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Two positive fixed points of nonlinear operators on ordered Banach spaces. (English) Zbl 1014.47025
This article deals with the following theorem: Let P be a cone in a real Banach space E, α and γ increasing nonnegative continuous and θ nonnegative continuous functionals on P with θ(0)=0, θ(λx)λθ(x) (0λ1), γθ(x)α(x) and xM γ (x) for all x{xP:γ(x)<c} ¯, and let a completely continuous operator A:{xP:γ(x)<c} ¯P satisfy, for some a,b,a<b<c, the following conditions: (i) γ(Ax)>c for all x{xP:γ(x)<c}; (ii) θ(Ax)<b for all x{xP:θ(x)<b}; (iii) {xP:α(x)<a} and α(Ax)>a for all x{xP:α(x)<a}. Then A has at least two fixed points x 1 and x 2 such that a<α(x 1 ), θ(x 1 )<b and b<θ(x 2 ), γ(x 2 )<c. As an application, the second order boundary value problem y '' +f(y)=0, y(0)=y(1)=0 with a continuous function f:[0,) is considered.

MSC:
47H07Monotone and positive operators on ordered topological linear spaces
34B15Nonlinear boundary value problems for ODE
47H10Fixed point theorems for nonlinear operators on topological linear spaces