Solvability of nonlinear systems including (γ, δ)-comparison pairs

Abstract

Let$\gamma ,\delta \in \mathbb{R}^n$ with$\gamma _j ,\delta _j \in \{ 0,1\}$. A comparison pair for a system of equations f_i(u₁,…,u_n)=0 (i=1,…,n) is a pair of vectors$v,w \in \mathbb{R}^n ,v \leqslant w$, such that

$$\begin{array}{*{20}c} {\gamma _i f_i (u_1 , \ldots ,u_{i - 1} ,v_i ,u_i + 1, \ldots ,u_n ) \leqslant 0,} \\ {\delta _i f_i (u_1 , \ldots ,u_{i - 1} ,w_i ,u_i + 1, \ldots ,u_n ) \geqslant 0} \\ \end{array} $$

for$\gamma _j u_j \geqslant v_j ,\delta _j u_j \leqslant w_j (j = 1, \ldots ,n)$. The presence of comparison pairs enables one to essentially weaken the assumptions of the existence theorem. Bibliography: 1 title.