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Solvability of a nonlinear fourth-order discrete problem at resonance. (English) Zbl 1193.39002

The authors consider the nonlinear discrete boundary value problem:

Δ 4 u(t-2)=λ 1 u(t0=f(t,u(t))+τφ(t)+h ¯(t),tΠ 2 ,
u(1)=u(T+1)=Δ 2 u(0)=Δ 2 u(T)=0,

where T is an integer with T5; Π 2 ={2,3,,T}; λ 1 is the first eigenvalue of the associated linear eigenvalue problem; φ(0) is the corresponding eigenfunction; f:Π 2 × is continuous and |f(t,s)A|s| B α , tΠ 2 , s for some 0α<1 and A,B[0,+); h ¯:Π 2 with s=2 T h ¯(t)φ(t)=0. The existence of the solution of the above problem is shown.

MSC:
39A12Discrete version of topics in analysis
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