In this paper, we use a Banach fixed point theorem to obtain stability results of the zero solution for a mixed linear Levin-Nohel integro-differential system. To be more precise, we are concerned with the following system x′ (t)+ ∫ t t−τ(t) C (t,s)x(s)ds+B(t)x(t−h(t)) = 0, where the importance of studying this system is that, generalizes a set of results at the same time, due to Burton [6], Becker and Burton [4], Jin and Luo [10] and Dung [9], from the one dimension to the n dimension. The last system with several delays terms is discussed as well.
{"title":"Stability conditions for a mixed linear Levin–Nohel integrodifferential system","authors":"M. Mesmouli, A. Ardjouni, A. Djoudi","doi":"10.1216/jie.2022.34.349","DOIUrl":"https://doi.org/10.1216/jie.2022.34.349","url":null,"abstract":"In this paper, we use a Banach fixed point theorem to obtain stability results of the zero solution for a mixed linear Levin-Nohel integro-differential system. To be more precise, we are concerned with the following system x′ (t)+ ∫ t t−τ(t) C (t,s)x(s)ds+B(t)x(t−h(t)) = 0, where the importance of studying this system is that, generalizes a set of results at the same time, due to Burton [6], Becker and Burton [4], Jin and Luo [10] and Dung [9], from the one dimension to the n dimension. The last system with several delays terms is discussed as well.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46042217","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}
{"title":"Existence and asymptotic stability for lattice stochastic integrodifferential equations with infinite delays","authors":"Nguyễn Như Quân","doi":"10.1216/jie.2022.34.357","DOIUrl":"https://doi.org/10.1216/jie.2022.34.357","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46054155","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}
A new measure of noncompactness (Mnc) and Darbo fixed point theorem are utilized on space L(R+) to prove the existence of solution for functional Volterra-Fredholm integral equations. An example is given to confirm the validity of results. Furthermore, we propound an iterative algorithm by Sinc interpolation to find the solution with an acceptable accuracy. In this algorithm, it does not need the problem is discretized to an algebraic system with unknown coefficients and we have an iterative processes to aproximate of solution with exponential convergence.
{"title":"Existence of solution of functional Volterra-Fredholm integral equations in space L∞(ℝ+) and sinc interpolation to find solution","authors":"R. Arab, M. Rabbani","doi":"10.1216/jie.2022.34.151","DOIUrl":"https://doi.org/10.1216/jie.2022.34.151","url":null,"abstract":"A new measure of noncompactness (Mnc) and Darbo fixed point theorem are utilized on space L(R+) to prove the existence of solution for functional Volterra-Fredholm integral equations. An example is given to confirm the validity of results. Furthermore, we propound an iterative algorithm by Sinc interpolation to find the solution with an acceptable accuracy. In this algorithm, it does not need the problem is discretized to an algebraic system with unknown coefficients and we have an iterative processes to aproximate of solution with exponential convergence.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46417064","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 article, using Petryshyn’s fixed point theorem associated with the measure of non-compactness, we discuss the existence result for functional integral equations in Banach algebra, which covers many existence results for functional integral equations as a particular case under some weaker conditions. Further, we provide some examples of functional integral equations to illustrate our analytical findings.
{"title":"An existence result for functional integral equations via Petryshyn’s fixed point theorem","authors":"A. Deep, Ashisha Kumar, Syed Abbas, B. Hazarika","doi":"10.1216/jie.2022.34.165","DOIUrl":"https://doi.org/10.1216/jie.2022.34.165","url":null,"abstract":"In this article, using Petryshyn’s fixed point theorem associated with the measure of non-compactness, we discuss the existence result for functional integral equations in Banach algebra, which covers many existence results for functional integral equations as a particular case under some weaker conditions. Further, we provide some examples of functional integral equations to illustrate our analytical findings.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42122932","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}
A probabilistic interpretation of the exact solution of a certain inhomogeneous Fredholm integral equation of the second kind is given. Then, this interpretation is used to obtain an approximate solution of a generalization of the integral equation. The approximate solutions are compared with the corresponding Neumann series solutions in various particular cases. AMS Subject Classification: Primary 45B05; Secondary 62M10, 93E20.
{"title":"Probabilistic solutions of integral equations from optimal control","authors":"M. Lefebvre","doi":"10.1216/jie.2022.34.215","DOIUrl":"https://doi.org/10.1216/jie.2022.34.215","url":null,"abstract":"A probabilistic interpretation of the exact solution of a certain inhomogeneous Fredholm integral equation of the second kind is given. Then, this interpretation is used to obtain an approximate solution of a generalization of the integral equation. The approximate solutions are compared with the corresponding Neumann series solutions in various particular cases. AMS Subject Classification: Primary 45B05; Secondary 62M10, 93E20.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44067086","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 are concerned with the robust stability of Volterra equations. We consider the conditions preserving the stability of these systems under perturbations. Also, we study the so-called Bohl-Perron type stability theorems.
