Year 2023

• A Generalized Version of the Lions-Type Lemma

  Publication

  • M. Chmara

  - Annales Mathematicae Silesianae - Year 2023

  In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz–Sobolev spaces. What matters here is the behavior of the integral, not the space

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Year 2021

• Existence of Two Periodic Solutions to General Anisotropic Euler-Lagrange Equations

  Publication

  • M. Chmara

  - TAIWANESE JOURNAL OF MATHEMATICS - Year 2021

  Abstract. This paper is concerned with the following Euler-Lagrange system d/dtLv(t,u(t), ̇u(t)) =Lx(t,u(t), ̇u(t)) for a.e.t∈[−T,T], u(−T) =u(T), Lv(−T,u(−T), ̇u(−T)) =Lv(T,u(T), ̇u(T)), where Lagrangian is given by L=F(t,x,v) +V(t,x) +〈f(t),x〉, growth conditions aredetermined by an anisotropic G-function and some geometric conditions at infinity.We consider two cases: with and without forcing termf. Using a general version...

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Year 2020

• Mountain pass solutions to Euler-Lagrange equations with general anisotropic operator

  Publication

  • M. Chmara
  • J. Maksymiuk

  - JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS - Year 2020

  Using the Mountain Pass Theorem we show that the problem \begin{equation*} \begin{cases} \frac{d}{dt}\Lcal_v(t,u(t),\dot u(t))=\Lcal_x(t,u(t),\dot u(t))\quad \text{ for a.e. }t\in[a,b]\\ u(a)=u(b)=0 \end{cases} \end{equation*} has a solution in anisotropic Orlicz-Sobolev space. We consider Lagrangian $\Lcal=F(t,x,v)+V(t,x)+\langle f(t), x\rangle$ with growth conditions determined by anisotropic G-function and some geometric conditions...

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Year 2019

• Mountain pass type periodic solutions for Euler–Lagrange equations in anisotropic Orlicz–Sobolev space

  Publication

  • M. Chmara
  • J. Maksymiuk

  - JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS - Year 2019

  Using the Mountain Pass Theorem, we establish the existence of periodic solution for Euler–Lagrange equation. Lagrangian consists of kinetic part (an anisotropic G-function), potential part and a forcing term. We consider two situations: G satisfying at infinity and globally. We give conditions on the growth of the potential near zero for both situations.

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Year 2017

• Anisotropic Orlicz–Sobolev spaces of vector valued functions and Lagrange equations

  Publication

  • M. Chmara
  • J. Maksymiuk

  - JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS - Year 2017

  In this paper we study some properties of anisotropic Orlicz and Orlicz–Sobolev spaces of vector valued functions for a special class of G-functions. We introduce a variational setting for a class of Lagrangian Systems. We give conditions which ensure that the principal part of variational functional is finitely defined and continuously differentiable on Orlicz–Sobolev space.

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