A General Fixed Point Theorem for Multivalued Mappings That Are Not Necessarily Contractions and Applications

and Applied Analysis 3 instead of x n when n k tends to ∞, we arrive at the following inequality:


Introduction
One of the most powerful results of functional analysis is the Banach contraction principle which states that if  is a contraction on a complete metric space (, ), that is, ((), ()) ≤ (, ) for every ,  ∈  and some fixed  ∈ (0, 1), then  has a unique fixed point.Moreover, that unique point can be approximately computed by a very simple iterative procedure.Namely, starting from any point  0 ∈ , the sequence obtained by  +1 = (  ) for  ≥ 0 converges to the fixed point.Numerous applications and generalizations of this principle are known in nonlinear analysis (see [1][2][3][4][5] and many references given in these).
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. [6] and Markin [7] are among the first efforts in this direction, in which the Hausdorff distance is used to define contraction set-valued mappings.Further significant generalizations are presented in [8][9][10][11][12][13][14][15][16][17][18][19] and many others.
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.

The Main Result
Throughout this section, we assume that (, ) is a complete metric space.Given a nonempty set  ⊆ , the distance from a point  ∈  to  is denoted by (, ) and defined by (, ) = inf ∈ (, ).Theorem 1.Let  be a set-valued map on  with values in the space of nonempty closed subsets of .Assume that the function (, ()) is lower semicontinuous on  and that there are positive valued functions  and  on [0, ∞) such that (i) lim sup  →  + (()/()) < 1 for each  ≥ 0.
We assume (A) first.Let us start with any point  0 ∈ .If  0 ∈ ( 0 ), we are done.If not, we choose  1 ∈  as given in (ii): Similarly, restarting from  1 we choose  2 ∈  satisfying the inequalities in (ii) and continue this process either to arrive at a fixed point of  or to obtain a sequence of    s such that   ∉ (  ) and for every  ≥ 1. Observe that ((  , (  ))) > 0 because, otherwise, in view of (3) one would have  +1 ∈ ( +1 ), which is a contradiction.Hence, ((  , (  ))) > 0. It follows that In view of (3), the sequence {(  , (  ))} ∞ =0 is decreasing and hence decreasingly converges to some limit  ≥ 0. Actually  = 0 because otherwise, by passing to the limit on both sides of (4) for    instead of   when  tends to ∞ and by (i), we would obtain which is a contradiction.The first part of hypothesis (A) and (3) imply lim We claim that this sequence is a Cauchy sequence.Indeed, by (i) and the first hypothesis of (A), there are some  ∈ (0, 1),  > 0, and  ≥ 1 such that Combining this with (3) yields for  ≥  and  ≥ 1.Since the sequence {(  , (  ))} ∞ =0 converges to 0 as  tends to ∞, we deduce that the sequence {  } ∞ =0 is Cauchy and hence it converges to some limit as requested.
Claim 1 ( = ).Suppose to the contrary that  < .Then  > 0. We choose a small  > 0 such that  +  <  − .Then there is some  > 1 such that This and (10) yield which implies that lim sup  → ∞ ((  ,  +1 )) < 1.By using the first part of (B) and by passing to the limit in (11) for    instead of   when   tends to ∞, we arrive at the following inequality: which is a contradiction.By this,  = .
It remains to apply the same argument as in the case of condition (A) to conclude the proof.
We close up this section by observing that in the literature on fixed points of contraction mappings it is frequently required that the element  in conditions (A) and (B) belongs to (), in which case the hypothesis (, ) ≥ (, ()) (condition (iii)) is evidently satisfied.The fact that  is allowed to be chosen outside () is extremely important in computing fixed points of mappings that are not contractive at certain points.Below is an example to illustrate this.
It is clear that  is not a contraction.If we start at  0 = 0 and apply the classical algorithm  +1 = (  ) for  ≥ 0, then it produces an infinite loop and we never get the fixed point.In order to avoid cycling, let us define two functions  and  on [0, ∞) by Now we start with  0 = 0.If we take  = ( 0 ), then neither (ii) nor (iv) is satisfied.Let us choose  1 = 0.75 the closest element to ( 0 ) for which condition (ii) of Theorem 1 is fulfilled.In the next iteration we take  2 = 0.5 = ( 1 ) that satisfies the above-mentioned condition too.This  2 is a fixed point of .

Particular Cases
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, [18]) which according to Suzuki [19] is a real generalization of Nadler's theorem [6].We recall that the Hausdorff metric induced by  is given by for any two subsets  and  of .