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Nonlinear boundary value problems for first-order differential equations with piecewise constant arguments. (English) Zbl 1057.34071

The authors consider the first-order nonlinear boundary value problem \[ x'(t)= f(t,x(t), x([t])),\quad t\in [0,T],\quad x(0)= h(x(T)). \] Here, \([t]\) denotes the largest integer smaller than or equal to \(t\). The function \(h\) is assumed to be continuously differentiable with positive derivative. Sufficient conditions for the existence and uniqueness of solutions are obtained. Comparison results are proved. The case of linear differential equation is studied separately.

MSC:

34K10 Boundary value problems for functional-differential equations
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