Existence of Iterative Cauchy Fractional Differential Equation

Our main aim in this paper is to use the technique of non expansive operators to more general iterative and non iterative fractional differential equations (Cauchy type ). The non integer case is taken in sense of Riemann-Liouville fractional operators. Applications are illustrated.

• Rabha W. Ibrahim

• Fractional calculus
• fractional differential equation
• Cauchy equation
• Riemann-Liouville fractional operators
• Volterra integral equation
• non-expansive mapping
• iterative differential equation.

<p>[1] R. Lewandowski, B. Chorazyczewski, Identification of the parameters of
the KelvinVoigt and the Maxwell fractional models, used to modeling of
viscoelastic dampers, Computers and Structures 88 (2010) 1-17.
[2] F. Yu, Integrable coupling system of fractional soliton equation hierarchy,
Physics Letters A 373 (2009) 3730-3733.
[3] K. Diethelm, N. Ford, Analysis of fractional differential equations, J.
Math. Anal. Appl., 265 (2002) 229-248.
[4] R. W. Ibrahim , S. Momani, On the existence and uniqueness of solutions
of a class of fractional differential equations, J. Math. Anal. Appl. 334
(2007) 1-10.
[5] S. M. Momani, R. W. Ibrahim, On a fractional integral equation of
periodic functions involving Weyl-Riesz operator in Banach algebras, J.
Math. Anal. Appl. 339 (2008) 1210-1219.
[6] B. Bonilla , M. Rivero, J. J. Trujillo, On systems of linear fractional
differential equations with constant coefficients, App. Math. Comp. 187
(2007) 68-78.
[7] I. Podlubny, Fractional Differential Equations, Acad. Press, London, 1999.
[8] S. Zhang, The existence of a positive solution for a nonlinear fractional
differential equation, J. Math. Anal. Appl. 252 (2000) 804-812.
[9] R. W. Ibrahim, M. Darus, Subordination and superordination for analytic
functions involving fractional integral operator, Complex Variables and
Elliptic Equations, 53 (2008) 1021-1031.
[10] R. W. Ibrahim, M. Darus, Subordination and superordination for univalent
solutions for fractional differential equations, J. Math. Anal. Appl.
345 (2008) 871-879.
[11] R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional
derivatives, J. Phys. Chem. Bio. 104(2000) 3914-3917.
[12] R. W. Ibrahim, Existence and uniqueness of holomorphic solutions for
fractional Cauchy problem, J. Math. Anal. Appl. 380 (2011) 232-240.
[13] A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and applications
of fractional differential equations. North-Holland, Mathematics Studies,
Elsevier 2006.
[14] V. Berinde, Iterative Approximation of Fixed Points,2nd Ed.,Springer
Verlag, Berlin Heidelberg New York, 2007.
[15] M. Edelstein, A remark on a theorem of M. A. Krasnoselskij, Amer.
Math. Monthly, 73(1966) 509-510.
[16] E. Egri, I. Rus, First order iterative functional-dierential equation with
parameter, Stud. Univ. Babes-Bolyai Math. 52 (2007) 67-80.
[17] C. Chidume, Geometric Properties of Banach spaces and nonlinear
Iterations, Springer Verlag, Berlin, Heidelberg, New York, 2009.
[18] Yang, D. and Zhang, W., Solution of equivariance for iterative differential
equations, Appl. Math. Lett. 17(2004) 759-765.
[19] A. Ronto, M. Ronto, Succsesive approximation method for some linear
boundary value problems for differential equations with a special type of
argument deviation, Miskolc Math. Notes, 10(2009) 69-95.
[20] V. Berinde, Existence and approximation of solutions of some first order
iterative differential equations, Miskolc Math. Notes,Vol. 11 (2010) pp.
1326.</p>