Fixed Points of Multivalued Quasi-nonexpansive Mappings Using a Faster Iterative Process

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1 Fixed Points of Multivalued Quasi-nonexpansive Mappings Using a Faster Iterative Process Safeer Hussain KHAN Department of Mathematics, Statistics and Physics, Qatar University, Doha 73, Qatar safeer@queduqa Mujahid ABBAS Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Abstract Pretoria 000, SouthAfrica mujahidabbas@yahoocom Sartaj ALI Department of Mathematics, National College of Business Administration and Economics, Lahore, Pakistan sartajali 00@yahoocom In this article, we prove some strong and weak convergence theorems for quasi-nonexpansive multivalued mappings in Banach spaces The iterative process used is independent of Ishikawa iterative process and converges faster Some examples are provided to validate our results Our results extend and unify some results in the contemporary literature Keywords Multivalued nonexpansive mapping, common fixed point, condition (I), weak and strong convergence MR(00) Subject Classification 7H0, 5H5 Introduction and Preliminaries Throughout this article, N denotes the set of positive integers Let E be a real Banach space A subset K is called proximinal if for each x E, there exists an element k K such that d(x, k) inf x y : y K} d(x, K) It is known that weakly compact convex subsets of a Banach space and closed convex subsets of a uniformly convex Banach space are proximinal We denote the family of nonempty proximinal bounded subsets of K by P(K) It is well known that if K is proximinal subset of E, then K is closed Consistent with [6], let CB(K) bethe class of all nonempty bounded and closed subsets of K Let H be a Hausdorff metric induced by the metric d of E, thatis, } H(A, B) max sup d(x, B), sup d(y, A) x A y B

2 for every A, B CB(E) Apointx K is called a fixed point of a multivlaued mapping T : K CB(K) ifx Tx A set of all fixed points of T is denoted by F (T ) Amultivalued mapping T : K CB(K) issaidtobe: (a)nonexpansive if H(Tx,Ty) x y for all x, y K; (b)quasi-nonexpansive mapping if H(Tx,p) x p for all x K and p F (T ) It is known that every nonexpansive multivalued map T with F (T ) is quasi nonexpansive but the converse is not true The study of fixed points for multivalued mappings using the Hausdorff metric was initiated by Markin [5] (see also [6]) Multivalued fixed point theory has applications in control theory, convex optimization, differential inclusion, and economics (see [] and references cited therein) The theory of multivalued mappings is harder than the corresponding theory of single-valued mappings Different iterative processes have been used to approximate the fixed points of multivalued mappings Among these iterative processes, Sastry and Babu [9] considered the following Let K be a nonempty convex subset of E, T : K P(K) a multivalued mapping with p Tp (i) The sequences of Mann iterates is defined by x K, x n+ ( a n )x n + a n y n, () where y n Tx n is such that y n p d(p, T x n )anda n } is a sequence of numbers in (0, ) satisfying lim a n 0and n a n (ii) Ishikawa iterative process is defined by starting with x K and defining y n ( b n )x n + b n z n, () x n+ ( a n )x n + a n u n, where z n Tx n,u n Ty n are such that z n p d(p, T x n )and u n p d(p, T y n ), and a n }, b n } are real sequences of numbers with 0 a n,b n < satisfying lim b n 0and an b n Panyanak [8] generalized the results proved by Sastry and Babu [9] The following lemma due to Nadler [6] is very useful Lemma Let A, B CB(E) and a A If η > 0, then there exists b B such that d(a, b) H(A, B)+η Based on the above lemma, Song and Wang [] modified the iterative process due to Panyanak [8] and improved the results presented there They used () but with a n [0, ]; b n [0, ] with lim b n 0 and n a nb n ; z n Tx n,u n Ty n with z n u n H(Tx n,ty n )+η n and z n+ u n H(Tx n+,ty n )+η n,whereη n (0, ) such that lim η n 0 It is to be noted that Song and Wang [] needed the condition Tp p} in order to prove their Theorem Actually, Panyanak [8] proved some results using Ishikawa type iterative process without this condition Song and Wang [] showed that without this condition his process was not well-defined They reconstructed the process using the condition Tp p} which made it well-defined Such a condition was also used by Jung [3] Later, Shazad and Zegeye [] got rid of this condition by using P T (x) y Tx : x y d(x, T x)} for a

