One way in which nonlinear descriptor systems of (index-k) naturally arise is through semiexplicit differential-algebraic equations. The study considers the nonbilinear dynamical systems which are described by the class of higher-index differential-algebraic equations (DAEs). Their nature is analysed both quantitatively and qualitatively, and stability characteristics are presented for their solution. Higher-index differential-algebraic systems seem to show inherent shaky around their solution manifolds. The often use of logarithmic norms is for the estimation of stability and perturbation bounds in linear ordinary differential equations (ODEs). The question of how to apply the notation of logarithmic norms to nonlinear DAEs has long been an open question. Other problem extensions including nonlinear dynamics and nonbilinear DAEs need subtle modification of the logarithmic norms. The logarithmic norm is combined by conceptual focus with the finite-time stability criterion in order to treat nonbilinear DAEs with the aim of covering some unbounded operators. This means we obtain the perturbation bounds from differential inequalities for a norm by the use of the relationship between Dini derivatives and semi-inner products. A numerical result obtained when tested on the nonbilinear mechanical system with a larger scale showed that the method was highly efficient and accurate and particularly suitable for nonbilinear DAEs.
{"title":"On Stabilizability of Nonbilinear Perturbed Descriptor Systems","authors":"Ghazwa F. Abd","doi":"10.1155/2023/5561224","DOIUrl":"https://doi.org/10.1155/2023/5561224","url":null,"abstract":"One way in which nonlinear descriptor systems of (index-k) naturally arise is through semiexplicit differential-algebraic equations. The study considers the nonbilinear dynamical systems which are described by the class of higher-index differential-algebraic equations (DAEs). Their nature is analysed both quantitatively and qualitatively, and stability characteristics are presented for their solution. Higher-index differential-algebraic systems seem to show inherent shaky around their solution manifolds. The often use of logarithmic norms is for the estimation of stability and perturbation bounds in linear ordinary differential equations (ODEs). The question of how to apply the notation of logarithmic norms to nonlinear DAEs has long been an open question. Other problem extensions including nonlinear dynamics and nonbilinear DAEs need subtle modification of the logarithmic norms. The logarithmic norm is combined by conceptual focus with the finite-time stability criterion in order to treat nonbilinear DAEs with the aim of covering some unbounded operators. This means we obtain the perturbation bounds from differential inequalities for a norm by the use of the relationship between Dini derivatives and semi-inner products. A numerical result obtained when tested on the nonbilinear mechanical system with a larger scale showed that the method was highly efficient and accurate and particularly suitable for nonbilinear DAEs.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41578882","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, oscillation and asymptotic behavior of three-dimensional third-order delay systems are discussed. Some sufficient conditions are obtained to ensure that every solution of the system is either oscillatory or nonoscillatory and converges to zero or diverges as t goes to infinity. A special technique is adopted to include all possible cases for all nonoscillatory solutions (NOSs). The obtained results included illustrative examples.
{"title":"Oscillation and Asymptotic Behavior of Three-Dimensional Third-Order Delay Systems","authors":"Ahmed Abdul Hasan Naeif, Hussain A. Mohamad","doi":"10.1155/2023/9939317","DOIUrl":"https://doi.org/10.1155/2023/9939317","url":null,"abstract":"In this paper, oscillation and asymptotic behavior of three-dimensional third-order delay systems are discussed. Some sufficient conditions are obtained to ensure that every solution of the system is either oscillatory or nonoscillatory and converges to zero or diverges as \u0000 \u0000 t\u0000 \u0000 goes to infinity. A special technique is adopted to include all possible cases for all nonoscillatory solutions (NOSs). The obtained results included illustrative examples.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42295606","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}
Ali El Mfadel, S. Melliani, A. Taqbibt, M. Elomari
This manuscript aims to highlight the existence and uniqueness results for the following Schrödinger problem in the extended Colombeau algebra of generalized functions.
In this article we study the existence and stability of bounded solutions for semilinear abstract dynamic equations on time scales in Banach spaces. In order to do so, we use the definition of the Riemann delta-integral to prove a result about closed operator in Banach spaces and then we just use the representation of bounded solutions as an improper delta-integral from minus infinite to t . We prove the existence, uniqueness, and exponential stability of such bounded solutions. As particular cases, we study the existence of periodic and almost periodic solutions as well. Finally, we present some equations on time scales where our results can be applied.
