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Conti-Opial type existence and uniqueness theorems for nonlinear singular boundary value problems. (English) Zbl 1048.34108
Summary: Conti-Opial-type conditions are found for the solvability as well as for the unique solvability of the nonlinear singular boundary value problem \[ x^{(n)}(t)= (t-a)^{-\alpha} (b-t)^{-\beta} f(x)(t),\;h_i(x)=0,\;i=1,\dots,n. \] Here, \(\alpha\) and \(\beta\in[0,n-1]\), \(f\) is the operator \((h_i,i=1,\dots,n\) are the operators) acting from some subspace of the space of \((n-1)\)-times continuously differentiable \(m\)-dimensional vector functions on the interval \(]a,b[\) into the space of integrable \(m\)-dimensional vector functions on \([a,b]\) (into the space \(\mathbb{R}^m)\).

MSC:
34K10 Boundary value problems for functional-differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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