Geometric Properties of Fixed Points and Simulation Functions
aa r X i v : . [ m a t h . M G ] F e b GEOMETRIC PROPERTIES OF FIXED POINTS ANDSIMULATION FUNCTIONS
NIHAL ¨OZG ¨UR ∗ AND NIHAL TAS¸
Abstract.
Geometric properties of the fixed point set
F ix ( f ) of a self-mapping f on a metric or a generalized metric space is an attractive issue. The set F ix ( f )can contain a geometric figure (a circle, an ellipse, etc.) or it can be a geometricfigure. In this paper, we consider the set of simulation functions for geometricapplications in the fixed point theory both on metric and some generalized met-ric spaces ( S -metric spaces and b -metric spaces). The main motivation of thispaper is to investigate the geometric properties of non unique fixed points ofself-mappings via simulation functions. Introduction
Recently, the set of simulation functions defined in [7] has been used for thesolutions of many recent problems such as fixed-circle problem (resp. fixed-discproblem) and Rhoades’ open problem on discontinuity (see [11, 18, 21]). On theother hand, simulation functions have been studied by various aspects in metricfixed-point theory (see for example [2, 6, 7, 8, 13, 22, 23]). For example, in [13],Olgun et al. gave a new class of Picard operators on complete metric spacesvia simulation functions. Simulation functions have been used to study the bestproximity points in metric spaces. For example, Kosti´c et al. presented severalbest proximity point results involving simulation functions (for more details see[6, 8] and the references therein).In [22], the set of simulation functions has been enlarged by A. F. Rold´an-L´opez-de-Hierro et al. Every simulation function in the original Khojasteh et al.’s senseis also a simulation function in A. F. Rold´an-L´opez-de-Hierro et al.’s sense but theconverse is not true (see [22] for more details). In this paper, we focus on the setof simulation functions in both sense and using their properties, we consider somerecent problems in fixed-point theory.
Date : Received: , Accepted: .
Mathematics Subject Classification.
Primary 54H25; Secondary 47H09, 47H10.
Key words and phrases.
Fixed circle, fixed disc, fixed ellipse, simulation function. ∗ Corresponding author: Balıkesir University, Department of Mathematics, 10145 Balıkesir,TURKEY, [email protected].
Recall that the function ζ : [0 , ∞ ) × [0 , ∞ ) → R is said to be a simulationfunction in the Khojasteh et al.’s sense, if the following hold:( ζ ) ζ (0 ,
0) = 0 , ( ζ ) ζ ( t, s ) < s − t for all s, t > ζ ) If { t n } , { s n } are sequences in (0 , ∞ ) such thatlim n →∞ t n = lim n →∞ s n > n →∞ ζ ( t n , s n ) < Z (see [2] and [7] for moredetails).In [22], Rold´an-L´opez-de-Hierro et al. modified this definition of simulationfunctions and so enlarged the family of all simulation functions. To do this, onlythe condition ( ζ ) was replaced by the following condition ( ζ ) ∗ as follows:( ζ ) ∗ If { t n } , { s n } are sequences in (0 , ∞ ) such thatlim n →∞ t n = lim n →∞ s n > t n < s n for all n ∈ N ,then lim sup n →∞ ζ ( t n , s n ) < ζ ) is not used. So, both definitions of thesimulation functions and examples can be used to study such kind applications.Some examples of simulation functions ζ : [0 , ∞ ) × [0 , ∞ ) → R are1) ζ ( t, s ) = λs − t , where λ ∈ [0 , ζ ( t, s ) = s − ϕ ( s ) − t , where ϕ : [0 , ∞ ) → [0 , ∞ ) is a continuous function suchthat ϕ ( t ) = 0 if and only if t = 0,3) ζ ( t, s ) = sφ ( s ) − t , where φ : [0 , ∞ ) → [0 ,
1) is a mapping such thatlim sup φ ( t ) t → r + < r > ζ ( t, s ) = η ( s ) − t , where ϕ : [0 , ∞ ) → [0 , ∞ ) be an upper semi-continuousmapping such that η ( t ) < t for all t > η (0) = 0, IMULATION FUNCTIONS 3 ζ ( t, s ) = s − t R ψ ( t ) dt , where ψ : [0 , ∞ ) → [0 ,
1) is a function such that t R ψ ( t ) dt exists and ε R ψ ( u ) du > ε for each ε > C , d ) with the metric defined for the complex numbers z = x + iy and z = x + iy as follows: d ( z , z ) = s ( x − x ) y − y ) ,where d is the metric induced by the norm function k z k = k x + iy k = q x + 4 y .Consider the circle C , and define the self-mapping g on C by gz = ( z ; x ≥ , y ≥ x ≤ , y ≤ z − z + z )+74 zz ; x < , y > x > , y < z = x + iy ∈ C , where z = x − iy is the complex conjugate of z . Then, itis easy to verify that the fixed point set of g contains the circle C , , that is, C , isa fixed circle of g . We use the notion of “inversion in an ellipse” to construct thisself-mapping (see Proposition 1 given in [25]).There are several papers for the cases a fixed circle and a fixed disc (see [1, 10,12, 17, 18, 19, 20, 24, 28] and the references therein). The fixed ellipse case is alsoconsidered in the recent studies [3] and [5]. In [3], the cases a fixed Apollonius circle,fixed hyperbola and fixed Cassini curve are considered extensively on metric andsome generalized metric spaces. Therefore, the study of geometric properties of thefixed point set of a self-mapping seems to be an interesting problem in case wherethe fixed point is non unique. In this paper, we study on the geometric propertiesof the fixed point set of a self-mapping via simulation functions on metric (resp. S -metric and b -metric) spaces. The relationships among a metric, an S -metric anda b -metric are well known, so we refer the reader to [14], [26] and [27] for moredetails. 2. Simulation functions and the geometry of fixed points
In this section, we study on geometric properties of the fixed point set
F ix ( f ) = { x ∈ X : f x = x } for a self-mapping f of a metric (resp. S -metric, b -metric) space.The set of simulation functions has been used in [11], [18] and [21] to obtain new N. ¨OZG ¨UR AND N. TAS¸ results on the fixed-circle (resp. fixed-disc) problem. In [21], together with someproperties of simulation functions, the numbers M ( x, y ) and ρ defined by M ( x, y ) = max ( ad ( x, f x ) + (1 − a ) d ( y, f y ) , (1 − a ) d ( x, f x ) + ad ( y, f y ) , d ( x,fy )+ d ( y,fx )2 ) , ≤ a <
1, (2.1) ρ = inf { d ( x, f x ) : x = f x, x ∈ X } , (2.2)and an auxiliary function ϕ : R + → R + satisfying ϕ ( t ) < t for each t >
0, wereused to get new fixed-circle (resp. fixed-disc) results. Using these numbers and anauxiliary function, we present new results on the geometric study of the fixed pointset of a self-mapping.2.1.
