Pub Date : 2022-08-03DOI: 10.21136/mb.2022.0096-21
N. Gürses, G. Y. Şentürk, S. Yüce
. We aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values α , β and p . Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given.
{"title":"Investigating generalized quaternions with dual-generalized complex numbers","authors":"N. Gürses, G. Y. Şentürk, S. Yüce","doi":"10.21136/mb.2022.0096-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0096-21","url":null,"abstract":". We aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values α , β and p . Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45494174","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 : 2022-08-03DOI: 10.21136/mb.2022.0166-21
D. O. Adams, M. Omeike, I. Osinuga, B. S. Badmus
. We consider certain class of second order nonlinear nonautonomous delay differential equations of the form where a , b , c , g , h , m and p are real valued functions which depend at most on the arguments displayed explicitly and r is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovskiˇı functional to establish our results. This work extends and improve on some results in the literature.
{"title":"Boundedness criteria for a class of second order nonlinear differential equations with delay","authors":"D. O. Adams, M. Omeike, I. Osinuga, B. S. Badmus","doi":"10.21136/mb.2022.0166-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0166-21","url":null,"abstract":". We consider certain class of second order nonlinear nonautonomous delay differential equations of the form where a , b , c , g , h , m and p are real valued functions which depend at most on the arguments displayed explicitly and r is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovskiˇı functional to establish our results. This work extends and improve on some results in the literature.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45989348","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 : 2022-08-02DOI: 10.21136/mb.2023.0108-22
I. Chajda, Miroslav Kolavr'ik, H. Langer
In their recent paper on posets with a pseudocomplementation denoted by * the first and the third author introduced the concept of a *-ideal. This concept is in fact an extension of a similar concept introduced in distributive pseudocomplemented lattices and semilattices by several authors, see References. Now we apply this concept of a c-ideal (dually, c-filter) to complemented posets where the complementation need neither be antitone nor an involution, but still satisfies some weak conditions. We show when an ideal or filter in such a poset is a c-ideal or c-filter, respectively, and we prove basic properties of them. Finally, we prove so-called Separation Theorems for c-ideals. The text is illustrated by several examples.
{"title":"c-ideals in complemented posets","authors":"I. Chajda, Miroslav Kolavr'ik, H. Langer","doi":"10.21136/mb.2023.0108-22","DOIUrl":"https://doi.org/10.21136/mb.2023.0108-22","url":null,"abstract":"In their recent paper on posets with a pseudocomplementation denoted by * the first and the third author introduced the concept of a *-ideal. This concept is in fact an extension of a similar concept introduced in distributive pseudocomplemented lattices and semilattices by several authors, see References. Now we apply this concept of a c-ideal (dually, c-filter) to complemented posets where the complementation need neither be antitone nor an involution, but still satisfies some weak conditions. We show when an ideal or filter in such a poset is a c-ideal or c-filter, respectively, and we prove basic properties of them. Finally, we prove so-called Separation Theorems for c-ideals. The text is illustrated by several examples.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49454673","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 : 2022-07-11DOI: 10.21136/mb.2022.0110-21
R. Shroff
. Absolute direct summand in lattices is defined and some of its properties in modular lattices are studied. It is shown that in a certain class of modular lattices, the direct sum of two elements has absolute direct summand if and only if the elements are relatively injective. As a generalization of absolute direct summand (ADS for short), the concept of Goldie absolute direct summand in lattices is introduced and studied. It is shown that Goldie ADS property is inherited by direct summands. A necessary and sufficient condition is given for an element of modular lattice to have Goldie ADS.
