New fixed-circle results related to Fc-contractive and Fc-expanding mappings on metric spaces
aa r X i v : . [ m a t h . GN ] J a n NEW FIXED-CIRCLE RESULTS RELATED TO F c -CONTRACTIVE AND F c -EXPANDING MAPPINGS ONMETRIC SPACES NABIL MLAIKI , N˙IHAL ¨OZG ¨UR and N˙IHAL TAS¸ Abstract.
The fixed-circle problem is a recent problem about the study ofgeometric properties of the fixed point set of a self-mapping on metric (resp.generalized metric) spaces. The fixed-disc problem occurs as a natural conse-quence of this problem. Our aim in this paper, is to investigate new classes ofself-mappings which satisfy new specific type of contraction on a metric space.We see that the fixed point set of any member of these classes contains a cir-cle (or a disc) called the fixed circle (resp. fixed disc) of the correspondingself-mapping. For this purpose, we introduce the notions of an F c -contractivemapping and an F c -expanding mapping. Activation functions with fixed circles(resp. fixed discs) are often seen in the study of neural networks. This showsthe effectiveness of our fixed-circle (resp. fixed-disc) results. In this context,our theoretical results contribute to future studies on neural networks. Introduction
In the last few decades, the Banach contraction principle has been generalizedand studied by different approaches such as to generalize the used contractivecondition (see [1], [4], [5], [6], [7], [8], [11], [18], [19] and [26] for more details) andto generalize the used metric space (see [2], [3], [9], [12], [14], [17], [25], [27] and[28] for more details). Recently, some fixed-circle theorems have been introducedas a geometrical direction of generalization of the fixed-point theorems (see [20],[21], [22] and [23] for more details).Let (
X, d ) be a metric space and f be a self-mapping on X . First, we recallthat the circle C u ,ρ = { u ∈ X : d ( u, u ) = ρ } is a fixed circle of f if f u = u for all u ∈ C u ,ρ (see [20]). Similarly, the disc D u ,ρ = { u ∈ X : d ( u, u ) ≤ ρ } iscalled a fixed disc of f if f u = u for all u ∈ D u ,ρ . There are some examples ofself-mappings such that the fixed point set of the self-mapping contains a circle(or a disc). For example, let us consider the metric space ( C , d ) with the metric d ( z , z ) = | x − x | + | y − y | + | x − x + y − y | , (1.1)defined for the complex numbers z = x + iy and z = x + iy . We note thatthe metric defined in (1.1) is the metric induced by the norm function k z k = k x + iy k = | x | + | y | + | x + y | , Mathematics Subject Classification.
Primary 47H10; Secondary 54H25.
Key words and phrases.
Fixed point, fixed circle, fixed disc. (see Example 2.4 in [24]). The circle C , is seen in the following figure which isdrawn using Mathematica [31]. Define the self-mapping f on C as follows: f z = z ; x ≤ , y ≥ x ≥ , y ≤ − y + + i (cid:0) − x + (cid:1) ; x > , y > − y − + i (cid:0) − x − (cid:1) ; x < , y < z = x + iy ∈ C , then clearly, the fixed point set of f contains thecircle C , , that is, C , is a fixed circle of f . Therefore, the study of geometricproperties of the fixed point set of a self-mapping seems to be an interestingproblem in case where the fixed point is non unique. - - - - Figure 1.
The graph of the circle C , . On the other hand, fixed points of self-mappings play an important role in thestudy of neural networks. For example, in [16], it was pointed out that fixed pointsof a neural network can be determined by fixed points of the employed activationfunction. If the global input-output relationship in a neural network can beconsidered in the framework of M¨obius transformations, then the existence of oneor two fixed points of the neural network is guaranteed (see [10] for basic algebraicand geometric properties of M¨obius transformations). Some possible applicationsof theoretical fixed-circle results to neural networks have been investigated in therecent studies [20] and [23].Next, we remind the reader of the following theorems on a fixed circle.
