Abstract
We give a definition of Green’s function of the general boundary value problems for non-self-adjoint second order differential equation with involution. The sufficient conditions for the basis property of system of eigenfunctions are established in the terms of the boundary conditions. Uniform equiconvergence of spectral expansions related to the second-order differential equations with involution:−y″(x)+αy″(−x)+qxyx=λyx,−1<x<1, with the boundary conditions y′−1+b1y−1=0,y′1+b2y1=0, is obtained. As a corollary, it is proved that the eigenfunctions of the perturbed boundary value problems form the basis in L2(−1,1) for any complex-valued coefficient q(x)∈L1(−1,1).
Highlights
On Eigenfunctions of the BoundaryIn this paper we consider in the Hilbert space L2 (−1, 1) a second-order differential operator L defined by differential expressionValue Problems for Second OrderDifferential Equations withInvolution
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affillq y = −y00 ( x ) + αy00 (− x ) + q( x )y( x ), −1 < x < 1, (1)
The spectral problems related to the second-order differential operators with involution have been studied in [10,11,12,13,14]
Summary
In this paper we consider in the Hilbert space L2 (−1, 1) a second-order differential operator L defined by differential expression. The spectral problems related to the second-order differential operators with involution have been studied in [10,11,12,13,14]. The Green’s function of the boundary value problems for the first order equations (and a system of equations) with involution have been derived in [3,15,16,17]. In [12,13,18], the Green’s functions of the second-order differential operators with involution have been investigated and theorems on basicity of eigenfunctions are proved. In order to study the basis property of system of eigenfunctions of the operator L (defined by (1), (2)), we construct the Green’s function G ( x, t, λ) of the problem l0 yq.
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