Existence of positive solutions for second-order boundary value problem with one parameter

Abstract

In the case of not requiring $f(t,u)$ to be nonnegative, by transforming the boundary value problem into the integral equation system, and applying the fixed point index theory, the author studies the following second-order boundary value problem with one parameter

$$\{\begin{array}{cc}-u^{″}+\lambda u=f(t,u),\hfill & 0<t<1,\hfill \\ u(0)=u(1)=0,\hfill \end{array}$$

and obtains the results on the existence of its positive solution.