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We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper.

We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper.

We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper.

We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper.

The paper presents the results of research related to the possibility of using an intelligent workpiece holder with adjustable stiffness, during end milling process. Machining a one side supported flexible workpiece will be performed with constant spindle speed and feed speed. In order to avoid hazardous vibration, stiffness of the especially designed spring (mounted in a workpiece holder) will be modified off-line. In order to predict the accuracy of the proposed method, appropriate simulations were performed. As a simplified model of the flat workpiece it was applied the Euler-Bernoulli bar with possibility of identifying the two lowest normal modes. On this basis and for approaching optimal spindle speeds, the mean of adjusting relevant stiffness of the holder is invented.

The paper presents the results of research related to the possibility of using an intelligent workpiece holder with adjustable stiffness, during end milling process. Machining a one side supported flexible workpiece will be performed with constant spindle speed and feed speed. In order to avoid hazardous vibration, stiffness of the especially designed spring (mounted in a workpiece holder) will be modified off-line. In order to predict the accuracy of the proposed method, appropriate simulations were performed. As a simplified model of the flat workpiece it was applied the Euler-Bernoulli bar with possibility of identifying the two lowest normal modes. On this basis and for approaching optimal spindle speeds, the mean of adjusting relevant stiffness of the holder is invented.

The paper presents the results of research related to the possibility of using an intelligent workpiece holder with adjustable stiffness, during end milling process. Machining a one side supported flexible workpiece will be performed with constant spindle speed and feed speed. In order to avoid hazardous vibration, stiffness of the especially designed spring (mounted in a workpiece holder) will be modified off-line. In order to predict the accuracy of the proposed method, appropriate simulations were performed. As a simplified model of the flat workpiece it was applied the Euler-Bernoulli bar with possibility of identifying the two lowest normal modes. On this basis and for approaching optimal spindle speeds, the mean of adjusting relevant stiffness of the holder is invented.

The paper presents the results of research related to the possibility of using an intelligent workpiece holder with adjustable stiffness, during end milling process. Machining a one side supported flexible workpiece will be performed with constant spindle speed and feed speed. In order to avoid hazardous vibration, stiffness of the especially designed spring (mounted in a workpiece holder) will be modified off-line. In order to predict the accuracy of the proposed method, appropriate simulations were performed. As a simplified model of the flat workpiece it was applied the Euler-Bernoulli bar with possibility of identifying the two lowest normal modes. On this basis and for approaching optimal spindle speeds, the mean of adjusting relevant stiffness of the holder is invented.

