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Multiple positive solutions of a discrete difference system. (English) Zbl 1030.39015
The authors study the discrete system $$\Delta^2u_1(k)+f_1(k,u_1(k),u_2(k))=0,\quad \Delta^2u_2(k)+f_2(k,u_1(k),u_2(k))=0,$$ $k\in[0,T]$, with the Dirichlet boundary conditions $$u_1(0)=u_1(T+2)=0,\quad u_2(0)=u_2(T+2)=0.$$ Sufficient conditions for existence of at least three positive solutions to the above boundary value problem are given. The proof is based on the fixed point theorem of {\it R. W. Leggett} and {\it L. R. Williams} [Indiana Univ. Math. J. 28, 673-688 (1979; Zbl 0421.47033)].

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis
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##### References:
 [1] Agarwal, P. R.; Lalli, B. S.: Discrete polynomial interpolation, Green’s functions, maximum principle, error bounds and boundary value problem. Comput. math. Appl. 25, 3-39 (1993) · Zbl 0772.65092 [2] Agarwal, P. R.; Wong, F. H.: Existence of positive solutions for nonpositive difference equations. Math. comput. Modell. 26, No. 7, 77-85 (1997) · Zbl 0889.39011 [3] Agarwal, P. R.; Henderson, J.: Positive solutions and nonlinear eigenvalue problems for third-order difference equations. Comput. math. Appl. 36, No. 10--12, 347-355 (1998) · Zbl 0933.39003 [4] Agarwal, P. R.; O’reagn, D.: A coupled system of difference equations. Appl. math. Comput. 114, 39-49 (2000) · Zbl 1023.39001 [5] Davis, J. M.; Eloe, P. W.; Henderson, J.: Triple positive solutions and dependence on high order derivatives. J. math. Anal. appl. 237, 710-720 (1999) · Zbl 0935.34020 [6] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045 [7] Leggett, R. W.; Williams, L. R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana univ. Math. J. 28, 673-688 (1979) · Zbl 0421.47033 [8] Shi, Y.; Chen, S.: Spectral theory of second order vector difference equations. J. math. Anal. appl. 239, 195-212 (1999) · Zbl 0934.39002 [9] Wong, P. J. Y.: Solutions of constant signs of system of Sturm--Liouville boundary value problems. Math. comput. Modell. 29, 27-38 (1999) · Zbl 1041.34015 [10] Wong, P. J. Y.; Agarwal, P. R.: On the existence of positive solutions of high order difference equations. Topol. meth. Nonlinear anal. 10, 339-351 (1997) · Zbl 0914.39005 [11] Wong, P. J. Y.; Agarwal, P. R.: On the existence of positive solutions of singular boundary value problems for high order difference equations. Nonlinear anal. TMA 28, No. 2, 277-287 (1997) · Zbl 0861.39002