Abstract
We consider a higher order functional difference equations on \({\mathbb Z}\) with an eigenvalue parameter \(\lambda \) in the equation. Sufficient conditions are obtained for the existence of at least one or two positive periodic solutions of the equation for different values of \(\lambda \). The nonlinear function in the equation is allowed to be sign-changing in some of our results. Our proofs utilize Krasnosel’skii’s fixed point theorem.
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Anuradha, V., Hai, D.D., Shivaji, R.: Existence results for superlinear semipositone BVP’s. Proc. Am. Math. Soc. 124, 757–763 (1996)
Aris, R.: Introduction to the analysis of chemical reactors. Prentice Hall, Englewood Cliffs (1965)
Cabada, A., Ferreiro, J.B.: Existence of positive solutions for \(n\)th-order periodic difference equations. J. Differ. Equ. Appl. 17, 935–954 (2011)
Graef, J.R., Kong, L.: Positive solutions for third order semipositone boundary value problems. Appl. Math. Lett. 22, 1154–1160 (2009)
Guo, D., Lakshmikantham, V.: Nonlinear problems in abstract cones. Academic Press, Orlanda (1988)
Henderson, J., Kunkel, C.J.: Singular discrete higher order boundary value problems. Int. J. Differ. Equ. 1, 119–133 (2006)
Henderson, J., Luca, R.: Positive solutions for a system of second-order multi-point discrete boundary value problems. J. Differ. Equ. Appl. 18, 1575–1592 (2012)
Jun, J., Yang, B.: Positive solutions of discrete third-order three-point right focal boundary value problems. J. Differ. Equ. Appl. 15, 185–195 (2009)
Jun, J., Yang, B.: Eigenvalue comparisons for a class of boundary value problems of discrete beam equation. Appl. Math. Comput. 218, 5402–5408 (2012)
Kelly, W.G., Peterson, A.C.: Difference equations, an introduction with applications, 2nd edn. Academic Press, New York (2001)
Kong, L., Kong, Q., Zhang, B.G.: Positive solutions for boundary value problems of third-order functional difference equations. Comput. Math. Appl. 44, 481–489 (2002)
Kocic, V.L., Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications. Kluwer Academic Publishers, Dordrecht (1993)
Lan, K.Q.: Multiple positive solutions of semi-positone Sturm–Liouville boundary value problems. Bull. Lond. Math. Soc. 38, 283–293 (2006)
Ma, R.: Multiple positive solutions for semipositone fourth-order boundary value problem. Hiroshima Math. J. 33, 217–227 (2003)
Ma, R.: Existence of positive solutions for superlinear semipositone \(m\)-point boundary value problems. Proc. Edinb. Math. Soc. 46, 279–292 (2003)
Tian, Y., Ge, W.: Existence of multiple positive solutions for discrete problems with p-Laplacian via variational methods. Electron. J. Differ. Equ. 2011(45), 1–8 (2011)
Tian, Y., Ge, W.: Two solutions for a discrete problem with a \(p\)-Laplacian. J. Appl. Math. Comput. 38, 353–365 (2012)
Wang, W., Chen, X.: Positive periodic solutions for higher order functional difference equations. Appl. Math. Lett. 23, 1468–1472 (2010)
Zhang, X., Liu, L., Wu, Y.: Positive solutions of nonresonance semipositone singular Dirichlet boundary value problems. Nonlinear Anal. 68, 97–108 (2008)
Acknowledgments
The research by Jacob D. Johnson, Michael G. Ruddy, and Alexander M. Ruys de Perez was conducted as part of a 2013 Research Experience for Undergraduates at the University of Tennessee at Chattanooga that was supported by NSF Grant DMS-1261308.
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Johnson, J.D., Kong, L., Ruddy, M.G. et al. Positive Periodic Solutions for a Higher Order Functional Difference Equation. Differ Equ Dyn Syst 23, 195–208 (2015). https://doi.org/10.1007/s12591-013-0192-4
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DOI: https://doi.org/10.1007/s12591-013-0192-4