Bifurcation conditions for a solution of an abstract wave equation

Бойчук, А. А.; Korostil, Igor Andriy; Fečkan, Michal

doi:10.1134/s0012266107040076

Cited by 5 publications

(4 citation statements)

References 5 publications

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“…By analogy with [4,6], we can prove that, for every fixed sufficiently small ε ∈ (0; ε 0 ] , series (15) converges for any t ∈ [a, b]. (11) is sufficient for the existence of solutions of the boundary-value problem (1), (2).…”

Section: Resultsmentioning

confidence: 99%

Conditions for bifurcation of solutions of degenerate boundary-value problems

Boichuk¹,

Shehda

2009

Nonlinear Oscill

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We obtain conditions for the bifurcation of solutions of linear degenerate Noetherian boundary-value problems with small parameter under the assumption that the unperturbed degenerate differential system can be reduced to the central canonical form. Statement of the Problem and Main AssumptionsWe consider the following linear inhomogeneous boundary value-problem with small parameter:where A(t), A 1 (t), B(t), and B 1 (t) are n × n matrices whose components are real functions continuously differentiable sufficiently many times onan n-dimensional column vector from the space C q−1 [a; b] (the value of q is determined according to Theorem 2.1 in [1]), α is an m-dimensional column vector of constants, α ∈ R m , and l and l 1 are linear vector functionals defined on the space of n-dimensional vector functions continuous on [a; b], l = col (l 1 , . . . , l m ) : C[a, b] → R m , l 1 = col l 1 1 , . . . , l 1 m : C[a, b] → R m , l i , l 1 i : C[a, b] → R. Assume that the generating degenerate boundary-value problemwhich is obtained from (1), (2) for ε = 0, does not have solutions for arbitrary inhomogeneities f (t) ∈ C q−1 [a, b] and α ∈ R m . Assume that, by a nondegenerate linear transformation, system (3) can be reduced to a central canonical form [1,2].Let us find conditions for the perturbing coefficients A 1 (t) and B 1 (t) in the differential system (1) and for l 1 in the boundary condition (2) under which the boundary-value problem (1), (2) has a solution.

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Section: Resultsmentioning

confidence: 99%

Conditions for bifurcation of solutions of degenerate boundary-value problems

Boichuk¹,

Shehda

2009

Nonlinear Oscill

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“…It is known (see [18]) that small perturbations preserve the Fredholm property of the operator, i.e., the operator U − n k=1 ε k U k is a Fredholm operator with nonzero index. This enables one to investigate equation ( 26) by the methods of the theory of perturbed operator boundary-value problems with Fredholm linear part (see, e.g., [4,2,7,22,8]) obtained as a generalization of the classical methods of the perturbation theory of periodic boundary-value problems in the theory of oscillations (see [19,15]). The analysis of the appearance of solutions of the equation ( 26) is closely connected with the ((r + d 1 ) × d 2 )-matrix…”

Section: Construction Of the Family Of Solutions Of The Perturbed Boumentioning

confidence: 99%

“…Reduction of the integral equation (1) to the Fredholm equation. In [3], we show that the boundary-value problem for the integral equation with unbounded kernel (3), (4) can be reduced to a boundary-value problem for the Fredholm integral equation and, in view of [2,7,5,6], establish the conditions of solvability and determine the general form of solutions to the generating problem (3), (4). By using the results obtained in [3], we show that the study of the problem of appearance of solutions of the boundary-value problem (1), (2) reduces to the corresponding task for the perturbed boundary-value problem for the Fredholm integral equation.…”

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confidence: 99%

Boundary-value problems for weakly singular integral equations

Boichuk¹,

Feruk²

2022

DCDS-B

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We consider a perturbed linear boundary-value problem for a weakly singular integral equation. Assume that the generating boundary-value problem is unsolvable for arbitrary inhomogeneities. Efficient conditions for the coefficients guaranteeing the appearance of the family of solutions of the perturbed linear boundary-value problem in the form of Laurent series in powers of a small parameter ε with singularity at the point ε = 0 are established.

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“…Similarly, conditions for the existence of solutions bounded on the whole line Z, which turn into one of generating solutions of system (8), for nonlinear difference system were derived. The proposed approach to the analysis of boundaryvalue problems for systems of ordinary differential equations can be also applied (with relevant modifications) to systems with delayed argument [3] and to finding conditions for the bifurcation of a weak T-periodic solution of the linear abstract wave equation and suggest an algorithm for constructing of such solution [4]. Weak solutions were studied and the Vishik-Lyusternik method has permitted one to find the desired weak T-periodic solution in the form of part of the Laurent series.…”

Section: X K C C and X Kmentioning

confidence: 99%

Discrete and Differential Equations in Applied Mathematics

Růžičková¹

2008

Komunikácie

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Bifurcation conditions for a solution of an abstract wave equation

Cited by 5 publications

References 5 publications

Conditions for bifurcation of solutions of degenerate boundary-value problems

Conditions for bifurcation of solutions of degenerate boundary-value problems

Boundary-value problems for weakly singular integral equations

Discrete and Differential Equations in Applied Mathematics

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