V literatuře je často studována Emden--Fowlerova nelineární diferenciální rovnice druhého řádu $$ y'' \pm x^\alpha y^m = 0, $$ kde $\alpha$ a $m$ jsou konstanty. V disertační práci je analyzována diskrétní analogie Emden-Fowlerovy diferenciální rovnice $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ kde $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ je nezávislá proměnná, $k_0$ je celé číslo a $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ je řešení. V této rovnici je $\Delta^2u(k)=\Delta(\Delta u(k))$, kde $\Delta u(k)$ je diference vpřed prvního řádu funkce $u(k)$, tj. $\Delta u(k) = u(k+1)-u(k)$ a $\Delta^2 (k)$ je její diference vpřed druhého řádu, tj. $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, a $\alpha$, $m$ jsou reálná čísla. Je diskutováno asymptotické chování řešení této rovnice a jsou stanoveny podmínky, garantující existence řešení s asymptotikou mocninného typu: $u(k) \sim {1}/{k^s}$, kde $s$ je vhodná konstanta. Je také zkoumána diskrétní analogie tzv. ``blow-up'' řešení (neohraničených řešení) známých v klasické teorii diferenciálních rovnic, tj. řešení pro která v některém bodě $x^*$ platí $\lim_{x \to x^*} y(x)= \infty$, kde $y(x)$ je řešení Emden-Fowlerovy diferenciální rovnice $$ y''(x) = y^s(x), $$ kde $s \ne 1$ je reálné číslo. Výsledky jsou ilustrovány příklady a porovnávány s výsledky doposud známými., In the literature a differential second--order nonlinear Emden--Fowler equation $$ y'' \pm x^\alpha y^m = 0, $$ where $\alpha$ and $m$ are constants, is often investigated. This thesis deals with a discrete equivalent of the second--order Emden-Fowler differential equation $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ where $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ is an independent variable, $k_0$ is an integer and $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ is an unknown solution. In this equation, $\Delta^2u(k)=\Delta(\Delta u(k))$, $\Delta u(k)$ is the the first-order forward difference of $u(k)$, i.e., $\Delta u(k) = u(k+1)-u(k)$, and $\Delta^2 (k)$ is its second--order forward difference, i.e., $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, $\alpha$, $m$ are real numbers. The asymptotic behaviour of the solutions to this equation is discussed and the conditions are found such that there exists a power-type asymptotic: $u(k) \sim {1}/{k^s}$, where $s$ is some constant. We also discuss a discrete analogy of so-called ``blow-up'' solutions in the classical theory of differential equations, i.e., the solutions for which there exists a point $x^*$ such that $\lim_{x \to x^*} y(x) = \infty$, where $y(x)$ is a solution of the Emden-Fowler differential equation $$ y''(x) = y^s(x), $$ with $s \ne 1$ being a real number. The results obtained are compared to those already known and illustrated with examples.

V literatuře je často studována Emden--Fowlerova nelineární diferenciální rovnice druhého řádu $$ y'' \pm x^\alpha y^m = 0, $$ kde $\alpha$ a $m$ jsou konstanty. V disertační práci je analyzována diskrétní analogie Emden-Fowlerovy diferenciální rovnice $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ kde $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ je nezávislá proměnná, $k_0$ je celé číslo a $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ je řešení. V této rovnici je $\Delta^2u(k)=\Delta(\Delta u(k))$, kde $\Delta u(k)$ je diference vpřed prvního řádu funkce $u(k)$, tj. $\Delta u(k) = u(k+1)-u(k)$ a $\Delta^2 (k)$ je její diference vpřed druhého řádu, tj. $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, a $\alpha$, $m$ jsou reálná čísla. Je diskutováno asymptotické chování řešení této rovnice a jsou stanoveny podmínky, garantující existence řešení s asymptotikou mocninného typu: $u(k) \sim {1}/{k^s}$, kde $s$ je vhodná konstanta. Je také zkoumána diskrétní analogie tzv. ``blow-up'' řešení (neohraničených řešení) známých v klasické teorii diferenciálních rovnic, tj. řešení pro která v některém bodě $x^*$ platí $\lim_{x \to x^*} y(x)= \infty$, kde $y(x)$ je řešení Emden-Fowlerovy diferenciální rovnice $$ y''(x) = y^s(x), $$ kde $s \ne 1$ je reálné číslo. Výsledky jsou ilustrovány příklady a porovnávány s výsledky doposud známými., In the literature a differential second--order nonlinear Emden--Fowler equation $$ y'' \pm x^\alpha y^m = 0, $$ where $\alpha$ and $m$ are constants, is often investigated. This thesis deals with a discrete equivalent of the second--order Emden-Fowler differential equation $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ where $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ is an independent variable, $k_0$ is an integer and $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ is an unknown solution. In this equation, $\Delta^2u(k)=\Delta(\Delta u(k))$, $\Delta u(k)$ is the the first-order forward difference of $u(k)$, i.e., $\Delta u(k) = u(k+1)-u(k)$, and $\Delta^2 (k)$ is its second--order forward difference, i.e., $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, $\alpha$, $m$ are real numbers. The asymptotic behaviour of the solutions to this equation is discussed and the conditions are found such that there exists a power-type asymptotic: $u(k) \sim {1}/{k^s}$, where $s$ is some constant. We also discuss a discrete analogy of so-called ``blow-up'' solutions in the classical theory of differential equations, i.e., the solutions for which there exists a point $x^*$ such that $\lim_{x \to x^*} y(x) = \infty$, where $y(x)$ is a solution of the Emden-Fowler differential equation $$ y''(x) = y^s(x), $$ with $s \ne 1$ being a real number. The results obtained are compared to those already known and illustrated with examples.

