Adrian Karpowicz
The method of quasilinearization for system of hyperbolic functional differential equations
Abstract. We deal with monotone iterative method for the Darboux problem for the system of hyperbolic partial functional-differential equations
( ∂2u
∂x∂y(x, y) = f (x, y, u(x,y), u(x, y)) a.e. in [0, a]× [0, b],
u(x, y) = ψ(x, y) on [−a0, a]× [−b0, b]\(0, a] × (0, b],
where the function u(x,y) : [−a0, 0]× [−b0, 0] → Rk is defined by u(x,y)(s, t) = u(s + x, t + y) for (s, t)∈ [−a0, 0]× [−b0, 0].
2000 Mathematics Subject Classification: 35L70; 35R10; 35R45.
Key words and phrases: Monotone iterative technique, Generalized quasilineariza- tion, Hyperbolic equations, Darboux problem, Functional differential inequalities.
1. Introduction. Put I = [0, a] × [0, b] , D = [−a0, 0]× [−b0, 0] , I∗ = [−a0, a]× [−b0, b] , I0 = I∗\I. We always assume that a, b > 0 and a0, b0 ∈ R+, where R+ = [0, +∞). We denote by C(D, Rk) and L1(D, Rk) the space of contin- uous functions and of Lebesgue integrable functions from D into Rk, respectively.
The norm | · | in Rk denotes the maximum norm. Moreover, ||w||0 denotes the usual supremum norm of w ∈ C(D, Rk). The inequality x < y in Rk means that xi < yi for each i ∈ {1, . . . , k}. Similarly for ”≥”,”>” and ”≤”. The function f = (f1, . . . , fk) : I × C(D, Rk) × Rk → Rk of the variables (x, y, ω, η) is said to be quasimonotonically nondecreasing with respect to η if each fi is nondecreas- ing with respect to all ηj for j 6= i. This function is said to be nondecreasing with respect to the functional argument ω if the inequality ω1 ≤ ω2 implies that f (x, y, ω1, η) ≤ f(x, y, ω2, η). Here ω1 ≤ ω2 means that ω1(s, t) ≤ ω2(s, t) for all (s, t) ∈ D. In this paper we shall discuss Carath´eodory solutions of hyperbolic functional differential equations. In order to define this solutions we need an appro- priately definition of an absolutely continuous function of two variables. For this
purpose, we first introduce suitable notation. Given a rectangle J = [a1, a2]×[b1, b2] contained in I and u : I → R, let
∆J(u) = u(a1, b1) − u(a2, b1) − u(a1, b2) + u(a2, b2).
A rectangle is called subrectangle of I if its sides are parallel to the coordinate axes. Let m denote Lebesgue measure on R2. We say that u : I → R is absolutely continuous if the following two conditions are satisfied:
a) Given > 0, there exists δ > 0 such that X
J∈J
|∆J(u)| <
whenever J is a finite collection of pairwise non-overlapping subrectangles of
I with X
J∈J
m(J) < δ.
b) The marginal functions u(·, b) and u(a, ·) are absolutely continuous functions of a single variable on [0, a] and [0, b], respectively.
Let AC(I, R) denote the set of absolutely continuous functions on I. In [1] we can find that the following statements are equivalent:
a) u ∈ AC(I, R),
b) There exist g ∈ AC([0, a], R), h ∈ AC([0, b], R) and L ∈ L1(I, R) such that u(x, y) = g(x) + h(y) +
Z x 0
Z y 0
L(s, t)dsdt
Note that if u ∈ AC(I, R) then ∂u∂x, ∂u∂y and ∂x∂y∂2u exist almost everywhere on I.
In this paper we consider the nonlinear problem (1)
( ∂2u
∂x∂y(x, y) = f(x, y, u(x,y), u(x, y)) a.e. in I,
u(x, y) = ψ(x, y) on I0,
where f : I × C(D, Rk) × Rk → Rk and ψ : I0→ Rk. We also consider the linear problem
(2)
( ∂2u
∂x∂y(x, y) = r(x, y) + P (x, y)u(x,y)+ C(x, y)u(x, y) a.e. in I,
u(x, y) = ψ(x, y) on I0,
where P (x, y) : C(D, Rk) → Rk is a linear operator for every (x, y) ∈ I, C is square k× k matrix and r ∈ L1(I, R).
