ON VON NEUMANN-JORDAN TYPE CONSTANT AND SUFFICIENT CONDITIONS FOR FIXED POINTS OF MULTIVALUED NONEXPANSIVE MAPPINGS

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1 Gulf Jounal of Mathematics Vol 4, Issue 06) - ON VON NEUMANN-JORDAN TYPE CONSTANT AND SUFFICIENT CONDITIONS FOR FIXED POINTS OF MULTIVALUED NONEXPANSIVE MAPPINGS MINA DINARVAND Abstact In the pesent pape, we give some geometic conditions in tems of the von Neumann-Jodan type constant and the Domínguez-Benavides coefficient which in tun ae sufficient fo the existence of fixed points of multivalued nonexpansive and χ-contactive nonself mappings satisfying the inwadness condition and nomal stuctue in a eflexive Banach space Ou main esults impove some well known esults in the ecent liteatue Intoduction Fixed point theoy is one of the most poweful tools of moden mathematics Not only is it used on a daily basis in pue and applied mathematics, but it also seves as a bidge between analysis and topology, and povides a vey fuitful aea of inteaction between the two In paticula, fixed point theoy fo multivalued mappings has many useful applications in applied sciences, such as, game theoy and mathematical economics Thus, it is natual to ty of extending the known fixed point esults fo singlevalued mappings to the setting of multivalued mappings In 969, S B Nadle [] extended the Banach Contaction Pinciple to multivalued contactive mappings in complete metic spaces Since then, the metic fixed point theoy of multivalued mappings has been apidly developed Some classical fixed point theoems fo singlevalued nonexpansive mappings have been extended to multivalued nonexpansive mappings Nevetheless, the fixed point theoy of multivalued nonexpansive mappings is much moe complicated and difficult than the coesponding theoy of singlevalued nonexpansive mappings and many poblems emain unsolved in it, fo instance, the possibility of extending the well known Kik s theoem [7], that is, do Banach spaces with weak nomal stuctue have the fixed point popety fo multivalued nonexpansive mappings? The concept of nomal stuctue plays an impotant ole in metic fixed point theoy fo nonexpansive mappings Since unde vaious geometic popeties of a Banach space often measued by diffeent geometic constants, nomal stuctue Date: Received: Oct, 05; Accepted: Ma 3, 06 Coesponding autho 00 Mathematics Subject Classification Pimay 47H0; Seconday 46B0 Key wods and phases Multivalued nonexpansive mapping, fixed point, nomal stuctue, von Neumann-Jodan type constant, Domínguez-Benavides coefficient

2 M DINARVAND of the space is guaanteed, it is natual to study if those popeties imply the fixed point popety fo multivalued mappings Recently, many geometic constants fo a Banach space have been investigated Both the James constant JX) and the von Neumann-Jodan constant C NJ X) play an impotant ole in the desciption of vaious geometic stuctues Theefoe, many ecent studies have focused on these constants see [3, 4, 5, 6, 7, 8, 0,, 5, 6, 0, 3, 5, 6]) S Dhompongsa et al [5] intoduced the Domínguez-Loenzo condition DL)-condition, in shot) which implies weak nomal stuctue of a Banach space and the fixed point popety fo multivalued nonexpansive mappings A possible appoach to the above poblem is to look fo geometic conditions in a Banach space X which imply the DL)-condition In this setting, the following esults have been obtained: i) B Gavia [, Theoem 3] showed that a Banach space X with JX) < + R, X) satisfies the DL)-condition ii) J Zhang and Y Cui [5, Theoem 3] poved that a Banach space X with ) + R,X) C NJ X) < implies the DL)-condition Following the above, we show some geometic conditions on a Banach space X concening the von Neumann-Jodan type constant and the coefficient R, X), which imply the existence of fixed points fo multivalued nonexpansive and χ-contactive nonself mappings satisfying the inwadness condition and nomal stuctue in a eflexive Banach space The esults obtained in this wok, impove some pevious esults in the ecent papes Peliminaies At fist, we ecall some concepts and notations which will be needed in this pape We shall assume thoughout this pape that X be a Banach space with the closed unit ball B X = {x X : x } and the unit sphee S X = {x X : x = } Recall that a Banach space X is unifomly nonsquae [4] if and only if thee exists δ > 0 such that x+y δ o x y δ fo all x, y B X Recently, Y Takahashi [3] has intoduced the James and von Neumann- Jodan type constants of Banach spaces Fo p [, + ) and t 0, the James type constant is defined by { ) sup x+ty p + x ty p p : x, y S X }, p, J X,p t) = sup { min x + ty, x ty ) } : x, y S X, p =