3 multivalued mapping T : K P(K) They proved a couple of strong convergence results using Ishikawa type iterative process Khan and Yildirim [] used the following iterative process using the method of direct construction of Cauchy sequence and without using the condition Tp p} for any p F (T ): x K, x n+ ( λ) v n + λu n, y n ( η)x n + ηv n,n N, where v n P T (x n ), u n P T (y n )and0<λ,η< Let us now construct the following iterative process for a multivalued mapping T : K P(K) with the help of P T defined above Let K be a nonempty convex subset of E, α n,β n,γ n [0, ] Start with choosing x K and u P T (x )andlet z ( γ )x + γ u Choose w P T (z ) such that y ( β )u + β w Choose v P T (y ) such that x ( α )v + α w Now choose u P T (x ) such that z ( γ )x + γ u Choose w P T (z ) such that y ( β )u + β w Choose v P T (y ) such that x 3 ( α )v + α w Next, choose u 3 P T (x 3 ) such that z 3 ( γ 3 )x 3 + γ 3 u 3 Choose w 3 P T (z 3 ) such that y 3 ( β 3 )u 3 + β 3 w 3 Choose v 3 P T (y 3 ) such that x ( α 3 )v 3 + α 3 w 3 Inductively, we obtain x n+ ( α n )v n + α n w n, y n ( β n )u n + β n w n, (3) z n ( γ n )x n + γ n u n, where u n P T (x n ),v n P T (y n ),w n P T (z n ) Its single-valued version was used by Abbas and Nazir [] 3

4 In this paper, we use the following simplified version of (3): x n+ ( α)v n + αw n, y n ( β)u n + βw n, () z n ( γ)x n + γu n, where α, β, γ [0, ],u n P T (x n ),v n P T (y n )andw n P T (z n ) Note that we are using α, β and γ only for the sake of simplicity and α n,β n and γ n could be used equally well under suitable conditions This scheme is independent of both Mann and Ishikawa iterative processes neither reduce to the other Moreover, it is faster than all of Picard, Mann and Ishikawa iterative processes in case of contractions [] Thus our results are independent but better (in the sense of speed of convergence of our iterative process) and more general (in view of more general class of mappings) than corresponding results of Shazad and Zegeye [], Khan and Yildirim [] and Song and Cho [3] and the results generalized therein At this stage, we recall the following A Banach space E is said to satisfy Opial s condition [7] if for any sequence x n } in E, x n ximplies that lim sup x n x < lim sup x n y for all y E with y x Examples of Banach spaces satisfying Opial s condition are Hilbert spaces and all l p spaces ( <p< ) On the other hand, L p [0, π] with<p fails to satisfy this condition A multivalued mapping T : K CB(K) is called demiclosed at y K if for any sequence x n } in K weakly convergent to x and y n Tx n strongly convergent to y, we have y Tx Now we state some useful lemmas Lemma ([3]) Let T : K P(K) be a multivalued mapping and P T (x) y Tx : x y d(x, T x)} Then the following are equivalent : () x F (T ); () P T (x) x}; (3) x F (P T ) Moreover, F (T )F(P T ) Lemma 3 ([0]) Let E be a uniformly convex Banach space and 0 <p t n q< for all n N Suppose that x n } and y n } are two sequences of E such that lim sup x n r, lim sup y n r and lim t n x n +( t n )y n r hold for some r 0 Then lim x n y n 0 Main Results Lemma Let E beanormedspaceandk be a nonempty closed convex subset of E Let T : K P(K) be a multivalued mapping such that F (T ) and P T be a quasi-nonexpansive mapping Let x n } be the sequence as defined in () Then lim x n p exists for all p F (T ) and lim d(x n,p T (x n )) 0 Proof To prove that lim x n p exists, we consider x n+ p ( α)v n + αw n p

5 ( α) v n p + α w n p ( α)h(p T (y n ),P T (p)) + αh(p T (z n ),P T (p)) ( α) y n p + α z n p Next y n p ( β)u n + βw n p ( β) u n p + β w n p ( β)h(p T (x n ),P T (p)) + βh(p T (z n ),P T (p)) ( β) x n p + β z n p And z n p ( γ)x n + γu n p ( γ) x n p + γ u n p ( γ) x n p + γh(p T (x n ),P T (p)) ( γ) x n p + γ x n p x n p () Thus y n p x n p () and hence x n+ p x n p This implies that lim x n p exists for each p F (T ) Suppose that where c 0 We now prove that lim x n p c, (3) lim d(x n,tx n )0 The case when c 0 is obvious We thus assume that c>0 Inasmuch as d(x n,tx n ) x n u n, it suffices to prove that lim x n u n 0 Now u n p H(P T (x n ),P T (p)) x n p implies that lim sup u n p c () From () and (), we obtain and Noting that lim sup z n p c, (5) lim sup y n p c (6) v n p H(P T (y n ),P T (p)) y n p x n p, 5

