关键词: 锥； 混合单调算子； 不动点

Abstract:

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.
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Key words: Cone, mixed monotone operator, fixed point