Ruling on killing of dogs that eat the sheep

Firstly: 

It is by the mercy of Allah, may He be exalted, to His slaves
that He has created for them what is in the heavens and on earth, and He has
subjugated it in their interests, for their livelihood. Included in that is
the fact that He has prescribed for them to ward off harm and aggression
from themselves and their property or wealth, even if that leads to killing
the aggressor or the one who is causing harm to them, such as wolves or
other predators. But with regard to animals that do not cause harm, it is
not permissible to kill them. 

See: al-Mughni, by Ibn Qudaamah, 4/324 

Secondly: 

Dogs are of two types: those which it is Sunnah to kill and
those which are not to be killed. As for those which it is Sunnah to kill,
it is the black dog because it is a devil, and the vicious dog because it is
harmful. All other dogs are not to be killed. 

If these dogs are attacking the sheep and killing and eating
some of them, then these are harmful dogs whose harm is to be warded off
even if that is by killing them; rather they should be killed because of the
mischief that they are causing. Al-Bukhaari (1829) and Muslim (1198)
narrated that ‘Aa’ishah (may Allah be pleased with her) said: I heard the
Messenger of Allah (blessings and peace of Allah be upon him) say: “There
are four (creatures), all of which are vermin and may be killed inside the
sanctuary and outside: kites, crows, rats and vicious dogs.” 

The scholars of the Standing Committee said: 

With regard to dogs that are harmful because they are vicious
or are predators or have rabies, and other kinds of aggression and fearful
matters, there is nothing wrong with killing them in ways that do not cause
harm to others that are not harmful. End quote from Fataawa al-Lajnah al-Daa’imah,
26/201 

The scholars of the Committee were asked: 

One day I caught a dog that had attacked one of the sheep but
not killed it, and I killed this dog. Please note that I do not know who
this flock belonged to; I carried the sheep and put it in a fenced area
nearby and left it there. 

They replied:

If the situation was as you describe, then what you did was
good and there is no blame on you, because you saved the property of your
Muslim brother from being destroyed and you will be rewarded for that, in
sha Allah.

End quote from Fataawa al-Lajnah al-Daa’imah, 26/205 

There is no doubt that it is worthwhile for the owner of the
sheep to appoint someone to guard them or a shepherd to look after them, but
that does not mean that it is not prescribed to kill these dogs, because
they have got used to attacking and if they fail to catch one thing they
will look for something else and attack it. Hence it is prescribed to kill
them. 

So these dogs are to be killed and the owner of the sheep
should be instructed to look after them and protect them. 

And Allah knows best.