The aim of this note is to point out an error in the proof of Theorem 1 in the paper entitled \u201cCoincidence theorems for some multivalued mappings\u201d by B. E. Rhoades, S. L. Singh and Chitra Kulshrestha [Internat. J. Math. & Math. Sci., 7 (1984), 429-434], and to indicate a way to repair it

Samet, B.

Vetro, C.

The aim of this note is to point out an error in the proof of Theorem 1 in the paper entitled “Coincidence theorems for some multivalued mappings” by B. E. Rhoades, S. L. Singh and Chitra Kulshrestha [Internat. J. Math. & Math. Sci., 7 (1984), 429-434], and to indicate a way to repair it

Samet, B

VETRO, Calogero

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F A S C I C U L I M A T H E M A T I C INr 47 2011Bessem Samet and Calogero Vetro ∗COMMENTS ON THE PAPER ”COINCIDENCETHEOREMS FOR SOME MULTIVALUED MAPPINGS”BY B. E. RHOADES, S. L. SINGH ANDCHITRA KULSHRESTHAAbstract. The aim of this note is to point out an error in theproof of Theorem 1 in the paper entitled ”Coincidence theoremsfor some multivalued mappings” by B. E. Rhoades, S. L. Singhand Chitra Kulshrestha [Internat. J. Math.& Math. Sci., 7(1984),429-434], and to indicate a way to repair it.Key words: coincidence point, multivalued mapping, Hausdorffdistance.AMS Mathematics Subject Classification: 47H10; 54H25.1. PreliminariesThe following definitions and the notations are the same as in [1]:Let (X, d) be a metric space. We denote by CL(X) the collection ofnon-empty closed subsets of X. Let A ∈ CL(X), we defineN(ε,A) = {x ∈ X : d(x, a) < ε for some a ∈ A, ε > 0},H(A,B) =inf{ε > 0 : A ⊆ N(ε,B)and B ⊆ N(ε,A)} if the infimum exists,∞ otherwise,for each A,B ∈ CL(X). H is called the generalized Hausdorff distancefunction for CL(X) induced by d. D(x,A) will denote the ordinary distancebetween x ∈ X and A, a non-empty subset of X.Let f : X → X and T : X → CL(X).The notation TX ⊆ fX means that Tx ⊆ fX for each x ∈ X.Definition 1. A point z ∈ X is said to be a coincidence point of f andT if fz ∈ Tz. We define C(f, T ) := {z ∈ X : fz ∈ Tz}.∗ The second author is supported by University of Palermo, Local University ProjectR. S. ex 60 %.90 Bessem Samet and Calogero VetroDefinition 2. If for a point x0 ∈ X there exists a sequence {xn} ⊂ Xsuch that fxn+1 ∈ Txn, n = 0, 1, 2, · · · , thenOf (x0) = {fxn : n = 1, 2, · · · }is an orbit for (T, f) at x0.Definition 3. If for x0 ∈ X there exists a sequence {xn} ⊂ X such thatfxn+1 ∈ Txn, n = 0, 1, 2, · · · and every Cauchy sequence of the form {fxni}converges in X, then X is called (T, f)-orbitally complete with respect to x0or simply (T, f, x0)-orbitally complete.Definition 4. T is said to be asymptotically regular at x0 ∈ X if, foreach sequence {xn} ⊂ X, xn ∈ Txn−1, limn→+∞ d(xn, xn+1) = 0.LetΨ = {φ : R+ → R+ such thatφ is upper semicontinuous and nondecreasing}.In this note, we focus on the following theorem of [1]:Theorem 1. Let T be a multi-valued mapping from a metric space (X, d)to CL(X) and φ ∈ Ψ. If there exists a mapping f : X → X such thatTX ⊆ fX, and for each x, y ∈ X,(1) H(Tx, Ty) ≤ φ(max{D(fx, Tx), D(fy, Ty), D(fx, Ty),D(fy, Tx), d(fx, fy)}),(2) φ(t) < qt for each t > 0, for some fixed 0 < q < 1,(3) there exists an x0 ∈ X such that T is asymptotically regular at x0,(4) X is (T, f, x0)-orbitally complete,then T and f have a coincidence point.We point out an error in the proof of Theorem 1 and indicate a way torepair it.2. Discussions and resultsIn the proof of Theorem 1, in order to prove the Cauchy character of thesequence {yn}, the constructive schema of the sequence is abusively used.To be precise, it is not true thatd(y2n(k)+1, y2m(k)) ≤ q−1H(Tx2n(k), Tx2m(k)−1),where y2n(k)+1 = fx2n(k)+1 ∈ Tx2n(k) and y2m(k) = fx2m(k) ∈ Tx2m(k)−1.In fact, if A,B ∈ CL(X) and a ∈ A, the inequalityd(a, b) ≤ q−1H(A,B)Comments on the paper ”Coincidence . . . 