For all problems we define u(x,y): D → Rk by the formula u(x,y)(s, t) = u(s + x, t + y) for (s, t)∈ D. By the solution of the problem we mean a function u : I∗ → Rk continuous on I∗ and absolutely continuous on I which satisfies the differential equation almost everywhere on I and the initial condition everywhere on I0. Now we give two examples of the operator P (x, y).
Example 1.1 Let ˜P (x, y) = (˜pij(x, y))i,j=1,...,k be a square k × k matrix, where
˜pij ∈ L1(I, R), γ1: I → [−a0, a], γ2: I → [−b0, b] and (γ1(x, y)−x, γ2(x, y)−y) ∈ D.
If for every (x, y) ∈ I, we define the operator P (x, y) : C(D, Rk) → Rk by the formula
P (x, y)w = ˜P (x, y)w(γ1(x, y) − x, γ2(x, y) − y), then
P (x, y)u(x,y)= ˜P (x, y)u(γ1(x, y), γ2(x, y)).
Consequently we get as a special case of (2) the following equation with a deviated argument
∂2u
∂x∂y(x, y) = r(x, y) + ˜P (x, y)u(γ1(x, y), γ2(x, y)) a.e. in I.
If we want to get an integro-differential equation then for every (x, y) ∈ I, we define
P (x, y)w = ˜P (x, y) Z 0
−a0
Z 0
−b0
w(s, t)dsdt, and consequently we have
P (x, y)u(x,y)= ˜P (x, y) Z x
x−a0
Z y y−b0
u(s, t)dsdt.
This paper is devoted to the study of the method of quasilinearization for prob- lem (1). Using the method of functional-differential inequalities we prove a theo- rem on the convergence of suitable sequences to the solution of problem (1). The convergence that we get is quadratic. The result is obtained when the function f : I × C(D, Rk) × Rk → Rk of the variables (x, y, ω, η) is only quasimonotone nondecreasing in η and not convex in η. However the function f + φ is convex in η for some convex function φ : I × Rk → Rk of the variables (x, y, η). In [7]
discusses the case of this method for the scalar equation, where the function f is nondecreasing with respect to functional and non-functional variable. This method has been applied by Lakshmikantham [11] to classical solutions of the Darboux problem for the scalar hyperbolic non-functional differential equations, where the function f : I × R → R of the variables (x, y, η) was nondecreasing in η. In [10] this method has been applied to classical solutions of the Cauchy problem for the system of ordinary non-functional differential equations. Most of the results presented in chapter have been proved in the paper [8] ( see also [9] ). Therefore we omit proofs of these theorems.
The method of generalized quasilinearization for ordinary non-functional differential equations can be found in [10, 12], [13] while for functional-differential equations in [6]. The Chaplyghin and Newton methods of approximating solutions of functional- differential Darboux problem has been studied by Cz lapi´nski [4, 5] while the method of approximating solutions of similar problem, but without linearization and conse- quently with slower convergence has been studied by Brzychczy and Janus in [2, 3].
In above-mentioned papers the authors study the classical solutions of Couchy or Darboux problems.
2. Inequalities and the existence theorem. We give theorems about functional-differential inequalities and the existence theorem for the Darboux prob- lem. Using these theorems we shall prove, in the next section, the existence of two monotone sequences convergent quadratically to the solution of the Darboux problem.
Assumption 1 Suppose that the function f : I× C(D, Rk) ×Rk→ Rk is such that for each A > 0 there exists a function l ∈ L1(I, R) such that
l(x, y)≥ Z x
0
l(z, y)dz Z y
0
l(x, z)dz a.e. in I,
|f(x, y, ω, η) − f(x, y, ω, η)| ≤ l(x, y)(||ω − ω||0+ |η − η|), f (x, y, ω, η)− f(x, y, ω, η) ≥
≥
− l(x, y) + Z x
0
l(z, y)dz Z y
0
l(x, z)dz
(η − η), for ||ω||0,||ω||0,|η|, |η| ≤ A and η ≤ η.
Assumption 2 Suppose that v, w are functions continuous on I∗ and absolute continuous on I. Furthermore,
∂2v
∂x∂y(x, y) ≤ f(x, y, v(x,y), v(x, y)) a.e. in I,
∂2w
∂x∂y(x, y) ≥ f(x, y, w(x,y), w(x, y)) a.e. in I, v(x, y)≤ w(x, y) on I0,
∂v
∂x(x, 0) ≤ ∂w
∂x(x, 0) on [0, a], ∂v
∂y(0, y) ≤ ∂w
∂y(0, y) on [0, b].
