This work is devoted to the solvability of the weighted Cauchy problem for fractional differential equations of arbitrary order, considering the Riemann-Liouville derivative. We show the equivalence between the weighted Cauchy problem and the Volterra integral equation in the space of Lebesgue integrable functions. Finally, we point out some discrepancies between the solutions for fractional and integer order case.
{"title":"Weighted Cauchy problem: fractional versus integer order","authors":"M. G. Morales, Z. Došlá","doi":"10.1216/jie.2021.33.497","DOIUrl":"https://doi.org/10.1216/jie.2021.33.497","url":null,"abstract":"This work is devoted to the solvability of the weighted Cauchy problem for fractional differential equations of arbitrary order, considering the Riemann-Liouville derivative. We show the equivalence between the weighted Cauchy problem and the Volterra integral equation in the space of Lebesgue integrable functions. Finally, we point out some discrepancies between the solutions for fractional and integer order case.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48533150","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}
Applying the two-operator approach, the mixed (Dirichlet-Neumann) boundary value problem for a second-order scalar elliptic differential equation with variable coefficient is reduced to several systems of Boundary Domain Integral Equations, briefly BDIEs. The two-operator BDIE sys- tem equivalence to the boundary value problem, BDIE solvability and invertibility of the boundary- domain integral operators are proved in the appropriate Sobolev spaces.
{"title":"Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side","authors":"T. Ayele, S. Mikhailov","doi":"10.1216/jie.2021.33.403","DOIUrl":"https://doi.org/10.1216/jie.2021.33.403","url":null,"abstract":"Applying the two-operator approach, the mixed (Dirichlet-Neumann) boundary value problem for a second-order scalar elliptic differential equation with variable coefficient is reduced to several systems of Boundary Domain Integral Equations, briefly BDIEs. The two-operator BDIE sys- tem equivalence to the boundary value problem, BDIE solvability and invertibility of the boundary- domain integral operators are proved in the appropriate Sobolev spaces.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42873150","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 study the unique continuation property for the N -dimensional BBM equations using Carleman estimates. We prove that if the solution of this equation vanishes in an open subset, then this solution is identically equal to zero in the horizontal component of the open subset.
{"title":"Carleman estimates and unique continuation property for N-dimensional Benjamin–Bona–Mahony equations","authors":"A. Esfahani, Y. Mammeri","doi":"10.1216/jie.2021.33.443","DOIUrl":"https://doi.org/10.1216/jie.2021.33.443","url":null,"abstract":"We study the unique continuation property for the N -dimensional BBM equations using Carleman estimates. We prove that if the solution of this equation vanishes in an open subset, then this solution is identically equal to zero in the horizontal component of the open subset.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47646849","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}
Summary: We study the uniqueness of solutions for certain partial integro-differential equations with the initial conditions in a Banach space. The results derived are new and based on Babenko’s approach, convolution and Banach’s contraction principle. We also include several examples for the illustration of main theorems.
{"title":"Uniqueness of the partial integro-differential equations","authors":"Chenkuan Li","doi":"10.1216/jie.2021.33.463","DOIUrl":"https://doi.org/10.1216/jie.2021.33.463","url":null,"abstract":"Summary: We study the uniqueness of solutions for certain partial integro-differential equations with the initial conditions in a Banach space. The results derived are new and based on Babenko’s approach, convolution and Banach’s contraction principle. We also include several examples for the illustration of main theorems.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" 54","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41252166","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 obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order integrals using Taylor and fractional Taylor series expansions and grouping terms in a novel manner. A fractional Taylor series expansion argument is utilized to provide fractional-order approximations for functions with minimal regularity. The resulting schemes allow for the approximation of functions in C [0, T ], where 0 < γ ≤ 5. A mild invertibility criterion is provided for the implicit schemes. Consistency and stability are proven separately for the whole-number-order approximations and the fractional-order approximations. The rate of convergence in the time variable is shown to be O(τ), 0 < γ ≤ 5 for u ∈ C [0, T ], where τ is the size of the partition of the time mesh. Crucially, the assumption of the integral kernel K being decreasing is not required for the scheme to converge in second-order and below approximations. Optimal convergence results are then proven for both sets of approximations, where fractional-order approximations can obtain up to whole-number rate of convergence in certain scenarios. Finally, numerical examples are provided that illustrate our findings. 2000 Mathematics Subject Classification. 26A33, 45D05, 65R20.