3 FIXED POINTS FOR MULTIVALUED NONEXPANSIVE MAPPINGS 3 As mentioned in [3], we could define the von Neumann-Jodan type constant C p X) by using the James type constant J X,p t) as { J } X,p t) C p X) = sup : 0 t + t In paticula, if t = o t =, p =, then we get C px) and C X), espectively It is obvious that C px) C p X) Note that the von Neumann- Jodan type constant includes some well known geometical constants, such as the von Neumann-Jodan constant C NJ X) see [3]), and the Zbăganu constant C Z X) see [4]) These constants ae defined by C NJ X) = C X) and C Z X) = C 0 X) The coefficient R, X) is defined by T Domínguez-Benavides [9] as { } R, X) = sup lim inf x n + x, n whee the supemum is taken ove all x X with x and all weakly null sequences {x n } in B X such that lim sup n lim sup x n x m ) m Obviously, R, X) We next eview some concepts and esults of multivalued mappings Fo moe details on this topic, the eade is efeed to [, 3] Let E be a nonempty subset of X We shall denote by F BE) the family of all nonempty bounded closed subsets of E, by F E) the family of all nonempty closed subsets of E, by F CE) the family of all nonempty closed convex subsets of E, and by KCE) the family of all nonempty compact convex subsets of E Let H, ) be the Hausdoff distance on F BX), ie, { } HA, B) = max sup inf x y, sup inf x y, A, B F BX) x A y B y B x A A multivalued mapping T : E F X) is said to be a contaction if thee exists a constant k [0, ) such that HT x, T y) k x y, x, y E ) In this case, we also say that T is k-contactive If ) is valid when k =, then T is called nonexpansive A point x is a fixed point fo a multivalued mapping T if x T x Recall that the Hausdoff measue of noncompactness of a nonempty bounded subset B of X ae defined as the numbe χb) = inf { d > 0 : B can be coveed by finitely many balls of adii d } A multivalued mapping T : E X is called χ-condensing espectively, χcontactive) if, fo each bounded subset B of E with χb) > 0, thee holds the inequality ) χt B)) < χb) espectively, χt B)) χb) Hee T B) = x B T x

4 4 M DINARVAND Definition [0]) Let E be a nonempty closed subset of a Banach space X The inwad set of E at x E is given by I E X) = { x + λy x) : λ, y E } In case E is a nonempty closed convex subset of a Banach space X, we have I E X) = { x + λy x) : λ 0, y E } A multivalued mapping T : E X is said to be inwad espectively, weakly inwad) on E if ) T x I E X) espectively, T x I E X) fo all x E Definition []) A nonempty bounded and convex subset K of a Banach space X is said to have nomal stuctue if fo evey convex subset E of K that contains moe than one point thee is a point x 0 E such that sup { x 0 y : y E } < sup { x y : x, y E } A Banach space X is said to have nomal stuctue if evey bounded convex subset of X has nomal stuctue A Banach space X is said to have weak nomal stuctue if fo each weakly compact convex set K of X that contains moe than one point has nomal stuctue It is clea that fo a eflexive Banach space, nomal stuctue and weak nomal stuctue coincide It was poved by W A Kik [7] that if a weakly compact convex subset E of X has nomal stuctue, then any nonexpansive mapping on E has a fixed point Since then, much attention has been focused on nomal stuctue Whethe o not a Banach space X has nomal stuctue depends on the geomety of the unit sphee S X, o closed unit ball B X Let {x n } be a bounded sequence in X The asymptotic adius E, {x n } ) and the asymptotic cente A E, {x n } ) of {x n } in E ae defined by and E, {x n } ) { = inf lim sup x n x : x E n A E, {x n } ) { = x E : lim sup x n x = E, {x n } )}, n espectively It is well known that A E, {x n } ) is a nonempty weakly compact convex set wheneve E is see [3]) The sequence {x n } is called egula with espect to E if E, {x n } ) = E, {x ni } ) fo all subsequences {x ni } of {x n }, and {x n } is called asymptotically unifom with espect to E if A E, {x n } ) = A E, {x ni } ) fo all subsequences {x ni } of {x n } Futhemoe, {x n } is called egula asymptotically unifom with espect to E if {x n } is egula and asymptotically unifom with espect to E Lemma 3 Let {x n } and E be as above Then the following assetions hold i) Goebel [], Lim [9]) Thee always exists a subsequence of {x n } which is egula with espect to E }