Theorem 1.1. [20]
Let ( X, d ) be a metric space and consider the map ϕ : X → [0 , ∞ ) , ϕ ( u ) = d ( u, u ) , (1.2) for all u ∈ X . If there exists a self-mapping f : X → X satisfying ( C d ( u, f u ) ≤ ϕ ( u ) − ϕ ( f u ) and ( C d ( f u, u ) ≥ ρ ,for each u ∈ C u ,ρ , then the circle C u ,ρ is a fixed circle of f . IXED-CIRCLE RESULTS 3
Theorem 1.2. [20]
Let ( X, d ) be a metric space and consider the map ϕ asdefined in ( ) . Also, assume that f : X → X satisfies the following conditions: ( C ∗ d ( u, f u ) ≤ ϕ ( u ) + ϕ ( f u ) − ρ and ( C ∗ d ( f u, u ) ≤ ρ ,for each u ∈ C u ,ρ , then the circle C u ,ρ is a fixed circle of f . Theorem 1.3. [20]
Let ( X, d ) be a metric space and consider the map ϕ asdefined in ( ) . Also, assume that f : X → X satisfies the following conditions: ( C ∗∗ d ( u, f u ) ≤ ϕ ( u ) − ϕ ( f u ) and ( C ∗∗ hd ( u, f u ) + d ( f u, u ) ≥ ρ ,for each u ∈ C u ,ρ and some h ∈ [0 , , then the circle C u ,ρ is a fixed circle of f . Theorem 1.4. [23]
Let ( X, d ) be a metric space and assume that the mapping ϕ ρ : R + ∪ { } → R be defined by ϕ ρ ( x ) = (cid:26) x − ρ ; x >
00 ; x = 0 , (1.3) for all x ∈ R + ∪ { } . If there exists a self-mapping f : X → X satisfying (1) d ( f u, u ) = ρ for each u ∈ C u ,ρ , (2) d ( f u, f v ) > ρ for each u, v ∈ C u ,ρ and u = v , (3) d ( f u, f v ) ≤ d ( u, v ) − ϕ ρ ( d ( u, f u )) for each u, v ∈ C u ,ρ ,then the circle C u ,ρ is a fixed circle of f . This manuscript is structured as follows; in Section 2, we give some generaliza-tions of Theorems 1.1, 1.2 and 1.3. In Section 3, we present the definitions of an“ F c -contraction” and an “ F c -expanding map” where we prove new theorems on afixed circle. In section 4, we consider the fixed point sets of some activation func-tions frequently used in the study of neural networks with a geometric viewpoint.This shows the effectiveness of our fixed-circle results. In section 5, we presentsome open problems for future works. Our results show the importance of thegeometry of fixed points of a self-mapping when the fixed point is non unique.2. New Fixed-Circle Theorems for Some Contractive Mappings
First, we give a fixed-circle theorem using an auxiliary function.
Theorem 2.1.
Let ( X, d ) be a metric space, f be a self-mapping on X and themapping θ ρ : R → R be defined by θ ρ ( x ) = (cid:26) ρ ; x = ρx + ρ ; x = ρ ,for all x ∈ R and ρ ≥ . Suppose that (1) d ( f u, u ) ≤ θ ρ ( d ( u, u )) + Ld ( u, f u ) for some L ∈ ( −∞ , and each u ∈ X , (2) ρ ≤ d ( f u, u ) for each u ∈ C u ,ρ , (3) d ( f u, f v ) ≥ ρ for each u, v ∈ C u ,ρ and u = v , N. MLAIKI, N. ¨OZG ¨UR and N. TAS¸ (4) d ( f u, f v ) < ρ + d ( v, f u ) for each u, v ∈ C u ,ρ and u = v ,then f fixes the circle C u ,ρ .Proof. Let u ∈ C u ,ρ be an arbitrary point. By the conditions (1) and (2), wehave d ( f u, u ) ≤ θ ρ ( d ( u, u )) + Ld ( u, f u ) = ρ + Ld ( u, f u )and so ρ ≤ d ( f u, u ) ≤ ρ + Ld ( u, f u ). (2.1)We have two cases. Case 1.
