Uniquely solvable problems for abstract Legendre equation
type: Item
instance of: scholarly article
title: Uniquely solvable problems for abstract Legendre equation
zbMATH Open document ID: 1405.34052
DOI: 10.3103/S1066369X18070010
author: A. V. Glushak
published in: Russian Mathematics
publication date: 2018-10-15T00:00:00Z
review text: Let \(A\) be a closed operator in a Banach space \(E\) with dense domain \(D(A)\) and suppose that \(A\) is the generator of a cosine operator family. Consider in \(E\) an abstract weakly loaded Legendre equation under the form \[ u''(t)+ k\coth(t)\left(u'(t)-\frac{\cosh^{2-k}(t/2)}{\cosh t} u'(0)\right)+\frac{k^2}{4}u(t)=Au(t) \] for \(t>0\). It is proved that for all \(u_0,u_1\in D(A)\) the Cauchy problem for the Legendre equation with the initial conditions \(u(0)=u_0,u'(0)=u_1\) has a unique solution and the explicit formula for the soluion is given. Under some additional conditions on the spectrum of \(A\) the solvability of the Legendre equation with the final condition \(u(1)=u_2,u'(1)=u_2\) on the interval \([0,1]\) is proved. The explicit formula for the solution is given. A result on the solvability of Legendre equation with non-local intergral condition and \(u'(0)=0\) is obtained as well.
reviewed by: Nikita V. Artamonov
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 6954488
zbMATH Keywords: abstract Legendre equation
zbMATH Keywords: Cauchy problem
zbMATH Keywords: nonlocal condition
zbMATH Keywords: boundary value problem
MaRDI profile type: Publication
cites work: Criterion for the solvability of the Cauchy problem for an abstract Euler-Poisson-Darboux equation
cites work: The Bessel operator function
cites work: Q5539141
cites work: Q3331943
cites work: Q5847436
cites work: Q3949414
cites work: The Cauchy problem for the Euler-Poisson-Darboux equation in a homogeneous symmetric Riemannian space. I
cites work: THE CAUCHY PROBLEM FOR THE EULER-POISSON-DARBOUX EQUATION IN A SYMMETRIC SPACE
cites work: The Legendre operator function
cites work: Semigroups of operators, cosine operator functions, and linear differential equations
cites work: Stabilization of the solution of the Cauchy problem for an abstract differential equation of first order
cites work: Q3320799
cites work: Q3141895
cites work: Q3999269
cites work: Q4235164
cites work: Uniqueness theorems for linear non-local problems for abstract differential equations
cites work: Abstract Euler-Poisson-Darboux equation with nonlocal condition
description: scientific article; zbMATH DE number 6954488