In this note, we study some approximation properties on a class of special Lototsky–Bernstein bases. We focus on approximation of $|x|$ on $[-1,1]$ by an approximation process generated from fixed points on Lototsky–Bernstein bases. Our first result shows that the approximation procedure $p_{n}(x)$ to $|x|$ preserves good shapes on $[-1,1]$ . Moreover, some convergence results and inequalities are derived. Our second main result states that the rate convergence of the approximation is $O(n^{-2})$ .
{"title":"A note on Lototsky–Bernstein bases","authors":"Xiao-Wei Xu, Xin Yu, Jia-Lin Cui, Qing-Bo Cai, Wen-Tao Cheng","doi":"10.1186/s13660-024-03076-7","DOIUrl":"https://doi.org/10.1186/s13660-024-03076-7","url":null,"abstract":"In this note, we study some approximation properties on a class of special Lototsky–Bernstein bases. We focus on approximation of $|x|$ on $[-1,1]$ by an approximation process generated from fixed points on Lototsky–Bernstein bases. Our first result shows that the approximation procedure $p_{n}(x)$ to $|x|$ preserves good shapes on $[-1,1]$ . Moreover, some convergence results and inequalities are derived. Our second main result states that the rate convergence of the approximation is $O(n^{-2})$ .","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"39 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-05DOI: 10.1186/s13660-024-03096-3
Guaiqi Tian, Yucheng An, Hongmin Suo
In this work, we study the following Schrödinger-Poisson system $$ textstylebegin{cases} -Delta _{H}u+mu phi u=lambda u^{-gamma}, &text{in } Omega , -Delta _{H}phi =u^{2}, &text{in } Omega , u>0, &text{in } Omega , u=phi =0, &text{on } partial Omega , end{cases} $$ where $Delta _{H}$ is the Kohn-Laplacian on the first Heisenberg group $mathbb{H}^{1}$ , and $Omega subset mathbb{H}^{1}$ is a smooth bounded domain, $mu =pm 1$ , $00$ are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for $mu =1$ and each $lambda >0$ . Multiple solutions of the system are also considered for $mu =-1$ and $lambda >0$ small enough using the critical point theory for nonsmooth functional.
{"title":"Multiple positive solutions for Schrödinger-Poisson system with singularity on the Heisenberg group","authors":"Guaiqi Tian, Yucheng An, Hongmin Suo","doi":"10.1186/s13660-024-03096-3","DOIUrl":"https://doi.org/10.1186/s13660-024-03096-3","url":null,"abstract":"In this work, we study the following Schrödinger-Poisson system $$ textstylebegin{cases} -Delta _{H}u+mu phi u=lambda u^{-gamma}, &text{in } Omega , -Delta _{H}phi =u^{2}, &text{in } Omega , u>0, &text{in } Omega , u=phi =0, &text{on } partial Omega , end{cases} $$ where $Delta _{H}$ is the Kohn-Laplacian on the first Heisenberg group $mathbb{H}^{1}$ , and $Omega subset mathbb{H}^{1}$ is a smooth bounded domain, $mu =pm 1$ , $0<gamma <1$ , and $lambda >0$ are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for $mu =1$ and each $lambda >0$ . Multiple solutions of the system are also considered for $mu =-1$ and $lambda >0$ small enough using the critical point theory for nonsmooth functional.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"606 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139690341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-01DOI: 10.1186/s13660-024-03095-4
Hadi Obaid AlShammari
The purpose of this paper is to introduce and study the structure of p-tuple of $(n,m)$ - $mathcal{D}$ -normal operators. This is a generalization of the class of p-tuple of n-normal operators. We consider a generalization of these single variable n- $mathcal{D}$ -normal and $(n,m)$ - $mathcal{D}$ -normal operators and explore some of their basic properties.
本文旨在介绍和研究 $(n,m)$ - $mathcal{D}$ 正则算子的 p-tuple 结构。这是 n 常算子 p 元组类的广义化。我们考虑了这些单变量 n- $mathcal{D}$ - 常算子和 $(n,m)$ - $mathcal{D}$ - 常算子的一般化，并探讨了它们的一些基本性质。
{"title":"Higher order ((n,m))-Drazin normal operators","authors":"Hadi Obaid AlShammari","doi":"10.1186/s13660-024-03095-4","DOIUrl":"https://doi.org/10.1186/s13660-024-03095-4","url":null,"abstract":"The purpose of this paper is to introduce and study the structure of p-tuple of $(n,m)$ - $mathcal{D}$ -normal operators. This is a generalization of the class of p-tuple of n-normal operators. We consider a generalization of these single variable n- $mathcal{D}$ -normal and $(n,m)$ - $mathcal{D}$ -normal operators and explore some of their basic properties.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"62 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139666551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-31DOI: 10.1186/s13660-024-03094-5
Lei Shi, Muhammad Abbas, Mohsan Raza, Muhammad Arif, Poom Kumam
In recent years, many subclasses of univalent functions, directly or not directly related to the exponential functions, have been introduced and studied. In this paper, we consider the class of $mathcal{S}^{ast}_{e}$ for which $zf^{prime}(z)/f(z)$ is subordinate to $e^{z}$ in the open unit disk. The classic concept of Hankel determinant is generalized by replacing the inverse logarithmic coefficient of functions belonging to certain subclasses of univalent functions. In particular, we obtain the best possible bounds for the second Hankel determinant of logarithmic coefficients of inverse starlike functions subordinated to exponential functions. This work may inspire to pay more attention to the coefficient properties with respect to the inverse functions of various classes of univalent functions.
