
In this work, we establish inequalities of Hermite-Hadamard-Mercer (HHM) type for convex functions by using generalized fractional integrals. The results of our paper are the extensions and refinements of Hermite-Hadamard (HH) and Hermite-Hadamard-Mercer (HHM) type inequalities. We discuss special cases of our main results and give new inequalities of HH and HHM type for different fractional integrals like, Riemann-Liouville (RL) fractional integrals, k-Riemann-Liouville (k-RL) fractional integrals, conformable fractional integrals and fractional integrals of exponential kernel.
Citation: Miguel Vivas-Cortez, Muhammad Aamir Ali, Artion Kashuri, Hüseyin Budak. Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions[J]. AIMS Mathematics, 2021, 6(9): 9397-9421. doi: 10.3934/math.2021546
[1] | Yaohong Li, Jiafa Xu, Honglin Luo . Approximate iterative sequences for positive solutions of a Hadamard type fractional differential system involving Hadamard type fractional derivatives. AIMS Mathematics, 2021, 6(7): 7229-7250. doi: 10.3934/math.2021424 |
[2] | Abdelouaheb Ardjouni . Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions. AIMS Mathematics, 2019, 4(4): 1101-1113. doi: 10.3934/math.2019.4.1101 |
[3] | Ahmed Hussein Msmali . Positive solutions for a system of Hadamard fractional (ϱ1,ϱ2,ϱ3)-Laplacian operator with a parameter in the boundary. AIMS Mathematics, 2022, 7(6): 10564-10581. doi: 10.3934/math.2022589 |
[4] | Hui Huang, Kaihong Zhao, Xiuduo Liu . On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses. AIMS Mathematics, 2022, 7(10): 19221-19236. doi: 10.3934/math.20221055 |
[5] | Yunzhe Zhang, Youhui Su, Yongzhen Yun . Existence and stability of solutions for Hadamard type fractional differential systems with p-Laplacian operators on benzoic acid graphs. AIMS Mathematics, 2025, 10(4): 7767-7794. doi: 10.3934/math.2025356 |
[6] | Xiaoxin Zuo, Weibing Wang . Existence of solutions for fractional differential equation with periodic boundary condition. AIMS Mathematics, 2022, 7(4): 6619-6633. doi: 10.3934/math.2022369 |
[7] | Sabbavarapu Nageswara Rao, Abdullah Ali H. Ahmadini . Multiple positive solutions for system of mixed Hadamard fractional boundary value problems with (p1,p2)-Laplacian operator. AIMS Mathematics, 2023, 8(6): 14767-14791. doi: 10.3934/math.2023755 |
[8] | Muath Awadalla, Mohammed S. Abdo, Hanan A. Wahash, Kinda Abuasbeh . Qualitative study of linear and nonlinear relaxation equations with ψ-Riemann-Liouville fractional derivatives. AIMS Mathematics, 2022, 7(11): 20275-20291. doi: 10.3934/math.20221110 |
[9] | Aslı Alkan, Halil Anaç . The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 2024, 9(9): 25333-25359. doi: 10.3934/math.20241237 |
[10] | Chengbo Zhai, Yuanyuan Ma, Hongyu Li . Unique positive solution for a p-Laplacian fractional differential boundary value problem involving Riemann-Stieltjes integral. AIMS Mathematics, 2020, 5(5): 4754-4769. doi: 10.3934/math.2020304 |
In this work, we establish inequalities of Hermite-Hadamard-Mercer (HHM) type for convex functions by using generalized fractional integrals. The results of our paper are the extensions and refinements of Hermite-Hadamard (HH) and Hermite-Hadamard-Mercer (HHM) type inequalities. We discuss special cases of our main results and give new inequalities of HH and HHM type for different fractional integrals like, Riemann-Liouville (RL) fractional integrals, k-Riemann-Liouville (k-RL) fractional integrals, conformable fractional integrals and fractional integrals of exponential kernel.
Taxis is understood as the motion of an organism towards or away from an external stimulus. In particular when the stimulus is a chemical, it is called chemotaxis. It seems natural to address the problem of driving the motion of the organism by modifying or applying a chemical gradient. In this paper we have in mind a process in which chemotaxis takes place, the tumor angiogenesis. However, we could extend the same idea for other biological processes where the chemotaxis is involved.
Tumor angiogenesis starts when, as a response to nutrient deprivation, cancer cells secrete a chemical factor known as Tumor Angiogenic Factors (TAF)s. (TAF)s diffuse in the extracellular matrix and activate endothelial cells which migrate, via chemotaxis, towards the source of TAF i.e., the tumor. When endothelial cells reach the tumor more nutrients are supplied to the tumor which grows further. See for instance [16] for additional details.
It is clear that angiogenesis is a crucial step in the development of the tumor. Therefore, if we act in the (TAF)s molecules or if we modify the response of the endothelial cells to the (TAF)s molecules we may either reduce angiogenesis or avoid it.
In this paper we consider two variables u, the density of endothelial cells and z the concentration of anti-angiogenic drug that modifies the sensitivity of a chemical gradient v (TAF) and the growth of the endothelial cells. We also assume that u, z and v are defined in a bounded domain Ω⊂IR3. Here we are interested in the minimum of the functional defined by
J(u,z)=a2∫Ωu(T)2+b2∫Ω×(0,T)z2, | (1.1) |
where a and b are positive parameters and (u,z) is a solution of the nonlinear differential equation
{ut−div(∇u−V(u)α(z)∇v)=β(z,v)u−u2inΩ×(0,T),∂nu−V(u)α(z)∂nv=0on∂Ω×(0,T),u(0,x)=u0(x)inΩ. | (1.2) |
and z∈Uad where
Uad:={z∈L2(Ω×(0,T)):z(x,t)≥0 for almost (x,t)∈Ω×(0,T)}. |
The functional has two terms; the first term refers to the amount of endothelial cells at the final stage and the second one can be seen as the cost of the drug over the time.
Minimizing a functional (1.1), subject to a differential problem (1.2) and a convex constraint, z∈Uad, is a problem of the optimal control theory. It is classical in this theory to call u state variable and to call z control variable, because it controls the state through the equation. The nonlinearity of (1.2) leads that the minimal problem is out of the convex framework. We will apply a generalization of the Lagrange multipliers theorem to obtain the optimality system which provides the necessary condition for the solution. This method, called Dubovitskii-Milyutin formalism (see [7]) has been mostly applied to ordinary differential equations (see [13]). In [6] it is applied to a linear partial differential problem with the feature that it is non well-posed. The optimal problem we study in this paper have the difficulty of the nonlinearity of the partial differential equation and the control constraint z∈Uad which has empty interior in L2(Q).
The formalism gives an optimality system constituted by two partial differential coupled equations, one is the state equation given in (1.2) and the other one is a linear equation for the adjoint variable, and a condition for the optimal control given by a projection operator. We will prove the uniqueness of solution of the optimality system for T small enough and this turns out the uniqueness of solution of the optimal problem.
The chemotaxis is described in (1.2) by the term div(V(u)α(z)∇v). A similar description of the chemotaxis was introduced in [12] where there is an additional equation for v. Since, the minimal system proposed in [12] could generate singularities in finite time (see [11]) then to avoid the singularities it is proposed a model with a bounded drift term in [9]. In the case of angiogenesis, one of the first continuous models related to angiogenesis is introduced in [1]. A similar model to the one in the paper, without the variable z but with an additional equation for chemoattractant is given in [4]. The case of a therapy z satisfying an additional parabolic equation is considered in [17] and [5].
The structure of the paper is as follows. In Section 2 we prove the existence of a unique weak solution of (1.2), global in time and positive, fixed z. The optimal control problem, the existence of the optimal control, its characterization and the uniqueness when T is small enough are studied in Section 3. In Section 4 we show some numerical simulations to illustrate the theoretical results.