{"title":"On Goldie absolute direct summands in modular lattices","authors":"R. Shroff","doi":"10.21136/mb.2022.0110-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0110-21","url":null,"abstract":". Absolute direct summand in lattices is defined and some of its properties in modular lattices are studied. It is shown that in a certain class of modular lattices, the direct sum of two elements has absolute direct summand if and only if the elements are relatively injective. As a generalization of absolute direct summand (ADS for short), the concept of Goldie absolute direct summand in lattices is introduced and studied. It is shown that Goldie ADS property is inherited by direct summands. A necessary and sufficient condition is given for an element of modular lattice to have Goldie ADS.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42393366","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 : 2022-07-10DOI: 10.21136/mb.2023.0162-22
U. M. Hanung
Various kinds of Stieltjes integrals using gauge integration have become highly popular in the field of differential equations and other applications. In the theories of integration and of ordinary differential equations, convergence theorems provide one of the most widely used tools. The Harnack extension principle, which discusses a sufficient condition for Kurzweil-Henstock integrable functions on particular subsets of $(a,b)$ to be integrable on $[a,b]$, is a key step to supply convergence theorems. The Kurzweil-Stieltjes integral reduces to the Kurzweil-Henstock integral whenever the integrator is an identity function. In general, if the integrator $F$ is discontinuous on $[c,d]subset[a,b]$, then the values of the Kurzweil-Stieltjes integrals $$int_c^d[dF]g, int_{[c,d]}[dF]g, int_{[c,d)}[dF]g, int_{(c,d]}[dF]g, {rm and} int_{(c,d)}[dF]g$$ need not coincide. Hence, the Harnack extension principle in the Kurzweil-Henstock integral cannot be valid any longer for the Kurzweil-type Stieltjes integrals with discontinuous integrators. The new concepts of equi-integrability and equiregulatedness are pivotal to the notion of the Harnack extension principle for the Kurzweil-Stieltjes integration. Moreover, the existence of the integral $int_a^b[dF]g$ does not (even in the case of the identity integrator) always imply the existence of the integral $int_{T}[dF]g$ for every subset $T$ of $[a,b]$. This follows from the well-known fact that, if e.g., $Tsubset[a,b]$ is not measurable, then the existence of the Lebesgue integral $int_a^b g [dt]$ does not imply that the integral $int_T g [dt]$ exists. Therefore, besides constructing the Harnack extension principle for the abstract Kurzweil-Stieltjes integral, the aim of this paper is also to demonstrate its role in guaranteeing the existence of the integrals $int_{T}[dF]g$ for arbitrary subsets $T$ of an elementary set $E$.
利用规范积分的各种Stieltjes积分在微分方程和其他应用领域得到了广泛的应用。在积分理论和常微分方程中，收敛定理是应用最广泛的工具之一。利用Harnack扩展原理，讨论了$(a,b)$的特定子集上的Kurzweil-Henstock可积函数在$[a,b]$上可积的充分条件，是给出收敛定理的关键步骤。当积分器是恒等函数时，Kurzweil-Stieltjes积分化为Kurzweil-Henstock积分。一般来说，如果积分器$F$在$[c,d]subset[a,b]$上是不连续的，那么Kurzweil-Stieltjes积分$$int_c^d[dF]g, int_{[c,d]}[dF]g, int_{[c,d)}[dF]g, int_{(c,d]}[dF]g, {rm and} int_{(c,d)}[dF]g$$的值不必重合。因此，对于具有不连续积分器的kurzweil型Stieltjes积分，Kurzweil-Henstock积分中的Harnack扩展原理不再有效。对于Kurzweil-Stieltjes积分的Harnack可拓原理的概念来说，等可积性和等正则性的新概念至关重要。此外，积分$int_a^b[dF]g$的存在性并不总是意味着对于$[a,b]$的每个子集$T$的积分$int_{T}[dF]g$的存在性(即使在单位积分器的情况下)。这源于一个众所周知的事实，即如果例如$Tsubset[a,b]$不可测量，那么勒贝格积分$int_a^b g [dt]$的存在并不意味着积分$int_T g [dt]$的存在。因此，本文除了构造抽象Kurzweil-Stieltjes积分的Harnack可拓原理外，还证明了它在保证初等集合$E$的任意子集$T$的积分$int_{T}[dF]g$存在性方面的作用。
{"title":"Role of the Harnack extension principle in the Kurzweil-Stieltjes integral","authors":"U. M. Hanung","doi":"10.21136/mb.2023.0162-22","DOIUrl":"https://doi.org/10.21136/mb.2023.0162-22","url":null,"abstract":"Various kinds of Stieltjes integrals using gauge integration have become highly popular in the field of differential equations and other applications. In the theories of integration and of ordinary differential equations, convergence theorems provide one of the most widely used tools. The Harnack extension principle, which discusses a sufficient condition for Kurzweil-Henstock integrable functions on particular subsets