A quadrature method for classes of multi-parameter two point boundary value problems. (English) Zbl 0838.34020
Summary: In the recent past many results have been established on non-negative solutions to boundary value problems of the form \(- u''(x)= \lambda f(u(x))\), \(0< x< 1\), \(u(0)= 0= u(1)\), where \(\lambda> 0\), \(f(0)< 0\) (semi-positone problems). In this paper, we discuss the existence of non-negative solutions when the boundary conditions are replaced by \(u(0)= 0= u(1)+ \alpha u'(1)\), where \(\alpha> 0\), via a quadrature method. In particular, we consider the case when \(f\) is superlinear. We also establish critical bounds for the bifurcation diagram of these non-negative solutions and study its evolution within these bounds as \(\alpha\) varies. We further discuss the existence of multiple positive solutions for ranges (which are independent of \(\alpha\)) of \(\lambda\). We discuss these results both for semi-positone as well as positone \((f(0)> 0)\) problems.
MSC:
34B15 | Nonlinear boundary value problems for ordinary differential equations |
34C23 | Bifurcation theory for ordinary differential equations |
37G99 | Local and nonlocal bifurcation theory for dynamical systems |
Keywords:
boundary value problems; non-negative solutions; quadrature method; bifurcation diagram; multiple positive solutionsReferences:
[1] | Alfonso Castro, Proc. Roy. Soc. Edin 108 pp 291– (1988) |
[2] | DOI: 10.1016/0096-3003(92)90076-D · Zbl 0765.34025 · doi:10.1016/0096-3003(92)90076-D |
[3] | Allegretto W., Nonlinear Anal.(to appear) |
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