On solution of nonlinear singular integral equations with shift in generalized Hölder space

Amer, S. M.

doi:10.1016/s0960-0779(00)00066-7

Cited by 12 publications

(7 citation statements)

References 4 publications

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“…Lemma 3.4 [20]. If the inequalities (3.9)-(3.11) are satisfied, then for every ) ( ) ( , where, 4 A is positive constant, therefore, we have , ) ( and put…”

Section: -Some Otatio S a D Auxiliary Resultsmentioning

confidence: 99%

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On the Application of New Convergence Criteria for Kantorovich Method to Nonlinear Singular Integral Equation with Shift

Allan

Dardery

2011

MCA

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Abstract-The paper is concerned with the applicability of some new conditions for the convergence of Newton-kantorovich approximations to solution of nonlinear singular integral equation with shift of Uryson type. The results are illustrated in generalized Holder space. Keywords: Newton-Kantorovich approximations, Nonlinear singular integral equations of Uryson type, Noether operator, Carleman shift. 1-I TRODUCTIOThe theory of approximation methods and its applications for the solution of linear and nonlinear singular integral equations (LSIE) and (NSIE) has been developed by many authors [5,11,12,17,21]. There is a literature on the successful development of the nonlinear singular integral equations with shift (NSIES) [1,3,4,15,18,20]. The Noether theory of singular integral operators with shift (SIOS) is developed for a closed and open contour ([2,10,13,14,16,18] and others). The theory of singular integral equations with shift (SIES) is an important part of integral equations because of its recent applications in many fields of physics and engineering, [6,14,16]. It is known [6,7], that Weiner-Hope equations are a natural apparatus for the solution of problems of synthesis of signais for linear systems with continuous time and stationary parameters. If the problem of synthesis is not stationary, then the Weiner-Hope method is not applicable and the problem is reduced to singular integral equation. In this paper, some new conditions for the convergence of Newton-Kantorovich approximations have been applied to solution of the following NSIES of Uryson type:

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“…Lemma 3.4 [20]. If the inequalities (3.9)-(3.11) are satisfied, then for every ) ( ) ( , where, 4 A is positive constant, therefore, we have , ) ( and put…”

Section: -Some Otatio S a D Auxiliary Resultsmentioning

confidence: 99%

“…There is a literature on the successful development of the nonlinear singular integral equations with shift (NSIES) [1,3,4,15,18,20]. The Noether theory of singular integral operators with shift (SIOS) is developed for a closed and open contour ([2,10,13,14,16,18] and others).…”

Section: -I Troductiomentioning

confidence: 99%

On the Application of New Convergence Criteria for Kantorovich Method to Nonlinear Singular Integral Equation with Shift

Allan

Dardery

2011

MCA

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“…and still get an existence result for problem (1). However, in this case the solvability of problem (1) also follows from generalized Kantorovich condition (15) which is a weaker condition than (19) since θ * (M, M 0 ) > 1.…”

Section: A Generalized Kantorovich Theorem 101mentioning

confidence: 93%

“…We are interested in the solvability of the equation f (u) = 0 (1) and the convergence of the method of Newton defined by the iterations…”

mentioning

confidence: 99%

“…The Kantorovich theorem has been used for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields such as semidefinite programming (see [13]), singularly perturbed multiparameter systems (see [12]), the design of optimal-complexity path following algorithms for linear complementarity problems ( [15]), the chaotic behaviour of maps ( [18]), nonlinear RiemannHilbert problems ( [6]), optimal control problems ( [9]), nonlinear elliptic problems ( [11]), integral equations ( [1,3,5]), nonlinear finite element analysis ( [20]) and flow problems (see [10]), to mention but a few of the more recent papers.…”

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confidence: 99%

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A generalized Kantorovich theorem on the solvability of nonlinear equations

Uko

Argyros

2009

Aequ. math.

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The Kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields. This theorem was weakened recently by Argyros who used a combination of Lipschitz and centerLipschitz conditions in place of the Lipschitz conditions of the Kantorovich theorem. In the present paper we use the Argyros theorem to formulate a generalized Kantorovich theorem that enables us deduce the solvability of equations and the convergence of Newton's method with minimal assumptions.Let X and Y be Banach spaces, let D be an open convex subset of X, and let f : D −→ Y be a Fréchet differentiable function, fixed all through this paper. In the sequel, given any u 0 ∈ D and r > 0, B[u 0 , r] will designate the set {x ∈ X : x − u 0 ≤ r}, and B(u 0 , r) will designate the interior of B[u 0 , r].We are interested in the solvability of the equationand the convergence of the method of Newton defined by the iterationsOne of the most important theorems in nonlinear analysis is the following result due -essentially -to Kantorovich (see [7]).

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On Approximate Solutions of Linear and Nonlinear Singular Integral Equations

Mustafa

Nezir

2019

Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

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On solution of nonlinear singular integral equations with shift in generalized Hölder space

Cited by 12 publications

References 4 publications

On the Application of New Convergence Criteria for Kantorovich Method to Nonlinear Singular Integral Equation with Shift

On the Application of New Convergence Criteria for Kantorovich Method to Nonlinear Singular Integral Equation with Shift

A generalized Kantorovich theorem on the solvability of nonlinear equations

On Approximate Solutions of Linear and Nonlinear Singular Integral Equations

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