A fixed point of a function is a value that, when applied as the input of the function, returns the same the little lisper pdf as its output. This is inconsistent in mathematical logic. The second variable may be used as a counter, or index. In the lambda calculus it is not possible to refer to the definition of a function in a function body.
Recursion may only be achieved by passing in a function as a parameter. Fixed-point combinators may also be easily defined in other functional and imperative languages. The implementation in lambda calculus is more difficult due to limitations in lambda calculus. Fixed point combinators may be applied to a range of different functions, but normally will not terminate unless there is an extra parameter. Even with lazy evaluation when the function to be fixed refers to its parameter, another call to the function is invoked.
The calculation never gets started. The extra parameter is needed to trigger the start of the calculation. The type of the fixed point is the return type of the function being fixed. This may be a real or a function or any other type. Alternately a function may be considered as a lambda term defined purely in lambda calculus. These different approaches affect how a mathematician and a programmer may regard a fixed point combinator. In contrast a person only wanting to apply a fixed point combinator to some general programming task may see it only as a means of implementing recursion.
Every expression has one value. This is true in general mathematics and it must be true in lambda calculus. This means that in lambda calculus, applying a fixed point combinator to a function gives you an expression whose value is the fixed point of the function. This equation has no solution in the real numbers. This demonstrates that there may be solutions to an equation in another domain.