The purpose of this paper is to prove a  fixed point theorem for multivalued operators and a fixed point theorem for  multivalued weakly Picard operators in the terms of τ -distance

Liliana Guran

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Surveys in Mathematics and its ApplicationsISSN 1842-6298 (electronic), 1843-7265 (print)Volume 5 (2010), 191 – 199FIXED POINTS FOR MULTIVALUEDCONTRACTIONS ON A METRIC SPACELiliana GuranAbstract. The purpose of this paper is to prove a ﬁxed point theorem for multivalued operatorsand a ﬁxed point theorem for multivalued weakly Picard operators in the terms of τ-distance.1 IntroductionIn 2001 T. Suzuki (see [14]) introduced the concept of τ-distance on a metricspace which is a generalization of both w-distance and Tataru’s distance. He gavesome examples of τ-distance and improve the generalization of Banach contractionprinciple, Caristi’s ﬁxed point theorem, Ekeland’s variational principle and theTakahashi’s nonconvex minimization theorem, see [14]. Also, some ﬁxed pointtheorems for multivalued operators on a complete metric space endowed with aτ-distance were established in T. Suzuki [15].The concept of multivalued weakly Picard operator (brieﬂy MWP operator) wasintroduced in close connection with the successive approximation method and thedata dependence phenomenon for the ﬁxed point set of multivalued operators oncomplete metric space, by I. A. Rus, A. Petru¸ sel and A. Sˆ ant˘ am˘ arian, see [11].Consider (X,d) be a complete metric space. Let T : X → P(X) be a multivaluedmapping. Deﬁne function f : X → R as f(x) = D(x,T(x)). For a positive constantb ∈ (0,1) deﬁne the set Ixb ⊂ X as:Ixb = {y ∈ T(x) | bd(x,y) ≤ D(x,T(x))}.In 2006 Y. Feng and S. Liu proved the following theorem, see [1].Let (X,d) be a complete metric space T : X → Pcl(X) be a multivalued mapping.If there exists a constant c ∈ (0,1) such that for any x ∈ X there is y ∈ Ixb satisfyingD(y,T(y)) ≤ cd(x,y). Then T has a ﬁxed point in X provided c < b and f is lowersemicontinuous on X.2010 Mathematics Subject Classiﬁcation: 47H10, 54H25.Keywords: Fixed point; τ-distance; Multivalued weakly Picard operator.******************************************************************************http://www.utgjiu.ro/math/sma192 L. GuranIn [2] and [3] we ﬁnd some generalizations of this theorem for a metric spacesendowed with a w-distance.The purpose of this paper is to extend the above ﬁxed point result for multivaluedoperators from [1] in terms of a τ-distance on a complete metric space.2 Notation and basic notionsLet (X,d) be a complete metric space. We will use the following notations:P(X) - the set of all nonempty subsets of X;Pcl(X) - the set of all nonempty closed subsets of X;Pb,cl(X) - the set of all nonempty bounded and closed, subsets of X;D : P(X) × P(X) → R+,D(Z,Y ) = inf{d(x,y) : x ∈ Z ,y ∈ Y }, Z ⊂ X - thegap between two nonempty sets.First we deﬁne the concept of L-space.Deﬁnition 1. Let X be a nonempty set and s(X) := {(xn)n∈N|xn ∈ X,n ∈ N}. Letc(X) ⊂ s(X) a subset of s(X) and Lim : c(X) → X an operator. By deﬁnition thetriple (X,c(X),Lim) is called an L-space if the following conditions are satisﬁed:(i) If xn = x, for all n ∈ N, then (xn)n∈N ∈ c(X) and Lim(xn)n∈N = x.