Fixed points for three point generalized orbital triangular contractions
Cristina Maria Pacurar Ovidiu Popescu

	
	AnoFPDM: Anomaly Segmentation with Forward Process of Diffusion Models for Brain MRI
    Yiming Che 1,2 Fazle Rafsani 1,2* Jay Shah 1,2* Md Mahfuzur Rahman Siddiquee 1,2 Teresa Wu 1,2
    April 29, 2024
==================================================================================================


	In this paper we introduce and study new classes of mappings in metric spaces. The main class of mappings is called generalized orbital triangular contractions and it generalizes some existing results (such as Banach contractions, mappings contracting perimeters of triangles). We prove that these contractions are not necessarily continuous and have a unique fixed point under certain conditions. Moreover, we extend our class to generalized orbital triangular Kannan contractions and generalized orbital triangular Chatterjea contractions.
	
	
	
	
	47H10, 54H25
	
	
	

§ INTRODUCTION




The research on fixed point theorems has been a central theme in mathematical analysis from the moment Banach introduced the fixed point result known as the contraction principle (see <cit.>). The continuous interest towards the subject is due to the numerous applications in various fields such as differential equations, dynamical systems, or the theory of computation. 

In light of the joint effort to obtain more general classes of mappings which have fixed points, very recently, Petrov introduced in <cit.> a new type of mappings in complete metric spaces, which can be characterized as mappings contracting perimeters of triangles. He showed that mappings contracting perimeters of triangles are continuous and proved the fixed point theorem for such mappings. His results were generalized in <cit.> for mappings contracting (a feature) of triangles, not necessarily perimeters. Also, related results to this new concept were obtained in <cit.>.

In this paper, we extend the classical frameworks of Banach contractions by introducing a new concept of generalized orbital triangular contractions in metric spaces. As opposed to traditional approaches that often require continuity conditions, our novel class of mappings are not necessarily continuous. For the class of generalized orbital triangular contractions, we establish the existence of unique fixed points. Moreover, we extend our results to Kannan orbital triangular contractions, which are a generalization of Kannan contractions, and to Chatterjea orbital triangular contractions, which are a generalization of Kannan contractions.



§ PRELIMINARIES




Let us recall some definitions of different types of contractions:


	Let (X,d) be a metric space. A mapping T:X → X is called:
	
		
  i) Banach contraction (see <cit.>) if there exists C ∈ [0,1) such that 
		
    d(Tx,Ty) ≤ C d(x,y),

		for all x,  y ∈ X;
		
  ii) Kannan contraction (see <cit.>) if there exists C ∈[0,1/2) such that 
		
    d(Tx,Ty) ≤ C [d(x,Tx) + d(y,Ty)],

		for all x,  y ∈ X;
		
  iii) Chatterjea contraction (see <cit.>) if there exists C ∈[0,1/2) such that 
		
    d(Tx,Ty) ≤ C [d(x,Ty)+d(y,Tx)],

		for all x,  y ∈ X;
		
  iv) mapping contracting perimeter of triangles on X (see <cit.>) if there exists C ∈ [0,1) such that the inequality 
		
    d(Tx,Ty) + d(Ty,Tz) + d(Tz,Tx) ≤ C [d(x,y)+d(y,z)+d(z,x)],

		holds for all three pairwise distinct points x,y,x ∈ X;
		
	



	Let (X,d) be a metric space and T:X→ X. A point x ∈ X is called periodic point of period n if T^nx =x. The least positive integer n for which T^n x=x is called the prime period of x.





§ GENERALIZED ORBITAL TRIANGULAR CONTRACTIONS




In this section we give the definition of generalized orbital triangular contraction, prove that they are not necessarily continuous and give a fixed point result for such mappings.



	Let (X,d) be a metric space. We shall say that T:X→ X is a generalized orbital triangular contraction on X if there exists α∈ [0,1) such that the inequality 
	
    d(Tx,T^2x)+d(T^2x,Ty)+d(Ty,Tx) ≤α [d(x,Tx)+d(Tx,y)+d(y,x)],

	holds for all x,y ∈ X, such that x≠ y ≠ Tx.
	









