Sentence examples for mapping principle see from inspiring English sources

(Contraction mapping principle, see [21]).

There are many papers in the literature that generalize the notion of metric spaces as well as the Banach contraction mapping principle (see [4 11] and the references therein).

There are many generalizations of the classical Banach's contraction mapping principle (see, e.g., [10, 11] and references therein), and these generalizations play an important role in research work about fixed points of nonlinear operators in partially ordered Banach spaces; see, for example, [1] and the proof of the following theorem.

After this remarkable paper, a number of authors have extended/generalized/improved the Banach contraction mapping principle in various ways in different abstract spaces (see, e.g., [2 22]).

We conclude the results from the characterization of the Banach contraction mapping principle in the context of b-metric space (see, e.g., [[14], Theorem 2]).

(Banach contraction mapping principle [6]).

Therefore, by the Banach contraction mapping principle, we see that (mathcal{F}) has a fixed point which is the unique solution of the boundary value problem (1.1).

Applying the contraction mapping principle, we see that there exists a solution (v z,u_{0},cdot)) to (3.7) and (A cdot)v z,u_{0},cdot)in mathcal{C}([0,T] L^{2} cap L^{q}((0,T L^{2sigma+2})).

Hence, by the contraction mapping principle, we see that Q has a unique fixed point x ( t ), which is a solution of (2.1) with x ( t ) = ψ ( t ) as t ∈ [ − τ, 0 ] and e α t E | x ( t ) | 1 2 → 0 as t → ∞. □.

Applying the Banach contraction principle (see Theorem 1.1) to the mapping G, we get the desired result.

Lemma 2.3 ((demiclosedness principle) (see [16]).