Problem with integral boundary condition for a system of differential equations with transformed independent variable (Q2768802)

The author deals with the system of differential equations \(\dot x(t)=f(t,x(t),x(\lambda(t)))\) with the integral boundary condition \(\int_{0}^{T}x(t) dt=d\), where \(t\in[0,T]\) is an independent variable; \(x,f,d\in \mathbb{R}^{n}\); \(\lambda:[0,T]\to[0,T]\) is a continuous mapping. Let us consider a sequence of functions \(x_0(t,x_0)=x_0\), NEWLINE\[NEWLINEx_{m}(t,x_0)=x_0+Lf(t,x_{m-1}(t,x_0),x_{m-1}(\lambda(t),x_0))+NEWLINE\]NEWLINE NEWLINE\[NEWLINE{2t\over T}\left[{1\over T}\left(d-\int_{0}^{T} Lf(t,x_{m-1}(t,x_0),x_{m-1}(\lambda(t),x_0))dt\right)-x_0\right], \quad m=1,2,\ldots,NEWLINE\]NEWLINE with \(x_0\in \mathbb{R}^{n}\), \(Lg(t)\equiv\int_0^{t} [g(t)-{1\over T}\int_0^{T}g(s) ds]dz\). The author obtains necessary and sufficient conditions under which the function \(x^{*}(t,x_0)=\lim_{m\to\infty}x_{m}(t,x_0)\) is a solution to the considered problem. A numerical algorithm for approximately solving the considered problem is presented.