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Détails du magazine
Format
Magazine
eISSN
2351-8227
Première publication
16 Apr 2015
Période de publication
3 fois par an
Langues
Anglais

Chercher

Volume 4 (2018): Edition 2 (December 2018)

Détails du magazine
Format
Magazine
eISSN
2351-8227
Première publication
16 Apr 2015
Période de publication
3 fois par an
Langues
Anglais

Chercher

10 Articles
access type Accès libre

Weighted Variable Exponent Sobolev spaces on metric measure spaces

Publié en ligne: 16 May 2019
Pages: 62 - 76

Résumé

Abstract

In this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.

Mots clés

  • weighted variable exponent Sobolev spaces on metric spaces
  • capacity
  • weighted variable exponent Sobolev with zero boundary values
  • Dirichlet energy

MSC 2010

  • Primary: 46E35
  • 31B15
  • 46E30
  • 42B25
  • 28A80
access type Accès libre

Weak solutions for generalized p-Laplacian systems via Young measures

Publié en ligne: 16 May 2019
Pages: 77 - 84

Résumé

Abstract

We prove the existence of weak solutions to a generalized p-Laplacian systems in degenerate form. The techniques of Young measure for elliptic systems are used to prove the existence result.

Mots clés

  • -Laplacian systems
  • Weak solutions
  • Young measures

MSC 2010

  • Primary: 35J50
  • 35J57
  • 35D30
  • 28Axx
access type Accès libre

The Upper and Lower Approximations in Rough Subgroupoid of a Groupoid

Publié en ligne: 16 May 2019
Pages: 85 - 93

Résumé

Abstract

In this article, we introduce the concept of rough subgroupoid of a groupoid as a generalization of a rough subgroup and give some features about the lower and the upper approximations in a groupoid [1]. We give some of the characterization of them.

Mots clés

  • rough set
  • rough groupoid
  • rough subgroupoid
access type Accès libre

Two-Point Quadrature Rules for Riemann–Stieltjes Integrals with Lp–error estimates

Publié en ligne: 16 May 2019
Pages: 94 - 109

Résumé

Abstract

In this work, we construct a new general two-point quadrature rules for the Riemann–Stieltjes integral abf(t)du(t)$\int_a^b {f(t)} \,du\,(t)$, where the integrand f is assumed to be satisfied with the Hölder condition on [a, b] and the integrator u is of bounded variation on [a, b]. The dual formulas under the same assumption are proved. Some sharp error Lp–Error estimates for the proposed quadrature rules are also obtained.

Mots clés

  • Quadrature formula
  • Riemann-Stieltjes integral
  • Ostrowski’s inequality

MSC 2010

  • Primary: 41A55
  • 65D30
  • 65D32
access type Accès libre

Opial type inequalities for double Riemann-Stieltjes integrals

Publié en ligne: 16 May 2019
Pages: 111 - 121

Résumé

Abstract

In this paper, we establish some Opial type inequalities for Riemann-Stieltjes integrals of functions with two variables. The obtained inequalities generalize those previously demonstrated (see [2])

Mots clés

  • Opial inequality
  • Hölder’s inequality

MSC 2010

  • Primary: 26D15
  • 26D10
  • 26B15
access type Accès libre

Harmonic numbers, harmonic series and zeta function

Publié en ligne: 16 May 2019
Pages: 122 - 157

Résumé

Abstract

This paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the Riemann hypothesis. We give an account on the theorem of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the series 𝒨is(z)=n=1μ(n)nszn$\mathcal{M}{i_s}(z) = \sum\limits_{n = 1}^\infty {{{\mu (n)} \over {{n^s}}}{z^n}} $. A significant result is that the real part f of

μ(n)ne2inπθ$$\sum {{{\mu (n)} \over n}{e^{2in\pi \theta }}}$$

is an example of a non-trivial real-valued continuous function f on the real line which is 1-periodic, is not odd and has the property h=1nf(h/k)=0$\sum\nolimits_{h = 1}^n {f(h/k) = 0}$ for every positive integer k.

