Articulo 3.dvi CUBO A Mathematical Journal Vol.12, No¯ 02, (29–42). June 2010 The method of Kantorovich majorants to nonlinear singular integral equations with Hilbert kernel M. H. Saleh, S. M. Amer 1 and M. H. Ahmed Departement of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt. email: amrsammer@hotmail.com ABSTRACT This paper concerned with applicability of the method of Kantorovich majorants to non- linear singular integral equations with Hilbert kernel . The results are illustrated in Hölder space. RESUMEN Este art́ıculo es concerniente a la aplicabilidad del método de mayorantes de Kantorovich para ecuaciones integrales singulares no lineales con núcleo de Hilbert. Los resultados son aplicaciones en espacios de Hölder. Key words and phrases: Nonlinear singular integral equations, Kantorovich majorants method, Hölder spaces. AMS 2000-Subject classification: 45F15, 45G10. 1. Introduction There is a large literature on nonlinear singular integral equations with Hilbert and Cauchy kernel and related Riemann boundary value problems for analytic functions,cf.the monograph by Pogorzel- 1Corresponding author 30 M. H. Saleh, S. M. Amer and M. H. Ahmed CUBO 12, 2 (2010) ski [16], Guseinov A.I. and Mukhtarov kh.sh. [9],Kantorovich L.V.[11],Muskhelishvill N.I.[14],and Mikhlin S.G.and Prossdorf S.[13].The method of singular integral equations on closed contour has been intensively investigated by many approximation methods, specially method of modified Newton- Kantorovich, reduction, collocation and mechanical quadratures, (see[1-6,10,12,15,17,19]). In this paper the method of Kantorovich majorants[7,18,20], has been applied to the following class of nonlinear singular integral equations with Hilbert kernel : ϕ(t) = λG(t, 1 2π ∫ 2π 0 g(t,σ,ϕ(σ)) cot σ − t 2 dσ), (1.1) where λ is a numerical parameter, where v(t) = Sg(t,σ,ϕ(σ)) = 1 2π ∫ 2π 0 g(t,σ,ϕ(σ)) cot σ − t 2 dσ, then equation (1.1) takes the form: ϕ(t) = λG(t,v(t)). Now, we consider the equation: B(ϕ) = 0, (1.2) where (Bϕ)(t) = ϕ(t) − λG(t,v(t)). (1.3) 2. Formulation of the problem Let f : S̄(ϕ 0 ,R) ⊂ X −→ Y be a nonlinear operator defined on the closure of a ball S(ϕ 0 ,R) = {ϕ : ϕ ∈ X,‖ ϕ − ϕ 0 ‖< R}, in a Banach space X into a Banach space Y. We give new conditions to ensure the convergence on Newton-Kantorovich