Let L = 0. Then we find d ( f u, u ) = ρ by (2.1), that is, we have f u ∈ C u ,ρ . Then d ( u, f u ) = 0 or d ( u, f u ) = 0. Assume d ( u, f u ) = 0 for u ∈ C u ,ρ . Since u = f u , from the condition (3), we obtain d ( f u, f u ) ≥ ρ . (2.2)Also using the condition (4), we get d ( f u, f u ) < ρ + d ( f u, f u )and hence d ( f u, f u ) < ρ .which contradicts the inequality (2.2). Therefore, it should be d ( u, f u ) = 0 whichimplies f u = u . Case 2.
Let L ∈ ( −∞ , d ( u, f u ) = 0 we get a contradiction by (2.1).Hence it should be d ( u, f u ) = 0.Thereby, we obtain f u = u for all u ∈ C u ,ρ , that is, C u ,ρ is a fixed circle of f .In other words, the fixed point set of f contains the circle C u ,ρ . (cid:3) Remark . Notice that, if we consider the case L ∈ ( −∞ ,
0) in the condition(1) of Theorem 2.1 for u ∈ C u ,ρ , then we get − Ld ( u, f u ) ≤ θ ρ ( d ( u, u )) − d ( f u, u ) = d ( u, u ) − d ( f u, u ) = ϕ ( u ) − ϕ ( f u )and hence − Ld ( u, f u ) ≤ ϕ ( u ) − ϕ ( f u ).For L = −
1, we obtain d ( u, f u ) ≤ ϕ ( u ) − ϕ ( f u ).This means that the condition ( C
1) (resp. the condition ( C ∗∗ ) is satisfied forthis case.Clearly, the condition (2) of Theorem 2.1 is the same as condition ( C C ∗∗ is satisfied. Consequently, Theorem 2.1 is a generalization of Theorem1.1 and Theorem 1.3 for the cases L ∈ ( −∞ , \ {− } . For the case L = − IXED-CIRCLE RESULTS 5
Example 2.3.
Let ( R , d ) be the metric space with the usual metric d ( x , x ) = | x − x | and consider the circle C , = {− , } . If we define the self-mapping f : R → R as f x = (cid:26) x + x − x ∈ {− , } x ∈ R , then it is not difficult to see that f satisfies the hypothesis ofTheorem 2.1 for the circle C , and L = − . Clearly, C , is the fixed circle of f . Example 2.4.
Consider ( R , d ) to be the usual metric space and the circle C , = {− , } . Define f : R → R by f x = x = − − x = 20 ; otherwise ,for each x ∈ R , then f does not satisfy the condition (1) of Theorem 2.1 for each x ∈ C , and for any L ∈ ( −∞ , f does not satisfy the condition (4)for each x ∈ C , and for any L ∈ ( −∞ , f does not fix C , and thisexample shows that the condition (4) is crucial in Theorem 2.1. Example 2.5.
Consider ( R , d ) to be the usual metric space and the circles C , = {− , } and C , = {− , } . If we define f : R → R as f x = (cid:26) x ; x ∈ C , ∪ C , x ∈ R , then f satisfies the hypothesis of Theorem 2.1 for each of thecircles C , and C , and for any L ∈ [ − , C , and C , are the fixedcircles of f .We give another fixed-circle result. Theorem 2.6.
Let ( X, d ) be a metric space, f be a self-mapping on X and themapping θ ρ : R → R be defined by θ ρ ( x ) = (cid:26) ρ ; x = ρx + ρ ; x = ρ ,for all x ∈ R and ρ ≥ . Suppose that (1) 2 d ( u, u ) − d ( f u, u ) ≤ θ ρ ( d ( u, u )) + Ld ( u, f u ) for some L ∈ ( −∞ , andeach u ∈ X , (2) d ( f u, u ) ≤ ρ for each u ∈ C u ,ρ , (3) d ( f u, f v ) ≥ ρ for each u, v ∈ C u ,ρ and u = v , (4) d ( f u, f v ) < ρ + d ( v, f u ) for each u, v ∈ C u ,ρ and u = v ,then the self-mapping f fixes the circle C u ,ρ . Proof.
