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November  2013, 12(6): 2753-2772. doi: 10.3934/cpaa.2013.12.2753

On general fractional abstract Cauchy problem

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China, China

2. 

Department of Basic Courses, Xi'an Technological University, North Institute of Information Engineering, Xi'an 710025, China

Received  October 2012 Revised  January 2013 Published  May 2013

This paper is concerned with general fractional Cauchy problems of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ in infinite-dimensional Banach spaces. A new notion, named general fractional resolvent of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ is developed. Some of its properties are obtained. Moreover, some sufficient conditions are presented to guarantee that the mild solutions and strong solutions of homogeneous and inhomogeneous general fractional Cauchy problem exist. An illustrative example is presented.
Citation: Zhan-Dong Mei, Jigen Peng, Yang Zhang. On general fractional abstract Cauchy problem. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2753-2772. doi: 10.3934/cpaa.2013.12.2753
References:
[1]

W. Arendt, C. Batty, M. Hiever and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems," Monogr. Math., vol. 96, Birh$\ddota$user, Basel, 2001.

[2]

E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," University Press Facilities, Eindhoven University of Technology, 2001.

[3]

M. Caputo, "Elasticita Dissipacione," Bologna: Zanichelli, 1969.

[4]

C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142. doi: 10.1007/s00233-009-9184-7.

[5]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255. doi: 10.1016/j.jde.2003.12.002.

[6]

A. Erdé, "Higher Transcendental Functions," vol. 3, McGraw-Hill, New Yourk, 1955.

[7]

R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives, J. Phys. Chem. B, 104 (2000), 3914-3917. doi: 10.1021/jp9936289.

[8]

R. Hilfer, Fractional time evolution, in "Applications of Fractional Calculus in Physics" (R. Hilfer ed.), World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87-130. doi: 10.1142/9789812817747_0002.

[9]

R. Hilfer, Fractional calculus and regular variation in thermodynamics, In "Applications of Fractional Calculus in Physics" (R. Hilfer ed.), World Scientific, Singapore (2000).

[10]

R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399-408. doi: 10.1016/S0301-0104(02)00670-5.

[11]

R. Hilfer, Y. Luchko and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12 (2009), 299-318.

[12]

K. X. Li and J. G. Peng, Fractional resolvents and fractional evolution equations, Applied Mathematics Letters, 25 (2012), 808-812. doi: 10.1016/j.aml.2011.10.023.

[13]

M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726. doi: 10.1016/j.jfa.2010.07.007.

[14]

M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy Problems on bounded domains, Ann. Anal., 37 (2009), 979-1007. doi: 10.1214/08-AOP426.

[15]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[16]

K. S. Miller and B. Ross, "An Introduction to the Fractional Differential Equations," New York: Wiley, 1993.

[17]

F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: a tutorial survey, Fract. Calc. Appl. Anal., 10 (2007), 269-308.

[18]

R. R. Nigmatullin, To the theoretical explanation of the "universal response", Phys. Sta. Sol. (b), 123 (1984), 739-745. doi: 10.1002/pssb.2221230241.

[19]

K. B. Oldham and J. Spanier, "The Fractional Calculus," New York: Academic, 1974.

[20]

I. Podlubny, "Fractional Differential Equations," Academic Press, New Yourk, 1999.

[21]

J. Prüs, "Evolutionary Integral Equations and Applications," Birkh$\ddota$ser, Basel, Berlin, 1993.

[22]

T. Sandev, R. Metzler and Ž. Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative, J. Phys. A: Math. Theor., 44 (2011), 255203 (21pp). doi: 10.1088/1751-8113/44/25/255203.

[23]

T. Sandev and Ž. Tomovski, The general time fractional Fokker-Planck equation with a constant external force, Proc. Symposium on Fractional Signals and Systems, Coimbra, Portugal, 4-5 November 2011, pages 27-39.

[24]

H. M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198-210. doi: 10.1016/j.amc.2009.01.055.

[25]

Ž. Tomovski, R. Hilferb and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms and Special Functions, 21 (2010), 797-814. doi: 10.1080/10652461003675737.

[26]

G. M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Phy. D., 76 (1994), 110-122. doi: 10.1016/0167-2789(94)90254-2.

show all references

References:
[1]

W. Arendt, C. Batty, M. Hiever and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems," Monogr. Math., vol. 96, Birh$\ddota$user, Basel, 2001.

[2]

E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," University Press Facilities, Eindhoven University of Technology, 2001.

[3]

M. Caputo, "Elasticita Dissipacione," Bologna: Zanichelli, 1969.

[4]

C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142. doi: 10.1007/s00233-009-9184-7.

[5]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255. doi: 10.1016/j.jde.2003.12.002.

[6]

A. Erdé, "Higher Transcendental Functions," vol. 3, McGraw-Hill, New Yourk, 1955.

[7]

R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives, J. Phys. Chem. B, 104 (2000), 3914-3917. doi: 10.1021/jp9936289.

[8]

R. Hilfer, Fractional time evolution, in "Applications of Fractional Calculus in Physics" (R. Hilfer ed.), World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87-130. doi: 10.1142/9789812817747_0002.

[9]

R. Hilfer, Fractional calculus and regular variation in thermodynamics, In "Applications of Fractional Calculus in Physics" (R. Hilfer ed.), World Scientific, Singapore (2000).

[10]

R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399-408. doi: 10.1016/S0301-0104(02)00670-5.

[11]

R. Hilfer, Y. Luchko and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12 (2009), 299-318.

[12]

K. X. Li and J. G. Peng, Fractional resolvents and fractional evolution equations, Applied Mathematics Letters, 25 (2012), 808-812. doi: 10.1016/j.aml.2011.10.023.

[13]

M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726. doi: 10.1016/j.jfa.2010.07.007.

[14]

M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy Problems on bounded domains, Ann. Anal., 37 (2009), 979-1007. doi: 10.1214/08-AOP426.

[15]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[16]

K. S. Miller and B. Ross, "An Introduction to the Fractional Differential Equations," New York: Wiley, 1993.

[17]

F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: a tutorial survey, Fract. Calc. Appl. Anal., 10 (2007), 269-308.

[18]

R. R. Nigmatullin, To the theoretical explanation of the "universal response", Phys. Sta. Sol. (b), 123 (1984), 739-745. doi: 10.1002/pssb.2221230241.

[19]

K. B. Oldham and J. Spanier, "The Fractional Calculus," New York: Academic, 1974.

[20]

I. Podlubny, "Fractional Differential Equations," Academic Press, New Yourk, 1999.

[21]

J. Prüs, "Evolutionary Integral Equations and Applications," Birkh$\ddota$ser, Basel, Berlin, 1993.

[22]

T. Sandev, R. Metzler and Ž. Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative, J. Phys. A: Math. Theor., 44 (2011), 255203 (21pp). doi: 10.1088/1751-8113/44/25/255203.

[23]

T. Sandev and Ž. Tomovski, The general time fractional Fokker-Planck equation with a constant external force, Proc. Symposium on Fractional Signals and Systems, Coimbra, Portugal, 4-5 November 2011, pages 27-39.

[24]

H. M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198-210. doi: 10.1016/j.amc.2009.01.055.

[25]

Ž. Tomovski, R. Hilferb and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms and Special Functions, 21 (2010), 797-814. doi: 10.1080/10652461003675737.

[26]

G. M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Phy. D., 76 (1994), 110-122. doi: 10.1016/0167-2789(94)90254-2.

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