sheet straps nany kinds https://www.google.com/search?q=sheet+straps&rlz=1C1CHBF_enUS750US750&oq=sheet&aqs=chrome.2.69i57j0j69i59j69i61j69i60l2.4958j0j7&sourceid=chrome&ie=UTF-8
Velcro could work! Here's a simple set of instructions:
I have used safety pins.
If sheets are too deep for your mattress, make corner pleats in all 4 corners and sew in place. Alternatively, measure depth of your mattress and look at Amazon.com website, where 6-9" depth sheets are sold.
If sheets are too shallow in depth, measure and sew onto each corner a length of 2" wide " underwear and swimsuit" elastic (available at Walmart, JoAnne fabric, etc. in their sewing/notions department), with edges cut on opposing diagonals. Pin the elastic 8-10" from each corner at both sides, to get a very secure fit.
I have things that slip right onto the sheet near pockets and they keep sheets in place. I think Wal Mart might carry them or a Bed Bath & Beyond.