Convergence Results for Contractive Type Set-Valued Mappings

: In this work, we study an iterative process induced by a contractive type set-valued mapping in a complete metric space and show its convergence, taking into account computational errors.


Introduction and Preliminaries
For more than sixty years, the fixed point theory has been an important area of nonlinear analysis.One of its main topics is the analysis of the existence of fixed points of contractive type maps.See, for example, .Many existence results can be found in [6,7,21] This topic is well developed for single-valued mappings [1,3,[8][9][10]17] as well as for set-valued mappings [14][15][16]20,22,26,27].In this work, we analyze an iterative process induced by a contractive type set-valued mapping in a complete metric space.More precisely, we analyze a fixed point problem corresponding to a contractive type set-valued mapping T acting in a complete metric space and study an iterative method which generates an approximate solution under the presence of computational errors which are always present in calculations.
Denote by Z the set of all integers, by N the set of all natural numbers and by R + the interval (0, ∞).Assume that (X, ρ) is a complete metric space.For every point u ∈ X and every a > 0, set B(u, a) = {y ∈ X : ρ(u, y) ≤ a}.
For every point u ∈ X and every nonempty set A ⊂ X, define ρ(u, A) = inf{ρ(u, y) : y ∈ A}.
Clearly, T is a contractive type set-valued mapping [17,21].We are interested in solving the problem Find z ∈ X satisfying z ∈ T(z).
In practice, we can only obtain an approximate solution of this problem z ∈ X such that ρ(z, T(z)) is small.In order to meet this goal, we use the following algorithm: Initialization: select an arbitrary point x 0 ∈ X and a small positive constant δ.
Iterative step: given a current iteration point x n , calculate the next iteration point x n+1 such that Clearly, at each iterative step x n+1 is an approximate solution of the problem Since the space X is not compact, a solution to the problem above does not exist in general.Note that we only assume the existence of the compact set X 0 but do not assume that it is given.
In this paper, we show that if δ is small enough, then our algorithm generates approximate solutions of the fixed point problem.
Proposition 1.There exists x T ∈ X 0 such that x T ∈ T(x T ).
Let us assume the contrary.Then, In view of (1), there is a positive number ϵ satisfying In view of (4), there exists By (7), there exists It follows from (1), (3), ( 4), ( 6), ( 8) and (9) that The inequality above contradicts relation (5).Therefore, Together with (4), this implies that the existence of {x n } ∞ n=0 ⊂ X 0 for which Since X 0 is a compact, it has a convergent subsequence.As usual, we may assume without the loss of generality that there is a limit By ( 3), ( 10) and (11), for every n ∈ N ∪ {0}, This implies that x T ∈ T(x T ).Proposition 1 is proved.
Proof.Assume that the proposition does not hold.Then, for every n ∈ N, there is Since X 0 is compact, {x n } ∞ n=1 has a convergent subsequence.Because, instead of the sequence, one can consider its convergent subsequence, we may assume without the loss of generality that there is a limit In view of (3) and ( 12), for every n ∈ N, This implies that x * ∈ T(x * ).
By (14), for all sufficiently large n ∈ N, This contradicts (13) and completes the proof of Proposition 2.

The First Main Result
We begin with the following result, which shows that the approximate fixed point of T is closed to its fixed points.
Theorem 1. Assume that ϵ ∈ R + .Then, there is δ ∈ R + such that for every x ∈ X which satisfies Proof.Proposition 2 implies the existence of such that the following is true: Choose a positive number By ( 16), there is for which ρ(x, x 0 ) < δ. (18) We show that ρ(x, X 0 ) < ϵ 1 .
Assume the contrary.Then, for each i ∈ {k, . . ., k + n 2 }, and, by property (d), Property (d), (61) and the relation above imply that and This contradicts (29) and proves that there exists Assume that i ≥ p is an integer and We show that ρ(u i+1 , u i+2 ) ≤ ϵ/2.
There are two cases: Assume that the first case holds.Then, in view of (30), ( 31), (51), the choice of p and property (a), Assume that the second case holds.Then, by our assumption and property (b), Thus, in both cases ρ(u i+1 , u i+2 ) ≤ ϵ/2.
Thus, we have shown that if i ≥ p is an integer and ρ(u i , u i+1 ) ≤ ϵ/2 holds, then the relation above is true.Combined with the choice of p, this implies that for each integer Property (a), (27), the choice of p and the equation above imply that for each integer i ≥ p, Let i ≥ p be an integer.By ( 27), (31), (38) and the equation above, Theorem 2 is proved.

Extensions
Theorems 1 and 2 easily imply the following result.
Theorem 3 easily implies the following result.
Then, there exists a unique compact set X T ⊂ X [22] such that T(A T ) = A T .

Conclusions
In the present paper, we analyze a fixed point problem corresponding to a contractive type set-valued mapping T acting in a complete metric space.We discuss a simple algorithm which generates an approximate solution under the presence of computational errors which are always present in calculations.Note that at any iterative step only a current iteration point x n and T(x n ) are known.