Journal of Integral Equations and Applications

Existence of mild solutions for fractional evolution equations

Abstract

In this paper, we study the nonlocal Cauchy problems of fractional evolution equations with Riemann-Liouville derivative by considering an integral equation which is given in terms of probability density. By using the theory of Hausdorff measure of noncompactness, we establish various existence theorems of mild solutions for the Cauchy problems in the cases $C_0$ semigroup is compact or noncompact.

Article information

Source
J. Integral Equations Applications, Volume 25, Number 4 (2013), 557-586.

Dates
First available in Project Euclid: 31 January 2014

https://projecteuclid.org/euclid.jiea/1391192606

Digital Object Identifier
doi:10.1216/JIE-2013-25-4-557

Mathematical Reviews number (MathSciNet)
MR3161625

Zentralblatt MATH identifier
1304.34013

Citation

Zhou, Yong; Zhang, Lu; Shen, Xiao Hui. Existence of mild solutions for fractional evolution equations. J. Integral Equations Applications 25 (2013), no. 4, 557--586. doi:10.1216/JIE-2013-25-4-557. https://projecteuclid.org/euclid.jiea/1391192606

References

• J. Bana\`s and K. Goebel, Measure of noncompactness in Banach spaces, Marcel Dekker Inc., New York, 1980.
• D. Bothe, Multivalued perturbation of $m$-accretive differential inclusions, Israel. J. Math. 108 (1998), 109-138.
• K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York, 1985.
• D.J. Guo, V. Lakshmikantham and X.Z. Liu, Nonlinear integral equations in abstract spaces, Kluwer Academic, Dordrecht, 1996.
• H.-P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonl. Anal.: TMA 7 (1983), 1351-1371.
• E. Hernandez, D. O'Regan and Krishnan Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonl. Anal. 73 (2010), 3462-3471.
• A.A. Kilbas, H.M. Srivastava and J. Juan Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 2006.
• S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Diff. Equat. 252 (2012), 6163-6174.
• V. Lakshmikantham and S. Leela, Nonlinear differential equations in abstract spaces, Pergamon Press, New York, 1969.
• K. Li, J. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal. 263 (2012), 476-510.
• L. Liu, F. Guo, C. Wu and Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005), 638-649.
• F. Mainardi, P. Paraddisi and R. Gorenflo, Probability distributions generated by fractional diffusion equations, in Econophysics: An emerging science, J. Kertesz and I. Kondor, eds., Kluwer, Dordrecht, 2000.
• H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonl. Anal.: TMA 4 (1980), 985-999.
• A. Pazy, Semigroups of linear operators and applications to partial differential equations, in Appl. Math. Sci. 44, Springer-Verlag, Berlin, 1983.
• X.B. Shu, Y.Z. Lai and Y.M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonl. Anal.: TMA 74 (2011), 2003-2011.
• A.Z.-A.M. Tazali, Local existence theorems for ordinary differential equations of fractional order, in Ordinary and partial differential equations, Lect. Notes Math. 964 (1982), Springer, Dundee.
• J. Wang, Y. Zhou and M. Fečkan, Abstract Cauchy problem for fractional differential equations, Nonl. Dynam. 71 (2013), 685-700.
• R. Wang, D. Chen and T.J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Diff. Equat. 252 (2012), 202-235.
• Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonl. Anal.: RWA 11 (2010), 4465-4475.
• L.P. Zhu and G. Li, Nonlocal differential equations with multivalued perturbations in Banach spaces, Nonl. Anal. 69 (2008), 2843-2850. \noindentstyle