Green functions and positive solutions for boundary value problems of third-order difference equations

In this paper, we investigate the boundary value problems for the third-order nonlinear difference equation @D^3u(t-2)+a(t)f(u(t))=0,t@?[2,T+2], satisfying five kinds of different boundary value conditions. The Green functions are constructed carefully and their bounds are evaluated. Some verifiable criteria for the existence of at least one positive solution and two positive solutions are obtained by using the Green functions and the evaluations of bounds. Some examples are given to demonstrate the applications of the theorems obtained.

[1]  Patricia J. Y. Wong,et al.  Positive solutions for a system of nonpositive difference equations , 2001 .

[2]  Johnny Henderson,et al.  Twin solutions of boundary value problems for ordinary differential equations and finite difference equations , 2001 .

[3]  Ravi P. Agarwal,et al.  Existence theorems for a system of difference equations with (n, p)-type conditions , 2001, Appl. Math. Comput..

[4]  Ferhan Merdivenci,et al.  Two positive solutions of a boundary value problem for difference equations , 1995 .

[5]  J. Yu,et al.  Global stability of a linear nonautonomous delay difference equation 1 , 1995 .

[6]  K. Deimling.  Nonlinear functional analysis , 1985 .

[7]  Patricia J. Y. Wong,et al.  Positive Solutions of Difference Equations with Two-Point Right Focal Boundary Conditions , 1998 .

[8]  Ravi P. Agarwal,et al.  Double positive solutions of (n, p) boundary value problems for higher order difference equations , 1996 .

[9]  Paul W. Eloe,et al.  A generalization of concavity for finite differences , 1998 .

[10]  Ravi P. Agarwal,et al.  Positive solutions and nonlinear eigenvalue problems for third-order difference equations , 1998 .

[11]  Tadayuki Hara,et al.  Global attractivity for a nonlinear difference equation with variable delay , 2001 .

[12]  Jianshe Yu,et al.  Asymptotic stability for a linear difference equation with variable delay , 1998 .

[13]  Shunian Zhang.  Boundedness of infinite delay difference systems , 1994 .

[14]  Ravi P. Agarwal,et al.  Multiple solutions for higher-order difference equations , 1999 .

[15]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[16]  V. Kocić,et al.  Global Behavior of Nonlinear Difference Equations of Higher Order with Applications , 1993 .