Pub Date : 2024-02-14DOI: 10.1186/s13660-024-03101-9
Shazia Kanwal, Hüseyin Işık, Sana Waheed
The main purpose of this research article is to generalize Kannan-type fixed-point (FP) theorems for single-valued mappings and Chatterjea-type FP result for fuzzy mappings (FMs) in the context of complete strong b-metric spaces (MSs). Moreover, fuzzy FPs are established for Suzuki-type fuzzy contraction in the setting of complete strong b-MSs. The conclusions are supported by nontrivial examples to enhance the validity of the results obtained in this study. In addition, previous findings have been made as corollaries from the relevant literature. The numerous implications that this technique has across the literature improve and integrate our findings. Applications of some of the results obtained are also incorporated.
{"title":"Generalized fixed points for fuzzy and nonfuzzy mappings in strong b-metric spaces","authors":"Shazia Kanwal, Hüseyin Işık, Sana Waheed","doi":"10.1186/s13660-024-03101-9","DOIUrl":"https://doi.org/10.1186/s13660-024-03101-9","url":null,"abstract":"The main purpose of this research article is to generalize Kannan-type fixed-point (FP) theorems for single-valued mappings and Chatterjea-type FP result for fuzzy mappings (FMs) in the context of complete strong b-metric spaces (MSs). Moreover, fuzzy FPs are established for Suzuki-type fuzzy contraction in the setting of complete strong b-MSs. The conclusions are supported by nontrivial examples to enhance the validity of the results obtained in this study. In addition, previous findings have been made as corollaries from the relevant literature. The numerous implications that this technique has across the literature improve and integrate our findings. Applications of some of the results obtained are also incorporated.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139773377","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 : 2024-02-14DOI: 10.1186/s13660-024-03097-2
Mi Zhou, Naeem Saleem, Mujahid Abbas
In this paper, we introduce two types of weak enriched contractions, namely weak enriched $mathcal{F}$ -contraction, weak enriched $mathcal{F^{prime}}$ -contraction, and k-fold averaged mapping based on Kirk’s iterative algorithm of order k. The types of contractions introduced herein unify, extend, and generalize several existing classes of enriched and weak enriched contraction mappings. Moreover, K-fold averaged mappings can be viewed as a generalization of the averaged mappings and double averaged mappings. We then prove the existence of a unique fixed point of the k-fold averaged mapping associated with weak enriched contractions introduced herein. We study necessary conditions that guarantee the equality of the sets of fixed points of the k-fold averaged mapping and weak enriched contractions. We show that an appropriate Kirk’s iterative algorithm can be used to approximate a fixed point of a k-fold averaged mapping and of the two weak enriched contractions. We also study the well-posedness, limit shadowing property, and Ulam–Hyers stability of the k-fold averaged mapping. We provide necessary conditions that ensure the periodic point property of each illustrated weak enriched contraction. Some examples are presented to show that our results are a potential generalization of the comparable results in the existing literature.
本文介绍了两种弱富集收缩，即弱富集 $mathcal{F}$ -收缩、弱富集 $mathcal{F^/{prime}}$ -收缩以及基于 Kirk 阶迭代算法的 k 折平均映射。本文介绍的收缩类型统一、扩展和概括了现有的几类富集和弱富集收缩映射。此外，K 折平均映射可以看作是平均映射和双平均映射的一般化。然后，我们证明了与本文引入的弱富集收缩相关的 K 折平均映射的唯一定点的存在性。我们研究了保证 k 折平均映射和弱充实收缩的定点集相等的必要条件。我们证明，可以使用适当的柯克迭代算法来逼近 k 折平均映射和两个弱充实收缩的定点。我们还研究了 k 折平均映射的好拟性、极限阴影特性和 Ulam-Hyers 稳定性。我们提供了确保每个图示弱增益收缩的周期点性质的必要条件。我们列举了一些例子来说明我们的结果是对现有文献中类似结果的潜在概括。
{"title":"Approximating fixed points of weak enriched contractions using Kirk’s iteration scheme of higher order","authors":"Mi Zhou, Naeem Saleem, Mujahid Abbas","doi":"10.1186/s13660-024-03097-2","DOIUrl":"https://doi.org/10.1186/s13660-024-03097-2","url":null,"abstract":"In this paper, we introduce two types of weak enriched contractions, namely weak enriched $mathcal{F}$ -contraction, weak enriched $mathcal{F^{prime}}$ -contraction, and k-fold averaged mapping based on Kirk’s iterative algorithm of order k. The types of contractions introduced herein unify, extend, and generalize several existing classes of enriched and weak enriched contraction mappings. Moreover, K-fold averaged mappings can be viewed as a generalization of the averaged mappings and double averaged mappings. We then prove the existence of a unique fixed point of the k-fold averaged mapping associated with weak enriched contractions introduced herein. We study necessary conditions that guarantee the equality of the sets of fixed points of the k-fold averaged mapping and weak enriched contractions. We show that an appropriate Kirk’s iterative algorithm can be used to approximate a fixed point of a k-fold averaged mapping and of the two weak enriched contractions. We also study the well-posedness, limit shadowing property, and Ulam–Hyers stability of the k-fold averaged mapping. We provide necessary conditions that ensure the periodic point property of each illustrated weak enriched contraction. Some examples are presented to show that our results are a potential generalization of the comparable results in the existing literature.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772243","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 : 2024-02-13DOI: 10.1186/s13660-024-03099-0
P. V. Ndlovu, L. O. Jolaoso, M. Aphane, H. A. Abass
In this article, we propose a viscosity extragradient algorithm together with an inertial extrapolation method for approximating the solution of pseudomonotone equilibrium and fixed point problem of a nonexpansive mapping in the setting of a Hadamard manifold. We prove that the sequence generated by our iterative method converges to a solution of the above problems under some mild conditions. Finally, we outline some implications of our results and present several numerical examples showing the implementability of our algorithm. The results of this article extend and complement many related results in linear spaces.
{"title":"Viscosity extragradient with modified inertial method for solving equilibrium problems and fixed point problem in Hadamard manifold","authors":"P. V. Ndlovu, L. O. Jolaoso, M. Aphane, H. A. Abass","doi":"10.1186/s13660-024-03099-0","DOIUrl":"https://doi.org/10.1186/s13660-024-03099-0","url":null,"abstract":"In this article, we propose a viscosity extragradient algorithm together with an inertial extrapolation method for approximating the solution of pseudomonotone equilibrium and fixed point problem of a nonexpansive mapping in the setting of a Hadamard manifold. We prove that the sequence generated by our iterative method converges to a solution of the above problems under some mild conditions. Finally, we outline some implications of our results and present several numerical examples showing the implementability of our algorithm. The results of this article extend and complement many related results in linear spaces.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772378","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}
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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