approximations toward a solution of f(ϕ) = 0, under the hypothesis that f is Frechet differentiable in S(ϕ 0 ,R), and that it’s derivative f̀ satisfies the local Lipschitz condition : ‖ f̀(ϕ 1 ) − f̀(ϕ 2 ) ‖≤ k(r) ‖ ϕ 1 − ϕ 2 ‖,ϕ 1 ,ϕ 2 ∈ S̄(ϕ 0 ,r),o < r < R, (2.1) where k(r)is a non decreasing function on the interval [0,R] and k(r) = sup{‖ f̀(ϕ1 ) − f̀(ϕ2 ) ‖ ‖ ϕ 1 − ϕ 2 ‖ ϕ 1 ,ϕ 2 ∈ S̄(ϕ 0 ,r)ϕ 1 6= ϕ 2 }. (2.2) Define a scalar function ψ : [0,R] → (0,∞) by ψ(r) = a + b ∫ r 0 w(t)dt − r, (2.3) CUBO 12, 2 (2010) The method of Kantorovich majorants to nonlinear singular integral equations with Hilbert kernel 31 using the function w(r) = ∫ r 0 k(t)dt, (2.4) and a =‖ f̀(ϕ 0 )−1f(ϕ 0 ) ‖, b =‖ f̀(ϕ 0 )−1 ‖ . (2.5) Theorem 2.1 [4,7] Suppose that the equation (2.3)has a unique positive root r ∗ in [0,R] and ψ(R) ≤ 0. Then the equation f(ϕ) = 0 has a unique solution ϕ ∗ in S(ϕ 0 ,R) and the Newton- Kantorovich approximations: ϕ n = ϕ n−1 − f̀(ϕ n−1 )−1f(ϕ n−1 ), n ∈ N, (2.6) are defined for all n ∈ N, belong to S(ϕ 0 ,r ∗ ) and converges to ϕ ∗ . Moreover , the following estimate holds ‖ ϕ n+1 − ϕ n ‖≤ r n+1 − r n , ‖ ϕ ∗ − ϕ n ‖≤ r ∗ − r n , (2.7) where the sequence (r n )n∈N converges to r∗ , is defined by the recurrence formula r 0 = 0, r n+1 = r n − ψ(rn ) `ψ(r n ) , n ∈ N. (2.8) In the present paper , we investigate some sufficient conditions , which ensure that the class of nonlinear singular integral equations (1.1) verifies the hypotheses of theorem (2.1). 3. Some auxiliary results Definition 3.1[9] We denote by H δ , 0 < δ < 1,the Hölder space of continuous functions , which satisfy the Hölder condition with exponent δ with norm ‖ ϕ ‖ δ =‖ ϕ ‖ c +Hδ(ϕ), (3.1) where ‖ ϕ ‖ c = max σ∈[0,2π] | ϕ(σ) |, and H δ(ϕ) = sup σ 1 6= σ 2 | ϕ(σ1) − ϕ(σ2) | | σ1 − σ2 |δ . Lemma 3.1 [9] Let the functions G(t,v(t)), g(t,σ,ϕ(σ)) and it’s partial derivatives up to second order, satisfy the following conditions | ∂ mG(t1,v(t1)) ∂vm − ∂ mG(t2,v(t2)) ∂vm |≤ cm(r){| t1 − t2 |δ + | v(t1 ) − v(t2 ) |}, (3.2) 32 M. H. Saleh, S. M. Amer and M. H. Ahmed CUBO 12, 2 (2010) and | ∂mg ϕ (t1,σ1,ϕ(σ1)) ∂ϕm − ∂mg ϕ (t2,σ2,ϕ(σ2)) ∂ϕm |≤ am(r){| t1 − t2 |δ + | σ1 − σ2 |δ + | ϕ(t1 ) − ϕ(t2 ) |} (3.3) where cm(r),am(r) are positive increasing functions m=0,1,2 and t i ,σ i ∈ [0, 2π], i = 