On the bari basis property of the eigenfunction system of a nonlinear integro-differential equation
This paper deals with the nonlinear integro-differential equation: \[ - u''(x)+ u(x) \int^1_0 k(x- y) u^2(y) dy= \lambda u(x) \] with boundary conditions \(u(0)= u(1)= 0\) and with a normalization condition \(\int^1_0 u^2(x) dx=1\). The existence of pairs of eigenvalues \(\lambda_n\) and the corresponding eigenfunctions \(u_n\) have been completely investigated in other works. With some hypothesis on \(k\), the author proves that exists a pair \((\lambda_n, u_n)\) such that: -- the zeros of \(u_n(x)\) lie exactly at the points \(k/(n+ 1)\), \(k= 0,\pm 1,\pm 2,\dots\); -- \(u_n\) is odd around each zero \(k/(n+ 1)\) and \(u_n(x)> 0\) for \(x\in (0,1/(n+ 1))\); -- \(u_n(x)\) is even round the midpoint of the intervals lying between any two consecutive zeros; -- the family \((u_n(x))_{n= 0,1,\dots}\) is uniformly bounded \((x\in\mathbb{R})\). A second theorem proves that an arbitrary sequence \((u_n)\) satisfying these four conditions is a Bari basis in \(L^2(0,1)\).
scientific article; zbMATH DE number 1916171