of $(a,b)$ to be integrable on $[a,b]$, is a key step to supply convergence theorems. The Kurzweil-Stieltjes integral reduces to the Kurzweil-Henstock integral whenever the integrator is an identity function. In general, if the integrator $F$ is discontinuous on $[c,d]subset[a,b]$, then the values of the Kurzweil-Stieltjes integrals $$int_c^d[dF]g, int_{[c,d]}[dF]g, int_{[c,d)}[dF]g, int_{(c,d]}[dF]g, {rm and} int_{(c,d)}[dF]g$$ need not coincide. Hence, the Harnack extension principle in the Kurzweil-Henstock integral cannot be valid any longer for the Kurzweil-type Stieltjes integrals with discontinuous integrators. The new concepts of equi-integrability and equiregulatedness are pivotal to the notion of the Harnack extension principle for the Kurzweil-Stieltjes integration. Moreover, the existence of the integral $int_a^b[dF]g$ does not (even in the case of the identity integrator) always imply the existence of the integral $int_{T}[dF]g$ for every subset $T$ of $[a,b]$. This follows from the well-known fact that, if e.g., $Tsubset[a,b]$ is not measurable, then the existence of the Lebesgue integral $int_a^b g [dt]$ does not imply that the integral $int_T g [dt]$ exists. Therefore, besides constructing the Harnack extension principle for the abstract Kurzweil-Stieltjes integral, the aim of this paper is also to demonstrate its role in guaranteeing the existence of the integrals $int_{T}[dF]g$ for arbitrary subsets $T$ of an elementary set $E$.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48524672","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 : 2022-06-27DOI: 10.21136/mb.2022.0069-21
Sungchol Kim, Dukman Ri
. This paper presents several sufficient conditions for the existence of weak solutions to general nonlinear elliptic problems of the type where Ω is a bounded domain of R n , n > 2. In particular, we do not require strict monotonicity of the principal part a ( x, z, · ), while the approach is based on the variational method and results of the variable exponent function spaces.
{"title":"Existence of weak solutions for elliptic Dirichlet problems with variable exponent","authors":"Sungchol Kim, Dukman Ri","doi":"10.21136/mb.2022.0069-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0069-21","url":null,"abstract":". This paper presents several sufficient conditions for the existence of weak solutions to general nonlinear elliptic problems of the type where Ω is a bounded domain of R n , n > 2. In particular, we do not require strict monotonicity of the principal part a ( x, z, · ), while the approach is based on the variational method and results of the variable exponent function spaces.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46707297","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 : 2022-06-23DOI: 10.21136/mb.2022.0061-21
Y. Akdim, M. Belayachi, H. Hjiaj
. This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by where Ω is a bounded open set of R N ( N > 2) with 1 < p < N and f ∈ L 1 (Ω) , under some growth conditions on the function b ( · ) and d ( · ) , where c ( · ) is assumed to be in L N/ ( p − 1) (Ω) . We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.
． 本文研究了一类非线性退化椭圆型方程，其原型为:Ω是R N (N > 2)的有界开集，1 < p < N, f∈l1 (Ω)，在函数b(·)和d(·)的某些生长条件下，假设c(·)在L N/ (p−1)(Ω)中。我们证明了该非强制椭圆方程重整解的存在性，并得到了一些正则性结果。
{"title":"Existence of renormalized solutions for some degenerate and non-coercive elliptic equations","authors":"Y. Akdim, M. Belayachi, H. Hjiaj","doi":"10.21136/mb.2022.0061-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0061-21","url":null,"abstract":". This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by where Ω is a bounded open set of R N ( N > 2) with 1 < p < N and f ∈ L 1 (Ω) , under some growth conditions on the function b ( · ) and d ( · ) , where c ( · ) is assumed to be in L N/ ( p − 1) (Ω) . We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45779416","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 : 2022-06-07DOI: 10.21136/mb.2022.0128-21
M. M. Chems-Eddin
Let p ≡ 1 (mod 8) and q ≡ 3 (mod 8) be two prime integers and let l 6∈ {−1, p, q} be a positive odd square-free integer. Assuming that the fundamental unit of Q (√ 2p ) has a negative norm, we investigate the unit group of the fields Q (√ 2, √ p, √ q, √ −l ) .