Let Ω⊂IRN be a regular bounded domain and T>0 a fixed number. We will denote Q=Ω×(0,T) and Γ=∂Ω×(0,T). In this paper, we consider the problem
{ut−div(∇u−V(u)α(z)∇v)=β(z,v)u−u2inQ,∂nu−V(u)α(z)∂nv=0onΓ,u(0,x)=u0(x)inΩ. | (2.1) |
Here, v∈W2,∞(Ω) is a known function, z∈L2(Q), and u0∈L2(Ω). The function V:[0,+∞)→[0,+∞) verify V(0)=0 and V∈C2([0,∞))∩L∞([0,∞)). If it is needed we will extend this function by zero for negative values. The functions α:[0,+∞)→[0,+∞) is in W1,∞([0,+∞)) and β:[0,+∞)×[0,+∞)→[0,+∞) is in C2([0,∞))∩L∞([0,∞)). We would like to point out here that the main difficulty of the problem is that z is not defined on ∂Ω.
Let T>0 and X is a Banach space, we define the space Lp(0,T;X) of equivalence classes of measurable functions u:I→X such that t∈(0,T)→‖u‖X belongs to Lp(I) which is a Banach space for the norm
‖u‖Lp(X)={(∫T0‖u‖pXdt)1/pif1≤p<∞Esssupt∈(0,T)‖u(t)‖Xifp=∞ |
For instance, we will use
‖u‖L2(L2)=(∫T0∫Ω|u(x)|2dxdt)1/2, |
‖u‖L2(H1)=(∫T0∫Ω|∇u(x)|2+|u(x)|2dxdt)1/2. |
The definition of a weak solution for the problem is
Definition 2.1. We say that u is a weak solution of (1.2) if the following conditions are verified
1) u∈W(0,T) where
W(0,T):={u∈L2(0,T;H1(Ω)) such that ut∈L2(0,T;(H1(Ω))′) }. |
2) ∀w∈C1([0,T]×ˉΩ):w(T,x)=0,∀x∈ˉΩ, it holds
∫T0⟨ut,w⟩(H1)′,H1+∫T0∫Ω∇u⋅∇w−∫T0∫ΩV(u)α(z)∇v⋅∇w==∫T0∫Ωβ(z,v)uw−∫T0∫Ωu2w+∫Ωu0(x)w(0,x). | (2.2) |
Theorem 2.2. If u0∈L∞(Ω), u0≥0, then there is a unique positive global weak solution to the problem (1.2) that also satisfies
u∈L∞(0,T;L∞(Ω)). |
Proof. We denote by
BR={u∈L2(0,T;L2(Ω)):‖u‖L2(L2)≤R}, |
and we define the mapping S:BR→L2(0,T;L2(Ω)) such that for each ˉu∈BR, S(ˉu)=u is the weak solution of the linear problem
{ut=div(∇u−V(ˉu)α(z)∇v)+β(z,v)u−Tk(ˉu)uinQ,∂nu−V(ˉu)α(z)∂nv=0onΓ,u(0,x)=u0(x)inΩ. | (2.3) |
where
Tk(φ)={kifφ>k,φ+ifφ≤k. |
We will prove the existence of a fixed point for S which is a weak solution of a truncated nonlinear problem. After that we will justify that the weak solution to the truncated problem is a solution of (1.2). At the end we will show the uniqueness of weak solution of (1.2).
Step1_ First of all we show that the operator S is well defined. The existence and the uniqueness of the weak solution for the linear problem (2.3) follow from [14]. More precisely, by [14], Theorem 6.39] the problem (see [14], p. 136])
{ut=div(∇u)+N∑i=1∂ifi(x,t)+c0(x,t)uinQ∂nu+N∑i=1fiνi=0onΓu(0,x)=u0(x)inΩ. | (2.4) |
(where fi=−V(ˉu)α(z)∂iv and c0(x,t)=β(z,v)−Tk(ˉu) in (2.3) and ν=(ν1,..,νN) is the outward normal to Ω), has a unique weak solution i.e. u∈L2(0,T;H1(Ω))∩L∞(0,T;L2(Ω)) such that
∫T0∫Ω(−uwt+|∇u|⋅|∇w|−N∑i=1fi∂iw−c0(x,t)uw)dxdt=∫Ωu0(x)v(0,x)dx, |
for each w∈C1([0,T]×ˉΩ) such that w(T,x)=0,∀x∈Ω. Note that, if ut∈L2(0,T;(H1(Ω))′) then the previous definition it is exactly the one given in Definition (2.1). Therefore we should show that ut∈L2(0,T;(H1(Ω)′).
⟨ut,φ⟩=∫Ω∇u⋅∇φdxdt+∫ΩV(ˉu)α(z)∇v⋅∇φdx+∫Ωc0(x,t)uφdx≤‖∇u‖L2‖∇φ‖L2+‖V(ˉu)α(z)∇v‖L2‖∇φ‖L2+‖c0(x,t)u‖L2‖φ‖L2≤(‖∇u‖L2+‖V(ˉu)α(z)∇v‖L2+‖c0(x,t)u‖L2)‖φ‖H1, |
where ⟨⋅,⋅⟩ stands for the duality pairing between (H1(Ω))′ and H1(Ω). Therefore
‖ut‖(H1(Ω))′=sup0≠φ∈H1(Ω)⟨ut,φ⟩‖φ‖H1≤‖∇u‖L2+‖V(ˉu)α(z)∇v‖L2+‖c0(x,t))u‖L2, |
and
‖ut‖L2(0,T;(H1(Ω))′)=(∫T0‖ut‖2(H1(Ω))′dt)1/2≤C | (2.5) |
i.e., ut∈L2(0,T;(H1(Ω))′). Hence, S is well defined.
In what follows we will apply the Schauder fixed point theorem to get the existence of a fixed point for S in BR.
Step2_ We claim that there exists R>0 and T>0 such that S(BR)⊂BR. In fact, multiplying the equation of (2.3) by u and integrating in Ω, it results
12ddt∫Ωu2=−∫Ω|∇u|2+∫Ωα(z)V(ˉu)∇u⋅∇v+∫Ωβ(z)u2−∫ΩTk(ˉu)u2≤−12∫Ω|∇u|2+12∫Ωα(z)2V(ˉu)2|∇v|2+‖β‖∞∫Ωu2≤‖β‖∞∫Ωu2+C. |
Thus for y(t)=∫Ωu2 we obtain the problem
{y′(t)≤ay(t)+b,y(0)=∫Ωu20, |
for some positive constants a,b. Consequently,
0≤y(t)≤(y0+ba)eat−ba. |
We can choose any R>y(0)=‖u0‖2 and determine T>0 such that
y(t)=∫Ωu2(x,t)dx≤R,∀t∈(0,T), | (2.6) |
and
‖u‖L2(L2)=(∫T0∫Ω|u(x,t)|2dxdt)1/2≤R1/2T1/2≤R, |
if we take R>1 and T<1. So, for R≥‖u0‖2+1 there exists T<1 such that S(BR)⊂BR.
On the other hand, taking into account that
y′(t)+12∫Ω|∇u|2≤ay(t)+b, |
we can choose R, determine T and integrate the inequality on the interval (0,T) to obtain
y(T)−y(0)+∫T0∫Ω|∇u|2dxdt≤a∫T0ydt+b∫T0dt, |
∫T0∫Ω|∇u|2dxdt≤y(0)+a∫T0∫Ωu2dxdt+bT≤‖u0‖2+aR2+bT. | (2.7) |
Consequently, ‖u‖L2(H1) is bounded.
