Is it possible to set up the screen memory after the cpu is already in protected mode without the use of the bios/uefi or vga stuff? Just so the cpu writes to the screen memory (after set up).
Thanks a lot in advance.

Sure. You just need to have documentation for your particular video card and find out which sequence of commands sent to it leads to the state you expect. Which might not be an easy thing (for example, even Intel HD Graphics documentation is quite a large set of information) and it might also be a good idea to have it separate from the kernel, as a driver.

there is a VESA operating mode that allows you to use it when you are in protected mode (since VESA 3.0) but to have tried it, I can say that with some card it does not work well

you can initialize a vesa video mode in real mode and use the linear frame buffer (alas it does not work for all video mode or all cards) in protected mode to change the display

personally I use a temporary return in real mode to use VESA (to have information on video modes, change video mode, change memory bank, ...) and until now it did not pose any problem

Thanks a lot for the help, i have a device with an powervr gx6430 and i dont know if it is vga compatible thats why i just planned to write to the screen using the cpu.
But i will check as adviced if i can find some documentation about it.

Please take a look at the classic MODES.ASM example. This is what I used as a template for VGA mode switching routines when I was trying my own OS development back in 1999.