In this paper, we prove some fixed point results for some classes of nonlinear mappings recently introduced by Okeke and Olaleru [5]. Our results improves several other known results in literature, including the results of Sahu et al. [8] and Sahu [7]

Okeke, G.A.

Olabiyi, M.A.

Covenant University Repository

Applied Mathematical Sciences, Vol. 9, 2015, no. 86, 4255 - 4260HIKARI Ltd, www.m-hikari.comhttp://dx.doi.org/10.12988/ams.2015.5128Existence of Fixed Points of Some Classes ofNonlinear Mappings in Spaces withWeak Uniform Normal StructureGodwin Amechi OkekeDepartment of Mathematics, College of Science and Technology,Covenant University, Canaanland, KM 10 Idiroko Road,P.M.B. 1023 Ota, Ogun State, NigeriaMufutau Adesina OlabiyiDepartment of MathematicsUniversity of Lagos, Akoka, Lagos, NigeriaCopyright c© 2015 Godwin Amechi Okeke and Mufutau Adesina Olabiyi. This articleis distributed under the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is properlycited.AbstractIn this paper, we prove some fixed point results for some classes ofnonlinear mappings recently introduced by Okeke and Olaleru [5]. Ourresults improves several other known results in literature, including theresults of Sahu et al. [8] and Sahu [7].Mathematics Subject Classification: 47H10, 47H09Keywords: Existence of fixed point, nonlinear mappings, φ-nearly Lips-chitzian mappings1 Introduction and PreliminariesLet C be a nonempty subset of a Banach spaceX and S : C → C a Lipschitzianmapping, we use the symbol σ(S) to denote the exact Lipschitz constant of S,4256 Godwin Amechi Okeke and Mufutau Adesina Olabiyii.e.,σ(S) = inf{k ∈ [0,∞] : ‖Sx− Sy‖ ≤ k‖x− y‖ for all x, y ∈ C}. (1.1)A mapping T : C → C is said to be(a) nonexpansive if σ(T ) = 1,(b) asymptotically nonexpansive if σ(T n) ≥ 1 for all n ∈ N and limn→∞ σ(T n) =1,(c) uniformly L-Lipschitzian if σ(T n) = L for all n ∈ N and for some L ∈(0,∞).Sahu [7] recently introduced the following classes of nonlinear mappings asintermediate classes between the class of asymptotically nonexpansive map-pings and that of mappings of asymptotically nonexpansive type (see, Goebeland Kirk [3], Kirk [4]).Definition 1.1 [7] Let C be a nonempty subset of a Banach space E andfix a sequence {an} in [0,∞) with an → 0. A mapping T : C → C will becalled nearly Lipschitzian with respect to {an} if for each n ∈ N, there existsa constant kn ≥ 0 such that‖T nx− T ny‖ ≤ kn(‖x− y‖+ an) ∀ x, y ∈ C. (1.2)The infimum of constants kn for which (2.18) holds will be denoted by η(Tn)and called nearly Lipschitz constant. Notice thatη(T n) = sup{‖T nx− T ny‖‖x− y‖+ an: x, y ∈ C, x 6= y}. (1.3)A nearly Lipschitzian mapping T with sequence {(an, η(T n))} is said to be(i) nearly contraction if η(T n) < 1 for all n ∈ N,(ii) nearly nonexpansive if η(T n) ≤ 1 for all n ∈ N,(iii) nearly asymptotically nonexpansive if η(T n) ≥ 1 for all n ∈ N andlimn→∞ η(Tn) ≤ 1,(iv) nearly uniformly k-Lipschitzian if η(T n) ≤ k for all n ∈ N,(v) nearly uniformly k-contraction if η(T n) ≤ k < 1 for all n ∈ N.Inspired by the facts above, Okeke and Olaleru [5] introduced the followingclasses of nonlinear mappings.Definition 1.2 Let C be a nonempty subset of a Banach space E, φ : R+ =[0,∞) → R+ be a continuous strictly increasing function such that φ(0) = 0,limt→∞ φ(t) = ∞ and fix a sequence {an} in [0,∞) with an → 0. A mappingT : C → C will be called φ-nearly Lipschitzian with respect to {an} if for eachn ∈ N, there exists a constant kn ≥ 0 such that‖T nx− T ny‖ ≤ kn.