{"title":"Bohl theorem for Volterra equation","authors":"Nguyen Thu Ha","doi":"10.1216/jie.2022.34.229","DOIUrl":"https://doi.org/10.1216/jie.2022.34.229","url":null,"abstract":"In this paper we are concerned with the robust stability of Volterra equations. We consider the conditions preserving the stability of these systems under perturbations. Also, we study the so-called Bohl-Perron type stability theorems.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46707325","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 establish that Nyström discretizations of linear Fredholm integral operators on Hölder spaces converge in the operator norm while preserving the consistency order of the quadrature or cubature rule. This allows to employ tools from classical perturbation theory, rather than collective compactness, when studying numerical approximations of integral operators, as well as applications in for instance the field of nonautonomous dynamical systems.
{"title":"Uniform convergence of Nyström discretization on Hölder spaces","authors":"C. Pötzsche","doi":"10.1216/jie.2022.34.247","DOIUrl":"https://doi.org/10.1216/jie.2022.34.247","url":null,"abstract":"We establish that Nyström discretizations of linear Fredholm integral operators on Hölder spaces converge in the operator norm while preserving the consistency order of the quadrature or cubature rule. This allows to employ tools from classical perturbation theory, rather than collective compactness, when studying numerical approximations of integral operators, as well as applications in for instance the field of nonautonomous dynamical systems.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45832631","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}
{"title":"Fixed point theorems in generalized convex metric space and an application to the solution of Volterra integral equations","authors":"Chao Wang, Xueli Li","doi":"10.1216/jie.2022.34.257","DOIUrl":"https://doi.org/10.1216/jie.2022.34.257","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48270100","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 consider the soulution of a class of nonlinear Volterra integral equations, x(t) = g(t) + ∫ t t0 k(t, s; x(s))ds, (t ≥ t0), where the kernel function k is finitely decomposable, and derive variation of parameters formulae that provides the solution of corresponding perturbed nonlinear equations. We attain this by relating the integral equations to a certain class of initial value problems for ODEs, for which variation of parameters are also formulated.
{"title":"On the solution of Volterra integral equations with decomposable kernel functions","authors":"E. Agyingi","doi":"10.1216/jie.2022.34.135","DOIUrl":"https://doi.org/10.1216/jie.2022.34.135","url":null,"abstract":"In this paper, we consider the soulution of a class of nonlinear Volterra integral equations, x(t) = g(t) + ∫ t t0 k(t, s; x(s))ds, (t ≥ t0), where the kernel function k is finitely decomposable, and derive variation of parameters formulae that provides the solution of corresponding perturbed nonlinear equations. We attain this by relating the integral equations to a certain class of initial value problems for ODEs, for which variation of parameters are also formulated.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48363781","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 investigate semi-linear evolution equations having the following form u(t) = U(t,s)u(s) + ∫ t s U(t,ξ ) f (ξ ,u(ξ ))dξ for t ≥ s and s ∈ R in Banach space X . Under the assumptions that the evolution family (U(t,s))t≥s has the exponential dichotomy and the function f : R×X → X has the Carathéodory property, we show that the semi-linear evolution equations on the line has a unique admissible solution, bounded solution, periodic solution when the function f satisfies the condition φ-Lipschitz and exists a periodic solution when the function f satisfies the condition ‖ f (t,x)‖ ≤ φ(t)(1+‖x‖) for all x ∈ X and almost everywhere t ∈ R.
{"title":"Admissible, bounded and periodic solutions of semilinear evolution equations on the line","authors":"Trinh Viet Duoc","doi":"10.1216/jie.2022.34.183","DOIUrl":"https://doi.org/10.1216/jie.2022.34.183","url":null,"abstract":"In this paper, we investigate semi-linear evolution equations having the following form u(t) = U(t,s)u(s) + ∫ t s U(t,ξ ) f (ξ ,u(ξ ))dξ for t ≥ s and s ∈ R in Banach space X . Under the assumptions that the evolution family (U(t,s))t≥s has the exponential dichotomy and the function f : R×X → X has the Carathéodory property, we show that the semi-linear evolution equations on the line has a unique admissible solution, bounded solution, periodic solution when the function f satisfies the condition φ-Lipschitz and exists a periodic solution when the function f satisfies the condition ‖ f (t,x)‖ ≤ φ(t)(1+‖x‖) for all x ∈ X and almost everywhere t ∈ R.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43303052","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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