Application of a numerical-analytic method for studying a three-point boundary value problem with parameters (Q2730547)

The author deals with the existence and construction of approximate solutions to a nonlinear system of first-order differential equations with the unknown scalar parameters \(\lambda_1,\ldots,\lambda_{k}\): \(dx/dt=f(t,x,\lambda_1,\ldots,\lambda_{k})\), with the three-point boundary condition \(Ax(0)+Bx(t_1)+Cx(T)=d\), and the initial condition \(x_{i}(0)=x_{0,i}, i=1,\ldots,k\), with \(x,f,d\in E_{n}\), \(n\geq k\); \(A,B,C\) are some \(n\)-dimensional matrices; \({\text det} (t_1 B/T+ C)\neq 0\). Using a modification of the numerical-analytic method of successive approximations proposed by A. M. Samojlenko the author constructs a sequence of vector-valued functions which satisfy the given initial and boundary conditions and converges uniformly to the solution to some perturbed system of differential equations.