The Cauchy problem for certain generalized differential equations of second order with singularity (Q2708130)

Let \(X\), \(Y\) be real Banach spaces and let \(U\) and \(V\) be open subsets of \(X\) and \(Y\), respectively. Let \(h_1\), \(h_2\), \(f\) be some mappings from \(U\) into \(X\) and \(Y\) a mapping from \(U\times V\) into \(Y\). One defines a derivative \(\nabla_{h_1} f\) of \(f\) in the direction of \(h_1\) and a second-order derivative \(\nabla^2_{(h_2,h_1)}f\) in the direction of the pair \((h_2, h_1)\). The author obtains some sufficient conditions for the existence of solutions for the Cauchy problem NEWLINE\[NEWLINE\nabla^2_{(h_2, h_1)} f(x)+ a(\nabla_{h_1}f)(x)= F(x,f(x))\;(x\in U),\quad f(0)= \Delta,\quad (\nabla_{h_1} f)(a)= 0,NEWLINE\]NEWLINE where \(a\geq 0\), with the assumption that \(0\) is a singular point (i.e. \(h_1(0)= 0\) and \(h_2(0)= 0\)).