Citation: Carla Marchetti. Green tea catechins and intracellular calcium dynamics in prostate cancer cells[J]. AIMS Molecular Science, 2021, 8(1): 1-12. doi: 10.3934/molsci.2021001
[1] | Saudia Jabeen, Bandar Bin-Mohsin, Muhammad Aslam Noor, Khalida Inayat Noor . Inertial projection methods for solving general quasi-variational inequalities. AIMS Mathematics, 2021, 6(2): 1075-1086. doi: 10.3934/math.2021064 |
[2] | S. S. Chang, Salahuddin, M. Liu, X. R. Wang, J. F. Tang . Error bounds for generalized vector inverse quasi-variational inequality Problems with point to set mappings. AIMS Mathematics, 2021, 6(2): 1800-1815. doi: 10.3934/math.2021108 |
[3] | Shujie Jing, Jixiang Guan, Zhiyong Si . A modified characteristics projection finite element method for unsteady incompressible Magnetohydrodynamics equations. AIMS Mathematics, 2020, 5(4): 3922-3951. doi: 10.3934/math.2020254 |
[4] | Muhammad Aslam Noor, Khalida Inayat Noor, Bandar B. Mohsen . Some new classes of general quasi variational inequalities. AIMS Mathematics, 2021, 6(6): 6406-6421. doi: 10.3934/math.2021376 |
[5] | Jamilu Abubakar, Poom Kumam, Jitsupa Deepho . Multistep hybrid viscosity method for split monotone variational inclusion and fixed point problems in Hilbert spaces. AIMS Mathematics, 2020, 5(6): 5969-5992. doi: 10.3934/math.2020382 |
[6] | Zuliang Lu, Xiankui Wu, Fei Huang, Fei Cai, Chunjuan Hou, Yin Yang . Convergence and quasi-optimality based on an adaptive finite element method for the bilinear optimal control problem. AIMS Mathematics, 2021, 6(9): 9510-9535. doi: 10.3934/math.2021553 |
[7] | Chunjuan Hou, Zuliang Lu, Xuejiao Chen, Fei Huang . Error estimates of variational discretization for semilinear parabolic optimal control problems. AIMS Mathematics, 2021, 6(1): 772-793. doi: 10.3934/math.2021047 |
[8] | Safeera Batool, Muhammad Aslam Noor, Khalida Inayat Noor . Merit functions for absolute value variational inequalities. AIMS Mathematics, 2021, 6(11): 12133-12147. doi: 10.3934/math.2021704 |
[9] | Zuliang Lu, Fei Cai, Ruixiang Xu, Chunjuan Hou, Xiankui Wu, Yin Yang . A posteriori error estimates of hp spectral element method for parabolic optimal control problems. AIMS Mathematics, 2022, 7(4): 5220-5240. doi: 10.3934/math.2022291 |
[10] | Satit Saejung . A counterexample to the new iterative scheme of Rezapour et al.: Some discussions and corrections. AIMS Mathematics, 2023, 8(4): 9436-9442. doi: 10.3934/math.2023475 |
This paper is concerned with the finite element approximation of system of J = 2 quasi-variational inequalities QVIs with term sources and obstacles depending on solution: Find a vector U=(u1,u2)∈(H10(Ω))2 satisfying
ai(ui,v−ui)≥(fi(ui),v−ui);v∈H10(Ω) | (1.1) |
v,ui≤Mui;ui≥0. |
Where Ω is a bounded smooth domain of RN with N≥ 1, each ai(.,.) is a continuous elliptic bilinear form, (.,.) is the inner product in L2(Ω) and each fi is a regular, nonlinear functional depending on solutions. The obstacle M provide the coupling between the unknowns u1; u2
Mui=k+infμ≠iuμ; |
k is a positive number. We point out that in the case where fi are independent of the solution, the system (1.1) coincides with that introduced by Bensoussan and Lions in [1] which arises in the management of energy production problems.
It is easy to note that the structure of system (1.1) is analogous to that of the classical obstacle problem [2] where the term source and obstacle are depending upon the solution sought. The terminology QVI being chosen is a result of this remark.
Numerical analysis of system of quasi-variational inequalities where term sources not depending on solutions were achieved in several works, we refer to [3,4,5,6,7,8] for system of quasi-variational inequalities with coercive or noncoercive operators.
For results on systems related to evolutionary Hamilton-Jacobi-Bellman equation we refer to [9,10,11].
The main objective of this paper is to show that problem (1.1) can be properly approximated by a finite element method and an optimal L∞-error estimates is derived, which coincides with the optimal convergence order of elliptic variational inequalities of an obstacle type problem [12].
The approximation is carried out by first introducing a modified Bensoussan-Lions type iterative scheme depending on parameters which is shown to converge geometrically to the continuous solution. By a symmetrical approach, using the standard finite element method and a discrete maximum principle (DMP), the geometric convergence of the discrete modified Bensoussan-Lions type iterative scheme depending upon parameters is given as well. An L∞-error estimates is then established combining the geometric convergence of both the continuous and discrete iterative schemes and the known uniform error estimates in elliptic VIs.
It is worth mentioning that even the guiding idea of this paper rests on the algorithmic approach followed in many papers cited above, the treatment of the geometric convergence of both continuous and discrete schemes is totally different because of the nonlinear nature of terms sources. Also, it is used for the first time for a system of QVIs.
