Pub Date : 2022-03-10DOI: 10.24193/subbmath.2022.1.04
Madalina Moga
"It is well known that of all the extensions of the Banach-Caccioppoli Contraction Principle, the most general result was established by '{C}iri'{c} in 1974. In this paper, we will present some results related to '{C}iri'{c} type operator in complete metric spaces. Existence and uniqueness are re-called and several stability properties (data dependence and Ostrowski stability property) are proved. Using the retraction-displacement condition, we will establish the well-posedness and the Ulam-Hyers stability property of the fixed point equation $x=f(x)$."
{"title":"On some qualitative properties of Ciric's fixed point theorem","authors":"Madalina Moga","doi":"10.24193/subbmath.2022.1.04","DOIUrl":"https://doi.org/10.24193/subbmath.2022.1.04","url":null,"abstract":"\"It is well known that of all the extensions of the Banach-Caccioppoli Contraction Principle, the most general result was established by '{C}iri'{c} in 1974. In this paper, we will present some results related to '{C}iri'{c} type operator in complete metric spaces. Existence and uniqueness are re-called and several stability properties (data dependence and Ostrowski stability property) are proved. Using the retraction-displacement condition, we will establish the well-posedness and the Ulam-Hyers stability property of the fixed point equation $x=f(x)$.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89874432","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 : 2022-03-10DOI: 10.24193/subbmath.2022.1.12
Zoubai Fayrouz, Merouani Boubakeur
"The paper deals with a nonlinear elasticity system with nonconstant coe cients. The existence and uniqueness of the solution of Neumann's problem is proved using Galerkin techniques and monotone operator theory, in Sobolev spaces with variable exponents."
{"title":"On a pure traction problem for the nonlinear elasticity system in Sobolev spaces with variable exponents","authors":"Zoubai Fayrouz, Merouani Boubakeur","doi":"10.24193/subbmath.2022.1.12","DOIUrl":"https://doi.org/10.24193/subbmath.2022.1.12","url":null,"abstract":"\"The paper deals with a nonlinear elasticity system with nonconstant coe cients. The existence and uniqueness of the solution of Neumann's problem is proved using Galerkin techniques and monotone operator theory, in Sobolev spaces with variable exponents.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83974273","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 : 2022-03-10DOI: 10.24193/subbmath.2022.1.05
C. Pintea
"In this work we first consider a certain monotonicity relative to some given one-to-one operator and prove the counterparts, adjusted to this new con- text, of most results obtained before in the joint work with G. Kassay [10]. For two operators with the same status relative to injectivity, such as two local in- jective operators, we de ne what we call mutual h-monotonicity and prove that every two mutual h-monotone local di eomorphisms can be obtained from each other via a composition with a h-monotone diffeomorphism."
{"title":"Relative and mutual monotonicity","authors":"C. Pintea","doi":"10.24193/subbmath.2022.1.05","DOIUrl":"https://doi.org/10.24193/subbmath.2022.1.05","url":null,"abstract":"\"In this work we first consider a certain monotonicity relative to some given one-to-one operator and prove the counterparts, adjusted to this new con- text, of most results obtained before in the joint work with G. Kassay [10]. For two operators with the same status relative to injectivity, such as two local in- jective operators, we de ne what we call mutual h-monotonicity and prove that every two mutual h-monotone local di eomorphisms can be obtained from each other via a composition with a h-monotone diffeomorphism.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76361422","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 : 2022-03-10DOI: 10.24193/subbmath.2022.1.03
M. Bianchi, N. Hadjisavvas, R. Pini
In this paper we study some properties of the adjusted normal cone operator of quasiconvex functions. In particular, we introduce a new notion of maximal quasimotonicity for set-valued maps different from similar ones recently appeared in the literature, and we show that it is enjoyed by this operator. Moreover, we prove the $stimes w^*$ cone upper semicontinuity of the normal cone operator in the domain of $f$ in case the set of global minima has non empty interior.
