Positive Solutions for a Class of Semipositone Discrete Boundary Value Problems with Two Parameters

Authors: Benshi Zhu

Abstract:

In this paper, the existence, multiplicity and noexistence of positive solutions for a class of semipositone discrete boundary value problems with two parameters is studied by applying nonsmooth critical point theory and sub-super solutions method.

Keywords: Discrete boundary value problems, nonsmoothcritical point theory, positive solutions, semipositone, sub-super solutions method

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1086223

References:

[1] G.A. Afrouzi and S.H. Rasouli, "Population models involving the p-Laplacian with indefinite weight and constant yield harvesting" , Chaos, Solit. Fract., vol. 31, pp. 404-408, Jan. 2007.
[2] R.P. Agarwal and D. O'Regan, "Nonpositone discrete boundary value problems", Nonlinear Anal., vol. 39, pp. 207-215, Jan. 2000.
[3] R.P. Agarwal, Difference Equations and Inequalities, in: Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc. New York, 2000, ch1.
[4] R. P. Agarwal, S. R. Grace and D. O'Regan, "Discrete semipositone higher-order equations", Comput. Math. Appl., vol. 45, pp. 1171-1179, Mar.-May 2003.
[5] R.P. Agarwal, S.R. Grace and D. O'Regan, "Semipositone higher-order differential equations", Appl. Math. Letters, vol. 17, pp. 201-207, Feb. 2004.
[6] J. Ali and R. Shivaji, "On positive solutions for a class of strongly coupled p-Laplacian systems", Electronic J. Diff. Eqns., conference 16, pp. 29-34, 2007.
[7] D.R. Anderson and F. Minhos, "A discrete fourth-order Lidstone problem with parameters", Appl. Math. Comput., vol. 214, pp. 523-533, Aug. 2009.
[8] A. Castro and R. Shivaji, "Nonnegative solutions for a class of nonpositone problems", Proc. Roy. Soc. Edin., vol. 108A, pp. 291-302, 1988.
[9] A. Castro, C. Maya and R. Shivaji, "Nonlinear eigenvalue problems with semipositone structure", Electronic J. Diff. Eqns. Conference 05, pp. 33-49, 2000.
[10] K.C. Chang, "Variational methods for non-differential functional and their applications to PDE", J. Math. Anal. Appl. vol. 80, pp. 102-129, Jan. 1981.
[11] D.S. Cohen and H.B. Keller, "Some positone problems suggested by nonlinear heat generation", J. Math. Mech. vol. 16, pp. 1361-1376, 1967.
[12] M. Chhetri and S.B. Robinson, "Existence and multiplicity of positive solutions for classes of singular elliptic PDEs", J. Math. Anal. Appl., vol. 357, pp. 176-182, Sep. 2009.
[13] D.G. Costa, H. Tehrani and J. Yang, "On a variational approach to existence and multiplicity results for semipositone problems", Electronic J. Diff. Eqns., vol. 2006, pp. 1-10, 2006.
[14] E.N. Dancer and Z. Zhang, "Critical point,anti-maximum and semipositone p-Laplacian problems", Dis. Con. Dyn. Sys., Supplement , pp. 209-215, 2005.
[15] D.D. Hai and R. Shivaji, "An existence result on positive solutions for a class of p-Laplacian systems", Nonlinear Anal., vol. 56, pp. 1007-1010, Mar. 2004.
[16] D.D. Hai and R. Shivaji, "Uniqueness of positive solutions for a class of semipositone elliptic systems", Nonlinear Anal., vol. 66, pp. 396-402, Jan. 2007.
[17] J. Jiang, L. Zhang, D. O'Regan and R.P. Agarwal, "Existence theory for single and multiple solutions to semipositone discrete Dirichlet boundary value problems with singular dependent nonlinearities", J. Appl. Math. Stoch. Anal., No. 1, pp. 19-31, 2003.
[18] P.L. Lion, "On the existence of positive solutions of semilinear elliptic equations", SIAM Rev. vol. 24, pp. 441- 467, Oct. 1982.
[19] H. Li and J. Sun, "Positive solutions of sublinear Sturm- CLiouville problems with changing sign nonlinearity", Comput. Math. Appl., vol. 58, pp. 1808-1815, Nov. 2009.
[20] M.R. Myerscough, B.F. Gray, W.L. Hogarth and J. Norbury, "An analysis of an ordinary differential equations model for a two species predator-prey system with harvesting and stocking", J. Math. Bio., vol. 30, pp. 389-411, Nov. 1992.
[21] R. Ma and B. Zhu, "Existence of positive solutions for a semipositone boundary value problem on the half-line", Comput. Math. Appl., vol. 58, pp. 1672-1686, Oct. 2009.
[22] J. Selgrade, "Using stocking and harvesting to reverse period-doubling bifurcations in models in population biology", J. Diff. Eqns. Appl., Vol. 4, pp. 163-183, Feb. 1998.
[23] Q. Yao, "Existence of n solutions and/or positive solutions to a semipositone elastic beam equation", Nonlinear Anal., vol. 66, pp. 138-150, Jan. 2007.
[24] N. Yebari and A. Zertiti, "Existence of non-negative solutions for nonlinear equations in the semi-positone case", Electronic J. Diff. Eqns., conference 14, pp. 249- 254, 2006.
[25] Z. Zhao, "Existence of positive solutions for 2nth-order singular semipositone differential equations with Sturm- CLiouville boundary conditions", Nonlinear Anal., vol. 72, pp. 1348-1357, Feb. 2010.