φ(‖x− y‖+ an) ∀ x, y ∈ C. (1.4)Existence of fixed points of some classes of nonlinear mappings 4257The infimum of constants kn for which (1.6) holds will be denoted by η(Tn)and called φ-nearly Lipschitz constant. Notice thatη(T n) = sup{‖T nx− T ny‖φ(‖x− y‖+ an): x, y ∈ C, x 6= y}. (1.5)A φ-nearly Lipschitzian mapping T with sequence {(an, η(T n))} is said to be(i) φ-nearly contraction if η(T n) < 1 for all n ∈ N,(ii) φ-nearly nonexpansive if η(T n) ≤ 1 for all n ∈ N,(iii) φ-nearly asymptotically nonexpansive if η(T n) ≥ 1 for all n ∈ N andlimn→∞ η(Tn) ≤ 1,(iv) φ-nearly uniformly k-Lipschitzian if η(T n) ≤ k for all n ∈ N,(v) φ-nearly uniformly k-contraction if η(T n) ≤ k < 1 for all n ∈ N.Observe that if φ is identity in Definition 1.2, then we obtain the conceptsintroduced by Sahu [7] (see Definition 1.1 above).Our purpose in this paper is to prove some fixed point results for the classesof nonlinear mappings defined by Okeke and Olaleru [5], as given in Definition1.2 above.The following definitions and lemma will be needed in this study.Definition 1.3 [7] Let C be a nonempty subset of a Banach space E andT : C → C a mapping. T is said to be demicontinuous if whenever a sequence{xn} in C converges strongly to x ∈ C, then {Txn} converges weakly to Tx.Definition 1.4 [2] The normal structure coefficient N(E) of a Banach spaceE is defined byN(E) = inf{diam(C)rC(C): C is nonempty bounded convex subset of E withdiam C > 0},where rC(C) = infx∈C{supy∈C ‖x − y‖} is the Chebyshev radius of C relativeto itself and diam (C) = supx,y∈C ‖x − y‖ is diameter of C. The space E issaid to have the uniform normal structure if N(E) > 1. A weakly convergentsequence coefficient of E is defined byWCS(E) = sup{k : k lim supn→∞ ‖xn‖ < diama({xn}) for all {xn} in E withxn ⇀ 0}.The space E is said to have the weak uniform normal structure ifWCS(E) > 1.Definition 1.5 [1] Let C be a nonempty subset of a Banach space E. Anonempty closed convex subset D of C is said to satisfy property (ω) withrespect to a mapping T : C → C ifωT (x) ⊂ D for every x ∈ D, (1.6)4258 Godwin Amechi Okeke and Mufutau Adesina Olabiyiwhere ωT (x) denotes the set of all weak subsequential limits of {T nx : n ∈ N}.Moreover, T is said to satisfy the (ω)-fixed point property if T has a fixed pointin every nonempty closed convex subset D of C which satisfies property (ω).Lemma 1.6 [8] Let C be a nonempty closed convex subset of a Banach spaceand T : C → C a mapping such that T nu → v as n → ∞ for some u, v ∈ C.Suppose that T is demicontinuous at v. Then v is a fixed point of T in C.2 Main ResultsTheorem 2.1 Let E be a Banach space with weak uniform normal structure, Ca nonempty weakly compact convex subset of E and T : C → C a φ-nearly Lip-schitzian mapping with sequence {(an, η(T n))} such that lim supn→∞ η(T n) <√WCS(E). Also suppose that there exists a nonempty closed convex subsetM of C which satisfies property (ω) with respect to T. Then(a) for an arbitrary x0 ∈ M, there exists an iterative sequence {xm} in Mdefined byxm = w − limn→∞T nxm−1 ∀m ∈ N, (2.1)(b) if T is asymptotically regular on C, then there exists an element v ∈Msuch that{xm} converges strongly to v ∈ M. Further, if T is demicontinuous atv, thenv ∈ F (T ).Proof. (a) We can easily construct a nonempty closed convex separable subsetC0 of C which is invariant under each Tn (i.e. T n(C0) ⊂ C0 for n = 1, 2, · · · ),we suppose that C itself is separable.Due to the separability of C0, we can select a subsequence {T nx} suchthat {T nx} is weakly convergent for each x ∈ C. For every x0 ∈ M ⊂ C, weconsider a sequence {T nx0} in C. Suppose that w − limn→∞ T nx0 = x1 ∈ C.Using property (ω) we have that x1 ∈ M. By induction, we can construct asequence {xm} in M defined by (2.1).(b) Suppose that T is asymptotically regular on C. The weak asymp-totic regularity of T ensures that xm = w − limn→∞ T n+rxm−1 for each r ∈N. We are to show that {xm} converges strongly to a fixed point T. Weset L := lim supn→∞ η(Tn), Dm := lim supn→∞ ‖xm − T nxm‖ and Rm :=lim supn→∞ ‖xm+1 − T nxm‖ for all m = 0, 1, 2, · · · Using the property ofWCS(E), we obtainRm = lim supn→∞‖xm+1 − T nxm‖ ≤1WCS(E)D[{T nxm}]. (2.2)Existence of fixed points of some classes of nonlinear mappings 4259Using the asymptotic regularity of T and the w-l.s.c. of the norm ‖.