Theorem 2.1 Suppose that the function f : I×C(D, Rk)×Rk→ Rkof the variables (x, y, ω, η) is nondecreasing with respect to ω and quasimonotone nondecreasing with respect to η. Moreover, assumptions 1 and 2 are satisfied. Then
v(x, y)≤ w(x, y) on I.
Assumption 3 Suppose that the function f is such that:
(A1) If u ∈ C(I, Rk) then f(x, y, u(x,y), u(x, y))∈ L1(I, Rk).
(A2) Uniform convergence ωn → ω in C(I∗,Rk) implies
f (x, y, ωn(x,y), ωn(x, y)) → f(x, y, ω(x,y), ω(x, y)) a.e. in I.
(A3) There is a function ˜l∈ L1(I, R) such that (3) |f(x, y, ω, η)| ≤ ˜l(x, y)
1 + ||ω||0+ |η| .
Assumption 4 Suppose that the function ψ : I0→ Rk is such that ψ ∈ C(I0,Rk), ψ(x, 0)∈ AC([0, a], Rk) and ψ(0, y) ∈ AC([0, b], Rk).
Theorem 2.2 Suppose that assumptions 3 and 4 are satisfied. Then problem (1) has a solution existing in I.
We omit proofs of the theorems 2.1 and 2.2. These theorems have been proved in the paper [8].
Theorem 2.3 Suppose that assumptions of theorems 2.1 and 2.2 are satisfied.
Moreover, functions v, w satisfy
v(x, y)≤ ψ(x, y) ≤ w(x, y) on I0,
∂v
∂x(x, 0) ≤ ∂ψ
∂x(x, 0) ≤ ∂w
∂x(x, 0) on [0, a],
∂v
∂y(0, y) ≤ ∂ψ
∂y(0, y) ≤ ∂w
∂y(0, y) on [0, b].
then problem (1) has a unique solution existing in I such that v(x, y)≤ u(x, y) ≤ w(x, y) on I.
Proof From theorem 2.2 it follows that problem (1) has a solution existing in I.
Let u1,u2 be the solutions of (1). Then from theorem 2.1 it is easily seen that u1 ≤ u2 on I and u2 ≤ u1 on I. Therefore (1) has a unique solution. The rest of
the proof follows by theorem 2.1.
3. The method of quasilinearization. We are now able to study the method of quasilinearization for the Darboux problem for system of hyperbolic functional- differential equations.
Assumption 5 Suppose that v0,w0 ∈ AC(I, Rk), v0 ≤ w0 on I and v0, w0 are lower and upper solution of (1), that is,
(4)
∂2v0
∂x∂y(x, y) ≤ f(x, y, v(x,y)0 , v0(x, y)) a.e. in I, v0(x, y) ≤ ψ(x, y) on I0,
∂v0
∂x(x, 0) ≤ ∂ψ∂x(x, 0), ∂v∂y0(0, y) ≤ ∂ψ∂y(0, y) for x ∈ [0, a], y ∈ [0, b],
(5)
∂2w0
∂x∂y(x, y) ≥ f(x, y, w(x,y)0 , w0(x, y)) a.e. in I, w0(x, y) ≥ ψ(x, y) on I0,
∂w0
∂x(x, 0) ≥ ∂ψ∂x(x, 0), ∂w∂y0(0, y) ≥ ∂ψ∂y(0, y) for x ∈ [0, a], y ∈ [0, b].
Assumption 6 Suppose that the function φ : I× Rk→ Rk of the variables (x, y, η) is such that
(B1) If u ∈ C(I, Rk) then φ(x, y, u(x, y)) ∈ L1(I, Rk).
(B2) Uniform convergence ηn → η in C(I, Rk) implies φ(x, y, ηn) → φ(x, y, η) a.e. in I.
Assumption 7 Suppose that:
(C1) For a.e. (x, y) ∈ I, η ∈ Λ = {η ∈ Rk: v0 ≤ η ≤ v0} the Frechet derivative
∂f
∂ω(x, y, ω, η) exists and is continuous linear operator for all ω ∈ Ω = {ω ∈ C(D,Rk): v0(x,y) ≤ ω ≤ v0(x,y)}. Moreover, ∂f∂ω(x, y, ·) is continuous operator and derivatives ∂f∂η, ∂φ∂η exist and are continuous.
