Lebesgue Measurable Function In Fractional Differential Equations
DOI:
https://doi.org/10.31642/JoKMC/2018/010303Keywords:
Ordinary Differential Equations, Lebesgue Measurable Function , Fractional Differential Equations.Abstract
Bassam, M.A. [1], proved some existence and uniqueness theorems for the following fractional linear differential equation.
            ..1
With the initial conditions
Â
Where a<x<b, 0< a£1, mk are real numbers, k=1,2,…,n,  pi(x) , F(x) are continuous functions defined on (a,b) such that p0(x)≠0, i=0,1…,n and y[(n-i) α] denotes the fractional derivative of order (n-i)α for the function y.
In this work we prove some theorems for equation (1), however for α=1. Equation (1) is an ordinary differential equation of order n, therefore all the theorems proved here will be reduced to well known result in the theory of ordinary differential equations. Moreover,
We give some examples and an application for equation (1).
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References
. Bassam, M. A., "Some Existence Theorems on Differential Equations of Generalized Order", Euilie Relve and
Aynowardt. Matreatio, (1965).
. Center, W., "On Fractional Differentiation", Cambridge and Dublin Math. J, 4, 21-26.(1849).
. Liouville, J., "Memoire Surle Changement de is Variable Dans ie Calcui des Different Lelles a Indices
Quelconques", Ecole polytech. 15, section 24, 17-45. (1835). DOI: https://doi.org/10.1016/S0140-6736(02)97304-6
. Oldham, Keith B. and Jerome Spanier, "The Fractional Calculus", Academic Press, Inc. New york and London, 1974.
. Davis, H. T., "The Application of fractional Operators to Functional Equations", presented to the Indiana Section of the Mathematical Association of America. May 8 1926.
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Copyright (c) 2015 Sabah Mahmood Shaker
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