带有超线性项的混合单调算子的不动点定理及其应用

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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

CLC Number:

47H07

LIU Jin-Sheng, LI Fu-Xi, DAI Li-Qing-. Fixed Point and Applications of Mixed Monotone Operator \=with Superlinear Nonlinearty[J].Acta mathematica scientia,Series A, 2003, 23(1): 19-24.

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

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