On the-approximate-solution-of-a-nonlinear-singular-integral-equation

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On the-approximate-solution-of-a-nonlinear-singular-integral-equation

  1. 1. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational Science and Engineering Vol:3 No:11, 2009 On the Approximate Solution of a Nonlinear Singular Integral Equation Nizami Mustafa, Cemal Ardil AbstractIn this study, the existence and uniqueness of the solution of a nonlinear singular integral equation that is defined on a region in the complex plane is proven and a method is given for finding the solution. KeywordsApproximate solution, Fixed-point principle, Nonlinear singular integral equations, Vekua integral operator International Science Index 35, 2009 waset.org/publications/358 I. INTRODUCTION A S it is known, the application area of nonlinear singular integral equation is so extensive such as the theories of elasticity, viscoelasticity, thermo elasticity, hydrodynamics, fluid mechanics and mathematical physics and many other fields [1]-[6]. Furthermore, the solution of the seismic wave equation that has a great importance in elastodynamics is investigated by reducing it to the solution of nonlinear singular integral equation by using Hilbert transformation [7]. All of these studies contribute to update of investigations about the solution of the nonlinear singular integral equation by using approximate and constructive methods. In this study, the approximate solution of nonlinear singular integral equations that is defined on a bounded region in complex plane is discussed. It is known that the investigation of the problems of the Dirichlet boundary-value problems for many nonlinear differential equation systems which have partial derivatives and defined on a planar region G ⊂ ℂ can be reduced to the investigation of the problem of a nonlinear singular integral equation which is in the following form [8], [9], [10]-[14] o ∂G is the boundary of the region G, G is the set of interior points of G and D = {( z , ϕ , t , s ) : z ∈ G , ϕ , t , s ∈ ℂ} = G × ℂ 3 are as given, f , g : D0 → ℂ and F : D → ℂ are the known functions, is any scalar parameter, λ ∈ℝ h ∈ H α (G )(0 < α < 1) and for z = x + iy, ζ = ξ + iη followings are the known Vekua integral operators [15]. TG h(.)( z ) = (−1 / π ).∫∫ h(ζ ).(ζ − z ) −1 dξdη , G Π G h(.)( z ) = (−1 / π ) ∫∫ h(ζ ).(ζ − z ) − 2 dξdη G In this study, by taking base, a more useful modified variant of Schauder and Banach fixed point principles, the existence and uniqueness of the solution of the Equation (1) is proven with more weak conditions on the functions f , g and F. Furthermore, iteration is given for the approximate solution of the Equation (1) and it is shown that the iteration is converging to the real solution. II. BASIC ASSUMPTIONS AND AUXILIARY RESULTS Further, throughout the study, if not the opposite said we take the set G ⊂ ℂ as bounded and simple connected region. o For G is to be the set of interior points of the region G and o ϕ ( z) = (1) λ.F ( z , ϕ ( z ), TG f (., ϕ (.))