Step3_ We claim that S is a continuous mapping. We will prove that
‖S(ˉu1)−S(ˉu2)‖L2(L2)≤Φ(k,‖ˉu1−ˉu2‖L2(L2))∀ˉu1,ˉu2∈BR, | (2.8) |
with Φ(k,s)→0 when s→0 for each fixed k, and T as in Step 2. If we denote u1=S(ˉu1) and u2=S(ˉu2), it holds
{(u1)t=div(∇u1−V(ˉu1)α(z)∇v)+β(z,v)u1−Tk(ˉu1)u1,(u2)t=div(∇u2−V(ˉu2)α(z)∇v)+β(z,v)u2−Tk(ˉu2)u2, |
and taking w=u1−u2=S(ˉu1)−S(ˉu2), we have
wt=Δw−div((V(ˉu1)−V(ˉu2)α(z)∇v)+β(z,v)w+(Tk(ˉu2)u2−Tk(ˉu1)u1)=Δw−div((V(ˉu1)−V(ˉu2)α(z)∇v)+β(z,v)w+Tk(ˉu1)w++(Tk(ˉu2)−Tk(ˉu1))u2. |
On multiplying the previous inequality by w and integrating on Ω we obtain
12ddt∫Ωw2=−∫Ω|∇w|2−∫Ω(V(ˉu1)−V(ˉu2))α(z)∇v⋅∇w++∫Ωβ(z,v)w2+∫ΩTk(ˉu1)w2+∫Ω(Tk(ˉu2)−Tk(ˉu1))u2w. | (2.9) |
Next, we provide a bound to the terms in the right hand side of (2.9)
|∫Ω(V(ˉu1)−V(ˉu2))α(z)∇v⋅∇w|≤12∫Ω|∇w|2+12∫Ω(V(ˉu1)−V(ˉu2))2α(z)2|∇v|2≤12∫Ω|∇w|2+C‖α‖2∞‖∇v‖2∞∫Ω|ˉu1−ˉu2|2, |
∫Ωβ(z,v)w2≤‖β‖∞∫Ωw2, |
∫ΩTk(ˉu1)w2≤k∫Ωw2, |
∫Ω(Tk(ˉu2)−Tk(ˉu1))u2w≤12∫Ωw2+12∫Ω(Tk(ˉu2)−Tk(ˉu1))2u22. |
The Sobolev embedding H1(Ω)↪L6(Ω) together with the estimate u2∈L2(0,T;H1(Ω)) implies that u2∈L2(0,T;L6(Ω)). On the other hand if we apply the Hölder inequality with exponents 3 and 3/2 to the last term of the above inequality we obtain
∫Ω(Tk(ˉu2)−Tk(ˉu1))2u22≤‖u22‖L3/2‖(Tk(ˉu2)−Tk(ˉu1))2‖L3=‖u2‖2L3‖(Tk(ˉu2)−Tk(ˉu1))‖2L6. |
Using the interpolation inequality,
‖u2‖2L3≤‖u2‖L2‖u2‖L6≤C(R)‖u2‖L6, |
and, thus
∫Ω(Tk(ˉu2)−Tk(ˉu1))2u22≤C(R)‖u2‖L6‖(Tk(ˉu2)−Tk(ˉu1))‖2L6. |
Then (2.9) becomes
12ddt∫Ωw2≤C1‖ˉu1−ˉu2‖2L2+C2‖w‖2L2+C(R)‖u2‖L6‖(Tk(ˉu2)−Tk(ˉu1))‖2L6. | (2.10) |
We denote by y(t)=∫Ωw(x,t)2dx. Since y(0)=0 it follows from (2.10) that
y(t)≤ect∫T0e−cs(C1‖ˉu1−ˉu2‖2L2+C(R)‖u2‖L6‖(Tk(ˉu2)−Tk(ˉu1))‖2L6)ds≤ecT(C3‖ˉu1−ˉu2‖L2(L2)+∫T0C(R)e−cs‖u2‖L6‖(Tk(ˉu2)−Tk(ˉu1))‖2L6ds). | (2.11) |
By the Hölder inequality it holds
∫T0C(R)e−cs‖u2‖L6‖(Tk(ˉu2)−Tk(ˉu1))‖2L6ds≤ |
≤(∫T0‖u2‖2L6)1/2(∫T0‖(Tk(ˉu2)−Tk(ˉu1)‖2L6)3)1/3(∫T0(C(R)e−cs)6ds)1/6. |
Since
(∫T0(‖Tk(ˉu2)−Tk(ˉu1)‖2L6)3)1/3=(∫T0∫Ω|Tk(ˉu2)−Tk(ˉu1)|6dxds)1/3= |
=(∫T0∫Ω|Tk(ˉu2)−Tk(ˉu1)|4|Tk(ˉu2)−Tk(ˉu1)|2dxds)1/3≤k4/3(∫T0‖Tk(ˉu2)−Tk(ˉu1)‖2L2ds)1/3≤k4/3(∫T0‖ˉu2−ˉu1‖2L2ds)1/3=k4/3‖ˉu2−ˉu1‖2/3L2(L2), |
it results from (2.11),
y(t)≤C4‖ˉu2−ˉu1‖2L2(L2)+C5k4/3‖ˉu2−ˉu1‖2/3L2(L2). | (2.12) |
Integrating on (0,T), we obtain
‖S(ˉu1)−S(ˉu2)‖L2(L2)≤T(C4‖ˉu2−ˉu1‖2L2(L2)+C5k4/3‖ˉu2−ˉu1‖2/3L2(L2)), |
which proves the desired continuity.
Step4_ We claim that S is a compact mapping in L2(Q). We know that for each ¯u∈BR S(¯u)=u is bounded in L2(0,T;H1(Ω)) and S(¯u)t=ut is bounded in L2(0,T;(H1(Ω))′) (see (2.7) and (2.5)) then by the Lions-Aubin Lemma (see for instance [18]) S(BR) is embedded compactly in L2(0,T;L2(Ω)).
Step5_ By the Schauder Theorem we have a solution to the problem
{ut=div(∇u−V(u)α(z)∇v)+β(z,v)u−Tk(u)uinQ,∂nu−V(u)α(z)∂nv=0onΓ,u(0,x)=u0(x)inΩ. | (2.13) |
We will prove that any solution of (2.13) is nonnegative. Let u−=min(u,0). Taking u− as a test function in the above equation we obtain
d2dt∫Ω(u−)2≤0. |
Integrating on the space variable infer
0≤∫Ωu−(t)2≤∫Ω((u0)−)2=0 |
Therefore, u−(t)=0 a.e. in Ω.
