ppp.dvi CUBO A Mathematical Journal Vol.12, No¯ 01, (59–66). March 2010 Weakly Picard Pairs of Multifunctions Valeriu Popa Department of Mathematics, University of Bacău, Str. Spiru Haret nr. 8, 600114 Bacău, Romania email : vpopa@ub.ro ABSTRACT The purpose of this paper is to present a general answer for the following problem: Let (X, d) be a metric space and T1, T2 : X → P (X) two multifunctions. Determine the metric conditions which imply that (T1, T2) is a weakly Picard pair of multifunctions and T1, T2 are weakly Picard multifunctions , for multifunctions satisfying an implicit contractive condition, generalizing some results from [6] and [7]. RESUMEN El proposito de este artículo es presentar una respuesta general para el siguiente problema: Sea (X, d) un espacio métrico y T1, T2 : X → P (X) dos multifunciones. Determine los condi- ciones metricas para las cuales (T1, T2) sea un par de multifunciones de Picard debil y T1, T2 sean multifunciones satisfaziendo una condición contractiva implícita, generalizando algunos resultados de [6] y [7]. Key words and phrases: Multifunction, fixed point, implicit relation, weakly Picard multifunc- tion, weakly Picard pair of multifunctions. Math. Subj. Class.: 47H10, 54H25. 60 Valeriu Popa CUBO 12, 1 (2010) 1 Introduction and Preliminaries Let X be a nonempty set. We denote P(X) the set of all nonempty subsets of X, i.e. P (X) = {Y : Φ 6= Y ⊂ X} . Let T : X → P (X) a multifunction. We denote by FT the set of fixed points of T, i.e. FT = {x ∈ X : x ∈ T (x)}. Let (X,d) be a metric space. We denote by Pcl(X) the set of all nonempty and closed sets of X. We also recall the functional D : P (X) × P (X) → R+, defined by D(A, B) = inf{d(a, b) : a ∈ A, b ∈ B} for each A, B ∈ P (X) and generalized Hansdorff-Pompeiu metric H : P (X) × P (X) → R+ ∪ {+∞} defined by H(A, B) = max {sup[D(a, B), a ∈ A], sup[D(A, b), b ∈ B]} for A, B ∈ P (X). The following property of H is well-known. Lemma 1.1. Let (X,d) be a metric space, A, B ∈ P (X) and q > 1. Then for every a ∈ A, there exists b ∈ B such that d(a, b) ≤ qH(A, B). Definition 1.1. Let (X,d) be a metric space and T : (X, d) → P (X) a multifunction. We say that T is a weakly Picard multifunction [3],[4] if for each x ∈ X and for every y ∈ T (x), there exists a sequence (xn)n∈N such that: (i) x0 = x, x1 = y; (ii) xn+1 ∈ T (xn) , for each n ∈ N ∗ ; (iii) the sequence (xn)n∈N is convergent and its limit is a fixed point of T. Definition 1.2. Let (X,d) be a metric space and T1, T2 : X → P (X) two multifunctions. We say that (T1, T2) is a weakly Picard pair of multifunctions if for each x ∈ X and for every y ∈ T1(x) ∪ T2(x), there exists a sequence (xn)n∈N such that (i) x0 = x, x1 = y; (ii) x2n+1 ∈ Ti(x2n) and x2n+2 ∈ Tj(x2n+1), for n ∈ N , where i, j ∈ {1, 2}, i 6= j ; (iii) the sequence (xn)n∈N