An outline of this paper is as follows: In section 2, we lay down some definitions and classical results related to variational inequalities and prove a Lipschitz continuous and discrete dependency with respect to the source term, the boundary condition and the obstacle. Section 3 discusses the continuous Bensoussan-Lions type iterative scheme and proves its geometrical convergence. In Section 4, we establish the finite element counter parts of the continuous system and the continuous Bensoussan-Lions type iterative scheme respectively and the geometrical convergence of the discrete scheme. Section 5 is devoted the L∞-error analysis of the method.
We are given functions aijk(x),aik(x),ai0(x),1≤i≤2 sufficiently smooth functions such that 1≤j,k≤N
∑1≤j,k≤Naijk(x)ξjξk≥α|ξ|2,ξ∈RN,α>0 |
ai0(x)≥βi>0,(x∈Ω) | (2.1) |
where βi is a positive constant. We define the bilinear forms: For all u,v∈H10(Ω)
ai(u,v)=∫Ω(∑1≤j,k≤Naijk(x)∂u∂xj∂v∂xk+∑Nk=1aik(x)∂u∂xkv+ai0(x)uv)dx | (2.2) |
We are given right-hand sides
fisuchthatfi∈L∞(Ω),fi≥f0>0, |
a nonlinear functional and Lipschitz continuous on R; that is
|fi(x)−fi(y)|≤ki|x−y|,∀x,y∈R, |
such that
αi=kiβi<1, | (2.3) |
where βi is a constant defined in (2.1). For W=(w1,w2)∈(L∞+(Ω))2 we introduce the norm
‖W‖∞=max1≤i≤2‖wi‖L∞(Ω). |
Let be Ω a bounded polyhedral domain of R2 or R3 with sufficiently smooth boundary ∂Ω. We consider the bilinear form of the same form of those defined in (2.2), the linear form
(f,v)=∫Ωf(x)v(x)dx, | (2.4) |
The right hand side
f∈L∞(Ω), | (2.5) |
the obstacle
ψ∈W2,∞(Ω)andψ≥0, | (2.6) |
the boundary condition g∈L∞(∂Ω) and the nonempty convex set
Kg={v∈H1(Ω)suchthatv=gon∂Ωandv≤ψonΩ}. | (2.7) |
We consider the variational inequality V.I.: Find u∈Kg such that
a(u,v−u)≥(f,v−u),∀v∈Kg. | (2.8) |
Proposition 1 Let (f,g,ψ); (˜f,˜g,˜ψ)be a pair of data and ζ=σ(f,g,ψ); ˜ζ=σ(˜f,˜g,˜ψ) the corresponding solution to (2.8). If f≤˜f in Ω, g≤˜g on ∂Ω and ψ≤˜ψ then, ζ≤˜ζ in Ω.
Proof. The proof is an adaptation of the proof of the monotonicity property of the solution of Ⅵ with nonlinear source term (see [13]). According to [14], ζ=max{ζ_} where {ζ_} is the set of all the subsolutions of ζ. Hence, ∀ζ_∈{ζ_}, ζ_ satisfies
a(ζ_,v)≤(f,v),∀v≥0withζ_≤ψandζ_≤g. |
By using the conditions f≤˜f in Ω, g≤˜g on ∂Ω and ψ≤˜ψ, we get
a(ζ_,v)≤(f,v)≤(˜f,v), |
with
ζ_≤ψ≤˜ψandζ_≤g≤˜gon∂Ω. |
Thus, ζ is a subsolution of ˜ζ=σ(˜f,˜g,˜ψ), that is ζ≤˜ζ in Ω.
This subsection is devoted to the establishment of a Lipschitz continuous dependence property of the solution with respect to the source term, the boundary condition and the obstacle by which we first, set out and demonstrate.
Proposition 2 Let (f,g,ψ); (˜f,˜g,˜ψ)be a pair of data andζ=σ(f,g,ψ); ˜ζ=σ(˜f,˜g,˜ψ) the corresponding solution to (2.8). Then, we have
‖ζ−˜ζ‖L∞(Ω)≤max{(1β)‖f−˜f‖L∞(Ω),‖g−˜g‖L∞(∂Ω),‖ψ−˜ψ‖L∞(Ω)}. | (2.9) |
Proof. The proof is an adaptation of the proof of a Lipschitz property of the solution of Ⅵ with nonlinear source term (see [13]). First, set
φ=max{(1β)‖f−˜f‖L∞(Ω),‖g−˜g‖L∞(∂Ω),‖ψ−˜ψ‖L∞(Ω)}. | (2.10) |
Then,
˜f≤f+‖f−˜f‖L∞(Ω) |
≤f+(1)‖f−˜f‖L∞(Ω) |
≤f+(a0(x)β)‖f−˜f‖L∞(Ω) |
≤f+a0(x)max{(1β)‖f−˜f‖L∞(Ω),‖g−˜g‖L∞(∂Ω),‖ψ−˜ψ‖L∞(Ω)}. |
So,
≤f+a0(x)φinΩ. | (2.11) |
Thus, for all 0<v,
(˜f,v)≤(f+a0(x)φ,v), |
with
˜ζ≤˜g≤g+φon∂Ω, |
˜ζ≤˜ψ≤ψ+φinΩ. |
So, according to the property ˜ζ is a subsolution ofσ(f+a0(x)φ,g+φ,ψ+φ)=σ(f,g,ψ)+φ, that is
˜ζ≤ζ+φin¯Ω |
or
˜ζ−ζ≤φin¯Ω. | (2.12) |
Similarly, interchanging the roles of the couples (f,g,ψ); (˜f,˜g,˜ψ), we obtain
ζ−˜ζ≤φin¯Ω, | (2.13) |
which completes the proof.
