International Journal of Analysis and Applications Volume 19, Number 4 (2021), 512-517 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-512 ON FIRMLY NON-EXPANSIVE MAPPINGS JOSEPH FRANK GORDON1,∗, ESTHER OPOKU GYASI2 1Department of Mathematics Education, Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development, Kumasi, Ghana 2Department of Mathematical Sciences, University of Mines and Technology, Tarkwa, Ghana ∗Corresponding author: josephfrankgordon@gmail.com Abstract. In this paper, it is shown that for a closed convex subset C and to every non-expansive mapping T : C → C, one can associate a firmly non-expansive mapping with the same fixed point set as T in a given Banach space. 1. Introduction The study of non-expansive mappings in the sixties have experimented a boost, basically motivated by Browder’s work on the relationship between monotone operators, non-expansive mappings [1–3, 3–5] and the seminal paper by Kirk [6], where the significance of the geometric properties of the norm for the existence of fixed points for non-expansive mappings was highlighted. Now the history of firmly non-expansive mappings goes back to the paper by Minty [7], where he implicitly used this class of mappings to study the resolvent of a monotone operator. Browder [3] first introduced firmly non-expansive mappings in the concept of Hilbert spaces H. That is, given a C closed convex subset of a Hilbert space H, a mapping F : C →H is firmly non-expansive if for all x,y ∈C ‖Fx−Fy‖2 ≤〈x−y,Fx−Fy〉.(1.1) Received March 13th, 2021; accepted April 20th, 2021; published May 11th, 2021. 2010 Mathematics Subject Classification. 47H10, 54H25. Key words and phrases. non-expansive mappings; firmly non-expansive mappings; metric projection; fixed points. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 512 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-512 Int. J. Anal. Appl. 19 (4) (2021) 513 In his study of non-expansive projections on subsets of Banach spaces, Bruck [8] defined a firmly non- expansive mapping F : C →X , where C is a closed convex subset of a real Banach space X , to be a mapping such that for all x,y ∈C and α ≥ 0, ‖Fx−Fy‖≤‖α(x−y) + (1 −α)(Fx−Fy)‖.(1.2) It is clear that equation (1.2) reduces to equation (1.1) when in Hilbert spaces and also when α = 1, F becomes a non-expansive mappings, that is, for each x and y in C, we have ‖Fx−Fy‖≤‖x−y‖. A trivial example of equation (1.2) is the identity mapping. A non-trivial example of equation (1.2) in a Hilbert space is given by the metric projection PCx = argminy∈C{‖x−y‖}.(1.3) To see this, recall that in a real Hilbert space H, ∀x,y ∈H, then 〈x,y〉≥ 0 if and only if ‖x‖≤‖x + ay‖,(1.4) for all a ≥ 0. Now equation (1.2) can be written as ‖Fx−Fy‖≤‖Fx−Fy + α(x−y −Fx + Fy)‖.(1.5) Now applying equation (1.4) on equation (1.5), we obtain the following 〈Fx−Fy,x−y −Fx + Fy〉≥ 0, 〈x−y −Fx + Fy,Fx−Fy〉≥ 0, 〈x−y − (Fx−Fy),Fx−Fy〉≥ 0, 〈x−y,Fx−Fy〉−〈Fx−Fy,Fx−Fy〉≥ 0, 〈x−y,Fx−Fy〉≥ 〈Fx−Fy,Fx−Fy〉, 〈x−y,Fx−Fy〉≥ ‖Fx−Fy‖2. Hence we have that in a real Hilbert space, a firmly non-expansive mapping F can be written as ‖Fx−Fy‖2 ≤〈x−y,Fx−Fy〉.(1.6) But in a real Hilbert space, equation (1.3) satisfies the following inequality ‖PCx−PCy‖2 ≤〈x−y,PCx−PCy〉.(1.7) This means that from equations (1.6) and (1.7), we can simply conclude that F = PC and so the metric projection PC is a firmly non-expansive mapping in a real Hilbert space. Int. J. Anal. Appl. 19 (4) (2021) 514 In this paper, we give a simple proof showing that to any non-expansive self-mapping T : C → C that has fixed points, one can associate a large family of firmly non-expansive mappings having the same fixed point set as T. That is, from the point of view of the existence of fixed points on closed convex sets, non-expansive and firmly non-expansive mappings exhibit a similar behavior. However, this is no longer true in non-convex domains [9]. 