Geometric study of fixed points on metric spaces.
Let (
X, d ) be ametric space and f : X → X be a self-mapping. First, we recall that the circle C x ,ρ = { x ∈ X : d ( x, x ) = ρ } (resp. the disc D x ,ρ = { x ∈ X : d ( x, x ) ≤ ρ } ) isa fixed circle (resp. a fixed disc) of f if f x = x for all x ∈ C x ,ρ (resp. for all x ∈ D x ,ρ ) (see [17], [18]). More generally, a geometric figure F (a circle, an ellipse,a hyperbola, a Cassini curve etc.) contained in the fixed point set F ix ( f ) is calleda fixed figure (a fixed circle, a fixed ellipse, a fixed hyperbola, a fixed Cassini curve,etc.) of the self-mapping f .Let E r ( x , x ) be the ellipse defined as E r ( x , x ) = { x ∈ X : d ( x, x ) + d ( x, x ) = r } .Clearly, we have r = 0 ⇒ x = x and E r ( x , x ) = C x ,r = { x } . Now, we use the set of simulation functions and the number ρ to obtain someresults for the case where the fixed point set F ix ( f ) contains an ellipse or an ellipsewith its interior. Theorem 2.1.
Let ( X, d ) be a metric space, f : X → X be a self-mapping, ζ ∈ Z be a simulation function and the number ρ be defined as in ( ) . If there existsome points x , x ∈ X such that ( a ) For all x ∈ E ρ ( x , x ) , there exists δ ( ρ ) > satisfying ρ ≤ M ( x, x ) + M ( x, x ) < ρ δ ( ρ ) = ⇒ d ( f x, x ) + d ( f x, x ) ≤ ρ , ( b ) For all x ∈ X , d ( f x, x ) > ⇒ ζ ( d ( f x, x ) , M ( x, x )) ≥ and ζ ( d ( f x, x ) , M ( x, x )) ≥ , IMULATION FUNCTIONS 5 ( c ) For all x ∈ X , d ( f x, x ) > ⇒ ζ (cid:18) d ( f x, x ) , d ( x, x ) + d ( f x, x ) + d ( x, x ) + d ( f x, x )2 (cid:19) ≥ ,then f x = x , f x = x and F ix ( f ) contains the ellipse E ρ ( x , x ) .Proof. We have M ( x , x ) = d ( f x , x ) and M ( x , x ) = d ( f x , x ).First, we show that f x = x and f x = x . If f x = x and f x = x then d ( f x , x ) > d ( f x , x ) >
0. Using the condition ( ζ ), we find ζ ( d ( f x , x ) , M ( x , x )) = ζ ( d ( f x , x ) , d ( f x , x )) < d ( f x , x ) − d ( f x , x ) = 0and ζ ( d ( f x , x ) , M ( x , x )) = ζ ( d ( f x , x ) , d ( f x , x )) < d ( f x , x ) − d ( f x , x ) = 0,which are contradictions with the condition ( b ). Hence it should be f x = x and f x = x .If ρ = 0, then we have E ρ ( x , x ) = C x ,ρ = { x } and x = x . Hence, the proofis completed.Now we assume ρ = 0. Let x ∈ E ρ ( x , x ) be any point such that f x = x . Then d ( x, f x ) > M ( x, x ) = max (cid:26) ad ( x, f x ) , (1 − a ) d ( x, f x ) , d ( x, x ) + d ( x , f x )2 (cid:27) and M ( x, x ) = max (cid:26) ad ( x, f x ) , (1 − a ) d ( x, f x ) , d ( x, x ) + d ( x , f x )2 (cid:27) .Using the condition ( a ), we get ρ ≤ M ( x, x ) + M ( x, x ) < ρ δ ( ρ ) = ⇒ d ( f x, x ) + d ( f x, x ) ≤ ρ . (2.3) N. ¨OZG ¨UR AND N. TAS¸
Now, using the inequality (2.3) and the conditions ( c ), ( ζ ), we obtain0 ≤ ζ (cid:18) d ( f x, x ) , d ( x, x ) + d ( f x, x ) + d ( f x, x ) + d ( x, x )2 (cid:19) < d ( x, x ) + d ( f x, x ) + d ( f x, x ) + d ( x, x )2 − d ( f x, x )= d ( x, x ) + d ( x, x )2 + d ( f x, x ) + d ( f x, x )2 − d ( f x, x ) ≤ ρ ρ − d ( f x, x ) = ρ − d ( f x, x )and hence d ( f x, x ) < ρ .This is a contradiction by the definition of the number ρ . Because of this contra-diction, it should be f x = x . Consequently, we have E ρ ( x , x ) ⊂ F ix ( f ). (cid:3) Remark 2.1. If x = x then we have E ρ ( x , x ) = C x , ρ and Theorem 2.1 isreduced to a fixed-circle theorem as follows : Theorem 2.2.
Let ( X, d ) be a metric space, f : X → X be a self-mapping, ζ ∈ Z be a simulation function and the number ρ be defined as in ( ) . If there existssome point x ∈ X such that ( a ) For all x ∈ C x ,ρ , there exists δ ( ρ ) > satisfying ρ ≤ M ( x, x ) < ρ δ ( ρ ) = ⇒ d ( f x, x ) ≤ ρ , ( b ) For all x ∈ X , d ( f x, x ) > ⇒ ζ ( d ( f x, x ) , M ( x, x )) ≥ ,then f x = x and the set F ix ( f ) contains the circle C x ,ρ .Proof. The proof follows by Theorem 2.1 and Remark 2.1. (cid:3)
Example 2.1.