( z ), Π G g (., ϕ (.))( z )). ∂G is to be the boundary of G let G = ∂G U G . If, for every z1 , z 2 ∈ G there exist H > 0 and α ∈ (0,1] numbers such that ϕ ( z1 ) − ϕ ( z 2 ) ≤ H . z1 − z 2 Here, let o   D0 = ( z , ϕ ) : z ∈ G = ∂G U G, ϕ ∈ ℂ   = G ×ℂ then it is said that the function α ϕ :G → ℂ satisfies the Holder condition on the set G with exponent α . We will show the set of all functions that satisfies Holder condition on the set G with exponent α with H α (G ) . Nizami Mustafa is with the Mathematics Department, University of Kafkas, Kars, 36100 Turkey (corresponding author to provide phone: +90474-2128898; fax +90-474-2122706; e-mail nizamimustafa@mynet.com) Cemal Ardil is with the National Academy Aviation, AZ 1056 Baku, Azerbaijan (e-mail: cemalardil@gmail.com). For (H 20 α ϕ ∈ H α (G ) ) and 0 < α < 1 , the vector space (G ); . α is a Banach space with the norm
  2. 2. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational Science and Engineering Vol:3 No:11, 2009 ϕ Here, ϕ = ϕ α ∞ Hα (G ) ≡ ϕ ∞ + H (ϕ , α ; G ). ϕ = max{ϕ ( z ) : z ∈ G }, H (ϕ , α ; G ) = sup z1 , z 2 ∈G , z1 ≠ z 2 where {ϕ ( z ) − ϕ ( z ) } 1 2   ∫∫ ϕ (ζ ) p G 1 < p, the vector space p with the following norm, ϕ Further, p = ϕ L p (G )   p ≡  ∫∫ ϕ (ζ ) dξdη    G  throughout the study International Science Index 35, 2009 waset.org/publications/358 + m 2 ϕ1 − ϕ 2 l1 z1 − z 2 (5) 2 2 +αp α and .ϕ p > 1 the αp 2 +αp p . (6) p ) = max{M 1 (α , p ), M 2 (α , p)} , −1 p M 1 (α , p ) = 2.( n(α , p )) + (π .( n(α , p )) ) , take (2) 2 M 2 (α , p ) = (2. 4 ).( 4 − 1) −1 .(n(α , p ))α , n(α , p ) = (α . p. π ) p −p 2 +α . p Lemma 2.3. For D0 R = {( z , ϕ ) ∈ D0 : z ∈ G , ϕ ≤ R}, R > 0, Bα (0; R) = { ∈ H α (G ) : ϕ ≤ R}, α ∈ (0,1) ϕ and ϕ ∈ Bα (0; R ) , f , g : D0 R → ℂ let f1 ( z ) = f ( z , ϕ ( z )), g 1 ( z ) = g ( z , ϕ ( z )), z ∈ G . Then for the functions f 1 , g1 : G → ℂ the following inequalities are true f 1 ( z ) ≤ m 0 + m 2 . R, z ∈ G , α f 1 ( z1 ) − f 1 ( z 2 ) ≤ (m1 + m2 .R). z1 − z 2 , (7) (3) + n 2 ϕ1 − ϕ 2 z 1 , z 2 ∈ G, g1 ( z ) ≤ n0 + n2 .R, z ∈ G, F ( z1 , ϕ1 , t1 , s1 ) − F ( z 2 , ϕ 2 , t 2 , s 2 ) ≤ α ∞ p g ( z1 , ϕ1 ) − g ( z 2 , ϕ 2 ) ≤ α ≤ M (α , p ). ϕ p f ( z1 , ϕ 1 ) − f ( z 2 , ϕ 2 ) ≤ n1 z1 − z 2 Here, M (α , . following inequalities be satisfied m1 z1 − z 2 ϕ α ∈ (0,1) α D = {( z , ϕ , t , s ) : z ∈ G , ϕ , t , s ∈ ℂ} = G × ℂ 3 . Besides, for every and z1 , z 2 ∈ G ( z k , ϕ k ) ∈ D0 , ( z k , ϕ k , t k , s k ) ∈ D , for k = 1,2 we suppose that the number α ∈ (0,1) and the positive constants m1 , m 2 , n1 , n 2 , l1 , l 2 , l 3 , l 4 are exist such that α + (πε 2 ) −1 / p . ϕ p , 1/ p we D0 = {( z , ϕ ) : z ∈ G , ϕ ∈ ℂ} = G × ℂ and α d = sup{ z1 − z 2 : z1 , z 2 ∈ G}.  dξdη < +∞ ,  (L (G ); . ) is a Banach space p ≤ 2.