Fractional order semilinear Volterra integrodifferential equations in Banach spaces

Kexue Li



Sufficient conditions are established for the existence results of fractional order semilinear Volterra integrodifferential equations in Banach spaces. Results are obtained by using the theory of fractional cosine families and fractional powers of operators.


Fractional integrodifferential differention; fractional cosine family; fractional powers of operators

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E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, PhD Thesis, Eindhoven University of Technology, 2001.

C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum 80 (2010), 121–142.

H. Engler, Weak solutions of a class of quasilinear hyperbolic integro-differential equations describing viscoelastic materials, Arch. Rational Mech. Anal. 113 (1991), 1–38.

W. Fitzgibbon, Semilinear integrodifferential equations in Banach space, Nonlinear Anal. 4 (1980), 745–760.

R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc. 273 (1982), 333–349.

E. Hernandez, C α -classical solutions for abstract non-autonomous integro-differential equations, Proc. Amer. Math. Soc. 139 (2011), 4307–4318.

A. Jawahdou, Mild solutions of functional semilinear evolution Volterra integrodifferential equations on an unbounded interval, Nonlinear Anal. 74 (2011), 7325–7332.

V. Keyantuo and C. Lizama, Hölder continuous solutions for integro-differential equations and maximal regularity, J. Differential. Equations 230 (2006), 634–660.

V. Keyantuo and C. Lizama, On a connection between powers of operators and fractiona Cauchy problems, J. Evol. Equ. 12 (2012), 245–265.

C. Lang and J. Chang, Local existence for nonlinear Volterra integrodifferential equations with infinite delay, Nonlinear Anal. 68 (2008), 2943–2956.

M. Li, C. Chen and F.B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal. 259 (2010), 2702–2726.

S.O. Londen, An existence result on a volterra equation in a Banach space, Trans. Amer. Math. Soc. 235 (1978), 285–304.

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London 2010.

E. Mainini and G. Mola, Exponential and polynomial decay for first order linear Volterra equations, Quart. Appl. Math. Vol. LXVII (2009), 93–111.

R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1–77.

M. Miklavĉiĉ, Applied Functional Analysis and Partial Differential Equations, World Scientific, Singapore, 1998.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York, 1983.

J. Peng and K. Li, A novel characteristic of solution operator for the fractional abstract Cauchy problem, J. Math. Anal. Appl. 385 (2012), 786–796.

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

J. Prüss, Bounded solutions of Volterra equations, SIAM J. Math. Anal. 19 (1988) 133–149.

J. Prüss, Positivity and regularity of hyperbolic Volterra equations in Banach spaces, Math. Ann. 279 (1987), 317–344.

J. Prüss, On linear Volterra equations of parabolic type in Banach spaces, Trans. Amer. Math. Soc. 301 (1987), 691–721.

C.C. Travis and G.F. Webb, An abstract second order semilinear volterra integrodifferential equation, SIAM J. Math. Anal. 10 (1979), 412–424.


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