In this paper, the notion of set-valued L -contractions is introduced, and a new fixed point theorem for such contractions is established. An example to illustrate main theorem is given.
本文引入了集值L-收缩的概念，并建立了一个新的集值L-压缩的不动点定理。给出了一个主要定理的例子。
{"title":"Fixed Point Theorems for Set-Valued \u0000 L\u0000 -Contractions in Branciari Distance Spaces","authors":"S. Cho","doi":"10.1155/2021/6384818","DOIUrl":"https://doi.org/10.1155/2021/6384818","url":null,"abstract":"In this paper, the notion of set-valued \u0000 \u0000 L\u0000 \u0000 -contractions is introduced, and a new fixed point theorem for such contractions is established. An example to illustrate main theorem is given.","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41668885","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}
In this article, we will recall the main properties of the Fourier transform on the Heisenberg motion group , where and denote the Heisenberg group. Then, we will present some uncertainty principles associated to this transform as Beurling, Hardy, and Gelfand-Shilov.
{"title":"Uncertainty Principles for Heisenberg Motion Group","authors":"Walid Amghar","doi":"10.1155/2021/3734817","DOIUrl":"https://doi.org/10.1155/2021/3734817","url":null,"abstract":"<jats:p>In this article, we will recall the main properties of the Fourier transform on the Heisenberg motion group <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <mi>G</mi>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>ℍ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msup>\u0000 <mo>⋊</mo>\u0000 <mi>K</mi>\u0000 </math>\u0000 </jats:inline-formula>, where <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <mi>K</mi>\u0000 <mo>=</mo>\u0000 <mi>U</mi>\u0000 <mfenced open=\"(\" close=\")\">\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula> and <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <msup>\u0000 <mrow>\u0000 <mi>ℍ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msup>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>ℂ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msup>\u0000 <mo>×</mo>\u0000 <mi>ℝ</mi>\u0000 </math>\u0000 </jats:inline-formula> denote the Heisenberg group. Then, we will present some uncertainty principles associated to this transform as Beurling, Hardy, and Gelfand-Shilov.</jats:p>","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45781238","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}
Solomon Gebregiorgis Teweldemedhin, Kidane Koyas Tola
In this paper, we introduce an almost generalized α -admissible Z -contraction with the help of a simulation function and study fixed point results in the setting of partially ordered b-metric spaces. The presented results generalize and unify several related fixed point results in the existing literature. Finally, we verify our results by using two examples. Moreover, one of our fixed point results is applied to guarantee the existence of a solution of an integral equation.
{"title":"Fixed Point Results for an Almost Generalized \u0000 α\u0000 -Admissible \u0000 Z\u0000 -Contraction in the Setting of Partially Ordered b-Metric Spaces","authors":"Solomon Gebregiorgis Teweldemedhin, Kidane Koyas Tola","doi":"10.1155/2021/5988007","DOIUrl":"https://doi.org/10.1155/2021/5988007","url":null,"abstract":"In this paper, we introduce an almost generalized \u0000 \u0000 α\u0000 \u0000 -admissible \u0000 \u0000 Z\u0000 \u0000 -contraction with the help of a simulation function and study fixed point results in the setting of partially ordered b-metric spaces. The presented results generalize and unify several related fixed point results in the existing literature. Finally, we verify our results by using two examples. Moreover, one of our fixed point results is applied to guarantee the existence of a solution of an integral equation.","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48800617","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}
Chaundy and Jolliffe proved that if is a nonnegative, nonincreasing real sequence, then series converges uniformly if and only if . The purpose of this paper is to show that if is nonincreasing and
Chaundy和Jolliffe证明了如果是非负的、不增加的实数序列，则级数∑a n sin nx一致收敛当且仅当n⟶ 0。本文的目的是证明如果n是非递增的并且n⟶ 0，则级数f x=∑an sin n x可以在c上逐项微分，d>0。然而f′0可能不存在。
{"title":"A Note on Derivative of Sine Series with Square Root","authors":"Sergiusz Kęska","doi":"10.1155/2021/7035776","DOIUrl":"https://doi.org/10.1155/2021/7035776","url":null,"abstract":"<jats:p>Chaundy and Jolliffe proved that if <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <mfenced open=\"{\" close=\"}\">\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula> is a nonnegative, nonincreasing real sequence, then series <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <mo>∑</mo>\u0000 <msub>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mi mathvariant=\"normal\">sin</mi>\u0000 <mfenced open=\"(\" close=\")\">\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mi>x</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula> converges uniformly if and only if <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <mi>n</mi>\u0000 <msub>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>⟶</mo>\u0000 <mn>0</mn>\u0000 </math>\u0000 </jats:inline-formula>. The purpose of this paper is to show that if <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\u0000 <mfenced open=\"{\" close=\"}\">\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <msub>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula> is nonincreasing and <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\"","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47847787","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}
Variational methods are used in order to establish the existence of nontrivial weak solution for superlinear second-order system with noninstantaneous impulses. The main result is obtained when a kind of definition of the weak solution for this system is introduced. Meanwhile, an example is presented to illustrate the main result.