5 FIXED POINTS FOR MULTIVALUED NONEXPANSIVE MAPPINGS 5 ii) Kik [8]) If E is sepaable, then {x n } contains a subsequence which is asymptotically unifom with espect to E Let C be a nonempty bounded subset of X The Chebyshev adius of C elative to E is defined by E C) = inf sup x E y C x y In 006, S Dhompongsa et al [5] intoduced the Domínguez-Loenzo condition DL)-condition, in shot) as follows Definition 4 [5]) A Banach space X is said to satisfy the DL)-condition if thee exists λ [0, ) such that fo evey weakly compact convex subset E of X and fo evey bounded sequence {x n } in E which is egula with espect to E, E A E, {xn } )) λ E, {x n } ) The next esults show that the DL)-condition is stonge than weak nomal stuctue and also implies the existence of fixed points fo multivalued nonexpansive and χ-contactive nonself mappings satisfying the inwadness condition in a Banach space Theoem 5 [5]) Let X be a Banach space satisfying the DL)-condition Then X has weak nomal stuctue Theoem 6 [5]) Let X be a eflexive Banach space satisfying the DL)- condition and suppose that E be a nonempty bounded closed convex sepaable subset of X Assume that T : E KCX) be a nonexpansive and χ-contactive mapping such that T E) is a bounded set which satisfies the inwadness condition: Then T has a fixed point T x I E X) fo all x E Theoem 7 [4, 5]) Let E be a nonempty weakly compact convex subset of a Banach space X which satisfies the DL)-condition and T : E KCE) be a nonexpansive mapping Then T has a fixed point In the sequel, we ecall some basic facts and concepts about ultapowes of Banach spaces which ae the main ingedient of ou esult Ultapowes ae poved to be useful in many banches of mathematics Many esults can be seen moe easily when teated in this setting Fo an in-depth discussion on the Banach space ultapowe constuction, the eade is diected to [3, ] Let F be a filte on N and let X be a Banach space A sequence {x n } in X conveges to x with espect to F, denoted by lim F x i = x, if fo each neighbohood U of x, {i N : x i U} F A filte U on N is called an ulteafilte if it is maximal with espect to set inclusion An ultafilte is called tivial if it is of the fom {A N : i 0 A} fo some fixed i 0 N, othewise, it is called nontivial Let l X) denotes the subspace of the poduct space n N X equipped with the nom x n ) := sup x n < n N

