The purpose of this paper is to present a partial answer to the 
following problem:
Let (X,d) be a metric space and $T_1,T_2:Xto P(X)$ two 
multivalued operators. Determine the metric conditions which 
imply that (T_1,T_2) is a weakly Picard pair of multivalued operators and T_1, T_2 are weakly Picard multivalued operators

A. Sîntărian

The purpose of this paper is to present a partial answer to the 

following problem:

Let (X,d) be a metric space and $T_1,T_2:X\to P(X)$ two 

multivalued operators. Determine the metric conditions which 

imply that (T_1,T_2) is a weakly Picard pair of multivalued operators and T_1, T_2 are weakly Picard multivalued operators

Sîntărian, A.

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Mathematical Communications 8(2003), 49-53 49Weakly Picard pairs of some multivalued operatorsAlina Sˆınta˘ma˘rian∗Abstract. The purpose of this paper is to present a partial answerto the following problem:Let (X, d) be a metric space and T1, T2 : X → P (X) two multivaluedoperators. Determine the metric conditions which imply that (T1, T2)is a weakly Picard pair of multivalued operators and T1, T2 are weaklyPicard multivalued operators.Key words: fixed point, common fixed point, weakly Picard multi-valued operator, weakly Picard pair of multivalued operatorsAMS subject classifications: 47H10, 54H25Received February 11, 2003 Accepted April 2, 20031. IntroductionLet X be a nonempty set. We denote by P (X) the set of all nonempty subsets ofX , i. e. P (X) := { Y | ∅ = Y ⊆ X }.Let T1, T2 : X → P (X) be two multivalued operators. We denote by GT1 thegraph of T1, i. e. GT1 := { (x, y) | x ∈ X, y ∈ T1(x) }, by FT1 the fixed pointsset of T1, i. e. FT1 := { x ∈ X | x ∈ T1(x) } and by (CF )T1,T2 the common fixedpoints set of T1 and T2.Let (X, d) be a metric space. Further on we shall need the following notationPcl(X) := { Y | Y ∈ P (X) and Y is a closed set }and the following functionalsD : P (X)× P (X)→ R+, D(A,B) = inf { d(a, b) | a ∈ A, b ∈ B },H : P (X)×P (X)→ R+ ∪{+∞}, H(A,B) = max{supa∈AD(a,B), supb∈BD(b, A)}.Definition 1 [[6], [7]]. Let (X, d) be a metric space and T : X → P (X) amultivalued operator. We say that T is a weakly Picard multivalued operator (brieflyw. P. m. o.) iff for each x ∈ X and for every y ∈ T (x), there exists a sequence(xn)n∈N such that :∗Department of Mathematics, Technical University of Cluj-Napoca, Str. C. Daicoviciu Nr. 15,3400 Cluj-Napoca, Romania, e-mail: Alina.Sintamarian@math.utcluj.ro50 A. Sˆınta˘ma˘rian(i) x0 = x, x1 = y;(ii) xn+1 ∈ T (xn), for each n ∈ N∗;(iii) sequence (xn)n∈N is convergent and its limit is a fixed point of T .Remark 1. A sequence (xn)n∈N which satisfies conditions (i) and (ii) fromDefinition 1 is, by definition, a sequence of successive approximations of T , startingfrom (x, y).For examples of w. P. m. o. see for instance [6], [7].Definition 2 [[7]]. Let (X, d) be a metric space and T : X → P (X) a w. P.m. o.. Then we define the multivalued operator T∞ : GT → P (FT ) by the formulaT∞(x, y) = { z ∈ FT | there exists a sequence of successive approximations of T ,starting from (x, y), that converges to z }, for each (x, y) ∈ GT .Definition 3 [[7]]. Let (X, d) be a metric space and T : X → P (X) a w. P.m. o.. Then T is a c-weakly Picard multivalued operator (c ∈ [0,+∞[) (briefly c-w.P. m. o.) iff there exists a selection t∞ of T∞ such thatd(x, t∞(x, y)) ≤ c d(x, y),for each (x, y) ∈ GT .Examples of c-w. P. m. o. are given in [7].Definition 4 [[10]]. Let (X, d) be a metric space and T1, T2 : X → P (X) twomultivalued operators. By definition, we say that the pair of multivalued operators(T1, T2) is a weakly Picard pair of multivalued operators (briefly w. P. p. m. o.) ifffor each x ∈ X and for every y ∈ T1(x) ∪ T2(x), there exists a sequence (xn)n∈Nsuch that :(i) x0 = x, x1 = y;(ii) x2n−1 ∈ Ti(x2n−2) and x2n ∈ Tj(x2n−1), for each n ∈ N∗, where i, j ∈ {1, 2},i = j;(iii) sequence (xn)n∈N is convergent and its limit is a common fixed point of T1and T2.Remark 2. A sequence (xn)n∈N which satisfies conditions (i) and (ii) fromDefinition 4 is a sequence of successive approximations for the pair (T1, T2), startingfrom (x, y).For examples of w. P. p. m. o. see [10].Definition 5 [[10]]. Let (X, d) be a metric space and T1, T2 : X → P (X) twomultivalued operators which form a w. P. p. m. o.. Then we define the multivaluedoperator (T1, T2)∞ : GT1 ∪ GT2 → P ((CF )T1,T2) by the formula (T1, T2)∞(x, y) ={ z ∈ (CF )T1,T2 | there exists a sequence of successive approximations for the pair(T1, T2), starting from (x, y), that converges to z }, for each (x, y) ∈ GT1 ∪GT2 .Definition 6 [[10]]. Let (X, d) be a metric space and T1, T2 : X → P (X) twomultivalued operators which form a w. P. p. m. o.. Then (T1, T2) is a c-weaklyPicard pair of multivalued operators (c ∈ [0,+∞[) (briefly c-w. P. p. m. o.) iff thereexists a selection (t1, t2)∞ of (T1, T2)∞ such thatd(x, (t1, t2)∞(x, y)) ≤ c d(x, y),Weakly Picard pairs of some multivalued operators 51for each (x, y) ∈ GT1 ∪GT2 .Examples of c-w. P. p. m. o. are given in [10].The purpose of this paper is to study the following problem.Problem 1. Let (X, d) be a metric space and T1, T2 : X → P (X) two mul-tivalued operators. Determine the metric conditions which imply that (T1, T2) is aweakly Picard pair of multivalued operators and T1, T2 are weakly Picard multivaluedoperators.2. Weakly Picard pairs of some multivalued operatorsThe following theorem was established by Sˆınta˘ma˘rian in [10] and it is a partialanswer to Problem 1.Theorem 1 [[10]]. Let (X, d) be a complete metric space and T1, T2 : X →Pcl(X) two multivalued operators for which there exists a ∈ [0, 1/2[ such thatH(T1(x), T2(y)) ≤ a [D(x, T1(x)) +D(y, T2(y))],for each x, y ∈ X.Then FT1 = FT2 ∈ Pcl(X), (T1, T2) is c-w. P. p. m. o. and T1 and T2 are c-w.P. m. o., with c = (1− a)/(1− 2a).Another partial answer to Problem 1 is the following result.Theorem 2. Let (X, d) be a complete metric space and T1, T2 : X → Pcl(X)two multivalued operators. We suppose that :(i) there exists a1 ∈ [0, 1/2[ such that for each x ∈ X, any ux ∈ T1(x) and for ally ∈ X, there exists uy ∈ T2(y) so thatd(ux, uy) ≤ a1 [d(x, ux) + d(y, uy)];(ii) there exists a2 ∈ [0, 1/2[ such that for each x ∈ X, any ux ∈ T2(x) and for ally ∈ X, there exists uy ∈ T1(y) so thatd(ux, uy) ≤ a2 [d(x, ux) + d(y, uy)].Then FT1 = FT2 ∈ Pcl(X) and (T1, T2) is c-w. P. p. m. o., with c = (1−a)/(1−2a),where a = max {a1, a2}.If in addition we have that 2max {a1, a2} + min {a1, a2} < 1, then Ti is ci-w.P. m. o., with ci = (1 − a1)(1− a2)/(1− 2ai − aj), i, j ∈ {1, 2}, i = j.Proof. First of all, we notice that from Theorem4.2 given by Latif-Beg in [1] itfollows that (CF )T1,T2 = ∅.From Theorem2.2 given by Sˆınta˘ma˘rian in [8] we have that FT1 = FT2 ∈ Pcl(X)and the fact that (T1, T2) is