	Every Banach contraction is a generalized orbital triangular contraction.
	
	Indeed, if T is a Banach contraction, then for x≠ y≠ Tx, by (<ref>) we have:
	
    d(Tx,T^2x) ≤ C · d(x,Tx),

	
    d(T^2x,Ty) ≤ C · d(Tx,y).

	
    d(Ty,Tx) ≤ C · d(y,x),

	where C ∈ [0,1).
	
	Adding inequalities (<ref>), (<ref>) and (<ref>) we obtain (<ref>), so the conclusion follows.



	If we impose the additional condition that x≠ Tx in Definition <ref>, then we obtain that every mapping contracting perimeters of triangles is a generalized orbital triangular contraction.
	
	Indeed, if T is a mapping contracting perimeters of triangles, then for x,Tx,y pairwise distinct, via (<ref>), we obtain (<ref>).
	
	



	Let (X,d) be a complete metric space and let T:X→ X be a generalized orbital triangular contraction on X such that T has no periodic points of prime period 2. Then, T has a unique fixed point.
	



	Let x_0 ∈ X, arbitrarily chosen, and 
    x_i+1=Tx_i
 for every i ≥ 0, the Picard iteration. 
	
	Now, suppose that x_i is not a fixed point of T, then x_i≠ x_i+1 = Tx_i. Moreover, since T does not have periodic points of prime period 2, we have x_i+2 = TTx_i≠ x_i for every i=0,1,… 
	
	Then we have x_i-1≠ x_i+1≠ x_i = Tx_i-1, and taking x=x_i-1 and y=x_i+1 in (<ref>), we obtain
	
    p_i = d(x_i,x_i+1)+d(x_i+1,x_i+2)+d(x_i+2,x_i) =
    
    	=d(Tx_i-1,T^2x_i-1)+d(T^2x_i-1,Tx_i+1)+d(Tx_i+1,Tx_i-1) ≤
    ≤α(d(x_i-1,Tx_i-1)+d(Tx_i-1,x_i+1)+d(x_i+1,x_i-1)) =
    
    	=α(d(x_i-1,x_i)+d(x_i,x_i+1)+d(x_i+1,x_i-1)) = α p_i-1,

	for every i≥ 1. So, we obtain 
	
    p_i ≤α p_i-1≤α^2 p_i-2≤…α^ip_0.

	
	Thus, for m ∈ℕ, by the triangle inequality we have 
	
    d(x_n,x_n+m)    ≤ d(x_n,x_n+1) + d(x_n+1,x_n+2)+ … + d(x_n+m-1,x_n+m) ≤
       ≤ p_n+m-2 + … +p_n-1≤
       ≤ p_0(α^n-1+α^n+…+α^n+m-2)
        =α^n-1·1-α^m1-αp_0
       ≤α^n-11-αp_0,

	so we obtain that {x_n} is Cauchy sequence and given the completeness of X, we have that {x_n} is convergent, so there exists x^* ∈ X such that x_n → x^* as n →∞. Moreover, there exists a subsequence {x_n(k)} such that x_n(k)≠ x^* ≠ Tx_n(k), so, by (<ref>) we obtain 
	
    d(Tx_n(k),T^2x_n(k))    + d(T^2x_n(k),Tx^*) + d(Tx^*,Tx_n(k)) ≤
       ≤α [d(x_n(k),Tx_n(k))+d(Tx_n(k),x^*)+d(x^*,x_n(k))].

	Hence, we get
	
    d(x_n(k)+1,x_n(k)+2)    + d(x_n(k)+1,Tx^*) + d(Tx^*,x_n(k)+2) ≤
       ≤α [d(x_n(k),x_n(k+1))+d(x_n(k+1),x^*)+d(x^*,x_n(k))],

	where taking the limit as k →∞ we get 
	
    2d(x^*,Tx^*) ≤ 0,

	by where d(x^*,Tx^*)=0, so x^* is a fixed point of T.
	
