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Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent
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Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data
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 |
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). Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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). Google Scholar |
[20] |
I. Podlubny, "Fractional Differential Equations,", Academic Press, (1999). Google Scholar |
[21] |
J. Prüs, "Evolutionary Integral Equations and Applications,", Birkh$\ddota$ser, (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).
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. 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.
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). Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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). Google Scholar |
[20] |
I. Podlubny, "Fractional Differential Equations,", Academic Press, (1999). Google Scholar |
[21] |
J. Prüs, "Evolutionary Integral Equations and Applications,", Birkh$\ddota$ser, (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).
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. 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.
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. |
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