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Zbl 1145.34007
Graef, John R.; Kong, Lingju
Existence of solutions for nonlinear boundary value problems.
(English)
[J] Commun. Appl. Nonlinear Anal. 14, No. 1, 39-60 (2007). ISSN 1074-133X

The authors study a one-dimensional $\Phi$-Laplacian equation of the form $$(\Phi(u'))'+f(t,u,u')=0, \quad 0<t<1\tag1$$ together with the boundary conditions $$u(0)=g(u(t_1),u'(t_1),\dots,u(t_m),u'(t_m)),\;u(1)=h(u(t_1),u'(t_1),\dots,u(t_m),u'(t_m)). \tag2$$ The numbers $t_i$ belong to $[0,1]$; $f$ is Carathéodory in $]0,1[\times{\Bbb R}^2$; $g$ and $h$ are continuous and nondecreasing in the $u$-variables; $\Phi$ is an odd homeomorphism of ${\Bbb R}$. They introduce the notion of coupled lower and upper solutions as a pair of functions $\alpha\le\beta$ of $C^1[0,1]$ satisfying, in addition to the standard differential inequalities, boundary inequalities that make sense by virtue of a Nagumo condition that $f$ verifies with respect to the pair $\alpha$, $\beta$. Under these conditions, they show that the problem (1)-(2) has at least one solution between $\alpha$ and $\beta$. \par Several consequences follow. In particular, if $$\vert f(t,x,y_1)-f(t,x,y_2)\vert \le \chi(t)\vert \Phi(y_1)-\Phi(y_2)\vert$$ with $\chi\in L^1(0,1)$ and $f$ is nondecreasing in $x$, there exists a unique solution. Also, an application is given to the existence of radial solutions in an annulus $R_1\le\vert z\vert \le R_2$ of the equation $$\text{ div} (A(\vert \nabla u\vert )\nabla u) +f(\vert z\vert ,u)=0$$ with $$u(R_i)=\sum_{j=1}^m(a_{ij}u(R_i)+g_{ij}(u'(R_i)))+\lambda_i,\quad i=1,\,2.$$ Here $A(\vert x\vert )x$ is an increasing homeomorphism of ${\Bbb R}$, the $a_{ij}$ are nonnegative, $\sum_{ij}a_{ij}<1$ and $\sup_{y\in{\Bbb R}^m}\left(\vert \sum g_{1j}(y_j)\vert +\vert \sum g_{2j}(y_j)\vert \right)<\infty$.
[Luis Sanchez (Lisboa)]
MSC 2000:
*34B15 Nonlinear boundary value problems of ODE
34B18 Positive solutions of nonlinear boundary value problems
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: boundary value problem; Nagumo condition; coupled lower and upper solutions; fixed point theorem

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