Fixed-point results for multi-valued operators in quasi-ordered metric spaces

Applied Mathematics Letters 25 (2012) 1856–1861

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Fixed-point results for multi-valued operators in quasi-ordered metric spaces F. Sabetghadam, H.P. Masiha ∗ Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran

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Article history: Received 25 July 2011 Received in revised form 19 February 2012 Accepted 19 February 2012 Keywords: Multi-valued operators Quasi-ordered metric space Hausdorff distance

abstract Nadler’s contraction principle (Nadler, 1969 [17]) has led to the fixed-point theory of setvalued contraction in nonlinear analysis. The purpose of this paper is to present some new fixed-point theorems involving multi-valued operators in the setting of quasi-ordered metric spaces. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction It is well known that the contractive conditions are very important in the study of fixed-point theory. The first important result on fixed points for contractive-type mappings was the well-known Banach–Caccioppoli theorem, published for the first time in 1922 in [1] and also found in [2]. There are a lot of generalizations of the Banach contraction mapping principle in the literature. Chatterjea, in [3], proved the following theorem. Theorem 1. Let (X , d) be a complete metric space, and let F : X → X be a C -contraction, i.e., d(Fx, Fy) ≤ α[d(x, Fy) + d(y, Fx)], where α ∈ 0,



1 2

for all x, y ∈ X ,



. Then F has a unique fixed point.

Also, Alber and Guerre-Delabriere, in [4], define weakly contractive maps. They confine their theorems to Hilbert spaces, but, in [5], Rhoades extends some results appearing in [4] to arbitrary Banach spaces. Now, let (X , d) be a metric space and let f : X → X . f is said to be weakly contractive if, for x, y ∈ X , d(fx, fy) ≤ d(x, y) − ϕ(d(x, y)), where ϕ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that it is positive in (0, ∞), ϕ(0) = 0, and limt →∞ ϕ(t ) = ∞. The existence of fixed points in partially ordered sets has been considered recently in [6–16]. In [13,14], Nieto and López used Tarski’s theorem to show the existence of solutions for fuzzy equations and fuzzy differential equations, respectively. The existence of solutions for matrix equations or ordinary differential equations by applying fixed-point theorems are presented in [9,10,16]. In [7,12], some fixed-point theorems are proved for a mixed monotone mapping in a metric space endowed with partial order, and the authors apply their results to problems of the existence and uniqueness of solutions

∗

Corresponding author. E-mail address: [email protected] (F. Sabetghadam).

0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.02.046

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for some boundary-value problems. Also, in [11], some new hybrid fixed-point theorems for multi-valued operators which satisfy weakly generalized contractive conditions in an ordered complete metric space are proved; as an application, the author gave an example of the existence of a solution for a perturbed impulsive hyperbolic differential inclusion with variable times. In this paper, we want to combine the concept of C -contraction and weak contraction and prove some fixed-point theorems for multi-valued operators in sequentially complete metric space. Let (X , d, ≼) be an quasi-ordered metric space, with an order ≼ as a quasi-order (preorder or pseudo-order; that is, a reflexive and transitive relation) and distance d(·, ·), and let N (X ) denote the class of all nonempty subsets of X . A function Hd : N (X ) × N (X ) → [0, +∞], defined by





Hd (A, B) = max sup D(a, B), sup D(b, A) , a∈A

b∈B

is said to be the Hausdorff metric on N (X ), where D(a, B) = D(B, a) = infb∈B d(a, b). A sequence {xn }n∈N is called ≼-nondecreasing (respectively, ≼-nonincreasing) if xn ≼ xn+1 (respectively, xn+1 ≼ xn ) for each n ∈ N. Definition 1. Let (X , d) be a metric space with a quasi-order ≼. We say that X is sequentially complete if every Cauchy sequence whose consecutive terms are comparable in X converges. Definition 2 ([11]). Let (X , d) be a metric space with a quasi-order ≼. A subset D ⊂ X is said to be approximative if the multi-valued mapping

PD (x) = {y ∈ D : d(x, y) = d(D, x)} ,

for all x ∈ X ,

has nonempty values. The multi-valued mapping M : X → N (X ) is said to have approximative values, AV for short, if Mx is approximative for each x ∈ X . The multi-valued mapping M : X → N (X ) is said to have comparable approximative values, CAV for short, if Mx has approximative values and, for each z ∈ X , there exists y ∈ PMz (x) such that y is comparable to z. The multi-valued mapping M : X → N (X ) is said to have upper comparable approximative values, UCAV for short (respectively, lower comparable approximative values, LCAV for short) if Mx has approximative values and, for each z ∈ X , there exists y ∈ PMz (x) such that y ≽ z (respectively, y ≼ z). It is clear that M has approximative values if it has compact values. In addition, if M is single valued, then UCAV (LCAV ) means that Mx ≽ x(Mx ≼ x) for x ∈ X . Definition 3. The multi-valued mapping M is said to have a fixed point if there is x ∈ X such that x ∈ Mx. Definition 4 ([11]). For two subsets A, B of X , we say that A ≼r B if, for each a ∈ A, there exists b ∈ B such that a ≼ b, and A ≼ B if each a ∈ A and each b ∈ B imply that a ≼ b. A multi-valued mapping M : X → N (X ) is said to be r-nondecreasing (r-nonincreasing) if x ≼r y implies that Mx ≼r My (My ≼r Mx) for all x, y ∈ X . M is said to be r-monotone if M is r-nondecreasing or r-nonincreasing. The notion of nondecreasing (nonincreasing) is similarly defined by writing ≼ instead of the notation ≼r . 2. Main results Theorem 2. Let (X , d, ≼) be a sequentially complete metric space with the following property (which appears in [13]). Any ≼-nondecreasing sequence {xn } with xn → x∗ implies that xn ≼ x∗ for each n ∈ N. Suppose that the multi-valued mapping M : X → N (X ) has UCAV and satisfies

Hd (Mx, My) ≤ α[D(x, My) + D(y, Mx)] − ϕ(D(x, My), D(y, Mx)),

(2.1)

for all x, y ∈ X which are comparable. Here, ϕ : [0, ∞) → [0, ∞) is a continuous function such that ϕ(x, y) = 0 if and only if  x = y = 0, and α ∈ 0, 21 . Then M has a fixed point x∗ ∈ X . Further, for each x0 ∈ X , some iterated sequence {xn } with xn ∈ Mxn−1 converges to a fixed point. 2

Proof. Given x0 ∈ X , if x0 ∈ Mx0 , our proof is complete. Otherwise, from the fact that Mx0 has UCAV , there exists x1 ∈ Mx0 with x1 ̸= x0 and x1 ≽ x0 such that d(x0 , x1 ) = inf d(x, x0 ) = D(Mx0 , x0 ). x∈Mx0

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We continue the procedure of constructing xn inductively. That is, either xn−1 ∈ Mxn−1 , in which case our proof is complete, or there exists xn ∈ Mxn−1 with xn ̸= xn−1 and xn ≽ xn−1 such that d(xn−1 , xn ) = D(Mxn−1 , xn−1 ),

n = 1, 2, . . . .

(2.2)

On the other hand, D(Mxn−1 , xn−1 ) ≤

sup D(Mxn−1 , x) ≤ Hd (Mxn−1 , Mxn−2 ). x∈Mxn−2

Therefore, d(xn , xn+1 ) ≤ Hd (Mxn−1 , Mxn ),

for n = 1, 2, . . . .

(2.3)

Now, by (2.2) and (2.3), we get d(xn , xn+1 ) ≤ Hd (Mxn−1 , Mxn )

≤ ≤ ≤ = Therefore d(xn , xn+1 ) ≤

α[D(xn−1 , Mxn ) + D(xn , Mxn−1 )] − ϕ(D(xn−1 , Mxn ), D(xn , Mxn−1 )) α[D(xn−1 , Mxn ) + D(xn , Mxn−1 )] α[d(xn−1 , xn ) + D(xn , Mxn )] α[d(xn−1 , xn ) + d(xn , xn+1 )]. α d(xn−1 , xn ). 1−α

α Set k = 1−α < 1. By repeating this procedure, we obtain

d(xn , xn+1 ) ≤ kd(xn−1 , xn ) ≤ · · · ≤ kn d(x0 , x1 ). Let m, n ∈ N, with n > m. Then, in virtue of the triangular inequality, we have d(xn , xm ) ≤ d(xm , xm+1 ) + d(xm+1 , xm+2 ) + · · · + d(xn−1 , xn )

=

n−1 

d(xi , xi+1 )

i=m

≤

n−1 

ki d(x0 , x1 ).

i=m

Let m, n → ∞. By the above inequality, it follows that {xn } is a ≼-nondecreasing Cauchy sequence. Since that X is sequentially complete, {xn } is convergent. We denote limn→∞ xn = x∗ for x∗ ∈ X . Finally, we prove that x∗ is a fixed point of M. Since xn ≼ x∗ , by (2.1), we have D(xn+1 , Mx∗ ) ≤ Hd (Mxn , Mx∗ )

≤ < ≤ =

α[D(xn , Mx∗ ) + D(x∗ , Mxn )] − ϕ(D(xn , Mx∗ ), D(x∗ , Mxn )) α[D(xn , Mx∗ ) + D(x∗ , Mxn )] α[D(xn , Mx∗ ) + d(xn , x∗ ) + D(xn , Mxn )] α[D(xn , Mx∗ ) + d(xn , x∗ ) + d(xn , xn+1 )].

Let n → ∞. Since α < 1, we get D(x∗ , Mx∗ ) ≤ α D(x∗ , Mx∗ ) < D(x∗ , Mx∗ ). This is a contradiction, so D(x∗ , Mx∗ ) = 0. Since Mx∗ is approximative, there exists y ∈ P(Mx∗ ) (x∗ ) such that d(y, x∗ ) = 0, i.e., y = x∗ . Hence x∗ ∈ Mx∗ . The proof is complete.  Similarly, we have the following. Theorem 3. Let (X , d, ≼) be a sequentially complete metric space with the following property (which appears in [13]). Any ≼-nonincreasing sequence {xn } with xn → x∗ implies that x∗ ≼ xn for each n ∈ N. Suppose that the multi-valued mapping M : X → N (X ) has LCAV and satisfies

Hd (Mx, My) ≤ α[D(x, My) + D(y, Mx)] − ϕ(D(x, My), D(y, Mx)),

(2.4)

for all x, y ∈ X which are comparable. Here, ϕ : [0, ∞) → [0, ∞) is a continuous function such that ϕ(x, y) = 0 if and only if  x = y = 0, and α ∈ 0, 12 . Then M has a fixed point x∗ ∈ X . Further, for each x0 ∈ X , some iterated sequence {xn } with xn ∈ Mxn−1 converges to a fixed point. 2

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Theorem 4. Let (X , d, ≼) be a sequentially complete metric space with the following property (which appears in [13]). Any ≼-nondecreasing sequence {xn } with xn → x∗ implies xn ≼ x∗ for each n ∈ N. Suppose that the multi-valued mapping M : X → N (X ) has AV , is nondecreasing, and satisfies

Hd (Mx, My) ≤ α[D(x, My) + D(y, Mx)] − ϕ(D(x, My), D(y, Mx)),

(2.5)

for all x, y ∈ X which are comparable. Here, ϕ : [0, ∞) → [0, ∞) is a continuous function such that ϕ(x, y) = 0 if and only if  x = y = 0, and α ∈ 0, 12 . If there exists x0 ∈ X such that {x0 } ≼ Mx0 , then M has a fixed point x∗ ∈ X . Further, for each x0 ∈ X , some iterated sequence {xn } with xn ∈ Mxn−1 converges to a fixed point. 2

Proof. If x0 ∈ Mx0 , then the proof is finished. Otherwise, for any x ∈ Mx0 , one has that x ≽ x0 . Since M has approximative values, there exists x1 ∈ Mx0 with x1 ≽ x0 and d(x0 , x1 ) = D(x0 , Mx0 ). We continue the procedure of constructing xn inductively; that is, either xn−1 ∈ Mxn−1 , in which case our proof is complete, or there exists xn ∈ Mxn−1 with xn ̸= xn−1 and xn ≽ xn−1 such that (2.2) and (2.3) hold. The rest of this proof is the same as that of Theorem 2.  Theorem 5. Let (X , d, ≼) be a sequentially complete metric space with the following property (which appears in [9]). If {xn } is a sequence whose consecutive terms are comparable such that xn → x as n → ∞ in X , then there exists a subsequence {xnk } of {xn } such that every term is comparable to the limit x. Suppose that the multi-valued mapping M : X → N (X ) has CAV and satisfies

Hd (Mx, My) ≤ α[D(x, My) + D(y, Mx)] − ϕ(D(x, My), D(y, Mx)),

(2.6)

for all x, y ∈ X which are comparable. Here, ϕ : [0, ∞)2 → [0, ∞) is a continuous function such that ϕ(x, y) = 0 if and only if  x = y = 0, and α ∈ 0, 12 . Then M has a fixed point x∗ ∈ X . Further, for each x0 ∈ X , some iterated sequence {xn } with xn ∈ Mxn−1 converges to a fixed point. Proof. As an analogue of the proof of Theorem 2, by means of the property that M has CAV , we obtain that a sequence {xn } whose consecutive terms are comparable satisfies (2.2), (2.3), and the following results: xn+1 ∈ Mxn ,

lim xn = x∗ .

n→∞

Now, we prove that x∗ is a fixed point of M. By using the above hypothesis, the fact that {xn } has a subsequence {xnk } whose every term is comparable to x∗ , and (2.6), we have D(xnk +1 , Mx∗ ) ≤ Hd (Mxnk , Mx∗ )

≤ α[D(xnk , Mx∗ ) + D(x∗ , Mxnk )] − ϕ(D(xnk , Mx∗ ), D(x∗ , Mxnk )) < α[D(xnk , Mx∗ ) + D(x∗ , Mxnk )] ≤ α[D(xnk , Mx∗ ) + d(x∗ , xnk ) + D(xnk , Mxnk )]. By letting k go to infinity, D(x∗ , Mx∗ ) ≤ α D(x∗ , Mx∗ ) < D(x∗ , Mx∗ ). This reduces to D(x∗ , Mx∗ ) = 0. Since Mx∗ is approximative, there exists y ∈ P(Mx∗ ) (x∗ ) such that d(y, x∗ ) = 0, i.e., y = x∗ . Hence x∗ ∈ Mx∗ . The proof is complete.  When M is a single-valued mapping, we obtain the following corollaries. Corollary 1. Let (X , d, ≼) be a sequentially complete metric space. Suppose that the single-valued map M satisfies d(Mx, My) ≤ α[d(x, My) + d(y, Mx)] − ϕ(d(x, My), d(y, Mx)),

(2.7)

for all x, y ∈ X which are comparable. Here, ϕ : [0, ∞)2 → [0, ∞) is a continuous function such that ϕ(x, y) = 0 if and only if  x = y = 0, and α ∈ 0, 12 . If the following conditions hold:

(c1 ) any ≼-nondecreasing (≼-nonincreasing) sequence {xn } with xn → x∗ implies xn ≼ x∗ (xn ≽ x∗ ) for each n ∈ N, (c2 ) Mx ≽ x (Mx ≼ x) for each x ∈ X , then M has a fixed point x∗ ∈ X . Furthermore, for each x0 ∈ X , the iterated sequence {xn } with xn = Mxn−1 converges to the fixed point x∗ .

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Proof. As an analogue of Theorem 2, we can obtain a sequence {xn } which satisfies xn ≼ M (xn ) = xn+1 for n = 1, 2, . . .. Moreover, d(xn , xn+1 ) = d(Mxn−1 , Mxn )

≤ α[d(xn−1 , Mxn ) + d(xn , Mxn−1 )] − ϕ(d(xn−1 , Mxn ), d(xn , Mxn−1 )) < α[d(xn−1 , Mxn ) + d(xn , Mxn−1 )] ≤ α[d(xn−1 , xn ) + d(xn , Mxn )]. Therefore, d(xn , xn+1 ) ≤

α 1−α

d(xn−1 , xn ).

α Set k = 1−α < 1. By repeating this procedure, we obtain

d(xn , xn+1 ) ≤ kd(xn−1 , xn ) ≤ · · · ≤ kn d(x0 , x1 ). Now, we show that {xn } is a Cauchy sequence. Let m, n ∈ N, with n > m. Then, in virtue of the triangular inequality, we have d(xn , xm ) ≤ d(xm , xm+1 ) + d(xm+1 , xm+2 ) + · · · + d(xn−1 , xn )

=

n−1 

d(xi , xi+1 )

i=m

≤

n−1 

ki d(x0 , x1 ).

i=m

Let m, n → ∞. By the above inequality, it follows that {xn } is a ≼-nondecreasing Cauchy sequence. Since X is sequentially complete, {xn } is convergent. We denote limn→∞ xn = x∗ for x∗ ∈ X . Finally, we prove that x∗ is a fixed point of M. To this end, we note that (c1 ) guarantees that xn is comparable to x∗ for n = 1, 2, . . .. Therefore, d(xn+1 , Mx∗ ) = d(Mxn , Mx∗ )

≤ α[d(xn , Mx∗ ) + d(x∗ , Mxn )] − ϕ(d(xn , Mx∗ ), d(x∗ , Mxn )) < α[d(xn , Mx∗ ) + d(x∗ , xn+1 )]. Letting n → ∞, we obtain d(x∗ , Mx∗ ) = 0. Hence, Mx∗ = x∗ .



Corollary 2. Let (X , d, ≼) be a sequentially complete metric space. Suppose that the single-valued map M satisfies d(Mx, My) ≤ α[d(x, My) + d(y, Mx)] − ϕ(d(x, My), d(y, Mx)),

(2.8)

for all x, y ∈ X which are comparable. Here, ϕ : [0, ∞) → [0, ∞) is a continuous function such that ϕ(x, y) = 0 if and only if  x = y = 0, and α ∈ 0, 12 . If the following conditions hold: 2

(c1 ) any ≼-nondecreasing (≼-nonincreasing) sequence {xn } with xn → x∗ implies that xn ≼ x∗ (xn ≽ x∗ ) for each n ∈ N, (c2 ) M is increasing and there exists x0 ∈ X such that x0 ≼ Mx0 , then M has a fixed point x∗ ∈ X . Proof. Taking x1 = Mx0 and xn+1 = Mxn for n = 1, 2, . . ., we easily see that {xn } is a ≼-nondecreasing sequence. The rest of this proof is the same as that in Corollary 1.  Corollary 3. Let (X , d, ≼) be a sequentially complete metric space. Suppose that the single-valued map M satisfies d(Mx, My) ≤ α[d(x, My) + d(y, Mx)] − ϕ(d(x, My), d(y, Mx)),

(2.9)

for all x, y ∈ X which are comparable. Here, ϕ : [0, ∞) → [0, ∞) is a continuous function such that ϕ(x, y) = 0 if and only if  x = y = 0, and α ∈ 0, 12 . If the following conditions hold: 2

(c1 ) if {xn } is a sequence whose consecutive terms are comparable such that xn → x as n → ∞ in X , then there exists a subsequence {xnk } of {xn } such that every term is comparable to the limit x, (c2 ) Mx is comparable to x for each x ∈ X , then M has a fixed point x∗ ∈ X . Furthermore, for each x0 ∈ X , the iterated sequence {xn } with xn = Mxn−1 converges to the fixed point x∗ .

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Proof. By using the proof of Corollary 1, we can prove that the sequence {xn+1 } with xn+1 = Mxn is convergent to x∗ in X . (c1 ) shows that there exists a subsequence {xnk } consisting of terms which are comparable to x∗ . Therefore, d(xnk +1 , Mx∗ ) = d(Mxnk , Mx∗ )

≤ α[d(xnk , Mx∗ ) + d(x∗ , Mxnk )] − ϕ(d(xnk , Mx∗ ), d(x∗ , Mxnk )) < α[d(xnk , Mx∗ ) + d(x∗ , xnk +1 )]. By letting k go to infinity, d(x∗ , Mx∗ ) ≤ α D(x∗ , Mx∗ ) < D(x∗ , Mx∗ ). This reduces to d(x∗ , Mx∗ ) = 0, i.e., x∗ = Mx∗ .



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