is convergent and its limit is a common fixed point of T1 and T2. Problem 1.1 [4]. Let (X,d) be a metric space and T1, T2 : (X, d) → P (X) two multifunctions. Determine the metric conditions which implies (T1, T2) is a weakly Picard pair of multifunctions and T1, T2 are weakly Picard multifunctions. Partial answers to Problem 1.1. was established by Sintămărian in [4]-[7]. In [1] and [2] is introduced the study of fixed point for mappings satisfying implicit relations. The purpose of this paper is to prove two general fixed points theorems for multifunctions which satisfy a new type of implicit contractive relation which generalize some results from [6], [7]. CUBO 12, 1 (2010) Weakly Picard Pairs of Multifunctions 61 2 Implicit Relations Let F be the set of all continuous multifunctions F (t1, ..., t6) : R6+ → R satisfying the following conditions: (F1): F is increasing in variable t1 and nonincreasing in variables t3, ..., t6; (F2): there exists k > 1, h ∈ [0, 1) and g ≥ 0 such that for every u ≥ 0, v ≥ 0, w ≥ 0, such that (Fa): u ≤ kt and F (t, v, v + w, u + w, u + v + w, w) ≤ 0, or (Fb): u ≤ kt and F (t, v, u + w, v + w, w, u + v + w) ≤ 0 implies u ≤ hv + gw. Example 2.1. F (t1, ..., t6) = t1 − at2 − b(t3 + t4) − c(t5 + t6), when 0 < a + 2b + 2c < 1. (F1): Obviously. (F2): F (t, v, v + w, u + w, u + v + w, w) = t − av − b(u + v + 2w) − c(u + v + 2w) ≤ 0, where 1 < k < 1 a+2b+2c . Then u ≤ kt ≤ k[av+b(u+v+2w)+c(u+v+2w)]. Hence u ≤ hv+gw, where 0 < h = k(a+b+c) 1−k(b+c) < 0 and g = 2k(b+c) 1−k(b+c) ≥ 0 Similarly, F (t, v, u + w, w + v, w, u + v + w ≤ 0 implies u ≤ hv + gw. Remark 2.1. If a + 4b + 4c < 1 and 1 < k < 1 a+4b+4c then h + g < 1. Example 2.2. F (t1, ..., t6) = t1 − mmax{t2, t3, t4, 12 (t5 + t6)} where 0 < m < 1 2 . (F1): Obviously. (F2): Let u ≥ 0, v ≥ 0, w ≥ 0, 1 < k < 12m and F (t, v, v + w, u + w, u + v + w, w) = t − mmax{v, u + w, u + v, 1 2 (u + v + 2w)} ≤ 0 which implies t ≤ m(u + v + w). Then u ≤ kt ≤ km(u + v + w). Hence, u ≤ hv + gw where 0 < h = km 1−km < 1 and g = km 1−km ≥ 0. Similarly, F (t, v, u + w, v + w, w, u + v + w) ≤ 0 implies u ≤ hv + gw . Remark 2.2. If 0 < m < 1 3 and 1 < k < 1 3m then h + g < 1. Example 2.3. F (t1, ..., t6) = t 2 1 − mmax{t22, t3t4, t5t6}, where 0 ≤ m < 14 . (F1): Obviously. (F2): Let u ≥ 0, v ≥ 0, w ≥ 0, 1 < k < 12√m and F (t, v, v + w, u + w, u + v + w, w) = t2 − mmax{v2, (v + w)(u + w), w(u + v + w)) ≤ 0 which implies t2 ≤ m(u + v + w)2 and t ≤ √ m(u + v + w). Then u ≤ kt ≤ k √ m(u + v + w). Hence, u ≤ hv + gw, where 0 ≤ h ≤ k √ m 1=k √ m < 1 and g = k √ m 1=k √ m ≥ 0. Similarly, F (t, v, u + w, v + w, w, u + v + w) ≤ 0 implies u ≤ hv + gw. Remark 2.3. If o ≤ m < 1 9 and 1 < k < 1 3 √ m then h + g < 1. Example 2.4. F (t1, ..., t6) = t 3 1 + t 2 2 + 1 1+t5+t6 − m(t22 + t23 + t24), where 0 < m < 112 . (F1): Obviously. (F2): Let u ≥ 0, v ≥ 0, w ≥ 0 and 1 < k < 12√m and F (t, v, v + w, u + w, u + v + w, w) = t3 + t2 + t 1+u+v+w − m(v2 + (v + w)2 + (u + w)2) ≤ 0 which implies 62 Valeriu Popa CUBO 12, 1 (2010) t2 ≤ m(v2 + (u + v)2 + (u + w)2) ≤ 3m(u + v + w)2 and t ≤ √ 3m(u + v + w). If u ≤ kt ≤ k √ 3m(u + v + w) then u ≤ hv + gw, where 0 ≤ h = k √ 3m 1−k √ 3m < 1 and g = k √ 3m 1−k √ 3m ≥ 0. Similarly, F (t, v, u + w, v + w, w, u + v + w) ≤ 0 implies u ≤ hv + gw. Remark 2.4. If 0 < m < 1 27 and 1 < k < 1 3 √ 3m then h + g < 1. 3 Main Results Theorem 3.1. Let T1, T2 : (X, d) → P cl(X) be two multifunctions. If the inequality (1) Φ(H(T1(x), T2(y)), d(x, y), D(x, T1(x)), d(y, T2(y)), D(x, T2(y)), D(y, T1(x)) ≤ 0 holds for all x, y ∈ X, where F ∈ F and FT1 6= Φ or FT2 6= Φ, then FT1 = FT2 . Proof. Let u ∈ FT1 , then u ∈ T1(u) and by (1) we have Φ(H(T1(u), T2(u)), d(u, u), d(u, T1(u)), D(u, T2(u)), D(u, T2(u)), D(u, T1(u)) ≤ 0 By D(u, T2(u)) ≤ H(T1(u), T2(u)) it follows that Φ(D(u, T2(u)), 0, 0, D(u, T2(u)), D(u, T2(u)), 0) ≤ 0 Since D(u, T2(u)) ≤ kD(u, T2(u)) by (Fa) we have that D(u, T2(u)) = 0. Since T2(u) is closed we obtain u ∈ T2(u) i.e. u ∈ FT2 and FT1 ⊂ FT2 . Similarly, by (Fb) we obtain FT2 ⊂ FT1 . Similarly, if u ∈ FT2 , then FT1 = FT2 . Theorem 3.2. Let (X,d) be a complete metric space and T1, T2 : (X, d) → P cl(X) two multifunc- tions. If (1) holds for all x, y ∈ X, where F ∈ F, then T1 and T2 have a common fixed point and FT1 = FT2 ∈ P cl(X). Proof. Let x0 ∈ X and x1 ∈ T1(x0). Then there exists x2 ∈ T2(x1) so that d(x1, x2) ≤ kH(T1(x0), T2(x1)) Suppose that x2, x3, ..., x2n−1, x2n, ... such that x2n−1 ∈ T1x2n−2, x2n ∈ T2x2n−1, n ∈ N ∗ and (2) d(x2n−1, x2n) ≤ kH(T1(x2n−2), T2(x2n−1)) , (3) d(x2n−2, x2n−1) ≤ kH(T1(x2n−2), T2(x2n−3)) . By (1) we have successively Φ(H(T1(x2n−2), T2(x2n−1)), d(x2n−2, x2n−1), D(x2n−2, T1(x2n−2)), D(x2n−1, T2(x2n−1)), D(x2n−2, T2(x2n−1)), D(x2n−1, T1(x2n−2)) ≤ 0 Φ(H(T1(x2n−2), T2(x2n−1)), d(x2n−2, x2n−1), d(x2n−1, x2n−2), d(x2n−1, x2n), d(x2n−2, x2n), 0) ≤ 0 (4) Φ(H(T1(x2n−2), T2(x2n−1)), d(x2n−2, x2n−1), d(x2n−2, x2n−1), d(x2n−1, x2n), d(x2n−2, x2n−1) + d(x2n−1, x2n), 0) ≤ 0 Since Φ ∈ F then by (2),(4) and (Fa) we obtain (5) d(x2n−1, x2n) ≤ hd(x2n−2, x2n−1) Similarly, by (3) and (Fb) we obtain (6) d(x2n−2, x2n−1) ≤ hd(x2n−2, x2n−3) CUBO 12, 1 (2010) Weakly Picard Pairs of Multifunctions 63 Then by a rutine calculation one can show that (xn)n∈N is a Cauchy sequence and since (X,d) is complete we have limxn = x for some x ∈ X. Now, if n ∈ N ∗, (1) implies Φ(H(T1(x), T2(x2n−1)), d(x, x2n−1), D(x, T1(x)), D(x2n−1, T2(x2n−1)), D(x, T2(x2n−1)), D(x2n−1, T1x) ≤ 0 As D(x2n, T1(x)) ≤ H(T2(x2n−1), T1(x)) we have Φ(D(x2n, T1(x)), d(x, x2n−1), D(x, T1(x)), d(x2n−1, x2n), d(x, x2n), D(x2n−1, T1(x)) ≤ 0 Letting n tend to infinity we obtain Φ(D(x, T1(x)), 