Binayak S Choudhury, Pranati Maity, Nikhilesh Metiya, Mihai Postolache
{"title":"Approximating distance between sets by multivalued coupling with application to uniformly convex Banach spaces.","authors":"Binayak S Choudhury, Pranati Maity, Nikhilesh Metiya, Mihai Postolache","doi":"10.1186/s13660-018-1720-0","DOIUrl":"https://doi.org/10.1186/s13660-018-1720-0","url":null,"abstract":"<p><p>In this paper, our aim is to ascertain the distance between two sets iteratively in two simultaneous ways with the help of a multivalued coupling define for this purpose. We define the best proximity points of such couplings that realize the distance between two sets. Our main theorem is deduced in metric spaces. As an application, we obtain the corresponding results in uniformly convex Banach spaces using the geometry of the space. We discuss two examples.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"130"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1720-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36422927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a class of N-dimensional anisotropic Sobolev inequalities.","authors":"Lirong Huang, Eugenio Rocha","doi":"10.1186/s13660-018-1754-3","DOIUrl":"https://doi.org/10.1186/s13660-018-1754-3","url":null,"abstract":"<p><p>In this paper, we study the smallest constant <i>α</i> in the anisotropic Sobolev inequality of the form <dispformula><math><msubsup><mrow><mo>∥</mo><mi>u</mi><mo>∥</mo></mrow><mi>p</mi><mi>p</mi></msubsup><mo>≤</mo><mi>α</mi><msubsup><mrow><mo>∥</mo><mi>u</mi><mo>∥</mo></mrow><mn>2</mn><mfrac><mrow><mn>2</mn><mo>(</mo><mn>2</mn><mi>N</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>+</mo><mo>(</mo><mn>3</mn><mo>-</mo><mn>2</mn><mi>N</mi><mo>)</mo><mi>p</mi></mrow><mn>2</mn></mfrac></msubsup><msubsup><mrow><mo>∥</mo><msub><mi>u</mi><mi>x</mi></msub><mo>∥</mo></mrow><mn>2</mn><mfrac><mrow><mi>N</mi><mo>(</mo><mi>p</mi><mo>-</mo><mn>2</mn><mo>)</mo></mrow><mn>2</mn></mfrac></msubsup><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><msubsup><mrow><mo>∥</mo><msubsup><mi>D</mi><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><msub><mi>∂</mi><msub><mi>y</mi><mi>k</mi></msub></msub><mi>u</mi><mo>∥</mo></mrow><mn>2</mn><mfrac><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow><mn>2</mn></mfrac></msubsup></math></dispformula> and the smallest constant <i>β</i> in the inequality <dispformula><math><msubsup><mrow><mo>∥</mo><mi>u</mi><mo>∥</mo></mrow><msub><mi>p</mi><mo>∗</mo></msub><msub><mi>p</mi><mo>∗</mo></msub></msubsup><mo>≤</mo><mi>β</mi><msubsup><mrow><mo>∥</mo><msub><mi>u</mi><mi>x</mi></msub><mo>∥</mo></mrow><mn>2</mn><mfrac><mrow><mn>2</mn><mi>N</mi></mrow><mrow><mn>2</mn><mi>N</mi><mo>-</mo><mn>3</mn></mrow></mfrac></msubsup><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><msubsup><mrow><mo>∥</mo><msubsup><mi>D</mi><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><msub><mi>∂</mi><msub><mi>y</mi><mi>k</mi></msub></msub><mi>u</mi><mo>∥</mo></mrow><mn>2</mn><mfrac><mn>2</mn><mrow><mn>2</mn><mi>N</mi><mo>-</mo><mn>3</mn></mrow></mfrac></msubsup><mo>,</mo></math></dispformula> where <math><mi>V</mi><mo>:</mo><mo>=</mo><mo>(</mo><mi>x</mi><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>y</mi><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>∈</mo><msup><mi>R</mi><mi>N</mi></msup></math> with <math><mi>N</mi><mo>≥</mo><mn>3</mn></math> and <math><mn>2</mn><mo><</mo><mi>p</mi><mo><</mo><msub><mi>p</mi><mo>∗</mo></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mo>(</mo><mn>2</mn><mi>N</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn><mi>N</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> . These constants are characterized by variational methods and scaling techniques. The techniques used here seem to have independent interests.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"163"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1754-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36422939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Windschitl type approximation formulas for the gamma function.","authors":"Zhen-Hang Yang, Jing-Feng Tian","doi":"10.1186/s13660-018-1870-0","DOIUrl":"10.1186/s13660-018-1870-0","url":null,"abstract":"<p><p>In this paper, we present four new Windschitl type approximation formulas for the gamma function. By some unique ideas and techniques, we prove that four functions combined with the gamma function and Windschitl type approximation formulas have good properties, such as monotonicity and convexity. These not only yield some new inequalities for the gamma and factorial functions, but also provide a new proof of known inequalities and strengthen known results.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"272"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6182422/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36664765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sobolev type inequalities for compact metric graphs.","authors":"Muhammad Usman","doi":"10.1186/s13660-018-1872-y","DOIUrl":"https://doi.org/10.1186/s13660-018-1872-y","url":null,"abstract":"<p><p>In this paper analogues of Sobolev inequalities for compact and connected metric graphs are derived. As a consequence of these inequalities, a lower bound, commonly known as Cheeger inequality, on the first non-zero eigenvalue of the Laplace operator with standard vertex conditions is recovered.