ε α . ϕ Lemma 2.2. For ϕ ∈ H α (G ) , following inequality is held are as given. For L p (G ) = ϕ : G → ℂ : ∞ + l 2 ϕ1 − ϕ 2 + α g1 ( z1 ) − g 2 ( z 2 ) ≤ (n1 + n2 .R ). z1 − z 2 , (8) (4) z 1 , z 2 ∈ G. l 3 t1 − t 2 + l 4 s1 − s 2 Here, m0 = max{ f ( z ,0) : z ∈ G }, We will denote the set of functions that satisfy the conditions (2), (3) and (4) with H α ,1 (m1 , m2 ; D0 ), H α ,1 (n1 , n2 ; D0 ) and H α ,1,1,1 (l1 , l 2 , l 3 , l 4 ; D) respectively. Now, let us give some supplementary lemmas for the theorems about the existence and uniqueness of the solution of the Equation (1). Lemma 2.1. If ϕ ∈ H α (G ) , α ∈ (0,1) then for every n0 = max{ g ( z ,0) : z ∈ G } . The proof of the Lemma 2.3 is obvious from the Inequalities (2), (3) and the assumptions of the lemma. Corollary 2.1. If the assumptions of the Lemma 2.3 are satisfied then for, f 1 , g 1 ∈ H α (G ), p > 1 and ε ∈ (0, d ) the following inequality is held 21 α ∈ (0,1) we have
  3. 3. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational Science and Engineering Vol:3 No:11, 2009 f1 α ≤ m0 + m1 + 2 Rm 2 , g1 α ≤ n0 + n1 + 2 Rn2 . F ( z ) ≤ l 0 + l 2 .R + l3 .L1 + l 4 .L2 , z ∈ G , (9) F ( z1 ) − F ( z 2 ) ≤ (l1 + l 2 .R + l 3 .L1 + l 4 .L2 ) × α Now, for f ∈ H α ,1 (m1 , m2 ; D0 ), g ∈ H α ,1 (n1 , n2 ; D0 ) and ϕ ∈ H α (G ) , α ∈ (0,1) let us define the functions ~ ~ f 1 , g 1 : G → ℂ as given below ~ f 1 ( z ) = TG f (., ϕ (.))( z ), ~ g ( z ) = Π g (., ϕ (.))( z ). 1 z1 − z 2 , z1 , z 2 ∈ G . Here, l 0 = F (.,0,0,0) ∞ . The proof of the Lemma 2.5 is obvious from the definition of (10) the function F : G → ℂ , the Inequality (13) and from the assumption of lemma. G Corollary 2.2. If the assumptions of the Lemma 2.5 are For the bounded operators TG , Π G : H α (G ) → H α (G ) , α ∈ (0,1) the following norms: TG International Science Index 35, 2009 waset.org/publications/358 ΠG α α { ≡ sup TG ϕ α : ϕ ∈ H α (G ), ϕ α { α : ϕ ∈ H α (G ), ϕ α F } ≤1 (11) ≡ sup Π Gϕ satisfied then for F ∈ H α (G ) we have let us define } ≤1 α ≤ L = 2{max(l 0 , l1 ) + l 2 .R + l3 .L1 + l 4 .L2 } . Corollary 2.3. If the assumptions of the Lemma 2.5 are satisfied then the operator A that is defined with A(ϕ )( z ) = (12) [11], [14]. λ.F ( z , ϕ ( z ), (TG o f1 )( z ), (Π G o g1 )( z ), z ∈ G Lemma 2.4. If (14) f ∈ H α ,1 (m1 , m2 ; D0 ), g ∈ H α ,1 (n1 , n2 ; D0 ) ϕ ∈ Bα (0; R ) , α ∈ (0,1) ~ ~ f 1 , g 1 ∈ H α (G ) and we have ~ f1 α ≤ L1 , then in and this transforms the sphere Bα (0; R ) to the sphere Bα (0; λ L ) . case Corollary 2.4. If the assumptions of the Lemma 2.5 are satisfied and if ~ g1 α ≤ L2 . (13) Here, L1 = (m0 + m1 + 2m2 .R ). TG , α L2 = (n0 + n1 + 2n2 .R ). Π G λ L ≤ R then the operator A that is defined with 14) transforms the sphere Bα (0; R ) to itself. Lemma 2.6. The sphere Bα (0; R ) , α set of the space ( H α (G ); . Proof. . ∞ ). From the definition of the sphere Bα (0; R ) , α ∈ (0,1) The proof of the Lemma 2.4 is