	Now, suppose that there exists another fixed point of T, y^*∈ X such that Ty^* = y^* and x^* ≠ y^*. Then Tx^* ≠ y^*, so, by (<ref>) we obtain 
	
    d(Tx^*,T^2x^*)+d(T^2x^*,Ty^*)+d(Ty^*,Tx^*)  ≤α [d(x^*,Tx^*)+d(Tx^*,y^*)+d(y^*,x^*)],

	so we get 
    2d(x^*,y^*) ≤ 2α d(x^*,y^*),
 which is a contradiction since α∈ [0,1). Thus, we obtain x^* =y^*, so T has a unique fixed point.		



	If we impose the additional condition that x≠ Tx in Definition <ref>, then T can have many fixed points.



	Generalized orbital triangular contractions on X are not necessarily continuous.
	
	Indeed, let X =[0,1] and T: X → X defined as 
    Tx=
    	0,    x∈ [0,1)
    14,    x =1.

	
	Obviously, T is not continuous at x=1, but T is a generalized orbital triangular contraction for α = 23 <1.
	
	Indeed, if x,y ∈ [0,1), then Tx=T^2x=Ty=0, so
	
    d(Tx,T^2x)+d(T^2x,Ty)+d(Ty,Tx) = 0 ≤23 (d(x,Tx)+d(Tx,y)+d(y,x)).

	
	If x=1, then Tx=14 and for y≠ x, we have T^2x=Ty=0, so 
	
    d(Tx,T^2x)+d(T^2x,Ty)+d(Ty,Tx) =  14 + 0+  14 =  12

	and 
    d(x,Tx)+d(Tx,y)+d(y,x) = 1-14+|14-y|+1-y =

	
    = 74-y+|14-y| ≥74-y ≥74-1 = 34,

	so for α =  23, since
	12 ≤23 · 34, (<ref>) is true.
	
	If y=1, then Ty=14 and since x ≠ y, Tx=T^2x=0 so we have 
	
    d(Tx,T^2x)+d(T^2x,Ty)+d(Ty,Tx) = 0+ 14 +  14 =  12

	and 
    d(x,Tx)+d(Tx,y)+d(y,x) = x+1+1-x =2,

	and since
	12 ≤23 · 2, we obtain that (<ref>) is true.
	
	So, T is generalized orbital triangular contraction which has a unique fixed point 0=T0.



	There exist generalized orbital triangular contractions which are not Banach contractions <cit.>, nor mappings contracting perimeters of triangles <cit.>, neither mappings contracting triangles <cit.>.
	
	Indeed, since generalized orbital triangular contractions are not necessarily continuous, the conclusion follows.





§ GENERALIZED ORBITAL TRIANGULAR KANNAN CONTRACTIONS




In this section, we give the definition of generalized orbital triangular Kannan contractions and prove a fixed point theorem for such mappings.


	Let (X,d) be a metric space. We shall say that T:X→ X is a generalized orbital triangular Kannan contraction on X if there exists β∈ [0,2/3) such that the inequality 
	
    d(Tx,T^2x)+d(T^2x,Ty)+d(Ty,Tx) ≤β [d(x,Tx)+d(y,Ty)+d(Tx,T^2x)],

	holds for all x,y ∈ X, such that x, y, Tx are pairwise distinct.



	Every Kannan contraction with C <13 is a generalized orbital triangular Kannan contraction.
	
	Indeed, if T is a Kannan contraction, then for x,y,Tx pairwise distinct, by (<ref>) we have:
	
    d(Tx,T^2x) ≤ C [d(x,Tx)+d(Tx,T^2x)],

	
    d(T^2x,Ty) ≤ C [d(Tx,T^2x)+d(y,Ty)].

	
    d(Ty,Tx) ≤ C [d(y,Ty)+d(x,Tx)].