{"title":"Stable and convergent difference schemes for weakly singular convolution integrals","authors":"W. Davis, R. Noren","doi":"10.1216/jie.2021.33.271","DOIUrl":"https://doi.org/10.1216/jie.2021.33.271","url":null,"abstract":"We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order integrals using Taylor and fractional Taylor series expansions and grouping terms in a novel manner. A fractional Taylor series expansion argument is utilized to provide fractional-order approximations for functions with minimal regularity. The resulting schemes allow for the approximation of functions in C [0, T ], where 0 < γ ≤ 5. A mild invertibility criterion is provided for the implicit schemes. Consistency and stability are proven separately for the whole-number-order approximations and the fractional-order approximations. The rate of convergence in the time variable is shown to be O(τ), 0 < γ ≤ 5 for u ∈ C [0, T ], where τ is the size of the partition of the time mesh. Crucially, the assumption of the integral kernel K being decreasing is not required for the scheme to converge in second-order and below approximations. Optimal convergence results are then proven for both sets of approximations, where fractional-order approximations can obtain up to whole-number rate of convergence in certain scenarios. Finally, numerical examples are provided that illustrate our findings. 2000 Mathematics Subject Classification. 26A33, 45D05, 65R20.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49243223","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}
Asymptotic stability, uniform stability, integrability, and boundedness of solutions of Volterra integro-differential equations with and without constant retardation are investigated using a new type of Lyapunov-Krasovskii functionals. An advantage of the new functionals used here is that they eliminate using Gronwall’s inequality. Compared to related results in the literature, the conditions here are more general, simple, and convenient to apply. Examples to show the application of the theorems are included.
{"title":"Asymptotic behavior of solutions of Volterra integro-differential equations with and without retardation","authors":"J. Graef, O. Tunç","doi":"10.1216/jie.2021.33.289","DOIUrl":"https://doi.org/10.1216/jie.2021.33.289","url":null,"abstract":"Asymptotic stability, uniform stability, integrability, and boundedness of solutions of Volterra integro-differential equations with and without constant retardation are investigated using a new type of Lyapunov-Krasovskii functionals. An advantage of the new functionals used here is that they eliminate using Gronwall’s inequality. Compared to related results in the literature, the conditions here are more general, simple, and convenient to apply. Examples to show the application of the theorems are included.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43577718","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":"Global solvability for nonlinear nonautonomous evolution inclusions of Volterra-type and its applications","authors":"Yanghai Yu, Zhongjie Ma","doi":"10.1216/jie.2021.33.381","DOIUrl":"https://doi.org/10.1216/jie.2021.33.381","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":"193 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41277062","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 following fractional Laplacian equation (−∆)u = a(x)|u(x)|q−2u(x) + λb(x)|u(x)|p−2u(x) in (−1, 1), u > 0 in (−1, 1), u = 0 in R (−1, 1), where s ∈ (0, 1), λ > 0. Using variational methods, we show existence and multiplicity of positive solutions.
