Positive solutions for nonlinear discrete second-order boundary value problems with parameter dependence. (English) Zbl 1260.39020

He, Tieshan; Xu, Yuantong

J. Math. Anal. Appl. 379, No. 2, 627-636 (2011).

The paper deals with the existence, multiplicity and nonexistence of positive solutions for the difference equation \[ \begin{aligned} &-\Delta\left[ p\left( t-1\right) \Delta u\left( t-1\right) \right]\\ &+q\left( t\right) u\left( t\right) =\lambda f\left( t,u\left( t\right) \right) ,\;t\in\mathbb{Z},\text{ }1\leq t\leq T, \end{aligned} \] subject to the boundary conditions \[ \begin{aligned} & u\left( 0\right) =u\left( T\right) ,\text{ }p\left( 0\right) \Delta\\ & u\left( 0\right) =p\left( T\right) \Delta u\left( T\right) , \end{aligned} \] where function \(f\) is continuous in the second variable, \(p,q\) are positive functions, \(\lambda\) is a positive parameter, and \(\Delta u\left( t\right) =u\left( t+1\right) -u\left( t\right) .\) For some values of \(\lambda,\) the above boundary value problem admits at least two positive solutions, for other values of \(\lambda\) it has at least one positive solutions, and for other values of \(\lambda,\) the problem admits no positive solution. The proofs employ the fixed point index theory.

Reviewer: N. C. Apreutesei (Iaşi)

MSC:

39A22	Growth, boundedness, comparison of solutions to difference equations
39A12	Discrete version of topics in analysis
39A10	Additive difference equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

difference equations; boundary value problem; fixed point index; Ambrosetti-Prodi type result; positive solutions

References:

[1]	Agarwal, R.P., Difference equations and inequalities: theory, methods, and applications, (2000), Marcel Dekker New York · Zbl 0952.39001
[2]	Agarwal, R.P.; Perera, K.; OʼRegan, D., Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear anal., 58, 69-73, (2004) · Zbl 1070.39005
[3]	Bereanu, C.; Mawhin, J., Existence and multiplicity results for periodic solutions of nonlinear difference equations, J. difference equ. appl., 12, 677-695, (2006) · Zbl 1103.39003
[4]	Bereanu, C.; Thompson, H.B., Periodic solutions of second order nonlinear difference equations with discrete ϕ-Laplacian, J. math. anal. appl., 330, 1002-1015, (2007) · Zbl 1127.39006
[5]	Atici, F.M.; Guseinov, G.Sh., Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. math. anal. appl., 232, 166-182, (1999) · Zbl 0923.39010
[6]	Atici, F.M.; Cabada, A., Existence and uniqueness results for discrete second-order periodic boundary value problems, Comput. math. appl., 45, 1417-1427, (2003) · Zbl 1057.39008
[7]	Cabada, A.; Iannizzotto, A.; Tersian, S., Multiple solutions for discrete boundary value problems, J. math. anal. appl., 356, 418-428, (2009) · Zbl 1169.39008
[8]	Cabada, A.; Dimitrov, N.D., Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities, J. math. anal. appl., 371, 518-533, (2010) · Zbl 1211.39009
[9]	Yu, J.S.; Guo, Z.M.; Zou, X.F., Periodic solutions of second order self-adjoint difference equations, J. lond. math. soc., 71, 146-160, (2005) · Zbl 1073.39009
[10]	Yu, J.S.; Guo, Z.M., On boundary value problems for a discrete generalized Emden-Fowler equation, J. differential equations, 231, 18-31, (2006) · Zbl 1112.39011
[11]	Guo, Z.M.; Yu, J.S., Multiplicity results for periodic solutions to second-order difference equations, J. dynam. differential equations, 18, 943-960, (2006) · Zbl 1109.39007
[12]	Zhou, Z.; Yu, J.S.; Chen, Y.M., Periodic solutions of a 2nth-order nonlinear difference equation, Sci. China ser. A, 53, 41-50, (2010)
[13]	Bai, D.Y.; Xu, Y.T., Positive solutions for semipositone BVPs of second-order difference equations, Indian J. pure appl. math., 39, 59-68, (2008) · Zbl 1148.39001
[14]	Lian, F.Y.; Xu, Y.T., Multiple solutions for boundary value problems of a discrete generalized Emden-Fowler equation, Appl. math. lett., 23, 8-12, (2010) · Zbl 1191.39007
[15]	Goldberg, S., Introduction to difference equations, (1960), John Wiley & Sons New York
[16]	Mawhin, J., Bounded solutions of some second order difference equations, Georgian math. J., 14, 315-324, (2007) · Zbl 1132.39007
[17]	Anderson, D.R.; Minhoʼs, F., A discrete fourth-order lidstone problem with parameters, Appl. math. comput., 214, 523-533, (2009) · Zbl 1172.39019
[18]	Lakshmikantham, V.; Trigiante, D., Theory of difference equations: numerical methods and applications, (1988), Academic Press Boston · Zbl 0683.39001
[19]	Elaydi, S.N., An introduction to difference equations, Undergrad. texts math., (1996), Springer-Verlag New York · Zbl 0840.39002
[20]	He, X.M.; Wu, X., Existence and multiplicity of solutions for nonlinear second order difference boundary value problems, Comput. math. appl., 57, 1-8, (2009) · Zbl 1165.39303
[21]	He, T.S.; Chen, W.G., Periodic solutions of second order discrete convex systems involving the p-Laplacian, Appl. math. comput., 206, 124-132, (2008) · Zbl 1158.39006
[22]	Wang, D.B.; Guan, W., Three positive solutions of boundary value problems for p-Laplacian difference equations, Comput. math. appl., 55, 1943-1949, (2008) · Zbl 1147.39008
[23]	Ma, M.J.; Tang, H.S.; Luo, W., Periodic solutions for nonlinear second-order difference equations, Appl. math. comput., 184, 685-694, (2007) · Zbl 1123.39005
[24]	Zhuang, W.; Chen, Y.; Cheng, S.S., Monotone methods for a discrete boundary problem, Comput. math. appl., 32, 12, 41-49, (1996) · Zbl 0872.39005
[25]	Ma, R.Y.; Ma, H.L., Positive solutions for nonlinear discrete periodic boundary value problems, Comput. math. appl., 59, 136-141, (2010) · Zbl 1197.39006
[26]	Zeidler, E., Nonlinear functional analysis and its applications, I. fixed-point theorems, (1985), Springer-Verlag New York
[27]	Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
[28]	Guo, D.J.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045