In this paper we give some fixed point theorems for nonself generalized contractions on a large Kasahara space, which generalize some results given by I.A. Rus and M.-A. c Serban (I. A. Rus, M.-A. c Serban, {it Some fixed point theorems for nonself generalized contractions}, Miskolc Math. Notes, {bf 17}(2016), no.2, 1021-1031) and by S. Reich and A.J. Zaslavski (S. Reich, A. J. Zaslavski, {it A note on Rakotch contractions}, Fixed Point Theory, {bf 9} (2008), no. 1, 267-273) in complete metric spaces. We prove our results without using the completeness of the metric structure.
{"title":"2022/07/26 \"Fixed point theorems for nonself generalized contractions on a large Kasahara space\"","authors":"A. Filip","doi":"10.37193/cjm.2022.03.23","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.23","url":null,"abstract":"In this paper we give some fixed point theorems for nonself generalized contractions on a large Kasahara space, which generalize some results given by I.A. Rus and M.-A. c Serban (I. A. Rus, M.-A. c Serban, {it Some fixed point theorems for nonself generalized contractions}, Miskolc Math. Notes, {bf 17}(2016), no.2, 1021-1031) and by S. Reich and A.J. Zaslavski (S. Reich, A. J. Zaslavski, {it A note on Rakotch contractions}, Fixed Point Theory, {bf 9} (2008), no. 1, 267-273) in complete metric spaces. We prove our results without using the completeness of the metric structure.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42022795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"Consider in a bounded domain $Omega subset mathbb{R}^N$, $Nge 2$, with smooth boundary $partial Omega$ the following nonlinear eigenvalue problem begin{equation*} left{begin{array}{l} -sum_{alpha in { p,q,r}}rho_{alpha}Delta_{alpha}u=lambda a(x) mid umid ^{r-2}u mbox{ in} ~ Omega,[1mm] big(sum_{alpha in {p,q,r}}rho_{alpha}mid nabla umid ^{alpha-2}big)frac{partial u}{partialnu}=lambda b(x) mid umid ^{r-2}u ~ mbox{ on} ~ partial Omega, end{array}right. end{equation*} where $p, q, rin (1, +infty),~q0.$ Such a triple-phase problem is motivated by some models arising in mathematical physics. If $r notin (q, p),$ we determine a positive number $lambda_r$ such that the set of eigenvalues of the above problem is precisely ${ 0} cup (lambda_r, +infty )$. On the other hand, in the complementary case $r in (q, p)$ with $r < q(N-1)/(N-q)$ if $q
{"title":"\"Eigenvalues of the (p, q, r)-Laplacian with a parametric boundary condition\"","authors":"L. Barbu, G. Moroşanu","doi":"10.37193/cjm.2022.03.03","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.03","url":null,"abstract":"\"Consider in a bounded domain $Omega subset mathbb{R}^N$, $Nge 2$, with smooth boundary $partial Omega$ the following nonlinear eigenvalue problem begin{equation*} left{begin{array}{l} -sum_{alpha in { p,q,r}}rho_{alpha}Delta_{alpha}u=lambda a(x) mid umid ^{r-2}u mbox{ in} ~ Omega,[1mm] big(sum_{alpha in {p,q,r}}rho_{alpha}mid nabla umid ^{alpha-2}big)frac{partial u}{partialnu}=lambda b(x) mid umid ^{r-2}u ~ mbox{ on} ~ partial Omega, end{array}right. end{equation*} where $p, q, rin (1, +infty),~q<p,~rnotin {p, q};$ $rho_p, rho_q, rho_r$ are positive constants; $Delta_{alpha}$ is the usual $alpha$-Laplacian, i.e., $Delta_alpha u=, mbox{div} , (|nabla u|^{alpha-2}nabla u)$; $nu$ is the unit outward normal to $partial Omega$; $ain L^{infty}(Omega),$ $bin L^{infty}(partialOmega)$ {are given nonnegative functions satisfying} $int_Omega a~dx+int_{partialOmega} b~dsigma >0.$ Such a triple-phase problem is motivated by some models arising in mathematical physics. If $r notin (q, p),$ we determine a positive number $lambda_r$ such that the set of eigenvalues of the above problem is precisely ${ 0} cup (lambda_r, +infty )$. On the other hand, in the complementary case $r in (q, p)$ with $r < q(N-1)/(N-q)$ if $q<N$, we prove that there exist two positive constants $lambda_*<lambda^*$ such that any $lambdain {0}cup [lambda^*, infty)$ is an eigenvalue of the above problem, while the set $(-infty, 0)cup (0, lambda_*)$ contains no eigenvalue $lambda$ of the problem.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49667513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On an arbitrary Banach space, assuming that a linear nonautonomous difference equation linebreak $x_{m+1} = A_m x_m$ admits a very general type of trichotomy, we establish conditions for the existence of global Lipschitz invariant center manifolds of the perturbed equation $x_{m+1} = A_m x_m + f_m(x_m)$. Our results not only improve results already existing in the literature, but also include new cases.
