Common fixed point theorems for multi-valued maps

Acta Mathematica Scientia 2012,32B(2):818–824 http://actams.wipm.ac.cn

COMMON FIXED POINT THEOREMS FOR MULTI-VALUED MAPS∗ Boˇsko Damjanovi´c Department of Mathematics, Faculty of Agriculture, Nemanjina 6, Belgrade, Serbia E-mail: [email protected]

Bessem Samet Universit´ e de Tunis, D´ epartement de Math´ ematiques, Ecole Sup´ erieure des Sciences et Techniques de Tunis E-mail: [email protected]

Calogero Vetro Department of Mathematics and Informatics, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy E-mail: [email protected]

Abstract We establish some results on coincidence and common fixed points for a twopair of multi-valued and single-valued maps in complete metric spaces. Presented theorems generalize recent results of Gordji et al [4] and several results existing in the literature. Key words Coincidence point; common fixed point; multi-valued maps 2000 MR Subject Classification

1

54H25; 47H10; 54E50

Introduction and Preliminaries

The Banach fixed-point theorem [1] (also known as the Banach contraction mapping theorem or the Banach contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points. Many generalizations of this famous theorem exist in the literature (cf. [1–8] and others). In [7], Nadler extended the Banach fixed-point theorem from the single-valued maps to the set-valued contractive maps. Before presenting this important theorem, we start with introducing some notations. Let (X, d) be a metric space. Denote by CB(X) the collection of non-empty closed bounded subsets of X. For A, B ∈ CB(X) and x ∈ X, define D(x, A) = inf d(x, a) a∈A

∗ Received

June 28, 2010; revised February 5, 2011. The first author is supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia; the third author is supported by Universit` a degli Studi di Palermo, Local project R. S. ex 60%.

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 H(A, B) = max sup D(a, B), sup D(b, A) . 

a∈A

b∈B

It is seen that H is a metric on CB(X). H is called the Hausdorff metric induced by d. It is well known that (CB(X), H) is a complete metric space, whenever (X, d) is a complete metric space. Definition 1.1 Let T : X → CB(X) be a multi-valued map. An element x ∈ X is said to be a fixed point of T if x ∈ T x. The Nadler’s fixed-point theorem [7] is the following. Theorem 1.1 Let (X, d) be a complete metric space, and T : X → CB(X) be a multivalued map satisfying H(T x, T y) ≤ qd(x, y) for all x, y ∈ X, where q is a constant such that q ∈ [0, 1). Then, T has a fixed point. Recently, an extension of Theorem 1.1 was obtained by Gordji et al [4]. They proved the following result. Theorem 1.2 Let (X, d) be a complete metric space, and T be a map from X into CB(X) such that H(T x, T y) ≤ αd(x, y) + β[D(x, T x) + D(y, T y)] + γ[D(x, T y) + D(y, T x)] for all x, y ∈ X, where α, β, γ ≥ 0 and α + 2β + 2γ < 1. Then, T has a fixed point. Note that Theorem 1.2 generalizes also other known results in the literature [5, 8, 9]. In this article, we establish some results on coincidence and common fixed points for a two-pair of multi-valued and single-valued maps in complete metric spaces. Presented theorems generalize the result given by Theorem 1.2 and other existing results in the literature. The following definitions will be used later. Definition 1.2 An element x ∈ X is said to be a coincidence point of T : X → CB(X) and f : X → X if f x ∈ T x. We denote C(f, T ) = {x ∈ X | f x ∈ T x}, the set of coincidence points of T and f . Definition 1.3 Maps f : X → X and T : X → CB(X) are weakly compatible if they commute at their coincidence points, that is, if f T x = T f x whenever f x ∈ T x. Definition 1.4 (see [6]) Let T : X → CB(X) be a multi-valued map and f : X → X be a single-valued map. The map f is said to be T -weakly commuting at x ∈ X if f f x ∈ T f x. Definition 1.5 An element x ∈ X is a common fixed point of T, S : X → CB(X) and f : X → X if x = f x ∈ T x ∩ Sx. Example 1.1 Consider X = [0, +∞) equipped with the metric d(x, y) = |x− y| for every x, y ∈ X. Define f : X → X and T : X → CB(X) as ⎧ ⎧ ⎨ {x} ⎨ 0 if x ∈ [0, 1), if x ∈ [0, 1), Tx = fx = ⎩ [1, 1 + 2x] if x ∈ [1, +∞). ⎩ 2x if x ∈ [1, +∞), We have

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• f 1 = 2 ∈ [1, 3] = T 1, that is, x = 1 is a coincidence point of f and T ; • f T 1 = [2, 6] = [1, 5] = T f 1, that is, f and T are not weakly compatible mappings; • f f 1 = 4 ∈ [1, 5] = T f 1, that is, f is T -weakly commuting at 1.

2

Main Results The following lemma (see [2, 3]) plays an important role in the proof of our results.

Lemma 2.1 If A, B ∈ CB(X) and a ∈ A, then for any fixed h > 1, there exists b = b(a) ∈ B such that d(a, b) ≤ hH(A, B). Our first result is the following. Theorem 2.1 Let (X, d) be a complete metric space. Let T, S : X → CB(X) be a pair of multi-valued maps and f, g : X → X a pair of single-valued maps. Suppose that H(Sx, T y) ≤ αd(f x, gy) + β[D(f x, Sx) + D(gy, T y)] + γ[D(f x, T y) + D(gy, Sx)],

(1)

for each x, y ∈ X, where α, β, γ ≥ 0 and 0 < α + 2β + 2γ < 1. Suppose also that (i) SX ⊆ gX, T X ⊆ f X, (ii) f (X) and g(X) are closed. Then, there exist points u and w in X, such that f u ∈ Su, gw ∈ T w, f u = gw

and Su = T w.

Proof As 0 < α + 2β + 2γ < 1, there exists r > 0, such that 0 < α + 2β + 2γ <

√

r < 1.

(2)

Let us denote α+β+γ . λ= √ r − (β + γ)

(3)

0 < λ < 1.

(4)

Clearly, from (2), it follows that

Let x0 ∈ X be arbitrary. Then, f x0 and Sx0 are well defined. From (i), there exists x1 ∈ X, √ such that gx1 ∈ Sx0 . Again from (i) and Lemma 2.1 with h = 1/ r, as gx1 ∈ Sx0 , there exists x2 ∈ X such that f x2 ∈ T x1 and 1 d(gx1 , f x2 ) ≤ √ H(Sx0 , T x1 ). r

(5)

From (1) and (5), we obtain α β d(gx1 , f x2 ) ≤ √ d(f x0 , gx1 ) + √ [D(f x0 , Sx0 ) + D(gx1 , T x1 )] r r γ + √ [D(f x0 , T x1 ) + D(gx1 , Sx0 )]. r

(6)

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In contrast, we have D(f x0 , Sx0 ) ≤ d(f x0 , gx1 ), D(gx1 , T x1 ) ≤ d(gx1 , f x2 ), D(gx1 , Sx0 ) = 0, D(f x0 , T x1 ) ≤ d(f x0 , f x2 ) ≤ d(f x0 , gx1 ) + d(gx1 , f x2 ).

(7)

From (6) and (7), we obtain α β d(gx1 , f x2 ) ≤ √ d(f x0 , gx1 ) + √ [d(f x0 , gx1 ) + d(gx1 , f x2 )] r r γ + √ [d(f x0 , gx1 ) + d(gx1 , f x2 )] r     β γ γ α β d(f x0 , gx1 ) + √ + √ d(gx1 , f x2 ). = √ +√ +√ r r r r r Hence,

√ [ r − (β + γ)]d(gx1 , f x2 ) ≤ (α + β + γ)d(f x0 , gx1 ).

Then, from (3), d(gx1 , f x2 ) ≤ λd(f x0 , gx1 ). Again, from (i) and Lemma 2.1, as f x2 ∈ T x1 , there exists x3 ∈ X such that gx3 ∈ Sx2 and 1 d(f x2 , gx3 ) ≤ √ H(Sx2 , T x1 ). r

(8)

By (1) and (8), we obtain α β d(f x2 , gx3 ) ≤ √ d(f x2 , gx1 ) + √ [D(f x2 , Sx2 ) + D(gx1 , T x1 )] r r γ + √ [D(f x2 , T x1 ) + D(gx1 , Sx2 )]. r

(9)

In contrast, we have D(f x2 , Sx2 ) ≤ d(f x2 , gx3 ), D(gx1 , T x1 ) ≤ d(gx1 , f x2 ), D(f x2 , T x1 ) = 0, D(gx1 , Sx2 ) ≤ d(gx1 , gx3 ) ≤ d(gx1 , f x2 ) + d(f x2 , gx3 ).

(10)

Similarly as above, from (9) and (10), we obtain d(f x2 , gx3 ) ≤ λd(gx1 , f x2 ). Continuing this process, we can construct a sequence {yn } in X, such that y0 = gx1 and, for each n ∈ N, y2n = gx2n+1 ∈ Sx2n ,

y2n+1 = f x2n+2 ∈ T x2n+1 ,

for each

n ∈ N,

(11)

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and d(y2n , y2n+1 ) = d(gx2n+1 , f x2n+2 ) ≤ λd(gx2n+1 , f x2n ), d(y2n−1 , y2n ) = d(f x2n , gx2n+1 ) ≤ λd(gx2n−1 , f x2n ). Therefore, we have d(yn , yn+1 ) ≤ λd(yn−1 , yn ),

for all n ≥ 1.

(12)

for all n ∈ N.

(13)

From (12), we get, by induction, d(yn , yn+1 ) ≤ λn d(y0 , y1 ),

Now, we shall show that {yn } is a Cauchy sequence. Let ε > 0 be arbitrary. We need to show that there is a positive integer n0 = n0 (ε) such that d(yn , yn+p ) < ε

for every n ≥ n0 , uniformly on p ∈ N.

(14)

By the triangular inequality, d(yn , yn+p ) ≤ d(yn , yn+1 ) + d(yn+1 , yn+2 ) + ··· + d(yn+p−1 , yn+p ). Thus, from (13), we obtain d(yn , yn+p ) ≤ λn d(y0 , y1 ) + λn+1 d(y0 , y1 ) + ··· + λn+p−1 d(y0 , y1 ) = λn (1 + λ + ··· + λp−1 )d(y0 , y1 ) ≤ λn (1 + λ + ··· + λp−1 + ···)d(y0 , y1 ). Hence, we obtain

λn d(y0 , y1 ) for all n ∈ N (15) 1−λ uniformly on p ∈ N. As 0 < λ < 1, it follows that λn → 0 as n → ∞. Thus there is a positive integer n0 , such that λn d(y0 , y1 ) < ε for all n ≥ n0 . (16) 1−λ From (15) and (16), we get (14). Thus, we have proved that {yn } is a Cauchy sequence. Now, as (X, d) is complete, {yn } converges to some y ∈ X. Therefore, d(yn , yn+p ) ≤

lim yn = lim gx2n+1 = lim f x2n+2 = y.

n→+∞

n→+∞

n→+∞

(17)

As y2n = gx2n+1 ; y2n+1 = f x2n+2 ; and f (X) and g(X) are closed, then, y ∈ f (X) and y ∈ g(X). So, there exist u, w ∈ X, such that f u = y and gw = y. Thus, we have proved that f u = gw. From the contraction type condition (1) and (11), we obtain D(f u, Su) ≤ d(f u, f x2n+2 ) + D(f x2n+2 , Su) ≤ d(f u, f x2n+2 ) + H(Su, T x2n+1 ) ≤ d(f u, f x2n+2 ) + αd(f u, gx2n+1 ) + β[D(f u, Su) + D(gx2n+1 , T x2n+1 )] +γ[D(f u, T x2n+1 ) + D(gx2n+1 , Su)] ≤ d(f u, f x2n+2 ) + αd(f u, gx2n+1 ) + β[D(f u, Su) + d(gx2n+1 , f x2n+2 )] +γ[d(f u, f x2n+2 ) + D(gx2n+1 , Su)].

(18)

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Letting n → +∞ in the above inequality and using (17) and (18), we obtain D(f u, Su) ≤ (β + γ)D(f u, Su). As β + γ < 1, it implies that D(f u, Su) = 0. Hence, as Su is closed, f u ∈ Su.

(19)

Similarly, we can prove that D(gw, T w) ≤ (β + γ)D(gw, T w). Hence, gw ∈ T w.

(20)

Su = T w.

(21)

Now, we have to prove that

Using (1), (18), (19), and (20), we obtain H(Su, T w) ≤ αd(f u, gw) + β[D(f u, Su) + D(gw, T w)] + γ[D(f u, T w) + D(gw, Su)] = α ·0 + β[0 + 0] + γ[D(gw, T w) + D(f u, Su)] = 0. Hence, Su = T w. Thus, by (18), (19), (20), and (21), we have proved that f u ∈ Su, gw ∈ T w, f u = gw

and Su = T w.

(22)

Example 2.1 Let X = [0, ∞) be the Euclidean space with the usual metric. Define S, T, f , and g on X as follows : Sx = x2 + 7/64,

T x = x3 + 7/64,

f x = 8x2

and

gx = 8x3 .

Then,

d(f x, gy) 8|x2 − y 3 | = . 4 4 Thus, (1) holds for all x, y ∈ X. Also, the other hypotheses (i) and (ii) are satisfied. It is seen that S(1/8) = f (1/8) = 1/8 and T (1/4) = g(1/4) = 1/8. Therefore, S and f have the coincidence at the point u = 1/8 , T and g at the point w = 1/4, and S(1/8) = T (1/4). If f = g in Theorem 2.1, then, we obtain the following coincidence result. Theorem 2.2 Let (X, d) be a complete metric space. Let T, S : X → CB(X) be multivalued maps and f : X → X be a single-valued map satisfying, for each x, y ∈ X, d(Sx, T y) = |x2 − y 3 | ≤

H(Sx, T y) ≤ αd(f x, f y) + β[D(f x, Sx) + D(f y, T y)] + γ[D(f x, T y) + D(f y, Sx)],

(23)

where α, β, γ ≥ 0 and 0 < α + 2β + 2γ < 1. If f X is a closed subset of X and T X ∪ SX ⊆ f X, then, f, T , and S have a coincidence in X. Moreover, if f is both T -weakly commuting and S-weakly commuting at each z ∈ C(f, T ), and f f z = f z, then, f, T , and S have a common fixed point in x.

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Proof If f = g in Theorem 2.1, we obtain that there exist points u and w in X such that f u ∈ Su, f w ∈ T w, f u = f w

and Su = T w.

As u ∈ C(f, T ), f is T -weakly commuting at u and f f u = f u. Set v = f u. Then, we have f v = v and v = f f u ∈ T (f u) = T v. Now, since also u ∈ C(f, S), then f is S-weakly commuting at u, and so we obtain v = f v = f f u ∈ S(f u) = Sv. Thus, we have proved that v = f v ∈ T v ∩ Sv, that is, v is a common fixed point of f, T and S. If f = g = IX (IX being the identity map on X) in Theorem 2.1, then, we obtain the following common fixed-point result. Corollary 2.1 Let (X, d) be a complete metric space. Let T, S : X → CB(X) be multi-valued maps satisfying, for each x, y ∈ X, H(Sx, T y) ≤ αd(x, y) + β[D(x, Sx) + D(y, T y)] + γ[D(x, T y) + D(y, Sx)], where α, β, γ ≥ 0 and 0 < α + 2β + 2γ < 1. Then, there exists a point z in X, such that z ∈ Sz ∩ T z and Sz = T z. Remark 1 If S = T in Corollary 2.1, then, we obtain Theorem 1.2 of Gordji et al [4]. Remark 2 If in Theorem 2.1: (i) β = γ = 0 and S = T ; f = g = IX , then, we obtain Theorem of Nadler [7]; (ii) if S = T and f = g = IX , then, we obtain the results of Reich [8, 9]. If S and T in Corollary 2.1 are single-valued maps, then, we obtain the following result. Corollary 2.2 Let (X, d) be a complete metric space. Let T, S : X → X be single-valued maps satisfying, for each x, y ∈ X, d(Sx, T y) ≤ αd(x, y) + β[d(x, Sx) + d(y, T y)] + γ[d(x, T y) + d(y, Sx)], where α, β, γ ≥ 0 and 0 < α + 2β + 2γ < 1. Then, S and T have a common fixed point in X, that is, there exists z ∈ X such that z = Sz = T z. Remark 3 If S = T in Corollary 2.2, then, we obtain the result of Hardy and Rogers [5]. References [1] Banach S. Sur les op´erations dans les ensembles absraites et leurs applications. Fund Math, 1992, 3: 133–181 ´ c Lj B. Fixed points for generalized multi-valued mappings. Mat Vesnik, 1972, 9(24): 265–272 [2] Ciri´ ´ c Lj B, Ume J S. Multi-valued non-self mappings on convex metric spaces. Nonlinear Anal, 2005, 60: [3] Ciri´ 1053–1063 [4] Gordji M E, Baghani H, Khodaei H, Ramezani M. A generalization of Nadler’s fixed point theorem. J Nonlinear Sci Appl, 2010, 3(2): 148–151 [5] Hardy G E, Rogers T D. A generalization of a fixed point theorem of Reich. Canad Math Bull, 1973, 16: 201–206 [6] Kamran T. Coincidence and fixed points for hybrid strict contractions. J Math Anal Appl, 2004, 299: 235–241 [7] Nadler S B Jr. Multi-valued contraction mappings. Pacific J Math, 1969, 30: 475–488 [8] Reich S. Kannan’s fixed point theorem. Boll Un Mat Ital, 1971, 4: 1–11 [9] Reich S. Fixed points of contractive functions. Boll Un Mat Ital, 1972, 5: 26–42