Pub Date : 2021-06-15DOI: 10.7494/opmath.2022.42.4.583
A. Karppinen
In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces (also known as Musielak-Orlicz spaces) for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation.
{"title":"Fractional operators and their commutators on generalized Orlicz spaces","authors":"A. Karppinen","doi":"10.7494/opmath.2022.42.4.583","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.4.583","url":null,"abstract":"In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces (also known as Musielak-Orlicz spaces) for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42139935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-24DOI: 10.7494/OPMATH.2021.41.2.187
Idowu Esther IJaodoro, E. Thiam
We consider a bounded domain $Omega$ of $mathbb{R}^N$, $Nge3$, $h$ and $b$ continuous functions on $Omega$. Let $Gamma$ be a closed curve contained in $Omega$. We study existence of positive solutions $u in H^1_0left(Omegaright)$ to the perturbed Hardy-Sobolev equation: $$ -Delta u+h u+bu^{1+delta}=rho^{-sigma}_Gamma u^{2^*_sigma-1} qquad textrm{ in } Omega, $$ where $2^*_sigma:=frac{2(N-sigma)}{N-2}$ is the critical Hardy-Sobolev exponent, $sigmain [0,2)$, $0
{"title":"Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve","authors":"Idowu Esther IJaodoro, E. Thiam","doi":"10.7494/OPMATH.2021.41.2.187","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.2.187","url":null,"abstract":"We consider a bounded domain $Omega$ of $mathbb{R}^N$, $Nge3$, $h$ and $b$ continuous functions on $Omega$. Let $Gamma$ be a closed curve contained in $Omega$. We study existence of positive solutions $u in H^1_0left(Omegaright)$ to the perturbed Hardy-Sobolev equation: $$ -Delta u+h u+bu^{1+delta}=rho^{-sigma}_Gamma u^{2^*_sigma-1} qquad textrm{ in } Omega, $$ where $2^*_sigma:=frac{2(N-sigma)}{N-2}$ is the critical Hardy-Sobolev exponent, $sigmain [0,2)$, $0<delta<frac{4}{N-2}$ and $rho_Gamma$ is the distance function to $Gamma$. We show that the existence of minimizers does not depend on the local geometry of $Gamma$ nor on the potential $h$. For $N=3$, the existence of ground-state solution may depends on the trace of the regular part of the Green function of $-Delta+h$ and or on $b$. This is due to the perturbative term of order ${1+delta}$.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46526871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-06DOI: 10.7494/opmath.2021.41.5.701
A. Verma, B. Gupta
In this paper we define a new class of continuous fractional wavelet transform (CFrWT) and study its properties in Hardy space and Morrey space. The theory developed generalize and complement some of already existing results.
{"title":"Certain properties of continuous fractional wavelet transform on Hardy space and Morrey space","authors":"A. Verma, B. Gupta","doi":"10.7494/opmath.2021.41.5.701","DOIUrl":"https://doi.org/10.7494/opmath.2021.41.5.701","url":null,"abstract":"In this paper we define a new class of continuous fractional wavelet transform (CFrWT) and study its properties in Hardy space and Morrey space. The theory developed generalize and complement some of already existing results.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42885051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7494/opmath.2021.41.4.465
L. Chergui
This paper is devoted to the Schrödinger-Choquard equation with linear damping. Global existence and scattering are proved depending on the size of the damping coefficient.
{"title":"Remarks on damped Schrödinger equation of Choquard type","authors":"L. Chergui","doi":"10.7494/opmath.2021.41.4.465","DOIUrl":"https://doi.org/10.7494/opmath.2021.41.4.465","url":null,"abstract":"This paper is devoted to the Schrödinger-Choquard equation with linear damping. Global existence and scattering are proved depending on the size of the damping coefficient.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7494/OPMATH.2021.41.2.227
S. Pedersen, Vincent T. Shaw
In this paper we consider a retained digits Cantor set (T) based on digit expansions with Gaussian integer base. Let (F) be the set all (x) such that the intersection of (T) with its translate by (x) is non-empty and let (F_{beta}) be the subset of (F) consisting of all (x) such that the dimension of the intersection of (T) with its translate by (x) is (beta) times the dimension of (T). We find conditions on the retained digits sets under which (F_{beta}) is dense in (F) for all (0leqbetaleq 1). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.