Mots clés

  • Harmonic numbers
  • Möbius function
  • Zeta functions
  • Explicit Formulas
  • Sums of squares

MSC 2010

  • Primary: 11E45
  • 11A25
  • 11A41
  • 11R47
access type Accès libre

A few results on some nonlinear parabolic problems in Orlicz-Sobolev spaces

Publié en ligne: 16 May 2019
Pages: 158 - 170

Résumé

Abstract

In this paper, we present our results (see our papers), which concern the existence of the renormalized solutions for equations of the type:

b(x,u)t-div(a(x,t,u,u))-div(Φ(x,t,u))=finQ=Ω×(0,T),$${{\partial b(x,u)} \over {\partial t}} - {\rm{div}}\left( {a(x,t,u,\nabla u)} \right) - {\rm{div}}\left( {\Phi \left( {x,t,u} \right)} \right) = f\,\,\,{\rm{in}}\,Q = \Omega \times (0,T),$$

where b(x, ·) is a strictly increasing C1-function for any x ∈ Δ, a(x, t, s, ξ) and Φ(x, t, s) are a Carathéodory functions. The function f is in L1(Q).

Mots clés

  • Parabolic equation
  • Orlicz-Sobolev spaces
  • Renormalized solutions
  • Unilateral problem

MSC 2010

  • Primary: 35K55
  • 47A15
  • 46A32
  • 47D20
access type Accès libre

Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

Publié en ligne: 16 May 2019
Pages: 171 - 188

Résumé

Abstract

In this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form

-i=1Niai(x,u,u)-i=1Niφi(u)=f,$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$

where the right hand side f belongs to L1(Ω). The operator -i=1Niai(x,u,u)$- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $ is a Leray-Lions anisotropic operator and ϕiC0(ℝ,ℝ).

Mots clés

  • Entropy solutions
  • Anisotropic elliptic equations
  • Anisotropic Sobolev space

MSC 2010

  • Primary: 35J60
  • 35J87
  • 35J66
access type Accès libre

Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces

Publié en ligne: 16 May 2019
Pages: 189 - 206

Résumé

Abstract

We prove existence of entropy solutions to general class of unilateral nonlinear parabolic equation in inhomogeneous Musielak-Orlicz spaces avoiding ceorcivity restrictions on the second lower order term. Namely, we consider

{uψinQT,b(x,u)t-div(a(x,t,u,u))=f+div(g(x,t,u))L1(QT).$$\left\{ \matrix{ \matrix{ {u \ge \psi } \hfill & {{\rm{in}}} \hfill & {{Q_T},} \hfill \cr } \hfill \cr {{\partial b(x,u)} \over {\partial t}} - div\left( {a\left( {x,t,u,\nabla u} \right)} \right) = f + div\left( {g\left( {x,t,u} \right)} \right) \in {L^1}\left( {{Q_T}} \right). \hfill \cr} \right.$$

The growths of the monotone vector field a(x, t, u, ᐁu) and the non-coercive vector field g(x, t, u) are controlled by a generalized nonhomogeneous N- function M (see (3.3)-(3.6)). The approach does not require any particular type of growth of M (neither Δ2 nor ᐁ2). The proof is based on penalization method.

Mots clés

  • Inhomogeneous Musielak-Orlicz space
  • Nonlinear Parabolic problems
  • Entropy solutions
  • Non-coercive Lower order term

MSC 2010

  • Primary 35K86
  • Secondary 35K55, 35A01
access type Accès libre

Extension of complete monotonicity results involving the digamma function

Publié en ligne: 06 Jun 2019
Pages: 207 - 212

Résumé

Abstract

By using some analytical techniques, we prove a complete monotonicity property of a certain function involving the (p, k)-digamma function. Subsequently, we derive some inequalities for the (p, k)- digamma function. As special cases of the established results, we deduce some new results concerning the p-digamma and the k-digamma functions. Our results are extensions of some previous results due to Qiu and Vuorinen, Mortici, and Merovci.