(ii) If (xn)n∈N ∈ c(X) and Lim(xn)n∈N = x, then for all subsequences,(xni)i∈N, of (xn)n∈N we have that (xni)i∈N ∈ c(X) and Lim(xni)i∈N = x.By the deﬁnition an element of c(X) is convergent and x := Lim(xn)n∈N is thelimit of this sequence and we can write xn → x as n → ∞.We will denote an L-space by (X,→).Let us give some examples of L-spaces, see [8].Example 2. (L-structures on Banach spaces) Let X be a Banach space. Wedenote by → the strong convergence in X and by * the weak convergence in X.Then (X,→), (X,*) are L-spaces.Example 3. (L-structures on function spaces) Let X and Y be two metricspaces. Let M(X,Y ) the set of all operators from X to Y . We denote byp→ the pointconvergence on M(X,Y ), byunif→ the uniform convergence and bycont → the convergencewith continuity. Then (M(X,Y ),p→), (M(X,Y ),unif→ ) and (M(X,Y ),cont → ) are L-spaces.Deﬁnition 4. Let (X,→) be an L-space. Then T : X → P(X) is a multivaluedweakly Picard operator (brieﬂy MWP operator) if for each x ∈ X and eachy ∈ T(x) there exists a sequence (xn)n∈N in X such that:1. x0 = x, x1 = y;******************************************************************************Surveys in Mathematics and its Applications 5 (2010), 191 – 199http://www.utgjiu.ro/math/smaFixed points for multivalued contractions on a metric space 1932. xn+1 ∈ T(xn), for all n ∈ N;3. the sequence (xn)n∈N is convergent and its limit is a ﬁxed point of T.Let us give some examples of MWP operators, see [8],[11].Example 5. Let (X,d) be a complete metric space and T : X → Pcl(X) be a Reichtype multivalued operator, i.e. there exists α,β,γ ∈ R+ withα + β + γ < 1 such thatH(T(x),T(y)) ≤ αd(x,y) + βD(x,T(x)) + γD(y,T(y)),for all x,y ∈ X. Then T is a MWP operator.Example 6. Let (X,d) be a complete metric space and T : X → Pcl(X) be aclosed multifunction for which there exists α,β ∈ R+ with α + β < 1 such thatH(T(x),T(y)) ≤ αd(x,y)+βD(y,T(y)), for every x ∈ X and every y ∈ T(x). ThenT is a MWP operator.Example 7. Let (X,d) be a complete metric space and T1,T2 : X → Pcl(X) forwhich there exists α ∈]0, 12[ such thatH(T1(x),T2(y)) ≤ α[D(x,T1(x)) + D(y,T2(y))],for each x,y ∈ X. Then T1 and T2 are a MWP operators.The concept of τ-distance was introduced by T. Suzuki (see[1]) as follows.Deﬁnition 8. Let (X,d) be a metric space, τ : X×X → [0,∞) is called τ−distanceon X if there exists a function η : X × R+ → R+ and the following are satisﬁed :(τ1) τ(x,z) ≤ τ(x,y) + τ(y,z), for any x,y,z ∈ X;(τ2) η(x,0) = 0 and η(x,t) ≥ t for all x ∈ X and t ∈ R+, and η is concave andcontinuous in its the second variable;(τ3) limn xn = x and limn sup{η(zn,τ(zn,xm)) : m ≥ n} = 0 imply τ(w,x) ≤liminfn(τ(w,xn)) for all w ∈ X;(τ4) limn sup{τ(xn,ym)) : m ≥ n} = 0 and limn η(xn,tn) = 0 imply limn η(yn,tn) =0;(τ5) limn η(zn,τ(zn,xn)) = 0 and limn η(zn,τ(zn,yn)) = 0 imply limn d(xn,yn) =0.We may replace (τ2) by the following (τ2)0:(τ2)0 inf{η(x,t) : t > 0} = 0 for all x ∈ X, and η is nondecreasing in the secondvariable.Let us give some examples of τ-distance (see[14]).Example 9. Let (X,d) be a metric space. Then the metric ”d” is a τ-distance onX.******************************************************************************Surveys in Mathematics and its Applications 5 (2010), 191 – 199http://www.utgjiu.ro/math/sma194 L. GuranExample 10. Let (X,d) be a metric space and w : X × X → R+ be a w-distanceon X. Then w is also a τ-distance on X.Example 11. Let (X,d) be a metric space and τ be a τ-distance on X, leth : R+ → R+ be a nondecreasing function such thatR ∞011+h(r)dr = ∞, and letz0 ∈ X be ﬁxed. Then a function q : X × X → R+ deﬁned by:q(x,y) =Z τ(z0,x)+τ(x,y)τ(z0,x)dr1 + h(r), for all x,y ∈ Xis a τ-distance.For the proof of the main result we need of the deﬁnition of the τ-Cauchysequence and the following lemmas (see [15]).Deﬁnition 12. Let (X,d) be a metric space and let τ be a τ-distance on X. Then asequence {xn} in X is called τ −Cauchy if there exists a function η : X ×[0,∞) →[0,∞) satisfying (τ2)-(τ5) and a sequence {zn} in X such that limn sup{η(zn,τ(zn,xm)) :m ≥ n} = 0.A crucial results in order to obtain ﬁxed point theorems by using τ-distance arethe following lemmas.Lemma 13. Let (X,d) be a metric space and let τ be a τ-distance on X. If asequence {xn} in X satisﬁes limn sup{τ(xn,xm) : m > n} = 0, then {xn} is a τ-Cauchy sequence. Moreover, if a sequence {yn} in X satisﬁes limn τ(xn,yn) = 0,then {yn} is also a τ-Cauchy sequence and limn d(xn,yn) = 0.Lemma 14. Let (X,d) be a metric space and let τ be a τ-distance on X. If asequence {xn} in X satisﬁes limn τ(z,xn) = 0 for z ∈ X then {xn} is a τ-Cauchysequence. Moreover, if a sequence {yn} in X also satisﬁes limn τ(z,yn) = 0, thenlimn d(xn,yn) = 0. In particular, for x,y,z ∈ X, τ(z,x) = 0 and τ(z,y) = 0 implyx = y.Lemma 15. Let (X,d) be a metric space and let τ be a τ-distance on X. If {xn} is aτ-Cauchy sequence, then {xn} is a Cauchy sequence. Moreover, if {yn} is a sequencesatisfying limn sup{τ(xn,ym) : m > n} = 0, then {yn} is a τ-Cauchy sequence andlimn d(xn,yn) = 0.3 Main resultsDeﬁnition 16. Let T : X → P(X) be a multivalued operator, τ : X × X → [0,∞)be a τ-distance on X. Deﬁne the function f : X → R as f(x) = Dτ(x,T(x)), whereDτ(x,T(x)) = inf{τ(x,y) | y ∈ T(x)}.For a positive constant b ∈ (0,1) deﬁne the set Ixb,τ ⊂ X as follows:Ixb,τ = {y ∈ T(x) | bτ(x,y) ≤ Dτ(x,T(x))}.******************************************************************************Surveys in Mathematics and its Applications 5 (2010), 191 – 199http://www.utgjiu.ro/math/smaFixed points for multivalued contractions on a metric space 195We will present now a ﬁxed point theorem for multivalued operators on a completemetric space endowed with a τ-distance.Theorem 17. Let (X,d) be a complete metric space, T : X → Pcl(X) a multivaluedoperator, τ : X × X → [0,∞) be a τ-distance on X and b ∈ (0,1).Suppose that(i) there exists c ∈ (0,b), such that for any x ∈ X there is y ∈ Ixb,τ satisfyingDτ(y,T(y)) ≤ cτ(x,y);(ii) f is lower semicontinuous, where f is previous deﬁned.Then T has a ﬁxed point in X. Moreover, if T(z) = z, then τ(z,z) = 0.Proof. For any initial point x0 ∈ X, there is x1 ∈ Ix0b,τ such that:Dτ(x1,T(x1)) ≤ cτ(x0,x1).For any x1 ∈ X there is x2 ∈ Ix1b,τ such that:Dτ(x2,T(x2)) ≤ cτ(x1,x2).We obtain an iterative sequence {xn}∞n=0 where xn+1 ∈ Ixnb,τ andDτ(xn+1,T(xn+1)) ≤ cτ(xn,xn+1),forn = 0,1,2,....We will verify that {xn}∞n=0 is a Cauchy sequence.Indeed:Dτ(xn+1,T(xn+1)) ≤ cτ(xn,xn+1),n = 0,1,2,... (3.1)Since xn+1 ∈ Ixnb,τ we obtain:bτ(xn,xn+1) ≤ Dτ(xn,T(xn)),n = 0,1,2,... (3.2)By (3.2) it follows : τ(xn,xn+1) ≤ 1bDτ(xn,T(xn)), n = 0,1,2,...Using (3.1)we obtain :Dτ(xn+1,T(xn+1)) ≤ c1bDτ(xn,T(xn)),n = 0,1,2,... (3.3)By (3.1) we have :Dτ(xn,T(xn)) ≤ cτ(xn−1,xn),n = 0,1,2,...Dτ(xn+1,T(xn+1)) ≤ cτ(xn,xn+1),n = 0,1,2,...We replace in (3.3) and we obtain :cτ(xn,xn+1) ≤cbcτ(xn−1,xn),n = 0,1,2,...******************************************************************************Surveys in Mathematics and its Applications 5 (2010), 191 – 199http://www.utgjiu.ro/math/sma196 L. GuranIf we divide by c we obtain :τ(xn,xn+1) ≤cbτ(xn−1,xn),n = 0,1,2,...then:τ(xn+1,xn+2) ≤cbτ(xn,xn+1),n = 0,1,2,... (3.4)By using an induction argument it is easy to see that:τ(xn,xn+1) ≤cnbnτ(x0,x1),n = 0,1,2,...Dτ(xn,T(xn)) ≤cnbnDτ(x0,T(x0)),n = 0,1,2,...Then for m,n ∈ N, m > n, a = cb and using the previous inequalities we have:τ(xn,xm) ≤ τ(xn,xn+1) + τ(xn+1,xn+2) + ··· + τ(xm−1,xm) ≤≤ anτ(x0,x1) + an+1τ(x0,x1) + ··· + am−1τ(x0,x1) ≤≤an1 − aτ(x0,x1)For n → ∞ we have an → 0, because a = cb with c < b. Thuslimn→∞τ(xn,xm) = 0, with m > n.Using Lemma 13 it follows that the sequence {xn}∞n=0 is a τ-Cauchy sequence andfrom the Lemma 15 result that this sequence is a Cauchy sequence. Since X is acomplete space, then the sequence {xn}∞n=0 converge to some point z ∈ X. Thesequence{f(xn)}∞n=0 = {Dτ(xn,T(xn))}∞n=0is decreasing and from the above construction it converges to 0.Since f is lower semicontinuous we have:0 ≤ f(z) ≤ limn→∞f(xn) = 0Thus f(z) = 0.Since f(z) = 0, then there exists a sequence (wn) ∈ T(z) such that τ(z,wn) → 0.Therefore,0 ≤ limnsup{τ(xn,wm) : m > n} ≤ limnsup{τ(xn,z) + τ(z,wm) : m > n} = 0.Thus, by Lemma 15 we obtain that wn → z and since T(z) is a closed set we concludethatz ∈ T(z).Since T(z) = z, hence Dτ(z,T(z)) = τ(z,z) ≥ 0 and since by hypothesis Izb,τ 6= ∅we know that bτ(z,z) ≤ τ(z,z) but 0 < b < 1, therefore τ(z,z) = 0.