	
	Adding inequalities (<ref>), (<ref>) and (<ref>) we obtain (<ref>), so the conclusion follows.



	Let (X,d) be a complete metric space and let T:X→ X be a generalized orbital triangular Kannan contraction on X such that T has no periodic points of prime period 2. Then, T has a fixed point.
	



	We suppose that T has no periodic points of prime period 2. And for x_0 ∈ X arbitrarily chosen, let the Picard iteration x_n=Tx_n-1 for all n ≥ 0. Suppose that x_n is not a fixed point of T for every n ≥ 0. Then, x_n,Tx_n=x_n+1 and T^2x_n=x_n+2 are pairwise distinct, and choosing x=x_n, y=T^2x_n in (<ref>) we have 
	
    d(Tx_n,T^2x_n)+d(T^2x_n,T^3x_n)+d(T^3x_n,Tx_n) ≤
    ≤β [d(x_n,Tx_n)+d(T^2x_n,T^3x_n)+d(Tx_n,T^2x_n)],

	thus, using the triangle inequality, we get 
	
    (1-β)[d(Tx_n,T^2x_n)+d(T^2x_n,T^3x_n)]+|d(Tx_n,T^2x_n)-d(T^2x_n,T^3x_n)| ≤
    ≤
    	(1-β)[d(Tx_n,T^2x_n)+d(T^2x_n,T^3x_n)]+d(T^3x_n,Tx_n) ≤
    ≤β [d(x_n,Tx_n)].

	
	Now, let d_n = d(x_n,x_n+1). By (<ref>), we obtain
	
    (1-β)(d_n+1+d_n+2)+|d_n+1-d_n+2| ≤β d_n.

	
	If d_n+1≥ d_n+2, (<ref>) becomes
	
    (2-β)d_n+1 - β d_n+2≤β d_n,
 
	by where
	
    (2-2β)d_n+1≤β d_n,

	so we obtain 
	
    d_n+1≤β2-2βd_n.

	
	If d_n+1 < d_n+2, then, (<ref>) becomes
	
    (2-β)d_n+2≤β (d_n+d_n+1),

	so we have
	
    (2-β)d_n+1≤(2-β)d_n+2≤β (d_n+d_n+1),

	and thus 
	
    d_n+1≤β2-2βd_n.

	
	By (<ref>) and (<ref>), since β < 23, λ=β2-2β <1 and we have 
	
    d_n+1≤λ d_n.

	
	Thus, for m ∈ℕ, by the triangle inequality and (<ref>), we have 
	
    d(x_n,x_n+m)    ≤ d(x_n,x_n+1) + d(x_n+1,x_n+2)+ … + d(x_n+m-1,x_n+m) = 
       = d_n + … +d_n+m-1≤
       ≤λ^n-1d_0+λ^nd_0+…+λ^n+m-1d_0
        =λ^n-1·1-λ^m1-λd_0
       ≤λ^n-11-λd_0,

	so, since λ <1, we obtain that {x_n} is Cauchy sequence and given the completeness of X, we have that {x_n} is convergent, so there exists x^* ∈ X such that x_n → x^* as n →∞.
	
	Moreover, there exists a subsequence {x_n(k)} such that x_n(k)≠ x^* ≠ Tx_n(k), so by (<ref>) we have 
	
    d(Tx_n(k),T^2x_n(k)) + d(T^2x_n(k),Tx^*) + d(Tx^*,Tx_n(k)) ≤
    ≤β [d(x_n(k),Tx_n(k))+d(x^*,Tx^*)+d(Tx_n(k),T^2x_n(k))],

	so
	
    d(x_n(k)+1,x_n(k)+2)    + d(x_n(k)+1,Tx^*) + d(x_n(k)+2,Tx^*) ≤
       ≤β [d(x_n(k),x_n(k)+1)+d(x^*,Tx^*)+d(x_n(k)+1,x_n(k)+2)],

	and taking the limit as k →∞ we get 
	
    2d(x^*,Tx^*) ≤β· d(x^*,T^*),

	by where d(x^*,Tx^*)=0, so x^* is a fixed point of T.