Let X = {− , − , , , , } with the metric d ( x, y ) = | x − y | .Define the self-mapping f : X → X by f x = ( x + 6 , x = 12 x , x ∈ {− , − , , , } . Then the self-mapping f satisfies the conditions of Theorem 2.1 for the points x = − and x = 1 and the simulation function ζ ( t, s ) = s − t . Indeed, we have ρ = min { d ( x, f x ) : x ∈ X, x = f x } = 6 and E ( − ,
1) = {− , } . IMULATION FUNCTIONS 7
For all x ∈ E ( − , , there exists δ ( ρ ) = 4 > satisfying ≤ M ( x, −
1) + M (3 , < ⇒ d ( f x, −
1) + d ( f x,
1) = 6 ≤ ρ ,hence the condition ( a ) is satisfied.For x = 12 , we have d ( x, f x ) = 0 , M (12 , −
1) = 16 , M (12 ,
1) = 14 and so, weobtain ζ ( d ( f x, x ) , M ( x, x )) = ζ (6 ,
16) = 162 −
6= 2 > and ζ ( d ( f x, x ) , M ( x, x )) = ζ (6 ,
14) = 142 −
6= 1 > . This shows that the condition ( b ) is also satisfied by f .Since we have d ( x, f x ) > for x = 12 , we find ζ (cid:18) d ( f x, x ) , d ( x, x ) + d ( f x, x ) + d ( f x, x ) + d ( x, x )2 (cid:19) = ζ (6 , − > ,hence the condition ( c ) is satisfied.Clearly, we have F ix ( f ) = {− , − , , , } and the ellipse E ( − ,
1) = {− , } is contained in the set F ix ( f ) . That is, the ellipse E ( − , is a fixed ellipse ofthe self-mapping f .On the other hand, it is easy to check that the self-mapping f satisfies the con-ditions of Theorem 2.2 for the point x = 3 and the simulation function ζ ( t, s ) = s − t . Clearly, the set F ix ( f ) contains the circle C , = {− } . Definition 2.1.
Let ζ ∈ Z be any simulation function. The self-mapping f issaid to be a Z E -contraction with respect to ζ if there exist x , x ∈ X such that thefollowing condition holds for all x ∈ X : d ( f x, x ) > ⇒ ζ ( d ( f x, x ) , d ( f x, x ) + d ( f x, x )) ≥ . If f is a Z E -contraction with respect to ζ , then we have d ( f x, x ) < d ( f x, x ) + d ( f x, x ) , (2.4)for all x ∈ X with f x = x or f x = x . Indeed, if f x = x then the inequality(2.4) is satisfied trivially. If f x = x then d ( f x, x ) > Z E -contraction and the condition ( ζ ), we obtain0 ≤ ζ ( d ( f x, x ) , d ( f x, x ) + d ( f x, x )) < d ( f x, x ) + d ( f x, x ) − d ( f x, x ) N. ¨OZG ¨UR AND N. TAS¸ and so Equation (2.4) is satisfied.Now we give the following theorem.
Theorem 2.3.
Let f be a Z E -contraction with respect to ζ with x , x ∈ X andconsider the set E ρ ( x , x ) = { x ∈ X : d ( x, x ) + d ( x, x ) ≤ ρ } .If the condition < d ( f x, x ) + d ( f x, x ) ≤ ρ holds for all x ∈ E ρ ( x , x ) − { x , x } then F ix ( f ) contains the set E ρ ( x , x ) .Proof. If ρ = 0, then we have E ρ ( x , x ) = D x ,ρ = { x } and this theorem coincideswith Theorem 2.2 in [18]. In this case, we have f x = x . Assume that ρ = 0. If x = x then E ρ ( x , x ) = D x , ρ and again this case is reduced to Theorem 2.2 in[18].Assume that x = x and let x ∈ E ρ ( x , x ) be such that f x = x . By thedefinition of ρ , we have 0 < ρ ≤ d ( x, f x ) and using the condition ( ζ ) we find ζ ( d ( f x, x ) , d ( f x, x ) + d ( f x, x )) < d ( f x, x ) + d ( f x, x ) − d ( f x, x ) ≤ ρ − d ( f x, x ) ≤ ρ − ρ = 0,a contradiction with the Z E -contractive property of f . This contradiction leads f x = x , so the set F ix ( f ) contains the set E ρ ( x , x ). (cid:3) Example 2.2.
Let us consider the self-mapping f defined in Example 2.1. f is an Z E -contraction with respect to ζ ( t, s ) = s − t with the points x = − and x = 1 .Indeed, we get ζ ( d ( f x, x ) , d ( f x, x ) + d ( f x, x )) = ζ (6 ,
19 + 17) = ζ (6 , − ≥ ,for x = 12 with d ( f x, x ) > . Also we have, < d ( f x, −
1) + d ( f x, ≤ ,for all x ∈ E ( − , − {− , } . Therefore, f satisfies the conditions of Theorem2.3. Notice that the set F ix ( f ) contains the set E ( − ,
1) = {− , − , , } . Let r ∈ [0 , ∞ ). Now we give a fixed-circle theorem using the auxiliary function ϕ r : R + ∪ { } → R defined by ϕ r ( u ) = ( u − r ; u >
00 ; u = 0 , (2.5)for all u ∈ R + ∪ { } (see [15]). IMULATION FUNCTIONS 9
Theorem 2.4.
Let ( X, d ) be a metric space, ζ ∈ Z be a simulation function and C x ,r be any circle on X . If there exists a self-mapping f : X → X satisfying i ) d ( x , f x ) = r for each x ∈ C x ,r , ii ) ζ ( r, d ( f x, f y )) ≥ for each x, y ∈ C x ,r with x = y , iii ) ζ ( d ( f x, f y ) , d ( x, y ) − ϕ r ( d ( x, f x ))) ≥ for each x, y ∈ C x ,r , iv ) f is one to one on the circle C x ,r ,then the circle C x ,r is a fixed circle of f .Proof. Let x ∈ C x ,r be an arbitrary point. By the condition ( i ), we have d ( x , f x ) = r , that is, f x ∈ C x ,r . Now we show that f x = x for all x ∈ C x ,r . Conversely,assume that x = f x for any x ∈ C x ,r . Then we have d ( x, f x ) > ii ), ( iv ) and ( ζ ), we get0 ≤ ζ (cid:0) r, d (cid:0) f x, f x (cid:1)(cid:1) < d (cid:0) f x, f x (cid:1) − r and so r < d (cid:0) f x, f x (cid:1) . (2.6)Using the definition of the function ϕ r and the conditions ( iii ), ( iv ) and ( ζ ), weobtain0 ≤ ζ (cid:0) d (cid:0) f x, f x (cid:1) , d ( x, f x ) − ϕ r ( d ( x, f x )) (cid:1) = ζ (cid:0) d (cid:0) f x, f x (cid:1) , d ( x, f x ) − ( d ( x, f x ) − r ) (cid:1) = ζ (cid:0) d (cid:0) f x, f x (cid:1) , r (cid:1) < r − d (cid:0) f x, f x (cid:1) and hence d (cid:0) f x, f x (cid:1) < r ,which is a contradiction with the inequality (2.6). Therefore it should be f x = x for each x ∈ C x ,r . Consequently, C x ,r is a fixed circle of f . (cid:3) Remark 2.2.