Let τh be a triangulation of Ω with meshsize h, Vh be the space of finite elements consisting of continuous piecewise linear functions v vanishing on ∂Ω and φs; s = 1, 2, …, m(h) be the basis functions of Vh.
The discrete counterpart of (2.8) consists of finding uh∈Kgh such that
a(uh,v−uh)≥(f,v−uh),∀v∈Kgh. | (2.14) |
Where
Kgh={v∈Vhsuchthatv=πhgon∂Ωandv≤rhψonΩ}, | (2.15) |
πh is an interpolation operator on ∂Ω and rhis the usual finite element restriction operator on Ω.
Theorem 3 (See [12] Under conditions (2.5) and (2.6), there exists a constant C independent of h such that
‖ζ−ζh‖L∞(Ω)≤Ch2|logh|2. | (2.16) |
Assuming that the DMP is satisfied, i.e. the matrix resulting from the finite element discretization is an M-matrix (see [15,16]), we prove the Lipschitz discrete dependence with respect to the boundary condition, the source term and the obstacle by a similar study to that undertaken previously for the Lipschitz continuous dependence property.
Proposition 4 Let (f,g,rhψ); (˜f,˜g,rh˜ψ)be a pair of data and ζh=σh(f,g,rhψ); ˜ζh=σh(˜f,˜g,rh˜ψ) the corresponding solution to (2.14). If f≤˜f in Ω, g≤˜g on ∂Ω and rhψ≤rh˜ψ then, ζh≤˜ζh in Ω.
Proof. The proof is similar to that of the continuous case.
The proposition below establishes a Lipschitz discrete dependence of the solution with respect to the data.
Proposition 5 Let the (d.m.p) holds. Then, we have
‖ζh−˜ζh‖L∞(Ω)≤max{(1β)‖f−˜f‖L∞(Ω),‖g−˜g‖L∞(∂Ω),‖rhψ−rh˜ψ‖L∞(Ω)} | (2.17) |
Proof. The proof is similar to that of the continuous case.
We define the following fixed-point mapping
T:(L∞+(Ω))2→(L∞+(Ω))2 |
Z=(z1,z2)→TZ=ζ=(ζ1,ζ2). |
Where ζi∈H10(Ω)∩L∞(Ω) is a solution to the following variational inequality
ai(ζi,v−ζi)≥(fi(zi),v−ζi);v∈H10(Ω) | (3.1) |
v,ζi≤Mζi=k+zj;ζi≥0withi≠j. |
Thanks to [1,2], ζi is the unique solution to coercive variational inequality (3.1).
Remark 1 We remark that the solution U=(u1,u2) of the system (1.1) is the fixed point of the mapping T; that isTU=U.
Starting from U0=(u1,0,u2,0) where ui,0; i = 1; 2 is solution of the variational equation
ai(ui,0,v)=(fi(ui,0),v),∀v∈H10(Ω), |
and for all 0<wi<1; i=1,2 we define the sequences (u1,n+1) and (u2,n+1) such that u1,n+1 and u2,n+1 the components of the vector Un+1, solve the following elliptic variational inequalities respectively
(u1,n+1,v−u1,n+1)≥(w1f1(u1,n+1)+(1−w1)f1(u1,n),v−u1,n+1) | (3.2) |
v,u1,n+1≤Mu1,n+1=k+u2,n, | (3.3) |
a2(u2,n+1,v−u2,n+1)≥(w2f2(u2,n+1)+(1−w2)f2(u2,n),v−u2,n+1) | (3.4) |
v,u2,n+1≤Mu2,n+1=k+u1,n+1. | (3.5) |
Theorem 2 The sequences (u1,n+1) and (u2,n+1) converge geometrically to the solution U=(u1,u2) of the system (1.1); there exist a positive real ρ∈(0,1) which depends on αi and wi such that for all n≥0
‖Un+1−U‖∞≤ρn+1‖U0−U‖∞ | (3.6) |
where
ρ=max1≤i≤2α1(1−w1)1−α1w1<1. | (3.7) |
Proof. The proof will carry out by induction.
● We first deal with the case
‖u1−u1,0‖L∞(Ω)=max1≤i≤2‖ui−ui,0‖L∞(Ω). | (3.8) |
● Indeed for n = 0; using (1.1), (3.2), (3.3) and (2.9), we have
‖u1−u1,1‖L∞(Ω)≤max{(1β1)‖f1(u1)−(w1f1(u1,1)+(1−w1)f1(u1,0))‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} |
≤max{(1β1)‖w1(f1(u1)−f1(u1,1))+(1−w1)(f1(u1)−f1(u1,0))‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} |
≤max{(k1β1)(w1‖u1−u1,1‖L∞(Ω)+(1−w1)‖u1−u1,0‖L∞(Ω));‖u2−u2,0‖L∞(Ω)}. |
So,
‖u1−u1,1‖L∞(Ω)≤max{α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} | (3.9) |
We distinguish two cases
max{α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} |
=α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω) | (3.10) |
or
max{α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} |
=‖u2−u2,0‖L∞(Ω) | (3.11) |
(3.9) in conjunction with case (3.10) implies
‖u1−u1,1‖L∞(Ω)≤α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω) | (3.12) |
with
‖u2−u2,0‖L∞(Ω)≤α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω), | (3.13) |
which implies
‖u1−u1,1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω). | (3.14) |
By replacing (3.14) in (3.13), we get
‖u2−u2,0‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω) |
≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω), |
which coincides with (3.8).
