Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.2.305
N. Tuan
The bi-parabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for bi-parabolic equation when the observed data are obtained in (L^p). We are interested in looking at three types of inverse problems. Regularization results in the (L^2) space appears in many related papers, but the survey results are rare in (L^p), (p neq 2). The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in (L^p) spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in (L^p), we obtain the approximated solution also in the space (L^p). Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space (L^p). This paper seems to generalize to previous results for bi-parabolic equation on this direction.
{"title":"On some inverse problem for bi-parabolic equation with observed data in L^{p} spaces","authors":"N. Tuan","doi":"10.7494/opmath.2022.42.2.305","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.2.305","url":null,"abstract":"The bi-parabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for bi-parabolic equation when the observed data are obtained in (L^p). We are interested in looking at three types of inverse problems. Regularization results in the (L^2) space appears in many related papers, but the survey results are rare in (L^p), (p neq 2). The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in (L^p) spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in (L^p), we obtain the approximated solution also in the space (L^p). Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space (L^p). This paper seems to generalize to previous results for bi-parabolic equation on this direction.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342490","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 : 2022-01-01DOI: 10.7494/opmath.2022.42.2.119
Huafei Di, Zefang Song
Considered herein is the global existence and non-global existence of the initial-boundary value problem for a quasilinear viscoelastic equation with strong damping and source terms. Firstly, we introduce a family of potential wells and give the invariance of some sets, which are essential to derive the main results. Secondly, we establish the existence of global weak solutions under the low initial energy and critical initial energy by the combination of the Galerkin approximation and improved potential well method involving with (t). Thirdly, we obtain the finite time blow-up result for certain solutions with the non-positive initial energy and positive initial energy, and then give the upper bound for the blow-up time (T^ast). Especially, the threshold result between global existence and non-global existence is given under some certain conditions. Finally, a lower bound for the life span (T^ast) is derived by the means of integro-differential inequality techniques.
{"title":"Global existence and blow-up phenomenon for a quasilinear viscoelastic equation with strong damping and source terms","authors":"Huafei Di, Zefang Song","doi":"10.7494/opmath.2022.42.2.119","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.2.119","url":null,"abstract":"Considered herein is the global existence and non-global existence of the initial-boundary value problem for a quasilinear viscoelastic equation with strong damping and source terms. Firstly, we introduce a family of potential wells and give the invariance of some sets, which are essential to derive the main results. Secondly, we establish the existence of global weak solutions under the low initial energy and critical initial energy by the combination of the Galerkin approximation and improved potential well method involving with (t). Thirdly, we obtain the finite time blow-up result for certain solutions with the non-positive initial energy and positive initial energy, and then give the upper bound for the blow-up time (T^ast). Especially, the threshold result between global existence and non-global existence is given under some certain conditions. Finally, a lower bound for the life span (T^ast) is derived by the means of integro-differential inequality techniques.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342136","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 : 2022-01-01DOI: 10.7494/opmath.2022.42.6.769
E. Attia, B. El-Matary, G. Chatzarakis
In this paper, we study the oscillatory behavior of the solutions of a first-order difference equation with non-monotone retarded argument and nonnegative coefficients, based on an iterative procedure. We establish some oscillation criteria, involving (lim sup), which achieve a marked improvement on several known conditions in the literature. Two examples, numerically solved in MAPLE software, are also given to illustrate the applicability and strength of the obtained conditions.
{"title":"New oscillation conditions for first-order linear retarded difference equations with non-monotone arguments","authors":"E. Attia, B. El-Matary, G. Chatzarakis","doi":"10.7494/opmath.2022.42.6.769","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.6.769","url":null,"abstract":"In this paper, we study the oscillatory behavior of the solutions of a first-order difference equation with non-monotone retarded argument and nonnegative coefficients, based on an iterative procedure. We establish some oscillation criteria, involving (lim sup), which achieve a marked improvement on several known conditions in the literature. Two examples, numerically solved in MAPLE software, are also given to illustrate the applicability and strength of the obtained conditions.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342815","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 : 2022-01-01DOI: 10.7494/opmath.2022.42.5.751
Robert Stegli�ski
{"title":"Notes on aplications of the dual fountain theorem to local and nonlocal elliptic equations with variable exponent","authors":"Robert Stegli�ski","doi":"10.7494/opmath.2022.42.5.751","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.5.751","url":null,"abstract":"","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342949","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 : 2022-01-01DOI: 10.7494/opmath.2022.42.4.549
B. Baculíková
In the paper, we study oscillation and asymptotic properties for even order linear functional differential equations [y^{(n)}(t)=p(t)y(tau(t))] with mixed deviating arguments, i.e. when both delayed and advanced parts of (tau(t)) are significant. The presented results essentially improve existing ones.
