三阶差分方程边值问题
摘要
本文首先研究了三阶非线性差分方程：
Δ~3u(t-2)+a(t)f(u(t))=0，t∈[2，T+2]，
满足多种边界条件时的正解存在性和二解性，其次研究了特征值问题：
Δ~3u(t-2)+λα(t)f(u(t))=0，t∈[2，T+2]，
在满足不同边界条件时正解的存在性及二解性。我们给出了各自边值问题的Green函数的具体表达式，使正解存在性及二解性的判断较为方便。并且分别举例说明定理的应用。
Firstly,some criteria for the existence of at least one positive solution and two positive solutions are obtained respectively for the third-order nonlinear difference equation:
satisfying several different types of boundary conditions. The second part is devoted to determine values of A for which there exist at least one positive solution and two positive solutions for
satisfying several different types of boundary conditions. We have presented different function expression of Green functions. So the judgement of the existence on one pos-itive solution and two positive solutions becomes convenient. At last,many examples have given to demonstrate the applications of our theorems.
Δ~3u(t-2)+a(t)f(u(t))=0，t∈[2，T+2]，
满足多种边界条件时的正解存在性和二解性，其次研究了特征值问题：
Δ~3u(t-2)+λα(t)f(u(t))=0，t∈[2，T+2]，
在满足不同边界条件时正解的存在性及二解性。我们给出了各自边值问题的Green函数的具体表达式，使正解存在性及二解性的判断较为方便。并且分别举例说明定理的应用。
Firstly,some criteria for the existence of at least one positive solution and two positive solutions are obtained respectively for the third-order nonlinear difference equation:
satisfying several different types of boundary conditions. The second part is devoted to determine values of A for which there exist at least one positive solution and two positive solutions for
satisfying several different types of boundary conditions. We have presented different function expression of Green functions. So the judgement of the existence on one pos-itive solution and two positive solutions becomes convenient. At last,many examples have given to demonstrate the applications of our theorems.
引文
[1] Agarwal R P. Difference Equations and Inequalities: Theory, Methods and Applications[M]. New York: Marcel Dekker, 1992.
[2] Erbe L H, Xia H, Yu J S. Global stability of a linear nonautononious delay difference equations[J]. J. Diff. Eqs. Appl., 1995(1): 151-161.
[3] Gy6ri I, Lades G. Oscillation Theory of Delay Differential Equations with Applicotions[M]. Oxford: Oxford University Press, 1991.
[4] Kocic V L, Lades G. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications[M]. Boston: Kluwer Academic Publishers, 1993.
[5] Hatsunaga H, Hare T, Sakata S. Global attractivity for a nonlinear difference equation with variable delay [J]. Computer Math. Applic., 2001: 41: 543-551.
[6] Zhang S N,Wu S J. Stability of first order autonomous difference equations[J]. Journal of Shanghai Jiaotong University, 2000, 8: 1119-1121.
[7] Yu J S. Asymptotic stability for a linear difference equation with variable delay[J]. Computers Math Applic, 1998, 36(10-12): 203-210.
[8] Zhang S N. Boundedness of finite delay difference systems[J]. Ann. of Diff. Eqs., 1993, 9: 107-115.
[9] Leela S. Monotone method for second order periodic boundary value problems[J]. Nonlinear Anal, 1983, 7: 349-355.
[10] Weng P X. Method of lower and upper solutions for fourth-order FDE boundary value problem[J]. Journal of South China Normal University, 2000, 3: 1-6.
[11] Cabada A, Nieto J J. A generation of the monotone iterative technique for nonlinear second-order periodic boundary value problems[J]. J. Math. AnaL. Appl., 1990, 151: 181-189.
[12] Cabada A. The method of lower and upper solutions for second, third, forth, and higher order boundary value problens[J]. J. Math. Anal. Appl., 1994, 185: 302-320.
[13] Weng P X. Boundary value problem for second order selfadjoint functional differential equation[J]. Ann. of Diff. Eqs., 1996, 1: 89-98.
[14] Weng P X. Boundary value problem for mixed type functional differential system[J]. Ann. of Diff. Eqs., 2000, 1: 98-110.
[15] Ge W G. On the existence of harmonic solutions of lienard system[J]. Nonlinear Anal., 1991, 16(2): 183-190.
