Sturm–Liouville problems with non-separated eigenvalue dependent boundary conditions

2000 ◽  
Vol 130 (2) ◽  
pp. 239-247 ◽  
Author(s):  
P. A. Binding ◽  
P. J. Browne
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Eigenvalue Dependent Boundary Conditions

Sturm–Liouville differential equations are studied under non-separated boundary conditions whose coefficients are first degree polynomials in the eigenparameter. Situations are examined where there are at most finitely many non-real eigenvalues and also where there are only finitely many real ones.

scholarly journals Coupled Systems of Sequential Caputo and Hadamard Fractional Differential Equations with Coupled Separated Boundary Conditions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 701 ◽  
Author(s):  
Suphawat Asawasamrit ◽  
Sotiris Ntouyas ◽  
Jessada Tariboon ◽  
Woraphak Nithiarayaphaks
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Coupled Systems ◽  
Uniqueness Of Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Hadamard Fractional Differential Equations ◽  
Special Cases ◽  
Separated Boundary Conditions ◽  
Fractional Differential

This paper studies the existence and uniqueness of solutions for a new coupled system of nonlinear sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions, which include as special cases the well-known symmetric boundary conditions. Banach’s contraction principle, Leray–Schauder’s alternative, and Krasnoselskii’s fixed-point theorem were used to derive the desired results, which are well-illustrated with examples.

scholarly journals A Note on Fractional Differential Equations with Fractional Separated Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Bashir Ahmad ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results ◽  
Separated Boundary Conditions ◽  
Fractional Differential

We consider a new class of boundary value problems of nonlinear fractional differential equations with fractional separated boundary conditions. A connection between classical separated and fractional separated boundary conditions is developed. Some new existence and uniqueness results are obtained for this class of problems by using standard fixed point theorems. Some illustrative examples are also discussed.

Factorization Ladders and Eigenfunctions

1949 ◽  
Vol 1 (4) ◽  
pp. 379-396 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Manufacturing Process ◽  
Boundary Value ◽  
Factorization Method ◽  
Sturm Liouville ◽  
The Subject

The eigenfunctions of a boundary value problem are characterized by two quite distinct properties. They are solutions of ordinary differential equations, and they satisfy prescribed boundary conditions. It is a definite advantage to combine these two requirements into a single problem expressed by a unified formula. The use of integral equations is an example in point. The subject of this paper, namely the Schrödinger-Infeld Factorization Method, which is applicable to certain restricted. Sturm-Liouville problems, is based upon another combination of the two properties. The Factorization Method prescribes a manufacturing process.

scholarly journals A solution to the inverse problem for the Sturm-Liouville-type equation with a delay

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1237-1245 ◽  
Author(s):  
Milenko Pikula ◽  
Vladimir Vladicic ◽  
Olivera Markovic
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Fourier Coefficients ◽  
Liouville Problem ◽  
Type Equation ◽  
Liouville Type ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Delay Factor ◽  
Spectral Assignment

The paper is devoted to study of the inverse problem of the boundary spectral assignment of the Sturm-Liouville with a delay. -y'(x) + q(x)y(? ? x) = ?y(x), q ? AS[0, ?], ? ? (0,1] (1) with separated boundary conditions: y(0) = y(?) = 0 (2) y(0) = y'(?) = 0 (3) It is argued that if the sequence of eigenvalues is given ?n(1) and ?n(2) tasks (1-2) and (1-3) respectively, then the delay factor ? ? (0,1) and the potential q ? AS[0, ?] are unambiguous. The potential q is composed by means of trigonometric Fourier coefficients. The method can be easily transferred to the case of ? = 1 i.e. to the classical Sturm-Liouville problem.

11.— A Left Definite Multiparameter Eigenvalue Problem in Ordinary Differential Equations

1976 ◽  
Vol 74 ◽  
pp. 145-155 ◽  
Author(s):  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Ellipticity Condition ◽  
Spectral Parameters ◽  
Multiparameter Eigenvalue Problem ◽  
Sturm Liouville ◽  
Image Position

SynopsisThe main result of this paper is to establish the completeness of the eigenfunctions for the multiparameter eigenvalue problem defined by the system of ordinary differential equations0 ≤ x, ≤ 1, r = 1, …, k, subject to the Sturm-Liouville boundary conditionsr = 1, …, k. In addition it is assumed that the coefficients ars of the spectral parameters λs, satisfy the ellipticity condition , s = 1, …, k, for all xrɛ[0, 1], r = 1, …, k, and some real k-tuple μ1, …, μk and where is the co-factor of asr in the determinant . The theory developed here contrasts with the results known when …k is assumed non-vanishing for all xrɛ[0,1].

X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

1971 ◽  
Vol 69 (2) ◽  
pp. 139-148
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter ◽  
General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

Darboux transformations and the factorization of generalized Sturm–Liouville problems

2010 ◽  
Vol 140 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Paul A. Binding ◽  
Patrick J. Browne ◽  
Bruce A. Watson
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Darboux Transformation ◽  
Darboux Transformations ◽  
Sturm Liouville ◽  
Eigenvalue Dependent Boundary Conditions

A version of the Darboux transformation is explored for Sturm-Liouville problems with eigenvalue-dependent boundary conditions, from differential-equation and operator-theoretic viewpoints. Some of the literature on Darboux's transformation is related in a historical introduction.

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