在本論文中，我們將探討以下型態的非線性邊界值問題正解之存在性與非存在性：
(*) u''(t)+f(t,u(t))=0, 0 <t <1;
u屬於B，其中B為適當的邊界條件。給予f(t,.)適當的條件，利用 Krasnoselskii 的固定點定理，我們將給出在幾種不同邊界值條件下的微分方程式多重正解的存在或非存在性。
經由(*) 問題的探討，我們將一般的常微分方程式推廣至延遲的微分方程式
u''(t)+f(t,u(t+s))=0, 0<t <1 , -r < s < a,
來討論其解的存在性。更經由上述的延遲方程式的研究，
我們發現在時標(time scale)所定義的測度鏈(measure chain)上的微分方程式，
(**)u''(t)+f(t,u(g(t)))=0, 0<t<1.
除了隱含方程式上的延遲性外，更可將一般的微分與差分方程做一個連結。
因此我們進一步討論(**)問題的正解存在性。

In this article, we concerned with the existence and nonexistence of positive
solutions of the following nonlinear boundary value problem of the form:
(*) u''(t)+f(t,u(t))=0, 0 <t <1.
Under the suitable condition f(t,.), by using Krasnoselskii''s fixed point theorem, we will give the existence and nonexistence of multiple positive solutions under several different boundary value conditions for the differential equations.
It follows from the boundary value problem (*), we can extend general ordinary diferential equation to the delay differential equations
u''(t)+f(t,u(t+s))=0, 0<t <1 , -r < s < a,
and consider the existence of positive solutions.
Moreover, it follows from above delay differential equations, we find that the
differential equation on a measure chain defined on time scale of the form:
(**) u''(t)+f(t,u(g(t)))=0, 0<t<1;
combine the difference and differential equations. So we deal with the existence of positive solutions of the problem (**).

1. R. P. Agarwal and M. Bohner, Basic calculus on tine scales and some of its applications, Results Math., 35 (1999), 3-22.
2. R. P. Agarwal, M. Bohner and P. J. Y. Wong, Positive solutions and eigenvalues of conjugate boundary value problems, Proc. Edinburgh Math. Soc., 42 (1999), 349-374.
3. R. P. Agarwal, M. Bohner and P. J. Y. Wong, Sturm-Liouville eigenvalue problems on time scales, Applied Mathematics and Computation, 99 (1999), 153-166.
4. R. P. Agarwal, M. Bohner and P. J. Y. Wong, Eigenvalues and eigenfunctions of discrete conjugate boundary value problems, Comput. Math. Appl., 38 (1999), 159-183.
5. R. P. Agarwal and F. H. Wong, Existence of positive solutions for higher order boundary value problems, Nolinear Stuies, 5 (1998), 15-24.
6. R. P. Agarwal, F. H. Wong and W. C. Lian, Existence of positive solutions of nonlinear second order differential systems, Appliciable Analysis, 63 (1996), 375-387.
7. R. P. Agarwal, F. H. Wong and W. C. Lian, Positive solutions for nonlinear singular boundary value problem, Appl. Math. Lett., 12 (1999), 115-120.
8. F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York -London 1964.
9. C. Bandle, C. V. Coffman and M. Marcus, Nonlinear elliptic problems in annular domains, J. Differential Equations, 69 (1987), 322-345.
10. C. Bandle and M. K. Kwong, Semilinear Elliptic problems in annular domains,
J. Appl. Math. Phys., 40 (1989), 245-257.
11. M. Bohner and A. Peterson, Dynamic Equations On Time Scales, Birkhauser, Boston, 2001.
12. C. J. Chyan and J. Handerson, Eigenvalue problems for nonlinear differential equations on a measure chain, J. Math. Anal. Appl., 245 (2000), 547-559.
13. C. J. Chyan and J. Henderson, Positive solutions for singular higher order nonlinear equations, Diff. Eqns. Dyn. Sys., 2 (1994), 153-160.
14. K. Deimling, Nonlinear Functional Analysis, Springer, New York, 1985.
15. L. H. Erbe, Boundary value problems for ordinary differential equations,
Rocky Mountain J. Math., 1 (1970), 709-729.
16. L. H. Erbe, S. Hu and H. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184 (1994), 640-648.
17. L. H. Erbe, S. Hilger, Sturm theory on measure chains, Differential Equations and Dynamical Systems, 1 (1993), 223-246.
