Pub Date : 2023-11-20DOI: 10.55630/serdica.2023.49.77-96
Pando Georgiev
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.
我们考虑了巴拿赫空间（E）中的参数化变分不等式（(A,Y)），该不等式由一个取决于参数的单调算子(A)定义在E的封闭、凸和有界子集(Y)上。我们证明，在合适的条件下，存在一个任意小的(A) 单调扰动，使得受扰动的变分不等式有一个解，这个解是参数的连续函数，并且接近给定的近似解。在非参数情况下，这可以看作是变分不等式的变分原理，即 Borwein-Preiss 平滑变分原理。 文中给出了一些应用：由部分单调算子定义的纳什均衡问题的类比，以及在宽松假设条件下，参数博尔文-普赖斯可变凸函数的变分原理的变种。 证明主要结果的工具是一个关于取决于参数的变分不等式的连续（varepsilon）解的存在性的有用lemma。它还具有独立的意义，可以直接证明 Ky Fan 单调算子不等式的类比，在此引入的 Ky Fan 不等式导致了在 Gâteaux smooth Banach 空间中 Schauder 定点定理的新证明。
{"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}
Pub Date : 2023-11-20DOI: 10.55630/serdica.2023.49.205-230
Tsvetomir Tsachev, I. Vassileva
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.
{"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}
Pub Date : 2023-11-20DOI: 10.55630/serdica.2023.49.1-8
R. T. Rockafellar
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.
{"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}
Pub Date : 2023-11-20DOI: 10.55630/serdica.2023.49.127-154
Martin Gugat, Jan Sokolowski
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.
{"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}
Pub Date : 2023-11-20DOI: 10.55630/serdica.2023.49.107-126
Stoyan Apostolov, Zhivko Petrov
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.
{"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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