{"title":"Invariant manifolds for difference equations with generalized trichotomies","authors":"A. J. G. Bento","doi":"10.37193/cjm.2022.03.25","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.25","url":null,"abstract":"On an arbitrary Banach space, assuming that a linear nonautonomous difference equation linebreak $x_{m+1} = A_m x_m$ admits a very general type of trichotomy, we establish conditions for the existence of global Lipschitz invariant center manifolds of the perturbed equation $x_{m+1} = A_m x_m + f_m(x_m)$. Our results not only improve results already existing in the literature, but also include new cases.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43398436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"For a class of semilinear control systems we get null controllability results and estimates for the minimum time function by considering appropriate feedback laws."
对于一类半线性控制系统，我们通过考虑适当的反馈律得到了零可控性结果和最小时间函数的估计。
{"title":"\"Feedback null controllability for a class of semilinear control systems\"","authors":"O. Cârją, A. Lazu","doi":"10.37193/cjm.2022.03.04","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.04","url":null,"abstract":"\"For a class of semilinear control systems we get null controllability results and estimates for the minimum time function by considering appropriate feedback laws.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44835905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"Two classes of fractional integro-differential inclusions with certain boundary conditions are studied. The existence of solutions is established in the case when the set-valued map has nonconvex values."
研究了两类具有一定边界条件的分数积分-微分包含。在集值映射具有非凸值的情况下，证明了解的存在性
{"title":"\"On certain boundary value problems associated to some fractional integro-differential inclusions\"","authors":"A. Cernea","doi":"10.37193/cjm.2022.03.11","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.11","url":null,"abstract":"\"Two classes of fractional integro-differential inclusions with certain boundary conditions are studied. The existence of solutions is established in the case when the set-valued map has nonconvex values.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45244234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"In a recent paper (I. A. Rus. Sets with structure, mappings and fixed point property: fixed point structures. Fixed Point Theory 23 (2022), No. 2) the author introduced, amongst others, the following notions: set-mapping pair and set-mapping pair bijection. In this paper we study the set-mapping pair bijections in connection with the isomorphisms structure, and their impact on the fixed point theory in a set-mapping pair. Some open problems are also formulated"
{"title":"\"A case study in set-mapping pair theory: the set-mapping pair bijections\"","authors":"A. Filip, I. Rus","doi":"10.37193/cjm.2022.03.19","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.19","url":null,"abstract":"\"In a recent paper (I. A. Rus. Sets with structure, mappings and fixed point property: fixed point structures. Fixed Point Theory 23 (2022), No. 2) the author introduced, amongst others, the following notions: set-mapping pair and set-mapping pair bijection. In this paper we study the set-mapping pair bijections in connection with the isomorphisms structure, and their impact on the fixed point theory in a set-mapping pair. Some open problems are also formulated\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48512248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"We modify the classical concept of an evolution semigroup associated to an evolution family on the half-line to fit to the general case when linear flows may not agree with the restricted hypothesis of uniform exponential growth. We study the connections between spectral properties of the corresponding generator and a wide class of behavior of the evolution family. As a consequence, we prove that the generalized exponential dichotomy of possible non-invertible evolution families persists under sufficiently small linear perturbations."
{"title":"\"Generalized exponential behavior on the half-line via evolution semigroups\"","authors":"Nicolae Lupa, L. Popescu","doi":"10.37193/cjm.2022.03.14","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.14","url":null,"abstract":"\"We modify the classical concept of an evolution semigroup associated to an evolution family on the half-line to fit to the general case when linear flows may not agree with the restricted hypothesis of uniform exponential growth. We study the connections between spectral properties of the corresponding generator and a wide class of behavior of the evolution family. As a consequence, we prove that the generalized exponential dichotomy of possible non-invertible evolution families persists under sufficiently small linear perturbations.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":"56 163","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41267760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"The aim of this paper is to provide new characterizations for polynomial dichotomies of discrete nonautonomous systems on the half-line. First, we establish equivalent structures for the ranges of projections for a polynomial dichotomy with respect to a sequence of norms. Next, we establish the connections between polynomial dichotomies and other dichotomic behaviors. We obtain for the first time a characterization of polynomial dichotomy with respect to a sequence of norms in terms of ordinary dichotomy and exponential dichotomy of suitable systems with respect to well-chosen sequences of norms. The results are obtained in the most general case, without any additional assumptions regarding the coefficients of the underlying systems."
{"title":"\"On Polynomial Dichotomies of Discrete Nonautonomous Systems on the Half-Line\"","authors":"D. Dragičević, A. L. Sasu, B. Sasu","doi":"10.37193/cjm.2022.03.12","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.12","url":null,"abstract":"\"The aim of this paper is to provide new characterizations for polynomial dichotomies of discrete nonautonomous systems on the half-line. First, we establish equivalent structures for the ranges of projections for a polynomial dichotomy with respect to a sequence of norms. Next, we establish the connections between polynomial dichotomies and other dichotomic behaviors. We obtain for the first time a characterization of polynomial dichotomy with respect to a sequence of norms in terms of ordinary dichotomy and exponential dichotomy of suitable systems with respect to well-chosen sequences of norms. The results are obtained in the most general case, without any additional assumptions regarding the coefficients of the underlying systems.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41929324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"This paper deals with the analysis of a discrete-time Cournot game with time delay, where the interactions of one public firm and n private firms on the market are considered. The production of the public firm is adjusted based on the past production levels of the private firms. At the same time, the productions of the private firms are updated with respect to the past production of the public firm. Two equilibrium points are determined for the discrete-time nonlinear mathematical model. The analysis of the stability reveals that the boundary equilibrium point is a saddle point, while the positive one, under some conditions, is asymptotically stable for any time delays. Numerical simulations illustrate the complex dynamic behaviour of the system."
{"title":"\"A dynamic Cournot mixed oligopoly model with time delay for competitors\"","authors":"Loredana Camelia Culda, E. Kaslik, M. Neamţu","doi":"10.37193/cjm.2022.03.13","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.13","url":null,"abstract":"\"This paper deals with the analysis of a discrete-time Cournot game with time delay, where the interactions of one public firm and n private firms on the market are considered. The production of the public firm is adjusted based on the past production levels of the private firms. At the same time, the productions of the private firms are updated with respect to the past production of the public firm. Two equilibrium points are determined for the discrete-time nonlinear mathematical model. The analysis of the stability reveals that the boundary equilibrium point is a saddle point, while the positive one, under some conditions, is asymptotically stable for any time delays. Numerical simulations illustrate the complex dynamic behaviour of the system.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49418855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"In this paper we provide a new approach in the study of a variational-hemivariational inequal- ity in Hilbert space, based on the theory of maximal monotone operators and the Banach fixed point theorem. First, we introduce the inequality problem we are interested in, list the assumptions on the data and show that it is governed by a multivalued maximal monotone operator. Then, we prove that solving the variational- hemivariational inequality is equivalent to finding a fixed point for the resolvent of this operator. Based on this equivalence result, we use the Banach contraction principle to prove the unique solvability of the problem. Moreover, we construct the corresponding Picard, Krasnoselski and Mann iterations and deduce their conver- gence to the unique solution of the variational-hemivariational inequality"
{"title":"\"A Fixed Point Approach of Variational-Hemivariational Inequalities\"","authors":"Rong Hu, M. Sofonea, Yi-bin Xiao","doi":"10.37193/cjm.2022.03.05","DOIUrl":"https://doi.org/10.37193/cjm.2022.03.05","url":null,"abstract":"\"In this paper we provide a new approach in the study of a variational-hemivariational inequal- ity in Hilbert space, based on the theory of maximal monotone operators and the Banach fixed point theorem. First, we introduce the inequality problem we are interested in, list the assumptions on the data and show that it is governed by a multivalued maximal monotone operator. Then, we prove that solving the variational- hemivariational inequality is equivalent to finding a fixed point for the resolvent of this operator. Based on this equivalence result, we use the Banach contraction principle to prove the unique solvability of the problem. Moreover, we construct the corresponding Picard, Krasnoselski and Mann iterations and deduce their conver- gence to the unique solution of the variational-hemivariational inequality\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48982908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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