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Regions of existence for a class of nonlinear diffusion type problem. (English) Zbl 1474.34151

Summary: The regions of existence are established for a class of two point nonlinear diffusion type boundary value problems (NDBVP) \[ \begin{aligned} & - s''(x)- ns'(x)- \frac{m}{x}s'(x) =f(x, s), \qquad m > 0,\, n \in \mathbb{R}, \qquad x \in (0,1),\\ & s'(0) = 0, \qquad a_1s(1) +a_2s'(1) =C, \end{aligned} \] where \(a_1>0\), \(a_2 \geq 0\), \(C \in \mathbb{R}\). These problems arise very frequently in many branches of engineering, applied mathematics, astronomy, biological system and modern science (see the existing literature of this paper). By using the concept of upper and lower solutions with monotone constructive technique, we derive some sufficient conditions for existence in the regions where \(\frac{\partial f} {\partial s} \geq 0\) and \(\frac{\partial f}{\partial s} \leq 0\). Theoretical methods are applied for a set of problems which arise in real life.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations

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