In this paper, we consider a third order singular differential operator L w + μ w = - w ′′′ + q ( x ) w + μ w in space L 2 ( R ) originally defined on the set C 0 ∞ ( R ) , where C 0 ∞ ( R ) is the set of infinitely differentiable compactly supported functions, μ≥0. Regarding the coefficient q ( x ) , we assume that it is a continuous function in R ( - ∞ , ∞ ) and can be a growing function at infinity. The operator L allows closure in the space L 2 ( R ) and the closure also be denoted by L. In the paper, under certain restrictions on q ( x ) , in addition to the above condition, the existence of the resolvent of the operator L and the existence of the estimate ‖ - w ′′′ ‖ L 2 ( R ) + ‖ q ( x ) w ‖ L 2 ( R ) ≤ c ( ‖ L w ‖ L 2 ( R ) + ‖ w ‖ L 2 ( R ) ) (0.1) have been proved, where c>0 is a constant. Example. Let q ( x ) = e 100 | x | , then the estimate (0.1) holds. The compactness of the resolvent is proved and two-sided estimates for singular numbers ( s-numbers) are obtained. Here we note that the estimates of singular numbers ( s-numbers) show the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. In the present paper, apparently for the first time, the nuclearity of the resolvent of the third-order differential operator and completeness of its root vectors are proved in the case of an unbounded domain with a greatly growing coefficient q ( x ) at infinity.

A theorem on the maximum regularity of solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space $L_2(R)\,(R=(-\infty,\infty))$ is proved in this paper. Two-sided 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 given the opportunity to choose approximation apparatus that guarantees the maximum possible error.

a) | k ( w ) | ≥ 0 is a piecewise continuous and bounded function in R = ( - ∞ , ∞ ) . The coefficients b ( w ) and q ( w ) are continuous functions in R and can be unbounded at infinity. The operator L admits closure in the space L 2 ( Ω ) and the closure is also denoted by L. Taking into consideration certain constraints on the coefficients b ( w ) q ( w ) , apart from the above-mentioned conditions, the existence of a bounded inverse operator is proved in this paper; a condition guaranteeing compactness of the resolvent kernel is found; and we also obtained two-sided estimates for singular numbers ( s-numbers). Here we note that the estimate of singular numbers ( s-numbers) shows the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. It is given an example of how the obtained estimates for the s-numbers enable to identify the estimates for the eigenvalues of the operator L. We note that the above results are apparently obtained for the first time for a mixed-type operator in the case of an unbounded domain with rapidly oscillating and greatly growing coefficients at infinity.