(C2) The function l ∈ L1(I, R) is such that
∂f
∂ω(x, y, ω, η)u ≤ l(x, y)||u||0,
∂f
∂η(x, y, ω, η) ≤
1 3l(x, y),
∂φ
∂η(x, y, η) ≤
1 3l(x, y) for all ω ∈ Ω, η ∈ Λ and u ∈ C(D, Rk).
(C3) For a.e. (x, y) ∈ I and all ˜v, ˜w∈ Ω, v, w ∈ Λ such that ˜v ≤ ˜w, v≤ w we have f (x, y, ˜w, w) + φ(x, y, w)≥ f(x, y, ˜v, v) + φ(x, y, v)+
(6) ∂f
∂ω(x, y, ˜v, v)( ˜w− ˜v) +
∂f
∂η(x, y, ˜v, v) +∂φ
∂η(x, y, v)
(w − v),
(7) φ(x, y, w)≥ φ(x, y, v) +∂φ
∂η(x, y, v)(w − v).
(C4) Uniform convergence ωn → ω in C(D, Rk) and ηn→ η in C(I, Rk) implies
∂f
∂ω(x, y, ωn, ηn)u → ∂f
∂ω(x, y, ω, η)u, ∂f
∂η(x, y, ωn, ηn) →∂f
∂η(x, y, ω, η),
∂φ
∂η(x, y, ηn) → ∂φ
∂η(x, y, η), a.e. in I, for all u ∈ C(D, Rk).
(C5) For a.e. (x, y) ∈ I and all η ∈ Λ operator ∂f∂ω(x, y, ω, η) from Ω into R is positive in the sense that u ≥ 0 implies ∂ω∂f(x, y, ω, η)u ≥ 0 for a.e. (x, y) ∈ I.
(C6) If ω1, ω2, ∈ Ω, η1, η2 ∈ Λ, ω1 ≤ ω2, η1≤ η2 and u ∈ C(D, Rk) is such that u≥ 0 then
∂f
∂ω(x, y, ω1, η1)u ≤ ∂f
∂ω(x, y, ω2, η2)u, ∂φ
∂η(x, y, η1) ≤∂φ
∂η(x, y, η2),
∂f
∂η(x, y, ω1, η1) +∂φ
∂η(x, y, η1) ≤ ∂f
∂η(x, y, ω2, η2) +∂φ
∂η(x, y, η2) for a.e. (x, y) ∈ I.
(C7) There is a function m ∈ L1(I, R) such that (8) ∂fi
∂ωj(x, y, ˜ω, ˜η) − ∂fi
∂ωj
(x, y, ω, η) ≤ m(x, y)
||˜ω − ω||0+ |˜η − η|,
(9) ∂ ˜fi
∂ηj(x, y, ˜ω, ˜η) − ∂ ˜fi
∂ηj
(x, y, ω, η)
≤ m(x, y)
||˜ω − ω||0+ |˜η − η|,
(10)
∂φi
∂ηj(x, y, ˜η) −∂φi
∂ηj
(x, y, η)
≤ m(x, y)|˜η − η|
for all i, j = {1, . . . , k}.
Remark 3.1 We note that if for a.e. (x, y)∈ I the function f is convex in ω and the function f + φ is convex in η, that is,
f (x, y, ˜w, v)≥ f(x, y, ˜v, v) + ∂f
∂ω(x, y, ˜v, v)( ˜w− ˜v), f (x, y, ˜w, w) + φ(x, y, w)≥ f(x, y, ˜w, v) + φ(x, y, v)
+∂f
∂η(x, y, ˜w, v) + ∂φ
∂η(x, y, v) (w − v)
for all ˜v, ˜w∈ Ω, v, w ∈ Λ such that ˜v ≤ ˜w, v≤ w, then condition (6) is satisfied.