0, D(x, T1(x)), 0, 0, D(x, T1(x)) ≤ 0 Since D(x, T1(x)) ≤ kD(x, T1(x)) by (Fb) we obtain D(x, T1(x)) = 0. Since T1(x) is closed, x ∈ T1(x). Hence x ∈ FT1 . By Theorem 3.1 FT1 = FT2 . Let us prove that FT1 = FT2 ∈ P cl(X). For this purpose that yn ∈ FT1 = FT2 for each n ∈ N such that yn → y∗ as n → ∞. For example yn ∈ T1(yn). Then by Lemma 1.1 there exists vn ∈ T2y∗ such that (7) d(yn, vn) ≤ kH(T1(yn), T2(y∗)) . By (1) and (F1) we have successively Φ(H(T1(yn), T2(y ∗)), d(yn, y ∗), D(yn, T1(yn)), D(y ∗, T2(y ∗)), D(yn, T2(y ∗)), D(y∗, T1(yn)) ≤ 0 Φ(H(T1(yn), T2(y ∗)), d(yn, y ∗), 0, d(y∗, vn), d(yn, vn), d(y ∗, yn)) ≤ 0 (8) Φ(H(T1(yn), T2(y ∗)), d(yn, y ∗), d(yn, y ∗) + d(yn, y ∗), d(y∗, yn) + d(yn, vn), d(yn, vn) + d(yn, y ∗) + d(yn, y ∗), d(yn, y ∗)) ≤ 0 Since Φ ∈ F by (7) and (8) it follows that d(yn, vn) ≤ hd(yn, y∗) + gd(y∗, yn) Using the triangle inequality we obtain d(y∗, vn) ≤ d(y∗, yn) + d(yn, vn) ≤ (1 + h + g)d(y∗, yn) Letting n tend to infinity we obtain that limvn = y ∗. Since vn ∈ T2(y∗), for each n ∈ N ∗ and T2(y ∗)is closed, it follows that y∗ ∈ T2(y∗), hence y∗ ∈ FT2 = FT1 and FT1 is closed. Therefore, FT1 = FT2 ∈ P cl(X). Theorem 3.3. Let (X,d) be a complete metric space and T1, T2 : (X, d) → P cl(X). If (1) holds for all x, y ∈ X, where Φ ∈ F, then FT1 = FT2 ∈ P cl(X) and (T1, T2) is a weakly Picard pair of multifunctions. If in adition we have that h + g < 1, then T1 and T2 are weakly Picard multifunctions. Proof. The first part it follows from Theorem 3.2. Let x0 ∈ X and x1 ∈ T1(x0). There exists y1 ∈ T2(x1) such that (9) d(x1, y1) ≤ kH(T1(x0), T2(x1)) By (1) and (F1) we have successively Φ(H(T1(x0), T2(x1)), d(x0, x1), D(x0, T1(x0)), D(x1, T2(x1)), D(x0, T2(x1)), D(x1, T1(x0)) ≤ 0 Φ(H(T1(x0), T2(x1)), d(x0, x1), d(x0, x1), d(x1, y1), d(x0, y1), 0) ≤ 0 64 Valeriu Popa CUBO 12, 1 (2010) (10) Φ(H(T1(x0), T2(x1)), d(x0, x1), d(x0, x1), d(x1, y1), d(x0, x1) + d(x1, y1), 0) ≤ 0 Since Φ ∈ F by (9) and (10) it follows that d(x1, y1) ≤ hd(x0, x1) Also, there exists x2 ∈ T1(x1) such that (11) d(x2, y1) ≤ kH(T1(x1), T2(x1)) By (1) we have successively Φ(H(T1(x1), T2(x1)), 0, D(x1, T1(x1)), D(x1, T2(x1)), D(x1, T2(x1)), D(x1, T1(x1)) ≤ 0 Φ(H(T1(x1), T2(x1)), 0, d(x1, x2), d(x1, y1), d(x1, y1), d(x1, x2)) ≤ 0 (12) Φ(H(T1(x1), T2(x1)), 0, d(x1, x2), d(x1, x2) + d(x2, y1), d(x1, x2) + d(x2, y1), d(x1, x2)) ≤ 0 Since Φ ∈ F by (11) and (12) it follows that d(y1, x2) ≤ gd(x1, x2) Using the triangle inequality we have d(x1, x2) ≤ d(x1, y1) + d(y1, x2) ≤ hd(x0, x1) + gd(x1, x2) which implies that d(x1, x2) ≤ h1−g d(x0, x1) Now, there exists y2 ∈ T2(x2) such that (13) d(x2, y2) ≤ kH(T1(x1), T2(x2)) By (1) we have successively Φ(H(T1(x1), T2(x2)), d(x1, x2), D(x1, T1(x1)), D(x2, T2(x2)), D(x1, T2(x2)), D(x2, T1(x1)) ≤ 0 Φ(H(T1(x1), T2(x2)), d(x1, x2), d(x1, x2), d(x2, y2), d(x1, y2), 