COLLOQUIUM MATHEMATICUM VOL. 109 2007 NO. 1 MINIMALITY OF THE SYSTEM OF ROOT FUNCTIONS OF STURM–LIOUVILLE PROBLEMS WITH DECREASING AFFINE BOUNDARY CONDITIONS BY Y. N. ALIYEV (Baku) Abstract. We consider Sturm–Liouville problems with a boundary condition linearly dependent on the eigenparameter. We study the case of decreasing dependence where non-real and multiple eigenvalues are possible. By determining the explicit form of a biorthogonal system, we prove that the system of root (i.e. eigen and associated) functions, with an arbitrary element removed, is a minimal system in L 2 (0, 1), except for some cases where this system is neither complete nor minimal. Introduction. Consider the following spectral problem: −y ′′ + q(x)y = λy, 0 <x< 1, (0.1) y ′ (0) sin β = y(0) cos β, (0.2) y ′ (1) = (aλ + b)y(1), (0.3) where a, b, β are real constants, 0 ≤ β<π, a< 0, λ is a spectral parameter and q(x) is a real-valued and continuous function over the interval [0, 1]. It was proved in [2] (see also [1]) that the eigenvalues of the boundary value problem (0.1)–(0.3) form an infinite sequence accumulating only at ∞ and only the following cases are possible: (a) all eigenvalues are real and simple; (b) all eigenvalues are real and all, except one double, are simple; (c) all eigenvalues are real and all, except one triple, are simple; (d) all eigen- values are simple and all, except a conjugate pair of non-real ones, are real. Let {v n } ∞ n=1 be a sequence of elements from L 2 (0, 1) and V k the closure (in the norm of L 2 (0, 1)) of the linear span of {v n } ∞ n=1,n=k . The system {v n } ∞ n=1 is called minimal in L 2 (0, 1) if v k / ∈ V k for all k =1, 2,... (see [9, Ch. I, §2]). The present article concerns the minimality in L 2 (0, 1) of the system of root functions of the boundary value problem (0.1)–(0.3). In cases (a) and (d), we complete the results of [2] by showing that the system of eigen- functions of (0.1)–(0.3), with an arbitrary element removed, is minimal in 2000 Mathematics Subject Classification : 34B24, 34L10. Key words and phrases : Sturm–Liouville, eigenparameter-dependent boundary condi- tions, minimal system, root functions, completeness, basis. [147] c  Instytut Matematyczny PAN, 2007