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Zbl 1068.45008
El-Sayed, Ahmed M.A.; Gaafar, Fatma M.; Hashem, Hind H.G.
On the maximal and minimal solutions of arbitrary-orders nonlinear functional integral and differential equations. (English)
[J] Math. Sci. Res. J. 8, No. 11, 336-348 (2004). ISSN 1537-5978

The authors study the functional differential equation with retarded argument having the form $$D^\alpha_ ax(t)= f(t,x(\varphi(t))),\tag1$$ where $D^\alpha_a$ denotes the fractional derivative. Moreover, the functional integral equation of fractional order of the form $$x(t)= P(t)+ (1/\Gamma(\alpha)) \int^t_0 (t- s)^{\alpha-1} f(s,x(s))\,ds\tag2$$ is also investigated. In (2) it is assumed that $f= f(t,x)$ satisfies the classical Carathéodory conditions and $P$ is a member of the space $C[0,b]$. A few theorems on the existence of solutions of (1) and (2) are established. Some results concerning the existence of the extremal solutions and comparison type theorems concerning (2) are also derived.
[J. Banaś (Rzeszów)]

MSC 2000:: *45G10 Nonsingular nonlinear integral equations
34K05 General theory of functional-differential equations

Keywords: functional differential equation; functional integral equation; fractional order; extremal solutions

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