Now, we define the sequences {vn}, {wn} by
(11)
( ∂2vn+1
∂x∂y (x, y) = F (x, y, vn+1; vn, wn) a.e. in I, vn+1(x, y) = ψ(x, y) on I0, and
(12)
( ∂2wn+1
∂x∂y (x, y) = G(x, y, wn+1; vn, wn) a.e. in I, wn+1(x, y) = ψ(x, y) on I0,
where for each n = 0, 1, . . . ,
(13) F (x, y, u; vn, wn) = f(x, y, v(x,y)n , vn(x, y))+
A(x, y, v˜ n)(u(x,y)− v(x,y)n ) + A(x, y, vn, wn)(u(x, y) − vn(x, y)),
(14) G(x, y, u; vn, wn) = f(x, y, wn(x,y), wn(x, y))+
A(x, y, v˜ n)(u(x,y)− wn(x,y)) + A(x, y, vn, wn)(u(x, y) − wn(x, y)), A(x, y, v˜ n) = ∂f
∂ω(x, y, v(x,y)n , vn(x, y)), A(x, y, vn, wn) = ∂ ˜f
∂η(x, y, v(x,y)n , vn(x, y)) −∂φ
∂η(x, y, wn(x, y)).
Here ˜f = f + φ.
Assumption 8 ˜A ={˜aij}, A = {aij} are k × k matrices such that ˜aij(x, y, v0) ≥ 0 for all i, j ∈ {1, . . . , k} and aij(x, y, v0, w0) ≥ 0 for i 6= j. Moreover, the function l∈ L(I, R) is such that
aii(x, y, v0, w0) ≥ −l(x, y) +Z x 0
l(z, y)dz Z y
0
l(x, z)dza.e. in I.
Remark 3.2 We note that from (C2) we have
|˜aij(x, y, v0)u| ≤ l(x, y)||u||0, |aij(x, y, v0, w0)| ≤ l(x, y), for all i, j ∈ {1, . . . , k} and u ∈ C(D, Rk).
Theorem 3.3 Suppose that assumptions 1-8 are satisfied, the function f is nonde- creasing in ω and quasimonotone nondecreasing in η, the function φ is nondecreas- ing in η for (x, y) ∈ I, ω ∈ Ω and η ∈ Λ. Moreover, the sequences {vn}, {wn} are defined by (11), (12). Then vn,wn → u in C(I, Rk), where u is the unique solution of problem (1) and the convergence is quadratic.
Proof From (4) and (13) we see that
∂2v0
∂x∂y(x, y) ≤ f(x, y, v0(x,y), v0(x, y)) ≡ F (x, y, v0; v0, w0).
From (5), (6), (13) and the fact that v0≤ w0 we have
∂2w0
∂x∂y(x, y) ≥ f(x, y, w(x,y)0 , w0(x, y)) ≥ f(x, y, v0(x,y), v0(x, y))
+φ(x, y, v0(x, y)) − φ(x, y, w0(x, y)) + ∂f
∂ω(x, y, v(x,y)0 , v0(x, y))(w0(x,y)− v(x,y)0 ) +
∂f
∂η(x, y, v0(x,y), v0(x, y)) +∂φ
∂η(x, y, v0(x,y), v0(x, y))
(w0(x, y) − v0(x, y)).
By the mean valued theorem, monotonicity of φ and the fact that v0 ≤ w0 we have
φ(x, y, v0(x, y)) − φ(x, y, w0(x, y))
= −Z 1 0
∂φ
∂η(x, y, tw0(x, y) + (1 − t)v0(x, y))(w0(x, y) − v0(x, y))dt
≥ −∂φ
∂η(x, y, w0(x, y))(w0(x, y) − v0(x, y)), and consequently,
∂2w0
∂x∂y(x, y) ≥ F (x, y, w0; v0, w0).
From (C5) we see that F (x, y, u; v0, w0) is nondecreasing and linear in u for a.e.
(x, y) ∈ I. Now, from (C2) and assumption 8 we see that F (x, y, v1; v0, w0) satisfies assumptions of theorem 2.3. Consequently, theorem 2.3 gives the existence of the unique solution v1∈ AC(I, Rk) of the problem
(15)
( ∂2v1
∂x∂y(x, y) = F (x, y, v1; v0, w0) a.e. in I, v1(x, y) = ψ(x, y) on I0,
such that v0 ≤ v1 ≤ w0 on I. A similar argument shows that w1 ∈ AC(I, R) is unique solution of the problem
(16)
( ∂2w1
∂x∂y(x, y) = G(x, y, w1; v0, w0) a.e. in I, w1(x, y) = ψ(x, y) on I0, such that v0≤ w1≤ w0on I.
