@ Appl. Gen. Topol. 15, no. 2(2014), 111-119doi:10.4995/agt.2014.2815 c© AGT, UPV, 2014 New common fixed point theorems for multivalued maps Raj Kamal a, Renu Chugh a, Shyam Lal Singh b,∗ and Swami Nath Mishrac a Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India. b 21, Govind Nagar, Rishikesh 249201, India. c Department of Mathematics, Walter Sisulu University, Mthatha, 5117, South Africa. Abstract Common fixed point theorems for a new class of multivalued maps are obtained, which generalize and extend classical fixed point theorems of Nadler and Reich and some recent Suzuki type fixed point theorems. 2010 MSC: 54H25; 47H10. Keywords: Fixed point; Banach contraction theorem; Hausdorff metric space 1. Introduction Let (X, d) be a metric space and CL(X) the family of all nonempty closed subsets of X. (CL(X), H) equipped with the generalized Hausdorff metric H defined by H(A, B) = max { sup x∈A d(x, B), sup y∈B d(y, A) } , where A, B ∈ CL(X) and d(x, K) = inf z∈K d(x, z), is called the generalized hy- perspace of X. ∗Corresponding author. Received November 2011 – Accepted May 2012 http://dx.doi.org/10.4995/agt.2014.2815 R. Kamal, R. Chugh, S. L. Singh and S. N. Mishra For any nonempty subsets A, B of X, d(A, B) denotes the gap between the subsets A and B, while ρ(A, B) = sup{d(a, b) : a ∈ A, b ∈ B}, BN(X) = {A : ∅ 6= A ⊆ X and the diameter of A is finite}. As usual, we write d(x, B) (resp. ρ(x, B)) for d(A, B) (resp. ρ(A, B)) when A = {x}. For x, y ∈ X, we follow the following notation, where S and T are maps to be defined specifically in a particular context: M(Sx, T y) = { d(x, y), d(x, Sx) + d(y, T y) 2 , d(x, T y) + d(y, Sx) 2 } . Recently Suzuki [23] obtained a forceful generalization of the famous Banach contraction theorem. Subsequently, a number of new fixed point theorems have been established and some applications have been discussed (see, for instance, [1, 5, 6, 7, 8, 9, 10, 13, 16, 20, 21, 22, 24]). The following result is essentially due to Kikkawa and Suzuki [8] (see also [22]) which generalizes the classical multivalued contraction theorem due to Nadler [11] (see also [2, 12, 14, 18]). Theorem 1.1. Let (X, d) be a complete metric space and let T : X → CL(X). Assume there exists r ∈ [0, 1) such that for every x, y ∈ X, d(x, T x) ≤ (1 + r)d(x, y) implies H(T x, T y) ≤ rd(x, y). Then there exists z ∈ X such that z ∈ T z. The following generalization of Theorem 1.1 is due to Singh and Mishra [20]. Theorem 1.2. Let X be a complete metric space and T : X → CL(X). Assume there exists r ∈ [0, 1) such that for every x, y ∈ X, d(x, T x) ≤ (1 + r)d(x, y) implies H(T x, T y) ≤ rM(T x, T y). Then there exists z ∈ X such that z ∈ T z. The following general common fixed point theorem is due to Sastry and Naidu [19]. Theorem 1.3. Let X be a complete metric space and S, T maps from X to itself. Assume there exists r ∈ [0, 1) such that for every x, y ∈ X, d(Sx, T y) ≤ r max { d(x, y), d(x, Sx), d(y, T y), d(x, T y) + d(y, Sx) 2 } .