带有超线性项的混合单调算子的不动点定理及其应用

该文讨论了带有齐次超线性项μC和线性项D的混合单调算子A=B+μC+D的不动点的存在性.在不假设耦合下上解存在的条件下,得到了算子A的一个不动点定理,并且将所获结果应用到常微分方程两点边值问题、积分方程和椭圆型方程边值问题中,得到了新的结论.因而本质上推广和改进了已有的混合单调算子和相应的增算子的不动点定理.

Fixed Point and Applications of Mixed Monotone Operator =with Superlinear Nonlinearty

In this paper, the existence of fixed point for a class operator $A=B+μC+D$ is established and is applied to Sturm Liouville two point boundary value problems, Hammerstein integral equations, and elliptic boundary value problems. Let E be a real Banach space, P a cone in E, $e∈P＼{θ}. P-e={x∈E:$ there exist positive numbers $λ,μ$ such that  $λe≤x≤μe}.$ Assume that (i) $B:P-e×P-e→P-e$ is mixed monotone, $B(tx,t+{-1}y)≥t(1+η(t))B(x,y),x,y∈P-e,t∈(0,1),$ and  $%{lim}%[DD(X]t→0++[DD)]η(t)=+∞;$ (ii) $C: P-e×P-e→P-e$ is mixed monotone and $β$ homogeneous operator, that is $C(tx,t+{-1}y)=t+β C(x,y),x,y∈P-e,t∈(0,+∞)$, and $%inf%〖DD(X〗t∈(0,1)〖DD)〗η(t)/(1t+{β-1})>0;$ (iii) $D:E→E$ is a positive linear operator, $D(P-e)P-e∪{θ}$, and $D$ has an eigenvector $h∈P-e$ respect with to an eigenvalue $λ∈［0,1).$ Then $A$ has a fixed point $x$ in $P-e$ for $μ≥0$ small enough.