本論文主要分兩個部分。首先，我們探討廣義區間上二階非線性常微分方程，搭配兩點邊界值條件之下正解的存在性，並進一步以劣線性及超線性的觀點提出關於外力項的限制條件。此外，將本文的結果應用於連續型的方程，也能將先前的研究結果加以推廣。
其次，我們探討類線性隱函數方程之零解的穩定性。我們先以指標的概念處理存在性，再進一步以Lyapunov函數討論其穩定性。

In this thesis,we give a criterion for the existence of positive solutions for nonlinear second order ordinary differential equations with two-point boundary value conditions on time scales.
  Moreover, for some source terms which are in the sense of sublinear or superlinear, we also formulate corollaries and examples for applications and we improve previous results.
  Second, we investigate the stability of the solution x=0 for a class of quasilinear implicit dynamic equations on time scales of the form $A_tx^{Delta}=f(t,x)$.
   We deal with an index concept to study the solvability and use Lyapunov functions as a tool to approach the stability problem.

1.Introduction.............................................1

2.Some Basic Notations of the Theory of the Analysis on Time Scales................................................3

3.Nonlinear two-point boundary value problems on time scales.....................................................7
3.1Introduction............................................7
3.2Some preliminaries and main result......................8
3.3Consequence and examples...............................11

4.Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales..................................16
4.1Introduction...........................................16
4.2Nonlinear Implicit Dynamic Equations on Time Scales....................................................18
4.3Quasilinear Implicit Dynamic Equations.................27
4.4Stability Theorem of Implicit Dynamic Equations........31
4.5Conclusion.............................................48

Reference.................................................49

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