Now, we shall show that v1≤ w1 on I. We note that from (6), (7), (C6) and the fact that v0≤ w1≤ w0on I we get
∂2w1
∂x∂y(x, y) = G(x, y, v1; v0, w0) ≥ f(x, y, w1(x,y), w1(x, y)) +φ(x, y, w1(x, y)) − φ(x, y, w0(x, y)) + ∂f
∂ω(x, y, w1(x,y), w1(x, y))(w(x,y)0 − w(x,y)1 ) +
∂f
∂η(x, y, w1(x,y), w1(x, y)) + ∂φ
∂η(x, y, w1(x, y))
(w0(x, y) − w1(x, y)) + ˜A(x, y, v0)(w1(x,y)− w(x,y)0 ) + A(x, y, v0, w0)(w1(x, y) − w0(x, y))
= f(x, y, w1(x,y), w1(x, y))
− Z 1
0
∂φ
∂η(x, y, tw0(x, y) + (1 − t)w1(x, y))(w0(x, y) − w1(x, y))dt +∂f
∂ω(x, y, w(x,y)1 , w1(x, y)) − ∂f
∂ω(x, y, v(x,y)0 , v0(x, y))
(w0(x,y)− w1(x,y))
+∂f
∂η(x, y, w(x,y)1 , w1(x, y)) +∂φ
∂η(x, y, w1(x, y))
−∂f
∂η(x, y, v0(x,y), v0(x, y)) −∂φ
∂η(x, y, v0(x, y))
(w0(x, y) − w1(x, y))
+∂φ
∂η(x, y, w0(x, y))(w0(x, y) − w1(x, y))
≥ f(x, y, w(x,y)1 , w1(x, y)) +
∂φ
∂η(x, y, w0(x, y))
−∂φ
∂η(x, y, w1(x, y))
(w0(x, y) − w1(x, y)) ≥ f(x, y, w1(x,y), w1(x, y)).
Similar arguments to those above show that
∂2v1
∂x∂y(x, y) ≤ f(x, y, v(x,y)1 , v1(x, y)).
By the above,
∂2v1
∂x∂y(x, y) ≤ f(x, y, v(x,y)1 , v1(x, y)) a.e. in I,
∂2w1
∂x∂y(x, y) ≥ f(x, y, w1(x,y), w1(x, y)) a.e. in I, v1(x, y) = ψ(x, y) = w1(x, y) on I0.
As v1(x, y) = ψ(x, y) = w1(x, y) we have∂v∂x1(x, 0) = ∂ψ∂x(x, 0) = ∂w∂x1(x, 0), ∂v∂y1(0, y) =
∂ψ
∂y(0, y) = ∂w∂y1(0, y) and finally from theorem 2.2 we get v1≤ w1on I. Summarize, we have
(17) v0≤ v1≤ w1≤ w0 on I.
From (C6) it is easily seen that aij(x, y, vn+1, wn+1) ≥ aij(x, y, vn, wn) for all i, j ∈ {1, . . . , k} and n = 0, 1, . . . . Now, analysis similar to that in the proof of inequalities (17) gives an implication if
vn−1≤ vn≤ wn≤ wn−1 on I, for some n ≥ 1, then
vn ≤ vn+1≤ wn+1≤ wn on I.
Hence, we have by induction
(18) v0≤ v1≤ . . . vn≤ wn≤ . . . w1≤ w0 on I,
for all n = 0, 1, . . . .
From (3) and (18) we have that
|f(x, y, ω, η)| ≤ ˜m(x, y) for all ω∈ Ω and η ∈ Λ, where ˜m(x, y) = ˜l(x, y)(1 +||w(x,y)0 ||0+ |w0(x, y)|).
For all n = 0, 1, . . . from the above and (C2) obtaining
|F (x, y, vn+1; vn, wn)| ≤ M(x, y),
|G(x, y, wn+1; vn, wn)| ≤ M(x, y), where M(x, y) = m(x, y) + l(x, y)( w(x,y)0 − v0(x,y)
0+ w0(x, y) − v0(x, y) ).
Of course, problems (11) and (12) are equivalent to problems
vn+1(x, y) = ψ(x, 0) + ψ(0, y) − ψ(0, 0) +Z x 0
Z y 0
F (s, t, vn+1; vn, wn)dsdt on I,
vn+1(x, y) = ψ(x, y) on I0, wn+1(x, y) = ψ(x, 0) + ψ(0, y) − ψ(0, 0) +Z x
0
Z y 0
G(s, t, wn+1; vn, wn)dsdt on I, vn+1(x, y) = ψ(x, y) on I0,
respectively.
