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

Abstract

It is known that the eigenvalues n(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 (*) n ! 0, when n ! 1, 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 = (􀀀1;1)) is proved. Twosided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear Sturm- Liouville 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.