6 6 M DINARVAND Let U be an ultafilte on N and let N U = { x n ) l X) : lim U x n = 0 } The ultapowe of X, denoted by X, is the quotient space l X) N U equipped with the quotient nom Wite x n ) U to denote the elements of the ultapowe It follows fom the definition of the quotient nom that xn ) U = limu x n Note that if U is nontivial, then X can be embedded into X isometically 3 Main esults We ae eady to state and pove ou main esults Theoem 3 Let E be a nonempty weakly compact convex subset of a Banach space X and suppose that {x n } be a bounded sequence in E egula with espect to E Then E A E, {xn } )) R, X) C px) R, X) + E, {x n } ) Poof Fo convenience, we denote = E, {x n } ) and A = A E, {x n } ) We can assume > 0 Since {x n } E is bounded and E is a weakly compact set, we can also assume, by passing though a subsequence if necessay, that {x n } is weakly convegent to a point x E and d := lim n m x n x m exists We note that since {x n } is egula with espect to E, passing though a subsequence does not have any effect to the asymptotic adius of the whole sequence {x n } Since the nom is weak lowe semicontinuity, it follows that lim inf n x n x lim inf n lim inf m x n x m = lim n m x n x m = d Let ε > 0 Taking a subsequence if necessay, we can assume that x n x < d+ε fo all n If z A, then lim sup n x n z = and x z lim inf n x n z Denote R = R, X) Since x n x) w 0 and by using the definition of R, we have R lim inf n x n x + z x = lim inf n x n x x z

7 FIXED POINTS FOR MULTIVALUED NONEXPANSIVE MAPPINGS 7 On the othe hand, obseve that the convexity of E implies that R x+ z R+ R+ E and by the weak lowe semicontinuity of the nom, we have lim inf x n z n + xn x R x z ) ) = lim inf n + x n R) R) + ) x R ) z R ) x + R R z + ) z R = + ) R R R + x + R + z z + ) ) E A), R lim inf x n z n xn x R x z ) ) = lim inf n x n x) R) + ) z x) R + ) z x + ) ) E A) R R Fo evey ε > 0, thee exists N N such that ) x N z + ε ) x N x) x z) R + ε) 3) R Rx N z) + x N x) x z) + ) ) ) E A) ε R 4) R Rx xn x) N z) x z)) + ) ) ) E A) ε R In the ultapowe X of X, we conside xn z ) ũ = B X, ṽ = + ε U R + ε) xn x) ) x z) U By applying the above estimates, we obtain ũ + ṽ = R + ε) Rx N z) + x N x) x z) + ) ) ) E A) ε, R + ε ũ ṽ = R + ε) Rx xn x) N z) x z)) + ) ) ) E A) ε R + ε B X

8 8 M DINARVAND Theefoe, by using the definition of C p X), we have C p X) ) ũ+ṽ p + ũ ṽ p p ) p + E A) R + R ) E A) ) p ) ε +ε ε +ε ) ) p ) p Since the above inequality is tue fo evey ε > 0 and C px) = C p X), we obtain R C px) E A) R + Coollay 3 Let E be a nonempty bounded closed convex sepaable subset of a Banach space X and suppose that T : E KCX) be a nonexpansive and χ-contactive mapping such that T E) is a bounded set which satisfies the inwadness condition: Assume that Then T has a fixed point T x I E X) fo all x E C px) < ) + R,X) Poof Obseve that C px) < since R, X) This implies that X is unifomly nonsquae, and consequently, X is eflexive So evey bounded closed convex set is weakly compact Now, since C px) < + R,X) ), it follows that X satisfies the DL)-condition by Theoem 3 Theefoe, T has a fixed point by Theoem 6 By applying Theoems 7 and 3, we immediately obtain the following esult Coollay 33 Let E be a nonempty bounded closed convex subset of a Banach space X such that ) + C px) R,X) < and T : E KCE) be a nonexpansive mapping Then T has a fixed point