obvious from the definitions of α ∈ (0,1) is a compact for ϕ ∈ Bα (0; R ) we have ϕ ∞ ≤ R, ~ ~ the functions f 1 , g 1 : G → ℂ and corollary 2.1. therefore, it is clear that the sphere Bα (0; R ) is uniform Lemma 2.5. Let bounded in the space ( H α (G ); ε > 0 if we take δ = (ε / R)1 / α then for every z1 , z 2 ∈ G and ϕ ∈ Bα (0; R ) when z1 − z 2 < δ we have f ∈ H α ,1 ( m1 , m 2 ; D0 ), g ∈ H α ,1 ( n1 , n 2 ; D0 ) , F ∈ H α ,1,1,1 (l1 , l 2 , l3 , l 4 ; DR ) and DR = {( z, ϕ , t , s) ∈ D 0 : z ∈ G , ϕ ≤ R, t ≤ R, s ≤ R} . F : G → ℂ that is defined as F ( z ) = F ( z , ϕ ( z ), (TG o f1 )( z ), (Π G o f1 )( z ) ), for the function z ∈ G and for ϕ ∈ Bα (0; R) , α ∈ (0,1) we have . ∞ ) . Furthermore, for every ϕ ( z1 ) − ϕ ( z 2 ) ≤ R. z1 − z 2 Then α < ε . From here we can see that the elements of the sphere Bα (0; R ) are continuous at same order. Thus, as required by the Arzela-Ascoli theorem about compactness the sphere Bα (0; R ) is a compact set of the space ( H α (G ); . ∞ ). Corollary 2.5. The sphere Bα (0; R ) , complete subspace of the space ( H α (G ); 22 α ∈ (0,1) is a . ∞).
  4. 4. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational Science and Engineering Vol:3 No:11, 2009 ~ l 2 ϕ ( z ) − ϕ ( z ) +    ~ ≤ λ .l3 TG ( f (., ϕ (.)) − f (., ϕ (.)))( z ) + .   ~ l 4 Π G ( g (., ϕ (.)) − g (., ϕ (.)))( z )  Now, let us define the following two transformations, which in the space ( H α (G ); . ∞ ) . ~ ∈ H (G ) , α ∈ (0,1) , let For ϕ , ϕ p >1 α ~ ~ ~ ~ d ∞ (ϕ , ϕ ) = ϕ − ϕ ∞ , d p (ϕ , ϕ ) = ϕ − ϕ p . Then it is are easy to show that the transformations d ∞ : H α (G ) → [0,+∞), d p : H α (G ) → [0,+∞) As for this, from the assumptions on the functions f , g and F and in accordance with Minkowski inequality we have, are   ~  ∫∫ A(ϕ )( z ) − A(ϕ )( z ) p dxdy    G  both defines a metric on the space H α (G ) , α ∈ (0,1) , ( H α (G ); . ∞ ) and consequently, it can be easily seen that Lemma 2.7.. The convergence is equivalent for the metrics d ∞ and d p in the subspace Bα (0; R ) , α ∈ (0,1) , p > 1 . [ International Science Index 35, 2009 waset.org/publications/358 inequalities, for all α ∈ (0,1) . ] .d ∞ (ϕ 0 , ϕ n ) ϕ 0 , ϕ n ∈ Bα (0; R), n = 1,2,... Now, for bounded operators  l 2 + m2 l3 . TG L (G ) p ≤ λ . n l . Π  2 4 G L (G ) p  TG , Π G : L p (G ) → L p (G ), p > 1 let us define the following norms { TG Lp (G ) ≡ sup TG ϕ ΠG L p (G ) ≡ sup{ Π G ϕ p :ϕ p p :ϕ } ≤1, p 1/ p l ϕ − ϕ + ~   2  p   ~ ≤ λ . l3 . TG L ( G ) . f (., ϕ (.)) − f (., ϕ (.)) p +  p   ~  l 4 . Π G L ( G ) . g (., ϕ (.)) − g (., ϕ (.)) p  p   Proof of Lemma 2.7 is direct result of (6) and d p (ϕ 0 , ϕ n ) ≤ π .(d / 2) ≤λ× p ~  l ϕ ( z ) − ϕ ( z ) +   2   ~  l T ( f (., ϕ (.)) − f (., ϕ (.)))( z ) +  dxdy    ∫∫  3 G  G  ~ (.)))( z )   l 4 Π G ( g (., ϕ (.)) − g (., ϕ      ( H α (G ); . p ) are metric spaces. 