{"title":"Multiplicity of nontrivial solutions for a class of fractional elliptic equations","authors":"T. Kenzizi","doi":"10.1216/jie.2021.33.315","DOIUrl":"https://doi.org/10.1216/jie.2021.33.315","url":null,"abstract":"In this paper, we are concerned with the following fractional Laplacian equation (−∆)u = a(x)|u(x)|q−2u(x) + λb(x)|u(x)|p−2u(x) in (−1, 1), u > 0 in (−1, 1), u = 0 in R (−1, 1), where s ∈ (0, 1), λ > 0. Using variational methods, we show existence and multiplicity of positive solutions.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47371061","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}
The existence of three solutions for perturbed systems of impulsive non-linear fractional di¤erential equations together with Lipschitz continuous non-linear terms is discussed in this paper. The idea relies on variational methods. Moreover, an example is presented in order to summarize the feasibility and e¤ectiveness of the main results. 1. Introduction In this work, we aim to study the perturbed impulsive fractional di¤erential system: (1.1) 8><>>>>: tD i T (ai (t) 0D i t ui (t)) = Fui (t; u) + Gui (t; u) + hi (ui (t)) ; t 2 [0; T ] ; t 6= tj 4 tD i 1 T ( c 0D i t ui) (tj) = Iij (ui (tj)) ; j = 1; 2; ;m ui (0) = ui (T ) = 0 ; for 1 i n; where u = (u1; u2; ; un) ; n 2; 0 < i 1 for 1 i n; > 0; > 0; T > 0; ai 2 L1 ([0; T ]) ; ai = ess inf t2[0;T ] ai (t) > 0 i.e kaik1 = inf C 2 R; jaij C ; 0D i t and tD i T designate the left and right RiemannLiouville fractional derivatives of order i respectively, F;G : [0; T ] R ! R are quanti able with respect to t, for all u 2 R; continuously di¤erentiable in u, for approximately every t 2 [0; T ] such that F1) For every > 0 and every 1 i n, n supj j & jF ( ; )j ; supj j & jG ( ; )j o 2 L ([0; T ]) ; for any > 0 with = ( 1; 2; ; n) and j j = qPn i=1 2 i : F2) F (t; 0; ; 0) = 0; for every t 2 [0; T ] : hi : R ! R is a Lipschitz continuous function with the Lipschitz constant Li > 0; i:e: Date : October 10th, 2020. 1991 Mathematics Subject Classi cation. 35J60, 35B30, 35B40. Key words and phrases. AMS subject classi cations: (2000) 35J60 35B30 35B40. *Corresponding author. 1 10 Oct 2020 10:16:52 PDT 200120-Boulaaras Version 4 Submitted to J. Integr. Eq. Appl. 2 FARES KAMACHE, SALAH BOULAARAS ,3 ; RAFIK GUEFAIFIA jhi ( 1) hi ( 2)j Li j 1 2j for every 1; 2 2 R; satisfying hi (0) = 0 for 1 i n; Iij 2 C (R;R) for i = 1; ; n; j = 1; ;m; 0 = t0 < t1 < t2 < < tm < tm+1 = T: The operator is de ned as 4 tD i 1 T ( c 0D i t ui) (tj) =t D i 1 T ( c 0D i t ui) t+j t D i 1 T ( c 0D i t ui) t j where tD i 1 T ( c 0D i t ui) t+j = lim t!t+j tD i 1 T ( c 0D i t ui) (t) and tD i 1 T ( c 0D i t ui) t j = lim t!t j tD i 1 T ( c 0D i t ui) (t) and 0D i T represents the left Caputo fractional derivatives with an i order. While, Fui and Gui respectively represent the partial derivatives of F and G following ui for 1 i n: Recently, Fractional Di¤erential Equations (FDEs) have emerged as important approaches that can be used to describe numerous phenomena in several elds of science such as physics and engineering. In fact, many applications related to viscoelasticity and porous media, as well as electrochemistry and electromagnetic can be modeled through the FDEs (see [15], [27], [33], [32] and [41] ). In the last few years, many papers focused on the search of solutions to boundary value problems for FDEs, we may refer to ([42], [4], [5], [13], [16], [18], [20], [21], [28], [34], [37], [38], [39] and [40]) and the references therein. It is worthy of note that Zhang et al. [46] have checked the existence of di¤erent solutions for a