{"title":"Inverse logarithmic coefficient bounds for starlike functions subordinated to the exponential functions","authors":"Lei Shi, Muhammad Abbas, Mohsan Raza, Muhammad Arif, Poom Kumam","doi":"10.1186/s13660-024-03094-5","DOIUrl":"https://doi.org/10.1186/s13660-024-03094-5","url":null,"abstract":"In recent years, many subclasses of univalent functions, directly or not directly related to the exponential functions, have been introduced and studied. In this paper, we consider the class of $mathcal{S}^{ast}_{e}$ for which $zf^{prime}(z)/f(z)$ is subordinate to $e^{z}$ in the open unit disk. The classic concept of Hankel determinant is generalized by replacing the inverse logarithmic coefficient of functions belonging to certain subclasses of univalent functions. In particular, we obtain the best possible bounds for the second Hankel determinant of logarithmic coefficients of inverse starlike functions subordinated to exponential functions. This work may inspire to pay more attention to the coefficient properties with respect to the inverse functions of various classes of univalent functions.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"32 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-31DOI: 10.1186/s13660-024-03090-9
H. M. Srivastava, Shahid Khan, Sarfraz Nawaz Malik, Fairouz Tchier, Afis Saliu, Qin Xin
Two new subclasses of the class of bi-Bazilevič functions, which are related to the Fibonacci-number series and the square-root functions, are introduced and studied in this article. Under a special choice of the parameter involved, these two classes of Bazilevič functions reduce to two new subclasses of star-like biunivalent functions related with the Fibonacci-number series and the square-root functions. Using the Faber polynomial expansion (FPE) technique, we find the general coefficient bounds for the functions belonging to each of these classes. We also find bounds for the initial coefficients for bi-Bazilevič functions and demonstrate how unexpectedly these initial coefficients behave in relation to the square-root functions and the Fibonacci-number series.
{"title":"Faber polynomial coefficient inequalities for bi-Bazilevič functions associated with the Fibonacci-number series and the square-root functions","authors":"H. M. Srivastava, Shahid Khan, Sarfraz Nawaz Malik, Fairouz Tchier, Afis Saliu, Qin Xin","doi":"10.1186/s13660-024-03090-9","DOIUrl":"https://doi.org/10.1186/s13660-024-03090-9","url":null,"abstract":"Two new subclasses of the class of bi-Bazilevič functions, which are related to the Fibonacci-number series and the square-root functions, are introduced and studied in this article. Under a special choice of the parameter involved, these two classes of Bazilevič functions reduce to two new subclasses of star-like biunivalent functions related with the Fibonacci-number series and the square-root functions. Using the Faber polynomial expansion (FPE) technique, we find the general coefficient bounds for the functions belonging to each of these classes. We also find bounds for the initial coefficients for bi-Bazilevič functions and demonstrate how unexpectedly these initial coefficients behave in relation to the square-root functions and the Fibonacci-number series.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"173 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-30DOI: 10.1186/s13660-024-03088-3
Pishtiwan Othman Sabir, Ravi P. Agarwal, Shabaz Jalil Mohammedfaeq, Pshtiwan Othman Mohammed, Nejmeddine Chorfi, Thabet Abdeljawad
Making use of the Hankel determinant and the Ruscheweyh derivative, in this work, we consider a general subclass of m-fold symmetric normalized biunivalent functions defined in the open unit disk. Moreover, we investigate the bounds for the second Hankel determinant of this class and some consequences of the results are presented. In addition, to demonstrate the accuracy on some functions and conditions, most general programs are written in Python V.3.8.8 (2021).
在这项工作中，我们利用汉克尔行列式和 Ruscheweyh 导数，考虑了定义在开放单位盘中的 m 折对称归一化双等价函数的一般子类。此外，我们还研究了该类函数的第二汉克尔行列式的边界，并给出了结果的一些后果。此外，为了证明某些函数和条件的准确性，我们使用 Python V.3.8.8 (2021) 编写了大部分通用程序。
{"title":"Hankel determinant for a general subclass of m-fold symmetric biunivalent functions defined by Ruscheweyh operators","authors":"Pishtiwan Othman Sabir, Ravi P. Agarwal, Shabaz Jalil Mohammedfaeq, Pshtiwan Othman Mohammed, Nejmeddine Chorfi, Thabet Abdeljawad","doi":"10.1186/s13660-024-03088-3","DOIUrl":"https://doi.org/10.1186/s13660-024-03088-3","url":null,"abstract":"Making use of the Hankel determinant and the Ruscheweyh derivative, in this work, we consider a general subclass of m-fold symmetric normalized biunivalent functions defined in the open unit disk. Moreover, we investigate the bounds for the second Hankel determinant of this class and some consequences of the results are presented. In addition, to demonstrate the accuracy on some functions and conditions, most general programs are written in Python V.3.8.8 (2021).","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"151 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139590184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-30DOI: 10.1186/s13660-024-03079-4
Maliha Rashid, Naeem Saleem, Rabia Bibi, Reny George
In this manuscript, we use the concept of multidimensional fixed point in a generalized space, namely, C-distance space with some nonlinear contraction conditions, such as Jaggi- and Dass-Gupta-type contractions. We provide results with a Jaggi-type hybrid contraction for the mentioned space. Moreover, we use control functions to get the desired results. After each theorem, we compare our results with previous ones to show that they are generalized. We provide examples to support our results. An application is also performed to solve the system of integral equations.
