I am having difficulty keeping the food down as 2 years ago I have a stomach surgery which removed most of it. Usually on non fasting days during the year I eat several meals to keep myself healthy and in good strength. During this Ramadan I find it really difficult to get through the day (as this is the first ramadan after the surgery, I wanted to make an effort to complete the whole month this year) . Unfortunately I find myself vomiting back all of the iftar. The next day I am really really weak. I did fast so far as I tried to "endure" it. Yesterday though after the iftar and vomiting I was so weak that I could not get up from the couch and when I tried to stand up I felt like I was about to faint. My question to you is the following: Is it permitted to skip fasting when I feel this way and give fidyah?

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According to the rule, if you can fast you keep your fasting and if you feel that you cant continue it, you may break your fast and no need to give any fidiya rather you have to do qadha later for that fast.
But if you are not in position to even begin your fast since fajr then do not keep fasting although if you know that InshAllah you will get back your health before next ramadan, you must do qadha for your lapsed fast. If you will not be able to fast qadha after Ramadan then you have to pay fidiya for your left fasts but not Kaffara.