Regions of existence and uniqueness for singular nonlinear diffusion problems

This paper presents a novel approach for constructing the lower and upper boundaries of closed regions where solutions to the singular nonlinear diffusion problems \(\begin{aligned} \begin{aligned} y''(x)+ \frac{m}{x}y'(x)= f(x,y(x)), \quad x \in (0,1], \quad m \ge 0 , \\ y'(0) = 0, \quad Ay(1)+By'(1) = C, \quad A>0, B \ge 0, C \ge 0 , \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{c}\hfill y^{\prime \prime }(x)+\frac{m}{x}{y}^{\prime }(x)=f(x,y(x)),x\in (0,1],m\ge 0,\\ \hfill y^{\prime }(0)=0,Ay(1)+B{y}^{\prime }(1)=C,A>0,B\ge 0,C\ge 0,\end{array}\end{array}$$ exist. This existence result is proved using the method of lower and upper solutions with monotone iterative technique under the restriction that f(x, y) is continuous in \(x \in [0,1]\) $x\in [0,1]$ and non-increasing in y in such regions. Additional uniqueness criteria is also established. The approach is illustrated on four singular nonlinear diffusion problems including some real life applications.