By the method of monotone operators we establish theorems on global existence and uniqueness, as well as estimats and methods of finding the solutions for various classes of nonlinear convolution type integral equations in the real space of 2πperiodic functions Lp(−π, π).
{"title":"Periodic solutions of convolution type equations with monotone nonlinearity","authors":"S. Askhabov","doi":"10.13108/2016-8-1-20","DOIUrl":"https://doi.org/10.13108/2016-8-1-20","url":null,"abstract":"By the method of monotone operators we establish theorems on global existence and uniqueness, as well as estimats and methods of finding the solutions for various classes of nonlinear convolution type integral equations in the real space of 2πperiodic functions Lp(−π, π).","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73162029","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}
. In the work we find the minimal value that can be taken by the type of an entire function of order 𝜌 ∈ (0 , 1) with zeroes of prescribed upper and lower densities and located in an angle of a fixed opening less than 𝜋 . The main theorem generalizes the previous result by the author (the zeroes lie on one ray) and by A.Yu. Popov (only the upper density of zeros was taken into consideration). We distinguish and study in detail the case when the an entire function has a measurable sequence of zeroes. We provide applications of the obtained results to the uniqueness theorems for entire functions and to the completeness of exponential systems in the space of analytic in a circle functions with the standard topology of uniform convergence on compact sets.
{"title":"Minimal value for the type of an entire function of order $rhoin(0,,1)$, whose zeros lie in an angle and have a prescribed density","authors":"V. Sherstyukov","doi":"10.13108/2016-8-1-108","DOIUrl":"https://doi.org/10.13108/2016-8-1-108","url":null,"abstract":". In the work we find the minimal value that can be taken by the type of an entire function of order 𝜌 ∈ (0 , 1) with zeroes of prescribed upper and lower densities and located in an angle of a fixed opening less than 𝜋 . The main theorem generalizes the previous result by the author (the zeroes lie on one ray) and by A.Yu. Popov (only the upper density of zeros was taken into consideration). We distinguish and study in detail the case when the an entire function has a measurable sequence of zeroes. We provide applications of the obtained results to the uniqueness theorems for entire functions and to the completeness of exponential systems in the space of analytic in a circle functions with the standard topology of uniform convergence on compact sets.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72543994","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}
In the paper we establish some tests for absolute Cesáro summability of the Fourier series for almost periodic functions in the Besicovitch space. We consider the case, when the Fourier exponents have a limiting point at zero and as a structure characteristics of the studied function, we use a high order averaging modulus.
{"title":"On absolute Cesáro summablity of Fourier series for almost-periodic functions with limiting points at zero","authors":"Y. Khasanov","doi":"10.13108/2016-8-4-144","DOIUrl":"https://doi.org/10.13108/2016-8-4-144","url":null,"abstract":"In the paper we establish some tests for absolute Cesáro summability of the Fourier series for almost periodic functions in the Besicovitch space. We consider the case, when the Fourier exponents have a limiting point at zero and as a structure characteristics of the studied function, we use a high order averaging modulus.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72417440","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}
In the paper we consider an universal solution to the KdV equation. This solution also satisfies a fifth order ordinary differential equation. We pose the problem on studying the behavior of this solution as t → ∞. For large time, the asymptotic solution has different structure depending on the slow variable s = x2/t. We construct the asymptotic solution in the domains s < −3/4, −3/4 < s < 5/24 and in the vicinity of the point s = −3/4. It is shown that a slow modulation of solution’s parameters in the vicinity of the point s = −3/4 is described by a solution to Painlevé IV equation.
本文考虑了KdV方程的一个通解。这个解也满足一个五阶常微分方程。我们提出了研究该解在t→∞时的行为的问题。对于大时间，随着慢变量s = x2/t的变化，渐近解具有不同的结构。构造了在s <−3/4，−3/4 < s < 5/24和点s =−3/4附近的渐近解。结果表明，在点s =−3/4附近溶液参数的缓慢调制可以用painlevev方程的解来描述。
{"title":"On simultaneous solution of the KdV equation and a fifth-order differential equation","authors":"R. Garifullin","doi":"10.13108/2016-8-4-52","DOIUrl":"https://doi.org/10.13108/2016-8-4-52","url":null,"abstract":"In the paper we consider an universal solution to the KdV equation. This solution also satisfies a fifth order ordinary differential equation. We pose the problem on studying the behavior of this solution as t → ∞. For large time, the asymptotic solution has different structure depending on the slow variable s = x2/t. We construct the asymptotic solution in the domains s < −3/4, −3/4 < s < 5/24 and in the vicinity of the point s = −3/4. It is shown that a slow modulation of solution’s parameters in the vicinity of the point s = −3/4 is described by a solution to Painlevé IV equation.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89935774","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}
We consider a degenerate fractional order differential equationDα t Lu(t) = Mu(t) in a Hausdorff sequentially complete locally convex space. Under the p-regularity of the operator pair (L,M), we find the phase space of the equation and the family of its resolving operators. We show that the identity image of the latter coincides with the phase space. We prove an unique solvability theorem and obtain the form of the solution to the Cauchy problem for the corresponding inhomogeneous equation. We give an example of application the obtained abstract results to studying the solvability of the initial boundary value problems for the partial differential equations involving entire functions on an unbounded operator in a Banach space, which is a specially constructed Frechét space. It allows us to consider, for instance, a periodic in a spatial variable x problem for the equation with a shift along x and with a fractional order derivative with respect to time t.
考虑Hausdorff序列完备局部凸空间中的退化分数阶微分方程d α t Lu(t) = Mu(t)。在算子对(L,M)的p正则性下，我们得到了方程的相空间及其解析算子族。我们证明了后者的恒等像与相空间重合。证明了相应的非齐次方程的柯西问题的唯一可解定理，得到了柯西问题的解的形式。我们给出了一个应用所得到的抽象结果在Banach空间(这是一个特殊构造的frech空间)上研究无界算子上包含整个函数的偏微分方程初边值问题的可解性的例子。它允许我们考虑，例如，一个关于空间变量x的周期问题对于一个方程，它沿着x移动并且对时间t有分数阶导数。
{"title":"Degenerate fractional differential equations in locally convex spaces with a $sigma$-regular pair of operators","authors":"M. Kostic, V. Fedorov","doi":"10.13108/2016-8-4-98","DOIUrl":"https://doi.org/10.13108/2016-8-4-98","url":null,"abstract":"We consider a degenerate fractional order differential equationDα t Lu(t) = Mu(t) in a Hausdorff sequentially complete locally convex space. Under the p-regularity of the operator pair (L,M), we find the phase space of the equation and the family of its resolving operators. We show that the identity image of the latter coincides with the phase space. We prove an unique solvability theorem and obtain the form of the solution to the Cauchy problem for the corresponding inhomogeneous equation. We give an example of application the obtained abstract results to studying the solvability of the initial boundary value problems for the partial differential equations involving entire functions on an unbounded operator in a Banach space, which is a specially constructed Frechét space. It allows us to consider, for instance, a periodic in a spatial variable x problem for the equation with a shift along x and with a fractional order derivative with respect to time t.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89513257","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}
In the work we study the properties of the resolvent of the Laplace-Beltrami operator on a two-dimensional sphere S2. We obtain the regularized trace formula for the Laplace-Beltrami operator perturbed by the operator of multiplication by a function in W 1 2 (S 2).
{"title":"Properties of the resolvent of the Laplace operator on a two-dimensional sphere and a trace formula","authors":"A. Atnagulov, V. Sadovnichii, Z. Fazullin","doi":"10.13108/2016-8-3-22","DOIUrl":"https://doi.org/10.13108/2016-8-3-22","url":null,"abstract":"In the work we study the properties of the resolvent of the Laplace-Beltrami operator on a two-dimensional sphere S2. We obtain the regularized trace formula for the Laplace-Beltrami operator perturbed by the operator of multiplication by a function in W 1 2 (S 2).","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73507464","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":"The problem of Steklov type in a half-cylinder with a small cavity","authors":"D. B. Davletov, D. V. Kozhevnikov","doi":"10.13108/2016-8-4-62","DOIUrl":"https://doi.org/10.13108/2016-8-4-62","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85528555","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}
. In this paper we study the solvability of one boundary value problem for an inhomogeneous polyharmonic equation. As a boundary operator, we consider a differen-tiation operator of fractional order in the Hadamard sense. The considered problem is a generalization of the known Neumann problem.
{"title":"On solvability of a boundary value problem for an inhomogeneous polyharmonic equation with a fractional order boundary operator","authors":"B. Turmetov","doi":"10.13108/2016-8-3-155","DOIUrl":"https://doi.org/10.13108/2016-8-3-155","url":null,"abstract":". In this paper we study the solvability of one boundary value problem for an inhomogeneous polyharmonic equation. As a boundary operator, we consider a differen-tiation operator of fractional order in the Hadamard sense. The considered problem is a generalization of the known Neumann problem.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82029911","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}
We show that the existence of unconditional exponential bases is not determined by the growth characteristics of a weight function. In order to do this, we construct examples of convex weights with arbitrarily slow growth near the boundary such that unconditional exponential bases do not exist in the corresponding space.
{"title":"On unconditional exponential bases in weak weighted spaces on segment","authors":"K. P. Isaev, A. Lutsenko, R. S. Yulmukhametov","doi":"10.13108/2016-8-4-88","DOIUrl":"https://doi.org/10.13108/2016-8-4-88","url":null,"abstract":"We show that the existence of unconditional exponential bases is not determined by the growth characteristics of a weight function. In order to do this, we construct examples of convex weights with arbitrarily slow growth near the boundary such that unconditional exponential bases do not exist in the corresponding space.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84267877","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}
We consider an elliptic operator in a multi-dimensional domain with frequent alternation of Dirichlet and Robin conditions. We study the case, when the homogenized operator has Robin condition with an additional coefficient generated by the geometry of the alternation. We prove the norm resolvent convergence of the perturbed operator to the homogenized one and obtain the estimate for the convergence rate. We construct the complete asymptotic expansion for the resolvent in the case, when it acts on sufficiently smooth functions.
{"title":"On resolvent of multi-dimensional operators with frequent alternation of boundary conditions: Critical case","authors":"T. F. Sharapov","doi":"10.13108/2016-8-2-65","DOIUrl":"https://doi.org/10.13108/2016-8-2-65","url":null,"abstract":"We consider an elliptic operator in a multi-dimensional domain with frequent alternation of Dirichlet and Robin conditions. We study the case, when the homogenized operator has Robin condition with an additional coefficient generated by the geometry of the alternation. We prove the norm resolvent convergence of the perturbed operator to the homogenized one and obtain the estimate for the convergence rate. We construct the complete asymptotic expansion for the resolvent in the case, when it acts on sufficiently smooth functions.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75161094","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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