(3.9) in conjunction with (3.11) implies
‖u1−u1,1‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω) | (3.15) |
with
α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω). | (3.16) |
‖u2−u2,0‖L∞(Ω) is bounded below by both ‖u1−u1,1‖L∞(Ω)
and
α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω). |
So,
‖u1−u1,1‖L∞(Ω)≤α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω) |
or
α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω). |
Then,
‖u1−u1,1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω) | (3.17) |
or
α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω). | (3.18) |
(3.15), (3.17) and (3.18) generate the following three possibilities
‖u1−u1,1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(Ω) |
or
‖u1−u1,1‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(Ω) |
or
α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(Ω). |
All possibilities are true in the same time because they coincide with (3.8). So, there is either a contradiction and thus case (3.11) is impossible or case (3.11) is possible if and only if
‖u1−u1,1‖L∞(Ω)=α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω). |
Hence, both cases (3.10) and (3.11) imply (3.14).
● Let us now discuss the second case
‖u2−u2,0‖L∞(Ω)=max1≤i≤2‖ui−ui,0‖L∞(Ω). | (3.19) |
(3.9) in conjunction with (3.10) implies (3.14) with
‖u2−u2,0‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω) |
≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω)<‖u2−u2,0‖L∞(Ω), |
which contradicts (3.19) which means that (3.10) is impossible. (3.9) in conjunction with (3.11) we get (3.17) and (3.18). So,
‖u1−u1,1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(Ω) |
or
α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(Ω). |
We remark that both alternatives are true in same time because both coincide with (3.19) which implies that in case (3.11), we must have
‖u1−u1,1‖L∞(Ω)=α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω). |
Hence, in both cases (3.8) and (3.19), we obtain (3.14). Hence,
‖u1−u1,1‖L∞(Ω)≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω). | (3.20) |
● As
U1=(u1,1,u2,1)andU=(u1,u2), |
we need to deal also with ‖u2−u2,1‖L∞(Ω), by following the same reasoning as that adopted for u1 and u1,1, we get
‖u2−u2,1‖L∞(Ω)≤max{α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω);‖u1−u1,1‖L∞(Ω)} | (3.21) |
Again we distinguish two possibilities
max{α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω);‖u1−u1,1‖L∞(Ω)} |
=α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω); | (3.22) |
or
max{α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω);‖u1−u1,1‖L∞(Ω)} |
=‖u1−u1,1‖L∞(Ω). | (3.23) |
(3.21) and (3.22) imply
‖u2−u2,1‖L∞(Ω)≤α2(1−w2)(1−α2w2)‖u2−u2,0‖L∞(Ω) | (3.24) |
with
‖u1−u1,1‖L∞(Ω)≤α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω). | (3.25) |
By substituting (3.24) in (3.25), we get
‖u1−u1,1‖L∞(Ω)≤α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω)≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω), |
which coincides with (3.20). (3.21) and (3.23) imply
‖u2−u2,1‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω), | (3.26) |
with
α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω). |
It is clear that ‖u1−u1,1‖L∞(Ω) is bounded below by both
‖u2−u2,1‖L∞(Ω) |
and
α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω), |
which leads us to distinguish the following possibilities
‖u2−u2,1‖L∞(Ω)≤α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω) |
or
α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω)≤‖u2−u2,1‖L∞(Ω). |
Then,
‖u2−u2,1‖L∞(Ω)≤α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω) | (3.27) |
or
α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω)≤‖u2−u2,1‖L∞(Ω). | (3.28) |
Thus, (3.26)-(3.28) imply that the three following alternatives are required
‖u2−u2,1‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω)≤α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω) |
or
‖u2−u2,1‖L∞(Ω)≤α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω) |
or
α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω)≤‖u2−u2,1‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω). |
It is clear that all alternatives coincide with (3.20). So, we must have
‖u2−u2,1‖L∞(Ω)=α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω). |
Thus, in both cases (3.22) and (3.23) we obtain (3.24). Hence,
‖u2−u2,1‖L∞(Ω)≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω). | (3.29) |
(3.20) and (3.29) imply
‖U1−U‖∞≤ρ‖U0−U‖∞. |
● Let us assume that, for n≥0
‖ui−ui,n‖L∞(Ω)≤ρnmax1≤i≤2‖ui−ui,0‖L∞(Ω),i=1,2. | (3.30) |
● We prove
‖ui−ui,n+1‖L∞(Ω)≤ρn+1max1≤i≤2‖ui−ui,n‖L∞(Ω),i=1,2. | (3.31) |
By adopting the same arguments for (1.1), (3.2), (3.3) and (2.9) as that applied for the previous iterates, we get