2 1/ p 1/ p + . ϕ − ϕ ~    p and therefore we obtain ≤ 1} 1/ p   ~  ∫∫ A(ϕ )( z ) − A(ϕ )( z ) p dxdy  ≤   G  ~ λ . l 2 + m 2 l 3 . TG L ( G ) + n 2 l 4 . Π G L ( G ) . ϕ − ϕ p . [11], [14]. ( Lemma 2.8. Let f ∈ H α ,1 ( m1 , m 2 ; D0 R ), g ∈ H α ,1 ( n1 , n 2 ; D0 R ) p ) p From this inequality, it is seen that the Inequality (15) is true. With this, the lemma is proved. F ∈ H α ,1,1,1 (l1 , l 2 , l3 , l 4 ; DR ) , α ∈ (0,1) , p > 1 . In ~ this case, for ϕ , ϕ ∈ B (0; R ) the operator A that is defined Lemma 2.9. If the assumptions of the Lemma 2.8 are satisfied with the Equality (14) satisfies the following inequality and and α ~ ~ d p ( A(ϕ ), A(ϕ )) ≤ λ M 3 ( p ).d ∞ (ϕ , ϕ ) . (15) ( Here, M 3 ( p ) = l 2 + m2 l 3 . TG (m(G) ) 1/ p L p (G ) + n2 l 4 . Π G L p (G ) )× Proof. For z ∈ G assumption of the lemma and definition of the operator A we have G G the operator n→∞ lim d ∞ ( A(ϕ n ), A(ϕ 0 )) = 0 . From the Inequality (6) n →∞ can write A(ϕ n ) − A(ϕ 0 ) ∞ ≤ 2 ~ A(ϕ )( z ) − A(ϕ )( z ) = λ × F ( z , ϕ ( z ), TG f (., ϕ (.))( z ), Π G g (., ϕ (.))( z ) ) − ~ ~ ~ F ( z , ϕ ( z ), T f (., ϕ (.))( z ), Π g (., ϕ (.))( z ) ) Then A : Bα (0; R) → Bα (0; R) , α ∈ (0,1) is a continuous operator for the metric d ∞ . Proof. Let ϕ 0 , ϕ n ∈ Bα (0; R ), n = 1,2,... α ∈ (0,1) and lim d ∞ (ϕ n , ϕ 0 ) = 0 . We want to show that . ~ and ϕ , ϕ ∈ Bα (0; R ) from the λ .L ≤ R. 2 M (α , p ) A(ϕ n ) − A(ϕ 0 ) α +αp × . A(ϕ n ) − A(ϕ 0 ) αp 2 +αp p Thus, due to the Inequality (15) we will have 23 we
  5. 5. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational Science and Engineering Vol:3 No:11, 2009 2 A(ϕ n ) − A(ϕ 0 ) ( λ .M 3 ( p )) αp 2 +αp ∞ ≤ M (α , p )(2 R) 2+αp × .(d ∞ (ϕ n , ϕ 0 )) αp 2 +αp . From this inequality, it can easily be seen that if lim d ∞ (ϕ n , ϕ 0 ) = 0 then n→∞ lim d ∞ ( A(ϕ n ), A(ϕ 0 )) = 0 . q ∈ [0,1) such that ρ 2 ( Ax, Ay ) ≤ q.ρ 2 ( x, y ) . In this case, the operator equation x = Ax has only unique solution x* ∈ X and for x 0 ∈ X to be any initial approximation, the velocity of the convergence of the sequence ( x n ) to the solution x* is determined by the following inequality ρ 2 ( x n , x* ) ≤ q n .(1 − q ) −1 .ρ 2 ( x1 , x0 ) n →∞ III. MAIN RESULTS Now, we can give the theorem about the existence of the solution of the Equation (1). Theorem 3.1. If f ∈ H α ,1 ( m1 , m 2 ; D0 R ), g ∈ H α ,1 ( n1 , n 2 ; D0 R ) and Here, the terms of the sequence are defined by xn = Ax n−1 , n = 1,2,... . On the basis of this theorem and preceding information, we can give the following theorem about the uniqueness of the solution of the Equation (1) and also about finding the solution. F ∈ H α ,1,1,1 (l1 , l 2 , l 3 , l 4 ; DR ) , α ∈ (0,1) and λ L ≤ R International Science Index 35, 2009 waset.org/publications/358 . Theorem 3.3. If then the nonlinear singular integral equation (1) has at least one solution in the sphere Bα (0; R ) . F ∈ H α ,1,1,1 (l1 , l 2 , l 3 , l 4 ; DR ) , α ∈ (0,1) , Proof. If f ∈ H α ,1 ( m1 , m 2 ; D0 R ), g ∈ H α ,1 ( n1 , n 2 ; D0 R ) , λ L ≤ R then the operator A which is defined λ L ≤ R , l = λ .