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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