A mapping T from a topological space X to a topological space Y is said to be compact if T(X) is contained in a compact subset of Y . The aim of this paper is to prove the existence of fixed points of a nonexpansive compact self-mapping defined on a closed subset having a contractive jointly continuous family when the underlying space is a metric space. The proved result generalizes and extends several known results on the subject

Narang, T. D.

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A fixed point theoremfor nonexpansive compact self-mapping

University of Maria Curie-Skłodowska (UMCS): Scientific e-Journals / Uniwersytet Marii Curie-Skłodowskiej: e-czasopisma naukowe

doi: 10.2478/umcsmath-2014-0005ANNALESUNIVERS ITAT I S MARIAE CUR IE - SKŁODOWSKALUBL IN – POLONIAVOL. LXVIII, NO. 1, 2014 SECTIO A 43–47T. D. NARANGA fixed point theoremfor nonexpansive compact self-mappingAbstract. A mapping T from a topological space X to a topological spaceY is said to be compact if T (X) is contained in a compact subset of Y . Theaim of this paper is to prove the existence of fixed points of a nonexpansivecompact self-mapping defined on a closed subset having a contractive jointlycontinuous family when the underlying space is a metric space. The provedresult generalizes and extends several known results on the subject.Finding sufficient conditions for the existence of fixed points of a self-mapping f defined on a topological space X is an interesting aspect of thetheory of fixed points. Several results are known in the literature on thissubject. In 1930, J. Schauder [11] proved that if K is a compact convexsubset of a Banach space X and T : K → K is a continuous map, then Thas at least one fixed point in K. The compactness condition on K is astrong one. Many problems in analysis do not have a compact setting andthe spaces are not Banach spaces. So, it is natural to prove the result byrelaxing the condition of compactness and considering spaces more generalthan Banach spaces. In this direction many results are available in theliterature (see e.g. [1]–[4], [6]–[8] and [10]). In this paper we also provea result on the existence of fixed points of a nonexpansive compact self-mapping defined on a closed subset having a contractive jointly continuous2000 Mathematics Subject Classification. 47H10, 54H25.Key words and phrases. Convex metric space, convex set, star-shaped set, nonexpan-sive map, compact map.The research has been sponsored by University Grants Commission, India under Emer-itus Fellowship.44 T. D. Narangfamily when the underlying space is a metric space. The proved resultgeneralizes and extends some of the results proved in [1], [2]–[4], [7]–[9].To start with, we recall few definitions and some related results.Let X and Y be topological spaces. A mapping T : X → Y is calledcompact if T (X) is contained in a compact subset of Y .If T is compact, then T (X) is compact.Let (X, d) be a metric space. A continuous mappingW : X×X× [0, 1] →X is said to be a convex structure on X if for all x, y ∈ X and λ ∈ [0, 1],d(u,W (x, y, λ)) ≤ λd(u, x) + (1− λ)d(u, y)holds for all u ∈ X. The metric space (X, d) together with a convex struc-ture is called a convex metric space [12].A subset K of a convex metric space (X, d) is said to be a convex set[12] if W (x, y, λ) ∈ K for all x, y ∈ K and λ ∈ [0, 1]. The set K is said tobe p-starshaped [8] if there exists a p ∈ K such that W (x, p, λ) ∈ K forall x ∈ K and λ ∈ [0, 1]. Clearly, each convex set is starshaped but notconversely.A convex metric space (X, d) is said to satisfy property (I) [8] if for allx, y, q ∈ X and λ ∈ [0, 1],d(W (x, q, λ),W (y, q, λ)) ≤ λd(x, y).A normed linear space and each of its convex subsets are simple examplesof convex metric spaces. There are many convex metric spaces which arenot normed linear spaces (see e.g. [8], [12]). Property (I) is always satisfiedin a normed linear space.Let K be a subset of a metric space (X, d) and F = {fα : α ∈ K} afamily of functions from [0, 1] into K, having the property fα(1) = α foreach α ∈ K. Such a family F is said to be(i) contractive if there exists a function φ : (0, 1) → (0, 1) such that for allα, β ∈ K and for t ∈ (0, 1), we haved(fα(t), fβ(t)) ≤ φ(t)d(α, β),(ii) jointly continuous if t → t0 in [0, 1] and α → α0 in K imply fα(t) →fα0(t0).In normed linear spaces these notions were discussed by Dotson [5].If K is a starshaped subset with star centre q of a convex metric space(X, d) with property (I), then the family F = {fx : x ∈ K} defined byfx(t) = W (x, q, t) satisfiesd(fx(t), fy(t)) = d(W (x, q, t),W (y, q, t)) ≤ t d(x, y).So taking φ(t) = t, 0 < t < 1, the family F is a contractive jointly con-tinuous family and therefore the class of subsets of X with the propertyof contractiveness and joint continuity contains the class of starshaped setswhich in turn contains the class of convex sets.A fixed point theorem for nonexpansive compact self-mapping 45Considering the existence of fixed points for nonexpansive mappings inconvex metric spaces, Beg and Abbas [2] proved that if K is a nonemptycompact and convex subset of a complete convex metric space (X, d) withproperty (I), then any nonexpansive mapping T : K → K has a fixed point.This result is not true in general for noncompact sets K (see [7], p. 12).Various extensions and generalizations of the results of Schauder [11] and ofBeg and Abbas [2] are known in the literature (see