V literatuře je často studována Emden--Fowlerova nelineární diferenciální rovnice druhého řádu $$ y'' \pm x^\alpha y^m = 0, $$ kde $\alpha$ a $m$ jsou konstanty. V disertační práci je analyzována diskrétní analogie Emden-Fowlerovy diferenciální rovnice $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ kde $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ je nezávislá proměnná, $k_0$ je celé číslo a $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ je řešení. V této rovnici je $\Delta^2u(k)=\Delta(\Delta u(k))$, kde $\Delta u(k)$ je diference vpřed prvního řádu funkce $u(k)$, tj. $\Delta u(k) = u(k+1)-u(k)$ a $\Delta^2 (k)$ je její diference vpřed druhého řádu, tj. $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, a $\alpha$, $m$ jsou reálná čísla. Je diskutováno asymptotické chování řešení této rovnice a jsou stanoveny podmínky, garantující existence řešení s asymptotikou mocninného typu: $u(k) \sim {1}/{k^s}$, kde $s$ je vhodná konstanta. Je také zkoumána diskrétní analogie tzv. ``blow-up'' řešení (neohraničených řešení) známých v klasické teorii diferenciálních rovnic, tj. řešení pro která v některém bodě $x^*$ platí $\lim_{x \to x^*} y(x)= \infty$, kde $y(x)$ je řešení Emden-Fowlerovy diferenciální rovnice $$ y''(x) = y^s(x), $$ kde $s \ne 1$ je reálné číslo. Výsledky jsou ilustrovány příklady a porovnávány s výsledky doposud známými., In the literature a differential second--order nonlinear Emden--Fowler equation $$ y'' \pm x^\alpha y^m = 0, $$ where $\alpha$ and $m$ are constants, is often investigated. This thesis deals with a discrete equivalent of the second--order Emden-Fowler differential equation $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ where $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ is an independent variable, $k_0$ is an integer and $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ is an unknown solution. In this equation, $\Delta^2u(k)=\Delta(\Delta u(k))$, $\Delta u(k)$ is the the first-order forward difference of $u(k)$, i.e., $\Delta u(k) = u(k+1)-u(k)$, and $\Delta^2 (k)$ is its second--order forward difference, i.e., $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, $\alpha$, $m$ are real numbers. The asymptotic behaviour of the solutions to this equation is discussed and the conditions are found such that there exists a power-type asymptotic: $u(k) \sim {1}/{k^s}$, where $s$ is some constant. We also discuss a discrete analogy of so-called ``blow-up'' solutions in the classical theory of differential equations, i.e., the solutions for which there exists a point $x^*$ such that $\lim_{x \to x^*} y(x) = \infty$, where $y(x)$ is a solution of the Emden-Fowler differential equation $$ y''(x) = y^s(x), $$ with $s \ne 1$ being a real number. The results obtained are compared to those already known and illustrated with examples.

