• Sonuç bulunamadı

Approximate solution of the double nonlinear singular integral equations with Hilbert Kernel by the method of contractive mappings

N/A
N/A
Protected

Academic year: 2021

Share "Approximate solution of the double nonlinear singular integral equations with Hilbert Kernel by the method of contractive mappings"

Copied!
7
0
0

Yükleniyor.... (view fulltext now)

Tam metin

(1)

Selçuk J. Appl. Math. Selçuk Journal of Vol. 13. No. 1. pp. 3-9, 2012 Applied Mathematics

Approximate Solution of the Double Nonlinear Singular Integral Equations with Hilbert Kernel by the Method of Contractive Mappings

Nushaba F. Gasimova

Ministry of Education of Azerbaijan Republic, Khatai(Xatai) prospect, 49, AZ 1008, Azerbaijan

e-mail: enusab e@ yaho o.com

Received Date: September 7, 2010 Accepted Date: March 8, 2011

Abstract. In this paper the double nonlinear singular integral equations with Hilbert kernel are solved by contractive mappings method and the rate of con-vergence of sequential approximations to exact solution is found.

Key words: Approximate solution; Singular integral equations; Bicylindrical domain; Superposition; Cfontractive mappings

2000 Mathematics Subject Classification: 45G05. 1. Introduction

Some notations and auxiliary facts

Let’s consider the following double nonlinear singular integral equation (NSIE) of the form (1.1) ϕ(x, y) = λ π Z −π π Z −π F [s, t, ϕ(s, t)]ctgs − x 2 ctg t − y 2 ds dt + f (x, y), where λ is a real parameter, F and f are the given functions, ϕ is the desired function. Equations of the form (1.1) are met by studying limit values on the frames of bicylinder of the function which is analytic in bicylindrical domain [1] and the theory of singular integral equations [2]. In this paper we’ll solve equation (1.1) by the contractive mappings method.

By C(T2) we denote a space of continuous functions on T2= [−π, π] × [−π, π]

and have 2π periodic by each of variables with the norm

(1.2) kfkC(T2)= max

(2)

Let

41,0h f (x, y) = f (x + h, y) − f(x, y), 4 0,1

η f (x, y) = f (x, y + η) − f(x, y),

(1.3) 41,1h,ηf (x, y) = f (x, y) − f(x + h, y) − f(x, y + η) + f(x + h, y + η). These quantities are called partial difference with respect to x with step h, with respect to c with step η and mixed difference in aggregate of variables with step h and η at the point (x, y).

Introduce the denotation: ω1,0f (δ) = sup |h|≤δk4 1,0 h f (x, y)kC(T2), ω0,1 f (η) = sup |h|≤ηk4 0,1 h f (x, y)kC(T2), (1.4) ω1,1f (δ, η) = sup |h1| ≤ δ |h2| ≤ η k41,1h1,h2f (x, y)kC(T2).

By means of these characteristics in the paper [3] we introduce the space (1.5)

Kα,β1,1=nf ∈ C(T2)¯¯¯ω1,0f (δ) = O(δα), ω0,1f (η) = O(ηβ), ω1,1f (δ, η) = O(δα·ηβ)o

0 < α, β ≤ 1 with finite norm

kfkKα,β1,1 = max ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ kfkC(T2), sup δ>0 ω1,0f (δ) δα , supη>0 ω0,1f (η) ηβ , sup δ > 0 η > 0 ω1,1f (δ) δα· ηβ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

and prove that the spaces Kα,β1,1 is a Banach space.

Let f ∈ C(T2). Let’s consider a double singular integral with Hilbert kernel

(1.6) f (x, y) =˜ 1 4π2 π Z −π π Z −π f (s, t)ctgs − x 2 ctg t − y 2 ds dt.

Note that integral (1.6) is understood in the sense of Cauchy’s principal value. From the estimates obtained in the papers [3], [4] it follows that the singular operator (SO)

(1.7) (Sf )(x, y) = ˜f (x, y)

(3)

In the space Kα,β1,1 we take a ball with center at zero of radius R Bα,β1,1(R) =nϕ ∈ Kα,β1,1 | kϕkK1,1

α,β ≤ R

o . The following statement was proved in the paper [5].

