Proof.
Before the proof, we need several facts about (1.3)-(1.4). First of all, it is easy to see that the general solution of (1.3)-(1.4) with α2≠1 is given by the following formula:
(2.1)u(x)=C(1+α cosλx1-α2+1-α sinλx1-α2).
Next, we observe that for α2≠1, γ2≠1 eigenvalues are equal to(2.2)λk=1-α2[kπ+arctan1-γ1+γ1+α1-α], k=0,±1,…
The related eigenfunctions are given by the following formula:
(2.3)uk(x)=1+α cos[kπ+arctan1-γ1+γ1+α1-α]x+1-α sin[kπ+arctan1-γ1+γ1+α1-α]x, k=0,±1,…
Observe also that for γ=1, the eigenvalues are
(2.4)λk=1-α2 kπ, k=0,±1,±2,…
The corresponding eigenfunctions are
(2.5)uk(x)=1+α cos(kπx)+1-α sin(kπx), k=0,±1,±2,…;
and for γ=-1, the eigenvalues are
(2.6)λk=1-α2(kπ+π2), k=0,±1,…,
and the corresponding eigenfunctions are
(2.7)uk(x)=1+α cos(kπ+π2)x+1-α sin(kπ+π2)x, k=0,±1,…
We introduce the differential operator L by
(2.8)Lu=u′(-x)+αu′(x), -1<x<1,
and by the boundary condition (1.4). Suppose that Lu belongs to the domain of L, Lu∈D(L). Then we consider
(2.9)L2u=L(u′(-x)+αu′(x))=-(1-α2)u′′(x).
From the boundary condition (1.4), for Lu we deduce that L2 is the second order differential operator generated by the following relations:
(2.10)L2u=-(1-α2)u′′(x), -1<x<1,(2.11)(α-γ)u′(-1)+(1-αγ)u′(1)=0,u(-1)-γu(1)=0.
Spectral problem (2.10)-(2.11) is a typical spectral problem for an ordinary second order differential operator. These problems are studied very well and they have numerous applications (see, for example, [12–14].) Recall [12, Chapter 2] that the following boundary conditions:
(2.12)a1u′(-1)+b1u′(1)+c1u(-1)+d1u(1)=0,c0u(-1)+d0u(1)=0,
for an ordinary second order differential operator are regular if b1c0+a1d0≠0; and they are strongly regular if additionally θ02≠4θ-1θ1, where
(2.13)θ-1=θ1=b1c0+a1d0,θ0=2(a1c0+b1d0).
Since a1=(α-γ), b1=(1-αγ), c0=1, d0=-γ, we obtain θ-1=θ1=γ2-2αγ+1, θ0=2(α-2γ+αγ2). It follows from α2≠1, γ2≠1 and γ≠α±α2-1 that the boundary conditions (2.11) are strongly regular. It is known [12, 13] that the eigenfunctions of an operator with strongly regular boundary conditions constitute a Riesz basis in L2(-1,1). By (2.2) numbers -λk cannot be eigenvalues of L, hence any eigenfunction of L2 which corresponds to λk2 will be an eigenfunction L which corresponds to λk as
(2.14)(L2-λk2E)uk=(L+λkE)(L-λkE)uk=0.
Finally, we deduce the assertion of Theorem 2.1 in the case γ2≠1.
For the case γ2=1 the explicit representations of eigenfunctions (2.5) and (2.7) give the Riesz basis property for these systems directly.
Proof.
The proof is analogous to the proof of Theorem 2.1. It uses the following spectral problem:
(2.16)-u′′(x)+α2u(x)=λu(x), -1<x<1,(2.17)γu′(-1)-u′(1)+α(γ2-1)u(1)=0,u(-1)-γu(1)=0.
Boundary conditions (2.17) are regular for γ2=-1, and nonregular for γ2=1. Then, basis property for eigenfunctions of an ordinary differential second order operator with constant coefficients gives the result for γ2=1.
For γ2≠±1, boundary conditions (2.17) are strongly regular and the proof terminates analogously to the proof of Theorem 2.1.