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Bounded solutions for singular boundary value problems. (English) Zbl 0642.34030
Let T be the set obtained by removing k points (counting multiplicities) from [0,1]. Let \(A_ i\) be an invertible matrix-valued function in T with real entries of class \(C^ i(T)\), \(i=1,...,k\), and let \(B_ j\) be a real continuous matrix-valued function in T for \(j=0,...,p\). Boundary value problems are considered for the vector-matrix differential equation \[ DA_ 1(t)... DA_ k(t)D^ px+\sum^{p}_{j=0}B_ j(t)D^ jx=f(t,x,...,x^{(p+q-1)}) \] in T, with homogeneous boundary conditions at p points in [0,1], where \(1\leq q\leq k\), \(D=d/dt\). Conditions are given for which such singular boundary value problems have solutions \(x\in C^{p+k}(T,R^ n)\) extendible to \(C^{p-1}([0,1],R^ n).\)
34C11 Growth and boundedness of solutions to ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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