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Existence of solutions for nonlinear boundary value problems. (English) Zbl 1145.34007

The authors study a one-dimensional \(\Phi\)-Laplacian equation of the form \[ (\Phi(u'))'+f(t,u,u')=0, \quad 0<t<1\tag{1} \]
together with the boundary conditions
\[ u(0)=g(u(t_1),u'(t_1),\dots,u(t_m),u'(t_m)),\;u(1)=h(u(t_1),u'(t_1),\dots,u(t_m),u'(t_m)). \tag{2} \]
The numbers \(t_i\) belong to \([0,1]\); \(f\) is Carathéodory in \(]0,1[\times{\mathbb R}^2\); \(g\) and \(h\) are continuous and nondecreasing in the \(u\)-variables; \(\Phi\) is an odd homeomorphism of \({\mathbb R}\).
They introduce the notion of coupled lower and upper solutions as a pair of functions \(\alpha\leq\beta\) 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 \(\beta\).
Several consequences follow. In particular, if
\[ | f(t,x,y_1)-f(t,x,y_2)| \leq \chi(t)| \Phi(y_1)-\Phi(y_2)| \]
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 \(R_1\leq| z| \leq R_2\) of the equation
\[ \text{ div} (A(| \nabla u| )\nabla u) +f(| z| ,u)=0 \]
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 \(A(| x| )x\) is an increasing homeomorphism of \({\mathbb R}\), the \(a_{ij}\) are nonnegative, \(\sum_{ij}a_{ij}<1\) and \(\sup_{y\in{\mathbb R}^m}\left(| \sum g_{1j}(y_j)| +| \sum g_{2j}(y_j)| \right)<\infty\).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
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