{"title":"Dimension of the intersection of certain Cantor sets in the plane","authors":"S. Pedersen, Vincent T. Shaw","doi":"10.7494/OPMATH.2021.41.2.227","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.2.227","url":null,"abstract":"In this paper we consider a retained digits Cantor set (T) based on digit expansions with Gaussian integer base. Let (F) be the set all (x) such that the intersection of (T) with its translate by (x) is non-empty and let (F_{beta}) be the subset of (F) consisting of all (x) such that the dimension of the intersection of (T) with its translate by (x) is (beta) times the dimension of (T). We find conditions on the retained digits sets under which (F_{beta}) is dense in (F) for all (0leqbetaleq 1). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7494/opmath.2021.41.5.613
V. Benekas, Ábel Garab, A. Kashkynbayev, I. Stavroulakis
We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay.
{"title":"Oscillation criteria for linear difference equations with several variable delays","authors":"V. Benekas, Ábel Garab, A. Kashkynbayev, I. Stavroulakis","doi":"10.7494/opmath.2021.41.5.613","DOIUrl":"https://doi.org/10.7494/opmath.2021.41.5.613","url":null,"abstract":"We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7494/opmath.2021.41.5.629
F. Boulahia, Slimane Hassaine
In the present paper, we give criteria for the existence of extreme points of the Besicovitch-Orlicz space of almost periodic functions equipped with Orlicz norm. Some properties of the set of attainable points of the Amemiya norm in this space are also discussed.
{"title":"Extreme points of the Besicovitch-Orlicz space of almost periodic functions equipped with Orlicz norm","authors":"F. Boulahia, Slimane Hassaine","doi":"10.7494/opmath.2021.41.5.629","DOIUrl":"https://doi.org/10.7494/opmath.2021.41.5.629","url":null,"abstract":"In the present paper, we give criteria for the existence of extreme points of the Besicovitch-Orlicz space of almost periodic functions equipped with Orlicz norm. Some properties of the set of attainable points of the Amemiya norm in this space are also discussed.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7494/opmath.2021.41.6.755
Ilwoo Cho
In this paper, we fix (N)-many (l^2)-Hilbert spaces (H_k) whose dimensions are (n_{k} in mathbb{N}^{infty}=mathbb{N} cup {infty}), for (k=1,ldots,N), for (N in mathbb{N}setminus{1}). And then, construct a Hilbert space (mathfrak{F}=mathfrak{F}[H_{1},ldots,H_{N}]) induced by (H_{1},ldots,H_{N}), and study certain types of operators on (mathfrak{F}). In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by (bigcup^N_{k=1} mathcal{B}_{k}), where (mathcal{B}_{k}) are the orthonormal bases of (H_{k}), for (k=1,ldots,N).
在本文中，我们修正了(N) -多个(l^2) -Hilbert空间(H_k)，其维度为(n_{k} in mathbb{N}^{infty}=mathbb{N} cup {infty})，对于(k=1,ldots,N)，对于(N in mathbb{N}setminus{1})。然后，构造由(H_{1},ldots,H_{N})引出的Hilbert空间(mathfrak{F}=mathfrak{F}[H_{1},ldots,H_{N}])，并研究(mathfrak{F})上的若干类型算子。我们特别感兴趣的是所谓的跳移算子。主要结果(i)表征了这些算子的光谱特性，(ii)显示了这些算子如何影响(bigcup^N_{k=1} mathcal{B}_{k})引起的半圆定律，其中(mathcal{B}_{k})是(k=1,ldots,N)的(H_{k})的标准正交基。
{"title":"Spectral properties of certain operators on the free Hilbert space mathfrak{F}[H_{1},...,H_{N}] and the semicircular law","authors":"Ilwoo Cho","doi":"10.7494/opmath.2021.41.6.755","DOIUrl":"https://doi.org/10.7494/opmath.2021.41.6.755","url":null,"abstract":"In this paper, we fix (N)-many (l^2)-Hilbert spaces (H_k) whose dimensions are (n_{k} in mathbb{N}^{infty}=mathbb{N} cup {infty}), for (k=1,ldots,N), for (N in mathbb{N}setminus{1}). And then, construct a Hilbert space (mathfrak{F}=mathfrak{F}[H_{1},ldots,H_{N}]) induced by (H_{1},ldots,H_{N}), and study certain types of operators on (mathfrak{F}). In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by (bigcup^N_{k=1} mathcal{B}_{k}), where (mathcal{B}_{k}) are the orthonormal bases of (H_{k}), for (k=1,ldots,N).","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.18154/RWTH-2021-04723
L. Volkmann
Let (G) be a graph with vertex set (V(G)). If (uin V(G)), then (N[u]) is the closed neighborhood of (u). An outer-independent double Italian dominating function (OIDIDF) on a graph (G) is a function (f:V(G)longrightarrow {0,1,2,3}) such that if (f(v)in{0,1}) for a vertex (vin V(G)), then (sum_{xin N[v]}f(x)ge 3), and the set ({uin V(G):f(u)=0}) is independent. The weight of an OIDIDF (f) is the sum (sum_{vin V(G)}f(v)). The outer-independent double Italian domination number (gamma_{oidI}(G)) equals the minimum weight of an OIDIDF on (G). In this paper we present Nordhaus-Gaddum type bounds on the outer-independent double Italian domination number which improved corresponding results given in [F. Azvin, N. Jafari Rad, L. Volkmann, Bounds on the outer-independent double Italian domination number, Commun. Comb. Optim. 6 (2021), 123-136]. Furthermore, we determine the outer-independent double Italian domination number of some families of graphs.
