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Positive solutions for nonlinear discrete periodic boundary value problems. (English) Zbl 1197.39006

The article deals with the following boundary value problem

-Δ[p(t-1)Δu(t-1)]+q(t)u(t)=rg(t)f(u(t)),t[1,T] ,
u(0)=u(T),p(0)Δu(0)=p(T)Δu(T),

where r is a positive parameter, T>2, fC(,), sf(s)>0 for s0 and there exist the limits

f 0 =lim |s|0 f(s) s,f =lim |s| f(s) s,

p,g:(0,) and q:[0,), q¬0, are T-periodic. The main result is the following:

The boundary value problem under consideration has two T-periodic solutions u + and u - , u + (t)>0 and u - (t)<0 for t(0,T), provided that either λ 1 /f <r<λ 1 /f 0 or λ 1 /f 0 <r<λ 1 /f , where λ 1 is the first eigenvalue of the linear eigenvalue problem

-Δ[p(t-1)Δu(t-1)]+q(t)u(t)=rg(t)u(t),t[1,T] ,
u(0)=u(T),p(0)Δu(0)=p(T)Δu(T)·

MSC:
39A23Periodic solutions (difference equations)
39A12Discrete version of topics in analysis
34B15Nonlinear boundary value problems for ODE
34C25Periodic solutions of ODE
39A22Growth, boundedness, comparison of solutions (difference equations)
34L05General spectral theory for OD operators
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