(1.1) Then S and T have a unique common fixed point. For an excellent discussion on several special cases and variants of Theo- rem 1.3, one may refer to Rus [18]. The generality of Theorem 1.3 may be appreciated from the fact that the condition (1.1) in Theorem 1.3 cannot be replaced by a slightly more general condition: d(Sx, T y) ≤ r max{d(x, y), d(x, Sx), d(y, T y), d(x, T y), d(y, Sx)}.(1.2) c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 112 New common fixed point theorems for multivalued maps See [19, Ex. 5]. Notice that the condition (1.2) with S = T is Ćirić’s quasi- contraction [4]. We remark that, in Rhoades’ comprehensive comparison of contractive conditions [15], the condition (1.2) with S = T is considered the most general contraction for a self-map of a metric space. A particular case of our main result (cf. Theorem 2.1) generalizes Theo- rems 1.1 and 1.2. Some other special cases are also discussed. 2. Main Results We shall need the following lemma essentially due to Nadler, Jr. [11] (see also [2], [3], [16, p. 4], [16, 17], [18, p. 76]). Lemma 2.1. If A, B ∈ CL(X) and a ∈ A, then for each ε > 0, there exists b ∈ B such that d(a, b) ≤ H(A, B) + ε. Theorem 2.2. Let X be a complete metric space and let S and T maps from X to CL(X). Assume there exists r ∈ [0, 1) such that for every x, y ∈ X, min{d(x, Sx), d(y, T y)} ≤ (1 + r)d(x, y) implies H(Sx, T y) ≤ rM(Sx, T y). Then there exists an element u ∈ X such that u ∈ Su ∩ T u. Proof. Obviously M(Sx, T y) = 0 iff x = y is a common fixed point of S and T . So we may assume that M(Sx, T y) > 0. Let ε > 0 be such that β = r + ε < 1. Let u0 ∈ X and u1 ∈ T u0. By Lemma 2.1, their exists u2 ∈ Su1 such that d(u2, u1) ≤ H(Su1, T u0) + M(Su1, T u0). Similarly, their exists u3 ∈ T u2 such that d(u3, u2) ≤ H(T u2, Su1) + εM(T u2, Su1). Continuing in this manner, we find a sequence {un} in X such that u2n+1 ∈ T u2n, u2n+2 ∈ Su2n+1 and d(u2n+1, u2n) ≤ H(T u2n, Su2n−1) + M(T u2n, Su2n−1), d(u2n+2, u2n+1) ≤ H(Su2n+1, T u2n) + εM(Su2n+1, T u2n). Now, we show that for any n ∈ N, d(u2n+1, u2n) ≤ βd(u2n−1, u2n).(2.1) Suppose if d(u2n−1, Su2n−1) ≥ d(u2n, T u2n), then min{d(u2n−1, Su2n−1)d(u2n, T u2n)} ≤ (1 + r)d(u2n−1, u2n). c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 113 R. Kamal, R. Chugh, S. L. Singh and S. N. Mishra Therefore by the assumption, d(u2n+1, u2n) ≤ H(Su2n−1, T u2n) ≤ rM(Su2n−1, T u2n) ≤ rM(Su2n−1, T u2n) + εM(Su2n−1, T u2n) = βM(Su2n−1, T u2n) = β max { d(u2n−1, u2n), d(u2n−1, Su2n−1) + d(u2n, T u2n) 2 , d(u2n−1, T u2n) + d(u2n, Su2n−1) 2 } ≤ β max d(u2n−1, u2n), d(u2n, u2n+1). This yields (2.1). Suppose, if d(u2n, T u2n) ≥ d(u2n−1, Su2n−1), then min{d(u2n−1, Su2n−1), d(u2n, T u2n)} ≤ (1 + r)d(u2n−1, u2n). Therefore by the assumption, d(u2n+1, u2n) ≤ H(Su2n−1, T u2n) ≤ rM(Su2n−1, T u2n) ≤ rM(Su2n−1, T u2n) + εM(Su2n−1, T u2n) = βM(Su2n−1, T u2n) = β max { d(u2n−1, u2n), d(u2n−1, Su2n−1) + d(u2n, T u2n) 2 , d(u2n−1, T u2n) + d(u2n, Su2n−1) 2 } ≤ β max{d(u2n−1, u2n), d(u2n, u2n+1)}. This prove (2.1). In an analogous manner, we show that d(u2n+2, u2n+1) ≤ βd(u2n+1, u2n).