If we consider the self-mapping f defined in Example 2.1, it is easyto verify that f satisfies the conditions of Theorem 2.1 and Theorem 2.3 for theellipse E ( − ,
3) = {− , − , , } with the simulation function ζ ( t, s ) = s − t .This shows that the fixed ellipse is not unique for the number ρ defined in ( ) .On the other hand, the fixed point set F ix ( f ) = {− , − , , , } contains also theellipses E ( − ,
1) = {− , − , } and E ( − ,
3) = {− , , } other than the ellipses E ( − , and E ( − , . We deduce that the number ρ defined in ( ) can notproduce all fixed ellipses ( resp. circles ) for a self-mapping f . This remark shows also that a fixed ellipse may not be unique. Now, we give ageneral result which ensure the uniqueness of a fixed geometric figure (for example,a circle, an Apollonius circle, an ellipse, a hyperbola, etc.) for a self-mapping of ametric space (
X, d ). Theorem 2.5. ( The uniqueness theorem ) Let ( X, d ) be a metric space, the number M ( x, y ) be defined as in ( ) and f : X → X be a self-mapping. Assume that thefixed point set F ix ( f ) contains a geometric figure F . If there exists a simulationfunction ζ ∈ Z such that the condition ζ ( d ( f x, f y ) , M ( x, y )) ≥ is satisfied by f for all x ∈ F and y ∈ X − F , then the figure F is the unique fixedfigure of f .Proof. Assume that F ∗ is another fixed figure of f . Let x ∈ F , y ∈ F ∗ with x = y be arbitrary points. Using the inequality (2.7) and the condition ( ζ ), we find0 ≤ ζ ( d ( f x, f y ) , M ( x, y )) = ζ ( d ( x, y ) , d ( x, y )) < d ( x, y ) − d ( x, y ) = 0,a contradiction. Hence, it should be x = y for all x ∈ F , y ∈ F ∗ . This shows theuniqueness of the fixed figure F of f . (cid:3) Now we give a condition which excludes the identity map I X : X → X definedby I X ( x ) = x for all x ∈ X from the above results. Theorem 2.6.
Let ( X, d ) be a metric space, f : X → X be a self-mapping and r ∈ [0 , ∞ ) be a fixed number. If there exists a simulation function ζ ∈ Z such thatthe condition d ( x, f x ) < ζ ( d ( x, f x ) , ϕ r ( d ( x, f x )) + r ) is satisfied by f for all x / ∈ F ix ( f ) if and only if f = I X .Proof. Let x ∈ X be an arbitrary point with x / ∈ F ix ( f ). Using (2.5) and thecondition ( ζ ), we find d ( x, f x ) ≤ ζ ( d ( x, f x ) , ϕ r ( d ( x, f x )) + r ) = ζ ( d ( x, f x ) , ( d ( x, f x ) − r ) + r )= ζ ( d ( x, f x ) , d ( x, f x )) < d ( x, f x ) − d ( x, f x ) = 0,a contradiction. Hence, it should be x ∈ F ix ( f ) for all x ∈ X , that is, F ix ( f ) = X .This shows that f = I X . Clearly, the identity map I X satisfies the condition of thehypothesis for any simulation function ζ ∈ Z . (cid:3) Geometric study of fixed points on S -metric and b -metric spaces. Atfirst, we recall the concept of an S -metric space. Definition 2.2. [26]
Let X be nonempty set and S : X → [0 , ∞ ) be a functionsatisfying the following conditions(1) S ( x, y, z ) = 0 if and only if x = y = z ,(2) S ( x, y, z ) ≤ S ( x, x, a ) + S ( y, y, a ) + S ( z, z, a ) , IMULATION FUNCTIONS 11 for all x, y, z, a ∈ X . Then S is called an S -metric on X and the pair ( X, S ) iscalled an S -metric space. Let (
X, d ) be a metric space. It is known that the function S d : X → [0 , ∞ )defined by S d ( x, y, z ) = d ( x, z ) + d ( y, z )for all x, y, z ∈ X is an S -metric on X [4]. The S -metric S d is called the S -metric generated by the metric d [14]. For example, let X = R and the function S : X → [0 , ∞ ) be defined by S ( x, y, z ) = | x − z | + | y − z | , (2.8)for all x, y, z ∈ R [27]. Then ( X, S ) is called the usual S -metric space. This S -metric is generated by the usual metric on R . The main motivation of thissubsection is the existence of some examples of S -metrics which are not generatedby any metric. For example, let X = R and the function S : X → [0 , ∞ ) bedefined by S ( x, y, z ) = | x − z | + | x + z − y | , (2.9)for all x , y and z ∈ R . Then, S is an S -metric on R , which is not generated byany metric, and the pair ( R , S ) is an S -metric space (see [14] for more details andexamples).Let ( X, S ) be an S -metric space and f : X → X be a self-mapping. In this sub-section, we give new solutions to the fixed-circle problem (resp. fixed-disc problemand fixed ellipse problem) for self-mappings of an S -metric space (resp. a b -metricspace). For the S -metric case, we use the following numbers µ = inf {S ( x, x, f x ) : x = f x, x ∈ X } , (2.10) M S ( x, y ) = max ( a S ( x, x, f x ) + (1 − a ) S ( y, y, f y ) , (1 − a ) S ( x, x, f x ) + a S ( y, y, f y ) , S ( x,x,fy )+ S ( y,y,fx )4 ) , ≤ a < S ( x, x, y ) = S ( y, y, x ) , (2.12)for all x, y ∈ X on an S -metric space ( X, S ). Before stating our results, we recallthe definitions of a circle, a disc and an ellipse on an S -metric space, respectively,as follows: C Sx ,r = { x ∈ X : S ( x, x, x ) = r } , D Sx ,r = { x ∈ X : S ( x, x, x ) ≤ r } and E Sr ( x , x ) = { x ∈ X : S ( x, x, x ) + S ( x, x, x ) = r } , where r ∈ [0 , ∞ ) [16], [26]. For a self-mapping f of an S -metric space, the definitionof a fixed figure (circle, disc, ellipse, etc.) can be given similar to the case introducedin the previous section (see [16] and [10] for the definitions of a fixed circle and afixed disc). Theorem 2.7.
