Does Rolls Royce produce any engine that matches the capabilities of the GE9x?
First let's talk about thrust.  Jet engines are designed for very specific thrusts for every specific applications.  For example, look at the GEnx-1B versus the GEnx-2B.  The Boeing 787-8 has GEnx-1B engines rated for ~70k lbf takeoff thrust.  At about the same time as Boeing was developing 787-8, it was also developing the 747-8.  That plane needed slightly less thrust per engine, about 5% less. Boeing could have just put the GEnx-1B on that plane.  But they didn't. Instead they had GE develop the GEnx-2B, rated for 66.5k lbf thrust.  They did that because an engine that had five percent too much thrust was a completely unfeasible option.  The GEnx-2B is smaller and several hundred pounds lighter.  So my point here is that RR is never going to develop an engine the size of the GE9X unless they have a very specific contract for a plane that needs exactly that much thrust.  Every engine RR makes is going to be sized exactly as big as it needs to be and no bigger.

RR certainly could make an engine that big if they had a contract to do so.  If you know how to make an engine that produces 70klbf of thrust, you know how to make an engine that produces 100 klbf of thrust.  Just make everything a little bit bigger. Turbomachinery is not perfectly scaleable, so just because you know how to make an engine with 100 lbf of thrust does not mean you know how to make an engine with 100,000 lbf of thrust.  But from 70k to 100k is not a big leap. Definitely within their capability.

As for everything else, I'd say just look at the market share. There's some interesting statistics here.  http://www.alafco-kw.com/upload/Media_Library__March_2016____Aircraft_Market_Report__March_2016__921.pdf

For example: RR has 59% of the A330 market, a plane on which there are 3 different engine options available (GE, RR, P&amp;W).  RR is certainly behind GE on the 787, but not by a lot.  And RR has an exclusive contract on the A350.  Overall RR is competitive in the market versus the other manufacturers. If RR was "losing out in the technological advancement", you would see this reflected in their market share, but this is not what is happening.