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This paper investigates some fixed point-related questions including the sequence boundedness and convergence properties of mappings

A

The study of geodesic paths is very relevant in spherical geometry, for instance, in the composition of trajectories through the Earth surface or in common planetary studies of distances. A metric space

Furthermore, it is well-known that

This paper has two subsequent body sections. Section

Let

Suppose a metric space

Definition

If a metric space

If

It is direct from (

Inequality (

If

Note from (

Suppose a metric space

Suppose a metric space

Since

Since the right-hand side of (

Assume that a metric space

The following technical definitions are of interest to characterize near-uniform convexity.

A family of points

A

It turns out that

Let

A convex metric space

A convex metric space

If a convex metric space

Since

A convex metric space

Set

Note that

If a convex metric space

Note that if (

If

Since

A metric space

In the paper, we write

Note the following from the basic results in Section

(1) A geodesic space is a CAT

(2) A CAT

(3) A CAT

(4) A CAT

A general technical result involving constructions with two self-mappings in a

Let a metric space

From (

From Lemma

Let a metric space

(i) Assume that

(ii) Assume that

either

Then,

(iii) If

It follows for constants

Now, assume that either

Property (iii) follows since (

Related to Lemma

Let a geodesic metric space

A geodesic space is a uniformly convex CAT(0) space if, for any

Let a geodesic metric space

Since the CAT(0) space satisfies Proposition

There are some particular results of Lemma

Let the metric space

(i)

(ii) If

If both

If

(iii) If

Equations (

Different classes of iterative schemes and their stability properties related to fixed point theory as Halpern, Jungck, Ishikawa, and many of its variants and extensions have been studied in a number of papers. See, for instance, [

Let the metric space

Then, the following properties hold:

(i) Assume that

if

if

(ii)

(iii) If

(iv) If there is a sequence

(v) Define the following sequences:

(vi) Property (v) also holds if the constraint

Note that, for any sequence

Property (ii) follows by direct calculation from (

From the fact that

To prove Property (v), first note that

Now, if

Assume that

Assume that

A potential application of the given results on motions of robots, or movable bodies in general, on surfaces subject to the definition of admissible obstacle-free corridors is now described. Let

if the convergence of the motion trajectory is suited as objective for any given

if one of the mappings

if both mappings are contractive and defined on a nonempty convex closed convex of

The author declares that there are no conflicts of interest.

The author is very grateful to the Spanish Government and European Fund of Regional Development FEDER for Grant DPI2015-64766-R and to UPV/EHU for Grant PGC 17/33.