Let ( X, S ) be an S -metric space, f : X → X be a self-mapping, ζ ∈ Z be a simulation function and the number µ be defined as in ( ) . If thereexist some points x , x ∈ X such that ( a ) For all x ∈ E Sρ ( x , x ) , there exists δ ( µ ) > satisfying µ ≤ M S ( x, x ) + M S ( x, x ) < µ δ ( µ ) = ⇒ S ( f x, f x, x ) + S ( f x, f x, x ) ≤ µ , ( b ) For all x ∈ X , S ( f x, f x, x ) > ⇒ ζ ( S ( f x, f x, x ) , M S ( x, x )) ≥ and ζ ( S ( f x, f x, x ) , M S ( x, x )) ≥ , ( c ) For all x ∈ X , S ( f x, f x, x ) > ⇒ ζ (cid:18) S ( f x, f x, x ) , S ( x, x, x ) + S ( f x, f x, x ) + S ( x, x, x ) + S ( f x, f x, x )2 (cid:19) ≥ ,then f x = x , f x = x and F ix ( f ) contains the ellipse E Sµ ( x , x ) .Proof. We have M S ( x , x ) = S ( x , x , f x ) and M S ( x , x ) = S ( x , x , f x ).First, we show that f x = x and f x = x . If f x = x and f x = x then S ( x , x , f x ) > S ( x , x , f x ) >
0. Using the symmetry condition (2.12)and the condition ( ζ ), we obtain ζ ( S ( f x , f x , x ) , M S ( x , x )) = ζ ( S ( f x , f x , x ) , S ( x , x , f x )) < S ( f x , f x , x ) − S ( x , x , f x ) = 0and ζ ( S ( f x , f x , x ) , M S ( x , x )) = ζ ( S ( f x , f x , x ) , d ( f x , x )) < d ( f x , x ) − d ( f x , x ) = 0,which are contradictions by the condition ( b ). Hence it should be f x = x and f x = x .If µ = 0, it is easy to check that E Sµ ( x , x ) = C Sx ,µ = { x } and x = x . Hence,the proof is completed.Assume that µ = 0. Let x ∈ E Sµ ( x , x ) be any point such that f x = x . Then S ( x, x, f x ) > M S ( x, x ) = max (cid:26) a S ( x, x, f x ) , (1 − a ) S ( x, x, f x ) , S ( x, x, f x ) + S ( x , x , f x )4 (cid:27) IMULATION FUNCTIONS 13 and M S ( x, x ) = max (cid:26) a S ( x, x, f x ) , (1 − a ) S ( x, x, f x ) , S ( x, x, f x ) + S ( x , x , f x )4 (cid:27) .Using the condition ( a ), we get µ ≤ M S ( x, x ) + M S ( x, x ) < µ δ ( µ ) = ⇒ S ( f x, f x, x ) + S ( f x, f x, x ) ≤ µ .(2.13)Now, using the inequality (2.13), the conditions ( c ), ( ζ ) and the symmetry condi-tion (2.12), we obtain0 ≤ ζ (cid:18) S ( f x, f x, x ) , S ( x, x, x ) + S ( f x, f x, x ) + S ( x, x, x ) + S ( f x, f x, x )2 (cid:19) < S ( x, x, x ) + S ( f x, f x, x ) + S ( x, x, x ) + S ( f x, f x, x )2 − S ( f x, f x, x )= S ( x, x, x ) + S ( x, x, x )2 + S ( f x, f x, x ) + S ( f x, f x, x )2 − S ( f x, f x, x ) ≤ µ µ − S ( f x, f x, x ) = µ − S ( f x, f x, x )and so S ( f x, f x, x ) < µ ,which is a contradiction by the definition of the number µ . This contradiction leadsto f x = x . Consequently, we have E Sµ ( x , x ) ⊂ F ix ( f ). (cid:3) If x = x then we have E Sµ ( x , x ) = C x , µ and Theorem 2.7 is reduced to afixed-circle theorem as follows: Theorem 2.8.
Let ( X, S ) be an S -metric space, f : X → X be a self-mapping andthe number µ be defined as in ( ) . If there exist a simulation function ζ ∈ Z and a point x ∈ X such that ( i ) For all x ∈ C Sx ,µ , there exists a δ ( µ ) > satisfying µ ≤ M S ( x, x ) < µ δ ( µ ) = ⇒ S ( f x, f x, x ) ≤ µ, ( ii ) For all x ∈ X , S ( f x, f x, x ) > ⇒ ζ ( S ( f x, f x, x ) , M S ( x, x )) ≥ , then x ∈ F ix ( f ) and the circle C Sx ,µ is a fixed circle of f .Proof. If f x = x then using the symmetry property (2.12) and the condition ( ζ ),we get M S ( x , x ) = S ( x , x , f x ) = S ( f x , f x , x ) and ζ ( S ( f x , f x , x ) , M S ( x , x )) = ζ ( S ( f x , f x , x ) , S ( f x , f x , x )) < S ( f x , f x , x ) − S ( f x , f x , x ) = 0.This last inequality is a contradiction by ( ii ). Therefore, it should be f x = x .This shows that the circle C Sx ,µ = { x } is a fixed circle of f when µ = 0. Now,let µ > x ∈ C Sx ,µ be any element. To show that f fixes the circle C Sx ,µ , wesuppose that f x = x for any x ∈ C Sx ,µ . Since x ∈ F ix ( f ), we have M S ( x, x ) = max (cid:26) a S ( x, x, f x ) , (1 − a ) S ( x, x, f x ) , S ( x, x, f x ) + S ( x , x , f x )4 (cid:27) = max (cid:26) a S ( x, x, f x ) , (1 − a ) S ( x, x, f x ) , µ + S ( x , x , f x )4 (cid:27) . Using the conditions ( i ), ( ii ), ( ζ ) and the symmetry property (2.12), we find0 < ζ ( S ( f x, f x, x ) , M S ( x, x )) < M S ( x, x ) − S ( f x, f x, x ) < µ − S ( f x, f x, x ) ,a contradiction by the definition of the number µ . Hence, we have f x = x . Conse-quently, f fixes the circle C Sx ,µ . (cid:3) Corollary 2.1.