Step6_ We prove that a solution to (2.3) satisfies u∈L∞(0,T;L∞(Ω)) independently of the k, as a consequence a solution of (2.13) is a solution to (1.2) for k sufficiently large. We multiply (2.3) by pTk(u)p−1 for p≥2, a cut off function that approximate to pup−1 (see for instance [3], p. 1196]) to get
ddt∫ΩTk(u)p+4(p−1)p∫Ω|∇Tk(u)p/2|2=p∫Ωβ(z,v)up−p∫ΩTk(u)p+1++2∫ΩTk(u)p/2−1V(u)∇(up/2). |
Since V(0)=0 and V is a Lipschitz function then,
ddt∫ΩTk(u)p+2∫Ω|∇Tk(u)p/2|2≤≤p‖β‖∞∫ΩTk(u)p+2∫ΩTk(u)p/2−1|V(u)−V(0)||∇(Tk(u)p/2)|≤p‖β‖∞∫ΩTk(u)p+2C∫ΩTk(u)p/2|∇(Tk(u)p/2)|≤p‖β‖∞∫ΩTk(u)p+2C(∫ΩTk(u)p)1/2(∫Ω|∇(Tk(u)p/2)|2)1/2 |
Adding on both sides of the above inequality ∫ΩTk(u)p we deduce that
ddt∫ΩTk(u)p+∫ΩTk(u)p+32∫Ω|∇(Tk(u)p/2)|2≤‖β‖∞(p+1+2C2‖β‖∞)∫ΩTk(u)p. | (2.14) |
At this point we will provide a bound for ∫Ωup. We claim that for any ϵ>0
∫ΩTk(u)p≤ϵ∫Ω|∇Tk(u)p/2|2+C(ϵ)(∫ΩTk(u)p/2)2. | (2.15) |
Next lines are devoted to the proof of the above inequality. Let ¯z=1|Ω|∫Ωz. We know that
∫Ωz2=∫Ω(z−¯z)2+1|Ω|(∫Ωz)2. |
On the other hand, by the Hölder inequality and the Poincare-Wintinger inequality we have
∫Ω(z−¯z)2≤‖z−¯z‖3/23‖z−¯z‖1/21=‖z−¯z‖3/23√2‖z‖1/21≤ϵ′‖z−¯z‖23+C(ϵ′)(∫Ω|z|)2≤ϵ∫Ω|∇z|2+C(ϵ′)(∫Ω|z|)2. |
Therefore if we take z=Tk(u)p/2 the claim easily follows. By (2.15) we get from (2.14) that
ddt∫ΩTk(u)p+∫ΩTk(u)p≤C(p+C)(∫ΩTk(u)p/2)2. |
For any 0≤t≤T<Tmax, where Tmax stand for the maximal existence time, the above inequality asserts
∫ΩTk(u)p(t)≤∫Ωup0e−t+C(p+C)sup0≤t≤T(∫ΩTk(u)p/2)2≤¯C(p+C)max{‖u0‖p∞,sup0≤t≤T(∫ΩTk(u)p/2)2}. | (2.16) |
Let us define
θ(p/2):=max{‖u0‖∞,sup0≤t≤T(∫ΩTk(u)p/2)2/p}. |
From (2.16), by a recursive procedure, we have
(∫ΩTk(u)p(t))1/p≤(¯C(p+k))1/pθ(p/2)≤(¯C(p+C))1/p(¯C(p/2+C))2/pθ(p/4) |
In particular, for p=2j we deduce
(∫ΩTk(u)2j(t))2−j≤¯Csjj∏i=0(2i+C)2−iθ(1), |
where sj=∑ji=02−i. Taking into account that
θ(1)≤max{‖u0‖∞,sup0≤t≤T∫Ωu}:=C1, |
Therefore we can take k→+∞ to get
(∫Ωu2j(t))2−j≤¯Csjj∏i=0(2i+C)2−iC1 |
Since
∞∑i=02−i≤C,j∏i=0(2i+C)2−i≤C |
we can take j→∞ to conclude that
‖u(t)‖∞≤C. |
Step7_ Uniqueness of solution for (1.2). Let be two solutions of (1.2), u1,u2, and we take w=u1−u2. We can argue as in Step 3 to obtain
12ddt‖w‖2L2≤a′‖w‖2L2+∫Ω(u1+u2)w2 | (2.17) |
for some a′>0. By the boundedness of u1+u2 we easily get the result by the Gronwall Lemma.
In this section we are interested in the minimum of the functional
J:W(0,T)×L2(Q)→IR+, |
defined by
J(u,z)=a2∫Ωu(T)2+b2∫Qz2, | (3.1) |
a and b are positive parameters and (u,z) has to be a solution of the nonlinear differential equation
{ut=∇⋅(∇u−α(z)V(u)∇v)+β(z,v)u−u2inQ,∂nu−α(z)V(u)∂nv=0inΓ,u(x,0)=u0(x)inΩ, | (3.2) |
and besides, we claim z that
z∈L2(Q), z≥0 for almost (x,t)∈Q (a convex constraint). | (3.3) |
The function u represents the endothelial cell (EC) density, the initial data u0∈L∞(Ω), u0≥0 and not identically zero, means that the angiogenesis process has already begun, the TAF concentration is given by v and it is known in the equation and the concentration of the anti-angiogenic drug is represented by z which is the control function. Minimizing J means to find the appropriated therapy z such that the total amount of EC at the end of the treatment, this is T, is the smallest as possible applying the least amount of drug possible. The term ∫Ωu(T)2 represents the total amount of EC in Ω at the final time T and the term ∫Qz2 plays the total amount of drug in Ω for the time interval (0,T).
The proof of the existence of solution of the optimal problem (3.1)–(3.3) is standard by a minimizing sequence. We will do the proof for the reader's convenience.
Theorem 3.1. The optimal control problem (3.1)–(3.3) has a solution (ˆu,ˆz) in W(0,T)×L2(Q).
Proof. Let be {un,zn} a minimizing sequence such that {J(un,zn)} is decreasing to ˆβ, which is the infimum of J(u,z) subject to (3.2) and (3.3). Then, {un(T)} is bounded in L2(Ω) and {zn} is bounded in L2(Q). So, there exist ˆz∈L2(Q) and a subsequence of {zn} which converges to ˆz in L2(Q)-weakly. Multiplying the differential equation by un and integrating by parts, we obtain
12ddt‖un‖2L2+‖∇un‖2L2=∫Ωα(zn)V(un)∇vun+∫Ωβ(zn,v)u2n−u3n. | (3.4) |
Using that α,V∈L∞(IR), β∈L∞(IR2) and un≥0, we have
12ddt‖un‖2L2≤C1+C2‖un‖2L2 |
and so,
‖un(t)‖2L2≤‖u0‖2L2+C1T+∫t0C2‖un(s)‖2L2ds. |
Applying Gronwall's Lemma we deduce that {un} is bounded in L∞(0,T;L2(Ω)). Using this fact and returning to (3.4) we obtain that {un} is bounded in L2(0,T;H1(Ω)). This boundness gives that {(un)t} is bounded in L2(0,T;(H1(Ω))′). Effectively, taking any w∈H1(Ω) and applying the duality product ⟨H1(Ω)′,H1(Ω)⟩ in the equation of (3.2) we have
⟨(un)t,w⟩=−∫Ω∇un⋅∇w+∫ΩV(un)α(z)∇v⋅∇w+∫Ωβ(z,v)unw−∫Ωu2nw. |
Since {un} is bounded in L∞(0,T;L2(Ω)), {un} is bounded in L2(0,T;H1(Ω)), applying the Hölder inequality, we get
|⟨(un)t,w⟩|≤‖∇un‖L2‖∇w‖L2+C1‖∇w‖L2+C2‖w‖L2. |
Then, we obtain
‖(un)t‖(H1(Ω))′≤‖∇un‖L2+C. |
So,
∫T0‖(un)t‖2(H1(Ω))′≤C1+C2‖un‖2L2(0,T;H1(Ω)), |
which proves that {(un)t} is bounded in L2(0,T;(H1(Ω))′).
And this gives that {un} is bounded in W(0,T). By the Aubin-Lions Lemma (see for instance [15]) we can assure that there exist a subsequence of {un} and a function ˆu in W(0,T) such that (still denoting by un the subsequence) {un} converges to ˆu in L2(Q) strongly.
The variational formulation of the problem (3.2) for the functions (un,zn) is
∫T0⟨(un)t,φ⟩+∫Q∇un⋅∇φ=∫Qα(zn)V(un)∇v⋅∇φ+∫Qβ(zn,v)unφ−∫Qu2nφ. | (3.5) |
The functions α and β are convex and continuous (for β it is only necessary to be convex and continuous in the first variable), V is a continuous function in IR. Then, using the convergence of the sequences, {un} and {zn}, it is possible to pass to the limit in (3.5) and so, we obtain that (ˆu,ˆz) solves the problem (3.2). Besides, as {zn} weakly converges to ˆz in L2(Q) and zn≥0, we have that ˆz≥0. This proves that (ˆu,ˆz) belongs to the admissible set of the optimal control problem. The functional J is continuous and convex, so
J(un,zn)→J(ˆu,ˆz), |
which gives that J(ˆu,ˆz)=ˆβ and then, (ˆu,ˆz) is a solution of the problem (3.1)–(3.3).
