MIXED MONOTONE ITERATIVE TECHNIQUE FOR HILFER FRACTIONAL EVOLUTION EQUATIONS WITH NONLOCAL CONDITIONS

Haide Gou; Yongxiang Li; Qixiang Li

doi:10.11948/20190211

2020 Volume 10 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(5): 1823-1847

Next Article Previous Article

Haide Gou, Yongxiang Li, Qixiang Li. MIXED MONOTONE ITERATIVE TECHNIQUE FOR HILFER FRACTIONAL EVOLUTION EQUATIONS WITH NONLOCAL CONDITIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1823-1847. doi: 10.11948/20190211

Citation:

Haide Gou, Yongxiang Li, Qixiang Li. MIXED MONOTONE ITERATIVE TECHNIQUE FOR HILFER FRACTIONAL EVOLUTION EQUATIONS WITH NONLOCAL CONDITIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1823-1847. doi: 10.11948/20190211

MIXED MONOTONE ITERATIVE TECHNIQUE FOR HILFER FRACTIONAL EVOLUTION EQUATIONS WITH NONLOCAL CONDITIONS

1.
Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

Corresponding authors: Email address:842204214@qq.com(H. Gou); Email address:liyxnwnu@163.com(Y. Li); Email address:liqixiang_19@163.com(Q. Li)
Fund Project: The authors were supported by National Natural Science Foundation of China (11661071)

Abstract

Abstract

The purpose of this paper is concerned with the existence of mild $ L $-quasi-solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach spaces $ E $. By employing mixed monotone iterative technique, measure of noncompactness and Sadovskii's fixed point theorem, we obtain the existence of mild $ L $-quasi-solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provide to illustrate the feasibility of our main results.
- Mixed monotone iterative technique /
- coupled L-quasi-upper and lower solutions /
- Hilfer fractional derivative /
- measure of noncompactness
MSC: 26A33, 34K30, 34K45, 47D06

FullText(HTML)

