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We study one-parameter families of quasi-periodically forced monotoneinterval maps and provide sufficient conditions for the existence of aparameter at which the respective system possesses a non-uniformly hyperbolicattractor. This is equivalent to the existence of a sink-source orbit, that is,an orbit with positive Lyapunov exponent both forwards and backwards in time.The attractor itself is a non-continuous invariant graph with negative Lyapunovexponent, often referred to as "SNA". In contrast to former results in thisdirection, our conditions are C^2-open in the fibre maps.  By applying a general result about saddle-node bifurcations in skew products,we obtain a conclusion on the occurrence of non-smooth bifurcations in therespective families. Explicit examples show the applicability of the derivedstatements.

Non-Smooth Saddle-Node Bifurcations of Forced Monotone Interval Maps I: Existence of an SNA