带有超线性项的混合单调算子的不动点定理及其应用

Fixed Point and Applications of Mixed Monotone Operator with Superlinear Nonlinearty

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 .

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\}\%t→0\++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 \%t∈(0,1)η(t)/(1-t\+\{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. [WTHZ]