本文讨论了带有Lipschitz连续非线性项的脉冲非线性分式微分方程扰动系统三个解的存在性。这个想法依赖于变分方法。并通过实例说明了主要结果的可行性和有效性。1.引言在这项工作中，我们旨在研究受扰动的脉冲分数微分系统：（1.1）8><>>:tD i T（ai（T）0D i T ui（T））=Fui（T；u）+Gui（T；u）+hi（ui（T））；t2[0]；t]；t6＝tj 4 tD i 1 t（c0D i t ui）（tj）＝Iij（ui（tj））；j＝1；2.m ui（0）=ui（T）=0；对于1 i n；其中u=（u1；u2；；un）；n2；0＜i 1表示1 i n；＞0；>0；T>0；ai 2 L1（[0；T]）；ai=ess inf t2[0]；T]ai（T）>0，即kaik1=inf C2 R；jaij C；0D i t和tD i t分别表示i阶的左和右Riemann-Liouville分数导数，F；G:[0]；T]R！对于所有u2R，R相对于t是可量化的；在u中连续可微分，大约每t2[0；t]，使得F1）对于每>0和每1 i n，n supj j&jF（；）j；supj j&jG（；）j o 2L（[0；T]）；对于任何>0，其中=（1；2；；n）和j j=qPn i=1 2 i:F2）F（t；0；；0）=0；对于每t 2[0]；t]：嗨：R！R是Lipschitz常数Li>0的Lipschitz-连续函数；i： e：日期：2020年10月10日。1991年数学学科分类。35J60、35B30、35B40。关键词和短语。AMS科目分类：（2000）35J60 35B30 35B40*通讯作者。1 2020年10月10日10:16:52 PDT 200120 Boulaaras版本4提交给J.Integr。Eq.Appl。2票价卡马切，萨拉赫博拉拉斯，3；RAFIK GUEFAIFIA jhi（1）hi（2）j Li j1 2j每1；2 2 R；对于1 i n，满足hi（0）＝0；Iij 2 C（R；R）对于i＝1；nj=1；m；0=t00；>0；0<1；0D吨；以及tD T是左和右Riemann-Liouville阶分数积分；F:[0]；T]R！R是一个给定的函数，rF（t；x）是F在x上的梯度，F（t：）分别是超二次函数、渐近二次函数和次二次函数。在[25]中，使用[[9]的轻版本，定理2.1]，使用非线性函数F的适当振荡响应，将有可能确定系数的精确集合，其中系统（1.1）将整体接受任何非负任意函数G:[0]；T]R的几个弱解（定理3.1）！在[0；T]中可测量的R和在nity中生长的C类R，当选择足够小的时候。改变在nity处的振荡形状条件，用在零处的类似条件，将有可能得到一系列收敛到零的成对不同弱解（定理3.4）。在[50]中，当=0时，问题（1.1）允许至少两个非平凡和非负解（定理3.2）：在这方面，我们的目的是通过这项工作来研究具有Lipschitz连续脉冲效应的系统（1.1）的非平凡和非负解的丰富性。使用某些逻辑10十月2020 10:16:52 PDT 200120 Boulaaras版本4提交给J.Integr。Eq.Appl。
{"title":"Existence of weak solutions for a new class of fractional boundary value impulsive systems with Riemann–Liouville derivatives","authors":"R. Guefaifia, S. Boulaaras, Fares Kamache","doi":"10.1216/jie.2021.33.301","DOIUrl":"https://doi.org/10.1216/jie.2021.33.301","url":null,"abstract":"The existence of three solutions for perturbed systems of impulsive non-linear fractional di¤erential equations together with Lipschitz continuous non-linear terms is discussed in this paper. The idea relies on variational methods. Moreover, an example is presented in order to summarize the feasibility and e¤ectiveness of the main results. 1. Introduction In this work, we aim to study the perturbed impulsive fractional di¤erential system: (1.1) 8><>>>>: tD i T (ai (t) 0D i t ui (t)) = Fui (t; u) + Gui (t; u) + hi (ui (t)) ; t 2 [0; T ] ; t 6= tj 4 tD i 1 T ( c 0D i t ui) (tj) = Iij (ui (tj)) ; j = 1; 2; ;m ui (0) = ui (T ) = 0 ; for 1 i n; where u = (u1; u2; ; un) ; n 2; 0 < i 1 for 1 i n; > 0; > 0; T > 0; ai 2 L1 ([0; T ]) ; ai = ess inf t2[0;T ] ai (t) > 0 i.e kaik1 = inf C 2 R; jaij C ; 0D i t and tD i T designate the left and right RiemannLiouville fractional derivatives of order i respectively, F;G : [0; T ] R ! R are quanti\u0085able with respect to t, for all u 2 R; continuously di¤erentiable in u, for approximately every t 2 [0; T ] such that F1) For every > 0 and every 1 i n, n supj j & jF ( ; )j ; supj j & jG ( ; )j o 2 L ([0; T ]) ; for any > 0 with = ( 1; 2; ; n) and j j = qPn i=1 2 i : F2) F (t; 0; ; 0) = 0; for every t 2 [0; T ] : hi : R ! R is a Lipschitz continuous function with the Lipschitz constant Li > 0; i:e: Date : October 10th, 2020. 1991 Mathematics Subject Classi\u0085cation. 35J60, 35B30, 35B40. Key words and phrases. AMS subject classi\u0085cations: (2000) 35J60 35B30 35B40. *Corresponding author. 