91is incorrect for arbitrary b ∈ B. As a counterexample, let X = R endowedwith the usual metric d(x, y) = |x − y| for all x, y ∈ R. Let A = [0, 9],B = [2, 10], a = 1 ∈ A and b = 9 ∈ B. Then,d(a, b) = 8 > q−1 · 2 = q−1H(A,B),for any 1/4 < q < 1.Certainly, the reader can explore the literature to find many papers withthe same mistake, but this enumeration is not the aim of the present note.As we have not found yet any counterexample to Theorem 1 or a newproof, we do not know whether it (and the related results in the literature)are true or not. Until a better solution is found, we suggest a correction byreplacing conditions (3) and (4) with TX is a closed subset of CL(X)” andcondition (0) below:(0) f and T satisfy the property (E.A).Definition 5 ([2]). Two mappings f : X → X and T : X → CL(X) aresaid to satisfy the property (E.A) if there exist a sequence {xn} ⊂ X, somet ∈ X and A ∈ CL(X) such thatlimn→+∞ fxn = t ∈ A = limn→+∞Txn.Example 1. Let X = [1,+∞) endowed with the usual metric d(x, y) =| x−y | for every x, y ∈ X. Clearly (X, d) is a complete metric space. Definef : X → X as fx = x2 and T : X → CL(X) as Tx = [1, 3 + x], for allx ∈ X. Now, for the sequence {xn} = {1 + 1/n}, we have:limn→+∞ fxn = 1 ∈ [1, 4] = limn→+∞Txn.Therefore f and T satisfy property (E.A).Remark 1. By virtue of the property (E.A), the proof of Theorem 1with condition (0) can be concluded following the lines in Theorem 3.4 of[2].Then, we give the proof of the following theorem.Theorem 2. Let T be a multi-valued mapping from a metric space X toCL(X). We assume that there exist φ ∈ Ψ and a mapping f : X → X suchthat TX ⊆ fX, and for each x, y ∈ X,(0) f and T satisfy the property (E.A),(1) H(Tx, Ty) ≤ φ(max{D(fx, Tx), D(fy, Ty), D(fx, Ty),D(fy, Tx), d(fx, fy)}),(2) φ(t) < qt for each t > 0, for some fixed 0 < q < 1.If TX is a closed subset of CL(X), then T and f have a coincidence point.92 Bessem Samet and Calogero VetroProof. By virtue of the property (E.A), there exist a sequence {xn} ⊂ X,some t ∈ X and A ∈ CL(X) such that limn→+∞ fxn = t ∈ A = limn→+∞Txn.Since TX is a closed subset of CL(X) and TX ⊆ fX, we have limn→+∞ fxn =fz for some z ∈ X. Thus t = fz ∈ A. We claim that fz ∈ Tz. If not, thencondition (1) impliesH(Txn, T z) ≤ φ(max{D(fxn, Txn), D(fz, Tz), D(fxn, T z),D(fz, Txn), d(fxn, fz)}).Taking the limit as n→ +∞, we obtainH(A, Tz) ≤ φ(max{D(fz,A), D(fz, Tz), D(fz, Tz),D(fz,A), d(fz, fz)}).Since φ(t) < qt for each t > 0, we getH(A, Tz) ≤ φ(D(fz, Tz)) < qD(fz, Tz).Now, by the definition of Hausdorff distance function, since fz ∈ A, itfollows thatD(fz, Tz) < qD(fz, Tz),which is a contradiction as 0 < q < 1. Therefore fz ∈ Tz. Remark 2. The conclusion of Theorem 2 remains true if we assume thatfX is closed instead of TX. In this case, we can remove that TX ⊆ fX.Example 2. Consider X = [1,∞) equipped with the usual metric d(x, y)=| x − y | for every x, y ∈ X. Clearly (X, d) is a complete metric space.Define f : X → X as fx = x2, T : X → CL(X) as Tx = [1, 1 + x/4],q = 2/3 and φ(t) = t/2 for each t > 0.Now, we