(l 2 + m2 l3 . TG with the Formula (14) transforms the convex, closed and compact set Bα (0; R ) to itself in the Banach space n2 l 4 . Π G ( H α (G ); . α ) , α ∈ (0,1) . Thus, the operator A is a L p (G ) + integral compact operator in the space H α (G ) . In addition to this, since the operator A is also continuous on the set Bα (0; R ) , it is completely continuous operator. In that case, due to the Schauder fixed-point principle the operator A has a fixed point in the set Bα (0; R ) . Consequently, the Equation (1) has a solution in the space H α (G ) , α ∈ (0,1) . With this, the theorem is proved. Now, we will investigate the uniqueness of the solution of the Equation (1) and the problem that how can we find the approximate solution. For this, we will use a more useful modified version of the Banach fixed-point principle for the uniqueness of the solution of operator equations [16]. Theorem 3.2. Suppose the following assumptions are satisfied: i. ( X , ρ 1 ) is a compact metric space; ii. In the space X, every sequence that is convergent for the metric ρ1 is also convergent for a second metric ρ 2 that is defined on X; iii. The operator A : X → X is a contraction mapping for the metric ρ 2 , that is, for L p (G ) < 1, p > 1 then the nonlinear singular equation (1) has only unique solution ϕ * ∈ Bα (0; R) and for ϕ 0 ∈ Bα (0; R) to be any initial approximation, this solution can be found as a limit of the sequence (ϕ n ), n = 1,2,... whose terms are defined as below  z , ϕ n −1 ( z ), TG f (., ϕ n −1 (.))( z ),  ,  Π G g (., ϕ n −1 (.))( z )   ϕ n ( z ) = λ .F   z ∈ G , n = 1,2,... . (16) Furthermore, for the velocity of the convergence the following evaluation is right, d p (ϕ n , ϕ * ) ≤ l n .(1 − l ) −1 .d p (ϕ1 , ϕ 0 ), n = 1,2,... . (17) Proof. In order to prove the first part of the theorem it is sufficient to show that the assumptions i-iii of the Theorem 3.2 is satisfied. In accordance with Lemma 2.6 and Corollary 2.5 ~ ~ for ϕ , ϕ ∈ Bα (0; R ) and d p (ϕ , ϕ ) = ∞ ~ ϕ −ϕ ∞ , (Bα (0; R); d ∞ ) is a compact metric space. Therefore, for X = Bα (0; R ) and ρ1 = d ∞ the assumption i. of the Theorem 3.2 is satisfied. If we take ρ 2 = d p , p > 1 , from the Lemma 2.7 it is obvious that the assumption ii. of the Theorem 3.2 is satisfied. every x, y ∈ X there exist a number 24
  6. 6. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational Science and Engineering Vol:3 No:11, 2009 Now, let us show that when l < 1 then the operator A is a contraction transformation with respect to the d p metric and so we have shown that the assumption iii. of the Theorem 3.2 is satisfied. For any ϕ1 , ϕ 2 ∈ Bα (0; R ) , same as the proof of the Lemma 2.8 the following inequality can be proven A(ϕ1 ) − A(ϕ 2 ) Here, l = λ .