[16] Mawhin J, Willem M. Multiple solutions of the periodic boundary value problem
for some forced pendulum-type equations[J]. J. Diff. Eqs., 1984, 52: 264-287.
[17] Zelati V C. Periodic solutions of dynamical systems with bounded potential[J]. J. Diff. Eqs., 1987, 67: 400-413.
[18] Lassoued L. Periodic solutions of a second order superquadratie system with a change of sign in potential[J]. J. Diff. Eqs., 1991, 93: 1-18.
[19] Krasnosel'skii M A. Positive Solutions of Operator Equations[M]. Noordhoff, Groningen, 1964.
[20] Guo D, and Lakshmikantham V. Nonlinear Problems in Abstract Cones[M]. Academic Press, San Diego, 1988.
[21] Figueiredo D G, Lions P L,and Nussbaum R D. A priors estimates and existence of positive solutions of semilinear elliptic equations[J]. J. Math. Puss Appl 1982, 61: 41-63.
[22] Fink A M, Gatica J A, and Hernandez G E. Eigenvalues of generalized Gelland models[J]. Nonlinear Anal. 1993, 20: 1453-1468.
[23] Kuiper H J. On positive solutions of nonlinear elliptic eigenvalue problems[J]. Rend. Mat. Cire. Palermo, Serie 11, Tom. 1979, 11: 113-158.
[24] Sanchez L. Positive solutions for a class of semilinear two-point boundary value problems[J]. Bull. Austral. Math. Soc., 1992, 45: 439-451.
[25] Erbe L H, and Wang H. On the existence of positive solutions of ordinary differential equations[J]. Proc. Amer. Math. Soc., 1994, 120: 743-748.
[26] Eloe P W, Henderson J. Positive solutions for higher order ordinary differential equations[J]. Electronic J. of Differential Equations, 1995, 3: 1-8.
[27] Eloe P W, Henderson J. Positive solutions for (n-1, 1) boundary value problems[J]. Nonlinear Analysis, 1997, 28: 1669-1680.
[28] Merdivenci F. Two positive solutions of a boundary value problem for difference equations[J]. J. Diff. Eqs. Appl, 1995, 1: 253-270.
[29] Merdivenei F. Green's matrices and positive solutions of a discrete boundary value problem[J]. Pan. Am. Math. J., 1995, 5: 25-42.
[30] Merdivenci F. Positive solutions for focal point problems for 2nih order difference equations[J]. Pan. Am. Math. J., 1995, 5: 71-82.
[31] Agarwal R P, Henderson J. Positive solutions and nonlinear problems for thirdorder difference equations[J]. Computers Math. Appl, 1998, 36: 347-355.
[32] Hao Z C. Nonnegative Solutions for semilinear third-order difference equation boundary value problems [J]. Acta Mathematica Scientia, 2001, 21A(2): 225-229.
[33] Guo D J. Nonlinear functional analysis[M]. Jinan:Shandong Press of Science
and Technoligy, 1985(in Chinese).
[34] Eloe P W. A generalization of concavity, for finite differences[J]. J Math Anal Applic, 1998, 36(10-12): 109-113.
[2] Erbe L H, Xia H, Yu J S. Global stability of a linear nonautononious delay difference equations[J]. J. Diff. Eqs. Appl., 1995(1): 151-161.
[3] Gy6ri I, Lades G. Oscillation Theory of Delay Differential Equations with Applicotions[M]. Oxford: Oxford University Press, 1991.
[4] Kocic V L, Lades G. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications[M]. Boston: Kluwer Academic Publishers, 1993.
[5] Hatsunaga H, Hare T, Sakata S. Global attractivity for a nonlinear difference equation with variable delay [J]. Computer Math. Applic., 2001: 41: 543-551.
[6] Zhang S N,Wu S J. Stability of first order autonomous difference equations[J]. Journal of Shanghai Jiaotong University, 2000, 8: 1119-1121.
[7] Yu J S. Asymptotic stability for a linear difference equation with variable delay[J]. Computers Math Applic, 1998, 36(10-12): 203-210.
[8] Zhang S N. Boundedness of finite delay difference systems[J]. Ann. of Diff. Eqs., 1993, 9: 107-115.
