Two-sides methods of integration of boundary value problems (Q2759410)

Two-sides methods for the approximate integration of boundary value problems are constructed and investigated for a system of nonlinear differential-functional equations \(L_{m}Y(x)=F[Y(x)]\), \(x\in (a,b)\) with double-point boundary conditions NEWLINE\[NEWLINE\sum_{s=0}^{m-1}[A_{v,s}Y^{(s)}(a)+B_{v,s}Y^{(s)}(b)]=0,\;v=1,\dots,m,NEWLINE\]NEWLINE where \(L_{m}\) is the differential operator generated by the differential expressionNEWLINENEWLINENEWLINE\(\sum_{k=0}^{m}P_{k}(x)Y^{(m-k)}(x)\); \(F[Y(x)]=(f_{i}[Y(x)]),\;i=1,\ldots,n\), NEWLINE\[NEWLINEf_{i}[Y(x)]= f_{i}(x,Y(x),\ldots,Y^{(m-2)}(x),Y(\theta_0(x)),\ldots, Y^{(m-2)}(\theta_{m-2}(x)),NEWLINE\]NEWLINE NEWLINE\[NEWLINEY(\lambda_0(x)),\ldots,Y^{(m-2)}(\lambda_{m-2}(x))),\;Y(x)=(y_{i}(x)),NEWLINE\]NEWLINE NEWLINE\[NEWLINEY^{(r)}(x)=(y_{i}^{(r)}(x)),\;Y^{(r)}(\theta_{r}(x))=(y_{i}^{(r)}(\theta_{i,r}(x))),\;Y^{(r)}(\lambda_{r}(x))=(y_{i}^{(r)}(\lambda_{i,r}(x))),NEWLINE\]NEWLINE \(\theta_{i,r}(x)=x-\tau_{i,r}(x)\), \(\lambda_{i,r}(x)=x+\rho_{i,r}(x)\), \(r=0,\ldots,m-2\), \(\rho_{i,r}(x)\geq 0\), \(\tau_{i,r}(x)\geq 0\) are known continuous functions; \(P_{k}(x)=(p_{i,j}^{k}(x)), i,j=1,\ldots,n\) are operator functions continuous on \(x\in [a,b]\), \(\det P_0(x)\neq 0\), \(x\in[a,b]\); \(A_{v,s}, B_{v,s}\) are fixed linear operators in \(R_{n}\). The multi-point boundary value problem is considered also.