Abstract
The paper is devoted to a question of existence and multiplicity of solutions of boundary value problems for a class of second order nonlinear difference equations with Jacobi operators. By using the critical point theory, some sufficient conditions are obtained.
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References
R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications (Marcel Dekker, New York, 1992).
C. D. Ahlbrandt, “Dominant and recessive solutions of symmetric three term recurrences”, J. Differential Equations, 107(2), 238–258, 1994.
V. Anuradha, C. Maya, R. Shivaji, “Positive solutions for a class of nonlinear boundary value problems with Neumann-Robin boundary conditions”, J. Math. Anal. Appl., 236(1), 94–124, 1999.
D. Arcoya, “Positive solutions for semilinear Dirichlet problems in an annulus”, J. Differential Equations, 94(2), 217–227, 1991.
M. Cecchi, M. Marini, G. Villari, “On the monotonicity property for a certain class of second order differential equations”, J. Differential Equations, 82(1), 15–27, 1989.
S. Z. Chen, “Disconjugacy, disfocality, and oscillation of second order difference equations”, J. Differential Equations, 107(2), 383–394, 1994.
P. Chen, H. Fang, “Existence of periodic and subharmonic solutions for second-order p-Laplacian difference equations”, Adv. Difference Equ., 2007, 1–9, 2007.
S. Elaydi, An Introduction to Difference Equation (Springer, New York, 1996).
P. Eloe, “A boundary value problem for a system of difference equations”, Nonlinear Anal., 7(8), 813–820, 1983.
Z.M. Guo, J. S. Yu, “Applications of critical point theory to difference equations”, Fields Inst. Commun., 42, 187–200, 2004.
Z. M. Guo, J. S. Yu, “The existence of periodic and subharmonic solutions for second-order superlinear difference equations”, Sci. China, Ser. A, 46, 506–515, 2003.
Z. M. Guo, J. S. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order difference equations”, J. LondonMath. Soc., 68, 419–430, 2003.
J. K. Hale, J. Mawhin, “Coincidence degree and periodic solutions of neutral equations”, J. Differential Equations, 15(2), 295–307, 1974.
J. Henderson, H. B. Thompson, “Existence of multiple solutions for second-order discrete boundary value problems”, Comput. Math. Appl., 43(10–11), 1239–1248, 2002.
W. G. Kelly, A. C. Peterson, Difference Equations: An Introduction with Applications. Academic Press, New York, 1991.
V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of High Order with Applications, Kluwer Academic Publishers, Boston, 1993.
V. A. Marchenko, Sturm-Liouville Operators and Applications (Birkháuser, Basel, 1986).
H. Matsunaga, T. Hara, S. Sakata, “Global attractivity for a nonlinear difference equation with variable delay”, Computers Math. Appl., 41, 543–551, 2001.
J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, (Springer, New York, 1989).
R. E. Mickens, Difference Equations: Theory and Application (Van Nostrand Reinhold, New York, 1990).
A. Pankov, N. Zakharchenko, “On some discrete variational problems”, Acta Appl. Math., 65, 295–303, 2001.
A. Peterson, “Boundary value problems for an n-th order linear difference equations”, SIAMJ. Math. Anal., 15(1), 124–132, 1984.
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations (Amer. Math. Soc., Providence, 1986).
Y. Rodrigues, “On nonlinear discrete boundary value problems”, J.Math. Anal. Appl., 114(2), 398–408, 1986.
A. N. Sharkovsky, Y. L. Maistrenko, E. Y. Romanenko, Difference Equations and Their Applications (Kluwer, Dordrecht, 1993).
D. Smets, M. Willem M, “Solitary waves with prescribed speed on infinite lattices”, J. Funct. Anal., 149, 266–275, 1997.
G. Teschl, Jacobi operators and completely integrable nonlinear lattices (Amer. Math. Soc., Providence, 2000).
H. Wang, “On the existence of positive solutions for semilinear elliptic equations in annulus”, J. Differential Equations, 109(1), 1–7, 1994.
J.S. Yu, Z. M. Guo, “Boundary value problems of discrete generalized Emden-Fowler equation”, Sci.China Math, 49(10), 1303–1314, 2006.
J. S. Yu, Z.M. Guo, “On boundary value problems for a discrete generalized Emden-Fowler equation”, J. Differential Equations, 231(1), 18–31, 2006.
W. M. Zou, M. Schechter, Critical Point Theory and Its Applications (Springer, New York, 2006).
Z. Zhou, J. S. Yu, Y. M. Chen, “Periodic solutions of a 2nth-order nonlinear difference equation”, Sci.China Math, 53(1), 41–50, 2010.
Z. Zhou, J. S. Yu, Y. M. Chen, “Homoclinic solutions in periodic difference equations with saturable nonlinearity”, Sci. China Math, 54(1), 83–93, 2011.
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Original Russian Text © X. Liu, Y. Zhang, H. Shi, X. Deng, 2013, published in Izvestiya NAN Armenii. Matematika, 2013, No. 6, pp. 123–137.
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Liu, X., Zhang, Y., Shi, H. et al. Boundary value problems of second order nonlinear difference equations with Jacobi operators. J. Contemp. Mathemat. Anal. 48, 273–284 (2013). https://doi.org/10.3103/S1068362313060046
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DOI: https://doi.org/10.3103/S1068362313060046