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1
Content available remote An operator-valued free Poincarè inequality
Ito Hyuga
Probability and Mathematical Statistics
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2023
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Vol. 43, Fasc. 2
241--246
EN
The purpose of this short note is to give an operator-valued free Poincarè inequality, which provides a simple proof to (an improvement of) a lemma of Voiculescu (2000) asserting that the kernel of the free difference quotient is exactly the coefficients.
2
Content available remote Density of smooth functions in Sobolev spaces "wth mixed functions"
Liskowski M.
Fasciculi Mathematici
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2014
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Nr 53
77--83
EN
The results presented in this paper concern approximation by smooth functions in the Sobolev spaces defined by means of a modular (1). These spaces can be a natural medium to study the partial differential equations with rapidly or slowly increasing coefficients (i.e. the coefficients are of a nonpolynomial type).
3
Content available remote Sobolev spaces "with mixed functions"
Liskowski M.
Fasciculi Mathematici
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2012
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Nr 48
73--82
EN
This paper describes some generalization of modular function spaces Lϕψ defined by a modular Iϕѱ(f) = ∫baϕ (x, ∫dc ψ (y, f (x,y))dy) dx, ([3]). The next part of this paper focuses on using of spaces, defined previously, to introduce Sobolev spaces as a vector subspace of the generalized space Lϕѱ. Some selected properties of these spaces are presented.
4
Content available Application of Green's operator to quadratic variational problems
Azbelev N. V., Tsalyuk V. Z.
Opuscula Mathematica
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2006
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Vol. 26, no. 2
243-256
EN
We use Green's function of a suitable boundary value problem to convert the variational problem with quadratic functional and linear constraints to the equivalent unconstrained extremal problem in some subspace of the space L2 of quadratically summable functions. We get the neccessary and sufficient criterion for unique solvability of the variational problem in terms of the spectrum of some integral Hilbert-Schmidt operator in L2 with symmetric kernel. The numerical technique is proposed to estimate this criterion. The results are demonstrated on examples: 1) a variational problem with deviating argument, and 2) the problem of the critical force for the vertical pillar with additional support point (the qualities of the pillar may vary discontinuously along the pillar's axis).
5
Content available remote Local existence of the solution to the initial-boundary value problem in nonlinear thermomicroelasticity theory
Gawinecki J., Łazuka J.
Biuletyn Wojskowej Akademii Technicznej
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2003
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Vol. 52, nr 7
5-29
EN
We prove a theorem about local existence (in time) of the solution to the first initial-boundary value problem for a nonlinear system of equation of the thermomicroelasticity theory. At first, we prove existence, uniqueness and regularity of the solution to this problem for the associated linearized system by using the method of semi-group theory. Next, basing on this theorem, we prove an energy estimate for the solution to the linearized system by applying the method of Sobolev space. At the end, using the banach fixed point theorem, we prove that the solution of our nonlinear problem exists and is unique.
PL
W pracy udowodniono twierdzenie o istnieniu i jednoznaczności rozwiązania zagadnienia brzegowo-początkowego dla nieliniowego układu równań mikropolarnej termosprężystości. W pierwszym etapie udowodniono twierdzenie o istnieniu, jednoznaczności i regularności rozwiązania zagadnienia brzegowo-początkowego dla zlinearyzowanego układu równań mikropolarnej termosprężystości stosując metody półgrup oraz metody przestrzeni Sobolewa. Następnie na podstawie udowodnionych twierdzeń pokazano nierówności energetyczne dla rozwiązania zlinearyzowanego układu równań. W ostatnim etapie stosując twierdzenie Banacha o punkcie stałym udowodniono, że rozwiązanie nieliniowego układu równań istnieje i że jest jedyne. W dowodzie zastosowano metody półgrup, metody przestrzeni Sobolewa oraz metody nierówności energetycznych.
6
Content available remote Existence and uniqueness of positive solution of a semilinear elliptic equation in Rn
Djellit A., Benouhiba N.
Demonstratio Mathematica
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2002
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Vol. 35, nr 1
61-73
EN
We study here semilinear elliptic indefinite-weight problems defined in Rn, n > 3. We show that under some conditions, these problems have exactly one positive solution.
7
Content available remote On nonlinear boundary value problem with a mixed nonhomogeneus condition: asymptotic behavior of solutions
Long N., Dung B., Nghia N., Thuyet T.
Demonstratio Mathematica
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2001
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Vol. 34, nr 3
609-618
8
Content available remote Existence of periodic solutions for quasilinear ordinary differential eqasilinear ordinary diefferential equations with discontinuities
Matzakos N., Papageorgiou N.
Demonstratio Mathematica
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2000
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Vol. 33, nr 4
753-770
EN
We consider a quasilinear differential equation with discontinuous right hand side and periodic boundary conditions. To obtain an existence theory we pass to a relevant multivalued variant of the original problem, which we solve. Our approach is a mixture of the variational method (for nonsmooth locally Lipschitz functionals) and of the method of upper and Iower solutions. The mixing of these two techniques is made possible by a nonresonance condition below the first nonzero eigenvalue of the one-dimensional p-Laplacian with periodic boundary conditions.
9
Content available remote The initial-boundary value problem for linearized system of equations of microelasticity theory
Gawinecki J., Kacprzyk P.
Biuletyn Wojskowej Akademii Technicznej
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2000
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Vol. 49, nr 1
5-17
EN
The aim of the paper is to prove the thorem about existence, uniqueness and regularity of the solution to the initial-boundary value problem for linearized system of equations of the microelasticity theory asociated to the non-linear one. Such a system of equations describes the microelastic medium in the three dimensional space. In our proof we used the method of semigroup theory and the method of Sobolev spaces.
PL
W pracy udowodniono twierdzenie o istnieniu, jednoznaczności i regulaności rozwiązania zgadnienia brzegowo-początkowego dla zlinearyzowanego układu równań mikropolarnej teorii sprężystości. Zastosowano metody teorii półgrup nieliniowych oraz metody przestrzeni Sobolewa.
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