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An adaptive output-feedback control approach is proposed for a class of uncertain non-linear systems with unknown time-varying delays in the strict-feedback form. The unknown time-varying delayed functions are bounded by non-linear functions related to all states variables. This is a key feature of this study because the previous output-feedback control results assume that the bounds of the delayed functions only depend on an output of the system. The authors first construct an observer to estimate state variables. Then, the adaptive controller is simply designed using the observer dynamic surface design technique where time-delay effects are compensated by using appropriate Lyapunov-Krasovskii functionals and the function approximation technique based on neural networks. In addition, it is shown that all signals in the closed-loop system are semi-globally uniformly bounded and the tracking error converges to an adjustable neighbourhood of the origin.