On the oscillation property of Green's function of a fourth-order discontinuous boundary-value problem
In this paper, the author gives a sufficient condition for the Green's function of the problem \[ (p(x)u'')''-(q(x)u')'+h(x)u=f(x),\quad x\in\Gamma\{a_i\}_{i=1}^n, \] \[ \Delta u(a_i)=0,\quad \Delta u'(a_i)=0,\quad \Delta(pu'')(a_i)=0, \] \[ \Delta[(p(a_i)u''(a_i))'-q(a_i)u'(a_i)]+\delta(a_i)u(a_i)=f(a_i), \] \[ u(a)+\alpha(a)[(p(a)u''(a))'-q(a)u'(a)]=0,\quad \vartheta(a)u'(a)-\beta(a)u''(a)=0, \] \[ u(b)-\alpha(b)[(p(b)u''(b))'-q(b)u'(b)]=0,\quad \vartheta(b)u'(b)+\beta(b)u''(b)=0, \] to have the oscillation property, where \(\Gamma\) is an interval \((a,b)\) with a fixed finite set of points \(a=a_0<a_1<\dots<a_n<a_{n+1}=b\), \(C[\Gamma]\) denotes the space of functions uniformly continuous on each of the intervals \((a_{i-1},a_i)\), \(C^k[\Gamma]\) denotes the space of functions from \(C[\Gamma]\) which have \(k\) derivatives from the class \(C[\Gamma]\), \(p\in C^2[\Gamma]\), \(\inf_{x\in\Gamma}p(x)>0\), \(q(x)\in C^1[\Gamma]\), \(q\geq 0\) on \(\Gamma\), \(h\in C[\Gamma]\), \(h(x)\geq 0\) on \(\Gamma\), \(f\in C[\Gamma]\), \(\alpha(.)\), \(\vartheta(.)\) and \(\beta(.)\) are nonnegative numbers such that \(\alpha(.)+\beta(.)\neq 0\), \(\delta(a_i)\geq 0\) for \(i=1,\dots,n\).
scientific article; zbMATH DE number 6682248
scientific article