Nontrivial solutions of boundary value problems of second-order difference equations. (English) Zbl 1113.39018

The authors are concerned with second order boundary value problem
\[ \begin{cases} \Delta^{2}u_{k-1}+\lambda h\left( k,u_{k}\right) =0,\quad k=1,2,\dots,T,\\ u_{0}=u_{T+1}=0, \end{cases} \]
where \(\Delta u_{k-1}=u_{k}-u_{k-1},\) \(\Delta^{2}u_{k-1}=\Delta\left( \Delta u_{k-1}\right) ,\) \(\lambda>0\) is a parameter. Using the critical point theory, the existence of nontrivial solutions is proved, together with their boundedness.


39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Kelley, W. G.; Peterson, A. C., Difference Equations (1991), Academic Press: Academic Press Boston
[2] Agarwal, R. P., Difference Equations and Inequalities (1992), Marcal Dekker: Marcal Dekker New York · Zbl 0784.33008
[3] Elaydi, S. N., An Introduction to Difference Equations, Undergrad. Texts Math. (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0930.39001
[4] Kocic, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher-Order with Applications (1993), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0787.39001
[5] Lakshmikantham, V.; Trigiante, D., Theory of Difference Equations: Numerical Methods and Applications (1988), Academic Press: Academic Press New York · Zbl 0683.39001
[6] Henderson, J.; Thompson, H. B., Existence of multiple solutions for second-order discrete boundary value problems, Comput. Math. Appl., 43, 1239-1248 (2002) · Zbl 1005.39014
[7] Agarwal, R. P.; Henderson, J., Positive solutions and nonlinear eigenvalue problems for third-order difference equations, Comput. Math. Appl., 36, 10-12, 347-355 (1998) · Zbl 0933.39003
[8] Avery, R. I.; Peterson, A. C., Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl., 42, 313-322 (2001) · Zbl 1005.47051
[9] Eloe, P., A boundary value problem for a system of difference equations, Nonlinear Anal., 7, 813-820 (1983) · Zbl 0518.39002
[10] Anderson, D.; Avery, R. I.; Peterson, A. C., Three positive solutions to a discrete focal boundary value problem, J. Comput. Appl. Math., 88, 103-118 (1998) · Zbl 1001.39021
[11] Agarwal, R. P.; Perera, K.; O’Regan, D., Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal., 58, 69-73 (2004) · Zbl 1070.39005
[12] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65 (1986), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0609.58002
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