Boundary-value problems for differential equations in a Banach space

Boichuk, О. A.; Panasenko, E. V.

doi:10.1007/s11072-009-0055-7

Cited by 10 publications

(21 citation statements)

References 4 publications

(11 reference statements)

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“…When considering linear boundary value problems of the form (1), (2) for the everywhere solvable differential operator Lz(t) = z (t) − A(t)z(t) that acts from the Banach space C 1 (I, B) of continuously differentiable functions ranging in the Banach space B into the Banach space C(I, B) of continuous vector functions with supremum norm, we obtain the well-known results in [15].…”

Section: Whereẑ(t) Is An Arbitrary Element Of the Banach Space L ∞ (Imentioning

confidence: 69%

“…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].…”

Section: Doi: 101134/s0012266114030057mentioning

confidence: 99%

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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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Section: Whereẑ(t) Is An Arbitrary Element Of the Banach Space L ∞ (Imentioning

confidence: 69%

Section: Doi: 101134/s0012266114030057mentioning

confidence: 99%

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

2014

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“…In the case where Lz.t / D z 0 .t/ A.t/z.t/ is an everywhere solvable differential operator, boundary-value problems of the form (2), (3) were considered in [10,11].…”

Section: Remarkmentioning

confidence: 99%

“…Following [11], we call the boundary-value problems described by Theorems 3 and 4 n-normally solvable and d -normally solvable.…”

Section: Remarkmentioning

confidence: 99%

“…In the case of finite-dimensional boundary-value problems for differential systems of equations in Banach spaces ..Lz/.t / Á z 0 .t / A.t /z.t //; problems of this type were considered in [11]. In this case, the generalized inverse operator L admits an integral representation, and the generalized Green operator has the form Consider the problem of necessary and sufficient conditions for the solvability of the linear inhomogeneous boundary-value problem (2), (3) and the structure of the set of its solutions z.t/ 2 l 1 .I; B 1 / .…”

Section: Remarkmentioning

confidence: 99%

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Boundary-value problems for linear equations with a generalized invertible operator in a Banach space with basis

Журавлев

2011

Nonlinear Oscill

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We consider linear boundary-value problems for operator equations with generalized invertible operators in Banach spaces that have bases. Using the technique of generalized inverse operators applied to generalized invertible operators in Banach spaces, we establish conditions for the solvability of linear boundary-value problems for these operator equations and obtain formulas for the representation of their solutions. We consider special cases of these boundary-value problems, namely, so-called n-and d -normally solvable boundary-value problems as well as normally solvable problems for Noetherian operator equations.

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Juvenile Probation Officer Self-Assessed Mental Health Competency as a Predictor of Case Management Practices

Holloway

Cruise

Downs

et al. 2016

Adm Policy Ment Health

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Justice-involved youth endorse high rates of mental health problems. Juvenile probation is the most common disposition in the justice system and juvenile probation officers (JPOs) are crucial for connecting justice-involved youth with appropriate care. We examined the role of mental health competency on the use of self-report case management strategy types (deterrence, restorative justice, and treatment) by JPOs and whether jurisdiction-level differences were relevant. Results suggest that mental health competency predicted use of restorative justice and treatment strategies and all three strategy types varied at the county level. The role of mental health competency in use of treatment strategies is relevant to connecting justice-involved youth to mental health care. Furthermore, a substantial amount of the variance predicting the use of all three strategies was accounted for at the county level.

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Boundary-value problems for differential equations in a Banach space

Cited by 10 publications

References 4 publications

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

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

Boundary-value problems for linear equations with a generalized invertible operator in a Banach space with basis

Juvenile Probation Officer Self-Assessed Mental Health Competency as a Predictor of Case Management Practices

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