{"title":"Some multidimensional fixed point theorems for nonlinear contractions in C-distance spaces with applications","authors":"Maliha Rashid, Naeem Saleem, Rabia Bibi, Reny George","doi":"10.1186/s13660-024-03079-4","DOIUrl":"https://doi.org/10.1186/s13660-024-03079-4","url":null,"abstract":"In this manuscript, we use the concept of multidimensional fixed point in a generalized space, namely, C-distance space with some nonlinear contraction conditions, such as Jaggi- and Dass-Gupta-type contractions. We provide results with a Jaggi-type hybrid contraction for the mentioned space. Moreover, we use control functions to get the desired results. After each theorem, we compare our results with previous ones to show that they are generalized. We provide examples to support our results. An application is also performed to solve the system of integral equations.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"6 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139590613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article considers and investigates a variational inequality problem and fixed-point problems in real Hilbert spaces endowed with graphs. A regularization method is proposed for solving a G-variational inequality problem and a common fixed-point problem of a finite family of G-nonexpansive mappings in the framework of Hilbert spaces endowed with graphs, which extends the work of Tiammee et al. (Fixed Point Theory Appl. 187, 2015) and Kangtunyakarn, A. (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112:437–448, 2018). Under certain conditions, a strong convergence theorem of the proposed method is proved. Finally, we provide numerical examples to support our main theorem. The numerical examples show that the speed of the proposed method is better than some recent existing methods in the literature.
本文考虑并研究了具有图的实希尔伯特空间中的变分不等式问题和定点问题。本文提出了一种正则化方法，用于解决禀赋有图的希尔伯特空间框架中的G变分不等式问题和有限族G-无穷映射的普通定点问题，该方法扩展了Tiammee等人（《定点理论应用》，187，2015年）和Kangtunyakarn，A. （Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112:437-448，2018年）的工作。在某些条件下，证明了所提方法的强收敛定理。最后，我们提供了数值示例来支持我们的主定理。数值实例表明，所提方法的速度优于近期文献中的一些现有方法。
{"title":"A regularization method for solving the G-variational inequality problem and fixed-point problems in Hilbert spaces endowed with graphs","authors":"Wongvisarut Khuangsatung, Akarate Singta, Atid Kangtunyakarn","doi":"10.1186/s13660-024-03089-2","DOIUrl":"https://doi.org/10.1186/s13660-024-03089-2","url":null,"abstract":"This article considers and investigates a variational inequality problem and fixed-point problems in real Hilbert spaces endowed with graphs. A regularization method is proposed for solving a G-variational inequality problem and a common fixed-point problem of a finite family of G-nonexpansive mappings in the framework of Hilbert spaces endowed with graphs, which extends the work of Tiammee et al. (Fixed Point Theory Appl. 187, 2015) and Kangtunyakarn, A. (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112:437–448, 2018). Under certain conditions, a strong convergence theorem of the proposed method is proved. Finally, we provide numerical examples to support our main theorem. The numerical examples show that the speed of the proposed method is better than some recent existing methods in the literature.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"90 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-25DOI: 10.1186/s13660-024-03093-6
Mohamed Houas, Mohammad Esmael Samei, Shyam Sundar Santra, Jehad Alzabut
In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes.
{"title":"On a Duffing-type oscillator differential equation on the transition to chaos with fractional q-derivatives","authors":"Mohamed Houas, Mohammad Esmael Samei, Shyam Sundar Santra, Jehad Alzabut","doi":"10.1186/s13660-024-03093-6","DOIUrl":"https://doi.org/10.1186/s13660-024-03093-6","url":null,"abstract":"In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"183 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-25DOI: 10.1186/s13660-024-03092-7
Muhammad Asim, Irshad Ayoob, Amjad Hussain, Nabil Mlaiki
In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces ${Mdot{K}^{alpha (cdot ),lambda}_{q,p(cdot)}(w)}$ . Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents.
{"title":"Weighted estimates for fractional bilinear Hardy operators on variable exponent Morrey–Herz space","authors":"Muhammad Asim, Irshad Ayoob, Amjad Hussain, Nabil Mlaiki","doi":"10.1186/s13660-024-03092-7","DOIUrl":"https://doi.org/10.1186/s13660-024-03092-7","url":null,"abstract":"In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces ${Mdot{K}^{alpha (cdot ),lambda}_{q,p(cdot)}(w)}$ . Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"61 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139553285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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