Positive solutions for nonlocal differential equations with concave and convex coefficients

In this paper, we study the positive solutions for nonlocal differential equations with concave and convex coefficients: \(\begin{aligned} -A\left( \int ^1_0 (u^p(s)+u^q(s))ds\right) u''(t)=f(t,u(t)),\quad t\in (0,1), \end{aligned}\) $$\begin{array}{c}\hfill -A\left({\int }_0^1,(u^p(s)+u^q(s)),d,s\right)u^{\prime \prime }(t)=f(t,u(t)),t\in (0,1),\end{array}$$ where \(0<p<1\le q.\) $0<p<1\le q.$ Using the fixed point index theory and fixed point theorems on cones, existence and multiplicity results are obtained, when the nonlinear term f(t, x) is continuous, has a singularity at \(x=0\) $x=0$ , changes sign, respectively.