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Zbl 1007.34019
Yazidi, Najla
Monotone method for singular Neumann problem. (English)
[J] Nonlinear Anal., Theory Methods Appl. 49, No.5, A, 589-602 (2002). ISSN 0362-546X

The existence of the extremal solutions to the BVP $$(1/A)(Au')'+ f(\cdot, u)= 0\quad\text{on }(0,\omega),\quad \lim_{t\to 0} A(t) u'(t)= a,\quad \lim_{t\to\omega} A(t) u'(t)= b,$$ is proved; here, $\omega\in (0,\infty)$, $A(t)\in C[0,\omega]$ is positive on $(0,\omega)$, $(a,b)\in \bbfR^2$ and $f\in C([0,\omega]\times \bbfR)$.\par The approach used relies on a preliminary study of appropriate eigenvalue problems and the monotone method.\par In the main result, the author assumes that the problem considered has a lower solution $\alpha$ and an upper solution $\beta$ such that $\alpha\le \beta$ and the map $x\to \lambda q x+ f(\cdot, x)$ is nonincreasing on $[\alpha,\beta]$ for some $\lambda$, where $q\in C(0,\omega)$ is a suitable function, and $\lambda$ is a $q$-eigenvalue of the operator $Lu= (1/A)(Au')'$, i.e. $\lambda$ is such that the problem $$(1/A)(Au')'= \lambda qu\quad\text{on }(0,\omega),\quad Au'(0)= 0,\quad u(\omega)= 0,$$ has a nontrivial solution in $C[0,\omega]\cap C^1(0,\omega)$.
[Petio S.Kelevedjiev (Sliven)]

MSC 2000:: *34B15 Nonlinear boundary value problems of ODE

Keywords: Neumann problem; existence; extremal solutions; eigenvalue problem

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