Hello, I am Dr. Ola and I am a licensed veterinarian. I have been in this situation and I know how frustrating it can be! Cats can be common culprits of "urine marking" and I find this especially so with male cats with odds going up even further if they are not neutered. The biggest issue is that when they choose one location in the house to urinate, you must completely alleviate the smell from this area or they will repeat the behavior. This is because cat urine contains a very high concentration of ammonia and common household cleaners, even bleach, will not completely clean the area. Often times when us humans can't smell it, we think it is taking care of when this isn't the case for our cats. You must look for an enzymatic cat urine cleaner which will break the chemical bonds forming the ammonia so it will be undetectable by your cats. If rugs and/or carpeting is involved, these may need to be removed. Since it is under the bed, remove all linen from the bed and clean the mattress thoroughly.
There may be a bit of a stress component to this too since you have recently introduced the cat door. While they like their new found freedom, they may be reluctant to the use the restroom outdoors, at least for now. I would keep the litter boxes in the house at least for a few months as they transition to this new change. For three cats, I would recommend you have 4 litter boxes as that is the the standard amount that you should need.
If one or more of your cats has been drinking more water to accompany this behavior, especially if he/she is older than 8 years old, you need to rule out a medical cause by scheduling an appointment with your veterinarian. Routine blood work and a urinalysis will rule out metabolic cause of inappropriate urination such as kidney disease, diabetes, urinary tract infection, or more commonly feline lower urinary tract disease.
I hope that my professional advice has been helpful to you. Please do not forget to rate my response which I hope you found to be excellent. If you have further question about this, please let me know:)