Statement 1. Let f ∈ Kα,β1,1 and 1 ≤ p < ∞. Then the inequality

(1.8) kfkC(T2)≤ lkfkγ K1,1α,β· kfk 1−γ Lp , where γ = 1 + p(α + β) (1 + αp)(1 + βp), (1.9) l = max ( (1 + αp)(1 + βp) (αβp)1−γ , p √ 4(1 + αp)(1 + βp) αβpπ1−γ ) is true.

Later on we’ll need the following statements proved in [10].

Statement 2. Let the function F (x, y, ϕ) : T2× [−R, R] → < satisfy the

conditions:

1) there exists a partial derivative F0

ϕ(x, y, ϕ) and there is C0 > 0 such

that for ∀ϕ1, ϕ2∈ [−R, R] |Fϕ0(x, y, ϕ1) − Fϕ0(x, y, ϕ2)| ≤ C0|ϕ1− ϕ2|; 2) ∃C1> 0, ∀x1, x2∈ [−π, π] |F (x1, y, ϕ) − F (x2, y, ϕ)| ≤ C1|x1− x2|α; 3) ∃C2> 0, ∀y1, y2∈ [−π, π] |F (x, y1, ϕ) − F (x, y2, ϕ)| ≤ C2|y1− y2|β; 4) ∃C3> 0, ∀x1, y1, x2, y2∈ [−π, π] |F (x1, y1, ϕ)−F (x1, y2, ϕ)−F (x2, y1, ϕ)+F (x2, y2, ϕ)| ≤ C3|x1−x2|α|y1−y2|β; 5) ∃C4> 0, ∀x1, x2∈ [−π, π], ∀ϕ1, ϕ2∈ [−R, R] |F (x1, y, ϕ1)−F (x1, y, ϕ2)−F (x2, y, ϕ1)+F (x2, y, ϕ2)| ≤ C4|x1−x2|α|ϕ1−ϕ2|; 6) ∃C5> 0, ∀y1, y2∈ [−π, π], ∀ϕ1, ϕ2∈ [−R, R] |F (x, y1, ϕ1) − F (x, y1, ϕ2) − F (x, y2, ϕ1) + F (x, y2, ϕ2)| ≤ C5|y1− y2|β|ϕ1− ϕ2|.

Then the operator of superposition F : ϕ(x, y) → F [x, y, ϕ(x, y)] acts from the ball Bα,β1,1(R) to the ball Bα,β1,1(R1) where radius R1 is uniquely determined by

initial data.

Statement 3.Let the function F (s, t, ϕ) : T2× [−R, R] → < satisfy conditions 1)- 6) and f ∈ Bα,β1,1(R0) (R0< R). Then for (1.10) λ < min ( 1 C∗kSkL 2→L2 , R − R 0 R1· kSkK1,1 α,β→K 1,1 α,β ) , where C∗= max x,y,ϕ|F 0 ϕ(x, y, ϕ)|, the operator (1.11) (Lϕ)(x, y) = λ π Z −π π Z −π F [s, t, ϕ(s, t)]ctgs − x 2 ctg t − y 2 ds dt + f (x, y)

(4)

Is a contractive map in the ball Bα,β1,1(R) in the metric of the space L2(T2).

2. Approximate Solution of NSIE (1.1)

From estimate (1.8) it follows that if a sequence of functions {fn} ⊂ B1,1α,β(R)

converges in the metric of the space L2(T2) to some function f0, it converges to

f0 in the metric of the space C(T2) as well.

It is valid.

Lemma 2.1. If the sequence {fu} ⊂ Bα,β1,1(R) converges in the metric of space

C¡T2¢to f

0, then f0∈ Bα,β1,1(R).

Proof. fn→ f0fn∈ B1,1α,β(R) . Then

(2.1) ∀ε > 0 ∃N(ε) ∀n > N(ε), ∀(x, y) ∈ T2|fn(x, y) − f(x, y)| < ε.

