Abstract This paper discusses the existence and uniqueness of solutions for a class of nonlinear fractional differential equations with a mixed fractional boundary value using Banach and Schaefer fixed-point theorems. The examples illustrating the main results are given.
{"title":"Nonlinear Katugampola Fractional Differential Equation with Mixed Boundary Conditions","authors":"Barbara Lupińska","doi":"10.2478/tmmp-2023-0013","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0013","url":null,"abstract":"Abstract This paper discusses the existence and uniqueness of solutions for a class of nonlinear fractional differential equations with a mixed fractional boundary value using Banach and Schaefer fixed-point theorems. The examples illustrating the main results are given.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"84 1","pages":"25 - 34"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46362789","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}
Abstract This article deals with the existence, uniqueness and Ulam-Hyers--Rassias stability results for a class of coupled systems for implicit fractional differential equations with Riesz-Caputo fractional derivative and boundary conditions. We will employ the Banach’s contraction principle as well as Schauder’s fixed point theorem to demonstrate our existence results. We provide an example to illustrate the obtained results.
{"title":"Existence, Uniqueness and Ulam-Hyers-Rassias Stability of Differential Coupled Systems with Riesz-Caputo Fractional Derivative","authors":"Abdelkrim Salim, J. Lazreg, M. Benchohra","doi":"10.2478/tmmp-2023-0019","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0019","url":null,"abstract":"Abstract This article deals with the existence, uniqueness and Ulam-Hyers--Rassias stability results for a class of coupled systems for implicit fractional differential equations with Riesz-Caputo fractional derivative and boundary conditions. We will employ the Banach’s contraction principle as well as Schauder’s fixed point theorem to demonstrate our existence results. We provide an example to illustrate the obtained results.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"84 1","pages":"111 - 138"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46317719","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}
Abstract We study the Volterra integro-differential equation on time scales and provide sufficient conditions for boundness of all solutions of considered equation. Using that result, we present the conditions for exponential stability of considered equation. All the results proved on the general time scale include results for both integral and discrete Volterra equations.
{"title":"Exponential Stability of Integro-Differential Volterra Equation on Time Scales","authors":"U. Ostaszewska, E. Schmeidel, M. Zdanowicz","doi":"10.2478/tmmp-2023-0017","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0017","url":null,"abstract":"Abstract We study the Volterra integro-differential equation on time scales and provide sufficient conditions for boundness of all solutions of considered equation. Using that result, we present the conditions for exponential stability of considered equation. All the results proved on the general time scale include results for both integral and discrete Volterra equations.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"84 1","pages":"77 - 86"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41362994","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}
Abstract We provide two sufficient criteria for the bifurcation of bounded entire or homoclinic solutions to nonautonomous difference equations depending on a single real parameter. Our analysis is based on a nonhyperbolic solution, whose variational equation possesses exponential dichotomies on semiaxes ensuring that the corresponding critical spectral interval of the dichotomy spectrum has strict multiplicity > 1. This extends earlier results on the fold bifurcation.
{"title":"Nonautonomous Fold Bifurcations from Spectral Intervals of Higher Strict Multiplicity","authors":"C. Pötzsche","doi":"10.2478/tmmp-2023-0018","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0018","url":null,"abstract":"Abstract We provide two sufficient criteria for the bifurcation of bounded entire or homoclinic solutions to nonautonomous difference equations depending on a single real parameter. Our analysis is based on a nonhyperbolic solution, whose variational equation possesses exponential dichotomies on semiaxes ensuring that the corresponding critical spectral interval of the dichotomy spectrum has strict multiplicity > 1. This extends earlier results on the fold bifurcation.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"84 1","pages":"87 - 110"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44553569","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}
Abstract We investigate the discrete equations of the form Δ(rnΔxn)=anf(xσ(n))+bn. Delta left( {{r_n}Delta {x_n}} right) = {a_n}fleft( {{x_{sigma left( n right)}}} right) + {b_n}. Using the Knaster-Tarski fixed point theorem, we study solutions with prescribed asymptotic behaviour. Our technique allows us to control the degree of approximation. In particular, we present the results concerning harmonic and geometric approximations of solutions.
{"title":"Asymptotic Properties of Solutions to Discrete Sturm-Liouville Monotone Type Equations","authors":"Janusz Migda, E. Schmeidel","doi":"10.2478/tmmp-2023-0014","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0014","url":null,"abstract":"Abstract We investigate the discrete equations of the form Δ(rnΔxn)=anf(xσ(n))+bn. Delta left( {{r_n}Delta {x_n}} right) = {a_n}fleft( {{x_{sigma left( n right)}}} right) + {b_n}. Using the Knaster-Tarski fixed point theorem, we study solutions with prescribed asymptotic behaviour. Our technique allows us to control the degree of approximation. In particular, we present the results concerning harmonic and geometric approximations of solutions.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"84 1","pages":"35 - 44"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48803205","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}
Abstract We investigate a discrete analogue of the polylogarithm function. Difference and summation relations are obtained, as well as its connection to the discrete hypergeometric series.
