^{1}

^{2}

^{1}

^{2}

Let

Let

Let

We mention that every compact and convex subset of a Banach space

In the current paper, we introduce a geometric notion of

In [

Let

An interesting feature about the above observation is that continuity of

Let

To describe our results, we need some definitions and notations. We shall say that a pair

The

A pair of sets

In [

A convex pair

It was announced in [

The next best proximity point theorem was established in [

Let

As a result of Theorem

Let

In this section motivated by Theorem

A convex pair

We note that the pair

Here, we state the main result of this section.

Let

Let

In what follows, we give a sufficient condition which ensures that every nonempty, bounded, closed, and convex pair of subsets of a uniformly convex Banach space has seminormal structure.

A nonempty, bounded, closed, and convex pair

Let

Suppose that

Let

Let

Let

The following corollary obtains from Theorem

Let

It is interesting to note that by admitting property

It is interesting to ask whether one considers a better condition than the property

Let us consider

If

If

In 1974, Lim proved the following common fixed point theorem.

Let

The following result is a generalized version of Lim’s theorem.

Let

At the end of this section, we consider cyclic mappings which do not increase large distances. Our purpose is not to seek fixed points but rather to determine what can be said about minimal displacement for such cyclic mappings.

Let

Our result is the following. The surprising aspect of the conclusion of the following theorem is the fact that cyclic relatively

Let

Proceeding in a similar way as in Theorem

Recently, Kosuru and Veeramani introduced a concept of

Let

The following best proximity point theorem was proved in [

Let

To establish our results, we introduce the following class of cyclic mappings.

Let

Obviously, every generalized pointwise cyclic contraction is a cyclic relatively nonexpansive mapping. Also, the class of generalized pointwise cyclic contractions contains the class of pointwise cyclic contractions as a subclass. The next example shows that the reverse implication does not hold.

Let

Here, we state the main result of this section.

Let

Since

Note that Theorem

Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Naseer Shahzad acknowledges with thanks DSR for financial support.