1, 2. If ϕ(σ) ∈ H δ , then G(t,v(t)), g(t,σ,ϕ(σ)) ∈ H δ . Lemma 3.2 If the functions G(t,v(t)) and g(t,σ,ϕ(σ)) satisfy the conditions of lemma(3.1), then the operator B(ϕ) has a Frechet derivative at every fixed point in the space H δ and its derivative is given by B̀(ϕ)h = h(t) − λGv(t,v(t))Sg ϕ (t,σ,ϕ(σ))h(σ). (3.4) Moreover it satisfies Lipschitz condition: ‖ B̀(ϕ 1 ) − B̀(ϕ 2 ) ‖≤ k(r) ‖ ϕ 1 − ϕ 2 ‖, (3.5) for all ϕ 1 ,ϕ 2 ∈ S(ϕ 0 ,r) and o < r < R. Proof Let ϕ(t)be any fixed point in the space 0,< δ < 1and h(t)be any arbitrary element in H δ , then we obtain : B(ϕ + h) − B(ϕ) = h(t) − λ[G(t,Sg(t,σ,ϕ(σ) + h(σ))) − G(t,Sg(t,σ,ϕ(σ)))] = B̀(ϕ)h + η(t,h), where 0 ≤ ξ ≤ 1 and η(t,h) = λ ∫ 1 0 (1 − ξ)[Gv2 (t,Sg(t,σ,ϕ(σ) + ξh(σ)))(Sg ϕ (t,σ,ϕ(σ) + ξh(σ))h(σ))2 + Gv(t,Sg(t,σ,ϕ(σ) + ξh(σ)))Sg ϕ2 (t,σ,ϕ(σ) + ξh(σ))h(σ)2]dξ. Now , we shall prove that lim ‖h‖→0 ‖ η(t,h) ‖ ‖ h ‖ = 0. Using the inequalities [9,13] ‖ ∫ b a y(s) s−x ds ‖≤ ρ 0 ‖ y ‖,where ρ0 is a positive constant ‖ uv ‖≤‖ u ‖‖ v ‖ for all u,v ∈ H δ        . (3.6) Now; ‖ η(t,h) ‖ ≤ ‖ h(σ)2 ‖ ρ 0 [ ‖ Gv2 (t,Sg(t,σ,ϕ(σ))) ‖‖ (g ϕ (t,σ,ϕ(σ)))2 ‖ + ‖ Gv(t,Sg(t,σ,ϕ(σ))) ‖‖ g ϕ2 (t,σ,ϕ(σ)) ‖ ]. Hence lim ‖h‖→0 ‖ η(t,h) ‖ ‖ h ‖ = 0, CUBO 12, 2 (2010) The method of Kantorovich majorants to nonlinear singular integral equations with Hilbert kernel 33 which prove the differentiability of B(ϕ) in the sense of Frechet and its derivative is given by (3.4). To prove the Frechet derivative B̀(ϕ) satisfies Lipschitz condition in the sphere S(ϕ 0 ,R) = {ϕ :‖ ϕ − ϕ 0 ‖< R}. We consider ‖ B̀(ϕ 1 )h − B̀(ϕ 2 )h ‖ = ‖ λGv(t,Sg(t,σ,ϕ1(σ)))Sg ϕ (t,σ,ϕ1(σ))h(σ) − λGv(t,Sg(t,σ,ϕ2(σ)))Sg ϕ (t,σ,ϕ2(σ))h(σ) ‖ ≤ | λ |‖ h ‖ [ ‖ Gv(t,v1(t)) ‖‖ Sg ϕ (t,σ,ϕ1(σ)) − Sg ϕ (t,σ,ϕ2(σ)) ‖ + ‖ Sg ϕ (t,σ,ϕ2(σ)) ‖‖ Gv(t,v1(t)) − Gv(t,v2(t)) ‖ ] ≤ ‖ h ‖ k(r) ‖ ϕ 1 − ϕ 2 ‖, where k(r) =| λ | ρ 0 [a1(r)D+ ‖ gϕ ‖ c1(r)a0(r)] , and D = max t | Gv(t,Sg(t,σ,ϕ(σ))) | then the lemma is proved. 4. Solution of linear singular integral equation To find the operator B̀(ϕ 0 )−1, we investigate the solution of the equation h(t) − λGv(t,v(t)) 2π ∫ 2π 0 g ϕ (t,σ,ϕ(σ)) h(σ) cot σ − t 2 dσ = f(t). (4.1) For this aim we introduce the following theorem: Theorem 4.1 If the functions G(t,v(t)) and g(t,σ,ϕ(σ)) satisfy the conditions of lemma(3.2), then the linear operator