c-w. P. p. m. o. follows from Theorem2.7 given bySˆınta˘ma˘rian in [10].Furthermore, we suppose that 2max {a1, a2} + min {a1, a2} < 1 and we shallprove that Ti is ci-w. P. m. o., i ∈ {1, 2}.Let i, j ∈ {1, 2}, i = j. Let x0 ∈ X and x1 ∈ Ti(x0). It follows that there existsy1 ∈ Tj(x1) such thatd(x1, y1) ≤ ai [d(x0, x1) + d(x1, y1)]52 A. Sˆınta˘ma˘rianand there exists x2 ∈ Ti(x1) such thatd(y1, x2) ≤ aj [d(x1, y1) + d(x1, x2)].From these, using the triangle inequality, we obtaind(x1, x2) ≤ d(x1, y1) + d(y1, x2)≤ d(x1, y1) + aj [d(x1, y1) + d(x1, x2)]= (1 + aj) d(x1, y1) + aj d(x1, x2)≤ (1 + aj)ai/(1− ai) d(x0, x1) + aj d(x1, x2).Sod(x1, x2) ≤ ai(1 + aj)/[(1− ai)(1 − aj)] d(x0, x1).Now, there exists y2 ∈ Tj(x2) such thatd(x2, y2) ≤ ai [d(x1, x2) + d(x2, y2)]and there exists x3 ∈ Ti(x2) such thatd(y2, x3) ≤ aj [d(x2, y2) + d(x2, x3)].From these we have thatd(x2, x3) ≤ ai(1 + aj)/[(1− ai)(1 − aj)] d(x1, x2).By induction, we obtain that there exists a sequence (xn)n∈N of successive approx-imations of Ti, starting from (x0, x1), with the property thatd(xn, xn+1) ≤ ai(1 + aj)/[(1− ai)(1− aj)] d(xn−1, xn),for each n ∈ N∗.It follows that (xn)n∈N is a convergent sequence, because (X, d) is a completemetric space and ai(1 + aj)/[(1− ai)(1 − aj)] < 1. Let x∗ = limn→∞ xn.From xn ∈ Ti(xn−1) we have that there exists un ∈ Tj(x∗) such thatd(xn, un) ≤ ai [d(xn−1, xn) + d(x∗, un)],for all n ∈ N∗.Using the triangle inequality we obtaind(x∗, un) ≤ (1 − ai)−1 [d(x∗, xn) + ai d(xn−1, xn)],for all n ∈ N∗.This implies that d(x∗, un) → 0, as n → ∞. Since un ∈ Tj(x∗), for all n ∈ N∗and Tj(x∗) is a closed set, it follows that x∗ ∈ Tj(x∗). So x∗ ∈ FTj = FTi .It is not difficult to verify thatd(xn, x∗) ≤ [ai(1+aj)(1−ai)−1(1−aj)−1]n(1−ai)(1−aj)/(1−2ai−aj) d(x0, x1),for each n ∈ N.For n = 0 we haved(x0, x∗) ≤ (1− ai)(1− aj)/(1− 2ai − aj) d(x0, x1),which means that Ti is ci-w. P. m. o., with ci = (1− ai)(1− aj)/(1− 2ai − aj). ✷Weakly Picard pairs of some multivalued operators 53References[1] A.Latif, I. Beg, Geometric fixed points for single and multivalued mappings,Demonstratio Math. 30(1997), 791-800.[2] I. A.Rus, Approximation of common fixed point in a generalized metric space,Anal. Nume´r. The´or. Approx. 8(1979), 83-87.[3] I. A.Rus, Basic problems of the metric fixed point theory revisited (II), StudiaUniv. Babes¸-Bolyai, Mathematica 36(1991), 81-99.[4] I. A.Rus, Picard operators and applications, Seminar on Fixed Point Theory,“Babes¸-Bolyai” Univ., Preprint Nr. 3(1996).[5] I. A.Rus, Generalized Contractions and Applications, Cluj University Press,Cluj-Napoca, 2001.[6] I. A.Rus, A. Petrus¸el, A. Sˆinta˘ma˘rian, Data dependence of the fixedpoints set of multivalued weakly Picard operators, Studia Univ. Babes¸-Bolyai,Mathematica, 46(2001), 111-121.[7] I. A.Rus, A. Petrus¸el, A. Sˆinta˘ma˘rian, Data dependence of the fixed pointset of some multivalued weakly Picard operators, Nonlinear Analysis TMA52(2003), 1947-1959.[8] A. Sˆinta˘ma˘rian, Common fixed point theorems for multivalued mappings, Se-minar on Fixed Point Theory Cluj-Napoca, 1(2000), 93-102.[9] A. Sˆinta˘ma˘rian, Picard pairs and weakly Picard pairs of operators, StudiaUniv. Babes¸-Bolyai, Mathematica, 47(2002), 89-103.[10] A. Sˆinta˘ma˘rian, Weakly Picard pairs of multivalued operators, Mathematica,Tome 43(2003), to appear.

Weakly Picard pairs of some multivalued operators

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