Let’s take arbitrary points (x1, y), (x2, y) ∈ T2 and arbitrary ε0 > 0 and fix

them. Take such ε > 0 that the inequality ε |x1− x2|α < ε0 be fulfilled. Then we have (2.2) |f0(x1,y)−f0(x2,y)| |x1−x2|α =

|f0(x1,y)−fn(x1,y)+fn(x1,y)−fn(x2,y)+fn(x2,y)−f0(x2,y)|

|x1−x2|α ≤ |f0(x1,y)−fn(x1,y)| |x1−x2|α + |fn(x1,y)−fn(x2,y)| |x1−x2|α + |fn(x2,y)−f0(x2,y)| |x1−x2|α < 2ε0+ R. The relation (2.3) f0(x, y1) − f0(x, y2) |y1− y2|β < 2ε0+ R

is proved similarly. Now, let’s fix the points (x1, y1), (x1, y2), (x2, y1), (x2, y2) ∈

T2and ε

0> 0. Take such ε > 0 that the relation

ε |x1− x2|α|y1− y2|β < ε0 be fulfilled. Then (2.4) |f0(x1,y1)−f0(x1,y2)−f0(x2,y1)+f0(x2,y2)| |x1−x2|α|y1−y2|β = (|f0(x1, y1) − fn(x1, y1) +fn(x1, y1) − f0(x1, y2) + fn(x1, y2) − fn(x1, y2) −f0(x2, y1) + fn(x2, y1) − fn(x2, y1) + f0(x2, y2) −fn(x2, y2) + fn(x2, y2)|) / |x1− x2|α|y1− y2|β ≤|f0(x1,y1)−fn(x1,y1)| |x1−x2|α|y1−y2|β + |f0(x1,y2)−fn(x1,y2)| |x1−x2|α|y1−y2|β + |f0(x2,y1)−fn(x2,y1)| |x1−x2|α|y1−y2|β +|f0(x2,y2)−fn(x2,y2)| |x1−x2|α|y1−y2|β + |fn(x1,y1)−fn(x1,y2)−fn(x2,y1)+fn(x2,y2)| |x1−x2|α|y1−y2|β < 4ε0+ R

(5)

It follows from estimates (2.2)-(2.4) that f0 ∈ B1,1α,β(R + 4ε). Since ε0 > 0 is

arbitrary, we get f0∈ B1,1α,β(R). The lemma is proved.

Now, let’s prove the main theorem:

Theorem 2.1.Let the function F (s, t, ϕ) : T2×[−R, R] → < satisfy conditions

1) - 6) and f ∈ B1,1α,β(R0) (R0< R). Then for

|λ| < min ( 1 C∗kSkL 2(T2) , R − R0 R1kSkK1,1α,β→Kα,β1,1 )

NSIE (1.1) has a unique solution ϕ∗ in the ball Bα,β1,1(R) and sequential ap-proximations ϕn = Lϕn−1 converge to this solution in the metric C(T2) with

rate kϕn− ϕ∗kC(T2)≤ M · ωnkϕ1− ϕ0k γ 1+γ L2(T2), where M is a constant, ω = {|λ|C∗kSkL2(T2)} γ 1+γ, γ = min{α, β}.

Proof. Under the conditions of the theorem Lis contractive map in the metric L2(T2). Then by contractive mappings principle we get

(2.6) kϕn− ϕ∗kL2(T 2)≤ 1 1 − ω0 ωn01− ϕ0kL 2(T2), where ω0= |λ|C ∗kSk L2(T2)

Estimate the norm kϕn− ϕ∗kC2(T2)by the norm kϕn− ϕ

k

L2(T2). By B((x, y); h)

we denote a circle of radius h > 0 and center at the point (x, y) ∈ T2. Later on,

let V2= V2h(x, y) = T2∩ B((x, y); h). It is clear that for the function g ∈ C(T2)

it holds the representation [9]:

g (x, y) = 1 mesV2 Z V2 Z g (s, t) ds dt −mesV1 2 Z V2 Z [g (s, t) − g (x, y)] ds dt Having taken g(x, y) = ϕ∗(x, y) − ϕ n(x, y) we get: (2.7) ϕ∗(x, y) − ϕ n(x, y) = 1 mesV2 R V2 R [ϕ∗(s, t) − ϕ n(s, t)]ds dt −mesV1 2 R V2 R [ϕ∗(s, t) − ϕ(x, y) − ϕ n(s, t) + ϕn(x, y)]ds dt

(6)

Since ϕ∗ϕn∈ Bα,β1,1(R), we have |ϕ∗(s, t) − ϕ(x, y) − ϕ n(s, t) + ϕn(x, y)| ≤ ≤ |ϕ∗(s, t) − ϕ(s, y) − ϕ(x, t) + ϕ(x, y)| + |ϕ∗(s, y) − ϕ∗(x, y)| + |ϕ∗(x, t) − ϕ∗(x, y)| − |ϕn(s, t) + ϕn(s, y) − ϕn(x, t) + ϕn(x, y)| + |ϕn(s, y) − ϕn(x, y)| + |ϕn(x, y) − ϕn(x, y)| ≤ 2M1(|s − x|α|t − y|β+ |s − x|α+ |t − y|β) ≤ 2M2hγ.

