In the forthcoming period, we plan to continue in the research devoted to various topics from thegeneral theory of differential equations and integration theory. In particular, we intend to workin the following fields:
- Generalized differential equations and integration theory
We shall conduct research in the integration theory and its applications to generalized differential equations. In particular, following similar ideas of Schwabik's last papers, we will investigate how to characterize the integrals which lie between the Lebesgue integral and the Henstock-Kurzweil integral by an extension process.
In anear future, M.Tvrdý and G.A.Monteiro (in cooperation with A.Slavík, Charles University) will finish their work on the monograph devoted to the abstract Kurzweil-Stieltjes integral and its applications (see [1]). Surely this will bring the need of solving some new and actual problems like e.g. [13, 14].
- Boundary value problems for ordinary differential equations
We shall study various boundary value problems for ordinary differential equations in both regular as well as singular cases (we shall deal with existence, uniqueness, and multiplicity of solutions, well-posedness, sign-constant solutions, theorems on differential inequalities, etc.).
We intend to follow our recent studies in the field of the existence of positive periodic solutions to second-order non-autonomous differential equations of Duffing type and their possible generalizations and extensions. Duffing type equations are characterized by super-linear non-linearities and, as far as we know, there are only afew results dealing with the non-autonomous case. Further, we plan to extend the results (if possible) to equations with damping terms as well as, to "full" two-dimensional systems.
We also plan focus on questions of solvability and unique solvability of second-order ordinary differential equations under mixed and Neumann boundary conditions and of higher-order ordinary differential equations under two-point conjugated and right-focal boundary conditions in theresonance case.
Furthermore, we shall continue in the research of singular problems describing the valveless pumping phenomena.
By using Lyapunov-Schmidt type reductions, non-linear systems possessing certain symmetries will be studied. We shall aim at discovering conditions guaranteeing existence and symmetry ofsolutions. In that case, we would like to establish certain efficient conditions for the stability and hyperbolicity. Averaging theory will also be addressed.
We mention here, for instance, the recent manuscripts [4, 5, 6, 7, 12] that are devoted to some of the above mentioned topics.
- Boundary value problems for functional differential equations
We intend to study various boundary value problems for functional differential equations in the regular as well as singular cases (existence, uniqueness, and multiplicity of solutions, well-posedness, sign-constant solutions, theorems on differential inequalities, etc.).
In particular, we plan to investigate strongly singular differential equations – we would like to generalize the results established for higher-order linear differential equations with deviating arguments as well as for linear functional differential equations of ageneral type.
- Asymptotic properties of solutions
Aparticular attention will be paid to establishing asymptotic formulas which refine information about behavior of solutions in some standard asymptotic classes. Along with the classical equations of Emden-Fowler type and half-linear differential equations, we shall also study in detail equations with more complicated types of nonlinearities (e.g., generalized p-Laplacian involving the mean curvature or relativity operator, nearly linear equations), differential equations with argument deviations, and partial differential equations. We shall examine asymptotic properties of linear and nonlinear difference and dynamic equations. For this purpose, the discrete regular variation and the theory of regularly varying functions will be further developed and applied. Our work will include both establishing discrete analogues of continuous results and purely discrete considerations; we shall try to explain possible discrepancies among the corresponding theories. Dynamic equations on time scales and higher order equations or systems will also be considered.
Some advances in the above issues have already been done very recently, see, e.g., [8-11]. Moreover, P. Řehák was invited byprof. V. Marić, one of the leading persons in the theory of differential equations in the framework of regular variation, to prepare anew monograph on this topic
- Oscillation theory
We shall look for (non)oscillation criteria for various linear and nonlinear difference equations, ordinary and functional differential equations, and dynamic equations on time scales. One of the approaches in nonlinear theory is based on utilizing information about behavior of solutions of associated (half-)linear equations. Aspecial attention will be devoted to certain delicate and difficult cases. Among the rest, we plan to study the role of the graininess in the oscillation theory of dynamic equations. Conjugacy of second order linear and half-linear equations (including the ones with deviated arguments) will be investigated. Since oscillation problems might be related to various special inequalities involving a function and its integral, this aspect will also be part of our study.
We mention here, for instance, the recent manuscripts [2, 3] which are devoted to some of the topics of this section.
- Constructive methods for boundary value problems
We shall investigate their implementation for wider classes of functional differential equations and the corresponding solvability analysis. Different assumptions on the non-linearity will be checked. We shall work on obtaining bifurcation-type results for certain classes of equations in this way. Among the rest, we plan to approach systems with discontinuous trajectories and study the peculiar
features arising in that case. In particular, we should like to approach systems with jumps at variable times and, in particular, with state-dependent jumps. Some discrete time systems will be considered
as well, and conditions necessary for the existence of asolution of agiven kind will be analysed. Methods using the monotonicity properties will also be applied in the existence analysis.
Manuscripts
[1]G.A.Monteiro, A. Slavík, M. Tvrdý, Kurzweil-Stiletjes Integral: Theory and Applications. World Scientific Publishing House, in preparation.
[2]Z. Opluštil, J. Šremr, Myshkis type oscillation criteria for second-order linear delay differential equations, Monatsh. Math., to appear.
[3]M. Dosoudilová, A. Lomtatidze, J. Šremr, Oscillatory properties of solutions to certain two-dimensional systems of non-linear ordinary differential equations, Nonlin. Anal. 120 (2015), 57-75.
[4]A. Lomtatidze, J. Šremr, On periodic solutions to second-order Duffing type equations, in preparation.
[5]A. Lomtatidze, J. Šremr, On the periodic problem for second-order Duffing type equations, in preparation.
[6]A. Lomtatidze, Z. Opluštil, Well-posedness of the second-order linear singular Dirichlet problem, Georgian Math. J., to appear.
[7]A. Lomtatidze, Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations, Mem. Differential Equations Math. Phys., submitted.
[8]S. Matucci, P. Řehák, Extremal solutions to a system of n nonlinear differential equations and regularly varying functions, Math. Nachr., to appear.
[9]P. Řehák, Exponential estimates for solutions of half-linear differential equations, Acta Math. Hungar., to appear
[10]P. Řehák, A few remarks on Poincaré-Perron solutions and regularly varying solutions, Math. Slov., to appear.
[11]P. Řehák, V. Taddei,Solutions of half-linear differential equations in the classes Gamma and Pi, submitted.
[12]J.Á.Cid, G.Infante, M.Tvrdý, M.Zima, A topological approach to periodic oscillations related to the Liebau phenomenon, J. Math. Anal. App. 423 (2), 2015, 1546-1556.
[13]G.A.Monteiro, M. Tvrdý, U.M.Hanung, Bounded convergence theorem for abstract Kurzweil-Stieltjes integral, submitted, preprint arXiv:1412.0993 [math.CA].
[14]G.A.Monteiro, On functions of bounded semivariation, Real Analysis Exchange, to appear.