(2.2) We conclude from (2.1) and (2.2) that for any n ∈ N, d(un+1, un) ≤ βd(un, un−1). Therefore {un} is a Cauchy sequence and has a limit in X. Call it u. Since un → u, there exists n0 ∈ N (natural numbers) such that d(u, un) ≤ 1 3 d(u, y) for y 6= u and all n ≥ n0. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 114 New common fixed point theorems for multivalued maps Then as in [23, p. 1862], (1 + r)−1d(u2n−1, Su2n−1) ≤ d(u2n−1, Su2n−1) ≤ d(u2n−1, u2n) ≤ d(u2n−1, u) + d(u, u2n) ≤ 2 3 d(y, u) = d(y, u) − 1 3 d(y, u) ≤ d(y, u) − d(u2n−1, u) ≤ d(u2n−1, y). Therefore d(u2n−1, Su2n−1) ≤ (1 + r)d(u2n−1, y).(2.3) Now either d(u2n−1, Su2n−1) ≤ d(y, T y) or d(y, T y) ≤ d(u2n−1, Su2n−1). In either case, by (2.3) and the assumption, d(u2n, T y) ≤ H(Su2n−1, T y) ≤ rM(Su2n−1, T y). ≤ r max { d(u2n−1, y), d(u2n−1, Su2n−1) + d(y, T y) 2 , d(u2n−1, T y) + d(y, Su2n−1) 2 } . Making n → ∞, d(u, T y) ≤ r max { d(u, y), d(u, u) + d(y, T y) 2 , d(u, T y) + d(y, u) 2 } ≤ r max { d(u, y), d(u, T y) + d(u, y) 2 } .(2.4) It is clear from (2.4) that d(u, T y) ≤ rd(u, y).(2.5) Now we show that H(Su, T y) ≤ r max { d(u, y), d(u, Su) + d(y, T y) 2 , d(u, T y) + d(y, Su) 2 } (2.6) Assume that y 6= u. Then for every n ∈ N, there exists zn ∈ T y such that d(u, zn) ≤ d(u, T y) + 1 n d(y, u). c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 115 R. Kamal, R. Chugh, S. L. Singh and S. N. Mishra So we have by (2.5), d(y, T y) ≤ d(y, zn) ≤ d(y, u) + d(u, zn) ≤ d(y, u) + d(u, T y) + 1 n d(y, u) ≤ d(y, u) + rd(u, y) + 1 n d(u, y) = ( 1 + r + 1 n ) d(y, u). Hence d(y, T y) ≤ (1 + r)d(y, u).(2.7) Now either d(u, Su) ≤ d(y, T y) or d(y, T y) ≤ d(u, Su). So in either case by (2.7) and the assumption, H(Su, T y) ≤ rM(Su, T y), which is (2.6). Now taking y = u2n in (2.6), we have d(Su, u2n+1) ≤ H(Su, T u2n) ≤ r max { d(u, u2n), d(u, Su) + d(u2n, u2n+1) 2 , d(u, u2n+1) + d(u2n, Su) 2 } . Passing to the limit this obtains d(Su, u) ≤ r 2 d(Su, u). So u ∈ Su, as Su is closed. In an analogous manner, we can show that u ∈ T u. � Corollary 2.3. Let X be a complete metric space and S, T : X → X. Assume there exists r ∈ [0, 1) such that for every x, y ∈ X, min{d(x, Sx), d(y, T y)} ≤ (1 + r)d(x, y) implies d(Sx, T y) ≤ rM(Sx, T y). Then S and T have a unique common fixed point. Proof. It comes from Theorem 2.2 that S and T have a common fixed point. The uniqueness of the common fixed point follows easily. � Corollary 2.4. Theorem 1.2. Corollary 2.5 ([20]). Let X be a complete metric space and T : X → X. Assume there exists r ∈ [0, 1) such that for every x, y ∈ X, d(x, T x) ≤ (1 + r)d(x, y) implies d(T x, T y) ≤ rM(T x, T y). Then T has a unique fixed point. Proof. It comes from Corollary 2.3 when S = T . � Now we give an application of Corollary 2.3. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 116 New common fixed point theorems for multivalued maps Theorem 2.6. Let P, Q : X → BN(X). Assume there exists r ∈ [0, 1) such that for every x, y ∈ X, min{ρ(x, Px), ρ(y, Qy)} ≤ (1 + r)d(x, y)(2.8) implies ρ(Px, Qy) ≤ r max { d(x, y), ρ(x, Px) + ρ(y, Qy) 2 , d(x, Qy) + d(y, Px) 2 } (2.9) Then there exsits a unique point z ∈ X such that z ∈ Pz ∩ Qz. Proof. Choose λ ∈ (0, 1). Define single-valued maps S, T : X → X as follows. For each x ∈ X, let Sx be a point of Px which satisfies d(x, Sx) ≥ rλρ(x, Px). Similarly, for each y ∈ X, let T y be a point of Qy such that d(y, T y) ≥ rλρ(y, Qy). Since Sx ∈ Px and T y ∈ Qy, d(x, Sx) ≤ ρ(x, Px) and d(y, T y) ≤ ρ(y, Qy). So (2.8) gives min{d(x, Sx), d(y, T y)} ≤ min{ρ(x, Px), ρ(y, Qy)} ≤ (1 + r)d(x, y),(2.10) and this implies (2.9). Therefore d(Sx, T y) ≤ ρ(Px, Qy) ≤ r.r−λ max { rλd(x, y), rλρ(x, Px) + rλρ(y, Qy) 2 , rλd(x, Qy) + rλd(y, Px) 2 } ≤ r1−λ max { d(x, y), d(x, Sx) + d(y, T y) 2 , d(x, T y) + d(y, Sx) 2 } . So (2.10), viz., min{d(x, Sx), d(y, T y)} ≤ (1 + r′)d(x, y) imlpies d(Sx, T y) ≤ r′ max { d(x, y), d(x, Sx) + d(y, T y) 2 , d(x, T y) + d(y, Sx) 2 } , where r′ = r1−λ < 1. Hence by Corollary 2.3, S and T have a unique point z ∈ X such that Sz = T z = z. This implies z ∈ Pz ∩ Qz. � The following result show that Theorem 2.6 is a generalization of the result of Singh and Mishra [20, Theorem 3.6]. Corollary 2.7. Let P : X → BN(X). Assume there exists r ∈ [0, 1) such that ρ(x, Px) ≤ (1 + r)d(x, y) c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 117 R. Kamal, R. Chugh, S. L. Singh and S. N. Mishra implies ρ(Px, Py) ≤ r max { d(x, y), ρ(x, Px) + ρ(y, Py) 2 , d(x, Py) + d(y, Px) 2 } . Then there exists a unique point z in X such that z ∈ Pz. Proof. It comes from Theorem 2.6 when Q = P . � We remark that Corollaries 2.5 and 2.7 generalize fixed point theorems from [11, 14, 18] and others. Now we give two examples to show the generality of our results. Example 2.8. Let X = {(0, 0), (4, 0), (0, 4), (4, 5), (5, 4)} and d be defined by d[(x1, x2), (y1, y2)] = |x1 − y1| + |x2 − y2|. Let S and T be such that S(x1, x2) = { (x1, 0) if x1 ≤ x2 (0, x2) if x1 > x2 and T (x1, x2) = { (0, x1) if x1 ≤ x2 (0, x2) if x1 > x2 Then maps S and T do not satisfy (1.1) of Theorem 1.3 (e.g. (x, y) = ((4, 5), (5, 4))). However, S and T satisfy all the hypotheses of Corollary 2.3. Example 2.9. Let X = {(1, 1), (4, 1), (1, 4), (4, 5), (5, 4)} and d be defined by d[(x1, x2), (y1, y2)] = |x1 − y1| + |x2 − y2| Let T be such that T (x1, x2) = { (x1, 1) if x1 ≤ x2 (1, x2) if x1 > x2 Then T satisfies all the hypotheses of Corollary 2.5, but does not satisfy Ciric’s quasi-contraction, viz. (1.2) with S = T (e.g.x = (4, 5), y = (5, 4)). We close this paper with the following. Question 2.10. 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