Are telescopes with a concave lens (instead of convex ones) ever useful for astronomy?
And if so, where are they used? Do they ever affect resolving power?
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If you only have concave lenses, you won't be able to focus. If you remember your elementary optics, a concave lense has a negative focal length.
They can be utilized as part of a series of lenses. A concave eyepiece can avoid image inversion, which many people find irritating. A concave lense mounted just inside of a normal eyepiece can be used to increase magnification. These are called Barlow lenses.
Divergent lenses or mirrors are used in telescopes all the time, just not alone. They are combined with convergent elements in order to fix some subtle issues of the convergent optics.
The achromatic refractor objective is a pair of lenses, one convergent, the other divergent.
The Cassegrain reflector telescope is made, again, with a convergent mirror and then a divergent smaller mirror.
Most eyepiece (ocular) designs these days include several lenses, usually between 4 and 7, sometimes more, and some of those are divergent.
A concave lens doesn't bring light to a focus, so is of no use as the objective lens of a telescope. Concave lenses were used as eyepieces by Galileo. Their main drawback is that give an extremely narrow field of view. They were soon replaced by Kepler's version: a simple double convex lens, and that was soon replaced by Huyghens' double lens design, still found today on cheap telescopes.
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