Mots clés

  • Complete monotonicity
  • ()-digamma function
  • Inequality

MSC 2010

  • Primary 33B15
  • 26A48
  • 26D07
10 Articles
access type Accès libre

Weighted Variable Exponent Sobolev spaces on metric measure spaces

Publié en ligne: 16 May 2019
Pages: 62 - 76

Résumé

Abstract

In this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.

Mots clés

  • weighted variable exponent Sobolev spaces on metric spaces
  • capacity
  • weighted variable exponent Sobolev with zero boundary values
  • Dirichlet energy

MSC 2010

  • Primary: 46E35
  • 31B15
  • 46E30
  • 42B25
  • 28A80
access type Accès libre

Weak solutions for generalized p-Laplacian systems via Young measures

Publié en ligne: 16 May 2019
Pages: 77 - 84

Résumé

Abstract

We prove the existence of weak solutions to a generalized p-Laplacian systems in degenerate form. The techniques of Young measure for elliptic systems are used to prove the existence result.

Mots clés

  • -Laplacian systems
  • Weak solutions
  • Young measures

MSC 2010

  • Primary: 35J50
  • 35J57
  • 35D30
  • 28Axx
access type Accès libre

The Upper and Lower Approximations in Rough Subgroupoid of a Groupoid

Publié en ligne: 16 May 2019
Pages: 85 - 93

Résumé

Abstract

In this article, we introduce the concept of rough subgroupoid of a groupoid as a generalization of a rough subgroup and give some features about the lower and the upper approximations in a groupoid [1]. We give some of the characterization of them.

Mots clés

  • rough set
  • rough groupoid
  • rough subgroupoid
access type Accès libre

Two-Point Quadrature Rules for Riemann–Stieltjes Integrals with Lp–error estimates

Publié en ligne: 16 May 2019
Pages: 94 - 109

Résumé

Abstract

In this work, we construct a new general two-point quadrature rules for the Riemann–Stieltjes integral abf(t)du(t)$\int_a^b {f(t)} \,du\,(t)$, where the integrand f is assumed to be satisfied with the Hölder condition on [a, b] and the integrator u is of bounded variation on [a, b]. The dual formulas under the same assumption are proved. Some sharp error Lp–Error estimates for the proposed quadrature rules are also obtained.

Mots clés

  • Quadrature formula
  • Riemann-Stieltjes integral
  • Ostrowski’s inequality

MSC 2010

  • Primary: 41A55
  • 65D30
  • 65D32
access type Accès libre

Opial type inequalities for double Riemann-Stieltjes integrals

Publié en ligne: 16 May 2019
Pages: 111 - 121

Résumé

Abstract

In this paper, we establish some Opial type inequalities for Riemann-Stieltjes integrals of functions with two variables. The obtained inequalities generalize those previously demonstrated (see [2])

Mots clés

  • Opial inequality
  • Hölder’s inequality

MSC 2010

  • Primary: 26D15
  • 26D10
  • 26B15
access type Accès libre

Harmonic numbers, harmonic series and zeta function

Publié en ligne: 16 May 2019
Pages: 122 - 157

Résumé

Abstract

This paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the Riemann hypothesis. We give an account on the theorem of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the series 𝒨is(z)=n=1μ(n)nszn$\mathcal{M}{i_s}(z) = \sum\limits_{n = 1}^\infty {{{\mu (n)} \over {{n^s}}}{z^n}} $. A significant result is that the real part f of

μ(n)ne2inπθ$$\sum {{{\mu (n)} \over n}{e^{2in\pi \theta }}}$$

is an example of a non-trivial real-valued continuous function f on the real line which is 1-periodic, is not odd and has the property h=1nf(h/k)=0$\sum\nolimits_{h = 1}^n {f(h/k) = 0}$ for every positive integer k.