6 we have Similarly, Moreover, lim sup v n p c (7) lim sup w n p c (8) lim x n+ p lim ( α)v n + αw n p From (7) (9) and Lemma 3, we have lim ( α)(v n p)+α(w n p) c (9) Together with this and lim v n w n 0 we obtain Using (7), we have In a way similar to above, it follows that lim inf x n+ p lim inf ( v n p + α v n w n ), c lim inf v n p (0) lim v n p c That is, lim z n p c () lim z n p ( γ)x n + γu n p Hence, from (), () and Lemma 3, we have which yields lim d(x n,p T x n ) 0 as desired ( γ)(x n p)+γ(u n p) c () lim x n u n 0, (3) We are now all set to go for our first strong convergence theorem Theorem Let E be a real Banach space and K be a nonempty compact convex subset of E Let T : K P(K) be a multivalued mapping such that F (T ) and P T be quasi-nonexpansive mapping Let x n } be the sequence as defined in () Then x n } converges strongly to a fixed point of T Proof We have proved in Lemma that lim x n p exists for all p F (T ) Now from the compactness of K, there exists a subsequence x nk } of x n } such that lim k x nk q 0 for some q K Then d(q, P T (q) d(x nk,q)+d(x nk,p T (x nk )+H(P T (x nk ),P T (q)) x nk q + x nk u nk + x nk q 0 as n, 6

7 because by Lemma, we have lim x nk u nk 0 That is, d(q, P T (q)) 0 Hence, q is a fixed point of P T Since the set of fixed points of P T is the same as that of T by Lemma, therefore x n } converges strongly to a fixed point of T Here is an example in support of the above theorem Example 3 Let (R, ) be a normed space with usual norm and K [0, ] Define T : K P(K) as [ ] x + Tx 0, Obviously, K is a compact convex subset of R Note that F T [0, ] Let α β γ Observe that P T (x) x} when x [0, ] In case x/ [0, ], ( [ ])} x + P T (x) y Tx : y x d x, 0, } y Tx : y x x + x x y Tx : y x x } } x + y We next prove that P T (x) is quasi-nonexpansive for all x K Thecaseof Thus we take x [, ] ( ) x + H(P T (x),p T (p)) H,p x + p x p [0, ]istrivial for all x [, ] Finally, we generate a sequence x n} as defined in () and show that it converges strongly to a fixed point of T Choose x K [0, ] Then P T (x ) x + ()+ + } and u P T (x ) + } That is, u + Then and z ( γ)x + γu + ( + ) } z + ( P T (z ) )+ } Choose w P T (z ) }, that is, w Then y ( β)u + βw ( + ) + ( + 3 ) 6 and } y + ( P T (y ) )+ } Choose v P T (y ) },v Then x ( α)v + αw ( ) } , + 7 } 6 ( + 3 ) < +, 7

8 } x + P T (x ) ( )+ Now choose u P T (x ) }, that is, u + 9 ( ( z ( γ)x + γu } z + P T (z ) ( + 57 ) + 5 )+ } Choose w P T (z ) }, that is, w Then + 9 } Then + 9 ) , } + 57 } 0 y ( β)u + βw ( + 9 ) + ( ) , } y + ( P T (y ) )+ } Choose v P T (y ) },v Then x 3 ( α)v + αw ( ) } ( + 57 ) < + 6 In a similar way, x < + 8,x 5 < + 0,,x n < + n This shows that x n} converges strongly to a point of F T [ 0, ] We now prove our strong convergence theorem using the following Condition (I) originally due to Senter and Dotson [] A multivalued nonexpansive mapping T : K CB(K) is said to satisfy Condition (I) if there exists a continuous nondecreasing function f :[0, [ [0, [ withf(0) 0,f(r) > 0for all r (0, ) such that d(x, T x) f(d(x, F (T )) for all x K Theorem Let E be a real Banach space, K a nonempty closed and convex subset of E, T : K P(K) a multivalued mapping satisfying Condition (I) such that F (T ) and P T be a quasi-nonexpansive mapping Then the sequence x n } as defined in () converges strongly to a fixed point p of T Proof From Lemma, lim x n p exists for all p F (T )F(P T ) If lim x n p 0, there is nothing to prove Thus we assume lim x n p c>0 From the same lemma, we know x n+ p x n p, so that d(x n+,f(t )) d(x n,f(t )) Hence, lim d(x n+,f(t )) exists We now prove that lim d(x n+,f(t )) 0 Suppose on contrary that lim d(x n+,f(t )) b>0 For all n N, take a n u n p x n p, b n x n p x n p Then b n and a n because u n p H(P T (x n ),P T (p)) x n p 8

