Sentence examples for contraction principles from inspiring English sources

Branciari [2] gave an integral version of the Banach contraction principles and proved fixed point theorem for a single-valued contractive mapping of integral type in metric space.

For applications of contraction principles to variational relations, the reader is referred to [4].

In 2002, Branciari [8] gave an integral version of the Banach contraction principles and showed the following fixed point theorem.

Liu and Xu in [1] reported the concept of cone metric space over Banach algebra (in short CMS-BA) and proved contraction principles in such space.

In this paper, we study some fixed point theorems for self-mappings satisfying certain contraction principles on a convex complete metric space.

Generalising the concept of cone metric space, Liu and Xu in [1] introduced a cone metric space over Banach algebra (in short CMS-BA) and proved contraction principles in such a space.

The real number λ is called a contractive constant of T. Theorem 11.1 (Banach contraction principle).

Suitable adaptation laws for unknown parameters are proposed based on the contraction principle.

(Banach's contraction principle).

(Banach contraction principle [1]).

(Multimapping contraction principle [15]).