Is it correct to say "A deterioration in my grandfather's health and memory has just begun"?. Or is it began?

Answer 1

Begun is correct. Began is past tense. Begun this the past participle, so "has begun" is correct. :)

Answer 2

"I Have Not Yet Begun to Fight," 1779 John Paul Jones http://www.eyewitnesstohistory.com/johnpauljones.htm

"We Have Only Just Begun to Fight". Franklin Roosevelt. October 31, 1936. https://patriotpost.us/documents/284