## Criteria for existence and nonexistence of positive solutions to a discrete periodic boundary value problem.(English)Zbl 1056.39016

The paper deals with a discrete nonlinear equation $-\Delta[p(n-1)\Delta u(n-1)] + q(n)u(n) = \lambda f(n,u(n)).$ The existence of positive solutions of a periodic boundary value problem $u(0)=u(N), \quad p(0)\Delta u(0) = p(N)\Delta u(N),$ where $$\{u(n)\}_{n=0}^{N+1}$$ is a desired solution, for the system is proved. Moreover, conditions for the nonexistence of positive solutions are defined.

### MSC:

 39A12 Discrete version of topics in analysis 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000)
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### References:

 [1] Atici F. M. Guseinov G. Sh. On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions J. Comput. Appl. Math. 132 2001 341 356 · Zbl 0993.34022 [2] Atici F. M. Cabada A. Existence and uniqueness results for discrete second order periodic boundary value problems Comput. Math. Appl. 2001 · Zbl 1057.39008 [3] Cabada A. Extremal solutions for the difference{$$\phi$$}Comput. Math. Appl. 42 2001 593 601 · Zbl 1001.39006 [4] Cabada A. Otero-Espinar V. Optimal existence results forn-th order periodic boundary value difference problems J. Math. Anal. Appl. 247 2000 67 86 · Zbl 0962.39006 [5] Deimling K. Nonlinear Functional Analysis Springer New York 1985 · Zbl 0559.47040 [6] Erbe L. H. Hu S. Wang H. Multiple positive solutions of some boundary value problems J. Math. Anal. Appl. 184 1994 640 648 · Zbl 0805.34021 [7] Erbe L. H. Tang M. Existence and multiplicity of positive solutions to nonlinear boundary value problems Diff. Eqn. Dyn. Syst. 4 1996 313 320 · Zbl 0868.35035 [8] Erbe L. H. Peterson A. Mathsen R. Existence, multiplicity and nonexistence of positive solutions to a differential equation on a measure chain J. Comput. Appl. Math. 113 2000 365 380 · Zbl 0937.34025 [9] Guo D. Lakshmikantham L. Nonlinear Problems in Abstract Cones Academic Press Orlanda, FL 1988 [10] Henderson J. Thompson H. B. Existence of multiple solutions for second order boundary value problems J. Diff. Eqn. 166 2 2000 443 454 · Zbl 1013.34017 [11] Krasnoselskii M. Positive Solutions of Operator Equations Noordhoff Groningen 1964 [12] Kong L. Jiang D. Multiple positive solutions of a nonlinear fourth order periodic boundary value problem Ann. Polon. Math. 69 3 1998 265 270 · Zbl 0918.34024 [13] Ma R. Multiplicity of positive solutions for second order three-point boundary value problems Comput. Math. Appl. 40 2000 193 204 · Zbl 0958.34019
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