Multiple positive solutions for a class of Neumann problems

We study the existence of multiple positive solutions of the Neumann problem \begin{equation*} \begin{split} -u''(x)&=\lambda f(u(x)), \qquad x\in(0,1),\\ u'(0)&=0=u'(1), \end{split} \end{equation*} where $\lambda$ is a positive parameter, $f\in C([0,\infty),\mathbb{R})$ and for some $\beta>0$ such that $f(0)=0$, $f(s)>0$ for $s\in(\beta,\infty)$, $f(s)\beta)$ is the unique positive zero of $F(s)=\int_0^sf(t)\,dt$. In particular, we prove that there exist at least $2n+1$ positive solutions for $\lambda\in \big(\frac{n^2\pi^2}{f'(\beta)},\infty\big)$, where $n\in \mathbb{N}$. The proof of our main result is based upon the bifurcation and continuation methods

Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary condition D0+αu(t)+a(t)f(t,u(t))=0, 0<t<1, u(0)=0, u(1)=∑i=1∞αiu(ξi) is considered, where 1<α≤2 is a real number, D0+α is the standard Riemann-Liouville differentiation, and ξi∈(0,1),  αi∈[0,∞) with ∑i=1∞αiξiα-1<1, a(t)∈C([0,1],[0,∞)),  f(t,u)∈C([0,1]×[0,∞),[0,∞))