×

zbMATH — the first resource for mathematics

Semigroups and some nonlinear fractional differential equations. (English) Zbl 1046.34079
Summary: Equations of the form \[ \frac{d^\alpha u(t)} {dt^\alpha}= Au(t)+F\bigl(t,B_1(t), \dots,B_r(t)u(t)\bigr) \] are considered, where \(0<\alpha\leq 1\), \(A\) is a closed linear operator defined on a dense set in a Banach space \(E\) into \(E\), \(\{B_i(t)\), \(i=1,\dots,r,\;t\geq 0\}\) is a family of linear closed operators defined on dense sets in \(E\) into \(E\) and \(F\) is a given abstract nonlinear function defined on \([0,T]\times E^r\) with values in \(E\), \(T>0\). It is assumed that \(A\) generates an analytic semigroup. Under suitable conditions on the family of operators \(\{B_i(t):i=1, \dots,r,\;t\geq 0\}\) and on \(F\), we study the existence and uniqueness of the solution of the Cauchy problem for the considered equation. Some properties concerning the stability of solutions are obtained. We also give an application for nonlinear partial differential equations of fractional orders.

MSC:
34G20 Nonlinear differential equations in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S.D. Zaidman, Abstract differential equations, Research Notes in Mathematics, Pitman Advanced Publishing Program, San Francisco, Londonm Melbourne, 1979 · Zbl 0465.34002
[2] M.M. El-Borai, Semigroups and incorrect problems for evolution equations, 16th IMACS WORLD CONGRESS LAUSANNE, August 2000
[3] El-Borai, M.M., Some probability densities and fundamental solutions of fractional evolution equation, Chaos, solitons & fractals, 14, 433-440, (2002) · Zbl 1005.34051
[4] El-Borai, M.M.; Moustafa, O.L.; Michael, F.H., On the correct for mulation of a nonlinear differential equations in Banach space, Int. J. math., 22, 1, (1999)
[5] El-Borai, M.M., On the initial value problem for a partial differential equation with operator coefficients, Int. J. math. math. sci., 3, (1980) · Zbl 0457.35086
[6] M.M. El-Borai, On the correct formulation of Cauchy problem, Vectnik Moscow University, l968
[7] Gelfand, I.M.; Shilov, G.E., Generalized functions, (1959), Nauka Moscow, vol. 1 · Zbl 0091.11102
[8] Schneider, W.R.; Wayes, W., Fractional diffusion and wave equation, J. math. phys., 30, (1989)
[9] Feller, W., An introduction to probability theory and its application, (1971), John Wiley New York, vol. II · Zbl 0219.60003
[10] Gorenflo, R.; Mainardi, F., Fractional calculus and stable probability distributions, Arch. mech., 50, (1995) · Zbl 0934.35008
[11] Mainradi, F., Fractional relaxation oscillation and fractional diffusion-wave phenomena, Chaos, solitions & fractals, 7, (1996)
[12] Mijatovie, M.; Pilipovie, S.; Vujzovie, F., α-timeas integrated semigroups, J. math. anal. appl., 210, (1997)
[13] Eidelman, S.D., On the fundamental solution of parabolic systems, Math. sobornik, 38, (1956)
[14] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag Berlin, New York · Zbl 0516.47023
[15] Vladimirove, V.S., Generalized functions in mathematical physics, (1979), Mir Moscow
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.