(l 2 + m2 l3 . TG Therefore, for p L p (G ) ϕ1 , ϕ 2 ∈ Bα (0; R) ≤ l . ϕ1 − ϕ 2 + n2 l 4 . Π G From the Inequality (19), it can be seen that the sequence (ϕ n ), n = 1,2,... is a Cauchy sequence with respect to the metric d p . Therefore, since (B (0; R); d p ) is a complete α metric space there exist the limit ϕ * ∈ Bα (0; R) such that lim ϕ n = ϕ * or lim d p (ϕ n , ϕ * ) = 0. n→∞ n→ ∞ From (18), for every natural number n since p d p (ϕ n +1 , A(ϕ * )) = d p ( A(ϕ n ), A(ϕ * )) ≤ . L p (G l.d p (ϕ n , ϕ * ) , ). we have we can write lim d p (ϕ n +1 , A(ϕ * )) = 0 , n→∞ d p ( A(ϕ1 ), A(ϕ 2 )) ≤ l.d p (ϕ1 , ϕ 2 ) . (18) International Science Index 35, 2009 waset.org/publications/358 Thus, when l < 1 it can be seen from the Inequality (18) that the operator A is a contraction mapping with respect to the d p metric that is defined on the Bα (0; R ) . Bα (0; R ) is a compact metric space; ii. In the space Bα (0; R ) , every sequence that is convergent for the metric d ∞ is also convergent for the metric d p that is defined on Bα (0; R ) and consequently, ϕ * = A(ϕ* ) . With this, it can be seen that the operator equation ϕ ( z ) = A(ϕ ( z )), z ∈ G , consequently, the solution of the Equation (1) is the limit of the sequence whose terms are defined by Thus, from the assumptions of the theorem; i. A : Bα (0; R ) → Bα (0; R) is a contraction mapping with respect to the norm d p . and iii. Therefore, because of the Theorem 3.2 the Equation (1) has only unique solution ϕ * ∈ Bα (0; R ) . Now, let us show that for ϕ 0 ∈ Bα (0; R ) to be an initial approximation point, this solution is the limit of the sequence (ϕ n ), n = 1,2,... whose terms are defined by (16) or by ϕ n ( z ) = A(ϕ n−1 ( z )), z ∈ G , n = 1,2,... . For every n = 1,2,... from the Inequality (18) we have d p (ϕ n +1 , ϕ n ) = d p ( A(ϕ n ), A(ϕ n −1 )) ≤ l.d p (ϕ n , ϕ n −1 ) so from here we can write ϕ n ( z ) = A(ϕ n−1 ( z )), z ∈ G , n = 1,2,... because z ∈ G , n = 1,2,... to the solution ϕ * ( z ) is given Corollary 3.1. In the space sequence (ϕ n (z ) ), z ∈ G , defined by (16) (or defined by Bα (0; R ), α ∈ (0,1) the n = 1,2,... whose terms are ϕ n ( z ) = A(ϕ n −1 ( z )), z ∈ G , n = 1,2,... ) also converges to the unique solution of the Equation (1) with respect to the metric d ∞ . ACKNOWLEDGMENT The authors would like to express sincerely thanks to the referees for their useful comments and discussions. REFERENCES [1] [2] [3] d p (ϕ m , ϕ 0 ) ≤ (l m −1 + l m− 2 + ... + l + 1)d p (ϕ1 , ϕ 0 ) we will have, m →∞ by the Inequality (17). From Theorem 3.3 and Lemma 2.7 we will have the following result. So, for any natural numbers m and n we can write Furthermore, because lim ϕ n + m ( z ) = ϕ * ( z ) and lim l m = 0 from the m →∞ (ϕ n ( z ) ), and from this inequality , we will have d p (ϕ n + m , ϕ n ) ≤ l n .d p (ϕ m , ϕ 0 ). . Furthermore, as Inequality (19) the convergence velocity of the sequence d p (ϕ n +1 , ϕ n ) ≤ l.d p (ϕ n , ϕ n −1 ) d p (ϕ n+1 , ϕ n ) ≤ l n .d p (ϕ1 , ϕ 0 ) . lim ϕ n +1 = A(ϕ * ) in other words n→∞ [4] d p (ϕ n + m , ϕ n ) ≤ (1 − l m ).