一类混合型分数阶半线性积分-微分方程解的存在性

数学物理学报 ›› 2019, Vol. 39 ›› Issue (6): 1334-1341.

一类混合型分数阶半线性积分-微分方程解的存在性

朱波^1,*(),韩宝燕²,刘立山³

¹ 山东财经大学数学与数量经济学院济南 250014
² 山东工艺美术学院公共教学部济南 250014
³ 曲阜师范大学数学科学学院山东曲阜 273165

收稿日期:2018-08-29 出版日期:2019-12-26 发布日期:2019-12-28
通讯作者: 朱波 E-mail:zhubo207@163.com
基金资助:
山东省高校科技计划项目(J16LI14);国家自然科学基金(11871302)

Existence of Mild Solutions for a Class of Fractional Semilinear Integro-Differential Equation of Mixed Type

Bo Zhu^1,*(),Baoyan Han²,Lishan Liu³

¹ School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014
² Common Course Teaching Department, Shandong University of Art and Design, Jinan 250014
³ School of Mathematical Sciences, Qufu Normal University, Shandong Qufu 273165

Received:2018-08-29 Online:2019-12-26 Published:2019-12-28
Contact: Bo Zhu E-mail:zhubo207@163.com
Supported by:
the PSDPHESTP(J16LI14);the NSFC(11871302)

摘要/Abstract

摘要：

该文利用非紧性测度、β-预解族、k-集压缩原理研究了一类混合型分数阶半线性积分-微分方程温和解的存在性.众所周知，利用k-集压缩原理证明解的存在性时需要单独给出附加条件来保证压缩系数小于1，而该文不需要单独附加保证压缩系数小于1的条件.在更一般的条件下证明了方程解的存在性.文章最后给出了一个例子说明该文主要结果的应用.

关键词: 混合型分数阶半线性积分-微分方程, k-集压缩, 非紧性测度, β-预解族

Abstract:

In this paper, the authors studied the existence results of the mild solutions for a class of fractional semilinear integro-differential equation of mixed type by using the measure of noncompactness, k-set contraction and β-resolvent family. It is well known that the k-set contraction requires additional condition to ensure the contraction coefficient 0 < k < 1. We don't require additional condition to ensure the contraction coefficient 0 < k < 1. An example is introduced to illustrate the main results of this paper.

Key words: Fractional semilinear integro-differential equation of mixed type, k-Set contraction, Measure of noncompactness, β-Resolvent family

中图分类号:

O175

朱波,韩宝燕,刘立山. 一类混合型分数阶半线性积分-微分方程解的存在性[J]. 数学物理学报, 2019, 39(6): 1334-1341.

Bo Zhu,Baoyan Han,Lishan Liu. Existence of Mild Solutions for a Class of Fractional Semilinear Integro-Differential Equation of Mixed Type[J]. Acta mathematica scientia,Series A, 2019, 39(6): 1334-1341.

