Boundary value problems of second order nonlinear functional difference equations. (English) Zbl 1198.39004

The author investigates the solvability of the nonlinear discrete boundary value problem with the Jacobi difference operator
\[ a_nu_{n+1}+b_nu_n+a_{n-1}u_{n-1}=f(n,u_{n+1},u_n,u_{n-1}), \quad n=0,1,\dots,N, \] and with the boundary condition \(\Delta u_0=A\), \(u_{N+1}=B\). It is supposed that the nonlinearity \(f\) is of “variational character” and the existence results are proved using the mountain pass theorem. Note that in contrast to some related papers, the nonlinearity of \(f\) is allowed to depend also on \(u_{n+1}\).


39A12 Discrete version of topics in analysis
34K10 Boundary value problems for functional-differential equations
39A10 Additive difference equations
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[1] Agarwal R.P., Difference Equations and Inequalities: Theory, Methods and Applications (1992) · Zbl 0925.39001
[2] DOI: 10.1155/ADE.2005.93 · Zbl 1098.39001
[3] DOI: 10.1006/jdeq.1994.1011 · Zbl 0797.93022
[4] DOI: 10.1006/jdeq.1994.1018 · Zbl 0791.39001
[5] Elaydi S., An Introduction to Difference Equation (1996) · Zbl 0840.39002
[6] Kelly W.G., Difference Equations: An Introduction with Applications (1991)
[7] Kocic V.L., Global Behavior of Nonlinear Difference Equations of Higher Order with Application (1993) · Zbl 0787.39001
[8] Mickens R.E., Difference Equations: Theory and Application (1990) · Zbl 0949.39500
[9] Rabinowitz P.H., Minimax Methods in Critical Point Theory with Applications to Differential Equations (1986) · Zbl 0609.58002
[10] Sharkovsky A.N., Difference Equations and Their Applications (1993)
[11] DOI: 10.1007/s11425-006-1999-z · Zbl 1117.39014
[12] DOI: 10.1016/j.jde.2006.08.011 · Zbl 1112.39011
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