Therefore, from the Lebesgue dominated convergence theorem we obtain vn → ρ, wn→ r uniform in C(I, Rk), where ρ,r ∈ AC(I, Rk) and ρ, r are solutions of (1).
Since ρ ≤ r, taking v = r, w = ρ and using theorem 2.2 we obtain r ≤ ρ on I, proving r = ρ = u is the unique solution of (1).
We shall show quadratic convergence of {vn}, {wn}, to the unique solution of (1).
Put pn+1= u − vn+1, qn+1= wn+1− u. Then, by the mean valued theorem and (C7), we have
∂2pn+1
∂x∂y (x, y) = ˜f (x, y, u(x,y), u(x, y))− ˜f (x, y, v(x,y)n , vn(x, y))
− ˜A(x, y, vn)(v(x,y)n+1 − v(x,y)n ) − A(x, y, vn, wn)(vn+1(x, y) − vn(x, y))
−φ(x, y, u(x, y)) + φ(x, y, vn(x, y))
=Z 1 0
∂f
∂ω x, y, tu(x,y)+ (1 − t)vn(x,y), u(x, y) pn(x,y)dt
+Z 1 0
∂ ˜f
∂η x, y, v(x,y)n , tu(x, y) + (1− t)vn(x, y)
pn(x, y)dt
− ˜A(x, y, vn) − pn+1(x,y)+ pn(x,y)
− A(x, y, vn, wn) − pn+1(x, y) + pn(x, y)
− Z 1
0
∂φ
∂η(x, y, tu(x, y) + (1 − t)vn(x, y))pn(x, y)dt
=Z 1 0
∂f
∂ω x, y, tu(x,y)+ (1 − t)v(x,y)n , u(x, y)
−∂f
∂ω x, y, v(x,y)n , vn(x, y) pn(x,y)dt
+Z 1 0
∂ ˜f
∂η x, y, vn(x,y), tu(x, y) + (1− t)vn(x, y)
−∂ ˜f
∂η x, y, vn(x,y), vn(x, y)
pn(x, y)dt
+∂f
∂ω(x, y, vn(x,y), vn(x, y))pn+1(x,y)+∂ ˜f
∂η(x, y, v(x,y)n , vn(x, y))pn+1(x, y)
−∂φ
∂η(x, y, wn(x, y))pn+1(x, y) +Z 1
0
∂φ
∂η x, y, wn(x, y)
−∂φ
∂η x, y, tu(x, y) + (1− t)vn(x, y)
pn(x, y)dt.
For all i, j ∈ {1, . . . , k} from (8) obtaining
∂fi
∂ωj x, y, tu(x,y)+ (1 − t)v(x,y)n , u(x, y)
− ∂fi
∂ωj x, y, vn(x,y), vn(x, y)
≤ m(x, y)
t pn(x,y) 0+ pn(x, y)
≤ (1 + t)m(x, y) pn(x,y) 0. For all i, j ∈ {1, . . . , k} from (9) obtaining
∂ ˜fi
∂ηj
x, y, vn(x,y), tu(x, y) + (1− t)vn(x, y)
− ∂ ˜fi
∂ηj
x, y, v(x,y)n , vn(x, y)
≤ tm(x, y) pn(x, y) . For all i, j ∈ {1, . . . , k} from (C6) and (9) obtaining
∂φi
∂ηj x, y, wn(x, y)
−∂φi
∂ηj x, y, tu(x, y) + (1− t)vn(x, y)
≤∂φi
∂ηj
x, y, wn(x, y)
−∂φi
∂ηj
x, y, vn(x, y)
≤ m(x, y) pn(x, y) + qn(x, y) .
From (C2) and the fact that pn+1(x, y) ≥ 0 on I∗ it follows that there exists a function l ∈ L1(I, R) such that
∂fi
∂ωj x, y, v(x,y)n , vn(x, y)
pn+1(x,y)≤ l(x, y) pn+1(x,y)
0,
∂ ˜fi
∂ηj
x, y, v(x,y)n , vn(x, y)
≤ l(x, y),
∂φi
∂ηj
x, y, wn(x, y)
≤ 1 3l(x, y), for all i, j ∈ {1, . . . , k}.