18. L. H. Erbe and Q. K. Kong, Boundary value problems for singular second-order functional differential equations, Comput. Math. Appl., 53 (1994), 377-388.
19. L. Erbe and A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain, Math. Comput. Modelling, 32 (2000), 571-585.
20. L. Erbe and A. Peterson, Eigenvalue conditions and positive solutions,
J. Differ. Equations Appl., 6 (2000), 165-191.
21. L. Erbe and A. Peterson, Green''s functions and comparison theorems for differential equations on measure chains, Dynamics of Continuous, discrete and Impulsive Systems, 6 (1999), 121-137.
22. L. H. Erbe and H. Wang, Existence and nonexistence of positive solutions for elliptic equaions in a annulus, World Sci. Publ. Comp., 3 (1994), 207-217.
23. L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), 743-748.
24. D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Poston 1988.
25. C. P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second ordinary differential equation, J. Math. Anal. Appl., 168 (1992), 540-551.
26. C. P. Gupta, A note on a second order three-point boundary value problem,
J. Math. Anal. Appl., 186 (1994), 277-281.
27. J. Henderson and W. N. Hudson, eigenvalue problems for nonlinear functional differential equations, Comm. Appl. Nonli. Anal., 3 (1996), 51-58.
28. J. Henderson and H. B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc., 128 (2000), 2373-2379.
29. S. Hilger, Analysis on measure chains$-$A unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18-56.
30. C. H. Hong, C. F. Lee. F. H. Wong and C. C. Yeh, Existence of positive solution for functional differential equations, Comput. Math. Appl., 40 (2000), 783-792.
31. C. H. Hong and C. C Yeh, Postive solutions for eigenvalue problems on a measure chain, to appear in Nonliear Analysis TM&A.
32. G. V. A. Il''in andE. I. Moiseev, Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finie difference aspects, Differential Equations, 23 (1987), 803-810.
33. G. V. A. Il''in andE. I. Moiseev, Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator in its differential and finie difference aspects, Differential Equations, 23(8) (1987), 979-987.
34. J. Kato, Functional Equations, 1974. (Japanese)
35. B. Kaymakcalan, V. Lakshmikantham and S. Sivasundaram, Dynamic Systems on a Measure Chain, Kluwer Academic Boston 1996.
36. M. A. Krasnoselskii, Positive solutions of operator equations, Noordhoff, Groningen 1964.
37. J. W. Lee and D. O''Regan, Existence results for differential delay equations-I, J. Diff. Eqns., 102 (1993), 342-359.
38. J. W. Lee and D. O''Regan, Existence results for differential delay equations-II, Nonlinear Analysis TM&A, 17 (1991), 683-702.
39. R. Leggett and L. Williams, Multiple positive fixed points of nonlinear operators on oredred Banach spaces, Indiana University Mathematics Journal, 28 (1979), 673-688.
40. W. C. Lian, C. C. Chou, C. T. Liu and F. H. Wong, Existence of solutions of nonlinear BVPs of second order differential equations on measure chains, Math. & Comp. Mod., 34 (2001), 821-837.
41. W. C. Lian, F. H. Wong and C. C. Yeh, On the positive solutions of nonlinear second-order differential equations, Proc. Amer. Math. Soc., 124 (1996), 1117-1126.
42. R. Ma, Existence theorems for a second-order three-point boundary value problems, J. Math. Anal. Appl., 212 (1997), 430-442.
43. R. Ma, Multiplicity of positive solutions for second-order three-point boundary value problems, Comput. Math. Appl., 40 (2000), 193-204.
44. S. K. Ntouyas, Y. G. Sficas and P. C. Tsamatos, An existence principle for boundary value problems for second order functional differential equations, Nonlinear Analysis TM&A, 20 (1993), 215-222.
45. D. Tanuton and William K. C. Yin, Existence of some functional differential equations, Comm. Appl. Nonli. Anal., 4 (1997), 31-43.
46. P. Weng and D. Jiang, Existence of positive solutions for boundary value problem of second-order FDE, Comput. Math. Appl., 37 (1999), 1-9.
47. M. R. Zhang and Y. R. Han, On the applications of Leray-Schauder continuation theorem to boundary value problems of semilinear differential equations, Ann. Diff. Eqs., 109 (1997) 189-206.