V literatuře je často studována Emden--Fowlerova nelineární diferenciální rovnice druhého řádu $$ y'' \pm x^\alpha y^m = 0, $$ kde $\alpha$ a $m$ jsou konstanty. V disertační práci je analyzována diskrétní analogie Emden-Fowlerovy diferenciální rovnice $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ kde $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ je nezávislá proměnná, $k_0$ je celé číslo a $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ je řešení. V této rovnici je $\Delta^2u(k)=\Delta(\Delta u(k))$, kde $\Delta u(k)$ je diference vpřed prvního řádu funkce $u(k)$, tj. $\Delta u(k) = u(k+1)-u(k)$ a $\Delta^2 (k)$ je její diference vpřed druhého řádu, tj. $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, a $\alpha$, $m$ jsou reálná čísla. Je diskutováno asymptotické chování řešení této rovnice a jsou stanoveny podmínky, garantující existence řešení s asymptotikou mocninného typu: $u(k) \sim {1}/{k^s}$, kde $s$ je vhodná konstanta. Je také zkoumána diskrétní analogie tzv. ``blow-up'' řešení (neohraničených řešení) známých v klasické teorii diferenciálních rovnic, tj. řešení pro která v některém bodě $x^*$ platí $\lim_{x \to x^*} y(x)= \infty$, kde $y(x)$ je řešení Emden-Fowlerovy diferenciální rovnice $$ y''(x) = y^s(x), $$ kde $s \ne 1$ je reálné číslo. Výsledky jsou ilustrovány příklady a porovnávány s výsledky doposud známými., In the literature a differential second--order nonlinear Emden--Fowler equation $$ y'' \pm x^\alpha y^m = 0, $$ where $\alpha$ and $m$ are constants, is often investigated. This thesis deals with a discrete equivalent of the second--order Emden-Fowler differential equation $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ where $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ is an independent variable, $k_0$ is an integer and $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ is an unknown solution. In this equation, $\Delta^2u(k)=\Delta(\Delta u(k))$, $\Delta u(k)$ is the the first-order forward difference of $u(k)$, i.e., $\Delta u(k) = u(k+1)-u(k)$, and $\Delta^2 (k)$ is its second--order forward difference, i.e., $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, $\alpha$, $m$ are real numbers. The asymptotic behaviour of the solutions to this equation is discussed and the conditions are found such that there exists a power-type asymptotic: $u(k) \sim {1}/{k^s}$, where $s$ is some constant. We also discuss a discrete analogy of so-called ``blow-up'' solutions in the classical theory of differential equations, i.e., the solutions for which there exists a point $x^*$ such that $\lim_{x \to x^*} y(x) = \infty$, where $y(x)$ is a solution of the Emden-Fowler differential equation $$ y''(x) = y^s(x), $$ with $s \ne 1$ being a real number. The results obtained are compared to those already known and illustrated with examples.

V literatuře je často studována Emden--Fowlerova nelineární diferenciální rovnice druhého řádu $$ y'' \pm x^\alpha y^m = 0, $$ kde $\alpha$ a $m$ jsou konstanty. V disertační práci je analyzována diskrétní analogie Emden-Fowlerovy diferenciální rovnice $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ kde $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ je nezávislá proměnná, $k_0$ je celé číslo a $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ je řešení. V této rovnici je $\Delta^2u(k)=\Delta(\Delta u(k))$, kde $\Delta u(k)$ je diference vpřed prvního řádu funkce $u(k)$, tj. $\Delta u(k) = u(k+1)-u(k)$ a $\Delta^2 (k)$ je její diference vpřed druhého řádu, tj. $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, a $\alpha$, $m$ jsou reálná čísla. Je diskutováno asymptotické chování řešení této rovnice a jsou stanoveny podmínky, garantující existence řešení s asymptotikou mocninného typu: $u(k) \sim {1}/{k^s}$, kde $s$ je vhodná konstanta. Je také zkoumána diskrétní analogie tzv. ``blow-up'' řešení (neohraničených řešení) známých v klasické teorii diferenciálních rovnic, tj. řešení pro která v některém bodě $x^*$ platí $\lim_{x \to x^*} y(x)= \infty$, kde $y(x)$ je řešení Emden-Fowlerovy diferenciální rovnice $$ y''(x) = y^s(x), $$ kde $s \ne 1$ je reálné číslo. Výsledky jsou ilustrovány příklady a porovnávány s výsledky doposud známými., In the literature a differential second--order nonlinear Emden--Fowler equation $$ y'' \pm x^\alpha y^m = 0, $$ where $\alpha$ and $m$ are constants, is often investigated. This thesis deals with a discrete equivalent of the second--order Emden-Fowler differential equation $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ where $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ is an independent variable, $k_0$ is an integer and $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ is an unknown solution. In this equation, $\Delta^2u(k)=\Delta(\Delta u(k))$, $\Delta u(k)$ is the the first-order forward difference of $u(k)$, i.e., $\Delta u(k) = u(k+1)-u(k)$, and $\Delta^2 (k)$ is its second--order forward difference, i.e., $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, $\alpha$, $m$ are real numbers. The asymptotic behaviour of the solutions to this equation is discussed and the conditions are found such that there exists a power-type asymptotic: $u(k) \sim {1}/{k^s}$, where $s$ is some constant. We also discuss a discrete analogy of so-called ``blow-up'' solutions in the classical theory of differential equations, i.e., the solutions for which there exists a point $x^*$ such that $\lim_{x \to x^*} y(x) = \infty$, where $y(x)$ is a solution of the Emden-Fowler differential equation $$ y''(x) = y^s(x), $$ with $s \ne 1$ being a real number. The results obtained are compared to those already known and illustrated with examples.

