Sentence examples for first fixed point from inspiring English sources

It is interesting to note that the two ways correspond to by-now standard tools, the so-called first fixed point theorem and second fixed point theorem of combinatory logic and lambda calculus (Barendregt 1984, p. 131 and p. 143; see also the distinction between the first recursion theorem and the second recursion theorem of classical recursion theory, in this SEP, entry recursive functions).

Historically, in 1886, Poincaré initiated first fixed point results.

Now we are ready to give the first fixed point result in this setting.

Before establishing our first fixed point result we recall that a point z ∈ X is said to be a fixed point of a multivalued map T : X → 2 X if z ∈ T z.

We say that T is a generalized -contractive mapping of type II if there exist two functions α : X × X → [ 0, ∞ ) and ψ ∈ Ψ such that α ( x, y ) d ( T x, T y ) ≤ ψ ( N ( x, y ) ), for all  x, y ∈ X, (2.3). where N ( x, y ) = max { d ( x, y ), d ( x, T x ) + d ( y, T y ) 2 }. (2.4). Now, we state our first fixed point result.

Also, T will be called strongly ρ-continuous if T is ρ-continuous and liminf n → ∞ ρ ( g - T ( f n ) ) = ρ ( g - T ( f ) ). for any sequence {f n } ⊂ C which ρ-converges to f and for any g ∈ C. The study of a common fixed point of a pair of commuting mappings was initiated as soon as the first fixed point result was proved.

Next we discuss our first common fixed point result of this work.

The first metric fixed point theorem was given by Banach in 1922.

Now, we can state and prove the first main fixed point theorem of this paper.

Recently, Matsushita and Takahashi [24] first investigated fixed point problems of relatively nonexpansive mappings based on hybrid projection methods.

In [10], Nakajo and Takahashi first investigated fixed point problems of nonexpansive mappings based on hybrid projection methods in the framework of Hilbert spaces.