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Eigenvalue problems for second-order nonlinear dynamic equations on time scales. (English) Zbl 1099.34026

The authors are concerned with the second-order nonlinear dynamic equation on time scales

u ΔΔ (t)+λa(t)f(u(σ(t)))=0,t[0,1],

satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=u Δ (σ(1))=0, where a and f are positive. The number of positive solutions of the above boundary value problem for λ belonging to the half-line (0,) and the dependence of positive solutions of the problem on the parameter λ are discussed. It is proved that there exists a λ * >0 such that the problem has at least two, one and no positive solution(s) for 0<λ<λ * ,λ=λ * and λ>λ * , respectively.

The main tool is a fixed-point index theorem on cones due to Guo-Lakshmikantham. Furthermore, by using the semi-order method on cones of Banach space, an existence and uniqueness criterion for a positive solution of the problem is established. In particular, such a positive solution u λ (t) of the problem depends continuously on the parameter λ, i.e., u λ (t) is nondecreasing in λ, lim λ0 + u λ =0 and lim λ+ u λ =+.

34B18Positive solutions of nonlinear boundary value problems for ODE
39A10Additive difference equations