• Previous Article
    Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data
  • CPAA Home
  • This Issue
  • Next Article
    Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent
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 & 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., (2001).

[2]

E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces,", University Press Facilities, (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. 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. doi: 10.1016/j.jde.2003.12.002.

[6]

A. Erdé, "Higher Transcendental Functions,", vol. 3, (1955).

[7]

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

[8]

R. Hilfer, Fractional time evolution,, in, (2000), 87. doi: 10.1142/9789812817747_0002.

[9]

R. Hilfer, Fractional calculus and regular variation in thermodynamics,, In, (2000).

[10]

R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials,, Chem. Phys., 284 (2002), 399. 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.

[12]

K. X. Li and J. G. Peng, Fractional resolvents and fractional evolution equations,, Applied Mathematics Letters, 25 (2012), 808. 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. 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. 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. 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.

[18]

R. R. Nigmatullin, To the theoretical explanation of the "universal response",, Phys. Sta. Sol. (b), 123 (1984), 739. 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, (1999).

[21]

J. Prüs, "Evolutionary Integral Equations and Applications,", Birkh$\ddota$ser, (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). 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, (2011), 4.

[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. 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. doi: 10.1080/10652461003675737.

[26]

G. M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos,, Phy. D., 76 (1994), 110. 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., (2001).

[2]

E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces,", University Press Facilities, (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. 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. doi: 10.1016/j.jde.2003.12.002.

[6]

A. Erdé, "Higher Transcendental Functions,", vol. 3, (1955).

[7]

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

[8]

R. Hilfer, Fractional time evolution,, in, (2000), 87. doi: 10.1142/9789812817747_0002.

[9]

R. Hilfer, Fractional calculus and regular variation in thermodynamics,, In, (2000).

[10]

R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials,, Chem. Phys., 284 (2002), 399. 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.

[12]

K. X. Li and J. G. Peng, Fractional resolvents and fractional evolution equations,, Applied Mathematics Letters, 25 (2012), 808. 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. 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. 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. 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.

[18]

R. R. Nigmatullin, To the theoretical explanation of the "universal response",, Phys. Sta. Sol. (b), 123 (1984), 739. 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, (1999).

[21]

J. Prüs, "Evolutionary Integral Equations and Applications,", Birkh$\ddota$ser, (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). 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, (2011), 4.

[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. 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. doi: 10.1080/10652461003675737.

[26]

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

[1]

Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control & Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016

[2]

Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025

[3]

Thaís Jordão, Xingping Sun. General types of spherical mean operators and $K$-functionals of fractional orders. Communications on Pure & Applied Analysis, 2015, 14 (3) : 743-757. doi: 10.3934/cpaa.2015.14.743

[4]

Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541

[5]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615

[6]

Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333

[7]

Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124

[8]

Claudianor O. Alves, Giovany M. Figueiredo, Gaetano Siciliano. Ground state solutions for fractional scalar field equations under a general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2199-2215. doi: 10.3934/cpaa.2019099

[9]

Imen Manoubi. Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837

[10]

Tatsien Li, Wancheng Sheng. The general multi-dimensional Riemann problem for hyperbolic systems with real constant coefficients. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 737-744. doi: 10.3934/dcds.2002.8.737

[11]

Daliang Zhao, Yansheng Liu, Xiaodi Li. Controllability for a class of semilinear fractional evolution systems via resolvent operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 455-478. doi: 10.3934/cpaa.2019023

[12]

Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541

[13]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067

[14]

Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947

[15]

Jean Daniel Djida, Juan J. Nieto, Iván Area. Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 609-627. doi: 10.3934/dcdss.2020033

[16]

Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605

[17]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062

[18]

Eddye Bustamante, José Jiménez Urrea, Jorge Mejía. The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1177-1203. doi: 10.3934/cpaa.2019057

[19]

Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709

[20]

Zhongyuan Liu. Concentration of solutions for the fractional Nirenberg problem. Communications on Pure & Applied Analysis, 2016, 15 (2) : 563-576. doi: 10.3934/cpaa.2016.15.563

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]