Global pointwise estimates of positive solutions to sublinear equations

Bilateral pointwise estimates are provided for positive solutions $u$ to the sublinear integral equation
$$ u = \mathbf{G}(\sigma u^q) + f \textrm{ in } \Omega, $$
for $0 < q < 1$, where $\sigma\ge 0$ is a measurable function or a Radon measure, $ f \ge 0$, and $\mathbf{G}$ is the integral operator associated with a positive kernel $G$ on $\Omega\times\Omega$. The main results, which include the existence criteria and uniqueness of solutions, hold true for quasi-metric, or quasi-metrically modifiable kernels $G$. As a consequence, bilateral estimates, are obtained, along with existence and uniqueness, for positive solutions $u$, possibly unbounded, to sublinear elliptic equations involving the fractional Laplacian,
$$ (-\Delta)^{\frac{\alpha}{2}} u = \sigma u^q + \mu \textrm{ in } \Omega, u=0 \textrm{ in } \Omega^c, $$
where $0<q<1$, and $\mu, \sigma \ge 0$ are measurable functions, or Radon measures, on a bounded uniform domain $\Omega \subset \mathbb{R}^n$ for $0 < \alpha \le 2$, or on the entire space $\mathbb{R}^n$, a ball or half-space, for $0 < \alpha <n$.