Abstract
We are concerned with the existence of mild solutions to the fractional differential equation \(D^{\alpha }_{t}u(t)=Au(t)+J_{t}^{1-\alpha }f(t,u(t)),\;\;0<t\le T,\) with nonlocal conditions \(u_0=u(0)+g(u)\), where \(0<\alpha <1\), \(D^{\alpha }_{t}\) is the Caputo derivative, \(J_{t}^{1-\alpha }h(t):=\frac{1}{\Gamma (1-\alpha )}\int _{0}^{t}\frac{h(s)}{(t-s)^{\alpha }}ds\) is the fractional integral of order \(\alpha \) of the function h, and \(A:D(A)\subset X\rightarrow X\) is a linear operator which generates a compact analytic resolvent family \((R_{\alpha }(t))_{t\ge 0}\), X being a Banach space. We obtain our results using the Krasnoleskii’s fixed point theorem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Advanced Fractional Differential Equations. Nova Science Publishers, New York (2015)
Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69, 3692–3705 (2008)
Deng, K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 179, 630–637 (1993)
Dong, X., Wang, J., Zhou, Y.: On local problems for fractional differential equations in Banach spaces. Opusc. Math. 31(3) (2011). https://doi.org/10.7494/OpMath.2011.31.341
Fan, Z.: Characterization of compactness for resolvents and its applications. Appl. Math. Comput. 232, 60–67 (2014)
Lizama, C., N’Guérékata, G.M.: Mild solutions for abstract fractional differential equations. Appl. Anal. 92(8), 1731–1754 (2013)
Lizama, C., Pereira, A., Ponce, R.: On the compactness of fractional resolvent operator functions. Semigroup Forum (2016). https://doi.org/10.1007/s00233-016-9788-7
Lizama, C., Pozo, J.C.: Existence of mild solutions for semilinear integrodifferential equations with nonlocal conditions. Abstr. Appl. Anal. 2012(2012) (Art. ID 647103), 15 (2012). https://doi.org/10.1155/2012/647103
Miller, K., Ross, B.: An Introduction to the Fractional Calulus and Fractional Differential Equations. Wiley, New York (1993)
Mophou, G., N’Guérékata, G.M.: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl. Math. Comput. 216, 61–69 (2010)
N’Guérékata, G.M.: A Cauchy problem for some abstract fractional differential equations with nonlocal conditions. Nonlinear Anal. 70, 11–14 (2009)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 11, 4465–4475 (2010)
Acknowledgements
We would like to thank the referee for his/her careful reading and valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
N’Guérékata, G.M. (2020). An Existence Result for Some Fractional Evolution Equation with Nonlocal Conditions and Compact Resolvent Operator. In: Roy, P., Cao, X., Li, XZ., Das, P., Deo, S. (eds) Mathematical Analysis and Applications in Modeling. ICMAAM 2018. Springer Proceedings in Mathematics & Statistics, vol 302. Springer, Singapore. https://doi.org/10.1007/978-981-15-0422-8_3
Download citation
DOI: https://doi.org/10.1007/978-981-15-0422-8_3
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-0421-1
Online ISBN: 978-981-15-0422-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)