The existence of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated. We develop duality and variational principles for this problem. Our variational approach enables us to approximate solutions and give a measure of a duality gap between the primal and dual functional for minimizing sequences.
We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.
We prove that the one-parameter group of holomorphic automorphisms induced on a strictly geometrically bounded domain by a biholomorphism with a model domain is parabolic. This result is related to the Greene-Krantz conjecture and more generally to the classification of domains having a non compact automorphisms group. The proof relies on elementary estimates on the Kobayashi pseudo-metric., François Berteloot, Ninh Van Thu., and Obsahuje seznam literatury
We classify all bundle functors $G$ admitting natural operators transforming connections on a fibered manifold $Y\rightarrow M$ into connections on $GY\rightarrow M$. Then we solve a similar problem for natural operators transforming connections on $Y\rightarrow M$ into connections on $GY\rightarrow Y$.
Let $\Omega $ be a bounded domain in ${\mathbb{R}}^n$ with a smooth boundary $\Gamma $. In this work we study the existence of solutions for the following boundary value problem: \[ \frac{\partial ^2 y}{\partial t^2}-M\biggl (\int _\Omega |\nabla y|^2\mathrm{d}x\biggr ) \Delta y -\frac{\partial }{\partial t}\Delta y=f(y) \quad \text{in} Q=\Omega \times (0,\infty ),.1 y=0 \quad \text{in} \Sigma _1=\Gamma _{\!1} \times (0,\infty ), M\biggl (\int _\Omega |\nabla y|^2\mathrm{d}x\biggr ) \frac{\partial y}{\partial \nu } +\frac{\partial }{\partial t}\Bigl (\frac{\partial y}{\partial \nu }\Bigr )=g \quad \text{in} \Sigma _0=\Gamma _{\!0} \times (0,\infty ), y(0)=y_0,\quad \frac{\partial y}{\partial t}\,(0)=y_1 \quad \text{in} \quad \Omega , \qquad \mathrm{(1)}\] where $M$ is a $C^1$-function such that $M(\lambda ) \ge \lambda _0 >0$ for every $\lambda \ge 0$ and $f(y)=|y|^\alpha y$ for $\alpha \ge 0$.
For any positive integer $D$ which is not a square, let $(u_1,v_1)$ be the least positive integer solution of the Pell equation $u^2-Dv^2=1,$ and let $h(4D)$ denote the class number of binary quadratic primitive forms of discriminant $4D$. If $D$ satisfies $2\nmid D$ and $v_1h(4D)\equiv0 \pmod D$, then $D$ is called a singular number. In this paper, we prove that if $(x,y,z)$ is a positive integer solution of the equation $x^y+y^x=z^z$ with $2\mid z$, then maximum $\max\{x,y,z\}<480000$ and both $x$, $y$ are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions $(x,y,z)$., Xiaoying Du., and Obsahuje bibliografické odkazy
A variant of Alexandrov theorem is proved stating that a compact, subadditive $D$-poset valued mapping is continuous. Then the measure extension theorem is proved for MV-algebra valued measures.
Exponential polynomials are the building bricks of spectral synthesis. In some cases it happens that exponential polynomials should be extended from subgroups to whole groups. To achieve this aim we prove an extension theorem for exponential polynomials which is based on a classical theorem on the extension of homomorphisms.