Fractional integrated semi groups and nonlocal Cauchy problem for abstract nonlinear fractional differential equations

Mahmoud Mohammed El-Borai; Khairia El-Said El-Nadi

Mahmoud Mohammed El-Borai Professor of Mathematics, Faculty of Science Department of Mathematics and Computer Science, Alexandria University, Egypt
Khairia El-Said El-Nadi Professor of Mathematics, Faculty of Science Department of Mathematics and Computer Science, Alexandria University, Egypt

Keywords: Integrated semi groups-Nonlinear fractional abstract differential equations- Nonlocal Cauchy problem.

Abstract

Some classes of fractional abstract differential equations with α-integrated semi groups are studied in Banach space. The existence of a unique solution of the nonlocal Cauchy problem is studied. Some properties are given.

Downloads

Download data is not yet available.

References

W. Arendt, Resolvent positive operators and integrated semi groups,

Proc.London.Math.Soc.54(1984),321-349.

M. Hieber, Integrated semi groups and differential operators in L$^{p}$Spaces,Math.Ann.201,(1991),1-16.

L. Hormander, Estimates for translation invariant operators in L$^{p}$ space , Act.Math.104,(1960),93-139.

WR.Schneider and W-wayes, Fractional diffusion and wave equation, J.Math.ics,30(1989),134-143.

W.Wayes, The fractional diffusion equation, J.Math.Phys.,27(1986),2782-2786.

O. Neuarander, Integrated semi group and their application to complete second order Cauchy problem, semi group Forum.,38 (1989),233-251.

R. Vggdalic, Representation theorem for integrated semi group, Sarajevo Journal of Mathematics, 14(2005),243-250.

A. Ducrot, P. Matal and K. Provost, Integrated semi groups and Parabolic equations . part I: linear perturbation of almost sectorial operators, J.Evau.10(2010),263-291.

Mahmoud M.El-Borai, Khairia El-Said El-Nadi, Integrated semi groups and Cauchy problem for some fractional abstract differential equations , Life Science Journal, 2013,10(3),793-795

Mahmoud M.El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. of Appl. Math.and Stochastic Analysis (JAMSA)2004,199,-211.

Mahmoud M.El-Borai, Khairia El-Said El-Nadi, Osama Labib, and Hamdy,M., Volterra equations with fractional Stochastic integrals, Mathematical problems in Engineering,5,(2004),453-468.

Mahmoud M. El-Borai, Khairia El-Said El-Nadi, Osama Labib and Hamdy M., Numerical merhods for some nonlinear stochastic differential equations, Applied math, and comp, 168, 2005 65-75.

Mahmoud M. El-Borai, On some fractional differential equations in the Hilbert space, Journal of Discrete and Continuous Dynamical Systems Series A, 2005, 233-241.

Mahmoud M. El-Borai, Exact solutions for some nonlinear fractional parabolic differential equations, Journal of Appl. Math. and Computation

(2008) 141 153.

Mahmoud M. El-Borai, Khairia El-Said El-Nadi and Hoda H. Foad, On some fractional stochastic delay differential equations, Computer and Mathematics with Applications, 59(2010) 11265- 1170 .

Mahmoud M. El-Borai, Khairia Ed-Said El-Nadi, and Eman G. El-Akabawy. On some fractional evolution equations, Computer and Mathematics with Applications, 59, (2010) 1352-1355.

K. Balachandran, K. Uchcana, Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach space, Proc. Indian Acad Sci, Math. Sci., 110, No. 2(2000).

L. Byszevaki, Theorems about the existence and uniqueness of solutions of semi linear evolution nonlocal Cauchy problem, J. Math. Anal. Appl, 162(1999), 494-505.

Mahmoud M. El-Borai, On the correct formulation of the Cauchy problem, Vecnik Moscow University (1968), 15-21.

Mahmoud M. El-Borai, On the initial value problem for a partial differential with operator coefficients, Int.J. Mat. Sc., 3(1980), 103-111.

Mahmoud M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equation, Chaos, Soliton and Fractals, 14(2002), 433-440

Mahmoud M. El-Borai, Semigroup and some nonlinear fractional differential equations, Appl. Math. and Computations, 149(2004), 823-831.

Mahmoud M. El-Borai, Evolution equations without semigroup, Appl. Math and Comp., 149(2004), 815-821.

Mahmoud M. El-Borai, The fundamental solution for fractional evolution equations of parabolic type, J. of Appl. Math. and Stot. Analysis, 3(2004), 197-211.

Mahmoud M. El-Borai, O. L. Moustafa, F.H. Mikhael, On the correct formulation of nonlinear differential equations in Banach space, Int. J. Math. 22, No. 1(1999) .

R. Huilgol, A second order fluid of differential type, Intern. J. Nonlinear Mech, 3(1968), 471-182.

D. Jackson, Existence and uniqueness of solutions to semi linear nonlocal parabolic equations, J. Math. Anal. Appl., 172(1993), 256-265.

A .C. Koll, Applications of fractional calculus to the theory of viscoelasticity, Trans. ASMi J.App. Each.,51(1984),307-317.

F. Lin, J. Veiu, Semi linear integrodifferential equations with non-locol Cauchy problem, Nonlinear Anal. TMA., 26(1996),1023-1033.