Solvability of nonlinear systems including $(\gamma,\delta)$-comparison pairs

M. N. Yakovlev

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part IX, Tome 202 (1992), pp. 185-189

Résumé

Let $\gamma,\delta\in R^n$ with $\gamma_j,\delta_j\in\{0,1\}$. A comparison pair for a system of equations $f_i(u_1,\dots,u_n)=0$ $(i=1,\dots,n)$ is a pair of vectors $v,w\in R^n$, $v\leqslant w$, such that \begin{gather*} \gamma_if_i(u_1,\dots,u_{i-1},v_i,u_{i+1},\dots,u_n)\leqslant0 \\ \delta_if_i(u_1,\dots,u_{i-1},w_i,u_{i+1},\dots,u_n)\geqslant0 \end{gather*} for $\gamma_ju_j\geqslant v_j$, $\delta_ju_j\leqslant w_j$ $(j=1,\dots,n)$. The presence of comparison pairs enables one to essentially weaken the assumptions of the existence theorem. Bibliography: 1 title.