Approximate Solution of Operator Equations

Krasnosel’skii, M. A.; Vainikko, Gennadi; Zabreiko, P. P.; Rutitskii, Ya. B.; Stetsenko, V. Ya.

doi:10.1007/978-94-010-2715-1

Cited by 508 publications

(368 citation statements)

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“…We leave the details to the interested reader. Further, we note that, according to [16,Chapter I,1.4], for a given norm · there exists an equivalent norm · * such that ρ(B) ≤ B * ≤ ρ(B) + ε, and a corresponding error estimate can be found (see [16, p. 19]). But, this norm is mainly of theoretical interest.…”

Section: The Jacobi Methods In Contemporary Numerical Mathematicsmentioning

confidence: 99%

Error estimate for the Jacobi method adapted to the weak row sum resp. weak column sum criterion

Kohaupt

2002

Elem. Math.

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Section: The Jacobi Methods In Contemporary Numerical Mathematicsmentioning

confidence: 99%

Error estimate for the Jacobi method adapted to the weak row sum resp. weak column sum criterion

Kohaupt

2002

Elem. Math.

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“…The monograph by Lakshmikantham and Leela [6] was one of important in this area. In the book by Krasnosel'skii et al [7], which was devoted to approximate methods for operator equations, property (1.6) ⇒ (1.7) was also essentially used. These ideas have been developing in scores of books on the monotone technique for approximate solution of boundary value problems for systems of differential equations.…”

Section: G(t S)f (S)ds + X(t)c T ∈ [0 ω]mentioning

confidence: 99%

Component-wise positivity of solutions to periodic boundary problem for linear functional differential system

2012

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The classical Ważewski theorem claims that the condition p ij ≤ 0, j ≠ i, i, j =1,...,n, is necessary and sufficient for non-negativity of all the components of solution vector to a system of the inequalitiesAlthough this result was extent on various boundary value problems and on delay differential systems, analogs of these heavy restrictions on non-diagonal coefficients p ij preserve in all assertions of this sort. It is clear from formulas of the integral representation of the general solution that these theorems claim actually the positivity of all elements of Green's matrix. The method to compare only one component of the solution vector, which does not require such heavy restrictions, is proposed in this article. Note that comparison of only one component of the solution vector means the positivity of elements in a corresponding row of Green's matrix. Necessary and sufficient conditions of this fact are obtained in the form of theorems about differential inequalities. It is demonstrated that the sufficient conditions of positivity of the elements in the nth row of Green's matrix, proposed in this article, cannot be improved in corresponding cases. The main idea of our approach is to construct a first order functional differential equation for the nth component of the solution vector and then to use assertions, obtained recently for first order scalar functional differential equations. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. Note that in some cases the sufficient conditions, obtained in the article, does not require any smallness of the interval [0, ω], where the system is considered.

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“…With the ntation here, the Kantorovich theorem can be established by taking Ptk [21] thus derived the a priori upper bounds of Gragg-Tapia with the use of (2.7), and recently, Potra-Ptk [17] Kusakin; see [8,Bibliography] for the references.…”

Section: Moreover If G(t)mentioning

confidence: 99%

Semilocal analysis of equations with smooth operators

Miel

1981

International Journal of Mathematics and Mathematical Sciences

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Approximate Solution of Operator Equations

Cited by 508 publications

References 0 publications

Error estimate for the Jacobi method adapted to the weak row sum resp. weak column sum criterion

Error estimate for the Jacobi method adapted to the weak row sum resp. weak column sum criterion

Component-wise positivity of solutions to periodic boundary problem for linear functional differential system

Semilocal analysis of equations with smooth operators

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