I am going to break down the phrase "failure on the part of loved ones to act" into its subparts and their meanings:

"Failure": this word has its normal dictionary meaning of "lack of success"

"On the part of": this can be replaced using the word "by", which has the same meaning.

"Loved ones": usually means family and/or close friends  

"to act": means "to do something" or to "do something about it"

So you could re-write this phrase and keep almost exactly the same meaning with:

"Failure by friends and family to do something"

I am going to break down the phrase "failure on the part of loved ones to act" into its subparts and their meanings:

• Failure: this word has its normal dictionary meaning of "lack of success,"
• On the part of: this can be replaced with the word "by," which has the same meaning,
• Loved ones: usually means family and/or close friends,
• To act: means "to do something" or "do something about it."

So you could rewrite this phrase as follows and keep the same meaning:

Failure by friends and family to do something.