Losing weight is simply a function of consuming fewer calories than your burn rate. To do that, you have to consume fewer calories and/or exercise more. However, that's easier said than done.
Fasting is not a healthy way to diet. The Atkins Diet can be effective but has risks if overdone (can increase cholesterol levels, risk of kidney stones, colon cancer) and if there are some existing underlying medical issues (for example, diminished kidney function). Extreme diets can also be effective ("give up all ...") but are also prone to snap back wait gain as well as nutritional deficiency.
The best way to lose weight and keep it off is to alter your diet combined with whatever amount of exercise you can manage on a regular basis. Find every way that you can to substitute complex carbs for simple carbs. IOW, more fruits (in moderation) and lots of vegetables with no candy, desserts, snack foods, sugar flavored anything, bread, etc. (a low Glycemic Index diet). Drink a minimum of 1/2 a gallon of water every day, always. More if you can. Graze every few hours rather than eating three large meals a day.
Only discipline and resolve will make this work. You can't behave for 6 days and fall off the wagon on the 7th day and eat a Mrs. Smith's pie for lunch. And if you want to take this past losing 20 pounds to achieving good nutrition, eat as little red meat, fast food and other cholesterol filled delights as possible.
Changing your lifestyle gradually allows your body to adapt to those changes. That acclimation makes the snap back weight gain less likely.
Good luck!