TL;DR: The answer will depend on 1) whether Japan manages to achieve higher growth rates in the coming years and 2) which economists you ask.
"Success" could have many definitions here, depending on what goals and objectives were are talking about and on what timescale. Since it is an ongoing policy, there is always the possibility that resulting performance will improve over time. Performance could be defined according to real GDP growth, employment, inflation targets, or a number of other criteria. And an improvement in performance alone may not imply a causal relation to the policies in place.
With all these complexities in mind, it is fair to say that while Abenomics may have been successful up to a point, there is some debate about this, and it is also not expected to resolve the relatively low economic growth and other problems faced by Japan.
Here is a paper from 2014:
We show that Abenomics ended deflation in 2013 and raised
long-run inflation expectations. Our estimates suggest that Abenomics
also raised 2013 output growth by 0.9 to 1.8 percentage
points. Monetary policy alone accounted for up to a percentage
point of growth, largely through positive effects on consumption. In
both the medium and the long run, Abenomics will likely continue to
be stimulative. However, the size of this effect, while highly
uncertain, thus far appears likely to fall short of Japan’s large output gap. In part this is because the Bank of Japan’s 2 percent inflation target
is not yet fully credible.
However another 2014 paper looks specifically at the history of disaster recovery and concludes that:
[...] government policies have failed to lift Japan's GDP to the
expected level. Even with the help of Abenomics, the gap remains in
the range of 6 to 14 trillion yen per year.
Another negative point from 2015:
Prime Minister Shinzo Abe’s economic stimulus package, Abenomics,
depreciated the yen sharply from the end of 2012, which was
expected to have a positive impact on Japan’s trade balance.
Contrary to the J-curve effect, however, Japan’s trade balance has
not shown any signs of improvement, even though two years have
passed. There is a growing concern that Japanese firms might lose their
export competitiveness in the global market.
Overall, here is a pretty good summary of where things stood as of 2016:
Has the Abenomics failed? Or can Abenomics eventually pull the
Japanese economy out of the low growth that has characterized
most of the last 20 years? The answer hinges on the success
of the growth strategy part of Abenomics, which is referred to as the
third arrow. Even if the first two arrows of Abenomics (bold monetary
policy and flexible fiscal policy) eventually turn out to be
successful, the best they can achieve is to fix the demand
shortage and eliminate the output gap. Indeed the expansionary
macroeconomic policies in the first three years of Abenomics have
significantly changed the inflation expectations and narrowed
the output gap, but the economy continued to stagnate. In order
for Japan to escape the long
- term stagnation and achieve a higher potential growth, the growth strategy part of Abenomics needs to be successful.