9 FIXED POINTS FOR MULTIVALUED NONEXPANSIVE MAPPINGS 9 Coollay 34 Let X be a Banach space such that ) C px) < + R,X) Then X has nomal stuctue Poof Because X is eflexive, it then implies that nomal stuctue and weak nomal stuctue ae the same Now, since C px) < + R,X) ), it follows that X satisfies the DL)-condition by Theoem 3 Hence, X has nomal stuctue by Theoem 5 Remak 35 Coollaies 3, 33, and 34 ae shap in the sense that thee is a Banach space X such that C px) = + R,X) ) and X does not satisfy the DL)- condition Conside the Bynum space l, defined as l, := l,, ) whee x, := max{ x +, x } with x + i) = max{xi), 0} fo each i and x = x + x We use the computation to conclude that the space l, is a limiting space fo Coollay 3, Coollay 33, and Coollay 34, ie, that coollaies ae shap It is known that Jl, ) = + see [5]) and C NJ l, ) = 3 see [5]) Repeating the aguments in [5], we can get that C NJ l, ) = 3+ 4 By the inequalities J X) C px) C NJ X) see [3]), we have C pl, ) = 3+ p ) It is easy to see that R, l 4, ) = see [9]) Thus, we have C pl, ) = 3 + = + R,l, ) ) 4 and l, fails to have weak nomal stuctue So l, does not satisfy the DL)- condition Theoem 36 Let E be a nonempty weakly compact convex subset of a Banach space X and suppose that {x n } be a bounded sequence in E egula with espect to E Then E A E, {xn } )) R, X) C X) R, X) + E, {x n } ) Poof Let, A, {x n }, x, z and R be as in the poof of Theoem 3 Repeating the aguments in the poof of Theoem 3, fo evey ε > 0, thee exists N N such that ) x N z + ε ) x N x) x z) R + ε) 3) Rx xn x) N z) + x z)) R + ) ) ε E A) 4) Rx xn x) N z) x z)) R + ) ) ε E A)

10 0 M DINARVAND In the ultapowe X of X, we conside ũ = R x N z ), ṽ = xn x) ) x z) U U By applying the above estimates, we obtain ũ R + ε), ṽ R + ε), and ũ + ṽ = Rx N z) + x N x) x z) R + ) ) ε E A), ũ ṽ = Rx xn x) N z) x z)) R + ) ) ε E A) Theefoe, by using the definition of C X), we have C X) min { ũ + ṽ, ũ ṽ } ũ + ṽ ) R + E A) ) ) ε R + ε ) Since the above inequality is tue fo evey ε > 0 and C X) = C X), we obtain E A) R C X) R + Coollay 37 Let E be a nonempty bounded closed convex sepaable subset of a Banach space X and suppose that T : E KCX) be a nonexpansive and χ-contactive mapping such that T E) is a bounded set which satisfies the inwadness condition: Assume that Then T has a fixed point T x I E X) fo all x E C X) < ) + R,X) Poof Obseve that C X) < since R, X) This implies that X is unifomly nonsquae, and consequently, X is eflexive So evey bounded closed convex set is weakly compact Now, since C X) < + R,X) ), it follows that X satisfies the DL)-condition by Theoem 36 Theefoe, T has a fixed point by Theoem 6

11 FIXED POINTS FOR MULTIVALUED NONEXPANSIVE MAPPINGS By vitue of Theoems 7 and 36, we immediately get the following esult Coollay 38 Let E be a nonempty bounded closed convex subset of a Banach space X such that ) + R,X) C X) < and T : E KCE) be a nonexpansive mapping Then T has a fixed point Coollay 39 Let X be a Banach space such that ) C X) < + R,X) Then X has nomal stuctue Poof Because X is eflexive, it then implies that nomal stuctue and weak nomal stuctue ae the same Now, since C X) < + R,X) ), it follows that X satisfies the DL)-condition by Theoem 36 Hence, X has nomal stuctue by Theoem 5 Remak 30 Coollaies 33 and 38 not only impove [5, Coollay 3], but also impove Coollay 36 of [5]; a Banach space X with ) + R,X) C Z X) < satisfies the DL)-condition Remak 3 It is well known that JX)) C px) C p X) Thus, it is easy to see that Theoems 3 and 36 include [, Theoem 3] and Coollaies 33 and 38 include [, Coollay ] Meanwhile, Coollaies 34 and 39 include [0, Coollay 4] Refeences J M Ayebe, T Domínguez Benavides and G López, Measues of Noncompactness in Metic Fixed Point Theoy, Linea Multilinea Algeba, Bikhäuse, 997 M S Bodskĭi and D P Mil man, On the cente of convex sets, Dokl Akad Nauk SSSR NS) ), in Russian) 3 J A Clakson, The von Neumann-Jodan constant fo the Lebesgue spaces, Ann of Math 38 ) 937), S Dhompongsa, T Domínguez Benavides, A Kaewchaoen, A Kaewkhao and B Panyanak, The Jodan-von Neumann constant and fixed points fo multivalued nonexpansive mappings, J Math Anal Appl ), S Dhompongsa, A Kaewchaoen and A Kaewkhao, The Domínguez-Loenzo condition and multivalued nonexpansive mappings, Nonlinea Anal ), M Dinavand, Hölde s means and absolute nomalized noms on R, Filomat, in pess 7 M Dinavand, The James and von Neumann-Jodan type constants and unifom nomal stuctue in Banach spaces, Int J Nonlinea Anal Appl, in pess

