Existence of solutions to a discrete fourth order boundary value problem
John R. GraefDepartment of Mathematics, University of Tennessee at Chattanooga , Chattanooga, TN, USAShapour HeidarkhaniFaculty of Sciences, Department of Mathematics, Razi University , Kermanshah, IranLingju KongDepartment of Mathematics, University of Tennessee at Chattanooga , Chattanooga, TN, USAMin WangDepartment of Mathematics, Rowan University , Glassboro, NJ, USA
2018en
ABI
Annotatsiya
Criteria are established for the existence of at least two nontrivial solutions to the discrete fourth order boundary value problem Δ4u(t-2)-αΔ2u(t-1)+βu(t)=f(t,u(t)),t∈[1,N]Z,u(-1)=Δu(-1)=0,u(N+1)=Δ2u(N)=0, where N≥1 is an integer, α,β≥0, and f:[1,N]Z×R→R is continuous in the second argument. Applications of the results to a related eigenvalue problem are also presented. The proofs are mainly based on the variational method and the classic mountain pass lemma of Ambrosetti and Rabinowitz. Examples are included to illustrate the applicability of the results.
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