In this paper we give some new results on complete abstract second order differential equations of elliptic type set in R+. In the framework of UMD spaces, we use the celebrated Dore-Venni Theorem to prove existence and uniqueness for the strict solution. We will use also the Da Prato-Grisvard Sum Theory to furnish results when the space is not supposed to be a UMD space.
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In this work we give an alternative approach to the study of some singular boundary value problems for a second order differential-operator equation in the space of Holder continuous functions. We prove that the solution can be represented explicitly as the sum u = uR + uS of a regular part and a singular part under some natural assumptions on the data. We then give a complete analysis of uR and uS by using the operational calculus.
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