Remark 1. The solution of the optimal problem might not be unique.
In order to obtain the first order necessary conditions of the optimal control problem, we enunciate the Duvobitskii-Milyutin theorem (see [7]), which is a generalization of the Lagrange's Multipliers theorem.
Theorem 3.2. Let X be a normed space. Assume that the functional J:X→IR has a local minimum with constraints Z=n+1⋂i=1Zi⊂X at a point x0∈Z. Assume that J is regularly decreasing at x0, with decreasing (and convex) cone DC0; the inequality constraints Zi, 1≤i≤n, are regular at x0, with feasible (and convex) cones FCi, 1≤i≤n; the equality constraint Zn+1 is also regular at x0, with tangent (and convex) cone TCn+1. Then, there exist continuous linear functionals f0∈DC∗0, fi∈FC∗i for 1≤i≤n and fn+1∈TC∗n+1 (we denote by ∗ the corresponding dual cone), not all identically zero, such that they satisfy the Euler-Lagrange equation:
f0+n∑i=1fi+fn+1=0in X′. |
The constraint (3.3) must be considered as an equality constraint, together with (3.2), because the convex U={z∈L2(Q): z≥0} is a set with empty interior in L2(Q). So, applying the Duvobitskii-Milyutin theorem to (3.1)–(3.3) there exist two functionals, f0∈(DC)∗, f∈(TC)∗ such that
f0+f=0 W(0,T)′×L2(Q)′, |
and they are not simultaneously identically zero.
The identification of the cones. Following [2], we are going to define the cones and their dual ones in a point. The functional f0∈(DC)∗ where (DC)∗ is the dual decreasing cone in (ˆu,ˆz), is given by
f0=−λJ′(ˆu,ˆz) with λ≥0. |
The computation of the tangent cone and its dual one is technically more complicated because there are two constraints, (3.2) and (3.3), considered like equalities generating a tangent cone. Each of these constraints generates, by itself, a tangent cone. We denote Z1 the constraint set given by (3.2) and TC1 the tangent cone associated to Z1 in (ˆu,ˆz). By Lyusternik's theorem (see [2]), this cone is the set
TC1={(u,z)∈W(0,T)×L2(Q): solution the problem |
{ut−Δu+∇⋅(α(ˆz)V′(ˆu)u∇v)−β(ˆz,v)u+2ˆuu++∇⋅(α′(ˆz)zV(ˆu)∇v)−∂1β(ˆz,v)zˆu=0,∂nu−α(ˆz)V′(ˆu)u∂nv−α′(ˆz)zV(ˆu)∂nv=0,u(0)=0. | (3.6) |
And (TC1)∗ is
(TC1)∗={⟨h,(u,z)⟩=0 ∀(u,z)∈TC1}. |
If we consider the convex constraint (3.3) and we call Z2 to this set, the dual tangent cone to Z2 in the point (ˆu,ˆz), denoted by (TC2)∗, is
(TC2)∗={(0,g): ⟨g,z−ˆz⟩L2(Q)′,L2(Q)≥0 ∀z≥0}. |
The question is if we could assure that the dual cone to the cone associated to the set Z1⋂Z2 in (ˆu,ˆz), denoted by (TC)∗, would be equal to (TC1)∗+(TC2)∗. We can always assure that (TC1)∗+(TC2)∗ is included in (TC)∗, but the converse inclusion will be true if we prove that (TC1)∗ and (TC2)∗ are a system of the same sense cones (see [19], Definition 2.2.1]). The definition of a system of the same sense cones (SSS) is the following (see [19]):
Definition 3.3. Let {Ci}ni=1 a system of cones in a normed space X. It is a (SSS) if for every M>0 there exist M1,...Mn such that, for each x=∑ni=1xi, xi∈Ci, the inequality ‖x‖≤M implies the inequalities ‖xi‖≤Mi, i=1,...,n.
This definition is not handle to determine if a system of cones is or not a (SSS). We enunciate a theorem that characterizes a (SSS) constituted by two cones when one of them is given by a linear continuous operator (see [19]).
Theorem 3.4. Let E=X×Y be the product space of two normed spaces and C1,C2 the following cones:
C1={(x,y)∈E: x=Ay}, |
where A is a linear continuous operator from Y to X, and
C2=X×~C2, |
and ~C2 is a cone in Y. If we denote by (Ci)∗ the dual cone to Ci, then
(C1)∗={(x∗,y∗)∈E∗, y∗=−A∗x∗} |
and
(C2)∗={(0,y∗)∈E∗, y∗∈~C2∗}. |
Besides, the system {(C1)∗,(C2)∗} is a (SSS).
In our case, C1=TC1 and C2=TC2. The tangent cone associated to (3.2), TC1, is given by a linear continuous operator, the operator of the differential equation (3.6). By this theorem, we have that (TC)∗=(TC1)∗+(TC2)∗.
In the following theorem we obtain the optimality system of the optimal control problem (3.1)–(3.3).
Theorem 3.5. If (ˆu,ˆz) is a solution of the optimal control problem (3.1)–(3.3), then there exists p∈W(0,T) such that (ˆu,ˆz,p) satisfies
{ˆut−Δˆu=−∇⋅(α(ˆz)V(ˆu)∇v)+β(ˆz,v)ˆu−ˆu2inΩ×(0,T),∂nˆu−α(ˆz)V(ˆu)∂nv=0in∂Ω×(0,T),ˆu(x,0)=u0(x)inΩ,−pt−Δp=α(ˆz)V′(ˆu)∇v⋅∇p+β(ˆz,v)p−2ˆupinΩ×(0,T),∂np=0in∂Ω×(0,T),p(T)=ˆu(T)inΩ,ˆz=[−ab(α′(ˆz)V(ˆu)∇v⋅∇p+∂1β(ˆz,v)ˆup)]+. | (3.7) |
Proof. The Euler-Lagrange equation
The Euler-Lagrange equation, (E-L), is
f0+f1+f2=0 in W(0,T)′×L2(Q)′, |
with f0∈(DC)∗, f1∈(TC1)∗ and f2∈(TC2)∗. We know that
f0=−λJ′(ˆu,ˆz), λ≥0 |
⟨f1,(u,z)⟩=0 ∀(u,z) solution of (3.6) |
f2=(0,˜f2) with <˜f2,z−ˆz>≥0 ∀z≥0. |
We will see that λ cannot be zero.