‖, we obtainD[{T nxm}] = lim supn→∞ (lim supr→∞ ‖T nxm − T rxm‖)≤ lim supn→∞(lim supr→∞(‖T nxm − T n+rxm‖+‖T n+rxm − T rxm‖))≤ lim supn→∞ (lim supr→∞(η(T n).φ(‖xm − T rxm‖+ an)))= L lim supr→∞ (φ(‖xm − T rxm‖))≤ L lim supr→∞ (φ(lim sups→∞(‖T sxm−1 − T rxm‖)))≤ L lim supr→∞(φ(lim sups→∞(‖T sxm−1 − T r+sxm−1‖+‖T r+sxm−1 − T rxm‖)))≤ L lim supr→∞(φ(lim sups→∞(‖T sxm−1 − T r+sxm−1‖+η(T r)(‖T sxm−1 − xm‖+ ar))))≤ L2 lim sups→∞(φ(‖T sxm−1 − xm‖)) = L2 × φ(Rm−1). (2.3)We set λ := L2WCS(E)< 1. Using (2.2), we haveφ(Rm) ≤ λ× φ(Rm−1) ≤ λ2 × φ(Rm−2) ≤ · · · ≤ λm × φ(R0)→ 0 (2.4)as m→∞. For each m ∈ N, we obtain‖xm+1 − xm‖ ≤ lim supn→∞(‖xm+1 − T nxm‖+ ‖T nxm − xm‖)≤ Rm + lim supn→∞(lim supr→∞ ‖T nxm − T rxm−1‖)≤ Rm + lim supn→∞(lim supr→∞ ‖T nxm − T n+rxm−1‖+‖T n+rxm−1 − T rxm−1‖))≤ Rm + lim supn→∞(φ(lim supr→∞(η(T n)×(‖xm − T rxm−1‖+ an))))≤ (λ+ L).φ(Rm−1)· · ·≤ (λ+ L)λm−1 × φ(R0). (2.5)We see that {xm} is a Cauchy sequence in M and hence there exists an elementv ∈M such that limm→∞ xm = v. Clearly,‖v − T nv‖ ≤ ‖v − xm+1‖+ ‖xm+1 − T nxm‖+ ‖T nxm − T nv‖≤ ‖v − xm+1‖+ ‖xm+1 − T nxm‖+ η(T n)×φ(‖xm − v‖+ an). (2.6)Taking limit superior as n→∞ on both sides, we obtainlim supn→∞‖v − T nv‖ ≤ ‖v − xm+1‖+ φ(Rm) + L‖xm − v‖ → 0,as m → ∞. Hence, we have that T nv → v as n → ∞. Furthermore, we as-sume that T is demicontinuous at v. Therefore, using Lemma 1.6, we obtainv ∈ F (T ). Remark 2.2 The results of Theorem 2.1 improves and generalizes severalother known results in literature, including the results of Sahu et al. [8] andSahu [7].4260 Godwin Amechi Okeke and Mufutau Adesina OlabiyiReferences[1] T. D. Benavides, P. L. Ramirez, Structure of the fixed point set and com-mon fixed points of asymptotically nonexpansive mappings, Proc. Amer.Math. Soc. 129 (2001) 3549-3557. http://dx.doi.org/10.1090/s0002-9939-01-06141-x[2] W. L. Bynum, Normal structure coefficient for Banach spaces, Pacific J.Math. 86 (1980) 427-436. http://dx.doi.org/10.2140/pjm.1980.86.427[3] K. Goebel, W. A. Kirk, A fixed point theorem for asymptoticallynonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972) 171-174.http://dx.doi.org/10.1090/s0002-9939-1972-0298500-3[4] W. A. Kirk, Fixed point theorems for non-Lipschitzian mappings ofasymptotically nonexpansive type, Israel J. Math. 17 (1974) 339-346.http://dx.doi.org/10.1007/bf02757136[5] G. A. Okeke, J. O. Olaleru, Fixed points of demicontinuous of φ-nearlyLipschitzian mappings in Banach spaces, (to appear).[6] G. A. Okeke, J. O. Olaleru, H. Akewe, Convergence theorems on asymp-totically generalized Φ-hemicontractive mappings in the intermediatesense, Int. Journal of Math. Analysis, Vol. 7, 2013, no. 40, 1991-2003.http://dx.doi.org/10.12988/ijma.2013.35117[7] D. R. Sahu, Fixed points of demicontinuous nearly Lipschitzian mappingsin Banach spaces, Comment. Math. Univ. Carolin. 46, 4 (2005) 653-666.[8] D. R. Sahu, Z. Liu, S. M. Kang, Existence and approximation of fixedpoints of nonlinear mappings in spaces with weak uniform normal struc-ture, Computers and Mathematics with Applications 64 (2012) 672-685.http://dx.doi.org/10.1016/j.camwa.2011.12.081Received: January 21, 2015; Published: June 6, 2015

Existence of Fixed Points of Some Classes of Nonlinear Mappings in Spaces with Weak Uniform Normal Structure

Godwin Amechi Okeke

Mufutau Adesina Olabiyi

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Applied Mathematical Sciences

Existence of fixed points of some classes of nonlinear mappings in spaces with weak uniform normal structure

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Existence of Fixed Points of Some Classes of Nonlinear Mappings in Spaces with Weak Uniform Normal Structure

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