Scientific Activity

Main directions of scientific research at the department

• Theory of oscillations

a) New classes of almost periodic functions and sequences have been constructed that do not coincide with the classes of almost periodic functions and sequences according to Bochner;

b) A theory of Amério-Favaro without H-classes has been developed for differential, differential-functional, functional, difference, and discrete equations;

c) New criteria for the existence of almost periodic, Poisson stable and bounded solutions of broad classes of linear and nonlinear evolutionary equations have been found;

d) In the general case, the representation of bounded solutions of linear non-autonomous discrete equations has been found; e) Conditions for the solvability of difference equations with non-uniformly compressing operators in the space of two-sided sequences have been established;

• Theory of stability of dynamic systems

a) Necessary and sufficient conditions for absolute instability of solutions of linear difference and differential-difference equations with self-adjoint operator coefficients have been found;

b) The instability of unbounded solutions of differential, difference, and differential-difference equations with operator coefficients that commute with rotation operators has been demonstrated;

c) Sufficient conditions for absolute instability of solutions of functional and differential-functional equations with operator coefficients have been established.

• New methods of nonlinear functional analysis

a) Necessary and sufficient conditions for the invertibility of nonlinear differentiable mappings acting in arbitrary Banach spaces have been obtained;

b) General theorems on fixed points for expansive operators have been developed.

c) A method of local linear approximation in the theory of nonlinear functional equations has been constructed, allowing for the establishment of conditions for the existence of solutions to these equations;

d) The concept of an L-injective operator has been introduced and conditions for the solvability of functional equations with a differentiable L-injective operator have been established.

• Celestial mechanics with finite speed of gravity

a) A mathematical model of the solar system has been constructed that takes into account the speed of gravity, which is more accurate than the classical Newtonian model;

b) A method for constructing models of stellar systems using differential equations with delayed arguments and functional equations has been developed;

c) The non-Keplerian nature and instability of the motion of two bodies caused by the finiteness of the speed of gravity have been shown, in particular, the law of increasing sector speed has been discovered;

d) It has been shown that Kepler's laws are not applicable in celestial mechanics with finite speed of gravity;

e) Systems of differential equations with delays and restrictions on delays and derivatives of solutions, used in celestial mechanics taking into account the speed of gravity, have been studied.

• Problems of teaching higher mathematics

a) An algorithm for rapid integration of rational functions without using classical methods of integrating these functions has been constructed;

b) A general integral criterion of convergence for arbitrary numerical, vector, and operator series has been established.

The results of scientific research at the Department of Higher Mathematics have been published in 32 articles in specialized publications and 5 articles in foreign publications (20 and 13 of them are included in the Scopus and Web of Science scientometric databases, respectively).

The department closely collaborates with leading educational and scientific institutions: Taras Shevchenko National University of Kyiv, Ivan Franko National University of Lviv, Yuriy Fedkovych Chernivtsi National University, Institute of Mathematics of the NAS of Ukraine, Institute of Mechanics of the NAS of Ukraine, and other higher educational institutions of Ukraine.