V literatuře je často studována Emden--Fowlerova nelineární diferenciální rovnice druhého řádu $$ y'' \pm x^\alpha y^m = 0, $$ kde $\alpha$ a $m$ jsou konstanty. V disertační práci je analyzována diskrétní analogie Emden-Fowlerovy diferenciální rovnice $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ kde $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ je nezávislá proměnná, $k_0$ je celé číslo a $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ je řešení. V této rovnici je $\Delta^2u(k)=\Delta(\Delta u(k))$, kde $\Delta u(k)$ je diference vpřed prvního řádu funkce $u(k)$, tj. $\Delta u(k) = u(k+1)-u(k)$ a $\Delta^2 (k)$ je její diference vpřed druhého řádu, tj. $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, a $\alpha$, $m$ jsou reálná čísla. Je diskutováno asymptotické chování řešení této rovnice a jsou stanoveny podmínky, garantující existence řešení s asymptotikou mocninného typu: $u(k) \sim {1}/{k^s}$, kde $s$ je vhodná konstanta. Je také zkoumána diskrétní analogie tzv. ``blow-up'' řešení (neohraničených řešení) známých v klasické teorii diferenciálních rovnic, tj. řešení pro která v některém bodě $x^*$ platí $\lim_{x \to x^*} y(x)= \infty$, kde $y(x)$ je řešení Emden-Fowlerovy diferenciální rovnice $$ y''(x) = y^s(x), $$ kde $s \ne 1$ je reálné číslo. Výsledky jsou ilustrovány příklady a porovnávány s výsledky doposud známými., In the literature a differential second--order nonlinear Emden--Fowler equation $$ y'' \pm x^\alpha y^m = 0, $$ where $\alpha$ and $m$ are constants, is often investigated. This thesis deals with a discrete equivalent of the second--order Emden-Fowler differential equation $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ where $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ is an independent variable, $k_0$ is an integer and $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ is an unknown solution. In this equation, $\Delta^2u(k)=\Delta(\Delta u(k))$, $\Delta u(k)$ is the the first-order forward difference of $u(k)$, i.e., $\Delta u(k) = u(k+1)-u(k)$, and $\Delta^2 (k)$ is its second--order forward difference, i.e., $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, $\alpha$, $m$ are real numbers. The asymptotic behaviour of the solutions to this equation is discussed and the conditions are found such that there exists a power-type asymptotic: $u(k) \sim {1}/{k^s}$, where $s$ is some constant. We also discuss a discrete analogy of so-called ``blow-up'' solutions in the classical theory of differential equations, i.e., the solutions for which there exists a point $x^*$ such that $\lim_{x \to x^*} y(x) = \infty$, where $y(x)$ is a solution of the Emden-Fowler differential equation $$ y''(x) = y^s(x), $$ with $s \ne 1$ being a real number. The results obtained are compared to those already known and illustrated with examples.