	Let X={A,B,C,D} and as in Figure <ref>:
	
    d(A,B)=d(A,C)=d(A,D)=4,    d(B,C)=d(C,D)=1,    d(B,D)=2,
 and
	T:X→ X defined as
	
    TA=TC=C,    TB=B,    TD=D.

	
	
	
	Since d(TB,TD) = 2 and d(B,TB)+d(D,TD) =0, T is not a Kannan contraction.
	
	Moreover, since 
	
    d(TA,TB)+d(TB,TD)+d(TD,TA)= 4 = d(A,TA)+d(B,TB)+d(D,TD),

	T is not a generalized Kannan mapping (see <cit.>).
	
	However, we have 
	
    d(TA,T^2A)+d(T^2A,TB)+d(TB,TA)= 2 ≤12 · 4 = 12[d(A,TA)+d(B,TB)+d(TA,T^2A)],

	
    d(TA,T^2A)+d(T^2A,TD)+d(TD,TA)= 2 ≤12 · 4 = 12[d(A,TA)+d(D,TD)+d(TA,T^2A)],

	so T is a generalized orbital triangular Kannan contraction.
	
	Let us note that T does not have periodic points of prime period 2, and T is not a Banach contraction either as d(TB,TD) = 2 = d(B,D).




§ GENERALIZED ORBITAL TRIANGULAR CHATTERJEA CONTRACTIONS




In this section, we give the definition of generalized orbital triangular Chatterjea contractions and prove a fixed point theorem for such mappings.


	Let (X,d) be a metric space. We shall say that T:X→ X is a generalized orbital triangular Chatterjea contraction on X if there exists γ∈[0,1/2) such that the inequality 
	
    d(Tx,T^2x)+d(T^2x,Ty)+d(Ty,Tx) ≤
    ≤γ [d(x,Ty)+d(y,Tx)+d(x,T^2x)+d(y,T^2x)+d(Tx,Ty)],

	holds for all x,y ∈ X, such that x≠ y ≠ Tx.



	Every Chatterjea contraction is a generalized orbital triangular Chatterjea contraction.
	
	Indeed, if T is a Chatterjea contraction, then there exists C ∈ [0,1/2) such that for x≠ y ≠ Tx by (<ref>) we have:
	
    d(Tx,T^2x) ≤ C [d(x,T^2x)+d(Tx,Tx)],

	
    d(T^2x,Ty) ≤ C [d(Tx,Ty)+d(y,T^2x)].

	
    d(Ty,Tx) ≤ C [d(x,Ty)+d(y,Tx)].

	
	Adding inequalities (<ref>), (<ref>) and (<ref>) we obtain (<ref>), so the conclusion follows.



	Let (X,d) be a complete metric space and let T:X→ X be a generalized orbital triangular Chatterjea contraction on X such that T has no periodic points of prime period 2. Then, T has a unique fixed point.
	



	Let x_i and the Picard iteration as in the proof of Theorem <ref> and 
    p_i = d(x_i,x_i+1)+d(x_i+1,x_i+2)+d(x_i+2,x_i),
 for every i ≥ 0. Then, by (<ref>), we have
	
    p_i = d(x_i,x_i+1)+d(x_i+1,x_i+2)+d(x_i+2,x_i) =
    
    	=d(Tx_i-1,T^2x_i-1)+d(T^2x_i-1,Tx_i+1)+d(Tx_i+1,Tx_i-1) ≤
    ≤γ[d(x_i-1,Tx_i+1)+d(x_i+1,Tx_i-1)+d(x_i-1,T^2x_i-1)+
    +d(x_i+1, T^2x_i-1)+d(Tx_i-1,Tx_i+1)] =
    