Suppose that λ=0. Then, the (E-L) equation is
f1+f2=0. |
Applying this equation to any (u,z)∈TC1 we have ⟨f2,(u,z)⟩=0 and so, ⟨˜f2,z⟩=0 for every z such that there exists u verifying (u,z)∈TC1. Since for every z∈L2(Q) there exists this u, we have that ˜f2≡0 and then, f2≡0 and f1≡0, which is impossible. So, λ≠0, we can rescale λ=1 and the (E-L) equation is
−J′(ˆu,ˆz)+f1+(0,˜f2)=0 in W(0,T)′×L2(Q)′. |
We have that
J′(ˆu,ˆz)(u,z)=(˜f2,z) ∀(u,z)∈TC1, |
this is
a∫Ωˆu(T)u(T)+b∫Qˆzz=(˜f2,z) ∀(u,z)∈TC1. | (3.8) |
In the following, we describe a standard procedure in order to rewrite the first term in (3.8) using the optimal control ˆz. This is done defining an adjoint problem. We multiply the differential equation of (3.6) by a function p, which will be the solution of the adjoint problem, and we integrate by parts. Taking account, we obtain
⟨u,−pt−Δp−α(ˆz)V′(ˆu)∇v⋅∇p−β(ˆz,v)p+2ˆup⟩+∫Ωu(T)p(T)− |
−⟨z,α′(ˆz)V(ˆu)∇v⋅∇p⟩−⟨z,∂1β(ˆz,v)ˆup⟩+∫T0∫∂Ωu∂np=0. |
We define the adjoint problem
{−pt−Δp=α(ˆz)V′(ˆu)∇v⋅∇p+β(ˆz,v)p−2ˆup∂np=0p(T)=ˆu(T). | (3.9) |
Then
∫Ωu(T)ˆu(T)−⟨z,α′(ˆz)V(ˆu)∇v⋅∇p+∂1β(ˆz,v)ˆup⟩=0, |
and replacing this equality in (3.8) we obtain
a⟨z,α′(ˆz)V(ˆu)∇v⋅∇p+∂1β(ˆz,v)ˆup⟩+b∫Qˆzz=(˜f2,z), |
for every z∈L2(Q) because, as we have already said, for every z∈L2(Q) there is a function u verifying (u,z)∈TC1. So, we have obtained the functional ˜f2:
˜f2=a(α′(ˆz)V(ˆu)∇v⋅∇p+∂1β(ˆz,v)ˆup)+bˆz. |
Since (˜f2,z−ˆz)≥0 for every z≥0, we have that
(a(α′(ˆz)V(ˆu)∇v⋅∇p+∂1β(ˆz,v)ˆup)+bˆz,z−ˆz)≥0 ∀z≥0, |
and this is equivalent to
ˆz=PU(−ab(α′(ˆz)V(ˆu)∇v⋅∇p+∂1β(ˆz,v)ˆup)), |
where U is the convex set {z∈L2(Q): z≥0} and P is the projection operator of L2(Q) on U.
In this case, because of the particular convex set we have, the operator P is well-known. It is
PUg=g+=max{g,0}, |
hence,
ˆz=[−ab(α′(ˆz)V(ˆu)∇v⋅∇p+∂1β(ˆz,v)ˆup)]+. | (3.10) |
We have just obtained the optimality system, given by the theorem.
Next, we are going to prove the uniqueness of solution of (3.7) for T small enough. To this aim it will be useful to obtain an uniform estimate in time for ∇p on the interval [0,T]. In order to get this estimate we will suppose additional restrictions on α and v. In particular, we require that α=α0 is a constant function and ∂nv=0. In this case the optimality system is given by the following equations:
ut−Δu=−∇⋅(α0V(u)∇v)+β(z,v)u−u2inΩ×(0,T), | (3.11) |
∂nu=0on∂Ω×(0,T), | (3.12) |
u(x,0)=u0(x)inΩ, | (3.13) |
−pt−Δp=α0V′(u)∇v⋅∇p+β(z,v)p−2upinΩ×(0,T), | (3.14) |
∂np=0on∂Ω×(0,T), | (3.15) |
p(T)=u(T)inΩ, | (3.16) |
z=[−ab(∂1β(z,v)up)]+. | (3.17) |
Lemma 3.6. We have that ‖u(T)‖W1,3(Ω)≤C.
Proof. Using the variations of constants formula in (3.11)–(3.13) we get
u(t)=e−tAu0+∫t0e−(t−s)Ah(u,v)ds, | (3.18) |
where A=−Δ+I with domain {ψ∈W2,3(Ω):∂nu=0} and
h(u,v):=−∇⋅(α0V(u)∇v)+(β(z,v)+1)u−u2. |
Let Xβ, β∈(0,1) the fractional powers of X. By [8], Theorem 1.6.1] we have that
Xβ↪W1,3, |
for any β∈(1/2,1). Taking the norm W1,3 in (3.18) and applying the above embedding in the right hand side we infer
‖u(T)‖W1,3(Ω)≤‖e−tAu0‖Xβ+∫T0‖e−(t−s)Ah‖Xβ. |
By [8], Theorem 1.4.3] we deduce
‖u(T)‖W1,3(Ω)≤Cβe−δtt−β‖u0‖L3+∫T0e−δ(t−s)(t−s)−β‖h(u,v)‖L3 |
Let us notice that by the regularity of β, V and v and the embedding W1,3(Ω)↪L6(Ω) we have
‖h(u,v)‖L3≤C‖u‖W1,3(Ω). |
As a consequence, we can conclude by the singular Gronwall Lemma (see [8], Section 1.2.1]).
In what follows we will show the regularity for the backward linear parabolic equation (3.14)–(3.16). The backward equation can be written in a forward way by the change of variable ¯p(t)=p(T−t). Therefore, the problem is
{¯pt−Δ¯p=α0V′(u)∇v⋅∇¯p+β(z,v)¯p−2u¯pin Ω×(0,T),∂n¯p=0on ∂Ω×(0,T),¯p(0)=u(T)in Ω. | (3.19) |
Lemma 3.7. We have that maxt∈[0,T]‖p(t)‖W1,3(Ω)≤C.
Proof. As in the previous Lemma we apply the variation of constants formula for ¯p to get
¯p(t)=e−tAu(T)+∫t0e−(t−s)Af(u,v,¯p)ds, |
where
f(u,v,¯p):=α0V′(u)∇v⋅∇¯p+(β(z,v)+1)¯p−2u¯p. |
We take W1,3(Ω) in the variations of constants formula for p and we apply [8], Theorem 1.4.3, Theorem 1.6.1] and ([10], p. 59]) to infer
‖¯p(t)‖W1,3(Ω)≤C‖u(T)‖W1,3(Ω)+∫T0(t−s)−βe−δ(t−s)‖f(u,v,¯p)‖L3ds |
with β∈(1/2,1). Since ‖f(u,v,¯p)‖L3≤C‖¯p‖W1,3(Ω) then by the Gronwall Lemma (see [20]) we can conclude the result.
Theorem 3.8. Let be α=α0 a constant function, the TAF concentration v satisfies that ∂nv=0 on Γ, the function β is twice differentiable with respect to the first variable and it verifies that ∂211β is bounded and its norm in L∞ is small. Then, the optimality system (3.7) has a unique solution when the time T is small enough.
Proof. Using the hypothesis α=α0, the optimality system is given by these equations:
z=[−ab(∂1β(z,v)up)]+. | (3.20) |
We call T to the following operator:
T:L2(L2)→L2(L2) |
z↦T(z)=[−ab∂1β(z,v)up]+, |
where L2(L2) is the space L2(0,T;L2(Ω)). The function u is obtained by the data z:
T1:L2(L2)→W(0,T) |
z↦u, the solution of (3.11), (3.12), (3.13) |
Having z and u, the function p can be solved:
T2:L2(L2)×W(0,T)→W(0,T) |
(z,T1(z))↦p, the solution of (3.14), (3.15), (3.16) |
With this notation, the operator T is
T(z)=[−ab∂1β(z,v)T1(z)T2(z,T1(z))]+. |
The existence of an optimal control is the existence of a fixed point of T. This is guaranteed by the Theorem 3.1. We want to prove that T is contractive. Then, there will be a unique fixed point, ˆz, and we can say that the optimality system has a unique solution and so, the optimal control problem has a unique optimal control.
Let be z1,z2∈L2(L2).
‖T(z1)−T(z2)‖L2(L2)≤ab‖∂1β(z1,v)u1p1−∂1β(z2,v)u2p2‖L2(L2), |
where we have called ui=T1(zi) and pi=T2(zi,T1(zi)), i=1,2. Adding and subtracting the appropriate terms and using the triangular inequality we have
‖T(z1)−T(z2)‖L2(L2)≤ab[‖∂211β‖L∞(L∞)‖z1−z2‖L2(L2)‖u1‖L∞(L∞)‖p1‖L∞(L∞)+ |
+‖∂1β‖L∞(L∞)‖u1−u2‖L2(L2)‖p1‖L∞(L∞)+‖∂1β‖L∞(L∞)‖p1−p2‖L2(L2)‖u2‖L∞(L∞)]. |
By Theorem 2.2, it is known that ‖u‖L∞(L∞), where u is a solution of (3.11), (3.12) and (3.13), and ‖p‖L∞(L∞), p is a solution of (3.14), (3.15) and (3.16), are bounded.
