Regions of existence for a class of nonlinear diffusion type problems

The regions of existence are established for a class of two point nonlinear diffusion type boundary value problems (NDBVP) \begin{eqnarray*} &&\label{abst-intr-1} -s''(x)-ns'(x)-\frac{m}{x}s'(x)=f(x,s), \qquad m>0,~n\in \mathbb{R},\qquad x\in(0,1),\\ &&\label{abst-intr-2} s'(0)=0, \qquad a_{1}s(1)+a_{2}s'(1)=C, \end{eqnarray*} where $a_{1}>0,$ $a_{2}\geq0,~ C\in\mathbb{R}$. These problems arise very frequently in many branches of engineering, applied mathematics, astronomy, biological system and modern science (see \cite{Gatica1989, GRAY1980, Baxley1991, Chandershekhar1939, Duggan1986, Chambre1952}). By using the concept of upper and lower solutions with monotone constructive technique, we derive some sufficient conditions for existence in the regions where $\frac{\partial f}{\partial s}\geq0$ and $\frac{\partial f}{\partial s}\leq0$. Theoretical methods are applied for a set of problems which arise in real life.