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

On general fractional abstract Cauchy problem

Zhan-Dong Mei 1, , Jigen Peng 1, and  Yang Zhang 2,

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.
Keywords: General fractional abstract Cauchy problem, general Riemann-Liouville fractional derivative, general fractional resolvent..
Mathematics Subject Classification: Primary: 34A08; Secondary: 47D0.
Citation: Zhan-Dong Mei, Jigen Peng, Yang Zhang. On general fractional abstract Cauchy problem. Communications on Pure & 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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

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