Existence results and approximation methods for functional ordinary differential equations with singular diffusion-type differential operators
In a previous paper, the first and the third author [Nonlinear Anal., theory Methods Appl. 42, No. 8, 1377-1796 (2000; Zbl 0964.34016)] proved the existence of extremal solutions \(u^*\) and \(u_*\) to an ordinary differential equation of the type \[ (k(u(x))u'(x))'=f(x,u(x))u'(x)+g(x,u(x))u(x), \quad \text{a.e. in } I=[0,\tau], \tag{1} \] with the initial data \[ u(0)=0,\quad k(u(0))u'(0)=0,\tag{2} \] where \(k\) is allowed to vanish for some values of \(u\), and therefore (1) is singular at these points. This paper continuous this research by considering the functional-differential equation \[ \frac{d}{dx}(k(u(x),u)u'(x))=f(x,u(x),u)u'(x)+g(x,u(x),u)u(x), \quad \text{a.e. in } I=[0,\tau],\tag{3} \] together with the initial condition \[ u(0)=0,\quad k(u(0),u)u'(0)=0.\tag{4} \] The authors give sufficient conditions for the existence of extremal solutions \(u^*\) and \(u_*\) to (3)--(4) and an approximate form of \(u^*\) and \(u_*\). The method which they use is the monotone iterative technique.
singular diffusion-type differential operators
scientific article; zbMATH DE number 1781556