‖u1−u1,n+1‖L∞(Ω)≤max{(1β1)‖f1(u1)−(w1f1(u1,n+1)+(1−w1)f1(u1,n))‖L∞(Ω);‖u2−u2,n‖L∞(Ω)} |
So,
‖u1−u1,n+1‖L∞(Ω)≤max{α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω);‖u2−u2,n‖L∞(Ω)} | (3.32) |
Also we distinguish two cases:
max{α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω);‖u2−u2,n‖L∞(Ω)} |
=α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω) | (3.33) |
or
max{α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω);‖u2−u2,n‖L∞(Ω)}=‖u2−u2,n‖L∞(Ω) | (3.34) |
(3.32) in conjunction with (3.33) implies
‖u1−u1,n+1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω), | (3.35) |
with
‖u2−u2,n‖L∞(Ω)≤α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω). | (3.36) |
By replacing (3.35) in (3.36) we get, according to (3.30); i = 1
‖u2−u2,n‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω)≤ρn+1max1≤i≤2‖ui−ui,0‖L∞(Ω) |
which matches with (3.30); i = 2. (3.32) in conjunction with (3.34) implies
‖u1−u1,n+1‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω) | (3.37) |
with
α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω). |
‖u2−u2,n‖L∞(Ω) is bounded below by both ‖u1−u1,n+1‖L∞(Ω)
and
α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω) |
So,
‖u1−u1,n+1‖L∞(Ω)≤α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω) |
or
α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω)≤‖u1−u1,n+1‖L∞(Ω). |
Thus,
‖u1−u1,n+1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω) |
or
α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω)≤‖u1−u1,n+1‖L∞(Ω). |
By taking into account (3.37), we get
‖u1−u1,n+1‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω) |
or
‖u1−u1,n+1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω) |
or
α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω)≤‖u1−u1,n+1‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω). |
Three possibilities are true because all coincide with (3.30). So, we necessarily get
‖u1−u1,n+1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω). |
Thus, both cases (3.33) and (3.34) imply (3.35). Hence, by using (3.30) we get (3.31) for i = 1. The proof for (3.31); i = 2 is obtain in similar way by using (3.31); i = 1 and (3.35) so, it will be omitted. The desired result (3.6) follows naturally from (3.31).
This section, we will handle the discrete problem by a perfect symmetry in the treatment of that the continuous one. Indeed, we define the discrete system of QVIs: Find a vector Uh=(u1h,u2h)∈(Vh)2 such that
{ai(uih,v−uih)≥(fi(uih),v−uih);v∈Vhv,uih≤rh(Muih)=rh(k+ujh);i≠j.uih≥0anduih=πhgon∂Ω. | (4.1) |
The related discrete fixed-point mapping
Th:(Vh)2→(Vh)2 |
Zh=(z1h,z2h)→ThZh=ζh=(ζ1h,ζ2h), |
where ζih∈Vh is the unique solution to the following discrete variational inequality
ai(ζih,v−ζih)≥(fi(zih),v−ζih);v∈Vh | (4.2) |
v,ζih≤rh(Mζih)=rh(k+zjh);ζih≥0withi≠jandζih=πhgon∂Ω. |
Remark 1 We remark that the solution Uh=(u1h,u2h) of the system (4.1) is the fixed point of the mapping Th; that is ThUh=Uh.
Starting from U0h=(u1,0h,u2,0h) where ui,0h=rhui,0;i=1,2 is the discrete analog of ui,0 then,
‖ui,0−ui,0h‖L∞(Ω)≤Ch2|logh|2. | (4.3) |
For all 0<wi<1;i=1,2 we define the discrete sequences (u1,n+1h) and such that and components of the vector solve discrete elliptic variational inequalities
(4.4) |
(4.5) |
(4.6) |
(4.7) |
Theorem 2 The discrete sequences and converge geometrically to the discrete solution of the system (4.1); there exist a positive real defined in (3.7) such that for all
(4.8) |
Proof. The proof is similar to that of the continuous case.
This section is devoted to the proof of the main result of this paper. For that purpose we need to introduce an auxiliary system.
Let be an initialization. For all we define the discrete sequences and such that and solve coercive variational inequalities
(5.1) |
(5.2) |
(5.3) |
(5.4) |
It is clear that components of the vector are finite element approximation of defined in (3.2)–(3.4). Thus, making use of (2.16); we get
(5.5) |
The algorithmic approach used in the present paper rests on the following crucial lemma, where the error estimate between the nth iterate and its discrete counter parts is established.
Lemma 1 Let and be the vectors whose components are sequences defined in (3.2)–(3.5) and (4.4)–(4.7) respectively. Then,
(5.6) |
Where
(5.7) |
Proof. The proof of the lemma rests on the discrete Lipschitz continuous dependency with respect to source term and obstacle and will carry out by induction.
● For n = 0, we have
(5.1), (5.2), (4.4), (4.5) and (2.17) imply
So,
Therefore,
(5.8) |
We distinguish two cases
(5.9) |
or
(5.10) |
(5.8) in conjunction with (5.9) imply
with
(5.11) |
So,
with (5.11). Then,
(5.12) |
By replacing (5.12) in (5.11) we obtain
According to (5.5) and (4.3) we get,
which coincides with (4.3).
