On the maximal and minimal solutions of arbitrary-orders nonlinear functional integral and differential equations.

*(English)*Zbl 1068.45008The authors study the functional differential equation with retarded argument having the form

$${D}_{a}^{\alpha}x\left(t\right)=f(t,x\left(\varphi \left(t\right)\right)),\phantom{\rule{2.em}{0ex}}\left(1\right)$$

where ${D}_{a}^{\alpha}$ denotes the fractional derivative. Moreover, the functional integral equation of fractional order of the form

$$x\left(t\right)=P\left(t\right)+(1/{\Gamma}\left(\alpha \right)){\int}_{0}^{t}{(t-s)}^{\alpha -1}f(s,x\left(s\right))\phantom{\rule{0.166667em}{0ex}}ds\phantom{\rule{2.em}{0ex}}\left(2\right)$$

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.

Reviewer: J. Banaś (Rzeszów)

##### MSC:

45G10 | Nonsingular nonlinear integral equations |

34K05 | General theory of functional-differential equations |