Solution estimates for a system of nonlinear integral equations arising in optometry (Q1751564)

This paper is devoted to a boundary value problem concerned to corneal topography modelling \[ -\left(\frac{h'}{\sqrt{1+h^\prime{^2}}}\right)'+ah=\frac{b}{\sqrt{1+h^\prime{{^2}}}}, \; h'(0)=0, \; h(1)=0, \tag{1} \] where \[ 0\leq t\leq 1, \; a,b\in\mathbb{R}_{+}. \] This Problem (1) is then transformed into the following system of integral equations \[ \begin{cases} x(t)=\sqrt{1+x^2(t)}\int\limits_0^t\left(ay(s)-\frac{b}{\sqrt{1+x^2(s)}}\right)ds, \\ y(t)=-\int\limits_t^1x(s)ds. \end{cases} \tag{2} \] The authors construct lower and upper estimates that bound the components of the exact solution to the System (2). These results generalize some of the recent work by other authors. In conclusion, some numerical examples of analytical estimates are given.