摘要研究了多对数函数的离散模拟。得到了离散超几何级数的差和关系及其与离散超几何级数的联系。
{"title":"Discrete Polylogarithm Functions","authors":"Tom Cuchta, Dallas D. Freeman","doi":"10.2478/tmmp-2023-0012","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0012","url":null,"abstract":"Abstract We investigate a discrete analogue of the polylogarithm function. Difference and summation relations are obtained, as well as its connection to the discrete hypergeometric series.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"84 1","pages":"19 - 24"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44898698","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}
Abstract We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales (a(t)xΔ2(t))Δ2=b(t)f(x(t))+c(t), {left( {aleft( t right){x^{{Delta ^2}}}left( t right)} right)^{{Delta ^2}}} = bleft( t right)fleft( {xleft( t right)} right) + cleft( t right), such that for a given function y : 𝕋 → ℝ there exists a solution x : 𝕋 → ℝ to considered equation with asymptotic behaviour x(t)=y(t)+o(1tβ) xleft( t right) = yleft( t right) + oleft( {{1 over {{t^beta }}}} right) . The presented result is applied to the study of solutions to the classical Euler–Bernoulli beam equation, which means that it covers the case 𝕋 = ℝ.
{"title":"Asymptotic Properties of Solutions to Fourth-Order Difference Equations on Time Scales","authors":"U. Ostaszewska, E. Schmeidel, M. Zdanowicz","doi":"10.2478/tmmp-2023-0016","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0016","url":null,"abstract":"Abstract We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales (a(t)xΔ2(t))Δ2=b(t)f(x(t))+c(t), {left( {aleft( t right){x^{{Delta ^2}}}left( t right)} right)^{{Delta ^2}}} = bleft( t right)fleft( {xleft( t right)} right) + cleft( t right), such that for a given function y : 𝕋 → ℝ there exists a solution x : 𝕋 → ℝ to considered equation with asymptotic behaviour x(t)=y(t)+o(1tβ) xleft( t right) = yleft( t right) + oleft( {{1 over {{t^beta }}}} right) . The presented result is applied to the study of solutions to the classical Euler–Bernoulli beam equation, which means that it covers the case 𝕋 = ℝ.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"84 1","pages":"61 - 76"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68923074","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}
Abstract The paper presents an analysis of modifications of the standard discrete Samuelson model with determination of the stability of the equilibrium point. The stability region of the equilibrium point depending on the parameters contained in each model is determined using the Schur-Cohn stability criterion.
{"title":"Modifications of the Samuelson Economic Cycle Model","authors":"Paulina Naumowicz, M. Ruzicková","doi":"10.2478/tmmp-2023-0015","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0015","url":null,"abstract":"Abstract The paper presents an analysis of modifications of the standard discrete Samuelson model with determination of the stability of the equilibrium point. The stability region of the equilibrium point depending on the parameters contained in each model is determined using the Schur-Cohn stability criterion.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"84 1","pages":"45 - 60"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44903507","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}
Abstract In this work, we consider a boundary value problem for a q-Dirac equation. We prove orthogonality of the eigenfunctions, realness of the eigenvalues, and we study asymptotic formulas of the eigenfunctions. We show that the eigenfunctions form a complete system, we obtain the expansion formula with respect to the eigenfunctions, and we derive Parseval’s equality. We construct the Weyl solution and the Weyl function. We prove a uniqueness theorem for the solution of the inverse problem with respect to the Weyl function.
{"title":"Uniqueness for an Inverse Quantum-Dirac Problem with Given Weyl Function","authors":"M. Bohner, Ayça Çetinkaya","doi":"10.2478/tmmp-2023-0011","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0011","url":null,"abstract":"Abstract In this work, we consider a boundary value problem for a q-Dirac equation. We prove orthogonality of the eigenfunctions, realness of the eigenvalues, and we study asymptotic formulas of the eigenfunctions. We show that the eigenfunctions form a complete system, we obtain the expansion formula with respect to the eigenfunctions, and we derive Parseval’s equality. We construct the Weyl solution and the Weyl function. We prove a uniqueness theorem for the solution of the inverse problem with respect to the Weyl function.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"84 1","pages":"1 - 18"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48085155","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}
Abstract Let 𝑥 be a Banach algebra. In this article, on the one hand, we proved some results concerning the continuous projection from 𝑥 to its center. On the other hand, we investigate the commutativity of 𝑥 under specific conditions. Finally, we included some examples and applications to prove that various restrictions in the hypotheses of our theorems are necessary.
{"title":"Commutativity Theorems and Projection on the Center of a Banach Algebra","authors":"Mohamed Moumen, L. Taoufiq","doi":"10.2478/tmmp-2023-0009","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0009","url":null,"abstract":"Abstract Let 𝑥 be a Banach algebra. In this article, on the one hand, we proved some results concerning the continuous projection from 𝑥 to its center. On the other hand, we investigate the commutativity of 𝑥 under specific conditions. Finally, we included some examples and applications to prove that various restrictions in the hypotheses of our theorems are necessary.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"83 1","pages":"119 - 130"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46650553","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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