We conclude from the above that there exists a function n ∈ L1(I, R) such that
(19) ∂2pn+1
∂x∂y (x, y) ≤ N(x, y) pn+1(x,y)
0+ pn+1(x, y) + + pn 2+ qn 2
a.e. in I, where N(x, y) =
n(x, y)
k×1.
Moreover, pn+1(x, y) = 0 on I0 and ∂p∂xn+1(x, 0) = ∂p∂yn+1(0, y) = 0 for x ∈ [0, a], y ∈ [0, b]. Let Q = 2 pn 2 + qn 2 and define s : I∗ → R by the formula s(x, y) = Q on I0 and s(x, y) = Qe3R0xR0yn(s,t)dsdt+x+y on I. Moreover, define S : I∗→ Rk by the formula S(x, y) = [s(x, y)]k×1 then it is easy to check that
(20) ∂2S
∂x∂y(x, y) ≥ 3n(x, y)S(x, y) ≥ N(x, y) s(x,y) 0+ s(x, y) +2 pn 2+ qn 2 a.e. in I.
(21) pn+1(x, y) ≤ s(x, y) on I0,
(22) ∂pn+1
∂x (x, 0) ≤ ∂s
∂x(x, 0), ∂pn+1
∂y (0, y) ≤ ∂s
∂y(0, y), for x ∈ [0, a] and y ∈ [0, b].
From (19)-(22) and theorem 2.1 we conclude that pn+1(x, y) ≤ s(x, y) on I. There- fore there is a constant C > 0 such that pn+1
≤ CQ = C2 pn 2+ qn 2. Analysis similar to that in the above shows that qn+1
≤ C
2 qn 2+ pn 2 which proves the quadratic convergence of vn, wn → u. ,
References
[1] E. Berkson and T. A. Gillespie, Absolutely continuous functions of two variables and well- bounded operators, J. Math. Soc. 30 (1984), 305–321.
[2] S. Brzychczy and J. Janus, Monotone iterative methods for nonlinear integro-differential hyperbolic equations, Univ. Iagell. Acta Math. 37 (1999), 246-261.
[3] S. Brzychczy and J. Janus, Monotone iterative methods for nonlinear hyperbolic differential- functional equations, Univ. Iagell. Acta Math. 38 (2000), 141-152.
[4] T. Cz lapi´nski, Iterative methods for Darboux problem for partial functional differential equa- tions, J. of Inequal. & Appl. 4 (1998), 141-161.
[5] T. Cz lapi´nski, Hyperbolic functional differential equations, Wydawnictwo Uniwersytetu Gda´nskiego, Gda´nsk, 1999.
[6] T. Jankowski, Functional differential equations, Czechoslovak Math. Jour. 52 (2000), 553- 563.
[7] A. Karpowicz, Monotone methods for hyperbolic functional differential equations, to appear.
[8] A. Karpowicz, The Darboux problem for hyperbolic functional differential equations and in- equalities in the sense of Carath´eodory, to appear.
[9] A. Karpowicz, Carath´eodory Solutions of Hyperbolic Functional Differential Inequalities with First Order Derivatives, Ann. Pol. Math. 94 (2008), 53-78.
[10] V. Lakshmikantham and A. S. Vatsala, Generalized Quasilinearization for Nonlinear Prob- lem, Kluwer Academi Publishers, The Netherlands, 1998.
[11] V. Lakshmikantham and S. K¨oksal, Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations, Taylor & Francis, London, 2003.
[12] V. Lakshmikantham and N. Shazad, Further generalization of generalized quasilinearization methods, J. of Appl. Math. and Stochastic Anal. 4 (1994), 545-552.
[13] R. N. Mohapatra, K. Vajravelu and Y. Yin, Extension of the method of quasilinearization and rapid convergence, J. of Optimization Theory and Appl. 96 (1998), 667-682.
[14] S. G Pandit, Monotone methods for systems of nonlinear hyperbolic problems in two inde- pendet variables, Nonlinear Anal., Theory, Meth. & Appl. 5 (1997), 2735-2742.
Adrian Karpowicz
Institute of Mathematics, University of Gda´nsk Wit Stwosz St. 57, 80-952 Gda´nsk, Poland E-mail: akarpowi@math.univ.gda.pl
(Received: 15.07.2008)
