This Licentiate thesis deals with the study of existence and uniqueness together with coercive estimates for solutions of certain differential equations. The thesis consists of four papers (papers A, B, C and D) and an introduction, which put these papers into a more general frame and which also serves as an overview of this interesting field of mathematics. In the text below the functions r(x), q(x), m(x) etc. are functions on (-∞,+∞), which are different but well defined in each paper. In paper A we study the separation and approximation properties for the differential operator ly=-y″+r(x)y′+q(x)y in the Hilbert space L2 :=L2(R), R=(-∞,+∞), as well as the existence problem for a second order nonlinear differential equation in L2 . Paper B deals with the study of separation and approximation properties for the differential operator ly=-y″+r(x)y′+s(x)‾y′ in the Hilbert spaceL2:=L2(R), R=(-∞,+∞), (here ¯y is the complex conjugate of y). A coercive estimate for the solution of the second order differential equation ly =f is obtained and its applications to spectral problems for the corresponding differential operatorlis demonstrated. Some sufficient conditions for the existence of the solutions of a class of nonlinear second order differential equations on the real axis are obtained. In paper C we study questions of the existence and uniqueness of solutions of the third order differential equation (L+λE)y:=-m(x)(m(x)y′)″+[q(x)+ir(x)+λ]y=f(x), (0.1) and conditions, which provide the following estimate: ||m(x)(m(x)y′)″||pp+||(q(x)+ir(x)+λ)y||pp≤ c||f(x)||pp for a solution y of (0.1). Paper D is devoted to the study of the existence and uniqueness for the solutions of the following more general third order differential equation with unbounded coefficients: -μ1(x)(μ2(x)(μ1(x)y′)′)′+(q(x)+ir(x)+λ)y=f(x). Some new existence and uniqueness results are proved and some normestimates of the solutions are given.

In this paper some new existence and uniqueness results are proved and maximal regularity estimates of solutions of third-order differential equation with unbounded coefficients are given.

A coercive estimate for a solution of a degenerate second order di fferential equation is installed, and its applications to spectral problems for the corresponding dif ferential operator is demonstrated. The suffi cient conditions for existence of the solutions of one class of the nonlinear second order diff erential equations on the real axis are obtained.

Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University.

Ospanov, K. N.

Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University.

Zulkhazhav, A.

Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University.

Compactness of the resolvent of one second order differential operator2014In: 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences (ICNPAA 2014) / [ed] Sivasundaram, Seenith, American Institute of Physics (AIP), 2014, Vol. 1637, no 1, p. 13-17, article id 13Conference paper (Refereed)

Abstract [en]

In this work a necessary and sufficient condition for the compactness of the resolvent of one second order degenerate differential operator in L-2 is obtained. We also discuss the two-sided estimates for the radius of fredholmness of this operator.

Department of Pure Mathematics, Eurasian National University.

Persson, Lars-Erik

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Department of Mathematics, Narvik University College.

Ospanov, K.N.

Department of Pure Mathematics, Eurasian National University.

Wall, Peter

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.

We consider the following third-order differential equation with unbounded coefficients:Some new existence and uniqueness results are proved, and precise estimates of the norms of the solutions are given. The obtained results may be regarded as a unification and extension of all other results of this type

Sufficient conditions for the invertibility and separability in L2(−∞,+∞) of the degenerate second order differential operator with complex-valued coefficients are obtained, and its applications to the spectral and approximate problems are demonstrated. Usinga separability theorem, which is obtained for the linear case, the solvability of nonlinear second order differential equation is proved on the real axis.

8.

Ospanov, K.N.

et al.

Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University.

Akhmetkaliyeva, Raya D.

Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University.

We study coercive estimates for some second-order degenerate and damped differential operators with unbounded coefficients. We also establish the conditions for invertibility of these operators.