(5.8) and (5.10) imply
(5.13) |
with
Then, multiplying (5.13) by and adding we obtain
We note that
is bounded by both
and
So,
or
Therefore, according to (5.5) and (4.3), we get
or
So, the last two alternatives are true at the same time because both coincide with (4.3). We necessarily deduce that
(5.14) |
By replacing (5.14) in (5.13); we get (5.12). Hence, in both cases (5.9) and (5.10); we can write
Thus,
(5.15) |
● In a similar way, that is by following the same steps as for and , and satisfy
So,
(5.16) |
We distinguish also two cases
(5.17) |
or
(5.18) |
(5.16) in conjunction with case (5.17); we get
+
with
(5.19) |
So,
(5.20) |
with, according to (5.20)
Then,
Therefore,
which coincides with (5.15). The conjunction of (5.16) with case (5.18), implies
(5.21) |
with
Then, by multiplying (5.21) by and adding, we obtain that the term is bounded by both
and
So, we distinguish again, the two following alternatives
or
We remark that both alternatives coincide with (5.15), which implies that case (5.18) is possible if and only if
(5.22) |
By substituting (5.22) in (5.21), we get (5.20). Hence, in both cases (5.17) and (5.18), we get
Thus,
(5.23) |
(5.15) and (5.23) imply
● Let us assume that for and i = 1, 2
(5.24) |
● And prove for i = 1, 2
(5.25) |
We operate in the same way as in iterate n = 0. Let us begin with case i = 1 in (5.25)
So, by applying (2.17), we get
(5.26) |
We distinguish again two cases
(5.27) |
or
(5.28) |
(5.26) in conjunction with case (5.27) implies
and
Then,
(5.29) |
with, according to (5.29)
(5.24) implies
with
Thus,
and as
Hence,
and
which corresponds with (5.24) for i = 2: Inequality (5.26) with (5.28) imply
(5.30) |
and
By multiplying (5.30) by and adding the term we get that the term
is bounded by the following two terms
and
So, we need to distinguish the followings possibilities
or
which implies
or
By using (5.24), we can write
or
Only the last alternative is true because it matches with (5.24) for i = 2. So, in (5.28) we get
(5.31) |
By replacing (5.31) in (5.30); we get (5.29). Hence, in both cases (5.27) and (5.28), we obtain
So,
Therefore,
(5.32) |
By using the last inequality (5.32) and by adopting the same reasoning we prove (5.25); i = 2, therefore, we get (5.6).
Theorem 2 Let and be the solution of systems (1.1) and (4.8), respectively. Then, there exists a constant C independent of h such that
Proof. Making use of (3.6), (5.6) and (4.8), we have
As and by using (5.5) we get (5.33).
In this work an optimal convergence order is derived for a class of system of two elliptic quasi-variational inequalities where terms sources and obstacles depend upon the solution, where the continuous and discrete Lipschitz dependence with respect to the terms sources, boundary condition and obstacles' played a leading role in obtaining the main result of this paper. As (1.1) plays a key role in solving Hamilton-Jacobi-Bellman equation the results obtained in this paper can give an optimal error estimate for HJB equation also even for . The approach used and the results obtained in this paper (optimal convergence order) remain valid when we deal with systems of quasi-variational inequalities with terms sources depends on solution and the obstacles i independent of the solution, that is systems of the form; Find a vector satisfying
The author states that no funding source or sponsor has participated in the realization of this work.
All authors declare no conflicts of interest in this paper.
[1] |
Berridge MJ, Bootman MD, Lipp P (1998) Calcium - A life and death signal. Nature 395: 645-648. doi: 10.1038/27094
![]() |
[2] |
Bootman MD, Bultynck G (2020) Fundamentals of cellular calcium signaling: A primer. Cold Spring Harb Perspect Biol 12: 1-16. doi: 10.1101/cshperspect.a038802
![]() |
[3] |
Prevarskaya N, Skryma R, Shuba Y (2004) Ca2+ homeostasis in apoptotic resistance of prostate cancer cells. Biochem Biophys Res Commun 322: 1326-1335. doi: 10.1016/j.bbrc.2004.08.037
![]() |
[4] |
Monteith GR, Prevarskaya N, Roberts-Thomson SJ (2017) The calcium-cancer signalling nexus. Nat Rev Cancer 17: 367-380. doi: 10.1038/nrc.2017.18
![]() |
[5] |
Vanoverberghe K, Vanden Abeele F, Mariot P, et al. (2004) Ca2+ homeostasis and apoptotic resistance of neuroendocrine-differentiated prostate cancer cells. Cell Death Differ 11: 321-330. doi: 10.1038/sj.cdd.4401375
![]() |
[6] |
Kucera R, Pecen L, Topolcan O, et al. (2020) Prostate cancer management: long-term beliefs, epidemic developments in the early twenty-first century and 3PM dimensional solutions. EPMA J 11: 399-418. doi: 10.1007/s13167-020-00214-1
