×

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 \(x\).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A36 Discontinuous ordinary differential equations
PDFBibTeX XMLCite