Let ( X, S ) be an S -metric space, f : X → X be a self-mappingand the number µ be defined as in ( ) . If there exist a simulation function ζ ∈ Z and a point x ∈ X such that ( i ) For all x ∈ D Sx ,µ , there exists a δ ( µ ) > satisfying µ ≤ M S ( x, x ) < r δ ( µ ) = ⇒ S ( f x, f x, x ) ≤ µ, ( ii ) For all x ∈ X , S ( f x, f x, x ) > ⇒ ζ ( S ( f x, f x, x ) , M S ( x, x )) ≥ , then the disc D Sx ,µ is a fixed disc of f . Example 2.3.
Let X = {− , − , , , , } with the S -metric defined in ( ) andconsider the self-mapping f defined in Example 2.1 on this S -metric space ( X, S ) .It is easy to check that the self-mapping f satisfies the conditions of Theorem 2.7with the points x = − and x = 1 , the simulation function ζ ( t, s ) = s − t andany a ∈ [0 , . We have µ = inf {S ( x, x, f x ) : x = f x, x ∈ X } = S (12 , ,
18) = 12 and clearly the set
F ix ( f ) contains the ellipse E S ( − ,
1) = {− , } . On the otherhand, the self-mapping f does not satisfy the condition ( b ) of Theorem 2.7 for the IMULATION FUNCTIONS 15 ellipse E S ( − ,
3) = {− , − , , } for any simulation function ζ and any a ∈ [0 , .Indeed, we have ζ ( S ( f x, f x, x ) , M S ( x, ζ ( S (18 , , , M S (12 , ζ (12 , < ,by the condition ( ζ ) for the point x = 12 with S (18 , , > . This shows thatthe converse statement of Theorem 2.7 is not true everwhen.The self-mapping f satisfies the conditions of Theorem 2.8 with the point x = − and the simulation function ζ ( t, s ) = s − t . The circle C S − , = { } is containedin the set F ix ( f ) . On the other hand, the self-mapping f does not satisfy thecondition ( ii ) of Theorem 2.8 with the point x = 3 for any simulation function ζ and any a ∈ [0 , . Since we have ζ ( S ( f x, f x, x ) , M S ( x, x )) = ζ ( S (18 , , , M S (12 , ζ (12 , < ,by the condition ( ζ ) for the point x = 12 . But, the circle C S , = {− } is a fixedcircle of f . This shows that the converse statement of Theorem 2.8 is not trueeverwhen. Remark 2.3. Example 2.3 shows the importance of the studies on an S -metricspace. Notice that the ellipse E ( − , and the circle C − , are empty sets in themetric space ( X, d ) in Example 2.1 but, if we consider the S -metric S ( x, y, z ) = | x − z | + | x + z − y | on X then the ellipse E S ( − , and the circle C S − , are notempty sets and both of them are contained in the set F ix ( f ) . Note that S -metric versions of Definition 2.1 and Theorem 2.3 can also begiven for self-mappings on an S -metric space. Now we give a fixed-circle theorem using the auxiliary function ϕ r : R + ∪{ } → R defined in (2.5). Theorem 2.9. ( X, S ) be an S -metric space, ζ ∈ Z be a simulation function and C Sx ,r be any circle on X with r > . If there exists a self-mapping f : X → X satisfying i ) S ( f x, f x, x ) = r for each x ∈ C Sx ,r , ii ) ζ ( r, S ( f x, f x, f y )) ≥ for each x, y ∈ C Sx ,r with x = y , iii ) ζ ( S ( f x, f x, f y ) , S ( x, x, y ) − ϕ r ( S ( x, x, f x ))) ≥ for each x, y ∈ C Sx ,r , iv ) f is one to one on the circle C Sx ,r ,then the circle C Sx ,r is a fixed circle of f .Proof. Let x ∈ C Sx ,r be an arbitrary point. By the condition ( i ), we have f x ∈ C Sx ,r .To show that f x = x for all x ∈ C Sx ,r , conversely, we assume that x = f x for any x ∈ C Sx ,r . Then we have S ( x, x, f x ) >
0. Using the conditions ( ii ), ( iv ) and ( ζ ),we obtain 0 ≤ ζ (cid:0) r, S (cid:0) f x, f x, f x (cid:1)(cid:1) < S (cid:0) f x, f x, f x (cid:1) − r and hence r < S (cid:0) f x, f x, f x (cid:1) . (2.14)Using the definition of the function ϕ r and the conditions ( iii ), ( iv ) and ( ζ ), weget 0 ≤ ζ (cid:0) S (cid:0) f x, f x, f x (cid:1) , S ( x, x, f x ) − ϕ r ( S ( x, x, f x )) (cid:1) = ζ (cid:0) S (cid:0) f x, f x, f x (cid:1) , S ( x, x, f x ) − ( S ( x, x, f x ) − r ) (cid:1) = ζ (cid:0) S (cid:0) f x, f x, f x (cid:1) , r (cid:1) < r − S (cid:0) f x, f x, f x (cid:1) and hence S (cid:0) f x, f x, f x (cid:1) < r ,which is a contradiction with the inequality (2.14). Consequently, it should be f x = x for each x ∈ C Sx ,r , that is, C Sx ,r is a fixed circle of f . (cid:3) Now we give a general result which ensure the uniqueness of a geometric figurecontained in the set
F ix ( f ) for a self-mapping of an S -metric space ( X, S ). Theorem 2.10. ( The uniqueness theorem ) Let ( X, S ) be an S -metric space, thenumber M S ( x, y ) be defined as in ( ) and f : X → X be a self-mapping. Assumethat the fixed point set F ix ( f ) contains a geometric figure F . If there exists asimulation function ζ ∈ Z such that the condition ζ ( S ( f x, f x, f y ) , M S ( x, y )) ≥ is satisfied by f for all x ∈ F and y ∈ X − F , then the figure F is the unique fixedfigure of f .Proof. On the contrary, we suppose that F ∗ is another fixed figure of the self-mapping f . Let x ∈ F , y ∈ F ∗ with x = y be arbitrary points. Using theinequality (2.15) and the condition ( ζ ), we get0 ≤ ζ ( S ( f x, f x, f y ) , M S ( x, y )) = ζ ( S ( x, x, y ) , S ( x, x, y )) < S ( x, x, y ) −S ( x, x, y ) = 0,a contradiction. Hence, it should be x = y for all x ∈ F , y ∈ F ∗ . This shows theuniqueness of the fixed figure F of f . (cid:3) We give a condition which excludes the identity map I X : X → X defined by I X ( x ) = x for all x ∈ X from the above results. IMULATION FUNCTIONS 17
Theorem 2.11.