{"title":"Continuity and maximal quasimonotonicity of normal cone operators","authors":"M. Bianchi, N. Hadjisavvas, R. Pini","doi":"10.24193/subbmath.2022.1.03","DOIUrl":"https://doi.org/10.24193/subbmath.2022.1.03","url":null,"abstract":"In this paper we study some properties of the adjusted normal cone operator of quasiconvex functions. In particular, we introduce a new notion of maximal quasimotonicity for set-valued maps different from similar ones recently appeared in the literature, and we show that it is enjoyed by this operator. Moreover, we prove the $stimes w^*$ cone upper semicontinuity of the normal cone operator in the domain of $f$ in case the set of global minima has non empty interior.","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76192643","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 : 2022-03-10DOI: 10.24193/subbmath.2022.1.14
A. Filip
"The aim of this paper is to study the existence and uniqueness of solutions for some Fredholm integral equation systems by applying the vectorial form of Maia's fixed point theorem. Some abstract Gronwall lemmas and an abstract comparison lemma are also obtained."
{"title":"Some applications of Maia's fixed point theorem for Fredholm integral equation systems","authors":"A. Filip","doi":"10.24193/subbmath.2022.1.14","DOIUrl":"https://doi.org/10.24193/subbmath.2022.1.14","url":null,"abstract":"\"The aim of this paper is to study the existence and uniqueness of solutions for some Fredholm integral equation systems by applying the vectorial form of Maia's fixed point theorem. Some abstract Gronwall lemmas and an abstract comparison lemma are also obtained.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85905139","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 : 2022-03-10DOI: 10.24193/subbmath.2022.1.08
J. B. G. Frenk, J. Gromicho, Shuzhong Zhang
"Since quasiconvex functions have convex lower level sets it is possible to minimize them by means of separating hyperplanes. An example of such a procedure, well-known for convex functions, is the subgradient method. However, to nd the normal vector of a separating hyperplane is in general not easy for the quasiconvex case. This paper attempts to gain some insight into the computational aspects of determining such a normal vector and the geometry of lower level sets of quasiconvex functions. In order to do so, the directional di erentiability of quasiconvex functions is thoroughly studied. As a consequence of that study, it is shown that an important subset of quasiconvex functions belongs to the class of quasidifferentiable functions. The main emphasis is, however, on computing actual separators. Some important examples are worked out for illustration."
{"title":"Quasiconvex functions: how to separate, if you must!","authors":"J. B. G. Frenk, J. Gromicho, Shuzhong Zhang","doi":"10.24193/subbmath.2022.1.08","DOIUrl":"https://doi.org/10.24193/subbmath.2022.1.08","url":null,"abstract":"\"Since quasiconvex functions have convex lower level sets it is possible to minimize them by means of separating hyperplanes. An example of such a procedure, well-known for convex functions, is the subgradient method. However, to nd the normal vector of a separating hyperplane is in general not easy for the quasiconvex case. This paper attempts to gain some insight into the computational aspects of determining such a normal vector and the geometry of lower level sets of quasiconvex functions. In order to do so, the directional di erentiability of quasiconvex functions is thoroughly studied. As a consequence of that study, it is shown that an important subset of quasiconvex functions belongs to the class of quasidifferentiable functions. The main emphasis is, however, on computing actual separators. Some important examples are worked out for illustration.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83076225","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 : 2022-03-10DOI: 10.24193/subbmath.2022.1.07
Mihaela Miholca
"In this paper we extend a concept of well-posedness for vector equilibrium problems to the more general framework of set-valued equilibrium problems in topological vector spaces using an appropriate reformulation of the concept of minimality for sets. Su cient conditions for well-posedness are given in the generalized convex settings and we are able to single out classes of well-posed set-valued equilibrium problems. On the other hand, in order to relax some conditions, we introduce a concept of minimizing sequences for a set-valued problem, in the set criterion sense, and further we will have a concept of well-posedness for the set-valued equilibrium problem we are interested in. Suficient results are also given for this well-posedness concept."