1 10 Oct 2020 10:16:52 PDT 200120-Boulaaras Version 4 Submitted to J. Integr. Eq. Appl. 2 FARES KAMACHE, SALAH BOULAARAS ,3 ; RAFIK GUEFAIFIA jhi ( 1) hi ( 2)j Li j 1 2j for every 1; 2 2 R; satisfying hi (0) = 0 for 1 i n; Iij 2 C (R;R) for i = 1; ; n; j = 1; ;m; 0 = t0 < t1 < t2 < < tm < tm+1 = T: The operator is de\u0085ned as 4 tD i 1 T ( c 0D i t ui) (tj) =t D i 1 T ( c 0D i t ui) t+j t D i 1 T ( c 0D i t ui) t j where tD i 1 T ( c 0D i t ui) t+j = lim t!t+j tD i 1 T ( c 0D i t ui) (t) and tD i 1 T ( c 0D i t ui) t j = lim t!t j tD i 1 T ( c 0D i t ui) (t) and 0D i T represents the left Caputo fractional derivatives with an i order. While, Fui and Gui respectively represent the partial derivatives of F and G following ui for 1 i n: Recently, Fractional Di¤erential Equations (FDEs) have emerged as important approaches that can be used to describe numerous phenomena in several \u0085elds of science such as physics and engineering. In fact, many applications related to viscoelasticity and porous media, as well as electrochemistry and electromagnetic can be modeled through the FDEs (see [15], [27], [33], [32] and [41] ). In the last few years, many papers focused on the search of solutions to boundary value problems for FDEs, we may refer to ([42], [4], [5], [13], [16], [18], [20], [21], [28], [34], [37], [38], [39] and [40]) and the references therein. It is worthy of note that Zhang et al. [46] have checked the existence of di¤erent solutions for a","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44185502","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 study a nonlinear Langevin equation involving a Caputo fractional derivatives of a function with respect to another function in a Banach space. Unlike previous papers, we assume the source function having a singularity. Under a regularity assumption of solution of the problem, we show that the problem can be transformed to a Volterra integral equation with two parameters Mittag-Leffler function in the kernel. Base on the obtained Volterra integral equation, we investigate the existence and uniqueness of the mild solution of the problem. Moreover, we show that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders, appropriate function, and friction constant. Meanwhile, a new Henry-Gronwall type inequality is established to prove the main results of the paper. Examples illustrating our results are also presented.
{"title":"Existence and continuity results for a nonlinear fractional Langevin equation with a weakly singular source","authors":"Nguyen Minh Dien","doi":"10.1216/jie.2021.33.349","DOIUrl":"https://doi.org/10.1216/jie.2021.33.349","url":null,"abstract":"We study a nonlinear Langevin equation involving a Caputo fractional derivatives of a function with respect to another function in a Banach space. Unlike previous papers, we assume the source function having a singularity. Under a regularity assumption of solution of the problem, we show that the problem can be transformed to a Volterra integral equation with two parameters Mittag-Leffler function in the kernel. Base on the obtained Volterra integral equation, we investigate the existence and uniqueness of the mild solution of the problem. Moreover, we show that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders, appropriate function, and friction constant. Meanwhile, a new Henry-Gronwall type inequality is established to prove the main results of the paper. Examples illustrating our results are also presented.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41521159","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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