0) ≤ 0 (14) Φ(H(T1(x1), T2(x2)), d(x1, x2), d(x1, x2), d(x2, y2), d(x1, x2) + d(x2, y2), 0) ≤ 0 Since Φ ∈ F by (13) and (14) it follows that d(x2, y2) ≤ hd(x1, x2) Also, there exists x3 ∈ T1(x2) such that (15) d(x3, y2) ≤ kH(T1(x2), T2(x2)) By (1) we have successively Φ(H(T1(x2), T2(x2)), 0, D(x2, T1(x2)), D(x2, T2(x2)), D(x2, T2(x2)), D(x2, T1(x2)) ≤ 0 Φ(H(T1(x2), T2(x2)), 0, d(x2, x3), d(x2, y2), d(x2, y2), d(x2, x3)) ≤ 0 (16) Φ(H(T1(x2), T2(x2)), 0, d(x2, x3), d(x2, x3) + d(x3y2), d(x2, x3) + d(x3, y2), d(x2, x3)) ≤ 0 Since Φ ∈ F by (15) and (16) it follows that d(x3, y2) ≤ gd(x2, x3) Using again the triangle inequality we obtain d(x2, x3) ≤ d(x2, y2) + d(y2, x3) ≤ hd(x1, x2) + gd(x2, x3) and so CUBO 12, 1 (2010) Weakly Picard Pairs of Multifunctions 65 d(x2, x3) ≤ h1−g d(x1, x2) By induction we obtain that there exists a sequence (xn)n∈N starting from x0, x1 with xn+1 ∈ T1(xn) such that d(xn, xn+1) ≤ h1−g d(xn−1, xn) for each n ∈ N ∗. Since h 1−g < 1 it follows that (xn)n∈N is a convergent sequence, because (X,d) is a complete metric space. Let x∗ = limxn. By (1) we have Φ(T1(xn), T2(x ∗)), d(x∗, xn), D(xn, T1(xn)), D(x ∗, T2(x ∗)), D(xn, T2(x ∗)), D(x∗, T1(xn))) ≤ 0 Since D(xn+1, T2(x ∗)) ≤ H(T1(xn), T2(x∗) we obtain Φ(D(x2n+1), T2(x ∗)), d(x∗, xn), d(xn, xn+1), D(x ∗, T2(x ∗)), D(xn, T2(x ∗)), D(x∗, xn+1))) ≤ 0 Letting n tend to infinity we obtain Φ(D(x∗, T2(x ∗)), 0, 0, D(x∗, T2(x ∗)), D(x∗, T2(x ∗)), 0) ≤ 0 Since D(x∗, T2(x ∗)) ≤ kD(x∗, T2(x∗)) and Φ ∈ F we obtain D(x∗, T2(x∗)) = 0 and since T2(x∗) is closed we have that x∗ ∈ T2(x∗) and x∗ ∈ FT2 = FT1 . Hence T1 is a weakly Picard multifunction. The fact that T2 is a weakly Picard multifunction is similar proved. Remark 3.1. By Theorems 2 and 3 and Ex. 2.1 we obtain generalizations of the results from Theorem 2.1 [6] and Theorem 2.1 [7]. By Ex. 2.2 -2.4 we obtain new results. Received: May, 2008. Revised: October, 2009. References [1] Popa, V., Some fixed point theorems for contractive mappings, Stud.Cerc.St.Ser. Mat. Univ. Bacău, 7(1997), 127–133. [2] Popa, V., Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstratio Math., 32(1999), 156–163. [3] Rus, I.A., Petruşel, A. and Sintămărian, A., Data dependence of the fixed points set of multivalued weakly Picard operators, Studia Univ. “Babeş Bolyai”, Mathematica, 46(2)(2001), 111–121. [4] Sintămărian, A., Weakly Picard pairs of multivalued operators, Mathematica, 45(2)(2003), 195–204. 66 Valeriu Popa CUBO 12, 1 (2010) [5] Sintămărian, A., Weakly Picard pairs of some multivalued operators, Mathematical Com- munications, 8(2003), 49–53. [6] Sintămărian, A., Pairs of multivalued operators, Nonlinear Analysis Forum, 10(1)(2005), 55–67. [7] Sintămărian, A., Some pairs of multivalued operators, Carpathian J. of Math. 21,1-2(2005), 115–125.