Let w=u1−u2. Writing the equation for w, multiplying by w and integrating in Ω we obtain the following estimate:
12ddt‖w‖2L2+12‖∇w‖2L2≤12∫Ωα20(V(u1)−V(u2))2|∇v|2+∫Ωβ(z1,v)w2+ |
+∫Ω(β(z1,v)−β(z2,v))u2w−∫Ω(u1+u2)w2. |
Applying the medium value theorem, V′, β, ∂1β and ‖u2‖L∞(Q) are bounded and the Young's inequality, we have
ddt‖w‖2L2≤A‖w‖2L2+B‖z1−z2‖2L2, A,B>0. |
Then,
‖w(t)‖2L2≤BeAt‖z1−z2‖2L2(L2). | (3.21) |
Therefore,
‖u1−u2‖2L2(L2)≤BA(eAT−1)‖z1−z2‖2L2(L2). |
We do a similar reasoning with p1−p2. Let η=p1−p2. Writing the equation for η, multiplying it by η and integrating in Ω, we get
−ddt∫Ωη2+∫Ω|∇η|2=α0∫ΩV′(u1)∇v⋅∇ηη+α0∫Ω(V′(u1)−V′(u2))∇v⋅∇p2η++∫Ωβ(z1,v)η2+∫Ω(β(z1,v)−β(z2,v))p2η−2∫Ωu1η2−2∫Ωwp2η. | (3.22) |
We multiply the previous equality by −1 and we apply the Young inequality and the Hölder inequality to obtain
ddt∫Ωη2+∫Ω|∇η|2≤ϵ∫Ω|∇η|2+C(ϵ)∫Ωη2++‖(u1−u2)∇v‖L2‖∇p2‖L3‖η‖L6+C‖z1−z2‖2L2+C∫Ωw2. |
Hence, the Sobolev embedding and (3.21) entails
ddt∫Ωη2≤C(ϵ)∫Ωη2+C(ϵ)BeAT‖z1−z2‖L2(L2)+C‖z1−z2‖L2. |
Next, we solve the differential equation to get
∫Ωη2≤eC(ϵ)t‖u1(T)−u2(T)‖L2(Ω)+ec(ϵ)tC(ϵ)BeAT‖z1−z2‖L2(L2)C(T)++eC(ϵ)t∫t0e−C(ϵ)sC‖z1−z2‖2L2. |
We apply (3.21) for t=T to obtain
∫Ωη2≤eC(ϵ)tC(T)‖z1−z2‖L2(L2). |
Therefore, after integration on the interval [0,T] we have
‖η‖L2(L2)≤C(T)eC(ϵ)T−1C(ϵ)‖z1−z2‖L2(L2). |
After all of these inequalities we get the following equation
‖T(z1)−T(z2)‖L2(L2)≤C(C(T)+‖∂211β‖L∞‖u1‖L∞‖p1‖L∞)‖z1−z2‖L2(L2), |
where C(T) goes to zero as T goes to zero. If we assume that ‖∂211β‖L∞ is small enough we can say that the operator T is contractive if T is small enough. Therefore, there exists a unique optimal control if T is small.
We are going to solve the optimality system in some particular cases. The program has been done in FreeFEM, using P1-Lagrange finit element method and Euler method to solve the problem of u and p. In the case of u, because of the nonlinearity of the equation, we have used the Newton's method, and for the problem of p we have to solve a backward equation. The optimality system is a recursive equation, by the hypothesis of Theorem 3.8, it is contractive for T small enough, so, we have chosen z, we get u and p, we obtain a new z and we repeat until the difference between this one and the previous one satisfies the stop test.
The domain Ω is plotted in the following figure:
![]() |
We have chosen the following data:
The coefficient for the chemotaxis term
α0=50, |
the time step,
dt=0.1 |
and a small final time, T,
T=0.5. |
We solve this problem to get v
−Δv+v=g, Ω∂v∂n=0, ∂Ω |
with
g=10.01+x2+y2. |
The graphic of the TAF concentration is shown in Figure 1.
The initial density of endothelial cells is
u0=x2+y2+100, |
and it is drawn in Figure 2:
The function V which appears in the chemotaxis term is given by
V(u)=u1+u2, |
and the function β which is the growth rate of the drug z, is
β(z,v)=kexp(−z)v1+v2, |
we have taken k=1. Finally, the coefficients, a and b are equal to one, that means that we optimize u(T) and z with the same weights. At the final time, the density of ECs is shown in Figure 3:
And the concentration of z at the final time is plotted in Figure 4.
The functional and norms are
J(ˆu,ˆz)=65.1, ‖u(T)‖L2(Ω)=6.27, ‖z‖L2(Q)=9.53 |
When the chemotaxis factor, α0, is smaller, the drug concentration is smaller too, it is necesary less chemical agent to control the angiogenesis process. We take
α0=5, |
instead of 50 and the rest of the data the same as before. The density of ECs at the final time is shown in Figure 5.
In this case, the value of J on the optimal and the norms of u(T) and z are
J(ˆu,ˆz)=66.46, ‖u(T)‖L2(Ω)=5.7, ‖z‖L2(Q)=10, |
and the concentration of drug at the final times is drawn in Figure 6. If we prioritize to minimize the term of u(T), taking
a=10, b=1 |
and we choose a growth function β smaller than the previous one, taking
k=0.1 |
the functions u(T) and z(T) decrease notably, as we can see in Figure 7 and Figure 8.
Now, the functional and the norms are
J(ˆu,ˆz)=70, ‖u(T)‖L2(Ω)=3,76, ‖z‖L2(Q)=0,46. |
The authors declare no conflict of interest.