{"title":"Existence of Solutions for Superlinear Second-Order System with Noninstantaneous Impulses","authors":"Yucheng Bu","doi":"10.1155/2021/5515575","DOIUrl":"https://doi.org/10.1155/2021/5515575","url":null,"abstract":"Variational methods are used in order to establish the existence of nontrivial weak solution for superlinear second-order system with noninstantaneous impulses. The main result is obtained when a kind of definition of the weak solution for this system is introduced. Meanwhile, an example is presented to illustrate the main result.","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46631711","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}
In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, , and mesh size, The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and - uniformly convergent for where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.
{"title":"Uniformly Convergent Nonpolynomial Spline Method for Singularly Perturbed Robin-Type Boundary Value Problems with Discontinuous Source Term","authors":"H. Debela, G. Duressa","doi":"10.1155/2021/7569209","DOIUrl":"https://doi.org/10.1155/2021/7569209","url":null,"abstract":"In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, , and mesh size, The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and - uniformly convergent for where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48775699","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}
This paper proposes a new fitted operator strategy for solving singularly perturbed parabolic partial differential equation with delay on the spatial variable. We decomposed the problem into three piecewise equations. The delay term in the equation is expanded by Taylor series, the time variable is discretized by implicit Euler method, and the space variable is discretized by central difference methods. After developing the fitting operator method, we accelerate the order of convergence of the time direction using Richardson extrapolation scheme and obtained uniform order of convergence. Finally, three examples are given to illustrate the effectiveness of the method. The result shows the proposed method is more accurate than some of the methods that exist in the literature.
{"title":"Numerical Treatment on Parabolic Singularly Perturbed Differential Difference Equation via Fitted Operator Scheme","authors":"D. Tefera, A. Tiruneh, G. A. Derese","doi":"10.1155/2021/1661661","DOIUrl":"https://doi.org/10.1155/2021/1661661","url":null,"abstract":"This paper proposes a new fitted operator strategy for solving singularly perturbed parabolic partial differential equation with delay on the spatial variable. We decomposed the problem into three piecewise equations. The delay term in the equation is expanded by Taylor series, the time variable is discretized by implicit Euler method, and the space variable is discretized by central difference methods. After developing the fitting operator method, we accelerate the order of convergence of the time direction using Richardson extrapolation scheme and obtained uniform order of convergence. Finally, three examples are given to illustrate the effectiveness of the method. The result shows the proposed method is more accurate than some of the methods that exist in the literature.","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":"2021 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41312566","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}
In this work, we consider the Darboux frame T , V , U of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a V -direction curve. Subsequently, a detailed study of these surfaces is made in the case where the directing vector of their generatrices is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.
{"title":"Some Special Ruled Surfaces Generated by a Direction Curve according to the Darboux Frame and their Characterizations","authors":"Nidal Echabbi, Amina Ouazzani Chahdi","doi":"10.1155/2021/8624794","DOIUrl":"https://doi.org/10.1155/2021/8624794","url":null,"abstract":"In this work, we consider the Darboux frame \u0000 \u0000 \u0000 \u0000 T\u0000 ,\u0000 V\u0000 ,\u0000 U\u0000 \u0000 \u0000 \u0000 of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a \u0000 \u0000 V\u0000 \u0000 -direction curve. Subsequently, a detailed study of these surfaces is made in the case where the directing vector of their generatrices is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41568277","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}
We present the concept of -contractive multivalued mappings in -metric spaces and prove some fixed point results for these mappings in this study. Our results expand and refine some of the literature’s findings in fixed point theory.
{"title":"Fixed Point Theorems for \u0000 \u0000 \u0000 α\u0000 ,\u0000 k\u0000 ,\u0000 θ\u0000 ,\u0000 φ\u0000 ","authors":"Youssef Errai, E. Marhrani, M. Aamri","doi":"10.1155/2021/9959374","DOIUrl":"https://doi.org/10.1155/2021/9959374","url":null,"abstract":"<jats:p>We present the concept of <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <mfenced open=\"(\" close=\")\">\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>,</mo>\u0000 <mi>k</mi>\u0000 <mo>,</mo>\u0000 <mi>θ</mi>\u0000 <mo>,</mo>\u0000 <mi>φ</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula>-contractive multivalued mappings in <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\u0000 <mi>b</mi>\u0000 </math>\u0000 </jats:inline-formula>-metric spaces and prove some fixed point results for these mappings in this study. Our results expand and refine some of the literature’s findings in fixed point theory.</jats:p>","PeriodicalId":7061,"journal":{"name":"Abstract and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42530044","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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