Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation

Abstract

It is known that the eigenvalues λn(n = 1, 2, ...) numbered in decreasing order and taking the multiplicity of the self-adjoint Sturm-Liouville operator with a completely continuous inverse operator L⁻¹ have the following property (^∗) λ_n → 0, when n → ∞, moreover, than the faster convergence to zero so the operator L⁻¹ is best approximated by finite rank operators.

The following question:
- Is it possible for a given nonlinear operator to indicate a decreasing numerical sequence characterized by the property (^∗)?

naturally arises for nonlinear operators. In this paper, we study the above question for the nonlinear Sturm-Liouville operator. To solve the above problem the theorem on the maximum regularity of the solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space L₂(R) (R = (−∞, ∞)) is proved. Twosided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear SturmLiouville equation are also obtained. As is known, the obtained estimates of Kolmogorov widths give the opportunity to choose approximation apparatus that guarantees the minimum possible error.