defined by (3.4) has a bounded inverse B̀(ϕ 0 )−1 for any fixed ϕ 0 ∈ H δ , (0 < δ < 1). Proof Let us transform the equation (4.1) by introducing new variables : s = eit,τ = eiσ,dτ = ieiσdσ, since 1 2 cot σ − t 2 dσ = ( 1 τ − s − 1 2τ )dτ, then equation (4.1) has the form h(s) − λXv(s,v(s)) πi ∫ γ ik ϕ (s,τ,ϕ(τ)) h(τ) ( 1 τ − s − 1 2τ )dτ = f(s), (4.2) where γ is a unit circle , Gv(t,v(t)) = Xv(s,v(s)) and g ϕ (t,σ,ϕ(σ)) = k ϕ (s,τ,ϕ(τ)). We introduce the sectionally holomorphic function of variable z as follows: H(z) = λXv(s,v(s)) 2πi ∫ γ ik ϕ (s,τ,ϕ(τ)) τ − z h(τ)dτ − C, (4.3) 34 M. H. Saleh, S. M. Amer and M. H. Ahmed CUBO 12, 2 (2010) and H(∞) = −C = −λXv(s,v(s)) 4π ∫ γ ik ϕ (s,τ,ϕ(τ)) τ h(τ)dτ = −iλGv(t,v(t)) 4π ∫ 2π 0 g ϕ (t,σ,ϕ(σ)) h(σ)dσ. According to Sokhotoski formulae[9], we have H±(s) = ±iλXv(s,v(s)) 2 k ϕ (s,s,ϕ(s))h(s) + λXv(s,v(s)) 2πi ∫ γ ik ϕ (s,τ,ϕ(τ)) τ − s h(τ)dτ − C. Therefore H+(s) − H−(s) = iλXv(s,v(s))k ϕ (s,s,ϕ(s))h(s) H+(s) + H−(s) = λXv (s,v(s)) πi ∫ γ ik ϕ (s,τ,ϕ(τ )) τ −s h(τ)dτ − 2C        . (4.4) Substituting from equation (4.4) into equation (4.2 )we have h(s) − f(s) + 2C = H+(s) + H−(s) + 2C. (4.5) Hence we get h(s) = H+(s) + H−(s) + f(s), (4.6) therefore from (4.4) and (4.6) we have, h(s)[1 ± iλXv(s,v(s))k ϕ (s,s,ϕ(s))] = 2H±(s) + f(s), since1 ± iλXv(s,v(s))k ϕ (s,s,ϕ(s)) 6= 0, then the last conditions equivalent to the following h(s) = 2H+(s) 1+iλXv (s,v(s))k ϕ (s,s,ϕ(s)) + f (s) 1+iλXv (s,v(s))k ϕ (s,s,ϕ(s)) , h(s) = 2H−(s) 1−iλXv (s,v(s))k ϕ (s,s,ϕ(s)) + f (s) 1−iλXv (s,v(s))k ϕ (s,s,ϕ(s))          . (4.7) By equating the right hand side of equation (4.7) we get the Riemann boundary value problem H +(s) = 1 + iλXv(s,v(s))k ϕ (s,s,ϕ(s)) 1 − iλXv(s,v(s))k ϕ (s,s,ϕ(s)) H −(s) + iλXv(s,v(s))k ϕ (s,s,ϕ(s)) 1 − iλXv(s,v(s))k ϕ (s,s,ϕ(s)) f(s). (4.8) It is well known that the index of equation (4.8) is zero[8],then 1 + iλXv(s,v(s))k ϕ (s,s,ϕ(s)) 1 − iλXv(s,v(s))k ϕ (s,s,ϕ(s)) = X+(s) X−(s) , where X(z) = 1 2π ∫ γ ln 1 + iλXv(s,v(s))ik ϕ (s,τ,ϕ(τ)) 1 − iλXv(s,v(s))k ϕ (s,τ,ϕ(τ)) dτ τ − z , CUBO 12, 2 (2010) The method of Kantorovich majorants to nonlinear singular integral equations with Hilbert kernel 35 the problem (4.8)can be written in the form H+(s) X+(s) − H −(s) X−(s) = iλXv(s,v(s))k ϕ (s,s,ϕ(s))f(s) 1 − iλXv(s,v(s))k ϕ (s,s,ϕ(s))X+(s) . Hence ,from [8], the boundary value problem (4.8) has the solution H(z) = X(z)[ λXv(s,v(s)) 2πi ∫ γ ik ϕ (s,τ,ϕ(τ))f(τ) X+(τ)(1 − iλXv(s,v(s))k ϕ (s,τ,ϕ(τ))) dτ τ − s − C]. By Sokhotski formulae H+(s) = iλXv(s,v(s))k ϕ (s,s,ϕ(s))f(s) 2(1 − iλXv(s,v(s))k ϕ (s,s,ϕ(s))) + λXv(s,v(s))X +(s) 2πi ∫ γ ik ϕ (s,τ,ϕ(τ))f(τ) X+(τ)(1 − iλXv(s,v(s))k ϕ (s,τ,ϕ(τ))) dτ τ − s − CX+(s). (4.9) Substituting from (4.9) into (4.7) we have, h(s) = f(s) u(s) + z(s)λXv(s,v(s)) u(s)πi ∫ γ ik ϕ (s,τ,ϕ(τ))f(τ) z(τ) dτ τ − s − 2Cz(s) u(s) , (4.10) where u(s) = 1 + λ2X2v (s,v(s))k 2 ϕ (s,s,ϕ(s)), z(s) = √ u(s)eΓ(s), and Γ(s) = 1 2πi ∫ γ ln 1 + iλXv(s,v(s))ik ϕ (s,τ,ϕ(τ)) 1 − iλXv(s,v(s))k ϕ (s,τ,ϕ(τ)) dτ τ − s , since dτ τ − s = 1 2 cot σ − t 2 + i 2 dσ. Hence z(eit) = z(s) = √ u(t)exp ( 1 4π ∫ 2π 0 ln 1 + iλGv(t,v(t))g ϕ (t,σ,ϕ(σ)) 1 − iλGv(t,v(t))g ϕ (t,σ,ϕ(σ)) dσ exp ( 1 4πi ∫ 2π 0 ln 1 + iλGv(t,v(t))g ϕ (t,σ,ϕ(σ)) 1 − iλGv(t,v(t))g ϕ (t,σ,ϕ(σ)) cot σ − t 2 dσ)). 36 M. H. Saleh, S. M. Amer and M. H. Ahmed CUBO 12, 2 (2010) Now we determine the constant C as follows C = iλGv(t,v(t)) 4π ∫ 2π 0 g ϕ (t,σ,ϕ(σ)) h(σ) dσ = = (1 + iz(t)λGv(t,v(t)) 2πu(t) ∫ 2π 0 g ϕ (t,σ,ϕ(σ))dσ)−1 [ iλGv(t,v(t)) 4π ∫ 2π 0 g ϕ (t,σ,ϕ(σ))[ f(t) u(t) + z(t) 2πu(t) ∫ 2π 0 g ϕ (ξ,σ,ϕ(σ))f(ξ) z(ξ) cot ξ − σ 2 dξ + iz(t) 2πu(t) ∫ 2π 0 λGv (ξ,v(ξ))g ϕ (ξ,σ,ϕ(σ))f(ξ) z(ξ) dξ] dσ.} Then h(t) = f(t) u(t) + λGv (t,v(t))z(t) 2πu(t) ∫ 2π 0 g ϕ (t,σ,ϕ(σ))f(σ) z(σ) cot σ − t 2 dσ + λGv(t,v(t))z(t) 2πu(t) ∫ 2π 0 g ϕ (t,σ,ϕ(σ))f(σ) z(σ) dσ − 2Cz(t) u(t) = B̀(ϕ 0 )−1f(t). We shall prove that the operator B̀(ϕ 0 )−1 is bounded. It is easy to prove that v(t), Γ(t) and z(t) ∈ H δ therefore by using inequality (3.6) we get ‖ B̀(ϕ 0 )−1 ‖δ ≤ ‖ 1 u ‖δ {1 + ρ 0 | λ |‖ z ‖δ‖ Gv(t,v(t)) ‖δ‖ g ϕ (t,t,ϕ(t)) ‖δ‖ 1 z ‖δ + ρ 1 | λ |‖ z ‖δ‖ Gv(t,v(t)) ‖δ +2C̃ ‖ z ‖δ}, (4.11) where ρ1 = 1 2π ∫ 2π 0 | g ϕ (t,σ,ϕ(σ)) z(σ) | dσ and C̃ = (1 + iz(t)λGv(t,v(t)) 2πu(t) ∫ 2π 0 g ϕ (t,σ,ϕ(σ))dσ)−1 [ iλGv(t,v(t)) 4π ∫ 2π 0 g ϕ (t,σ,ϕ(σ))[ 1 u(t) + z(t) 2πu(t) ∫ 2π 0 g ϕ (ξ,σ,ϕ(σ)) z(ξ) cot ξ − σ 2 dξ + iz(t) 2πu(t) ∫ 2π 0 λGv(ξ,v(ξ))g ϕ (ξ,σ,ϕ(σ)) z(ξ) dξ]] dσ. We determine the norm of each term in right hand side of the above inequality. From definition (3.1) we have ‖ 1 u ‖ c =‖ 1 1 + λ2G2v(t,v(t))g 2 ϕ (t,t,ϕ(t)) ‖c≤ 1, CUBO 12, 2 (2010) The