{"title":"On units of some fields of the form $mathbb{Q}big(sqrt2, sqrt{p}, sqrt{q}, sqrt{-ell}big)$","authors":"M. M. Chems-Eddin","doi":"10.21136/mb.2022.0128-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0128-21","url":null,"abstract":"Let p ≡ 1 (mod 8) and q ≡ 3 (mod 8) be two prime integers and let l 6∈ {−1, p, q} be a positive odd square-free integer. Assuming that the fundamental unit of Q (√ 2p ) has a negative norm, we investigate the unit group of the fields Q (√ 2, √ p, √ q, √ −l ) .","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44744636","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 : 2022-06-06DOI: 10.21136/mb.2022.0197-20
Santosh Kumar, Johnson Allen Kessy
. The notions of compatible mappings play a crucial role in metrical fixed point theory. Partial metric spaces are a generalization of the notion of a metric space in the sense that distance of a point from itself is not necessarily zero. In this paper, we prove coincidence and fixed point theorems for a pair of single-valued and multi-valued weak compatible mappings on a complete partial metric space. Our main results generalize, in particular, the results of Kaneko and Sessa (1989), Pathak (1995) and Kessy, Kumar and Kakiko (2017). Examples that illustrate the generality of our results are also provided.
{"title":"Fixed point theorems for hybrid pair of weak compatible mappings\u0000 \u0000in partial metric spaces","authors":"Santosh Kumar, Johnson Allen Kessy","doi":"10.21136/mb.2022.0197-20","DOIUrl":"https://doi.org/10.21136/mb.2022.0197-20","url":null,"abstract":". The notions of compatible mappings play a crucial role in metrical fixed point theory. Partial metric spaces are a generalization of the notion of a metric space in the sense that distance of a point from itself is not necessarily zero. In this paper, we prove coincidence and fixed point theorems for a pair of single-valued and multi-valued weak compatible mappings on a complete partial metric space. Our main results generalize, in particular, the results of Kaneko and Sessa (1989), Pathak (1995) and Kessy, Kumar and Kakiko (2017). Examples that illustrate the generality of our results are also provided.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47492304","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 : 2022-06-02DOI: 10.21136/mb.2022.0100-21
Figen Takil Mutlu, A. Tercan, R. Yaşar
In this article, we study modules with the weak FI-extending property. We prove that if M satisfies weak FI-extending, pseudo duo, C3 properties and M/SocM has finite uniform dimension then M decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the weak FI-extending, pseudo duo, C3 properties and ascending (or descending) chain condition on essential submodules then M = M1 ⊕ M2 for some semisimple submodule M1 and Noetherian (or Artinian, respectively) submodule M2. Moreover, we show that a nonsingular weak CS (or weak C 11, or weak FI) module has a direct summand which essentially contains the socle of the module and is a CS (or C11, or FI-extending, respectively) module.
{"title":"Eventually semisimple weak $FI$-extending modules","authors":"Figen Takil Mutlu, A. Tercan, R. Yaşar","doi":"10.21136/mb.2022.0100-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0100-21","url":null,"abstract":"In this article, we study modules with the weak FI-extending property. We prove that if M satisfies weak FI-extending, pseudo duo, C3 properties and M/SocM has finite uniform dimension then M decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the weak FI-extending, pseudo duo, C3 properties and ascending (or descending) chain condition on essential submodules then M = M1 ⊕ M2 for some semisimple submodule M1 and Noetherian (or Artinian, respectively) submodule M2. Moreover, we show that a nonsingular weak CS (or weak C 11, or weak FI) module has a direct summand which essentially contains the socle of the module and is a CS (or C11, or FI-extending, respectively) module.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43808788","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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