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On the existence of positive solutions of fourth-order difference equations. (English) Zbl 1068.39008
Let $T\ge 1$ fixed, $f\in C(\bbfR^+,\bbfR^+)$, $a:\{1,2,\dots, T+1\}\subset\bbfZ\to \bbfR^+$ and $\lim_{x\to 0,x> 0}\,x^{-1} f(x)$ and $\lim_{x\to+\infty}\, x^{-1} f(x)$ be limits that exist. The authors determine eigenvalues $\lambda$ for which there exist positive solutions $u$ of the fourth-order difference equation $$\Delta^4 u(t- 2)- \lambda a(t) f(u(t))= 0,\quad t\in \{2,3,\dots, T+ 2\}\subset\bbfZ$$ satisfying the boundary conditions $$u(0)= \Delta^2 u(0)= u(T+ 2)= \Delta^2 u(T)= 0$$ or $$u(0)= \Delta^2 u(0)= \Delta u(t+ 1)= \Delta^3 u(T- 1)= 0$$ by means of Krasnosel’skij’s fixed point theorem.

MSC:
39A11Stability of difference equations (MSC2000)
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References:
[1] Agarwal, R. P.; Henderson, J.: Positive solutions and nonlinear eigenvalue problems for third-order difference equations. Comp. math. Appl 36, No. 11--12, 347-355 (1998) · Zbl 0933.39003
[2] Henderson, J.: Positive solutions for nonlinear difference equations. Nonlinear stud 4, No. 1, 29-36 (1997) · Zbl 0883.39002
[3] Merdivenci, F.: Two positive solutions of a boundary value problem for difference equations. J. diff. Eqns. appl 1, 263-270 (1995) · Zbl 0854.39001
[4] Agarwal, R. P.; O’regan, D.: A fixed-point approach for nonlinear discrete boundary value problems. Comp. math. Appl 36, No. 10--12, 115-121 (1998) · Zbl 0933.39004
[5] Henderson, J.; Wang, H. Y.: Positive solutions for nonlinear eigenvalue problems. J. math. Anal. appl 208, 252-259 (1997) · Zbl 0876.34023
[6] Zhang, B. G.; Kong, L. J.; Sun, Y. J.; Deng, X. H.: Existence of positive solutions for BVPs of fourth-order difference equations. Appl. math. Comput 131, 583-591 (2002) · Zbl 1025.39006
[7] Ma, R. Y.; Wang, H. Y.: On the existence of positive solutions of fourth-order ordinary differential equations. Appl. anal 59, 225-231 (1995) · Zbl 0841.34019
[8] Graef, J. R.; Yang, B.: On a nonlinear boundary value problem for fourth order equations. Appl. anal 72, 439-448 (1999) · Zbl 1031.34017
[9] Krasnosel’skii, M. A.: Positive solutions of operator equations. (1964)
[10] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040