# Ordinary differential equations - boundary value problems and asymptotic theory

**Members: **

Alexander Lomtatidze, Zdeněk Opluštil, Jiří Šremr

**Research Focus: **

We focus on the qualitative properties of solutions to ordinary differential equations. In particular, we study the existence and uniqueness of a solution to various boundary value problems both in linear and non-linear cases as well as dependence of solutions on parameters leading to bifurcation problems. We are also interested in the asymptotic properties of solutions to ordinary differential equations, for instance, stability and oscillations of solutions. The equations considered in our research are motivated by the problems appearing in mathematical models of the processes studied in mechanics and other technical disciplines, for example, Duffing type equations, equations with a relativistic differential operator, equations with the so-called \(p\)-laplacian, etc. The main tool in our research is methods of mathematical analysis, which are adapted for the study of the particular problems.

**Collaboration Opportunities: **

The group is open to collaboration in all of the above-mentioned areas. Several tasks we have been working on recently are listed below.

Currently, there is a collaboration with the Viktor Kaplan Department of Fluid Engineering of the Energy Institute, BUT.

**Some Areas of Research, Outcomes:**

The scientific results of the team members are published in the international scientific journals and are presented at scientific conferences and seminars both in the Czech Republic and abroad.

Currently, we are interested in the following problems:

- The existence, multiplicity, and structure of the solution set of the periodic solutions to non-autonomous pendulum equation $$\small u''=p(t)\sin u+q(t),$$ where \(\small p,q\) are arbitrary \(\omega\)-periodic locally lebesgue integrable functions, and the generalization for the relativistic-type equation $$\small \left(\frac{u'}{\sqrt{1-[u']^2}}\right)'=p(t)\sin u+q(t),$$ At this moment, we work on the publication::
- Lomtatidze, A.
*On periodic solutions to relativistic pendulum-type equation.*

- Lomtatidze, A.
- Maximum/anti-maximum principles and their relationship with stability of the linear differential equation $$\small u''+g(t)u'+p(t)u=0$$ with non-constant periodic coefficients. Although this equation seems to be very simple, the properties mentioned are only investigated in detail in the case of a constant damping coefficient \(\small g\). In the case of a non-constant damping coefficient, a lot of interesting questions are still open.
- Existence and the exact multiplicity of periodic solutions to the Duffing-type second-order differential equation. Such equations are usually obtained after approximating nonlinearities in the equations of motion of various one-degree-of-freedom oscillators by using Taylor polynomials of higher degrees.
Our results obtained for equations with non-constant coefficients generalize the properties of autonomous equations. In the autonomous case, i.e., for equations with constant coefficients, the properties mentioned can be usually derived quite easily (e.g., using the behaviour of phase trajectories around equilibria), but in the non-autonomous case, different techniques of the qualitative theory of differential equations need to be employed (theorems on differential inequalities, upper and lower function of the boundary value problems, Fredholm's theorems, Sturm's separation and comparison theorems, etc.). As an example, let us show the periodic boundary value problem $$\small u''=p(t)u+h(t)u^3+\mu f(t),\quad u(0)=u(\omega),\ u'(0)=u'(\omega),$$ where \(\small p,h,f\) are functions lebesgue integrable on the interval \(\small [0,\omega]\) and \(\small\mu\) is a real parameter. Some of our results concerning the existence and multiplicity of positive solutions to this problem depending on the choice of the parameter \(\small\mu\) can be found in the papers:
- Šremr, J.
*Parameter-dependent periodic problems for non-autonomous Duffing equations with sign-changing forcing term.*Electronic Journal of Differential Equations, 2023, roč. 2023, č. 65, s. 1-23. ISSN: 1072-6691. - Šremr, J.
*Bifurcation of positive periodic solutions to non-autonomous undamped Duffing equations.*Mathematics for applications, 2021, roč. 10, č. 1, s. 79-92. ISSN: 1805-3610.

- Lomtatidze, A., Šremr, J.
*On a peridic problem for second order Duffing type equation.*

- Šremr, J.
- We have recently concluded a series of our publications concerning oscillations and non-oscillations of solutions to the half-linear system of ordinary differential equations $$\small \begin{array}{l}u'=g(t)|v|^{1/\alpha}\mathrm{sgn}\,v,\\v'=-p(t)|u|^{\alpha}\mathrm{sgn}\,u,\end{array}$$ where \(\small p,g\) are functions locally lebesgue integrable on the interval \(\small [0,+\infty[\). We may consider the paper
- Lomtatidze, A., Dosoudilová, M., Šremr, J.
*Oscillatory properties of solutions to certain two-dimensional systems of non-linear ordinary differential equations.*Nonlinear Analysis, Theory, Methods and Applications, 2015, roč. 2015(120), č. 1, s. 57-75. ISSN: 0362-546X,

- Opluštil, Z.
*On non-oscillation for certain system of non- linear ordinary differential equations.*Georgian Mathematical Journal, 2017, roč. 24, č. 2, s. 277-286. ISSN: 1072-947X. - Opluštil, Z.
*Oscillatory properties of certain system of non-linear ordinary differential equations.*Miskolc Mathematical Notes (electronic version), 2018, roč. 19, č. 1, s. 439-459. ISSN: 1787-2413. - Opluštil, Z.
*On non-oscillation for two dimensional systems of non-linear ordinary differential equations.*Miskol Mathematical Notes. Currently being printed.

- Lomtatidze, A., Dosoudilová, M., Šremr, J.
- We also continue in the study of the oscillatory properties of solutions to the linear second-order differential equations with a deviating argument. The technique developed for delayed equations in
- Opluštil, Z., Šremr, J.
*Myshkis type oscillation criteria for second-order linear delay differential equations.*Monatshefte fur Mathematik, 2015, roč. 2015 (1), č. 1, s. 143-161. ISSN: 0026-9255

- Opluštil, Z.
*Oscillation criteria for the second-order linear advanced differential equations.*Applied Mathematics Letters.

- Opluštil, Z., Šremr, J.