******************************************************************************Surveys in Mathematics and its Applications 5 (2010), 191 – 199http://www.utgjiu.ro/math/smaFixed points for multivalued contractions on a metric space 197The second main result of this paper is a ﬁxed point theorem for MWP operatorson metric spaces endowed with a τ-distance.Corollary 18. Let (X,d) be a complete metric space, T : X → Pcl(X) a multivaluedoperator, τ : X × X → [0,∞) be a τ-distance on X and b ∈ (0,1). Suppose thatthere exists c ∈ (0,1), with c < b, such that for any x ∈ X there is y ∈ Ixb,τ satisfyingDτ(x,T(x)) ≤ cτ(x,y). Then T is a MWP operator.Proof. As in the proof of the previous theorem we construct inductively a sequence{xn}∞n=0 such that:1. xn+1 ∈ T(xn), for every n ∈ N;2. τ(xn,xn+1) ≤ cnbnτ(x0,x1), for every n ∈ N;3. Dτ(xn,T(xn)) ≤ cnbnDτ(x0,T(x0)), for every n ∈ N.Thus x0 = x, x1 = y and xn+1 ∈ T(xn). For m,n ∈ N with m > n and a = cb wehave the inequalityτ(xn,xm) ≤an1 − aτ(x0,x1).Using Lemma 13 it follows that the sequence {xn}∞n=0 is a τ-Cauchy sequence. FromLemma 15 we have that the sequence {xn}∞n=0 is a Cauchy sequence. Since (X,d)is a complete metric space then there exists z ∈ X such that xnd → z.For n ∈ N, from (τ3) we get that:τ(xn,z) ≤ limm→∞inf τ(xn,xm) ≤an1 − aτ(x0,x1).By hypothesis, for a = cb, we also have wn ∈ T(z) such thatτ(xn,wn) ≤ aτ(xn−1,z) for every n ∈ N.So, we havelimn→∞supτ(xn,wn) ≤ limn→∞supaτ(xn−1,z) ≤ limn→∞an1 − aτ(x0,x1) = 0.By Lemma 13 result that wn converge to z. Since T(z) is closed we obtain thatz ∈ T(z). Then T is MWP operator.******************************************************************************Surveys in Mathematics and its Applications 5 (2010), 191 – 199http://www.utgjiu.ro/math/sma198 L. GuranReferences[1] Y. Feng and S. Liu, Fixed point theorems for multivalued contractive mappingsand multi-valued Caristi type mappinggs, J. Math. Anal. Appl. 317 (2006),103-112.[2] J. Garcia-Falset, L. Guran and E. Llorens-Fuster, Fixed points for multivaluedcontractions with respect to a w-distance, Scientiae Math. Japonicae, 71, 1(2010), 83-91. Zbl pre05689237.[3] L. 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Petru¸ sel, Fixed point theory 1950-2000:Romanian contributions, House of the Book of Science, Cluj-Napoca, 2002.MR1947195(2003h:47104). Zbl 1171.54034.[14] T. Suzuki, Generalized Distance and Existence Theorems in Complete MetricSpaces, J. Math. Anal. Appl. 253 (2001), 440-458. MR1808147(2002f:49038).Zbl 0983.54034.[15] T. Suzuki, Several Fixed Point Theorems Concerning τ-distance, Fixed PointTheory and Application, 3 (2004), 195-209. MR2096951. Zbl 1076.54532.Liliana GuranDepartment of Accounting and Managerial Information System,Faculty of Economic Sciences, Titu Maiorescu University,Calea V˘ ac˘ are¸ sti, nr. 189, 040056, sector 4, Bucharest, Romania.e-mail: gliliana.math@gmail.com******************************************************************************Surveys in Mathematics and its Applications 5 (2010), 191 – 199http://www.utgjiu.ro/math/sma

Fixed points for multivalued contractions on a metric space

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