![]() |
[7] |
Siegel RL, Miller KD, Jemal A (2020) Cancer statistics, 2020. CA Cancer J Clin 70: 7-30. doi: 10.3322/caac.21590
![]() |
[8] |
Saraon P, Jarvi K, Diamandis EP (2011) Molecular Alterations during Progression of Prostate Cancer to Androgen Independence. Clin Chem 57: 1366-1375. doi: 10.1373/clinchem.2011.165977
![]() |
[9] |
Howard N, Clementino M, Kim D, et al. (2019) New developments in mechanisms of prostate cancer progression. Semin Cancer Biol 57: 111-116. doi: 10.1016/j.semcancer.2018.09.003
![]() |
[10] |
Boutin B, Tajeddine N, Monaco G, et al. (2015) Endoplasmic reticulum Ca2+ content decrease by PKA-dependent hyperphosphorylation of type 1 IP3 receptor contributes to prostate cancer cell resistance to androgen deprivation. Cell Calcium 57: 312-320. doi: 10.1016/j.ceca.2015.02.004
![]() |
[11] |
Flourakis M, Lehen'kyi V, Beck B, et al. (2010) Orai1 contributes to the establishment of an apoptosis-resistant phenotype in prostate cancer cells. Cell Death Dis 1: e75. doi: 10.1038/cddis.2010.52
![]() |
[12] |
Dubois C, Vanden Abeele F, Lehen'kyi V, et al. (2014) Remodeling of Channel-Forming ORAI Proteins Determines an Oncogenic Switch in Prostate Cancer. Cancer Cell 26: 19-32. doi: 10.1016/j.ccr.2014.04.025
![]() |
[13] |
Perrouin Verbe MA, Bruyere F, Rozet F, et al. (2016) Expression of store-operated channel components in prostate cancer: The prognostic paradox. Hum Pathol 49: 77-82. doi: 10.1016/j.humpath.2015.09.042
![]() |
[14] |
Kampa M, Nifli AP, Notas G, et al. (2007) Polyphenols and cancer cell growth. Rev Physiol Biochem Pharmacol 159: 79-113. doi: 10.1007/112_2006_0702
![]() |
[15] |
Jian L, Xie LP, Lee AH, et al. (2004) Protective effect of green tea against prostate cancer: A case-control study in southeast China. Int J Cancer 108: 130-135. doi: 10.1002/ijc.11550
![]() |
[16] |
Bettuzzi S, Brausi M, Rizzi F, et al. (2006) Chemoprevention of human prostate cancer by oral administration of green tea catechins in volunteers with high-grade prostate intraepithelial neoplasia: A preliminary report from a one-year proof-of-principle study. Cancer Res 66: 1234-1240. doi: 10.1158/0008-5472.CAN-05-1145
![]() |
[17] |
Brausi M, Rizzi F, Bettuzzi S (2008) Chemoprevention of Human Prostate Cancer by Green Tea Catechins: Two Years Later. A Follow-up Update. Eur Urol 54: 472-473. doi: 10.1016/j.eururo.2008.03.100
![]() |
[18] |
Yuan JM (2013) Cancer prevention by green tea: evidence from epidemiologic studies. Am J Clin Nutr 98: 1676S-1681S. doi: 10.3945/ajcn.113.058271
![]() |
[19] | Perletti G, Magri V, Vral A, et al. (2019) Green tea catechins for chemoprevention of prostate cancer in patients with histologically-proven HG-PIN or ASAP. Concise review and meta-analysis. Arch Ital di Urol e Androl 91: 153-156. |
[20] |
Cui K, Li X, Du Y, et al. (2017) Chemoprevention of prostate cancer in men with high-grade prostatic intraepithelial neoplasia (HGPIN): a systematic review and adjusted indirect treatment comparison. Oncotarget 8: 36674-36684. doi: 10.18632/oncotarget.16230
![]() |
[21] |
Guo Y, Zhi F, Chen P, et al. (2017) Green tea and the risk of prostate cancer. Medicine (Baltimore) 96: e6426. doi: 10.1097/MD.0000000000006426
![]() |
[22] |
Kurahashi N, Sasazuki S, Iwasaki M, et al. (2008) Green tea consumption and prostate cancer risk in Japanese men: A prospective study. Am J Epidemiol 167: 71-77. doi: 10.1093/aje/kwm249
![]() |
[23] | Filippini T, Malavolti M, Borrelli F, et al. (2020) Green tea (Camellia sinensis) for the prevention of cancer. Cochrane Database Syst Rev 3: CD005004. |
[24] |
Tauber AL, Schweiker SS, Levonis SM (2020) From tea to treatment; epigallocatechin gallate and its potential involvement in minimizing the metabolic changes in cancer. Nutr Res 74: 23-36. doi: 10.1016/j.nutres.2019.12.004
![]() |
[25] |
Miyata Y, Shida Y, Hakariya T, et al. (2019) Anti-cancer effects of green tea polyphenols against prostate cancer. Molecules 24: 193-212. doi: 10.3390/molecules24010193
![]() |
[26] |
Sang S, Lambert JD, Ho CT, et al. (2011) The chemistry and biotransformation of tea constituents. Pharmacol Res 64: 87-99. doi: 10.1016/j.phrs.2011.02.007
![]() |
[27] |
Gan RY, Li HB, Sui ZQ, et al. (2018) Absorption, metabolism, anti-cancer effect and molecular targets of epigallocatechin gallate (EGCG): An updated review. Crit Rev Food Sci Nutr 58: 924-941. doi: 10.1080/10408398.2016.1231168
![]() |
[28] |
