Properties in L p of root functions for a nonlocal problem with involution

Abstract

The spectral problem −u ′′ (x) + αu ′′ (−x) = λu(x) , −1 < x < 1 , with nonlocal boundary conditions u(−1) = βu(1) , u ′ (−1) = u ′ (1) , is studied in the spaces Lp(−1, 1) for any α ∈ (−1, 1) and β ̸ = ±1. It is proved that if r = √ (1 − α)/(1 + α) is irrational then the system of its eigenfunctions is complete and minimal in Lp(−1, 1) for any p > 1 , but does not form a basis. In the case of a rational value of r , the way of supplying this system with associated functions is specified to make all the root functions a basis in Lp(−1, 1) .

Direct and inverse problems for nonlocal heat equation with boundary conditions of periodic type

“…Spectral problems related to our topic were considered in the works [27][28][29][30][31][32]. In [27] a problem with the nonlocal conditions y(-1) = 0, y (-1) = y (1)…”

Inverse spectral problems for functional-differential operators with involution

“…There is also a number of studies devoted to the first-order (see [8][9][10][11][12][13][14][15]) and second-order FDO with involution (see [16][17][18][19][20][21][22]). The majority of results of the mentioned papers are concerned with the basis property of eigenfunctions, eigenvalue asymptotics, and other issues of direct spectral theory.…”

On Eigenfunctions and Eigenvalues of a Nonlocal Laplace Operator with Multiple Involution

“…The monographs of D. Przeworska-Rolewicz [3] and J. Wiener [4] are devoted to the theory of solvability of various differential equations with involution. In papers [5][6][7][8][9][10][11][12][13][14], spectral problems for differential operators of the first and second orders with involution were studied. In [15][16][17][18][19][20][21][22], the results of studying spectral problems with involution are used to solve inverse problems.…”