(1 − l ) −1 .l n .d p (ϕ1 , ϕ 0 ). (19) 25 E.G. Ladopoulas, “On the numerical evaluation of the general type of finite-part singular integrals and integral equations used in fracture mechanics”, J. Engng. Fract. Mech. 31, 1988, pp.315-337. E.G. Ladopoulas, “On a new integration rule with the Gegenbauer polynomials for singular integral equations used in the theory of elasticity”, Ing. Archiv 58, 1988, pp.35-46. E.G. Ladopoulas, “On the numerical solution of the finite-part singular integral equations of the first and the second kind used in fracture mechanics”, Comput. Methods Appl. Mech. Engng. 65, 1987, pp.253266. E.G. Ladopoulas, “On the solution of the finite-part singular integrodifferential equations used in two-dimensional aerodynamics”, Arch. Mech. 41, 1989, pp.925-936.
  7. 7. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational Science and Engineering Vol:3 No:11, 2009 [5] [6] [7] [8] [9] [10] [11] [12] [13] International Science Index 35, 2009 waset.org/publications/358 [14] [15] [16] E.G. Ladopoulas, “The general type of finite-part singular integrals and integral equations with logarithmic singularities used in fracture mechanics”, Acta Mech. 75, 1988, pp.275-285. E.G. Ladopoulas, V.A. Zisis, “Nonlinear finite-part singular integral equations arising in two-dimensional fluid mechanics”, J. Nonlinear Analysis, 42, 2000, pp.277-290. E.G. Ladopoulas, “Nonlinear singular integral equations elastodynamics by using Hilbert Transformations”, J. Nonlinear Analysis: Real World Applications 6 2005, pp.531-536. L. Bers and L. Nirenberg , “On a representation for linear elliptic systems with discontinuous coefficients and its applications”, Convegno intern sulle equaz. lin. alle deriv. parz; Triest, 1954, pp.111-140. B.V. Bojarskii, “Quasiconformal mappings and general structural properties of system of nonlinear elliptic in the sense of Lavrentev”, Symp. Math.18, 1976, pp.485-499. E. Lanckau and W. Tutschke, Complex analysis, methods, trends and applications. Pergamon Press, London, 1985. V.N. Monahov, Boundary value problems with free boundaries for elliptic systems. Nauka, Novosibirsk , 1977. A.S. Mshim Ba and W. Tutschke, Functional-analytic methods in complex analysis and applications to partial differential equations. World Scientific, Singapore, New Jersey, London, 1990. W. Tutschke,“ Lözung nichtlinearer partieller Differentialgleichungssysteme erster Ordnung inder Ebene cluch verwendung einer komplexen Normalform“, Mat. Nachr. 75, 1976, pp.283-298. I.N. Vekua, Generalized analytic functions. Pergamon Press, London, 1962. A. Zygmund and A.P. Calderon A.P, “On the existence of singular integrals”, Acta Mathematica 88, 1952, pp.85- 139. A.I. Gusseinov and H.S. Muhtarov, Introduction to the theory of nonlinear singular integral equations. Nauka, Moscow, 1980. 26

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