Ordinary differential equations in the complex domain via topological transversality
In this paper the author studies the differential equation \(y^{(n)}=f(t,y,y',...,y^{(n-1)}),\) \(t\in U_ T\), with initial conditions \(y(0)=\alpha_ 0,...,v_{(0)}^{(n-1)}=\alpha_{n-1}\), where \(\alpha_ j\in {\mathbb{C}}\), \(j=0,1,...,n-1\), and \(f: \overline{U}_ T\times {\mathbb{C}}^ n\to {\mathbb{C}}\) is analytic in \(t,y,...,y^{(n-1)}\) for \(t\in U_ T\) and continuous in \(t,y,...,y^{(n-1)}\) for \(t\in \overline{U}_ T\), where \(U_ T=\{z: | z| <T\}\), \(T\geq 0\). For a given integer \(k\geq 0\) and a constant \(T>0\), let \(A^ k(\overline{U}_ T)\) denote the Banach space of functions g: \(\overline{U}_ T\to {\mathbb{C}}\) which are analytic on \(U_ T\) with \(g,...,g^{(k)}\) continuous on \(\overline{U}_ T\) with the norm \(\| g\|_ k\) defined as \(\max \{\| g\|_ 0,\| g'\|_ 0,...,\| g^{(k)}\|_ 0\}\), where \(\| g\|_ 0=\sup_{t\in \overline{U}_ T}| g(t)|\). Imposing upon \(| f(t,y,...,y^{(n-1)}|\) different types of growth conditions the author shows how to determine \(T>0\) such that the above initial value problem has at least one solution y in \(A^ n(\overline{U}_ T)\). In the case \(\alpha_ j=0\), \(j=0,...,n-1\), the author shows that if \[ | f(t,u_ 0,u_ 1,...,u_{n-1})| \leq A_ 0(t)| u_ 0| +...+A_{n-1}(t)| u_{n-1}| +C(t), \] where \(A_ 0(t),...,A_{n-1}(t)\), C(t)\(\geq 0\) are functions bounded on bounded sets, then the initial value problem has at least one solution in \(A^ n(\overline{U}_ T)\) provided \(0<T<T_ 0\), where \(T_ 0\) is the smallest positive root of \(A_ 0T^ n+A_ 1T^{n-1}+...+A_{n-1}T- 1=0\) and \(A_ 0,...,A_{n-1}\) are the upper bounds on \(\overline{U}_ T\) for \(A_ 0(t),...,A_{n-1}(t)\), respectively. The method used is extended also to the case where the values of \(y(t),y'(t),...,y^{(n- 1)}(t)\) are prescribed at different points as exemplified by the following result for \(n=2:\) The problem \[ y''=f(t,y,y'),\quad t\in U_ T,\quad y'(0)=0,\quad y(c)=0,\quad c\in \overline{U}_ T \] has at least one solution in \(A^ 2(\overline{U}_ T)\) provided the function f(t,u,p) satisfies \(| f(t,u,p)| \leq A(t)| u|^ 2+B(t)| p|^ 2+C(t),\) where A(t), B(t), C(t)\(\geq 0\) are bounded functions on bounded sets, and \(0<T<(-BC+(B^ 2C^ 2+2AC)^{1/2}/4AC)^{1/2},\) where A, B and C are the upper bounds of A(t), B(t) and C(t) for \(t\in \overline{U}_ T\), respectively.
scientific article; zbMATH DE number 4084074