Quasilinearization for integro-differential equations with retardation and anticipation (Q2866648)

The authors investigate the existence and uniqueness of solutions of integro-differential equations with retardation and anticipation NEWLINE\[NEWLINE\begin{cases} x'=f(t,x,Sx,x_t,x^t),\;t\in J=[t_0,T],\\ x_{t_0}=\Phi_0,\;x^T=\Psi_0,\end{cases}NEWLINE\]NEWLINE where \(\Phi_0\in C_1\), \(\Psi_0\in C_2\), \(f\in C(J\times \mathbb{R}\times \mathbb{R}\times C_1\times C_2,\mathbb{R})\), \(Sx(t)=\int_{t_0}^tK(t,s)x(s)\,ds\) for all \(t\in J\), \(K\in C[J\times J,\mathbb{R}_+]\), \(C_1=C([-h_1,0],\mathbb{R})\), \(C_2=C([0,h_2],\mathbb{R})\), \(x_t=x_t(s)=x(t+s)\) for \(s\in [-h_1,0]\), and \(x^t=x^t(\sigma)=x(t+\sigma)\) for \(\sigma\in [0,h_2]\). They present a technique of quasilinearization for the problem above by using the method of lower and upper solutions, and the monotone iterative technique.