On the continuous dependence of solutions of nonlinear equations (Q1910840)

The author considers the boundary value problem (1) \(x' (t)= f(x, t)+ y(t)\), \(L(x)= r\), where \(f: \mathbb{R}^m \times [0, 1]\to \mathbb{R}^m\), \(L: C^0 ([ 0,1], \mathbb{R}^m)\to \mathbb{R}^m\) and \(y: [0, 1]\to \mathbb{R}^m\) are continuous, \(r\in \mathbb{R}^m\). The existence, the uniqueness and the continuous dependence of solutions to (1) is studied here. The proof is based on the Leray-Schauder degree theory. The case of the linear boundary value conditions either of the type \(x(1)- x(0) =r\) or \(x(1)+ x(0)= r\) are investigated in detail as well. Finally, problems for partial differential equations and functional differential equations are presented to illustrate the possibility of application of this approach for more general nonlinear equations.