Solutions of Neumann Boundary Value Problems for Higher Order Nonlinear Functional Difference Equations With $p$-Laplacian
DOI:
https://doi.org/10.5644/SJM.03.1.06Keywords:
Solutions, higher order functional difference equation with $p$-Laplacian, Neumann boundary value problems, fixed-point theorem, growth conditionAbstract
Sufficient conditions for the existence of at least one solution of Neumann boundary value problems for higher order nonlinear functional difference equations with $p$-Laplacian are established. We allow $f$ to be at most linear, superlinear or sublinear in the obtained results.
2000 Mathematics Subject Classification. 34B10, 34B15
Downloads
References
R. P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, Kluwer, Dordrecht, 1998.
R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999.
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
J. Henderson and H. B. Thompson, Existence of multiple solutions for second order discrete boundary value problems, Comput. Math. Appl., 43 (2002), 1239–1248.
L. Kong, Q. Kong, and B. Zhang, Positive solutins of boundary value problems for third order functional difference equations, Comput. Math. Appl., 44 (2002), 481–489.
R. P. Agarwal and J. Henderson, Positive solutions and nonlinear eigenvalue problems for third order difference equations, Comput. Math. Appl., 36 (1998), 347–355.
Y. Liu and W. Ge, Positive solutions of boundary value problems for finite difference equations with p-Laplacian operator, J. Math. Anal. Appl., 278 (2003), 551–561.
D. R. Anderson, Existence of solutions to higher order discrete three points problems, Electron. J. Differential Equations, 40 (2003), 1–7.
R. I. Avery, J. M. Davis and J. Henderson, Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem, Electron. J. Differential Equations, 40 (2000), 1–15.
D. R. Anderson, Discrete third-order three-point right focal boundary value problems, Comput. Math. Appl., 45 (2003), 861–871.
R. I. Avery and A. C. Peterson, Multiple positive solutions of a discrete second order conjugate problem, Panamer. Math. J., 8 (1998), 1–12.
J. M. Davis, P. W. Eloe and J. Henderson, Comparison of engenvalues for discrete Lidstone boundary value problems, Dynam. Systems Appl., 8 (1999), 381–387.
R. I. Avery, C. J. Chyan and J. Henderson, Twin positive solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. Math. Appl., 42 (2001), 695–704.
Y. Wang and Y. Shi, Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions, J. Math. Anal. Appl., 309 (2005), 56–69.
F. M. Atici and A. Cabada, Existence and uniqueness results for discrete second order periodic boundary value problems, Comput. Math. Appl., 45 (2003), 1417–1427.
F. M. Atici and G.Sh. Gusenov, Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. Math. Anal. Appl., 232 (1999), 166–182.
R. E. Mickens, Periodic solutions of second order nonlinear difference equations containing a small parameter-IV. Multi-discrete time method, J. Franklin Inst., 324 (1987), 263–271.
R. E. Mickens, Periodic solutions of second order nonlinear difference equations containing a small parameter-III. Perturbation theory, J. Franklin Inst., 321 (1986), 39–47.
R. E. Mickens, Periodic solutions of second order nonlinear difference equations containing a small parameter-II. Equivalent linearization, J. Franklin Inst., 320 (1985), 169–174.
Y. Li and L. Lu, Positive periodic solutions of higher-dimensional nonlinear functional difference equations, J. Math. Anal. Appl., 309 (2005), 284–293.
H. Sun and Y. Shi, Eigenvalues of second order difference equations with coupled boundary value conditions, Linear Algebra Appl., 414 (2006), 361–372.
J. Ji and B. Yang, Eigenvalue comparisons for boundary problems for second order difference equations, J. Math. Anal. Appl., In press, to appear.
N. Aykut, Existence of positive solutions for boundary value problems of second order functional difference equations, Comput. Math. Appl., 48 (2004), 517–527.
A. Cabada and V. Otero-Espinar, Fixed sign solutions of second order difference equations with Neumann boundary conditions, Comput. Math. Appl., 45 (2003), 1125–1136.
R. P. Agarwal, H. B.Thompson and C. C. Tisdell, Three-point boundary value problems for second order discrete equations, Comput. Math. Appl., 45 (2003), 1429–1435.
H. B. Thompson and C. C. Tisdell, The existence of spurious solutions to discrete, two-point boundary value problems, Appl. Math. Letters, 16 (2003), 79–84.
A. Cabada and V. Otero-Espinar, Existence and comparison results for difference φ−Laplacian boundary value problems with lower and upper solutions in reverse order, J. Math. Anal. Appl., 267 (2002), 501–521.
A. Cabada, Extremal solutions for the difference φ−Laplacian problem with functional boundary value conditions, Comput. Math. Appl., 42 (2001), 593–601.