{"title":"Oscillation of even order linear functional differential equations with mixed deviating arguments","authors":"B. Baculíková","doi":"10.7494/opmath.2022.42.4.549","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.4.549","url":null,"abstract":"In the paper, we study oscillation and asymptotic properties for even order linear functional differential equations [y^{(n)}(t)=p(t)y(tau(t))] with mixed deviating arguments, i.e. when both delayed and advanced parts of (tau(t)) are significant. The presented results essentially improve existing ones.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342145","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 : 2022-01-01DOI: 10.7494/opmath.2022.42.1.65
T. Madaras, Alfréd Onderko, Thomas Schweser
We explore four kinds of edge colorings defined by the requirement of equal number of colors appearing, in particular ways, around each vertex or each edge. We obtain the characterization of graphs colorable in such a way that the ends of each edge see (not regarding the edge color itself) (q) colors (resp. one end sees (q) colors and the color sets for both ends are the same), and a sufficient condition for 2-coloring a graph in a way that the ends of each edge see (with the omission of that edge color) altogether (q) colors. The relations of these colorings to (M_q)-colorings and role colorings are also discussed; we prove an interpolation theorem for the numbers of colors in edge coloring where all edges around each vertex have (q) colors.
{"title":"Edge homogeneous colorings","authors":"T. Madaras, Alfréd Onderko, Thomas Schweser","doi":"10.7494/opmath.2022.42.1.65","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.1.65","url":null,"abstract":"We explore four kinds of edge colorings defined by the requirement of equal number of colors appearing, in particular ways, around each vertex or each edge. We obtain the characterization of graphs colorable in such a way that the ends of each edge see (not regarding the edge color itself) (q) colors (resp. one end sees (q) colors and the color sets for both ends are the same), and a sufficient condition for 2-coloring a graph in a way that the ends of each edge see (with the omission of that edge color) altogether (q) colors. The relations of these colorings to (M_q)-colorings and role colorings are also discussed; we prove an interpolation theorem for the numbers of colors in edge coloring where all edges around each vertex have (q) colors.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342408","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 : 2022-01-01DOI: 10.7494/opmath.2022.42.4.635
Michal Sta�, M�ria �vecov�
{"title":"The crossing numbers of join products of paths with three graphs of order five","authors":"Michal Sta�, M�ria �vecov�","doi":"10.7494/opmath.2022.42.4.635","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.4.635","url":null,"abstract":"","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342635","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 : 2022-01-01DOI: 10.7494/opmath.2022.42.6.793
Imed Bachar, Habib M�agli, H. Eltayeb
In this paper, we obtain sufficient conditions for the existence of a unique nonnegative continuous solution of semipositone semilinear elliptic problem in bounded domains of (mathbb{R}^n) ((ngeq 2)). The global behavior of this solution is also given.
{"title":"Nonnegative solutions for a class of semipositone nonlinear elliptic equations in bounded domains of R^{n}","authors":"Imed Bachar, Habib M�agli, H. Eltayeb","doi":"10.7494/opmath.2022.42.6.793","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.6.793","url":null,"abstract":"In this paper, we obtain sufficient conditions for the existence of a unique nonnegative continuous solution of semipositone semilinear elliptic problem in bounded domains of (mathbb{R}^n) ((ngeq 2)). The global behavior of this solution is also given.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342878","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 : 2022-01-01DOI: 10.7494/opmath.2022.42.6.833
I. Dzhalladova, Miroslava R��i�kov�
A stochastic model describing the concentration of the drug in the body during its IV-administration is discussed. The paper compares a deterministic model created with certain simplifications with the stochastic model. Fluctuating and irregular patterns of plasma concentrations of some drugs observed during intravenous infusion are explained. An illustrative example is given with certain values of drug infusion rate and drug elimination rate.
{"title":"Stochastic model of drug concentration level during IV-administration","authors":"I. Dzhalladova, Miroslava R��i�kov�","doi":"10.7494/opmath.2022.42.6.833","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.6.833","url":null,"abstract":"A stochastic model describing the concentration of the drug in the body during its IV-administration is discussed. The paper compares a deterministic model created with certain simplifications with the stochastic model. Fluctuating and irregular patterns of plasma concentrations of some drugs observed during intravenous infusion are explained. An illustrative example is given with certain values of drug infusion rate and drug elimination rate.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342896","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 : 2022-01-01DOI: 10.7494/opmath.2022.42.3.361
Shunya Adachi
We study the monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities. The moduli space of our monodromy representations can be realized by certain affine cubic surface. In this paper we characterize the irreducible monodromies having the non-degenerate invariant Hermitian forms in terms of that cubic surface. The explicit forms of invariant Hermitian forms are also given. Our result may bring a new insight into the study of the Painlev� differential equations.
{"title":"Monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities","authors":"Shunya Adachi","doi":"10.7494/opmath.2022.42.3.361","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.3.361","url":null,"abstract":"We study the monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities. The moduli space of our monodromy representations can be realized by certain affine cubic surface. In this paper we characterize the irreducible monodromies having the non-degenerate invariant Hermitian forms in terms of that cubic surface. The explicit forms of invariant Hermitian forms are also given. Our result may bring a new insight into the study of the Painlev� differential equations.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342368","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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