Consider u ∈ C u ,ρ to be an arbitrary point. Using the conditions (1) and(2), we get 2 d ( u, u ) − d ( f u, u ) ≤ d ( u, u ) + Ld ( u, f u ),2 ρ − d ( f u, u ) ≤ ρ + Ld ( u, f u ) N. MLAIKI, N. ¨OZG ¨UR and N. TAS¸ and ρ ≤ d ( f u, u ) + Ld ( u, f u ) ≤ ρ + Ld ( u, f u ). (2.3)Similarly to the arguments used in the proof of Theorem 2.1, a direct computationshows that the circle C u ,ρ is fixed by f . (cid:3) Remark . Notice that, if we consider the case L = − u ∈ C u ,ρ then we get d ( u, f u ) ≤ θ ρ ( d ( u, u ))+ d ( f u, u ) − d ( u, u ) = ρ + d ( f u, u ) − ρ = ϕ ( u )+ ϕ ( f u ) − ρ .Hence the condition ( C ∗ is satisfied. Also, the condition (2) of Theorem 2.6is contained in the condition ( C ∗ . Therefore, Theorem 2.6 is a special caseof Theorem 1.2 in this case. For the cases L ∈ ( −∞ , Example 2.8.
Consider the usual metric space ( R , d ) and the circle C , = {− , } . Define the map f : R → R as f x = (cid:26) x ; u ∈ {− , } x ; otherwise ,for each x ∈ R , hence f satisfies the hypothesis of Theorem 2.6 for L = − .Clearly, C , is the fixed circle of f . It is easy to check that f does not satisfythe condition (1) of Theorem 2.1 for any L ∈ ( −∞ , Example 2.9.
Consider the usual metric space ( R , d ) and the circles C , = {− , } and C , = {− , } . Define the self-mapping f : R → R as f x = (cid:26) x ; x ∈ C , ∪ C , αx ; otherwise ,for each x ∈ R and α ≥
2, then f satisfies the hypothesis of Theorem 2.6 for L = 0 and for each of the circles C , and C , . Clearly, C , and C , are thefixed circles of f . Notice that the fixed circles C , and C , are not disjoint.Considering Example 2.5 and Example 2.9, we deduce that a fixed circle neednot to be unique in Theorem 2.1 and Theorem 2.6. If a fixed circle is non uniquethen two fixed circle of a self-mapping can be disjoint or not. Next, we prove atheorem where f fixes a unique circle. Theorem 2.10.
Let ( X, d ) be a metric space and f : X → X be a self-mappingwhich fixes the circle C u ,ρ . If the condition d ( f u, f v ) < max { d ( v, f u ) , d ( v, f v ) } , (2.4) is satisfied by f for all u ∈ C u ,ρ and v ∈ X \ C u ,ρ , then C u ,ρ is the unique fixedcircle of f .Proof. Let C u ,µ be another fixed circle of f . If we take u ∈ C u ,ρ and v ∈ C u ,µ with u = v , then using the inequality (2.4), we obtain d ( u, v ) = d ( f u, f v ) < max { d ( v, f u ) , d ( v, f v ) } = d ( u, v ), IXED-CIRCLE RESULTS 7 a contradiction. We have u = v for all u ∈ C u ,ρ , v ∈ C u ,µ hence f only fixes thecircle C u ,ρ . (cid:3) In the following example, we show that the converse of Theorem 2.10 is nottrue in general.
Example 2.11.
Consider the usual metric space ( C , d ) and the circle C , . Define f on C as follows: f z = (cid:26) z if z = 00 if z = 0 ,for z ∈ C , where z denotes the complex conjugate of z . It is not difficult to seethat C , is the unique fixed circle of f where f does not satisfy the hypothesisof Theorem 2.10.Now, we give the following example as an illustration of Theorem 2.10. Example 2.12.