Albrecht DS, Clubbs EA, Ferruzzi M, et al. (2008) Epigallocatechin-3-gallate (EGCG) inhibits PC-3 prostate cancer cell proliferation via MEK-independent ERK1/2 activation. Chem Biol Interact 171: 89-95. doi: 10.1016/j.cbi.2007.09.001
![]() |
[29] |
Gupta S, Hussain T, Mukhtar H (2003) Molecular pathway for (−)-epigallocatechin-3-gallate-induced cell cycle arrest and apoptosis of human prostate carcinoma cells. Arch Biochem Biophys 410: 177-185. doi: 10.1016/S0003-9861(02)00668-9
![]() |
[30] |
Hagen RM, Chedea VS, Mintoff CP, et al. (2013) Epigallocatechin-3-gallate promotes apoptosis and expression of the caspase 9a splice variant in PC3 prostate cancer cells. Int J Oncol 43: 194-200. doi: 10.3892/ijo.2013.1920
![]() |
[31] |
Ju J, Lu G, Lambert JD, et al. (2007) Inhibition of carcinogenesis by tea constituents. Semin Cancer Biol 17: 395-402. doi: 10.1016/j.semcancer.2007.06.013
![]() |
[32] |
Yang CS, Wang H, Chen JX, et al. (2014) Effects of Tea Catechins on Cancer Signaling Pathways. Enzymes Academic Press, 195-221. doi: 10.1016/B978-0-12-802215-3.00010-0
![]() |
[33] |
Kim HS, Quon MJ, Kim JA (2014) New insights into the mechanisms of polyphenols beyond antioxidant properties; lessons from the green tea polyphenol, epigallocatechin 3-gallate. Redox Biol 2: 187-195. doi: 10.1016/j.redox.2013.12.022
![]() |
[34] |
Hastak K, Agarwal MK, Mukhtar H, et al. (2005) Ablation of either p21 or Bax prevents p53-dependent apoptosis induced by green tea polyphenol epigallocatechin-3-gallate. FASEB J 19: 1-19. doi: 10.1096/fj.04-2226fje
![]() |
[35] |
Van Aller GS, Carson JD, Tang W, et al. (2011) Epigallocatechin gallate (EGCG), a major component of green tea, is a dual phosphoinositide-3-kinase/mTOR inhibitor. Biochem Biophys Res Commun 406: 194-199. doi: 10.1016/j.bbrc.2011.02.010
![]() |
[36] |
Siddiqui IA, Asim M, Hafeez BB, et al. (2011) Green tea polyphenol EGCG blunts androgen receptor function in prostate cancer. FASEB J 25: 1198-1207. doi: 10.1096/fj.10-167924
![]() |
[37] |
Morré DJ, Morré DM, Sun H, et al. (2003) Tea Catechin Synergies in Inhibition of Cancer Cell Proliferation and of a Cancer Specific Cell Surface Oxidase (ECTO-NOX). Pharmacol Toxicol 92: 234-241. doi: 10.1034/j.1600-0773.2003.920506.x
![]() |
[38] |
Marchetti C, Gavazzo P, Burlando B (2020) Epigallocatechin-3-gallate mobilizes intracellular Ca2+ in prostate cancer cells through combined Ca2+ entry and Ca2+-induced Ca2+ release. Life Sci 258: 118232. doi: 10.1016/j.lfs.2020.118232
![]() |
[39] |
Van Bokhoven A, Varella-Garcia M, Korch C, et al. (2003) Molecular Characterization of Human Prostate Carcinoma Cell Lines. Prostate 57: 205-225. doi: 10.1002/pros.10290
![]() |
[40] |
Tai S, Sun Y, Squires JM, et al. (2011) PC3 is a cell line characteristic of prostatic small cell carcinoma. Prostate 71: 1668-1679. doi: 10.1002/pros.21383
![]() |
[41] | Grynkiewicz G, Poenie M, Tsien RY (1985) A new generation of Ca2+ indicators with greatly improved fluorescence properties. J Biol Chem 25: 3440-3450. |
[42] |
Friedman M, Mackey BE, Kim HJ, et al. (2007) Structure-activity relationships of tea compounds against human cancer cells. J Agric Food Chem 55: 243-253. doi: 10.1021/jf062276h
![]() |
[43] |
Lewandowska U, Gorlach S, Owczarek K, et al. (2014) Synergistic interactions between anticancer chemotherapeutics and phenolic compounds and anticancer synergy between polyphenols. Postepy Hig Med Dosw 68: 528-540. doi: 10.5604/17322693.1102278
![]() |
[44] |
Kennedy DO, Matsumoto M, Kojima A, et al. (1999) Cellular thiols status and cell death in the effect of green tea polyphenols in Ehrlich ascites tumor cells. Chem Biol Interact 122: 59-71. doi: 10.1016/S0009-2797(99)00114-3
![]() |
[45] |
Clapham DE, Runnels LW, Strübing C (2001) The trp ion channel family. Nat Rev Neurosci 2: 387-396. doi: 10.1038/35077544
![]() |
[46] |
Thebault S, Roudbaraki M, Sydorenko V, et al. (2003) α1-adrenergic receptors activate Ca2+-permeable cationic channels in prostate cancer epithelial cells. J Clin Invest 111: 1691-1701. doi: 10.1172/JCI16293
![]() |
[47] |
Miller M, Shi J, Zhu Y, et al. (2011) Identification of ML204, a Novel Potent Antagonist That Selectively Modulates Native TRPC4/C5 Ion Channels. J Biol Chem 286: 33436-33446. doi: 10.1074/jbc.M111.274167
![]() |
[48] |
Sun YH, Gao X, Tang YJ, et al. (2006) Androgens induce increases in intracellular calcium via a G protein-coupled receptor in LNCaP prostate cancer cells. J Androl 27: 671-678. doi: 10.2164/jandrol.106.000554
![]() |
[49] | Loughlin KR (2014) Calcium channel blockers and prostate cancer. Urol Oncol Semin Orig Investig 32: 537-538. |