{"title":"Variational principles for monotone variational inequalities: The single-valued case","authors":"Pando Georgiev","doi":"10.55630/serdica.2023.49.77-96","DOIUrl":"https://doi.org/10.55630/serdica.2023.49.77-96","url":null,"abstract":"We consider a parameterized variational inequality ((A,Y)) in a Banach space (E) defined on a closed, convex and bounded subset (Y) of (E) by a monotone operator (A) depending on a parameter. We prove that under suitable conditions, there exists an arbitrarily small monotone perturbation of (A) such that the perturbed variational inequality has a solution which is a continuous function of the parameter, and is near to a given approximate solution. In the nonparametric case this can be considered as a variational principle for variational inequalities, an analogue of the Borwein-Preiss smooth variational principle. Some applications are given: an analogue of the Nash equilibrium problem, defined by a partially monotone operator, and a variant of the parametric Borwein-Preiss variational principle for Gâteaux differentiable convex functions under relaxed assumtions. The tool for proving the main result is a useful lemma about existence of continuous (varepsilon)-solutions of a variational inequality depending on a parameter. It has an independent interest and allows a direct proof of an analogue of Ky Fan's inequality for monotone operators, introduced here, which leads to a new proof of the Schauder fixed point theorem in Gâteaux smooth Banach spaces.","PeriodicalId":509503,"journal":{"name":"Serdica Mathematical Journal","volume":"854 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139256089","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}
{"title":"Fiscal policy options for reducing informality in an economy with fiscal rules","authors":"Tsvetomir Tsachev, I. Vassileva","doi":"10.55630/serdica.2023.49.205-230","DOIUrl":"https://doi.org/10.55630/serdica.2023.49.205-230","url":null,"abstract":"We use a dynamic general equilibrium model to examine the effect of various fiscal policies in an economy with a large informal sector. More specifically, we simulate (i) a reduction of the tax rate on personal income and social security contributions and (ii) an increase in the efforts of the tax administration to uncover shadow activities in a model, calibrated to the Bulgarian economy. While both policies encourage total economic activity and discourage shadow practices, they worsen the inequalities in the economy. Due to the application of fiscal rules in the model, the tax cut leads to a permanent increase in public debt and lower public investment, while the strengthened control over tax collection has a negative impact on employment, but, nonetheless, does not deepen inequalities as much as the tax cut.","PeriodicalId":509503,"journal":{"name":"Serdica Mathematical Journal","volume":"4 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139257417","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}
{"title":"Metric regularity properties of monotone mappings","authors":"R. T. Rockafellar","doi":"10.55630/serdica.2023.49.1-8","DOIUrl":"https://doi.org/10.55630/serdica.2023.49.1-8","url":null,"abstract":"The theory of metric regularity deals with properties of set-valued mappings that provide estimates useful in solving inverse problems and generalized equations. Maximal monotone mappings, which dominate applications related to convex optimization, have valuable special features in this respect that have not previously been recorded. Here it is shown that the property of strong metric subregularity is generic in an almost everywhere sense. Metric regularity not only coincides with strong metric regularity but also implies local single-valuedness of the inverse, rather than just of a graphical localization of the inverse. Consequences are given for the solution mapping associated with a monotone generalized equation.","PeriodicalId":509503,"journal":{"name":"Serdica Mathematical Journal","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139255417","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}
{"title":"An aspect of the turnpike property. Long time horizon behavior","authors":"Martin Gugat, Jan Sokolowski","doi":"10.55630/serdica.2023.49.127-154","DOIUrl":"https://doi.org/10.55630/serdica.2023.49.127-154","url":null,"abstract":"The turnpike phenomenon concerns the structure of the optimal control and the optimal state of dynamic optimal control problems for long time horizons. The focus is regularly placed on the study of the interior of the time interval. Classical turnpike results state how the solution of the dynamic optimal control problems approaches the solution of the corresponding static optimal control problem in the interior of the time interval. In this paper we look at a new aspect of the turnpike phenomenon. We show that for problems without explicit terminal condition, for large time horizons in the last part of the time interval the optimal state approaches a certain limit trajectory that is independent of the terminal time exponentially fast. For large time horizons also the optimal state in the initial part of the time interval approaches exponentially fast a limit state.","PeriodicalId":509503,"journal":{"name":"Serdica Mathematical Journal","volume":"82 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139259287","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}
{"title":"On sequences which are not uniformly converging on any open subset","authors":"Stoyan Apostolov, Zhivko Petrov","doi":"10.55630/serdica.2023.49.107-126","DOIUrl":"https://doi.org/10.55630/serdica.2023.49.107-126","url":null,"abstract":"We consider the property of nonuniform convergence to 0 of a sequence of functions on any open subset of a metric space. We consider three examples with respect to three different characteristics. Next we show that the three characteristics cannot be present simultaneously. For this purpose we introduce the so-called height function, which we use to quantify how far is a sequence of functions from satisfying any of the third characteristic. Moreover, we study properties of the height function and its relation to uniform convergence. Finally, we show that this quantification is precise.","PeriodicalId":509503,"journal":{"name":"Serdica Mathematical Journal","volume":"14 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139255058","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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