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Článek nabízí stručný přehled témat a osobností, které formovaly rozvoj matematiky na Masarykově univerzitě v Brně od jejího založení v roce 1919. Vývoj vědních oborů sledujeme ve čtyřech obdobích historie univerzity.

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$x^{\text{'}\text{'}\text{'}}\left(t\right)+q\left(t\right)x^{\text{'}}\left(t\right)+r\left(t\right){\left|x\right|}^{\lambda }\left(t\right)\mathrm{sgn}x\left(t\right)=0,t\ge 0.$$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $\lambda \le 1$ and if the corresponding second order differential equation $h^{\text{'}\text{'}}+q\left(t\right)h=0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.

In this paper we introduce the definition of coupled point with respect to a (scalar) quadratic functional on a noncompact interval. In terms of coupled points we prove necessary (and sufficient) conditions for the nonnegativity of these functionals.

Singular quadratic functionals of one dependent variable with nonseparated boundary conditions are investigated. Necessary and sufficient conditions for nonnegativity of these functionals are derived using the concept of and . The paper also includes two comparison theorems for coupled points with respect to the various boundary conditions.

We give an equivalence criterion on property A and property B for delay third order linear differential equations. We also give comparison results on properties A and B between linear and nonlinear equations, whereby we only suppose that nonlinearity has superlinear growth near infinity.

The second order linear difference equation $$\Delta \left(r_k▵x_k\right)+c_k{x}_{k+1}=0,\left(1\right)$$ where $r_k\ne 0$ and $k\in \mathbb{Z}$, is considered as a special type of symplectic systems. The concept of the phase for symplectic systems is introduced as the discrete analogy of the Borůvka concept of the phase for second order linear differential equations. Oscillation and nonoscillation of (1) and of symplectic systems are investigated in connection with phases and trigonometric systems. Some applications to summation of number series are given, too.

A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation $$\Delta \left(a_n{\left|\Delta x_n\right|}^{\alpha }sgn\Delta x_n\right)+b_n{\left|x_{n+1}\right|}^{\beta }sgn{x}_{n+1}=0$$ is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.

Consider the third order differential operator $L$ given by $$L(·)\equiv \frac{1}{a_3\left(t\right)}\frac{\text{d}}{\text{d}t}\frac{1}{a_2\left(t\right)}\frac{\text{d}}{\text{d}t}\frac{1}{a_1\left(t\right)}\frac{\text{d}}{\text{d}t}(·)$$ and the related linear differential equation $L\left(x\right)\left(t\right)+x\left(t\right)=0$. We study the relations between $L$, its adjoint operator, the canonical representation of $L$, the operator obtained by a cyclic permutation of coefficients $a_i$, $i=1,2,3$, in $L$ and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).

Some asymptotic properties of principal solutions of the half-linear differential equation $${\left(a\left(t\right)\Phi \left(x^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right)\Phi \left(x\right)=0,(*)$$ $\Phi \left(u\right)={\left|u\right|}^{p-2}u$, $p>1$, is the $p$-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of (), which completes previous results, are presented as well.

We investigate two boundary value problems for the second order differential equation with $p$-Laplacian $${\left(a\left(t\right){\Phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}=b\left(t\right)F\left(x\right),t\in I=[0,\infty ),$$ where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: $$\mathrm{i})x\left(0\right)=c>0,\underset{t\to \infty }{lim}x\left(t\right)=0;\mathrm{ii})x^{\text{'}}\left(0\right)=d<0,\underset{t\to \infty }{lim}x\left(t\right)=0.$$