Let ( X, S ) be an S -metric space, f : X → X be a self-mappingand r ∈ [0 , ∞ ) be a fixed number. If there exists a simulation function ζ ∈ Z suchthat the condition S ( x, x, f x ) < ζ ( S ( x, x, f x ) , ϕ r ( S ( x, x, f x )) + r ) is satisfied by f for all x / ∈ F ix ( f ) if and only if f = I X .Proof. The proof is similar to the proof of Theorem 2.6. (cid:3)
Remark 2.4. Let ( X, S ) be an S -metric space. Suppose that the S -metric S isgenerated by a metric d . Then, for ≤ a < we have M S ( x, x ) = max ( a S ( x, x, f x ) + (1 − a ) S ( x , x , f x ) , (1 − a ) S ( x, x, f x ) + a S ( x , x , f x ) , S ( x,x,fx )+ S ( x ,x ,fx )4 ) = max ( ad ( x, f x ) + 2(1 − a ) d ( x , f x ) , − a ) d ( x, f x ) + 2 ad ( x , f x ) , d ( x,fx )+ d ( x ,fx )2 ) and so M ( x, x ) ≤ M S ( x, x ) .Consequently, Theorem 2.8 ( resp. Corollary 2.1 ) is a generalization of Theorem . resp. Corollary . given in [21] . Similar definition of the notion of a fixed figure ( circle, disc, ellipse and soon ) can be given for a self-mapping of a b -metric space. From [27] , we know that the function defined by d S ( x, y ) = S ( x, x, y ) = 2 d ( x, y ) ,for all x, y ∈ X , defines a b -metric on an S -metric space ( X, S ) with b = . Ifwe consider Theorem 2.8 and Theorem 2.9 together with this fact, then we get thefollowing fixed-circle ( resp. fixed-disc ) results on a b -metric space using the number M d S ( x, y ) = max ( ad S ( x, f x ) + (1 − a ) d S ( y, f y ) , (1 − a ) d S ( x, f x ) + a (1 − a ) d S ( y, f y ) , d S ( x,fy )+ d S ( y,fx )2 ) , ≤ a < . (2.16) Theorem 2.12.
Let ( X, d S ) be a b -metric space, f : X → X a self-mapping and µ be defined as µ = inf (cid:8) d S ( x, f x ) : x = f x, x ∈ X (cid:9) . (2.17) If there exist a simulation function ζ ∈ Z and a point x ∈ X such that ( i ) For all x ∈ C d S x ,µ , there exists a δ ( µ ) > satisfying µ ≤ M d S ( x, x ) < µ δ ( µ ) = ⇒ d S ( f x, x ) ≤ µ , where C d S x ,µ = (cid:8) x ∈ X : d S ( x, x ) = µ (cid:9) and M d S ( x, x ) = max ( ad S ( x, f x ) + (1 − a ) d S ( x , f x ) , (1 − a ) d S ( x, f x ) + ad S ( x , f x ) , d S ( x,fx )+ d S ( x ,fx )4 ) , ≤ a < , ( ii ) For all x ∈ X,d S ( f x, x ) > ⇒ ζ (cid:0) d S ( f x, x ) , M d S ( x, x ) (cid:1) ≥ , then x ∈ F ix ( f ) and the circle C d S x ,µ is a fixed circle of f . Corollary 2.2.
Let ( X, d S ) be a b -metric space, f : X → X be a self-mapping andthe number µ be defined as in ( ) . If there exist a simulation function ζ ∈ Z and a point x ∈ X such that ( i ) For all x ∈ D d S x ,µ , there exists a δ ( µ ) > satisfying µ ≤ M d S ( x, x ) < µ δ ( µ ) = ⇒ d S ( f x, x ) ≤ µ ,where D d S x ,µ = (cid:8) x ∈ X : d S ( x, x ) ≤ µ (cid:9) , ( ii ) For all x ∈ X , d S ( f x, x ) > ⇒ ζ (cid:0) d S ( f x, x ) , M d S ( x, x ) (cid:1) ≥ , then the disc D d S x ,µ is a fixed disc of f . Theorem 2.13.
Let ( X, d S ) be a b -metric space, ζ ∈ Z be a simulation functionand C d S x ,r be any circle on X with r > . If there exists a self-mapping f : X → X satisfying i ) d S ( f x, x ) = r for each x ∈ C d S x ,r , ii ) ζ (cid:0) r, d S ( f x, f y (cid:1) ) ≥ for each x, y ∈ C d S x ,r with x = y , iii ) ζ (cid:0) d S ( f x, f y ) , d S ( x, y ) − ϕ r (cid:0) d S ( x, f x (cid:1)(cid:1) ) ≥ for each x, y ∈ C d S x ,r , iv ) f is one to one on the circle C d S x ,r ,then the circle C d S x ,r is a fixed circle of f . Theorem 2.14.
Let ( X, d S ) be a b -metric space, the number M d S ( x, y ) be definedas in ( ) and f : X → X be a self-mapping. Assume that the fixed point set F ix ( f ) contains a geometric figure F . If there exists a simulation function ζ ∈ Z such that the condition ζ (cid:0) d S ( f x, f y ) , M d S ( x, y ) (cid:1) ≥ is satisfied by f for all x ∈ F and y ∈ X − F , then the figure F is the unique fixedfigure of f . IMULATION FUNCTIONS 19
Theorem 2.15.
