On neutral differential equations and the monotone iterative method. (English) Zbl 1264.34132
The authors study the initial value problems
\[
\left\{\begin{aligned} x^{\prime }\left( t\right)& = f\left( t,x^{\prime }\left( t\right) ,x^{\prime }\left( \beta \left( t\right) \right) ,x\left( t\right) ,x\left( \alpha \left( t\right) \right) \right) ,\quad t\in J=[0,T], \\ x\left( 0\right) &= 0 \end{aligned}\right.\tag{1}
\]
and
\[
\left\{\begin{aligned} x^{\prime }\left( t\right)& = g\left( t,x^{\prime }\left( t\right) ,x^{\prime }\left( \beta \left( t\right) \right) ,x\left( t\right) ,x\left( \alpha \left( t\right) \right) \right) ,\quad t\in J=[0,T], \\ x\left( T\right)& = 0. \end{aligned}\right.\tag{2}
\]
Applying the monotone iterative method to the integral equations equivalent to (1) and (2), they prove some existence results for extremal solutions and some existence and uniqueness results for the solution to (1) and (2).
Reviewer: M. Serban (Cluj-Napoca)
MSC:
34K07 | Theoretical approximation of solutions to functional-differential equations |
34K05 | General theory of functional-differential equations |
34K40 | Neutral functional-differential equations |