Mots clés

  • Harmonic numbers
  • Möbius function
  • Zeta functions
  • Explicit Formulas
  • Sums of squares

MSC 2010

  • Primary: 11E45
  • 11A25
  • 11A41
  • 11R47
access type Accès libre

A few results on some nonlinear parabolic problems in Orlicz-Sobolev spaces

Publié en ligne: 16 May 2019
Pages: 158 - 170

Résumé

Abstract

In this paper, we present our results (see our papers), which concern the existence of the renormalized solutions for equations of the type:

b(x,u)t-div(a(x,t,u,u))-div(Φ(x,t,u))=finQ=Ω×(0,T),$${{\partial b(x,u)} \over {\partial t}} - {\rm{div}}\left( {a(x,t,u,\nabla u)} \right) - {\rm{div}}\left( {\Phi \left( {x,t,u} \right)} \right) = f\,\,\,{\rm{in}}\,Q = \Omega \times (0,T),$$

where b(x, ·) is a strictly increasing C1-function for any x ∈ Δ, a(x, t, s, ξ) and Φ(x, t, s) are a Carathéodory functions. The function f is in L1(Q).

Mots clés

  • Parabolic equation
  • Orlicz-Sobolev spaces
  • Renormalized solutions
  • Unilateral problem

MSC 2010

  • Primary: 35K55
  • 47A15
  • 46A32
  • 47D20
access type Accès libre

Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

Publié en ligne: 16 May 2019
Pages: 171 - 188

Résumé

Abstract

In this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form

-i=1Niai(x,u,u)-i=1Niφi(u)=f,$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$

where the right hand side f belongs to L1(Ω). The operator -i=1Niai(x,u,u)$- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $ is a Leray-Lions anisotropic operator and ϕiC0(ℝ,ℝ).

Mots clés

  • Entropy solutions
  • Anisotropic elliptic equations
  • Anisotropic Sobolev space

MSC 2010

  • Primary: 35J60
  • 35J87
  • 35J66
access type Accès libre

Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces

Publié en ligne: 16 May 2019
Pages: 189 - 206

Résumé

Abstract

We prove existence of entropy solutions to general class of unilateral nonlinear parabolic equation in inhomogeneous Musielak-Orlicz spaces avoiding ceorcivity restrictions on the second lower order term. Namely, we consider

{uψinQT,b(x,u)t-div(a(x,t,u,u))=f+div(g(x,t,u))L1(QT).$$\left\{ \matrix{ \matrix{ {u \ge \psi } \hfill & {{\rm{in}}} \hfill & {{Q_T},} \hfill \cr } \hfill \cr {{\partial b(x,u)} \over {\partial t}} - div\left( {a\left( {x,t,u,\nabla u} \right)} \right) = f + div\left( {g\left( {x,t,u} \right)} \right) \in {L^1}\left( {{Q_T}} \right). \hfill \cr} \right.$$

The growths of the monotone vector field a(x, t, u, ᐁu) and the non-coercive vector field g(x, t, u) are controlled by a generalized nonhomogeneous N- function M (see (3.3)-(3.6)). The approach does not require any particular type of growth of M (neither Δ2 nor ᐁ2). The proof is based on penalization method.

Mots clés

  • Inhomogeneous Musielak-Orlicz space
  • Nonlinear Parabolic problems
  • Entropy solutions
  • Non-coercive Lower order term

MSC 2010

  • Primary 35K86
  • Secondary 35K55, 35A01
access type Accès libre

Extension of complete monotonicity results involving the digamma function

Publié en ligne: 06 Jun 2019
Pages: 207 - 212

Résumé

Abstract

By using some analytical techniques, we prove a complete monotonicity property of a certain function involving the (p, k)-digamma function. Subsequently, we derive some inequalities for the (p, k)- digamma function. As special cases of the established results, we deduce some new results concerning the p-digamma and the k-digamma functions. Our results are extensions of some previous results due to Qiu and Vuorinen, Mortici, and Merovci.

Mots clés

  • Complete monotonicity
  • ()-digamma function
  • Inequality

MSC 2010

  • Primary 33B15
  • 26A48
  • 26D07

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