{"title":"Remarks on the outer-independent double Italian domination number","authors":"L. Volkmann","doi":"10.18154/RWTH-2021-04723","DOIUrl":"https://doi.org/10.18154/RWTH-2021-04723","url":null,"abstract":"Let (G) be a graph with vertex set (V(G)). If (uin V(G)), then (N[u]) is the closed neighborhood of (u). An outer-independent double Italian dominating function (OIDIDF) on a graph (G) is a function (f:V(G)longrightarrow {0,1,2,3}) such that if (f(v)in{0,1}) for a vertex (vin V(G)), then (sum_{xin N[v]}f(x)ge 3), and the set ({uin V(G):f(u)=0}) is independent. The weight of an OIDIDF (f) is the sum (sum_{vin V(G)}f(v)). The outer-independent double Italian domination number (gamma_{oidI}(G)) equals the minimum weight of an OIDIDF on (G). In this paper we present Nordhaus-Gaddum type bounds on the outer-independent double Italian domination number which improved corresponding results given in [F. Azvin, N. Jafari Rad, L. Volkmann, Bounds on the outer-independent double Italian domination number, Commun. Comb. Optim. 6 (2021), 123-136]. Furthermore, we determine the outer-independent double Italian domination number of some families of graphs.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67703713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7494/opmath.2021.41.4.571
Amit Verma, Nazia Urus, R. Agarwal
This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as [begin{gathered} -u''(x)=psi(x,u,u'), quad xin (0,1), u'(0)=lambda_{1}u(xi), quad u'(1)=lambda_{2} u(eta),end{gathered}] where (I=[0,1]), (0ltxileqetalt 1) and (lambda_1,lambda_2gt 0). The nonlinear source term (psiin C(Itimesmathbb{R}^2,mathbb{R})) is one sided Lipschitz in (u) with Lipschitz constant (L_1) and Lipschitz in (u') such that (|psi(x,u,u')-psi(x,u,v')|leq L_2(x)|u'-v'|). We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter (k) equivalent to (max_ufrac{partial psi}{partial u}). We compute the range of (k) for which iterative sequences are convergent.
{"title":"Region of existence of multiple solutions for a class of Robin type four-point BVPs","authors":"Amit Verma, Nazia Urus, R. Agarwal","doi":"10.7494/opmath.2021.41.4.571","DOIUrl":"https://doi.org/10.7494/opmath.2021.41.4.571","url":null,"abstract":"This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as [begin{gathered} -u''(x)=psi(x,u,u'), quad xin (0,1), u'(0)=lambda_{1}u(xi), quad u'(1)=lambda_{2} u(eta),end{gathered}] where (I=[0,1]), (0ltxileqetalt 1) and (lambda_1,lambda_2gt 0). The nonlinear source term (psiin C(Itimesmathbb{R}^2,mathbb{R})) is one sided Lipschitz in (u) with Lipschitz constant (L_1) and Lipschitz in (u') such that (|psi(x,u,u')-psi(x,u,v')|leq L_2(x)|u'-v'|). We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter (k) equivalent to (max_ufrac{partial psi}{partial u}). We compute the range of (k) for which iterative sequences are convergent.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71341614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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