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Existence of positive solution for nonlinear fourth-order difference equations. (English) Zbl 1220.39008

Let T5 be an integer 𝕋 0 ={0,,T+2}, 𝕋 2 ={2,,T} and let f:𝕋 2 ×[0,)[0,) be a continuous function. The author gives some sufficient conditions under which the difference problem

Δ 4 u(t-2)-λf(t,u(t))=0,T𝕋 2 (1)
u(1)=u(T+1)=Δ 2 u(0)=Δ 2 u(T)=0,(2)

where λ>0 is a parameter, has at least two positive solutions.

Moreover, the author presents two theorems that describe conditions such that there exists a sequence {u n } of positive solutions of (1), (2) for which

u n :=max{|u n (j)|:j𝕋 0 }·

MSC:
39A12Discrete version of topics in analysis
39A22Growth, boundedness, comparison of solutions (difference equations)
39A10Additive difference equations
34B15Nonlinear boundary value problems for ODE
References:
[1]Gupta, C. P.: Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. anal. 26, No. 4, 289-304 (1988) · Zbl 0611.34015 · doi:10.1080/00036818808839715
[2]Gupta, C. P.: Existence and uniqueness results for the bending of an elastic beam equation at resonance, J. math. Anal. appl. 135, No. 1, 208-225 (1988) · Zbl 0655.73001 · doi:10.1016/0022-247X(88)90149-7
[3]Aftabizadeh, A. R.: Existence and uniqueness theorems for fourth-order boundary value problems, J. math. Anal. appl. 116, No. 2, 415-426 (1986) · Zbl 0634.34009 · doi:10.1016/S0022-247X(86)80006-3
[4]Yang, Yisong: Fourth-order two-point boundary value problems, Proc. amer. Math. soc. 104, No. 1, 175-180 (1988) · Zbl 0671.34016 · doi:10.2307/2047481
[5]Del Pino, M. A.; Manásevich, R. F.: Multiple solutions for the p-Laplacian under global nonresonance, Proc. amer. Math. soc. 112, No. 1, 131-138 (1991) · Zbl 0725.34021 · doi:10.2307/2048489
[6]Ma, Ruyun; Wang, Haiyan: On the existence of positive solutions of fourth-order ordinary differential equations, Appl. anal. 59, No. 4, 225-231 (1995) · Zbl 0841.34019 · doi:10.1080/00036819508840401
[7]Ma, Ruyun: Existence of positive solutions of a four-order boundary value problem, Appl. math. Comput. 168, 1219-1231 (2005)
[8]Bai, Zhanbing; Wang, Haiyan: On positive solutions of some nonlinear fourth-order beam equations, J. math. Anal. appl. 270, No. 2, 357-368 (2002) · Zbl 1006.34023 · doi:10.1016/S0022-247X(02)00071-9
[9]Chai, Guoqing: Existence of positive solutions for fourth-order boundary value problem with variable parameters, Nonlinear anal. 26, 289-304 (2007)
[10]Yao, Q. L.; Bai, Z. B.: Existence of solutions of BVP for u(4)(t)-λh(t)f(u(t))=0, Chinese ann. Of math. Ser. A 20, 575-578 (1999)
[11]Li, Y. X.: On the existence of positive solutions for the bending elastic beam equations, Appl. math. Comput. 189, No. 1, 821-827 (2007) · Zbl 1118.74032 · doi:10.1016/j.amc.2006.11.144
[12]Zhang, Binggen; Kong, Lingju; Sun, Yijun; Deng, Xinghua: Existence of positive solutions for BVPs of fourth-order difference equations, Appl. math. Comput. 131, 583-591 (2002) · Zbl 1025.39006 · doi:10.1016/S0096-3003(01)00171-0
[13]He, Zhimin; Yu, Jianshe: On the existence of positive solutions of fourth-order difference equations, Appl. math. Comput. 161, 139-148 (2005) · Zbl 1068.39008 · doi:10.1016/j.amc.2003.12.016
[14]Eloe, P. W.; Henderson, J.: Positive solutions and nonlinear multipoint conjugate eigenvalue problems, Electron. J. Differential equations 03, 11 (1997) · Zbl 0888.34013 · doi:emis:journals/EJDE/Volumes/1997/03/abstr.html
[15]Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones, (1988)
[16]G.B. Gustafson, K. Schmitt, Method of nonlinear analysis in the theory of differential equations, Lecture Notes, University of Utah, 1975.
[17]Kelley, W. G.; Peterson, A. C.: Difference equations. An introduction with applications, (2001) · Zbl 0970.39001