e.g. [1], [3], [4], [6]–[10]).In this direction, we also prove the following:Theorem 1. Let (X, d) be a metric space and K a non-empty closed subsetwith a contractive jointly continuous family {fα : α ∈ K}. If T : K → K isa nonexpansive compact mapping, then T has a fixed point.Proof. Let kn = nn+1 , n = 1, 2, 3, . . . . Define Tn : K → K asTnx = fTx(kn), x ∈ K.Since T (K) ⊂ K and kn < 1, Tn : K → K is well defined. Considerd(Tnx, Tny) = d(fTx(kn), fTy(kn))≤ φ(kn)d(Tx, Ty)≤ φ(kn)d(x, y), x, y ∈ Kand so each Tn is a contraction mapping on K. Since cl Tn(K) is Tn-invariant and also compact for each n and hence complete, by Banach con-traction principle, each Tn has a unique fixed point xn ∈ K i.e. Tnxn = xn.Since T (K) lies in a compact subset ofK, (Txn) has a subsequence (Txni)such that (Txni) → x0 ∈ K.Now xni = Tni(xni) = fTxni (kni) → fx0(1) = x0. Since T is a continuous,(Txni) → Tx0 and hence x0 = Tx0 i.e. x0 is a fixed point of T . If K is a starshaped subset with a star centre p, of a convex metricspace (X, d) with property (I), then the family {fα : α ∈ K} defined byfα(t) = W (α, p, t) is contractive if we take φ(t) = t for 0 < t < 1, and isjointly continuous. So, we haveCorollary 1 ([3]). Let (X, d) be a convex metric space with property (I)and K a non-empty closed starshaped subset of X. If T : K → K is anonexpansive compact map, then T has a fixed point.Since every normed linear space is a convex metric with property (I), asan immediate consequence of Corollary 1, we get:Corollary 2 ([9]). If T : K → K is a nonexpansive, where K is a closedand starshaped subset of a normed linear space E, and cl[T (K)] is compact,then T has a fixed point.Since every convex set is starshaped, we have46 T. D. NarangCorollary 3. Let K be a non-empty closed convex subset of a convex metricspace (X, d) with property (I), then any nonexpansive compact mapping T :K → K has a fixed point.Since for a compact set K, any continuous mapping T : K → K is acompact mapping, we haveCorollary 4 ([2]). Let K be a compact subset of a complete convex metricspace (X, d). Suppose there exists a contractive, jointly continuous family ofmaps associated with K, then any nonexpansive mapping T of K into itselfhas a fixed point in K.Note. We do not require the completeness of the space in Corollary 4proved by Beg and Abbas [2].Corollary 5 ([2]). Let K be a non-empty compact convex (or starshaped)subset of a convex metric space (X, d) with property (I), then any nonex-pansive mapping T : K → K has a fixed point.Corollary 6 ([8]). Let X be a compact starshaped metric space satisfyingproperty (I). Let T : X → X be a nonexpansive mapping. Then T has afixed point in X.Since every normed linear space is a convex metric space with property(I), we haveCorollary 7 ([7], Theorem 3.1, p. 28). If K is a non-empty compact andconvex subset of a Banach space, then any nonexpansive mapping T : K →K has a fixed point.Corollary 8 ([1]). Let K be a closed starshaped or convex subset of a Ba-nach space X and f : K → K a nonexpansive map with f(K) contained ina compact set K. Then f has a fixed point.Corollary 9 ([4]). Let X be a Banach space and K be a nonempty compactstarshaped subset of X. Let T : K → K be a nonexpansive mapping. ThenT has a fixed point in K.Acknowledgement. The author is thankful to the learned referee for thecritical comments and valuable suggestions leading to an improvement ofthe paper.References[1] Agarwal, R. P., Meehan, M., O’Regan, D., Fixed Point Theory and Applications,Cambridge University Press, Cambridge, 2001.[2] Beg, I., Abbas, M., Fixed points and best approximation in Menger convex metricspaces, Arch. Math. (Brno) 41 (2005), 389–397.[3] Beg, I., Shahzad, N., Iqbal, M., Fixed point theorems and best approximation inconvex metric spaces, J. Approx. Theory 8 (1992), 97–105.A fixed point theorem for nonexpansive compact self-mapping 47[4] Dotson Jr., W. G., Fixed-point theorems for nonexpansive mappings on starshapedsubset of Banach spaces, J. London Math. Soc. 2 (1972), 408–410.[5] Dotson Jr., W. G., On fixed points of nonexpansive mappings in nonconvex sets,Proc. Amer. Math. Soc. 38 (1973), 155–156.[6] Dugundji, J., Granas, A., Fixed Point Theory, PWN-Polish Sci. Publ., Warszawa,1982.[7] Goebel, K., Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge UniversityPress, Cambridge, 1990.[8] Guay, M. D., Singh, K.L., Whitfield, J. H. M., Fixed point theorems for nonexpansivemappings in convex metric spaces, Proc. Conference on nonlinear analysis (Ed. S.P.Singh and J.H. Bury), Marcel Dekker, New York, 1982, 179–189.[9] Habiniak, L., Fixed point theory and invariant approximations, J. Approx. Theory56 (1989), 241–244.[10] Singh, S., Watson, B., Srivastava, P., Fixed Point Theory and Best Approximation:The KKM-map Principle, Kluwer Academic Publishers, Dordrecht, 1997.[11] Schauder, J., Der fixpunktsatz in funktionaraumen, Studia Math. 2 (1930), 171–180.[12] Takahashi, W., A convexity in metric space and nonexpansive mappings, I, KodaiMath. Sem. Rep. 22 (1970), 142–149.T. D. NarangDepartment of Mathematics GuruNanak Dev UniversityAmritsar -143005Indiae-mail: tdnarang1948@yahoo.co.inReceived October 11, 2011

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A fixed point theoremfor nonexpansive compact self-mapping

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A fixed point theoremfor nonexpansive compact self-mapping

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