method of Kantorovich majorants to nonlinear singular integral equations with Hilbert kernel 37 | 1 u(t1) − 1 u(t2) | ≤ | u(t1) − u(t2) |≤| λ2 || G2v(t1,v(t1))g2 ϕ (t1, t1,ϕ(t1)) − G2v(t2,v(t2))g2 ϕ (t2, t2,ϕ(t2)) | ≤ λ2[| Gv(t1,v(t1))g ϕ (t1, t1,ϕ(t1)) − Gv(t2,v(t2))g ϕ (t2, t2,ϕ(t2)) |] [ 2 ‖ Gv(t,v(t)) ‖c‖ g ϕ (t,t,ϕ(t)) ‖c], since ‖ Gv(t,v(t)) ‖c≤ c1(r) ‖ v ‖c + ‖ Gv(t, 0) ‖c, similarly ‖ g ϕ (t,t,ϕ(t)) ‖c≤ a1(r) ‖ ϕ ‖c + ‖ g ϕ (t,t, 0) ‖c, using conditions(3.2)and (3.3)we have | g ϕ (t1, t1,ϕ(t1)) − g ϕ (t2, t2,ϕ(t2)) |≤ a1(r)(2 + Hδ(ϕ)) | t1 − t2 |δ, | Gv(t1,v(t1)) − Gv(t2,v(t2)) |≤ c1(r)(1 + Hδ(v))) | t1 − t2 |δ . and | Gv(t2,v(t2)) |≤| Gv(t2, 0)) | +c1(r)(| v(t2) |), similarly | g ϕ (t1, t1,ϕ(t1)) |≤ a1(r) | ϕ | + | g ϕ (t1, t1, 0) | . Hence | 1 u(t1) − 1 u(t2) |≤ λ2β. So ‖ 1 u ‖ δ ≤ R1, (4.12) where R1 = 1 + λ 2β and β = [(| g ϕ (t1, t1, 0) | +a1(r) | ϕ |)(c1(r)(1 + Hδ(v)) | t1 − t2 |δ) + (| Gv(t2, 0) | +c1(r) | v |)(a1(r)(2 + Hδ(ϕ)) | t1 − t2 |δ)] [ (c1(r) ‖ v ‖c + ‖ Gv(t, 0) ‖c)(a1(r) ‖ ϕ ‖c + ‖ g ϕ (t,t, 0) ‖c)], To determine ‖ z ‖δ we get ‖ z ‖δ≤‖ √ u ‖δ (1+ ‖ Γ ‖δ)e‖Γ‖δ, (4.13) since ‖ u ‖ c ≤ √ 1 + λ2(c1 ‖ v ‖c + ‖ Gv(t, 0) ‖c)2(a1 ‖ ϕ ‖c + ‖ g ϕ (t,t, 0) ‖c)2. By 38 M. H. Saleh, S. M. Amer and M. H. Ahmed CUBO 12, 2 (2010) applying Lagrange theorem: | √ u(t1) − √ u(t2) | = | 1 2 (1 + θ)−1/2 λ2 [G2v(t1,v(t1)) g 2 ϕ (t1, t1,ϕ(t1)) − G2v(t2,v(t2)) g2 ϕ (t2, t2,ϕ(t2))] | ≤ λ2β, where θ between λGv(t1,v(t1)) g ϕ (t1, t1,ϕ(t1)) and λGv(t2,v(t2)) g ϕ (t2, t2,ϕ(t2)). Then ‖ √ u ‖δ≤ R2, (4.14) where R2 = √ 1 + (c1 ‖ v ‖c + ‖ Gv(t, 0) ‖c)2(a1 ‖ ϕ ‖c + ‖ g ϕ (t,t, 0) ‖c)2 + λ2β. Also, we determine ‖ Γ ‖δ, since Γ(t) = 1 2π ∫ 2π 0 arctgλ Gv(t,v(t)) g ϕ (t,σ,ϕ(σ)) cot σ − t 2 dσ + Q, where Q = 1 4π ∫ 2π 0 ln 1 + iλGv(t,v(t))g ϕ (t,σ,ϕ(σ)) 1 − iλGv(t,v(t))g ϕ (t,σ,ϕ(σ)) dσ, by using (3.6)we have ‖ Γ ‖c≤ ρ 0 ‖ arctgλ Gv(t,v(t)) g ϕ (t,t,ϕ(t)) ‖c + | Q |≤ ρ 0 π 2 + | Q |, | arctgλ Gv(t1,v(t1)) g ϕ (t1, t1,ϕ(t1)) − arctgλ Gv(t2,v(t2)) g ϕ (t2, t2,ϕ(t2)) | ≤ | λ 1 + θ21 [Gv(t1,v(t1)) g ϕ (t1, t1,ϕ(t1)) − Gv(t2,v(t2)) g ϕ (t2, t2,ϕ(t2))] | ≤ | λ | [(| g ϕ (t1, t1, 0) | +a1(r) | ϕ |)(c1(r)(1 + Hδ(v)) | t1 − t2 |δ) + (| Gv(t2, 0) | +c1(r) | v |)(a1(r)(2 + Hδ(ϕ)) | t1 − t2 |δ)], where θ1 between λGv(t1,v(t1)) g ϕ (t1, t1,ϕ(t1)) and λGv(t2,v(t2)) g ϕ (t2, t2,ϕ(t2)). Therefore ‖ Γ ‖δ≤ R3, (4.15) where R3 = ρ 0 π 2 + | Q | + | λ | [(| g ϕ (t1, t1, 0) | + a1(r) | ϕ | (c1(r)(1 + Hδ(v)) | t1 − t2 |δ) + (| Gv(t2, 0) | +c1(r) | v |)(a1(r)(2 + Hδ(ϕ)) | t1 − t2 |δ)]. Substituting from (4.14) and (4.15) into (4.13) we have ‖ z ‖δ≤ R2(1 + R3)eR3. (4.16) CUBO 12, 2 (2010) The method of Kantorovich majorants to nonlinear singular integral equations with Hilbert kernel 39 From (4.14) we can determine ‖ 1 z ‖δ, ‖ 1 z ‖δ≤ 1 ‖ √ u ‖δ (1+ ‖ Γ ‖δ)e‖Γ‖δ. But ‖ 1√ u ‖c≤‖ 1 √ 1 + λ2G2v(t2,v(t2)) g 2 ϕ (t2, t2,ϕ(t2)) ‖c≤ 1 and | 1√ u(t1) − 1√ u(t2) |≤| √ u(t1) − √ u(t2) |≤ λ2β then ‖ 1√ u ‖δ≤ R4, where R4 = (1 + λ 2 β). So that ‖ 1 z ‖δ≤ R4(1 + R3)eR3. (4.17) Then: ‖ B̀(ϕ 0 )−1 ‖≤ α0, where α0 = R1(1 + ρ | λ | R2(1 + R3)eR3 )(‖ Gv(t, 0) ‖c + c1(r)(1+ ‖ v ‖)(a1(r)(2+ ‖ ϕ ‖) + ‖ g ϕ (t,t, 0) ‖c)(R4(1 + R3)eR3 ) + | ρ 1 || λ | R2(1 + R3)eR3 )(‖ Gv(t, 0) ‖c +c1(r)(1+ ‖ v ‖) + 2C̃R2(1 + R3)e R3, Hence the theorem is proved. Now ,we determine ‖ B̀(ϕ 0 )−1B(ϕ 0 ) ‖ as follows: ‖ B̀(ϕ 0 )−1B(ϕ 0 ) ‖≤ α0 ‖ B(ϕ0 ) ‖≤ µ0, (4.18) where µ0 = α0(‖ ϕ0 ‖ + | λ | c0(r)(1+ ‖ v ‖)+ ‖ G(t, 0) ‖c), Since a =‖ B̀(ϕ 0 )−1B(ϕ 0 ) ‖, hence a ≤ b[‖ ϕ 0 ‖ + | λ | c0(r)(1+ ‖ v ‖)+ ‖ G(t, 0) ‖c], and b ≤ α0 therefore , the following theorem is valid. 40 M. H. Saleh, S. M. Amer and M. H. Ahmed CUBO 12, 2 (2010) Theorem 4.2 Suppose that the equation (2.3)has a unique positive root r ∗ in [0,R] and ψ(R) ≤ 0. Then the equationB(ϕ) = 0has a unique solution ϕ ∗ in S(ϕ 0 ,R) and the Newton- Kantorovich approximations: ϕ n = ϕ n−1 − B̀(ϕ n−1 )−1B(ϕ n−1 ), n ∈ N, are defined for all n ∈ N, belong to S(ϕ 0 ,r ∗ ) and converges to ϕ ∗ . Moreover , the following estimate holds ‖ ϕ n+1 − ϕ n ‖≤ r n+1 − r n , ‖ ϕ ∗ − ϕ n ‖≤ r ∗ − r n , where the sequence (r n )n∈N converges to r∗ , is defined by the recurrence formula r 0 = 0, r n+1 = r n − ψ(rn ) `ψ (r n ) , n ∈ N. We will illustrate the theorem 4.2 by the following example. Consider the nonlinear function f(u) = 1 6 u3 + 1 6 u2 − 5 6 u + 1 3 , with derivative f̀(u) = 1 2 u 2 + 1 3 u − 5 6 , it’s clear that ‖ f̀(u1) − f̀(u2) ‖ ‖ u1 − u2 ‖ ≤ 1 6 [‖ 3(u1 + u2) ‖ +2] ≤ r + 1 3 , therefore we get k(r) = r + 1 3 . Obviously, the scaler equation (2.3) takes the form ψ(r) = a + b 6 r3 + b 6 r2 − r. 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