[1] | J. Hadamard, Étude sur les propriétés des fonctions entières en particulier d'une fonction considérée par Riemann, J. Math. Pures Appl., 58 (1893), 171–215. |
[2] |
M. A. Ali, H. Budak, Z. Zhang, H. Yildrim, Some new Simpson's type inequalities for co-ordinated convex functions in quantum calculus, Math. Meth. Appl. Sci., 44 (2021), 4515–4540. doi: 10.1002/mma.7048
![]() |
[3] |
M. A. Ali, H. Budak, M. Abbas, Y. M. Chu, Quantum Hermite-Hadamard-type inequalities for functions with convex absolute values of second qb-derivatives, Adv. Differ. Equ., 2021 (2021), 7. doi: 10.1186/s13662-020-03163-1
![]() |
[4] |
M. A. Ali, M. Abbas, H. Budak, P. Agarwal, G. Murtaza, Y. M. Chu, New quantum boundaries for quantum Simpson's and quantum Newton's type inequalities for preinvex functions, Adv. Differ. Equ., 2021 (2021), 64. doi: 10.1186/s13662-021-03226-x
![]() |
[5] |
M. A. Ali, Y. M. Chu, H. Budak, A. Akkurt, H. Yildrim, Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables, Adv. Differ. Equ., 2021 (2021), 25. doi: 10.1186/s13662-020-03195-7
![]() |
[6] |
M. A. Ali, N. Alp, H. Budak, Y. M. Chu, Z. Zhang, On some new quantum midpoint type inequalities for twice quantum differentiable convex functions, Open Math., 19 (2021), 427–439. doi: 10.1515/math-2021-0015
![]() |
[7] |
M. A. Ali, H. Budak, A. Akkurt, Y. M. Chu, Quantum Ostrowski type inequalities for twice quantum differentiable functions in quantum calculus, Open Math., 19 (2021), 440–449. doi: 10.1515/math-2021-0020
![]() |
[8] | M. A. Ali, M. Abbas, M. Sehar, G. Murtaza, Simpson's and Newton's type quantum integral inequalities for preinvex functions, Korean J. Math., 29 (2021), 193–203. |
[9] |
H. Budak, M. A. Ali, M. Tarhanaci, Some new quantum Hermite-Hadamard-like inequalities for coordinated convex functions, J. Optim. Theory Appl., 186 (2020), 899–910. doi: 10.1007/s10957-020-01726-6
![]() |
[10] | H. Budak, S. Erden, M. A. Ali, Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Meth. Appl. Sci., 44 (2020), 378–390. |
[11] | S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. |
[12] |
A. Guessab, G. Schmeisser, Sharp integral inequalities of Hermite-Hadamard type, J. Apprx. Theory, 115 (2002), 260–288. doi: 10.1006/jath.2001.3658
![]() |
[13] | A. Guessab, G. Schmeisser, Convexity results and sharp error estimates in approximate multivariate integration, Math. Comput., 73 (2004), 1365–1384. |
[14] |
A. Guessab, G. Schmeisser, Sharp error estimates for interpolatory approximation on convex polytope, SIAM J. Numer. Anal., 43 (2005), 909–923. doi: 10.1137/S0036142903435958
![]() |
[15] | A. Guessab, Direct and converse results for generalized multivariate Jensen-type inequalities, J. Nonlinear Convex Anal., 13 (2012), 777–797. |
[16] |
M. Z. Sarikaya, M. E. Kiris, Some new inequalities of Hermite-Hadamard type for s-convex functions, Miskolc Math. Notes, 16 (2015), 491–501. doi: 10.18514/MMN.2015.1099
![]() |
[17] |
P. O. Mohammed, Some new Hermite-Hadamard type inequalities for MT-convex functions on differentiable coordinates, J. King Saud Univ. Sci., 30 (2018), 258–262. doi: 10.1016/j.jksus.2017.07.011
![]() |
[18] | K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. |
[19] | K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, San Diego, 1974. |
[20] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Taylor Francis, London, 2002 [orig. ed. in Russian; Nauka i Tekhnika, Minsk, 1987]. |
[21] | M. Z. Sarikaya, F. Ertugral, On the generalized Hermite-Hadamard inequalities, An. Univ. Craiova Ser. Mat. Inform., 47 (2020), 193–213. |
[22] |
M. Z. Sarikaya, A. Karaca, On the k-Riemann-Liouville fractional integral and applications, IJSM, 1 (2014), 033–043. doi: 10.26524/jms.2012.4
![]() |
[23] |
T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Differ. Equ., 2017 (2017), 313. doi: 10.1186/s13662-017-1285-0
![]() |
[24] |
T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl., 279 (2015), 57–66. doi: 10.1016/j.cam.2014.10.016
![]() |
[25] |
M. Z. Sarikaya, E. Set, H. Yaldiz, N. Başak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403–2407. doi: 10.1016/j.mcm.2011.12.048
![]() |
[26] |
M. Z. Sarikaya, H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17 (2017), 1049–1059. doi: 10.18514/MMN.2017.1197
![]() |
[27] |
P. O. Mohammed, I. Brevik, A New Version of the Hermite-Hadamard Inequality for Riemann-Liouville Fractional Integrals, Symmetry, 12 (2020), 610. doi:10.3390/sym12040610. doi: 10.3390/sym12040610
![]() |
[28] | A. Fernandez, P. Mohammed, Hermite-Hadamard inequalities in fractional calculus defined using Mittag-Leffler kernels, Math. Meth. Appl. Sci., 2020, 1–18. Available from: https://doi.org/10.1002/mma.6188. |
[29] |
P. O. Mohammed, T. Abdeljawad, Modification of certain fractional integral inequalities for convex functions, Adv. Differ. Equ., 2020 (2020), 69. doi: 10.1186/s13662-020-2541-2
![]() |
[30] | P. O. Mohammed, Hermite-Hadamard inequalities for Riemann-Liouville fractional integrals of a convex function with respect to a monotone function, Math. Meth. Appl. Sci., (2019), 1–11. Available from: https://doi.org/10.1002/mma.5784. |
[31] |
P. O. Mohammed, M. Z. Sarikaya, Hermite-Hadamard type inequalities for F-convex function involving fractional integrals, J. Inequal. Appl., 2018 (2018), 359. doi: 10.1186/s13660-018-1950-1
![]() |
[32] | P. O. Mohammed, M. Z. Sarikaya, D. Baleanu, On the Generalized Hermite-Hadamard Inequalities via the Tempered Fractional Integrals, Symmetry, 12 (2020), 595. Available from: http://doi.org/10.3390/sym12040595. |
[33] |
F. Qi, P. O. Mohammed, J. C. Yao, Y. H. Yao, Generalized fractional integral inequalities of Hermite-Hadamard type for (α,m)-convex functions, J. Inequal. Appl., 2019 (2019), 135. doi: 10.1186/s13660-019-2079-6
![]() |
[34] |
P. O. Mohammed, M. Z. Sarikaya, On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Math., 372 (2020), 112740. doi: 10.1016/j.cam.2020.112740
![]() |
[35] |
J. Han, P. O. Mohammed, H. Zeng, Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function, Open Math., 18 (2020), 794–806. doi: 10.1515/math-2020-0038
![]() |
[36] |
D. Baleanu, P. O. Mohammed, S. Zeng, Inequalities of trapezoidal type involving generalized fractional integrals, Alex. Eng. J., 59 (2020), 2975–2984. doi: 10.1016/j.aej.2020.03.039
![]() |
[37] | P. M. Vasić, J. E. Pečarić, On the Jensen inequality, Univ. Beograd. Publ. Elektrotehn Fak. Ser. Mat. Fis., 634–677 (1979), 50–54. |
[38] |
A. Matković, J. Pečarić, I. Perić, A variant of Jensens inequality of Mercers type for operators with application, Linear Al. Appl., 418 (2006), 551–564. doi: 10.1016/j.laa.2006.02.030
![]() |
[39] | M. Kian, M. S. Moslehian, Refinements of the operator Jensen-Mercer inequality, Electron J. Linear Al., 26 (2013), 50. |
[40] | A. McD Mercer, A variant of Jensens inequality, J. Ineq. Pure Appl. Math., 4 (2003), 73. |
[41] |
A. M. Fink, M. Jodeit Jr, Jensen inequalities for functions with higher monotonicities, Aequations Math., 40 (1990), 26–43. doi: 10.1007/BF02112278
![]() |
[42] | M. Kian, M. S. Moslehian, Refinements of the operator Jensen-Mercer inequality, Electron. J. Linear Algebra,, 26 (2013), 50. |
[43] | H. Öǧülmüs, M. Z. Sarikaya, Hermite-Hadamard-Mercer type inequalities for fractional integrals, Available from: DOI: 10.13140/RG.2.2.30669.79844. |
[44] | G. Farid, A. U. Rehman, M. Zahra, On Hadamard inequalities for k-fractional integrals, Nonlinear Funct. Anal. Appl., 21 (2016), 463–478. |
[45] | M. A. Khan, T. Ali, S. S. Dragomir, M. Z. Sarikaya, Hermite-Hadamard type inequalities for conformable fractional integrals, RACSAM Rev. R. Acad. A, 112 (2018), 1033–1048. |
[46] |
B. Ahmad, A Alsaedi, M. Kirane, B. T. Torebek, Hermite-Hadamard, Hermite-Hadamard-Fejér, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals, J. Comput. Appl. Math., 353 (2019), 120–129. doi: 10.1016/j.cam.2018.12.030
![]() |
[47] | S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Let., 11 (1998), 91–95. |
[48] | U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147 (2004), 137–146. |
1. | V. N. Deiva Mani, S. Karthikeyan, L. Shangerganesh, S. Marshal Anthoni, Solvability of the acid-mediated tumor growth model with nonlinear acid production term, 2023, 9, 2296-9020, 887, 10.1007/s41808-023-00227-7 |