Study on monotonicity of boundary value problems of differential equations

This article presents two new test methods to decide monotonic increasing or decreasing properties of differential equations. For simple problems, the eigenfunction test method can be applied to obtain their exact monotonicity or their loosest conditional expression, and can contribute to probes into accumulated effect of various terms in equations on their monotonicity. Secondly, the trial function test method sets relatively strict conditions for discrimination of monotonicity, but it can verify complex non-linear problems in a simple and rapid way. Therefore, these two similar test methods will help to further understand theories on monotonicity, thus making it possible to apply monotonic equations more widely to other problems.