Generalized Contraction and Invariant Approximation Results on Nonconvex Subsets of Normed Spaces

Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. In this paper, we introduce a notion of generalized F -contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are also deduced. Our results extend, unify, and generalize comparable results in the literature.


Introduction and Preliminaries
One of the most basic and important results in metric fixed point theory is the Banach contraction principle due to Banach [1]. It states that if ( , ) is a complete metric space and : → satisfies for all , ∈ , with ∈ (0, 1), then has a unique fixed point. This theorem that has been extended in many directions (see, e.g., [2][3][4][5][6]) has many applications in mathematics and other related disciplines as well (see, e.g., [7][8][9]). Meinardus [10] and Brosowski [11] employed fixed point theory to obtain invariant approximation results in normed linear spaces. A number of authors generalized their results (see [12][13][14][15][16][17][18] and the references therein). On the other hand, Dotson [19] extended Banach's contraction principle for nonexpansive mappings on star-shaped subsets of Banach spaces and proved Brosowski-Meinardus type theorems on invariant approximations. L. A. Khan and A. R. Khan [20] generalized Dotson's results on star shaped subsets of pnormed spaces.
Recently, Wardowski [21] introduced a new contractive mapping called an -contraction and proved some fixed point theorems in complete metric spaces. In this paper we introduce a notion of generalized F-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are deduced. Our results extend, unify, and generalize comparable results in [10,11,19,20]. Some illustrative examples are also presented.
Next we give some definitions which will be used in the sequel. The letters R + , R will denote the set of positive real numbers and the set of real numbers, respectively.
for all ∈ .
Definition 4. Let ( , ) be a metric space and ∈ ϝ. A mapping : → is said to be -nonexpansive if for all , ∈ .
If we take 2 ( ) = ln( ) + , it is clear that 2 ∈ ϝ, and then (2) becomes whenever ( , 2 ) > 0, which implies that that is, Definition 8. Let be a closed subset of metric space ( , ). Then : → is called compact if for every bounded subset of , ( ) is compact in .
( ) is called the set of best approximations of from . If for each ∈ , ( ) is nonempty, then is called proximal. Observe that if is closed, then ( ) is also closed.
Definition 11. Let be a linear space over R. A subset of is called star-shaped if there exists at least one point ∈ such that + (1 − ) ∈ for all ∈ and 0 < < 1. In this case is called a star centre of .
Let ( , ) be a metric space, a closed subset of , and : → a self-mapping. For each ∈ , the set ( ) = { , , . . . , , . . .} is called the orbit of (compare [23]). The mapping is called orbitally continuous at if , and is orbitally continuous on a set if is orbitally continuous for all ∈ .

Main Results
In the following a normed linear space ( , ‖ ⋅ ‖) will be simply denoted by if no confusion arises. Furthermore, by a complete subset of a normed linear space we will mean a subset of such that the restriction to of the metric induced on by its norm is complete. Of course, every complete subset of a normed linear space is closed, and every closed subset of a Banach space is complete.
Our main result (Theorem 13 below) will be proved with the help of the following re-formulation of Theorem 4 of [22].

Theorem 13.
Let be a normed linear space, a complete and star-shaped subset of , and ∈ ϝ. If : → is an -nonexpansive mapping and ( ) is compact, then has a fixed point.
Proof. We first note that, by Remark 5, is nonexpansive on , so it is continuous on . Now let be a star centre of . For each ≥ 1, define : → by for all ∈ , where 0 < < 1 and lim → ∞ = 1. From the fact that is continuous on it immediately follows that each is continuous on .
For any fixed ≥ 1 and any ∈ , we have Since is strictly increasing, with < 1 for each ≥ 1, and is -nonexpansive, we have Hence This implies that there exists > 0, such that Therefore, Hence, is a generalized -contraction for each ≥ 1. By Theorem 12, for each ≥ 1 there exists ∈ such that = . Since ( ) is compact, there exist a subsequence { } ≥1 of the sequence { } ≥1 , and an ∈ ( ) such that In fact ∈ because ( ) ⊆ and is closed. Since lim → ∞ = 1 and = for all ≥ 1, we deduce that Therefore, We conclude that = .
Remark 14. Note that, by Remark 5, we can restate the preceding theorem as follows. Let be a normed linear space (Banach space, resp.) and a complete (closed, resp.) and star-shaped subset of . If : → is a nonexpansive mapping and ( ) is compact, then has a fixed point.
The following is an example where we can apply Theorem 13 to every ∈ ϝ.
Let be the closed unit ball of (ℓ 1 , ‖ ⋅ ‖ 1 ); that is, It is well known that is a (noncompact) closed subset of (ℓ 1 , ‖ ⋅ ‖ 1 ). Moreover is star-shaped with z = 0 a star center of . Now let ∈ (0, 1] be constant, and define : → as 4 Abstract and Applied Analysis for all x := ( ) ≥1 ∈ ℓ 1 . Clearly is nonexpansive on , and hence it is -nonexpansive for any ∈ ϝ, by Remark 5. Furthermore, ( ) is homeomorphic to the compact subset of R 2 , so that ( ) is compact. Hence ( ) = ( ) and thus ( ) is compact. We have shown that all conditions of Theorem 13 (compare Remark 14) are satisfied. Thus has a fixed point. In fact, the fixed points of are the elements x = { } ≥1 of such that = 0 whenever ≥ 2.
Now we give an example of a discontinuous self-mapping on a compact metric space which is a generalizedcontraction but not an -contraction. So the class of generalized -contraction mappings is bigger than the class of -contraction. Let : R + → R be defined by = ln for ∈ (0, 1). Note that is satisfied for all ∈ whenever ( , 2 ) > 0. Hence, is a generalized -contraction. Clearly is not continuous at = 1 and at = 0. Hence, is not an -contraction (see Remark 2.1 in [21] ) or let = 0 and = 1/2; then while and thus that is is orbitally continuous at 1. Hence, by Theorem 12, has a fixed point (in fact = 1 is the only fixed point of ).
In the last part of the paper we discuss nonemptiness and existence of fixed points for the set of best approximations of closed subsets of metric spaces and of normed spaces, respectively.

Theorem 17. Let ( , ) be a metric space. Let
∈ ϝ be a continuous mapping and let : → be -nonexpansive with a fixed point ∈ . If is a closed -invariant subset of such that is compact on , then the set ( ) of best approximations is nonempty.
As an application of Theorems 13 and 17, we deduce the following.
Theorem 18. Let be a normed linear space. Let ∈ ϝ be a continuous mapping and let : → be -nonexpansive with a fixed point ∈ . If is a complete -invariant subset of such that is compact on , and ( ) is a star-shaped set, then has a fixed point in ( ).
Abstract and Applied Analysis 5 Remark 19. As in the case of Theorem 13 (see Remark 14), the preceding theorem can be restated as follows. Let be a normed linear space (Banach space, resp.) and let : → be nonexpansive with a fixed point ∈ . If is a complete (closed, resp.) -invariant subset of such that is compact on and ( ) is a star-shaped set, then has a fixed point in ( ).
We conclude the paper illustrating Theorem 18 with the following example.
As in Example 15, let be the closed unit ball of (ℓ 1 , ‖ ⋅ ‖ 1 ). We know that is a closed and thus complete, -invariant subset of such that is compact on .