9 Now, by Condition (I) Since f is continuous, b n a n x n p x n p u n p x n p x n u n x n p d(x n,tx n ) x n p f(d(x n,f(t ))) x n p lim inf b n a n f(b) > 0 for all n N n c We have already established lim x n p c and lim z n p c in Lemma Using these two, we have lim ( γ)b n + γa n lim ( γ) x n p x n p + γ u n p x n p lim ( γ)x n + γu n p x n p lim z n p lim x n p c c That is, lim ( γ)b n + γa n Now Lemma 3 implies that lim b n a n 0, a contradiction to lim inf n b n a n > 0 Thus we have lim d(x n+,f(t )) 0, and so lim x n p 0 Hence, the sequence x n } converges strongly to a fixed point p of T To testify our above theorem, we give the following example Example 5 Consider the Banach space (R, ) and K [, ) Obviously, K is a nonempty closed and convex subset of R Define T : K P(K) as [ Tx, + x ] Then F T [, ] Let α β γ Define a continuous and nondecreasing function f :[0, ) [0, ) byf(r) r First, we show that d(x, T x) f(d(x, F T )) for all x K and Indeed, when x F T [, ], d(x, T x) 0f(d(x, F T )) When x (, ), ( [ d(x, T x) d x,, + x ]) (+ x x ) x, f(d(x, F (T ))) f(d(x, [, ])) f( x ) x Hence, d(x, T x) f(d(x, F T )) for all x K 9

10 Note that P T (x) x} when x [, ] If x (, ), then ( [ P T (x) y Tx : y x d x,, + x ])} y Tx : y x (+ x x ) } x y Tx : y x x } y + x } Next, P T is quasi-nonexpansive for all x K The case of [0, ] is trivial Thus we take x> ( H(P T (x),p T (p)) H + x } ), p} +x p x p Finally, we generate a sequence x n } as defined in () and show that it converges strongly to a fixed point of T Choose x 3 K [, ), P T (x )+ 3 5 } and u P T (x ) 5 } That is, u 5 Then z ( γ)x + γu (3) + ( ) 5 and P T (z ) Choose w P T (z ) 9 8 } That is, w + z } + y ( β)u + βw 9 8 Then ( 5 and P T (y ) + y } 39 } } Choosing v P T (y ) 7 3 },wegetx ( 7 P T (x ) } + x 7 } } } ) + 3 )+ ( 9 8 } 9 8 ( ) , ) < + and Continuing in this way, we get x n < + n This shows that x n} converges strongly to a point of F T [, ] Now we approximate fixed points of the mapping T through weak convergence of the sequence x n } defined in () Theorem 6 Let E be a uniformly convex Banach space satisfying Opial s condition and K be a nonempty closed convex subset of E Let T : K P(K) be a multivalued mapping such that F (T ) and P T be a quasi-nonexpansive mapping Let x n } be the sequence as defined in () Let I P T be demiclosed with respect to zero Then x n } converges weakly to a fixed point of T Proof Let p F (T )F(P T ) From the proof of Lemma, lim x n p exists for all p Now we prove that x n } has a unique weak subsequential limit in F (T ) To prove this, let z and z be weak limits of the subsequences x ni } and x nj } of x n }, respectively By (3), there exists u n Tx n such that lim x n u n 0 Since I P T is demiclosed with respect to zero, we obtain z F (P T )F(T) Inthesameway,wecanprovethatz F (T ) 0

11 Next, we prove the uniqueness For this, suppose that z z Then by Opial s condition, we have lim x n z lim x n n i i z < lim n i x n i z lim x n z lim n j x n j z < lim n j x n j z lim x n z, which is a contradiction Hence, x n } converges weakly to a point in F (T ) References [] Abbas, M, Nazir, T: A new faster iteration process applied to constrained minimization and feasibility problems Mathematicki Vesnik, 06, (03) [] Gorniewicz, L: Topological Fixed Point Theory of Multivalued Mappings, Kluwer Academic Pub, Dordrecht, Netherlands, 999 [3] Jung, J S: Strong convergence theorems for multivalued nonexpansive nonself mappings in Banach spaces Nonlinear Anal, 66, (007) [] Khan, S H, Yildirim, I: Fixed points of multivalued nonexpansive mappings in Banach spaces Fixed Point Theory Appl, 0, 73, 9pp (0) [5] Markin, J T: Continuous dependence of fixed point sets ProcAmerMathSoc, 38, (973) [6] Nadler, S B, Jr: Multivalued contraction mappings Pacific J Math, 30, (969) [7] Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings Bull Amer Math Soc, 73, (967) [8] Panyanak, B: Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces Comp Math Appl, 5, (007) [9] Sastry, K P R, Babu, G V R: Convergence of Ishikawa iterates for a multivalued mapping with a fixed point Czechoslovak Math J, 55, (005) [0] Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings Bull Austral Math Soc, 3, (99) [] Senter, H F, Dotson, W G: Approximatig fixed points of nonexpansive mappings Proc Amer Math Soc, (), (97) [] Shahzad, N, Zegeye, H: On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces Nonlinear Anal, 7(3 ), (009) [3] Song, Y, Cho, Y J: Some notes on Ishikawa iteration for multivalued mappings Bull Korean Math Soc, 8(3), (0) [] Song, Y, Wang, H: Erratum to Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces [Comp Math Appl, 5, (007)] Comp Math Appl, 55, (008)