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Periodic boundary value problems of first order ordinary Carathéodory and discontinuous differential equations. (English) Zbl 1171.34005
The author considers the periodic problem for a first order scalar differential equation $$\frac{d}{dt}\left(\frac{x-k(t,x)}{f(t,x)}\right)=g(t,x),\quad x(0)=x(T),\tag 1$$ where $k,\,f$ are continuous, $T$-periodic in $t$, satisfy some assumptions of Lipschitz type, and $g$ is Carathéodory. Several results of existence are given on the basis of abstract fixed point theorems for equations of type $$Ax\cdot Bx+Cx=x,$$ where $A,\,B,\, C$ are certain operators in a Banach algebra. Equation (1) is reduced to an equation of this type via Green’s function. In the presence of lower and upper solutions it is shown that there are minimal and maximal solutions. In some cases $g$ may have discontinuities provided that $g$ and a certain function formed on the basis of $f,\,g,\,k$ is increasing in the variable $x$.

34B15Nonlinear boundary value problems for ODE
34A36Discontinuous equations