And here M2 is a constant and γ = min {α, β} . Then it follows from (2.7) that

(2.8) (x, y) − ϕn(x, y)| ≤ 1 mesV2 An+ M2hγ≤ M3h−1An+ M2hγ, where An = kϕn− ϕ∗kL2(T2). If we take h = A 1 1+γ n , we have from (2.8) |ϕ∗(x, y) − ϕ n(x, y)| ≤ M4A γ 1+γ n = M4kϕn− ϕ∗k γ 1+γ L2(T2)⇒ ⇒ |ϕ∗− ϕ n|C(T2)≤ M4n− ϕ∗k γ 1+γ L2(T2).

Taking into account the last inequality and taking M = M4

µ 1 1 − ω0 ¶ γ 1+γ we get the affirmation of the theorem.

The theorem is proved. References

1. Kakichev V.A., Schwartz integral and Hilbert formula for analytical functions of many complex variables, Izv. Vuzov. matem., No 2, (1959), 80-93.

2. Gabdulkhaev B.G., Approximate solutions of multidimensional singular integral equations, II. Izv. Vuzov. matem., No 1, (1976), 30-41.

3. Musayev B.I., Salayev V.V., On conjugated functions of many groups of variables, II. Uchen.zap. Azerb. Univ. Ser. Fiz. Mat., No 4, (1978), 5-18.

4. Jvarsheishvili A.G., On A.Zigmund’s inequality for functions of two variables, Soobsh.AN. Gruz. SSR., v.15, No 9, (1954), 9-15.

5. Abdullayev F.A., On some multiplicative inequalities for functions of two variables, Azerb. Resp. "Tahsil" , No 2, (2004), 56-60 (Russian).

6. Zhizhiashvili L.V., On some problems from the theory of prime and multiple trigono-metric and orthogonal series, Uspekhi mat. Nauk, v.28, No 2, (1973), 65-119.

7. Babayev A.A., On structure of a linear operator and its application, Uchen.Zap.AGU, ser.fiz.-mat.nauk, No 4, (1961), 13-16.

8. Huseynov A.I., Mukhtarov Kh.Sh., Introduction to the theory of nonlinear singular integral equations, "Nauk" (1979).

(7)

9. Il’in V.P. Some inequalities in functional spaces and their application to research of convergence of variational processes, Trudy Inst. Matem. AN SSSR. v.53, (1959), 64-121.

10. Gasimova N.F. On a nonlinear singular integral operator with Hilbert kernel, Baku Universitetinin Khabarleri. Fiz.mat.ser., No 4, (2005), 56-63.

Referanslar

Benzer Belgeler

We also propose two different energy efficient routing topology construction algorithms that complement our sink mobility al- gorithms to further improve lifetime of wireless

Hence the focus on the shifting landscapes of postapartheid Johannesburg, the locale of the texts I discuss below: Ivan Vladislavic’s experimental urban chronicle Portrait with

4.7 The effect of palmitoleate on palmitate-induced inactivation of 5 ’ AMP activated protein kinase...54 4.8 The effect of PERK and IRE1 branches of the Unfolded Protein Response

Unlike the late Ottoman period, the early Republic was characterized by civilian supremacy and the relegation of the military into a secondary position vis-a`-vis the..

Tam köprü yumuşak anahtarlamalı güç kaynağına ait çıkış gerilimi ve doğrultucu çıkışı 75KHz anahtarlama frekansı ve 100V giriş gerilimi için şekil

This study was conducted to compare the cyclic fatigue resistance of VDW.ROTATE, TruNatomy Prime, HyFlex CM, and 2Shape nickel-titanium (NiTi) rotary instruments in double-

Baseline scores on the QLQ-C30 functioning scales from patients in both treat- ment arms were comparable to available reference values for patients with ES-SCLC; however, baseline

In a trial conducted by Metcalfe (16) et al., rate of ath- erosclerotic renal artery disease in patients with PAD in ≥ 3 segments (43,4%) was found to be higher than in patients