Positive Solutions for a Class of Semipositone Discrete Boundary Value Problems with Two Parameters
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:64988", author = "Benshi Zhu", title = "Positive Solutions for a Class of Semipositone Discrete Boundary Value Problems with Two Parameters", 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", volume = "4", number = "10", pages = "1388-7", }