Let Y = {− , , } and the metric d : Y × Y → [0 , ∞ ) be definedby d ( u, v ) = (cid:26) u = v | u | + | v | ; u = v ,for all u ∈ Y . If we consider the self-mapping f : Y → Y defined by f u = 0,for any u ∈ Y , then C , = { } is the unique fixed circle of f . Next, we present the following interesting theorem that involves the identitymap I X : X → X defined by I X ( u ) = u for all u ∈ X. Theorem 2.13.
Let ( X, d ) be a metric space. Consider the map f from X toitself with the fixed circle C u ,ρ . The self-mapping f satisfies the condition d ( u, f u ) ≤ α [max { d ( u, f u ) , d ( u , f u ) } − d ( u , f u )] , (2.5) for all u ∈ X and some α ∈ (0 , if and only if f = I X .Proof. Let u ∈ X with f u = u . By inequality (2.5), if d ( u, f u ) ≥ d ( u , f u ), thenwe get d ( u, f u ) ≤ α [ d ( u, f u ) − d ( u , f u )] ≤ αd ( u, f u ),which leads us to a contradiction due to the fact that α ∈ (0 , d ( u, f u ) ≤ d ( u , f u ), then we find d ( u, f u ) ≤ α [ d ( u , f u ) − d ( u , f u )] = 0 . Hence, f u = u and that is f = I X since u is an arbitrary in X .Conversely, I X satisfies the condition (2.5) clearly. (cid:3) Corollary 2.14.
Let ( X, d ) be a metric space and f : X → X be a self-mapping.If f satisfies the hypothesis of Theorem 2.1 ( resp. Theorem 2.6 ) but the condition ( ) is not satisfied, then f = I X . Now, we rewrite the following theorem given in [20].
N. MLAIKI, N. ¨OZG ¨UR and N. TAS¸
Theorem 2.15. [20]
Let ( X, d ) be a metric space. Consider the map f from X to itself which have a fixed circle C u ,ρ and ϕ as in ( ) . Then f satisfies thecondition d ( u, f u ) ≤ ϕ ( u ) − ϕ ( f u ) h , (2.6) for every u ∈ Y and h > if and only if f = I X . Theorem 2.16.
Let ( X, d ) be a metric space. Consider the map f from X toitself which have a fixed circle C u ,ρ and ϕ as in ( ) . Then f satisfies ( ) ifand only if f satisfies ( ) .Proof. The proof follows easily. (cid:3) F c - contractive and F c - expanding mappings in metric spaces In this section, we use a different approach to obtain new fixed-circle results.First, we recall the definition of the following family of functions which was in-troduced by Wardowski in [30].
Definition 3.1. [30] Let F be the family of all functions F : (0 , ∞ ) → R suchthat( F ) F is strictly increasing,( F ) For each sequence { α n } in (0 , ∞ ) the following holdslim n →∞ α n = 0 if and only if lim n →∞ F ( α n ) = −∞ ,( F ) There exists k ∈ (0 ,
1) such that lim α → + α k F ( α ) = 0.Some examples of functions that satisfies the conditions ( F ), ( F ) and ( F )of Definition 3.1 are F ( u ) = ln( u ), F ( u ) = ln( u ) + u , F ( u ) = − √ u and F ( u ) =ln( u + u ) (see [30] for more details).At this point, we introduce the following new contraction type. Definition 3.2.
Let (
X, d ) be a metric space and f be a self-mapping on X . Ifthere exist t > F ∈ F and u ∈ X such that d ( u, f u ) > ⇒ t + F ( d ( u, f u )) ≤ F ( d ( u , u )),for all u ∈ X , then f is called as an F c -contraction.We note that the point u mentioned in Definition 3.2 must be a fixed point ofthe mapping f . Indeed, if u is not a fixed point of f , then we have d ( u , f u ) > d ( u , f u ) > ⇒ t + F ( d ( u , f u )) ≤ F ( d ( u , u )).This is a contradiction since the domain of F is (0 , ∞ ). Consequently, we obtainthe following proposition as an immediate consequence of Definition 3.2. Proposition 3.3.
