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The purpose of the paper is to investigate the global existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis. More precisely, it deals with the initial value problemwhere 0 < α < 1, D α 0+ denotes the Riemann-Liouville fractional derivative of order α, and f : (0, ∞) × R → R is a continuous function. Unlike all the previous papers dealing with the problem of existence of solutions to (*), this problem is solved here by constructing a special locally convex space which is metrizable and complete. Then Schauder's fixed point theorem enables to provide sufficient conditions on f , ensuring that (*) possesses at least one solution. The growth conditions imposed to f are weaker than other similar conditions already used in the literature.MSC 2010 : Primary 34A08; Secondary 26A33, 34A12

Abstract. We prove certain new inequalities for special means of two arguments, including the identric, arithmetic, and geometric means.

The paper deals with the polynomial-like iterative functional equationBy using Schauder's fixed point theorem and a version of the uniform boundedness principle for families of convex (respectively higher order convex) functions as basic tools, the existence of nondecreasing convex (respectively higher order convex) solutions to this equation on open (possibly unbounded) intervals is investigated. The results of the paper complement similar ones established by other authors, concerning the existence of monotonic or convex solutions to the above equation on compact intervals. Some examples illustrating their applicability are provided.Mathematics Subject Classification (2010). Primary 39B12; Secondary 26A18, 47H10.