reproducible]. (73)

By (17) and (19) we derive

[mathematical expression not reproducible]. (74)

Setting [Z.sub.h] = [z.sub.h] - [z.sub.h](u) and [P.sub.h] = [p.sub.h] - [p.sub.h](u) and choosing [[psi].sub.h] = [Z.sub.h] and [[phi].sub.h] = [P.sub.h] yield

-([Z.sub.h,t], [Z.sub.h]) + A([P.sub.h], [P.sub.h]) + D([Z.sub.h], [Z.sub.h]) = ([Y.sub.h], [Z.sub.h]). (75)

Furthermore, we derive by Young inequality

[mathematical expression not reproducible]. (76)

Similar to the proof of Lemmas 1 and 2 we have

[mathematical expression not reproducible]. (77)

Then the theorem result follows from (73) and (77).

Lemma 6. Assume that (y, q, z, P, u) and ([y.sub.h], [q.sub.h], [z.sub.h], [p.sub.h], [u.sub.h]) be the solutions of (11) and (14), respectively. Let

[J'.sub.h](u)(v - u) = [[integral].sup.T.sub.0] ([gamma]u + [z.sub.h] (u), v - u), (78)

where [z.sub.h](u) is the solution of (19). Then the following estimate holds:

[mathematical expression not reproducible]. (79)

Proof. By the definitions of ([y.sub.h](u), [q.sub.h](u)), ([z.sub.h](u), [p.sub.h](u)) we have

[mathematical expression not reproducible]. (80)

Note that

[mathematical expression not reproducible]. (81)

By setting [w.sub.h] = [z.sub.h](v) - [z.sub.h](u), [v.sub.h] = [p.sub.h](v) - [p.sub.h](u) and [[psi].sub.h] = [y.sub.h](v) - [y.sub.h](u), [[phi].sub.h] = [q.sub.h](v) - [q.sub.h](u) in the above equations, respectively, we can prove

[mathematical expression not reproducible]. (82)

Since ([y.sub.h](v) - [y.sub.h](u))(0) = 0, ([z.sub.h](u))(T) = 0, and

[mathematical expression not reproducible] (83)

we can prove that

[[integral].sup.T.sub.0] ([z.sub.h](v) - [z.sub.h](u), v - u) = [[integral].sup.T.sub.0] ([y.sub.h](v) - [y.sub.h](v) - [y.sub.h](v) - [y.sub.h](u)) [greater than or equal to] 0.

Therefore, we derive

[mathematical expression not reproducible]. (85)

Lemma 7. Let (y, q, z, p, u) and ([y.sub.h], [q.sub.h], [z.sub.h], [p.sub.h], [u.sub.h]) be the solutions of (11) and (14), respectively. Then we have

[mathematical expression not reproducible]. (86)

Proof. By Lemma 6 we have

[mathematical expression not reproducible]. (87)

Using the result of Theorem 4 yield

[mathematical expression not reproducible]. (88)

Theorem 8. Let (y, q, z, p) and ([y.sub.h](u), [q.sub.h](u), [z.sub.h](u), [p.sub.h](u)) be the solutions of (12) and (14), respectively. Then we have

[mathematical expression not reproducible]. (89)

Proof. By Lemma 2 and (88) we derive

[mathematical expression not reproducible]. (90)

Combining Lemma 5 and (90) yields

[mathematical expression not reproducible]. (91)

Combining (88)-(91) we can derive the theorem result.

4. Fully Discrete Error Estimate

In this section we will prove the error estimates for the fully discrete scheme. For this purpose we firstly introduce the following auxiliary problems:

[mathematical expression not reproducible]. (92)

Lemma 9. Let (y, q, z, p) and ([y.sup.n.sub.h](u), [q.sup.n.sub.h](u), [z.sup.n-1.sub.h](u), [p.sup.n-1.sub.h](u)) be the solutions of (12) and (92), respectively. Then we have

[[absolute value of ([parallel](y - [y.sub.h](u), q - [q.sub.h](u))[parallel])].sub.[??]] [less than or equal to] C([h.sup.1/2] + k), (93)

where

[mathematical expression not reproducible]. (94)

Proof. Using (12) and (92) we derive the following error equations:

[mathematical expression not reproducible]. (95)

Setting [mathematical expression not reproducible] and combining the definition of y and q we have

[mathematical expression not reproducible]. (96)

Testing (96) with [w.sub.h] = [[chi].sup.n] and [v.sub.h] = on yields

[mathematical expression not reproducible]. (97)

By Taylor expansion with integral reminder we deduce that

[mathematical expression not reproducible]. (98)

By Holder inequality we obtain

[mathematical expression not reproducible]. (99)

Then by Young inequality we derive

[mathematical expression not reproducible]. (100)

Choosing [delta] to be small enough leads to

[mathematical expression not reproducible]. (101)

Multiplying (114) by 2k and summing up with respect to n from 1 to m yield

[mathematical expression not reproducible]. (102)

Multiplying (101) by 2k and summing up with respect to n from 1 to N lead to

[mathematical expression not reproducible]. (103)

Collecting above estimates we have

[mathematical expression not reproducible]. (104)

Using above estimates, Lemmas 1 and 2, and triangle inequality we deduce that

[mathematical expression not reproducible]. (105)

Lemma 10. Let (y, q, z, p) and ([y.sup.n.sub.h](u), [q.sup.n.sub.h](u), [z.sup.n-1.sub.h](u), [p.sup.n-1.sub.h](u)) be the solutions of (12) and (92), respectively. Then we have

[[absolute value of ([parallel](z - [z.sub.h](u), p - [p.sub.h](u))[parallel])].sub.[??]] [less than or equal to] C([h.sup.1/2] + k). (106)

Proof. Using (12) and (92) we derive the following error equations:

[mathematical expression not reproducible]. (107)

Setting [mathematical expression not reproducible] and using the definitions of p and p give

[mathematical expression not reproducible]. (108)

Choosing [w.sub.h] = [[xi].sup.n-1] and [v.sub.h] = [[sigma].sup.n] in (108) yields

[mathematical expression not reproducible]. (109)

Using Holder inequality and Young inequality we obtain

[mathematical expression not reproducible]. (110)

Then by setting [delta] small enough we have

[mathematical expression not reproducible]. (111)

Multiplying (111) by and summing up with respect to n from m to N yield

[mathematical expression not reproducible]. (112)

Multiplying (111) by 2k and summing up with respect to n from 1 to N give

[mathematical expression not reproducible]. (113)

Collecting (112) and (113) we obtain

[mathematical expression not reproducible]. (114)

By Lemmas 1 and 2 and triangle inequality we derive

[mathematical expression not reproducible]. (115)

Lemma 11. Let ([y.sup.n.sub.h], [q.sup.n.sub.h], [z.sup.n-1.sub.h], [p.sup.n-1.sub.h]) and ([y.sup.n.sub.h](u), [q.sup.n.sub.h](u), [z.sup.n-1.sub.h](u), [p.sup.n-1.sub.h](M)) be the solutions of (12) and (92), respectively. Then we have

[mathematical expression not reproducible]. (116)

Proof. Using (92) along with (17) leads to

[mathematical expression not reproducible]. (117)

Choosing [w.sub.h] = [y.sup.n.sub.h] - [y.sup.n.sub.h](u) and [v.sub.h] = [q.sup.n.sub.h] - [q.sup.n.sub.h](u) in the above equations and adding the resulting equations together yield

[mathematical expression not reproducible]. (118)

Setting [Y.sub.h] = [y.sub.h] - [y.sub.h](u) and [Q.sub.h] = [q.sub.h] - [q.sub.h](u) in the above equation leads to

[mathematical expression not reproducible]. (119)

and then

[mathematical expression not reproducible]. (120)

Furthermore, we have

[mathematical expression not reproducible]. (121)

Similar to the proof of Lemmas 9 and 10 we have

[mathematical expression not reproducible], (122)

Similarly, by (17) and (92) we derive

[mathematical expression not reproducible]. (123)

Taking [[psi].sub.h] = [z.sup.n-1.sub.h] - [z.sup.n-1.sub.h](u) and [[phi].sub.h] = [p.sup.n-1.sub.h] - [p.sup.n- 1.sub.h](u) in the above equation, setting [Z.sup.n-1.sub.h] = [z.sup.n-1.sub.h] - [z.sup.n-1.sub.h](u) and [P.sup.n-1.sub.h] = [p.sup.n-1.sub.h] - [p.sup.n-1.sub.h](w), and using similar argument to (124) give

[mathematical expression not reproducible]. (124)

Then the theorem result follows from (122) and (124).