References

[1]	H. M. Ahmed, M. M. EI-Borai, Hilfer fractional stochastic integro-differential equations, Appl. Math. Comput., 2018, 331, 182-189. Google Scholar
[2]	H. M. Ahmed, M. M. EI-Borai, H. M. EI-Owaidy, A. S. Ghanem, Impulsive Hilfer fractional differential equations, Advances in Difference Equations., 2018, 226. Google Scholar
[3]	S. Agarwal, D. Bahuguna, Existence of solutions to Sobolev-type paritial neutral differential equations, J. Appl. Math. Stoch. Anal., 2006, Art. ID 16308, 10pp. Google Scholar
[4]	K. Balachandran, S. Kiruthika, J. J. Trujillo, On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Comput. Math. Appl., 2011, 62, 1157-1165. doi: 10.1016/j.camwa.2011.03.031 CrossRef Google Scholar
[5]	K. Balachandran, J. P. Dauer, Controllability of functional differential systems of Sobolev type in Banach spaces, Kybernetika., 1998, 34, 349-357. Google Scholar
[6]	P. Chen, Y. Li, Mixed Monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces, Nonlinear Analysis, 2011, 74, 3578-3588. doi: 10.1016/j.na.2011.02.041 CrossRef Google Scholar
[7]	L. Debnath, D. Bhatta, Intergral transforms and their applications, Second edition. Chapman Hall CRC. Boca Raton, FL, 2007. Google Scholar
[8]	Y. Du, Fixed points of increasing operators in order Banach spaces and applications. Appl. Anal., 1990, 38, 1-20. doi: 10.1080/00036819008839957 CrossRef Google Scholar
[9]	K. M. Furati, M. D. Kassim, N.e-. Tatar, Existence and uniqueness for a problem involving Hilfer factional derivative, Comput. Math. Appl., 2012, 64, 1616-1626. doi: 10.1016/j.camwa.2012.01.009 CrossRef Google Scholar
[10]	D. Guo, J. Sun, Ordinary Differential Equations in Abstract Spaces. Shandong Science and Technology, Jinan, 1989.(in Chinese) Google Scholar
[11]	H. Gu, J. J. Trujillo, Existence of mild solution for evolution equation with Hilfre fractional derivative, Appl. Math. Comput., 2015, 257, 344-354. Google Scholar
[12]	H. Gou, B. Li, Study on the mild solution of Sobolev type Hilfer fractional evolution equations with boundary conditions, Chaos, Solitons Fractals., 2018, 112, 168-179. doi: 10.1016/j.chaos.2018.05.007 CrossRef Google Scholar
[13]	H. R. Heinz, On the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 1983, 71, 1351-1371. Google Scholar
[14]	R. Hilfer, Applications of Fractional Caiculus in Physics, World Scientific, Singapore, 2000. Google Scholar
[15]	R. Hilfer, Fractional Time Evolution, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. Google Scholar
[16]	T. D. Ke, C. T. Kinh, Generalized cauchy problem involving a class of degenerate fractional differential equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis., 2014, 1, 1-24. Google Scholar
[17]	Y. Li, The positive solutions of abstract semilinear evolution equations and their applications, Acta Math. Sin., 1996, 39(5), 666-672. (in Chinese) Google Scholar
[18]	L. S. Liu, F. Guo, C. X. Wu, Y. H. Wu, Existence theorems of global solutions for nonlinear Volterra type integral eqautions in Banach spaces, J. Math. Anal. App., 2005, 309, 638-649. doi: 10.1016/j.jmaa.2004.10.069 CrossRef Google Scholar
[19]	F. Li, J. Liang, H. Xu, Existence of mild solutions for fractioanl integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl., 2012, 391, 510-525. doi: 10.1016/j.jmaa.2012.02.057 CrossRef Google Scholar
[20]	J. Liang, H. Yang, Controllability of fractional integro-differential evolution equations with nonlocal conditions, Appl. Math. Comput., 2015, 254, 20-29. Google Scholar
[21]	J. Mu, Monotone iterative technique for fractional evolution equations in Banach spaces, J. Appl. Math., 2011, Art. ID 767186, 13 pp. Google Scholar
[22]	F. Mainardi, P. Paradisi, R. Corenflo, Probability distributions generated by fractional diffusion equations, in: J. Kertesz, I. Kondor (Eds.), Econophysics: An Emerging Science, Kluwer, Dordrecht, 2000. Google Scholar
[23]	J. Mu, Y. Li, Monotone interative technique for impulsive fractional evolution equations, Journal of Inequalities and Applications., 2011, 125. Google Scholar
[24]	J. Mu, Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions, Boundary Value Problem., 2012, 71. Google Scholar
[25]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, Berlin, 1983. Google Scholar
[26]	X. B. Shu, F. Xu, Upper and lower solution method for fractional evolution equations with order $1 < \alpha < 2$, J. Korean Math. Soc., 2014, 51(6), 1123-1139. doi: 10.4134/JKMS.2014.51.6.1123 CrossRef $1 < \alpha < 2$" target="_blank">Google Scholar
[27]	V. Singh, D. N. Pandey, A study od Sobolev Trpe Fractional Impulsive Differential System with Fractional Nonlocal Conditions, Int. J. Appl. Comput. Math., 2018, 4:12. doi: 10.1007/s40819-017-0453-y CrossRef Google Scholar
[28]	J. Wang, Y. Zhou, M. Fečkan, Abstract Cauchy problem for fractional differential equations, Nonlinear Dyn., 2013, 74, 685-700. doi: 10.1007/s11071-013-0997-2 CrossRef Google Scholar
[29]	J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal., 12(2011), 262-272. Google Scholar
[30]	M. Yang, Q. Wang, Existence of mild solutions for a class of Hilfer fractional evolution eqautions with nonlocal conditions, Fract. Calc. Appl. Anal., 2017, 20(3), 679-705. Google Scholar
[31]	H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its applications to a fractional differential equation, J. Math. Anal. Appl., 2007, 328, 1075-1081. doi: 10.1016/j.jmaa.2006.05.061 CrossRef Google Scholar
[32]	Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput Math Appl., 2010, 59, 1063-1077. doi: 10.1016/j.camwa.2009.06.026 CrossRef Google Scholar