have:H(Tx, Ty) =14| x− y |≤ 12| x2 − y2 |= φ(d(fx, fy)).Then, all the conditions of Theorem 2 are easily verified. Therefore f1 ∈ T1,that is, f and T have a coincidence point.By definition, an element x ∈ X is said to be a common fixed point of fand T iff x = fx ∈ Tx. Moreover, as 1 = f1 ∈ T1, then x = 1 is commonfixed point of f and T .We recall the following notion [2].Two mappings f : X → X and T : X → CL(X) are noncompatibleif fTx ∈ CL(X) for all x ∈ X and there exists at least one sequence{xn} ⊂ X such that limn→+∞Txn = A ∈ CL(X) and limn→+∞ fxn = t ∈ A butlimn→+∞H(fTxn, T fxn) 6= 0 or the limit does not exist.Comments on the paper ”Coincidence . . . 93Remark 3. It is immediate to see that if f and T are noncompatible,then f and T satisfy the property (E.A).We state the following corollary.Corollary 1. Let T be a multi-valued mapping from a metric space Xto CL(X). We assume that there exists a mapping f : X → X such that,for each x, y ∈ X,(0) f and T are noncompatible,(1) H(Tx, Ty) ≤ φ(max{D(fx, Tx), D(fy, Ty), D(fx, Ty),D(fy, Tx), d(fx, fy)}),(2) φ(t) < qt for each t > 0, for some fixed 0 < q < 1, φ ∈ Ψ.If fX is a closed subset of X, then T and f have a coincidence point.Finally, motivated by [3], we state and prove the following theorem.Theorem 3. Let T be a multivalued mapping from a metric space X toCL(X). We assume that there exists a mapping f : X → X such that, foreach x, y ∈ X,(0) f and T satisfy the property (E.A),(1) H(Tx, Ty) ≤ φ(max{D(fx, Tx), D(fy, Ty), D(fx, Ty),D(fy, Tx), d(fx, fy)}),(2) φ(t) < qt for each t > 0, for some fixed 0 < q < 1, φ ∈ Ψ.If fX is a closed subset of X, then T and f have a common fixed point,provided that ffz = fz, for each z ∈ C(f, T ).Proof. Essentially the same reasoning as in Theorem 2, keeping in mindRemark 2, establishes t = fz ∈ Tz. From this and ffz = fz for z ∈ C(f, T ),we have t = fz = ffz = ft ∈ Tz. We claim that z = t. If not, thencondition (1) impliesH(Tt, Tz) ≤ φ(max{D(ft, T t), D(fz, Tz), D(ft, Tz),D(fz, T t), d(ft, fz)}).Since φ(t) < qt for each t > 0, we getH(Tt, Tz) ≤ φ(D(ft, T t)) < qD(ft, T t).Now, by the definition of Hausdorff distance function, since ft ∈ Tz, itfollows thatD(ft, T t) < qD(ft, T t),which is a contradiction as 0 < q < 1. Therefore ft ∈ Tt. Hence z = t andso t = ft ∈ Tt. This makes end to the proof. 94 Bessem Samet and Calogero VetroReferences[1] Rhoades B.E., Singh S.L., Kulshrestha C., Coincidence theorems forsome multivalued mappings, Internat. J. Math. & Math. Sci., 7(1984) 429-434.[2] Kamran T., Coincidence and fixed points for hybrid strict contractions, J.Math. Anal. Appl., 299(2004) 235-241.[3] Sintunavarat W., Kumam P., Coincidence and common fixed points for hy-brid strict contractions without the weakly commuting condition, Appl. Math.Lett., 22(2009) 1877-1881.Bessem SametUniversite´ de Tunis, Ecole Supe´rieure des Scienceset Techniques de TunisDe´partement de Mathe´matiques5, Avenue Taha Hussein-TunisB.P.:56, Bab Menara-1008, Tunisiee-mail: bessem.samet@gmail.comCalogero VetroDepartment of Mathematics and InformaticsUniversity of PalermoVia Archirafi 34, 90123 Palermo, Italye-mail: cvetro@math.unipa.itReceived on 14.11.2010 and, in revised form, on 04.01.2011.

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Comments on the paper "COINCIDENCE THEOREMS FOR SOME MULTIVALUED MAPPINGS" by B. E. RHOADES, S. L. SINGH AND CHITRA KULSHRESTHA

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