{"title":"Well-posedness for set-valued equilibrium problems","authors":"Mihaela Miholca","doi":"10.24193/subbmath.2022.1.07","DOIUrl":"https://doi.org/10.24193/subbmath.2022.1.07","url":null,"abstract":"\"In this paper we extend a concept of well-posedness for vector equilibrium problems to the more general framework of set-valued equilibrium problems in topological vector spaces using an appropriate reformulation of the concept of minimality for sets. Su cient conditions for well-posedness are given in the generalized convex settings and we are able to single out classes of well-posed set-valued equilibrium problems. On the other hand, in order to relax some conditions, we introduce a concept of minimizing sequences for a set-valued problem, in the set criterion sense, and further we will have a concept of well-posedness for the set-valued equilibrium problem we are interested in. Suficient results are also given for this well-posedness concept.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82515932","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 : 2022-03-10DOI: 10.24193/subbmath.2022.1.01
Kay Barshad, S. Reich, A. Zaslavski
"The notion of porosity is well known in Optimization and Nonlinear Analysis. Its importance is brought out by the fact that the complement of a -porous subset of a complete pseudo-metric space is a residual set, while the existence of the latter is essential in many problems which apply the generic approach. Thus, under certain circumstances, some re nements of known results can be achieved by looking for porous sets. In 2001 Gabour, Reich and Zaslavski developed certain generic methods for solving stochastic feasibility problems. This topic was further investigated in 2021 by Barshad, Reich and Zaslavski, who provided more general results in the case of unbounded sets. In the present paper we introduce and examine new generic methods that deal with the aforesaid problems, in which, in contrast with previous studies, we consider sigma-porous sets instead of meager ones."
{"title":"Porosity-based methods for solving stochastic feasibility problems","authors":"Kay Barshad, S. Reich, A. Zaslavski","doi":"10.24193/subbmath.2022.1.01","DOIUrl":"https://doi.org/10.24193/subbmath.2022.1.01","url":null,"abstract":"\"The notion of porosity is well known in Optimization and Nonlinear Analysis. Its importance is brought out by the fact that the complement of a -porous subset of a complete pseudo-metric space is a residual set, while the existence of the latter is essential in many problems which apply the generic approach. Thus, under certain circumstances, some re nements of known results can be achieved by looking for porous sets. In 2001 Gabour, Reich and Zaslavski developed certain generic methods for solving stochastic feasibility problems. This topic was further investigated in 2021 by Barshad, Reich and Zaslavski, who provided more general results in the case of unbounded sets. In the present paper we introduce and examine new generic methods that deal with the aforesaid problems, in which, in contrast with previous studies, we consider sigma-porous sets instead of meager ones.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82444652","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 : 2022-03-10DOI: 10.24193/subbmath.2022.1.09
T. N. Hai, N. T. Thuong
"In this paper, we discuss a new splitting algorithm for solving equilibrium problems arising from Nash-Cournot oligopolistic equilibrium problems in electricity markets with non-convex cost functions. Under the strong pseudomonotonicity of the original bifunction and suitable conditions of the component bifunctions, we prove the strong convergence of the proposed algorithm. Our results improve and develop previously discussed extragradient-like splitting algorithms and general extragradient algorithms. We also present some numerical experiments and compare our algorithm with the existing ones."
{"title":"A new splitting algorithm for equilibrium problems and applications","authors":"T. N. Hai, N. T. Thuong","doi":"10.24193/subbmath.2022.1.09","DOIUrl":"https://doi.org/10.24193/subbmath.2022.1.09","url":null,"abstract":"\"In this paper, we discuss a new splitting algorithm for solving equilibrium problems arising from Nash-Cournot oligopolistic equilibrium problems in electricity markets with non-convex cost functions. Under the strong pseudomonotonicity of the original bifunction and suitable conditions of the component bifunctions, we prove the strong convergence of the proposed algorithm. Our results improve and develop previously discussed extragradient-like splitting algorithms and general extragradient algorithms. We also present some numerical experiments and compare our algorithm with the existing ones.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91159385","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 : 2022-02-06DOI: 10.24193/subbmath.2022.2.09
M. Elin, Fiana Jacobzon
"We study the FeketeSzego problem on the open unit ball of a complex Banach space. Namely, the FeketeSzego inequalities are proved for the class of spirallike mappings relative to an arbitrary strongly accretive operator, and some of its subclasses. Next, we consider families of non-linear resolvents for holomorphically accretive mappings vanishing at the origin. We solve the Fekete- Szego problem over these families."
{"title":"The Fekete-Szego problem for spirallike mappings and non-linear resolvents in Banach spaces","authors":"M. Elin, Fiana Jacobzon","doi":"10.24193/subbmath.2022.2.09","DOIUrl":"https://doi.org/10.24193/subbmath.2022.2.09","url":null,"abstract":"\"We study the FeketeSzego problem on the open unit ball of a complex Banach space. Namely, the FeketeSzego inequalities are proved for the class of spirallike mappings relative to an arbitrary strongly accretive operator, and some of its subclasses. Next, we consider families of non-linear resolvents for holomorphically accretive mappings vanishing at the origin. We solve the Fekete- Szego problem over these families.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75613866","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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