{"title":"On the Existence and Stability of Bounded Solutions for Abstract Dynamic Equations on Time Scales","authors":"C. Duque, H. Leiva, R. Gallo, A. Tridane","doi":"10.1155/2023/8489196","DOIUrl":"https://doi.org/10.1155/2023/8489196","url":null,"abstract":"In this article we study the existence and stability of bounded solutions for semilinear abstract dynamic equations on time scales in Banach spaces. In order to do so, we use the definition of the Riemann delta-integral to prove a result about closed operator in Banach spaces and then we just use the representation of bounded solutions as an improper delta-integral from minus infinite to \u0000 \u0000 t\u0000 \u0000 . We prove the existence, uniqueness, and exponential stability of such bounded solutions. As particular cases, we study the existence of periodic and almost periodic solutions as well. Finally, we present some equations on time scales where our results can be applied.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46373733","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}
Nonlinear partial differential equations are considered as an essential tool for describing the behavior of many natural phenomena. The modeling of some phenomena requires to work in Sobolev spaces with constant exponent. But for others, such as electrorheological fluids, the properties of classical spaces are not sufficient to have precision. To overcome this difficulty, we work in the appropriate spaces called Lebesgue and Sobolev spaces with variable exponent. In recent works, researchers are attracted by the study of mathematical problems in the context of variable exponent. This great interest is motivated by their applications in many fields such as elastic mechanics, fluid dynamics, and image restoration. In this paper, we combine the technic of monotone operators in Banach spaces and approximation methods to prove the existence of renormalized solutions of a class of nonlinear anisotropic problem involving p ⟶ . − Leray–Lions operator, a graph, and L 1 data. In particular, we establish the uniqueness of the solution when the graph data are considered a strictly increasing function.
{"title":"Existence and Uniqueness of Renormalized Solution to Nonlinear Anisotropic Elliptic Problems with Variable Exponent and <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <msup>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 <","authors":"Ibrahime Konaté, Arouna Ouédraogo","doi":"10.1155/2023/9454714","DOIUrl":"https://doi.org/10.1155/2023/9454714","url":null,"abstract":"Nonlinear partial differential equations are considered as an essential tool for describing the behavior of many natural phenomena. The modeling of some phenomena requires to work in Sobolev spaces with constant exponent. But for others, such as electrorheological fluids, the properties of classical spaces are not sufficient to have precision. To overcome this difficulty, we work in the appropriate spaces called Lebesgue and Sobolev spaces with variable exponent. In recent works, researchers are attracted by the study of mathematical problems in the context of variable exponent. This great interest is motivated by their applications in many fields such as elastic mechanics, fluid dynamics, and image restoration. In this paper, we combine the technic of monotone operators in Banach spaces and approximation methods to prove the existence of renormalized solutions of a class of nonlinear anisotropic problem involving \u0000 \u0000 \u0000 p\u0000 ⟶\u0000 \u0000 \u0000 \u0000 .\u0000 \u0000 \u0000 −\u0000 \u0000 Leray–Lions operator, a graph, and \u0000 \u0000 \u0000 \u0000 L\u0000 \u0000 \u0000 1\u0000 \u0000 \u0000 \u0000 data. In particular, we establish the uniqueness of the solution when the graph data are considered a strictly increasing function.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48884709","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}
The boundedness of Bessel–Riesz operators defined on Lebesgue spaces and Morrey spaces in measure metric spaces is discussed in this research study. The maximal operator and traditional dyadic decomposition are used to study the Bessel-Riesz operators. We investigate the interaction between the kernel and space parameters to get the results and see how this affects kernel-bound operators.
{"title":"Bessel-Riesz Operators on Lebesgue Spaces and Morrey Spaces Defined in Measure Metric Spaces","authors":"Saba Mehmood, Eridani, Fatmawati, Wasim Raza","doi":"10.1155/2023/3148049","DOIUrl":"https://doi.org/10.1155/2023/3148049","url":null,"abstract":"The boundedness of Bessel–Riesz operators defined on Lebesgue spaces and Morrey spaces in measure metric spaces is discussed in this research study. The maximal operator and traditional dyadic decomposition are used to study the Bessel-Riesz operators. We investigate the interaction between the kernel and space parameters to get the results and see how this affects kernel-bound operators.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45167865","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}
Most nonlinear partial differential equations have many applications in the physical world. Finding solutions to nonlinear partial differential equations is not easily solvable and hence different modified techniques are applied to get solutions to such nonlinear partial differential equations. Among them, we considered the modified Korteweg–de Vries third order using the balance method and constructing its models using certain parameters. The method is successfully implemented in solving the stated equations. We obtained kind one and two soliton solutions and their graphical models are shown using mathematical software-12. The obtained results lead to shallow wave models. A few illustrative examples were presented to demonstrate the applicability of the models. Furthermore, physical and geometrical interpretations are considered for different parameters to investigate the nature of soliton solutions to their models. Finally, the proposed method is a standard, effective, and easily computable method for solving the modified Korteweg–de Vries equations and determining its perspective models.
