On a Gronwall-type inequality for the general fractional integrals with the Sonin kernels and its applications

IF 3.4 2区数学 Q1 MATHEMATICS, APPLIED

Communications in Nonlinear Science and Numerical Simulation Pub Date : 2025-06-09 DOI:10.1016/j.cnsns.2025.108985

Mohammed Al-Refai , Mohammadkheer Al-Jararha , Yuri Luchko

{"title":"On a Gronwall-type inequality for the general fractional integrals with the Sonin kernels and its applications","authors":"Mohammed Al-Refai , Mohammadkheer Al-Jararha , Yuri Luchko","doi":"10.1016/j.cnsns.2025.108985","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, for the first time, we formulate and prove a Gronwall-type inequality for the general fractional integrals with the Sonin kernels and with variable coefficients. Most of the Gronwall-type inequalities for the Fractional Calculus operators introduced in the literature so far are particular cases of this inequality. In the second part of the paper, the Gronwall-type inequality for the general fractional integrals with the Sonin kernels is applied for investigation of initial value problems for the non-linear fractional differential equations with the general fractional derivatives in the Riemann–Liouville sense and with the 1st level general fractional derivatives. This analysis covers the equations with both the Riemann–Liouville and Hilfer fractional derivatives and with several other types of fractional derivatives. In particular, we derive the uniqueness and continuous dependence of solutions to these problems on the initial data.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"150 ","pages":"Article 108985"},"PeriodicalIF":3.4000,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Nonlinear Science and Numerical Simulation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S100757042500396X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

In this paper, for the first time, we formulate and prove a Gronwall-type inequality for the general fractional integrals with the Sonin kernels and with variable coefficients. Most of the Gronwall-type inequalities for the Fractional Calculus operators introduced in the literature so far are particular cases of this inequality. In the second part of the paper, the Gronwall-type inequality for the general fractional integrals with the Sonin kernels is applied for investigation of initial value problems for the non-linear fractional differential equations with the general fractional derivatives in the Riemann–Liouville sense and with the 1st level general fractional derivatives. This analysis covers the equations with both the Riemann–Liouville and Hilfer fractional derivatives and with several other types of fractional derivatives. In particular, we derive the uniqueness and continuous dependence of solutions to these problems on the initial data.

查看原文本刊更多论文

带Sonin核的一般分数阶积分的gronwall型不等式及其应用

本文首次构造并证明了一类带Sonin核变系数的一般分数阶积分的gronwall型不等式。到目前为止，文献中介绍的分数阶微积分算子的大多数gronwall型不等式都是该不等式的特殊情况。本文的第二部分，利用Sonin核一般分数阶积分的gronwall型不等式，研究了一类具有Riemann-Liouville意义和一级一般分数阶导数的非线性分数阶微分方程的初值问题。这一分析涵盖了黎曼-刘维尔和希尔弗分数阶导数和其他几种类型的分数阶导数的方程。特别地，我们得到了这些问题的解对初始数据的唯一性和连续依赖性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.