Implicit function theorem via the DSM (Q2654015)

The article deals with the following solvability problem \[ F(u) = f; \quad F: H_a \to H_{a + \delta}, \quad u \in B(U,R) = B_a(U,R),\tag{1} \] where \(B_a(U,R) = \{u: \|u - U\|_a \leq R\}\), \(\delta\) is a positive constant, and the operator \(F: H_a \to H_{a+\delta}\) is continuous. It is assumed that \(A(u) = F'(u)\) exists, is an isomorphism of \(H_a\) onto \(H_{a+\delta}\) (\(c_0\|v\|_a \leq \|A(u)v\|_{a+\delta} \leq c_0'\|v\|_a\) for \(u,v \in B(U,R)\)), and satisfies the inequalities \[ \|A^{-1}(v)A(w)\|_a \leq c, \quad v, w \in B(U,R), \] \[ \|A^{-1}(u)(A(u) - A(v))\|_a \leq c\|u - v\|_a, \quad u, v \in B(U,R), \] (\(c_0, c_0', c\) are positive constants). The main result is the solvability theorem for (1) provided that \(\|f - F(U)\|_{a+\delta}\) is sufficiently small. The method of the proof is based on the analysis of the Cauchy problem \[ \dot u = - [F'(u)]^{-1}(F(u) - h), \quad u(0) = u_0. \] As an illustration of this solvability result, the integral equation \[ \int_0^x u^2(s) \, ds = h(x) \] is considered.