Let ( X, d ) be a metric space. If f is an F c -contraction with u ∈ X then we have f u = u . Using this new type contraction we give the following fixed-circle theorem.
IXED-CIRCLE RESULTS 9
Theorem 3.4.
Let ( X, d ) be a metric space and f be an F c -contraction with u ∈ X . Define the number σ by σ = inf { d ( u, f u ) : u = f u, u ∈ X } .Then C u ,σ is a fixed circle of f . In particular, f fixes every circle C u ,r where r < σ .Proof. If σ = 0 then clearly C u ,σ = { u } and by Proposition 3.3, we see that C u ,σ is a fixed circle of f . Assume σ > u ∈ C u ,σ . If f u = u , then bythe definition of σ we have d ( u, f u ) ≥ σ . Hence using the F c -contractive propertyand the fact that F is increasing, we obtain F ( σ ) ≤ F ( d ( u, f u )) ≤ F ( d ( u , u )) − t < F ( d ( u , u )) = F ( σ ),which leads to a contradiction. Therefore, we have d ( u, f u ) = 0, that is, f u = u .Consequently, C u ,σ is a fixed circle of f .Now we show that f also fixes any circle C u ,r with r < σ . Let u ∈ C u ,r andassume that d ( u, f u ) >
0. By the F c -contractive property, we have F ( d ( u, f u )) ≤ F ( d ( u , u )) − t < F ( r ).Since F is increasing, then we find d ( u, f u ) < r < σ. But σ = inf { d ( u, f u ) : for all u = f u } , which leads us to a contradiction. Thus, d ( u, f u ) = 0 and f u = u . Hence, C u ,r is a fixed circle of f . (cid:3) Remark .
1) Notice that, in Theorem 3.4, the F c -contraction f fixes the disc D u ,σ . Therefore, the center of any fixed circle is also fixed by f . In Theorem 1.4,the self-mapping f maps C u ,ρ into (or onto) itself, but the center of the fixedcircle need not to be fixed by f .2) Related to the number of the elements of the set X , the number of the fixedcircles of an F c -contractive self-mapping f can be infinite (see Example 3.8).We give some illustrative examples. Example 3.6.
Let X = { , , e , − e , e − , e + 1 } be the metric space withthe usual metric. Define the self-mapping f : X → X as f u = (cid:26) u = 0 u ; otherwise ,for all u ∈ X . Then the self-mapping f is an F c -contractive self-mapping with F = ln u , t = 1 and u = e . Using Theorem 3.4, we obtain σ = 1 and f fixes the circle C e , = { e − , e + 1 } . Clearly, C fixes the disc D e , = { u ∈ Y : d ( u, e ) ≤ } = { e , e − , e + 1 } . Notice that f fixes also the circle C ,e = {− e , e } . The converse statement of Theorem 3.4 is not always true as seen in the fol-lowing example.
Example 3.7.
Let (
X, d ) be a metric space, u ∈ X any point and the self-mapping f : X → X defined as f u = (cid:26) u ; d ( u, u ) ≤ µu ; d ( u, u ) > µ ,for all u ∈ X with any µ >
0. Then it can be easily seen that f is not an F c -contractive self-mapping for the point u but f fixes every circle C u ,r where r ≤ µ . Example 3.8.
Let ( C , d ) be the usual metric space and define the self-mapping f : C → C as f u = (cid:26) u ; | u | < u + 1 ; | u | ≥ u ∈ C . We have σ = min { d ( u, f u ) : u = f u } = 1. Then f is an F c -contractive self-mapping with F = ln u , t = ln 2 and u = 0 ∈ C . Evidently,the number of the fixed circles of f is infinite.Now, to obtain a new fixed-circle theorem, we use the well-known fact that ifa self-mapping f on X is surjective, then there exists a self mapping f ∗ : X → X such that the map ( f ◦ f ∗ ) is the identity map on X . Definition 3.9.