    	=γ[d(x_i-1,x_i+2)+d(x_i+1,x_i)+d(x_i-1,x_i+1)+d(x_i,x_i+2)] ≤
    ≤γ[d(x_i-1,x_i)+d(x_i,x_i+1)+d(x_i+1,x_i-1) +
    + d(x_i,x_i+1)+d(x_i+1,x_i+2)+d(x_i+2,x_i)]=
    =
    	γ·(p_i-1+p_i),

	for every i ≥ 1, so we obtain 
    p_i ≤γ1-γ p_i-1
 for every i ≥ 1. Thus, since γ1-γ < 1, as in the proof of Theorem <ref> we obtain that {x_n} is a Cauchy sequence, so it is a convergent sequence to an x^* ∈ X. 
	
	To prove that x^* is a fixed point of T, notice that since {x_n} is convergent and x_n ≠ x_n+1≠ x_n+2, there exists a subsequence {x_n(k)} such that x_n(k)≠ Tx_n(k)≠ x^*, so, by (<ref>) we obtain 
	
    d(Tx_n(k),T^2x_n(k)) + d(T^2x_n(k),Tx^*) + d(Tx^*,Tx_n(k)) ≤
    ≤γ [d(x_n(k),Tx^*)+d(x^*,Tx_n(k))+d(x_n(k),T^2x_n(k))+
    +d(x^*,T^2x_n(k))+d(Tx_n(k),Tx^*)],

	so
	
    d(x_n(k)+1,x_n(k)+2) + d(x_n(k)+1,Tx^*) + d(x_n(k)+2,Tx^*) ≤
    ≤γ [d(x_n(k),Tx^*)+d(x^*,x_n(k+1))+d(x_n(k),x_n(k+2))+
    +d(x^*,x_n(k+2))+d(x_n(k)+1,Tx^*)],

	and taking the limit as k →∞ we get 
	
    2d(x^*,Tx^*) ≤ 2γ d(x^*,T^*),

	by where d(x^*,Tx^*)=0, so x^* is a fixed point of T.
	
	Let us suppose that there exists another fixed point of T, y^* ∈ X for which Ty^*=y^*≠ x^* ≠ Tx^*. Then, by (<ref>) we get 
	
    d(Tx^*,T^2x^*)+d(T^2x^*,Ty^*)+d(Ty^*,Tx^*) ≤
    ≤γ [d(x^*,Ty^*)+d(y^*,Tx^*)+d(x^*,T^2x^*)+d(y^*,T^2x^*)+d(Tx^*,Ty^*)],

	so 
    2d(x^*,y^*) ≤ 4γ d(x^*,y^*),
 which is a contradiction since γ < 12.



	Let X={0,1,2,3} endowed with the distance d(x,y) =|x-y| and let T:X→ X defined as 
    T0=T1=T2=0,    T3=2.

	Let us first note that T has a unique fixed point 0=T0 and T does not possess any periodic points of prime period 2 since
	
    T^21=T^22=T^23=0.

	
	Moreover, T is not a Chatterjea contraction since we have 
	
    d(T2,T3) = |0-2|=2
 and 
    d(2,T3)+d(3,T2) = |2-2|+|3-0|=3.

	
	However, for every x,y∈ X such that x≠ y≠ Tx, letting 
    L(x,y) := d(Tx,T^2x)+d(T^2x,Ty)+d(Ty,Tx)
 and 
    R(x,y):=d(x,Ty)+d(y,Tx)+d(x,T^2x)+d(y,T^2x)+d(Tx,Ty)
 we have 
	
    L(0,1)=L(0,2)=L(1,2)=L(2,1) = 0,

	
    L(0,3) = L(3,0) = L(1,3)=L(3,1) = L(2,3)= 4

	and 
	
    R(0,1)= 3,    R(0,2)= 4,    R(1,2)=R(2,1) = 6,

	
    R(0,3) = R(3,0) = R(1,3)=R(3,1) = R(2,3) =10.

	
	Thus, T is a generalized orbital triangular Chatterjea contraction with γ=25.






9
	
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