Boundary-Value problems for systems of integro-differential equations with Degenerate Kernel

Самойленко, А. М.; Boichuk, О. A.; Krivosheya, S. A.

doi:10.1007/bf02529500

Cited by 36 publications

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“…By using the theory of pseudoinverse Moore-Penrose matrices [1][2][3], we investigate the conditions of solvability and propose an iterative algorithm for the construction of solutions of weakly nonlinear systems of integrodifferential equations with small parameter " …”

Section: Statement Of the Problem And Auxiliary Resultsmentioning

confidence: 99%

Weakly Nonlinear Systems of Integrodifferential Equations

Boichuk

Golovats’ka

2014

J Math Sci

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517.929We establish necessary and sufficient conditions for the existence of the solution of a weakly nonlinear system of integrodifferential equations. We propose a convergent iterative procedure for finding solutions of the system and establish a relationship between the necessary and sufficient conditions. Statement of the Problem and Auxiliary ResultsBy using the theory of pseudoinverse Moore-Penrose matrices [1-3], we investigate the conditions of solvability and propose an iterative algorithm for the construction of solutions of weakly nonlinear systems of integrodifferential equations with small parameter "We seek conditions for the existence of a solutionIn what follows, this solution is called a generating solution of the nonlinear system (1). Here, A.t / and B.t / are .m ⇥ n/ matrices,ˆ.t / is an .n ⇥ m/ matrix, f .t / is an .n ⇥ 1/ matrix, and K.t; s/ is an .n ⇥ n/ matrix whose components belong to the space L 2 OEa; bç; the columns of the matrixˆ.t / are linearly independent of OEa; bç; and Z.x.t; "/; t; "/ is an n -dimensional vector function nonlinear with respect to the first component and such that Z.�; t; "/ 2 C 1OEkx � x 0 k  qç; Z.x.�; "/; �; "/ 2 L 2 OEa; bç; Z.x.t; �/; t; �/ 2 C OE0; " 0 ç:First, we present the well-known criterion of solvability of the generating linear system (2) of integrodifferential equations [1]. Theorem 1. System (2) is solvable in the space D 2 OEa; bç of absolutely continuous vector functions on OEa; bç whose derivative belongs to the space L 2 OEa; bç if and only if the inhomogeneity f .t / 2 L 2 OEa; bç satisfies the condition

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Section: Statement Of the Problem And Auxiliary Resultsmentioning

confidence: 99%

Weakly Nonlinear Systems of Integrodifferential Equations

Boichuk

Golovats’ka

2014

J Math Sci

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“…In S.G. Krein's terminology [18, p. 8], such problems are everywhere solvable. However, there exist boundary value problems for which the original operator equation is not everywhere solvable, for example, problems for integro-differential equations [19] and problems for singular differential systems [20][21][22]. In this connection, it is topical to study general boundary value problems for operator equations in a Banach space that are not everywhere solvable, and this is what we deal with in the present paper.…”

Section: Doi: 101134/s0012266114030057mentioning

confidence: 99%

“…where c is an arbitrary element of the space c and (20), we obtain the general solution of the boundary value problem (17), (18) Therefore, the boundary value problem (17), (18) is solvable if and only if conditions (19) and (23) are satisfied, and, in this case, it has the family of solutions…”

Section: )mentioning

confidence: 99%

Linear boundary value problems for normally solvable operator equations in a Banach space

2014

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Abstract-We consider linear boundary value problems for operator equations with generalized-invertible operator in a Banach or Hilbert space. We obtain solvability conditions for such problems and indicate the structure of their solutions. We construct a generalized Green operator and analyze its properties and the relationship with a generalized inverse operator of the linear boundary value problem. The suggested approach is illustrated in detail by an example. DOI: 10.1134/S0012266114030057The classical theory of periodic problems for various classes of differential equations [1][2][3] arose well before the development of methods of functional analysis. Its further development [4][5][6][7][8] used functional analysis as a technique for studying general boundary value problems. For example, on the basis of the theory of the generalized inversion of matrices [9], methods of the investigation of periodic boundary value problems were extended to general boundary value problems for various classes of functional-differential equations: ordinary differential systems in critical (resonance) cases [10]; differential systems with retarded argument [11]; ordinary impulsive differential systems [12,13]. General theorems on the solvability and the representation of solutions of critical boundary value problems were proved for various classes of linear and nonlinear equations, and spaces in which these boundary value problems were considered were generalized.Further development was given by the theory of boundary value problems for ordinary differential equations in a Banach space with the finite-dimensional Euclidean space of values of the unknown function replaced by a more general Banach space [14]. For example, a criterion for the existence of solutions of linear boundary value problems was obtained in [15] for ordinary differential equations in a Banach space in the critical case. Conditions for the existence of periodic solutions of differential and difference equations in the Banach space m of bounded numerical sequences were considered in the monograph [16, p. 266]. Conditions for the existence and bifurcation of bounded, on the entire real line R = (−∞, +∞), solutions of a weakly perturbed differential equation in a Banach space were obtained in [17].From the viewpoint of the theory of operators in function spaces, the above-listed boundary value problems have the following specific features: the original equations in these boundary value problems have solutions for an arbitrary right-hand side. In S.G. Krein's terminology [18, p. 8], such problems are everywhere solvable. However, there exist boundary value problems for which the original operator equation is not everywhere solvable, for example, problems for integro-differential equations [19] and problems for singular differential systems [20][21][22]. In this connection, it is topical to study general boundary value problems for operator equations in a Banach space that are not everywhere solvable, and this is what we deal with in the present paper. STATEMENT O...

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“…8], такие задачи являются везде разрешимыми. Однако существуют краевые задачи, у кото-рых исходное операторное уравнение не является везде разрешимым, например, задачи для интегро-дифференциальных уравнений [19], задачи для сингулярных дифференциальных сис-тем [20][21][22]. В связи с этим актуальной остается задача рассмотрения общих краевых задач для не везде разрешимых операторных уравнений в банаховых пространствах, чему и посвящена настоящая работа.…”

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