[9] Leela S. Monotone method for second order periodic boundary value problems[J]. Nonlinear Anal, 1983, 7: 349-355.
[10] Weng P X. Method of lower and upper solutions for fourth-order FDE boundary value problem[J]. Journal of South China Normal University, 2000, 3: 1-6.
[11] Cabada A, Nieto J J. A generation of the monotone iterative technique for nonlinear second-order periodic boundary value problems[J]. J. Math. AnaL. Appl., 1990, 151: 181-189.
[12] Cabada A. The method of lower and upper solutions for second, third, forth, and higher order boundary value problens[J]. J. Math. Anal. Appl., 1994, 185: 302-320.
[13] Weng P X. Boundary value problem for second order selfadjoint functional differential equation[J]. Ann. of Diff. Eqs., 1996, 1: 89-98.
[14] Weng P X. Boundary value problem for mixed type functional differential system[J]. Ann. of Diff. Eqs., 2000, 1: 98-110.
[15] Ge W G. On the existence of harmonic solutions of lienard system[J]. Nonlinear Anal., 1991, 16(2): 183-190.
[16] Mawhin J, Willem M. Multiple solutions of the periodic boundary value problem
for some forced pendulum-type equations[J]. J. Diff. Eqs., 1984, 52: 264-287.
[17] Zelati V C. Periodic solutions of dynamical systems with bounded potential[J]. J. Diff. Eqs., 1987, 67: 400-413.
[18] Lassoued L. Periodic solutions of a second order superquadratie system with a change of sign in potential[J]. J. Diff. Eqs., 1991, 93: 1-18.
[19] Krasnosel'skii M A. Positive Solutions of Operator Equations[M]. Noordhoff, Groningen, 1964.
[20] Guo D, and Lakshmikantham V. Nonlinear Problems in Abstract Cones[M]. Academic Press, San Diego, 1988.
[21] Figueiredo D G, Lions P L,and Nussbaum R D. A priors estimates and existence of positive solutions of semilinear elliptic equations[J]. J. Math. Puss Appl 1982, 61: 41-63.
[22] Fink A M, Gatica J A, and Hernandez G E. Eigenvalues of generalized Gelland models[J]. Nonlinear Anal. 1993, 20: 1453-1468.
[23] Kuiper H J. On positive solutions of nonlinear elliptic eigenvalue problems[J]. Rend. Mat. Cire. Palermo, Serie 11, Tom. 1979, 11: 113-158.
[24] Sanchez L. Positive solutions for a class of semilinear two-point boundary value problems[J]. Bull. Austral. Math. Soc., 1992, 45: 439-451.
[25] Erbe L H, and Wang H. On the existence of positive solutions of ordinary differential equations[J]. Proc. Amer. Math. Soc., 1994, 120: 743-748.
[26] Eloe P W, Henderson J. Positive solutions for higher order ordinary differential equations[J]. Electronic J. of Differential Equations, 1995, 3: 1-8.
[27] Eloe P W, Henderson J. Positive solutions for (n-1, 1) boundary value problems[J]. Nonlinear Analysis, 1997, 28: 1669-1680.
[28] Merdivenci F. Two positive solutions of a boundary value problem for difference equations[J]. J. Diff. Eqs. Appl, 1995, 1: 253-270.
[29] Merdivenei F. Green's matrices and positive solutions of a discrete boundary value problem[J]. Pan. Am. Math. J., 1995, 5: 25-42.
[30] Merdivenci F. Positive solutions for focal point problems for 2nih order difference equations[J]. Pan. Am. Math. J., 1995, 5: 71-82.
[31] Agarwal R P, Henderson J. Positive solutions and nonlinear problems for thirdorder difference equations[J]. Computers Math. Appl, 1998, 36: 347-355.
[32] Hao Z C. Nonnegative Solutions for semilinear third-order difference equation boundary value problems [J]. Acta Mathematica Scientia, 2001, 21A(2): 225-229.
[33] Guo D J. Nonlinear functional analysis[M]. Jinan:Shandong Press of Science
and Technoligy, 1985(in Chinese).
[34] Eloe P W. A generalization of concavity, for finite differences[J]. J Math Anal Applic, 1998, 36(10-12): 109-113.