{"title":"Solving Nonlinear Partial Differential Equations of Special Kinds of 3rd Order Using Balance Method and Its Models","authors":"Daba Meshesha Gusu, Wakjira Gudeta","doi":"10.1155/2023/7663326","DOIUrl":"https://doi.org/10.1155/2023/7663326","url":null,"abstract":"Most nonlinear partial differential equations have many applications in the physical world. Finding solutions to nonlinear partial differential equations is not easily solvable and hence different modified techniques are applied to get solutions to such nonlinear partial differential equations. Among them, we considered the modified Korteweg–de Vries third order using the balance method and constructing its models using certain parameters. The method is successfully implemented in solving the stated equations. We obtained kind one and two soliton solutions and their graphical models are shown using mathematical software-12. The obtained results lead to shallow wave models. A few illustrative examples were presented to demonstrate the applicability of the models. Furthermore, physical and geometrical interpretations are considered for different parameters to investigate the nature of soliton solutions to their models. Finally, the proposed method is a standard, effective, and easily computable method for solving the modified Korteweg–de Vries equations and determining its perspective models.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48869829","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, a fractional order of a modified Leslie–Gower predator-prey model with disease and the double Allee effect in predator population is proposed. Then, we analyze the important mathematical features of the proposed model such as the existence and uniqueness as well as the non-negativity and boundedness of solutions to the fractional-order system. Moreover, the local and global asymptotic stability conditions of all possible equilibrium points are investigated using Matignon’s condition and by constructing a suitable Lyapunov function, respectively. Finally, numerical simulations are presented to verify the theoretical results. We show numerically the occurrence of two limit cycles simultaneously driven by the order of the derivative, the bistability phenomenon for both the weak and strong Allee effect cases, and more dynamic behaviors such as the forward, backward, and saddle-node bifurcations which are driven by the transmission rate. We have found that the risk of extinction for the predator with a strong Allee effect is much higher when the spread of disease is relatively high.
{"title":"A Fractional-Order Eco-Epidemiological Leslie–Gower Model with Double Allee Effect and Disease in Predator","authors":"Emli Rahmi, I. Darti, A. Suryanto, T. Trisilowati","doi":"10.1155/2023/5030729","DOIUrl":"https://doi.org/10.1155/2023/5030729","url":null,"abstract":"In this paper, a fractional order of a modified Leslie–Gower predator-prey model with disease and the double Allee effect in predator population is proposed. Then, we analyze the important mathematical features of the proposed model such as the existence and uniqueness as well as the non-negativity and boundedness of solutions to the fractional-order system. Moreover, the local and global asymptotic stability conditions of all possible equilibrium points are investigated using Matignon’s condition and by constructing a suitable Lyapunov function, respectively. Finally, numerical simulations are presented to verify the theoretical results. We show numerically the occurrence of two limit cycles simultaneously driven by the order of the derivative, the bistability phenomenon for both the weak and strong Allee effect cases, and more dynamic behaviors such as the forward, backward, and saddle-node bifurcations which are driven by the transmission rate. We have found that the risk of extinction for the predator with a strong Allee effect is much higher when the spread of disease is relatively high.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41592550","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}
This paper deals with a class of perturbation planar Keplerian Hamiltonian systems, by exploiting the nondegeneracy properties of the circular solutions of the planar Keplerian Hamiltonian systems, and by applying the implicit function theorem, we show that noncollision periodic solutions of such perturbed system bifurcate from the manifold of circular solutions for the Keplerian Hamiltonian system.
{"title":"Perturbed Keplerian Hamiltonian Systems","authors":"Riadh Chteoui","doi":"10.1155/2023/3575701","DOIUrl":"https://doi.org/10.1155/2023/3575701","url":null,"abstract":"This paper deals with a class of perturbation planar Keplerian Hamiltonian systems, by exploiting the nondegeneracy properties of the circular solutions of the planar Keplerian Hamiltonian systems, and by applying the implicit function theorem, we show that noncollision periodic solutions of such perturbed system bifurcate from the manifold of circular solutions for the Keplerian Hamiltonian system.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43945786","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, a model of the rates of enzyme-catalyzed chemical reactions in the sense of Caputo–Fabrizio a fractional derivative was investigated. Its existence and uniqueness as a solution of the model was proved by setting different criteria. An iterative numerical scheme was provided to support the findings. In order to verify the applicability of the result, numerical simulations using the MATLAB software package that confirms the analytical result was lucidly shown.
{"title":"Existence and Uniqueness Solution of the Model of Enzyme Kinetics in the Sense of Caputo–Fabrizio Fractional Derivative","authors":"G. K. Edessa","doi":"10.1155/2022/1345919","DOIUrl":"https://doi.org/10.1155/2022/1345919","url":null,"abstract":"In this paper, a model of the rates of enzyme-catalyzed chemical reactions in the sense of Caputo–Fabrizio a fractional derivative was investigated. Its existence and uniqueness as a solution of the model was proved by setting different criteria. An iterative numerical scheme was provided to support the findings. In order to verify the applicability of the result, numerical simulations using the MATLAB software package that confirms the analytical result was lucidly shown.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42012248","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：