Delay integro-differential inequalities with initial time difference and applications
The author considers initial value problems of differential equations relative to the changes in the initial time. He introduces the operator \[ Fx(t;t_0)=f(t,x(t),x(\alpha(t-t_0)+t_0))+\int^t_{t_0}H(t,s,x(s),x(\beta(s-t_0)+t_0))\,ds \] and formulates two problems: \[ x'(t)=Fx(t;t_0),\quad t\in [t_0,t_0+T],\;x(t_0)=\overline\kappa \] \(x'(t)=Fx(t;t_0)\), \(t\in[\tau_0,\tau_0+T]\), \(x(\tau_0)=\overline\kappa\) for some fixed positive \(T>0\), where the functions \(f,F,\alpha,\beta\) are prescribed. In Section 2 some comparison results are formulated to compare any two solutions of corresponding delay integro-differential inequalities with initial time difference. In Section 3, the author uses the monotone iterative technique to delay integro-differential equations. The existence of extremal solutions ist obtained. Section 5 contains some existence results when there are several delayed arguments on the right hand side of \(F\). Illustrative examples are presented in Section 4.
scientific article; zbMATH DE number 2075646