Sentence examples for multidimensional fixed points from inspiring English sources

We also underline that one can easily state the same theorem in the frame of multidimensional fixed points.

Afterward, many results on multidimensional fixed points have been established (see, e.g., [13 18]).

Using Theorem 3.1, we obtain the following result on multidimensional fixed points, which generalizes Theorem 3.1 of Wang [26].

Therefore, our results can be applied directly to the coupled fixed points of mixed monotone operators and multidimensional fixed points theorems [26 31].

Recently, the notion of coupled fixed point was extended to the higher dimensions by defining tripled, quadrupled and, hence, multidimensional fixed points [2 5].

Once the notion of coupled fixed point was given by Gnana Bhaskar and Lakshmikantham in [1], the theory of multidimensional fixed points has attracted much attention (see, for instance, [2 8]), specially in the tripled case (see [9 17]).

It is very natural to extend the definition of two-dimensional fixed point (coupled fixed point), three-dimensional fixed point (tripled fixed point) and so on to multidimensional fixed point (n-tuple fixed point) (see, e.g., [9 17]).

The following multidimensional fixed point theorem is an immediate consequence of Theorems 3.3 and 3.4.

Multidimensional fixed point theory was initiated in 2006 by Gnana Bhaskar and Lakshmikantham [1].

In [20], the authors proved that the initial multidimensional fixed point result, Theorem 9 in [11], can be derived from Theorem 2.1 in [21] either.

Further, by using our main results, we prove some results about multidimensional common fixed points.