参考文献

1	Kamrani M Jamshidi N Implicit Euler approximation of stochastic evolution equations with fractional Brownian motion Commun Nonlinear Sci Numer Simulat 2017, 44: 1- 10
2	Wang J R Stability of noninstantaneous impulsive evolution equations Appl Math Lett 2017, 73: 157- 162
3	Vázquez J The Dirichlet problem for the fractional $p$-Laplacian evolution equation J Diff Equ 2016, 260: 6038- 6056
4	Ge F Zhou H Kou C Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique Appl Math Comput 2016, 275: 107- 120
5	Zhu B Liu L S Wu Y H Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations Fract Calcu Appl Anal 2017, 20: 1338- 1355
6	Yang D Wang J R O'Regan D On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses CR Math 2018, 356 (2): 150- 171
7	Zhu B Liu L S Wu Y H Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay Comput Math Appl 2019, 78: 1811- 1818
8	Zhang X G Liu L S Wu Y H Wiwatanapataphee B The spectral analysis for a singular fractional differential equation with a signed measure Aplied Math Comput 2015, 257: 252- 263
9	Zhu B Liu L S Wu Y H Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay Appl Math Lett 2016, 61: 73- 79
10	Yan Z M Lu F X Approximate controllability of a multi-valued fractional impulsive stochastic partial integro-differential equation with infinite delay Appl Math Comput 2017, 292: 425- 447
11	Liang J Yang H Controllability of fractional integro-differential evolution equations with nonlocal conditions Appl Math Comput 2015, 254: 20- 29
12	Liu L S Wu C X Guo F Existence theorems of global solutions of initial value problems for nonlinear integro-differential equations of mixed type in banach spaces and applications Comput Math Appl 2004, 47: 13- 22
13	Liu L S Guo F Wu C X Wu Y H Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces J Math Anal Appl 2005, 309: 638- 649
14	Shu X B Wang Q Q The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order $1 < \alpha < 2$ Comput Math Appl 2012, 64: 2100- 2109
15	Rashid M Al-Omari A Local and global existence of mild solutions for impulsive fractional semi-linear integro-differential equation Commun Nonlinear Sci Numer Simul 2011, 16: 3493- 3503
16	Gou H D Li B L Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup Commun Nonlinear Sci Numer Simulat 2017, 42: 204- 214
17	Gou H D Li B L Existence of mild solutions for fractional nonautonomous evolution equations of Sobolev type with delay J Inequ Appl 2017, 2017: 252
18	Araya D Lizama C Almost automorphic mild solutions to fractional differential equations Nonlinear Anal 2008, 69: 3692- 3705
19	Debbouche A Baleanu D Controllability of Fractional Evolution Nonlocal Impulsive Quasilinear Delay Integro-Differential Systems Comput Math Appl 2011, 62: 1442- 1450
20	Pazy A. Semigroup of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983
21	Guo D J, Lakshmikantham V, Liu X Z. Nonlinear Integral Equations in Abstract Spaces. Dordrecht: Kluwer Academic, 1996
22	Liu L S Iterative method for solutions and coupled quasi-solutions of nonlinear Fredholm integral equations in ordered Banach spaces Indian J Pure Appl Math 1996, 27: 959- 972
23	Liu L S Wu Y H Zhang X G On well-posedness of an initial value problem for nonlinear second-order impulsive integro-differential equations of Volterra type in Banach spaces J Math Anal Appl 2006, 317: 634- 649

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[6]	夏治南. Volterra-Stieltjes型泛函积分方程解的存在性及渐近行为[J]. 数学物理学报, 2016, 36(1): 130-143.
[7]	谢胜利. 有阻尼的二阶脉冲无穷时滞泛函微分方程解的存在性[J]. 数学物理学报, 2015, 35(1): 97-109.
[8]	尹婷婷, 胡长松, 吴智宇. 非扩张映射的不动点定理[J]. 数学物理学报, 2013, 33(5): 906-911.
[9]	李烨, 刘立山. Banach空间非线性脉冲Volterra型积分方程的可解性[J]. 数学物理学报, 2011, 31(4): 1097-1104.
[10]	张玲忠, 李永祥. Banach空间非线性Sturm-Liouville边值问题的正解[J]. 数学物理学报, 2009, 29(3): 784-793.
[11]	张新光;刘立山. Banach 空间中一阶脉冲积分-微分方程初值问题整体解的存在性[J]. 数学物理学报, 2008, 28(2): 383-392.
[12]	谢胜利. Banach空间二阶非线性脉冲积分-微分方程初值问题[J]. 数学物理学报, 2007, 27(1): 138-145.
[13]	陈芳启1,2 陈予恕2 吴志强2. Banach空间一类非线性微分积分方程的整体解[J]. 数学物理学报, 2000, 20(4): 506-512.
[14]	路慧芹刘立山. Banach空间非线性脉冲Volterra型积分方程整体解的存在性定理及应用[J]. 数学物理学报, 2000, 20(1): 101-108.
[15]	王建国. 二阶非线性积-微分方程的初值问题的上下解方法[J]. 数学物理学报, 1999, 19(5): 589-596.