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Fixed point theorem of cone expansion and compression of functional type. (English) Zbl 1013.47019

This article deals with a modification of the classical Krasnosel’skij fixed point theorem about compressions and expansions of a cone P. The authors formulate conditions of compression and expansion in terms of two nonnegative continuous functionals α,γ:P[0,) and the sets

P(γ,R)={xP:γ(x)<R},P(γ,α,r,R)={xP:r<α(x)andγ(x)<r}

under the assumptions that P(α,R) ¯P(γ,R) and inf xP(γ,α,r,R) Ax>0, the operator A:P(γ,α,r,R) ¯P has at least one positive fixed point x * such that rα(x * ) and γ(x * )R, if one of the two conditions is satisfied:

(H1) α(Ax)r for xP(α,r), γ(Ax)R for xP(γ,R), and, in addition, α(λy)λα(y), γ(μz)μγ(z) (yP(α,r), zP(γ,R), λ1, μ(0,1]), α(0)=0;

(H2) α(Ax)r for xP(α,r), γ(Ax)R for xP(γ,R), and, in addition, α(λy)λα(y), γ(μz)μγ(z) (yP(α,r), zP(γ,R), λ(0,1], μ1), γ(0)=0.

As applications, the authors consider the existence problem of a positive solution to the following discrete second-order conjugate boundary value problem:

Δ 2 (t-1)+f(x(t))=0forallt[a+1,b+1],x(a)=x(b+2)=0·


MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
39A10Additive difference equations
47H07Monotone and positive operators on ordered topological linear spaces