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We prove a general theorem on fixed points of multivalued mappings that are not necessarily contractions and derive a number of recent contributions on this topic for contraction mappings.

One of the most powerful results of functional analysis is the Banach contraction principle which states that if

Since the publication of the Banach principle, there have been a huge number of research papers devoted to its generalization. Among them, the extension to set-valued mappings receives a lot of attention. The works by Nadler Jr. [

The aim of the present paper is to give a general condition for existence of fixed points of set-valued mappings that are not necessarily contractions. The novelty of our approach is the relaxation of requirements for a point to be chosen at current iteration to lie in the image of the point at the preceding iteration during the construction of a sequence of points that converges to a fixed point. Another novelty resides in the use of two different functions to estimate the distance between two consecutive points of the procedure, which makes our result flexible and allows us to deduce a number of important theorems of the aforementioned works for contraction mappings.

Throughout this section, we assume that

Let

(i)

(A)

(ii)

(B)

(iii)

(iv)

We wish to construct a Cauchy sequence

We assume (A) first. Let us start with any point

We now assume (B). By the same argument as mentioned above, we may find either a fixed point of

We close up this section by observing that in the literature on fixed points of contraction mappings it is frequently required that the element

Consider a discrete metric space

In this section we deduce a number of results in recent publications from the main theorem given in the preceding section. The first corollary is Mizoguchi-Takahashi’s theorem (Theorem 5, [

Let

Our aim is to apply Theorem

Second, for every

Third, by the definition of

And finally, the first inequality of (iv) of Theorem

As far as we know, most important generalizations of fixed point conditions for contraction mappings without using Hausdorff distance belong to Ciric in his recent works [

Let

For every

For every

In this condition

Under (C1) and (C2), set

Other important results such as Theorems 6 and 7 of [

We close up this section by discussing the following very recent result of Du and Khojasteh [

If

There is

One of the following conditions holds:

the graph of

the function

for every sequence

In order to apply Theorem

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

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (130-85-D1435). The authors, therefore, acknowledge with thanks DSR technical and financial support.