A self-mapping f on a metric space X is called as an F c -expanding map if there exist t < F ∈ F and u ∈ X such that d ( u, f u ) > ⇒ F ( d ( u, f u )) ≤ F ( d ( u , f u )) + t ,for all u ∈ X . Theorem 3.10.
Let ( X, d ) be a metric space. If f : X → X is a surjective F c -expanding map with u ∈ X , then f has a fixed circle in X. Proof.
Since f is surjective, we know that there exists a self-mapping f ∗ : X → X, such that the map ( f ◦ f ∗ ) is the identity map on X . Let u ∈ X be such that d ( u, f ∗ u ) > z = f ∗ u . First, notice the following fact f z = f ( f ∗ u ) = ( f ◦ f ∗ ) u = u .Since d ( z, f z ) = d ( f z, z ) > F c -expanding property of f we get F ( d ( z, f z )) ≤ F ( d ( u , f z )) + t and F ( d ( f ∗ u, u )) ≤ F ( d ( u , u )) + t .Therefore, we obtain − t + F ( d ( f ∗ u, u )) ≤ F ( d ( u , u )).Consequently, f ∗ is an F c -contraction on X with u as − t >
0. Then by Theorem3.4, f ∗ has a fixed circle C u ,σ . Let v ∈ C u ,σ be any point. Using the fact that f v = f ( f ∗ v ) = v , IXED-CIRCLE RESULTS 11 we deduce that f v = v , that is v is a fixed point of f , which implies that f alsofixes C u ,σ , as required. (cid:3) Example 3.11.
Let X = { , , , , } with the usual metric. Define the self-mapping f : X → X by f u = u = 11 ; u = 2 u ; u ∈ { , , } . f is a surjective F c -expanding map with u = 4, F ( u ) = ln u and t = − ln 2.We have σ = min { d ( u, f u ) : u = f u, u ∈ X } = 1and the circle C , = { , } is the fixed circle of f . Remark . If f is not a surjective map, then the result in Theorem 3.10 is nottrue everywhen. For example, let X = { , , , } with the usual metric d. Definethe self-mapping f : X → X by f u = u ∈ { , } u = 24 ; u = 4 .Then, it is easy to check that f satisfies the condition d ( u, f u ) > ⇒ F ( d ( u, f u )) ≤ F ( d ( u , f u )) + t for all u ∈ X , with F ( u ) = ln u , u = 4 and t = − ln 2. Therefore, f satisfiesall the conditions of Theorem 3.10, except that f is not surjective. Notice that σ = 1 and f does not fix the circle C , .4. Fixed point sets of activation functions
Activation functions are the primary neural networks decision-making units ina neural network and hence it is critical to choose the most appropriate activationfunction for neural network analysis [29]. Characteristic properties of activationfunctions play an important role in learning and stability issues of a neural net-work. A comprehensive analysis of different activation functions with individualreal-world applications was given in [29]. We note that the fixed point sets ofcommonly used activation functions (e.g. Ramp function, ReLU function, LeakyReLU function) contain some fixed discs and fixed circles. For example, let usconsider the Leaky ReLU function defined by f ( x ) = max( kx, x ) = (cid:26) kx ; x ≤ x ; x > k ∈ [0 , ρ = u ∈ (0 , ∞ ) be any positive numberand consider the circle C u ,ρ = { , u } . Then it is easy to check that the function f ( x ) satisfies the conditions of Theorem 2.1 for the circle C u ,ρ with L = 0. Clearly, the circle C u ,ρ is a fixed circle of f ( x ) and the center of the fixed circleis also fixed by f ( x ).On the other hand, theoretic fixed point theorems have been extensively used inthe study of neural networks. For example, in [15], the existence of a fixed pointfor every recurrent neural network was shown and a geometric approach was usedto locate where the fixed points are. Brouwer’s Fixed Point Theorem was usedto ensure the existence of a fixed point. This study shows the importance of thegeometric viewpoint and theoretic fixed point results in applications. Obviously,our fixed circle and fixed disc results are important for future studies in the studyof neural networks. 5. Conclusion and future works
