Long-time solvability of functional differential equations with "supremum" involving increasing nonlinearity at infinite time

In this paper, we investigate a class of initial value problems of nonlinear fractional differential equations with state deviating arguments, delayed impulses and supremum on the half line. A global existence-uniqueness result is obtained using the theory of fixed points in uniform spaces, which is different from what is commonly used in such studies. Our result is obtained in a setting where classical variants of the fixed point theorems frequently used in the literature are inapplicable, and moreover without any bounding restriction with respect to time, neither on the solution nor on the reaction part of the problem. Two examples illustrating our main findings are also given.