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"271"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1872-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36664774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Well-posedness for a class of generalized variational-hemivariational inequalities involving set-valued operators.","authors":"Caijing Jiang","doi":"10.1186/s13660-018-1776-x","DOIUrl":"https://doi.org/10.1186/s13660-018-1776-x","url":null,"abstract":"<p><p>The aim of present work is to study some kinds of well-posedness for a class of generalized variational-hemivariational inequality problems involving set-valued operators. Some systematic approaches are presented to establish some equivalence theorems between several classes of well-posedness for the inequality problems and some corresponding metric characterizations, which generalize many known results. Finally, the well-posedness for a class of generalized mixed equilibrium problems is also considered.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"187"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1776-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36419176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spectral properties of an impulsive Sturm-Liouville operator.","authors":"Elgiz Bairamov, Ibrahim Erdal, Seyhmus Yardimci","doi":"10.1186/s13660-018-1781-0","DOIUrl":"https://doi.org/10.1186/s13660-018-1781-0","url":null,"abstract":"<p><p>This work is devoted to discuss some spectral properties and the scattering function of the impulsive operator generated by the Sturm-Liouville equation. We present a different method to investigate the spectral singularities and eigenvalues of the mentioned operator. We also obtain the finiteness of eigenvalues and spectral singularities with finite multiplicities under some certain conditions. Finally, we illustrate our results by a detailed example.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"191"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1781-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36419180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Smoothing approximation to the lower order exact penalty function for inequality constrained optimization.","authors":"Shujun Lian, Nana Niu","doi":"10.1186/s13660-018-1723-x","DOIUrl":"https://doi.org/10.1186/s13660-018-1723-x","url":null,"abstract":"<p><p>For inequality constrained optimization problem, we first propose a new smoothing method to the lower order exact penalty function, and then show that an approximate global solution of the original problem can be obtained by solving a global solution of a smooth lower order exact penalty problem. We propose an algorithm based on the smoothed lower order exact penalty function. The global convergence of the algorithm is proved under some mild conditions. Some numerical experiments show the efficiency of the proposed method.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"131"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1723-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36422929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Convergence analysis on a modified generalized alternating direction method of multipliers.","authors":"Sha Lu, Zengxin Wei","doi":"10.1186/s13660-018-1721-z","DOIUrl":"10.1186/s13660-018-1721-z","url":null,"abstract":"<p><p>The alternating direction method of multipliers (ADMM) is one of the most powerful and successful methods for solving convex composite minimization problem. The generalized ADMM relaxes both the variables and the multipliers with a common relaxation factor in <math><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math> , which has the potential of enhancing the performance of the classic ADMM. Very recently, two different variants of semi-proximal generalized ADMM have been proposed. They allow the weighting matrix in the proximal terms to be positive semidefinite, which makes the subproblems relatively easy to evaluate. One of the variants of semi-proximal generalized ADMMs has been analyzed theoretically, but the convergence result of the other is not known so far. This paper aims to remedy this deficiency and establish its convergence result under some mild conditions in the sense that the relaxation factor is also restricted into <math><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math> .</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"129"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5993865/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36424090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An extended reverse Hardy-Hilbert's inequality in the whole plane.","authors":"Qiang Chen, Bicheng Yang","doi":"10.1186/s13660-018-1706-y","DOIUrl":"https://doi.org/10.1186/s13660-018-1706-y","url":null,"abstract":"<p><p>Using weight coefficients, a complex integral formula, and Hermite-Hadamard's inequality, we give an extended reverse Hardy-Hilbert's inequality in the whole plane with multiparameters and a best possible constant factor. Equivalent forms and a few particular cases are considered.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"115"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1186/s13660-018-1706-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36114853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New bounds for the exponential function with cotangent.","authors":"Ling Zhu","doi":"10.1186/s13660-018-1697-8","DOIUrl":"10.1186/s13660-018-1697-8","url":null,"abstract":"<p><p>In this paper, new bounds for the exponential function with cotangent are found by using the recurrence relation between coefficients in the expansion of power series of the function [Formula: see text] and a new criterion for the monotonicity of the quotient of two power series.</p>","PeriodicalId":49163,"journal":{"name":"Journal of Inequalities and Applications","volume":"2018 1","pages":"106"},"PeriodicalIF":1.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5940775/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36105458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}