In this section, we want to bring to the reader’s attention in connection withthe investigation of some open questions. Concerning the geometry of non uniquefixed points of a self-mapping on a metric space, we have obtained new geometric(fixed-circle or fixed-disc) results. To do this, we use two different approaches.One of them is to measure whether a given circle is fixed or not by a self-mapping.Another approach is to find which circle is fixed by a self-mapping under somecontractive or expanding conditions. The investigation of new conditions whichensure a circle or a disc to be fixed by a self-mapping can be considered as afuture problem. For a self-mapping of which fixed point set contains a circle or adisc, new contractive or expanding conditions can also be investigated.On the other hand, there are some examples of self-mappings which have acommon fixed circle. For example, let ( R , d ) be the usual metric space andconsider the circle C , = {− , } . We define the self-mappings f : R → R and f : R → R as f x = (cid:26) x ; x ∈ {− , } f x = 5 x + 33 x + 5 ,for each x ∈ R , respectively. Then both the self-mappings f and f fixes thecircle C , = {− , } , that is, the circle C , = {− , } is a common fixed circleof the self-mappings f and f . At this point, the following question can be leftas a future study. Question 5.1.
What is (are) the condition(s) to make any circle C u ,ρ as thecommon fixed circle for two (or more than two) self-mappings?Finally, the problems considered in this paper can also be studied on somegeneralized metric spaces. For example, the notion of an M s -metric space wasintroduced in [13]. Notation 5.2.
We use the following notations.1. m s u,v,z := min { m s ( u, u, u ) , m s ( v, v, v ) , m s ( z, z, z ) } M s u,v,z := max { m s ( u, u, u ) , m s ( v, v, v ) , m s ( z, z, z ) } Definition 5.3. An M s -metric on a nonempty set Y is a function m s : Y → R + if for all u, v, z, t ∈ Y we have(1) m s ( u, u, u ) = m s ( v, v, v ) = m s ( z, z, z ) = m s ( u, v, z ) ⇐⇒ u = v = z, IXED-CIRCLE RESULTS 13 (2) m s u,v,z ≤ m s ( u, v, z ) , (3) m s ( u, u, v ) = m s ( v, v, u ) , (4)( m s ( u, v, z ) − m s u,v,z ) ≤ ( m s ( u, u, t ) − m s u,u,t )+ ( m s ( v, v, t ) − m s v,v,t ) + ( m s ( z, z, t ) − m s z,z,t ) . Then the pair (
Y, m s ) is called an M s -metric space.One can consult [13] for some examples and basic notions of an M s -metricspace.In M s -metric spaces we define a circle as follow; C u ,ρ = { u ∈ Y | m s ( u , u, u ) − m s u ,u,u = ρ } . Question 5.4.
Let (
Y, m s ) be an M s -metric space, k > f be a surjectiveself-mapping on Y . Let we have m s ( u, f u, f u ) ≤ km s ( u , u, f u ) , for every u ∈ Y and some u ∈ Y . Does f have point circle on Y ? Question 5.5.
Let (
Y, m s ) be an M s -metric space, t > F ∈ F and f be asurjective self-mapping on Y . Let we have m s ( u, f u, f u ) > ⇒ F ( m s ( u, f u, f u )) ≥ F ( m s ( u , u, f u )) + t, for every u ∈ Y and some u ∈ Y . Does f have a fixed circle on Y ? Acknowledgements
The first author would like to thank Prince Sultan University for funding thiswork through research group Nonlinear Analysis Methods in Applied Mathemat-ics (NAMAM) group number RG-DES-2017-01-17.
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