Lemma 12. Let (y, q, z, p, m) and ([y.sup.n-1.sub.h], [q.sup.n-1.sub.h], [z.sup.n-1.sub.h], [p.sup.n-1.sub.h], [u.sup.n.sub.h]) be the solutions of (12) and (92), respectively. Let

[[??].sub.h](u)(v - u) = k [N.summation over (n=1)] ([gamma][u.sup.n] + [z.sup.n-1.sub.h](u), [v.sup.n] - [u.sup.n]), (125)

where [z.sup.n-1.sub.h](w) is the solution of (92). Then the following estimate holds:

[mathematical expression not reproducible]. (126)

Here [mathematical expression not reproducible].

Proof. Note that

[mathematical expression not reproducible]. (127)

Setting [Y.sup.n.sub.h] = [y.sup.n.sub.h](v) - [y.sup.n.sub.h](u), [Q.sup.n.sub.h] = [q.sup.n.sub.h](v) - [q.sup.n.sub.h](u), [Z.sup.n-1.sub.h] = [z.sup.n-1.sub.h](v) - [z.sup.n-1.sub.h](u), and [p.sup.n-1.sub.h] = [p.sup.n-1.sub.h](v) - [p.sup.n-1.sub.h](u), by the definition of [y.sup.n.sub.h](w) and [p.sup.n-1.sub.h](w) we have

[mathematical expression not reproducible]. (128)

Choosing [w.sub.h] = [Z.sup.n-1.sub.h], [v.sub.h] = [P.sup.n-1.sub.h] and [[psi].sub.h] = [Y.sup.n.sub.h], [[phi].sub.h] = [Q.sup.n.sub.h] in the above equations, respectively, we can easily prove that

k [N.summation over (n=1)]([z.sup.n-1.sub.h](v) - [z.sup.n-1.sub.h](u), [v.sup.n] - [u.sup.n]) [greater than or equal to] 0. (129)

Therefore, we derive

[mathematical expression not reproducible]. (130)

Lemma 13. Let (y, q, z, p, m) and ([y.sup.n.sub.h], [q.sup.n.sub.h], [z.sup.n-1.sub.h], [p.sup.n-1.sub.h], [u.sup.n.sub.h]) be the solutions of (12) and (92), respectively. Then we have

[mathematical expression not reproducible]. (131)

Proof. Following Lemma 12 we derive

[mathematical expression not reproducible]. (132)

Using Young inequality we obtain

[mathematical expression not reproducible]. (133)

Theorem 14. Let (y, q, z, p, u) and ([y.sup.n.sub.h], [q.sup.n-1.sub.h], [z.sup.n-1.sub.h], [p.sup.n-1.sub.h], [u.sup.n.sub.h]) be the solutions of (11) and (17), respectively. Then we have

[mathematical expression not reproducible]. (134)

Proof. By Lemma 9 and using triangle inequality we derive

[mathematical expression not reproducible]. (135)

By Lemma 10 we deduce that

[mathematical expression not reproducible]. (136)

Combining Lemma 13, (135), and (136) we arrive at

[mathematical expression not reproducible], (137)

which completes the theorem.

http://dx.doi.org/ 10.1155/2017/6901467

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge the support of Natural Science Foundation of Shandong Province (no. ZR2016JL004).

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Qingjin Xu and Zhaojie Zhou

College of Mathematical Sciences, Shandong Normal University, Jinan, China

Correspondence should be addressed to Zhaojie Zhou; zhouzhaojie@sdnu.edu.cn

Received 4 November 2016; Accepted 9 January 2017; Published 21 February 2017

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Title Annotation:Research Article
Author:Xu, Qingjin; Zhou, Zhaojie
Publication:Journal of Mathematics
Article Type:Report
Date:Jan 1, 2017
Words:4712
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