We prove existence of solutions, and particularly positive solutions, of initial value problems (IVPs) for nonlinear fractional differential equations involving the Caputo differential operator of order\documentclass[12pt]{minimal}
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				\begin{document}$$\alpha \in (0,1)$$\end{document}α∈(0,1). One novelty in this paper is that it is not assumed thatfis continuous but that it satisfies an\documentclass[12pt]{minimal}
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				\begin{document}$$L^{p}$$\end{document}Lp-Carathéodory condition for some\documentclass[12pt]{minimal}
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				\begin{document}$$p>\frac{1}{\alpha }$$\end{document}p>1α(detailed definitions are given in the paper). We prove existence on an interval [0,T] in cases whereTcan be arbitrarily large, called global solutions. The necessarya prioribounds are found using a new version of the Bihari inequality that we prove here. We show that global solutions exist whenf(t,u) grows at most linearly inu, and also in some cases when the growth is faster than linear. We give examples of the new results for some fractional differential equations with nonlinearities related to some that occur in combustion theory. We also discuss in detail the often used alternative definition of Caputo fractional derivative and we show that it has severe disadvantages which restricts its use. In particular we prove that there is a necessary condition in order that solutions of the IVP can exist with this definition, which has often been overlooked in the literature.