Let ( X, d S ) be a b -metric space, f : X → X be a self-mappingwith the fixed point set F ix ( f ) and r ∈ [0 , ∞ ) be a fixed number. If there exists asimulation function ζ ∈ Z such that the condition d S ( x, f x ) < ζ (cid:0) d S ( x, f x ) , ϕ r (cid:0) d S ( x, f x ) (cid:1) + r (cid:1) is satisfied by f for all x / ∈ F ix ( f ) if and only if f = I X . Conclusions and Future Works
In this paper, we have obtained new results on the study of geometric propertiesof the fixed point set of a self-mapping on a metric (resp. S -metric, b -metric)space via the properties of the set of simulation functions. As a future work,the determination of new conditions which ensure a geometric figure to be fixedby a self-mapping can be considered using similar approaches. Further possibleapplications of our theoretic results can be done on the applied sciences using thegeometric properties of fixed points. For example, in [9], the existence of a fixedpoint for every recurrent neural network was shown using Brouwer’s Fixed PointTheorem and a geometric approach was used to locate where the fixed points are(see [9] for more details). Therefore, theoretic fixed figure results are important inthe study of neural networks. References [1] H. Aydi, N. Ta¸s, N. Y. ¨Ozg¨ur and N. Mlaiki, Fixed-discs in rectangular metric spaces,Symmetry 11 (2019), no. 2, 294.[2] A. Chanda, D. Dey, K. Lakshmi and S. Radenovi´c, Simulation functions: a survey of recentresults, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3,2923-2957.[3] G. Z. Er¸cınar, Some geometric properties of fixed points, Ph.D. Thesis, Eski¸sehir OsmangaziUniversity, 2020.[4] N. T. Hieu, N. T. Thanh Ly and N. V. Dung, A generalization of ´Ciri´c quasi-contractionsfor maps on S -metric spaces. Thai J. Math. 13 (2015), no. 2, 369-380.[5] M. Joshi, A. Tomar and S. K. Padaliya, Fixed Point to Fixed Ellipse in Metric Spaces andDiscontinuous Activation Function, To appear in Applied Mathematics E-Notes.[6] E. Karapınar and F. Khojasteh, An approach to best proximity points results via simulationfunctions, J. Fixed Point Theory Appl. 19 (2017), no. 3, 1983-1995.[7] F. Khojasteh, S. Shukla and S. Radenovi´c, A new approach to the study of fixed point theoryfor simulation functions, Filomat 29 (6) (2015), 1189-1194.[8] A. Kosti´c, V. Rakoˇcevi´c and S. Radenovi´c, Best proximity points involving simulation func-tions with w -distance, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 113(2019), no. 2, 715-727. [9] L. K. Li, Fixed point analysis for discrete-time recurrent neural networks, In: Proceedings1992 IJCNN International Joint Conference on Neural Networks, Baltimore, MD, USA, 1992,pp. 134-139 vol.4, doi: 10.1109/IJCNN.1992.227277.[10] N. Mlaiki, U. C¸ elik, N. Ta¸s, N. ¨Ozg¨ur and A. Mukheimer, Wardowski type contractions andthe fixed-circle problem on S -metric spaces, J. Math. 2018, Art. ID 9127486, 9 pp.[11] N. Mlaiki, N. Y. ¨Ozg¨ur and N. Ta¸s, New fixed-point theorems on an S -metric space viasimulation functions, Mathematics, 7 (2019), no. 7, 583.[12] N. Mlaiki, N. ¨Ozg¨ur and N. Ta¸s, New fixed-circle results related to F c -contractive and F c -expanding mappings on metric spaces, arXiv:2101.10770.[13] M. Olgun, ¨O. Bi¸cer and T. Alyıldız, A new aspect to Picard operators with simulationfunctions, Turkish J. Math. 40 (2016), no. 4, 832-837.[14] N. Y. ¨Ozg¨ur, N. Ta¸s, Some new contractive mappings on S -metric spaces and their relation-ships with the mapping ( S S -metric spaces with a geometric viewpoint,Facta Universitatis. Series: Mathematics and Informatics 34 (2019), no. 3, 459-472.[17] N. Y. ¨Ozg¨ur and N. Ta¸s, Some fixed-circle theorems on metric spaces, Bull. Malays. Math.Sci. Soc. 42 (2019), no. 4, 1433-1449.[18] N. ¨Ozg¨ur, Fixed-disc results via simulation functions, Turkish J. Math. 43 (2019), no. 6,2794-2805.[19] R. P. Pant, N. Y. ¨Ozg¨ur and N. Ta¸s, On discontinuity problem at fixed point, Bull. Malays.Math. Sci. Soc. 43 (2020), no. 1, 499-517.[20] R. P. Pant, N. Y. ¨Ozg¨ur and N. Ta¸s, Discontinuity at fixed points with applications, Bull.Belg. Math. Soc. Simon Stevin 26 (2019), no. 4, 571-589.[21] R. P. Pant, N. ¨Ozg¨ur, N. Ta¸s, A. Pant and M. Joshi, New results on discontinuity at fixedpoint, J. Fixed Point Theory Appl. 22 (2020), no. 2, 39.[22] A. F. Rold´an-L´opez-de-Hierro, E. Karapınar, C. Rold´an-L´opez-de-Hierro and J. Mart´ınez-Moreno, Coincidence point theorems on metric spaces via simulation functions, J. Comput.Appl. Math. 275 (2015), 345-355.[23] A. F. Rold´an L´opez de Hierro and N. Shahzad, New fixed point theorem under R -contractions, Fixed Point Theory Appl. 2015, 2015:98, 18 pp.[24] N. Ta¸s, N. Y. ¨Ozg¨ur and N. Mlaiki, New types of F c -contractions and the fixed-circle problem,Mathematics 6 (2018), 188.[25] J. L. Ramirez, Inversions in an ellipse, Forum Geom. 14 (2014), 107-115.[26] S. Sedghi, N. Shobe, A. Aliouche, A generalization of fixed point theorems in S -metric spaces,Mat. Vesnik 64 (2012), no. 3, 258-266.[27] S. Sedghi, N. V. Dung, Fixed point theorems on S -metric spaces, Mat. Vesnik, 66 (2014),no. 1, 113-124.[28] N. Ta¸s, Bilateral-type solutions to the fixed-circle problem with rectified linear units appli-cation, Turkish J. Math. 44 (2020), no. 4, 1330-1344. IMULATION FUNCTIONS 21
Balıkesir University, Department of Mathematics, 10145 Balıkesir, Turkey
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