2. Main Results Let T be a non-expansive mapping defined on a closed convex subset C of a normed space X , thus, T : C →C. For a fixed r ∈ R>1, we can define the following mapping Tr : C →C by x 7→ ( 1 − 1 r ) x + 1 r T(Trx).(2.1) Now we observe that equation (2.1) (the new mapping Tr) always exist. To see this, one can create an internal contraction F : C →C such that F(y) = ( 1 − 1 r ) x + 1 r Ty, where x is fixed. Now ‖F(y) − F(z)‖ = 1 r ‖Ty − Tz‖ ≤ 1 r ‖y − z‖. Hence F is a contraction mapping and by the Banach contraction mapping theorem [10], there exists u ∈C such that F(u) = u, thus, u = (1 − 1 r )x + 1 r Tu. Since for every x ∈C, we can find a unique u such that u = Trx, then equation (2.1) always exists. Now we have the following claims. Claim 1: Tr is a non-expansive mapping. To see this, we have the following: ‖Trx−Try‖ = ‖(1 − 1 r )(x−y) + 1 r (T(Trx) −T(Try))‖, ≤ (1 − 1 r )‖x−y‖ + 1 r ‖T(Trx−T(Try))‖, ≤ (1 − 1 r )‖x−y‖ + 1 r ‖Trx−Try)‖, ‖Trx−Try‖− 1 r ‖Trx−Try‖≤ (1 − 1 r )‖x−y‖, (1 − 1 r )‖Trx−Try‖≤‖(1 − 1 r )‖‖x−y‖. So we have that Tr is a non-expansive mapping since r > 1. Claim 2: Now we prove that Trx is a firmly non-expansive mapping. Now for r > 1,α ∈ (0, 1) and β > 0, we have the following evaluation: ‖Trx−Try‖ = ‖β[α(x−y) + (1 −α)(Trx−Try)] −βα(x−y) + (Trx−Try) −β(1 −α)(Trx−Try)‖. Int. J. Anal. Appl. 19 (4) (2021) 515 But (Trx−Try) −β(1 −α)(Trx−Try) = ( 1 − 1 r ) (x−y) + 1 r ( T(Trx) −T(Try) ) −β(1 −α) [( 1 − 1 r ) (x−y) + 1 r ( T(Trx) −T(Try) )] , = ( 1 − 1 r ) (x−y) −β(1 −α) ( 1 − 1 r ) (x−y) + 1 r ( T(Trx) −T(Try) ) −β(1 −α) 1 r ( T(Trx) −T(Try) ) , = ( 1 − 1 r ) (x−y)[1 −β(1 −α)] + 1 r ( 1 −β(1 −α) ) (T(Trx) −T(Try)). Hence ‖Trx−Try‖ = ‖β[α(x−y) + (1 −α)(Trx−Try)] −βα(x−y) + ( 1 − 1 r ) (x−y)[1 −β(1 −α)] + 1 r ( 1 −β(1 −α) ) (T(Trx) −T(Try))‖, = ‖β[α(x−y) + (1 −α)(Trx−Try)] + [−βα + ( 1 − 1 r ) (1 −β(1 −α))](x−y) + 1 r ( 1 −β(1 −α ) (T(Trx) −T(Try))‖. Now let −βα + ( 1 − 1 r ) (1 −β(1 −α)) = 0. This implies that β = r − 1 αr + (α− 1)(r − 1) , 1 r (1 −β(1 −α)) = α αr + (1 −α)(r − 1) . Hence we have ‖Trx−Try‖≤ β‖α(x−y) + (1 −α)(Trx−Try)‖ + α αr + (1 −α)(r − 1) ‖T(Trx) −T(Try)‖. So by the non-expansiveness of T ,The above inequality becomes ‖Trx−Try‖≤ β‖α(x−y) + (1 −α)(Trx−Try)‖ + α αr + (1 −α)(r − 1) ‖Trx−Try‖, = r − 1 αr + (1 −α)(r − 1) ‖α(x−y) + (1 −α)(Trx−Try)‖ + α αr + (1 −α)(r − 1) ‖Trx−Try‖. Int. J. Anal. Appl. 19 (4) (2021) 516 After simplifying we obtain the following results ‖Trx−Try‖− α αr + (1 −α)(r − 1) ‖Trx−Try‖≤ r − 1 αr + (1 −α)(r − 1) ‖α(x−y) + (1 −α)(Trx−Try)‖, [1 − α αr + (1 −α)(r − 1) ]‖Trx−Try‖≤ r − 1 αr + (1 −α)(r − 1) ‖α(x−y) + (1 −α)(Trx−Try)‖, r − 1 αr + (1 −α)(r − 1) ‖Trx−Try‖≤ r − 1 αr + (1 −α)(r − 1) ‖α(x−y) + (1 −α)(Trx−Try)‖, ‖Trx−Try‖≤‖α(x−y) + (1 −α)(Trx−Try)‖, since β = r−1 αr+(1−α)(r−1) > 0. Hence Tr is firmly non-expansive mapping. Claim 3: We have that z is a fixed point of T if and only if it is also a fixed point of Tr. Proof. Now suppose that Tz = z. Then we have the following evaluation: ‖Trz −z‖ = ∥∥∥(1 − 1 r ) z + 1 r T(Trz) −z ∥∥∥, = ∥∥∥1 r T(Trz) − 1 r z ∥∥∥, = 1 r ∥∥∥T(Trz) −z∥∥∥, = 1 r ∥∥∥T(Trz) −Tz∥∥∥, ≤ 1 r ∥∥∥Trz −z∥∥∥. Hence ‖Trz−z‖≤ 1r‖Trz−z‖ which is not possible since r > 1. it is possible when ‖Trz−z‖ = 0 ⇒ Trz = z. So z is a fixed point of Trz. On the other hand, let us suppose that Trz = z, that is z is a fixed point of Tr. Then z = Trz, = ( 1 − 1 r ) z + 1 r T(Trz), = ( 1 − 1 r ) z + 1 r T(z). This gives us [ 1 − ( 1 − 1 r )] z = 1 r T(z). Hence z is a fixed point of T and that concludes our main result. � Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. Int. J. Anal. Appl. 19 (4) (2021) 517 References [1] F.E. Browder, Nonlinear monotone operators and convex sets in banach spaces. Bull. Amer. Math. Soc. 71(5) (1965), 780–785. [2] F.E. Browder, Nonexpansive nonlinear operators in a Banach space. Proc. Nat. Acad. Sci. 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