12 M DINARVAND 8 M Dinavand, On some Banach space popeties sufficient fo nomal stuctue, Filomat, in pess 9 T Domínguez Benavides, A geometical coefficient implying the fixed point popety and stability esults, Houston J Math 4) 996), T Domínguez Benavides and B Gavia, The fixed point popety fo multivalued nonexpansive mappings, J Math Anal Appl 38 ) 007), B Gavia, Some geometic conditions which imply the fixed point popety fo multivalued nonexpansive mappings, J Math Anal Appl ), K Goebel, On a fixed point theoem fo multivalued nonexpansive mappings, Ann Univ Maiae Cuie-Sklodowska 9 975), K Goebel and W A Kik, Topics in Metic Fixed Point Theoy, Cambidge Univesity Pess, Cambidge, R C James, Unifomly non-squae Banach spaces, Ann of Math 80 ) 964), A Jiménez-Melado, E Lloens-Fuste and S Saejung, The von Neumann-Jodan constant, weak othogonality and nomal stuctue in Banach spaces, Poc Ame Math Soc 34 ) 006), A Kaewkhao, The James constant, the Jodan-von Neumann constant, weak othogonality and fixed points fo multivalued mappings, J Math Anal Appl 333 ) 007), W A Kik, A fixed point theoem fo mappings which do not incease distances, Ame Math Monthly 7 965), W A Kik, Nonexpansive mappings in poduct spaces, set-valued mappings, and k-unifom otundity, in: FE Bowde Ed), Nonlinea Functional Analysis and Its Applications, Ame Math Soc Symp Pue Math ), T C Lim, A fixed point theoem fo multivalued nonexpansive mappings in a unifomly convex Banach space, Bull Ame Math Soc ), E M Mazcuñán-Navao, Banach space popeties sufficient fo nomal stuctue, J Math Anal Appl ), 97 8 S B Nadle J, Multivalued contaction mappings, Pacific J Math ), B Sims, Ulta-Techniques in Banach Space Theoy, Queen s Papes in Pue and Applied Mathematics, Vol 60, Queen s Univesity, Kingston, 98 3 Y Takahashi, Some geometic constants of Banach Spaces-a unified appoach, Poc of nd Intenational Symposium on Banach and Function Spaces II, Yokohama Publishes, Yokohama, 008, pp G Zbăganu, An equality of M Rădulescu and S Rădulescu which chaacteizes the inne poduct spaces, Rev Roumaine Math Pues Appl 47 ) 00), J Zhang and Y Cui, On some geometic constants and the fixed point popety fo multivalued nonexpansive mappings, Fixed Point Theoy Appl 00 00), Aticle ID 59695, 6 Z Zuo, Some fixed point popety fo multivalued nonexpansive mappings in Banach spaces, J Math Inequal 7 ) 03), 9 37 Faculty of Mathematics, K N Toosi Univesity of Technology, PO Box , Tehan, Ian addess: dinavand mina@yahoocom

SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS

Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal