Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric

Nonlinear Analysis 74 (2011) 1804–1813

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Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric✩ Mircea-Dan Rus ∗ Department of Mathematics, Faculty of Automation and Computer Science, Technical University of Cluj-Napoca, 400027 Cluj-Napoca, Romania

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Article history: Received 27 July 2010 Accepted 26 October 2010

abstract In this paper, we study the existence and uniqueness of (coupled) fixed points for mixed monotone mappings in partially ordered metric spaces with semi-monotone metric. As an application, we prove the existence and uniqueness of the solution for a first-order differential equation with periodic boundary conditions. © 2010 Elsevier Ltd. All rights reserved.

MSC: 47H10 34B15 Keywords: Partially ordered set Semi-monotone metric Mixed monotone mapping Fixed point Coupled fixed point Symmetric Meir–Keeler type contraction Coupled upper and lower solution Periodic boundary value problem

1. Introduction In recent years, there has been an increasing interest in the study of fixed points for mappings that possess monotonicity type properties, in the context of partially ordered metric spaces (cf. [1–18]). The used approach was to combine some contraction principle (e.g., Banach’s contraction principle, or any of its many generalizations) with the method of monotone iterations and the method of upper and lower solutions, while weakening the contraction condition. Following this trend, Gnana Bhaskar and Lakshmikantham [3], Drici et al. [6], Lakshmikantham and Ćirić [10], Samet [16], Choudhury and Kundu [13] investigated the existence and the uniqueness of fixed points and coupled fixed points of mixed monotone mappings in partially ordered metric spaces and obtained important results which were then applied to the study of several nonlinear problems. Recall that if (X , ≼) is a partially ordered set, a bivariate mapping A : X × X → X is said to be mixed monotone (or is said to have the mixed monotone property) (cf. [19,3]) if A is nondecreasing in the first argument and nonincreasing in the second argument, i.e., x1 , x2 , y ∈ X ,

x1 ≼ x2 ⇒ A(x1 , y) ≼ A(x2 , y)

x, y1 , y2 ∈ X ,

y1 ≼ y2 ⇒ A(x, y1 ) ≽ A(x, y2 ).

and

✩ This paper is supported by the Project PN2—Partnership No. 11018 MoDef.

∗

Tel.: +40 745708961; fax: +40 364814987. E-mail addresses: [email protected], [email protected].

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.10.053

M.-D. Rus / Nonlinear Analysis 74 (2011) 1804–1813

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A pair (x, y) ∈ X × X is called a coupled fixed point of a bivariate mapping A : X × X → X (cf. [19,3]) if A(x, y) = x and

A(y, x) = y.

Also, a point x ∈ X is called a fixed point of A (cf. [19]) if A(x, x) = x. In [10], Lakshmikantham and Ćirić proved the following existence result for the (coupled) fixed points of mixed monotone mappings that satisfy a generalized φ -contraction condition and also discussed the uniqueness of a (coupled) fixed point. Theorem 1.1 (Lakshmikantham and Ćirić [10]). Let (X , ≼) be a partially ordered set and d a complete metric on X . Let A : X × X → X be a mixed monotone mapping and assume there exists a function φ : [0, +∞) → [0, +∞) with

φ(t ) < t and

lim φ(s) < t s→t

for all t > 0,

such that d(A(x, y), A(u, v)) ≤ φ



d(x, u) + d(y, v)



2

for all x, y, u, v ∈ X with x ≼ u, y ≽ v.

(1)

Also suppose either: (a) A is continuous, or (b) X has the following properties: (i) if (xn ) is a nondecreasing sequence that is convergent to x, then xn ≼ x for all n; (ii) if (yn ) is a nonincreasing sequence that is convergent to y, then yn ≽ y for all n. If there exist x0 , y0 ∈ X such that x0 ≼ A(x0 , y0 ) and

y0 ≽ A(y0 , x0 ),

then A has a coupled fixed point. In addition, if x0 , y0 are comparable, then A has a fixed point. In [16], Samet obtained a similar result for mixed strict monotone mappings that satisfy a generalized Meir–Keeler type contraction condition in place of the generalized φ -contraction condition (1). Recall the following definitions from [16]: Definition 1.1 (Samet [16]). Let (X , ≼) be a partially ordered set. A mapping A : X × X → X is said to be mixed strict monotone (or is said to have the mixed strict monotone property) if A is increasing in the first argument and decreasing in the second argument, i.e., x1 , x2 , y ∈ X ,

x1 ≺ x2 ⇒ A(x1 , y) ≺ A(x2 , y)

x, y1 , y2 ∈ X ,

y1 ≺ y2 ⇒ A(x, y1 ) ≻ A(x, y2 ).

and

Definition 1.2 (Samet [16]). Let (X , ≼) be a partially ordered set and d a metric on X . A mapping A : X × X → X is called a generalized Meir–Keeler type function if for each ε > 0 there exists δ = δ(ε) > 0 such that x, y, u, v ∈ X ,

x ≼ u, y ≽ v,

ε≤

d(x, u) + d(y, v) 2

< ε + δ ⇒ d(A(x, y), A(u, v)) < ε.

(2)

The main results in [16] (Theorems 2.1–2.5) are combined in the following theorem. Note that, in order to simplify the statement of the theorem, we will call a partial ordered set (X , ≼) quasi-directed if every pair of elements from X has a lower bound or an upper bound. A simple argument shows (see [2, p. 231]) that X is quasi-directed if and only if for every x, y ∈ X there exists z ∈ X such that z is comparable with both x and y. Also, when referring to the partial order on the product space X × X , we understand the following partial order:

(x, y), (u, v) ∈ X × X ,

(x, y) . (u, v) ⇔ x ≼ u, y ≽ v.

Theorem 1.2 (Samet [16]). Let (X , ≼) be a partially ordered set and d a complete metric on X . Let A : X × X → X be a generalized Meir–Keeler type function that has the mixed strict monotone property. Suppose either: (a) A is continuous, or (b) X has the following properties: (i) if (xn ) is an increasing sequence that is convergent to x, then xn ≺ x for all n; (ii) if (yn ) is a decreasing sequence that is convergent to y, then yn ≻ y for all n. If there exist x0 , y0 ∈ X such that x0 ≺ A(x0 , y0 ) and

y0 ≽ A(y0 , x0 ),

then A has a coupled fixed point.

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M.-D. Rus / Nonlinear Analysis 74 (2011) 1804–1813

In addition, 1. if x0 , y0 are comparable, then A has a fixed point; 2. if (X , ≼) is quasi-directed, then each coupled fixed point of A has equal components (hence, A has a fixed point); 3. if (X × X , .) is quasi-directed, then A has a unique coupled fixed point (hence, a unique fixed point). In this paper, we extend and complement the results of Lakshmikantham and Ćirić [10] (Theorem 1.1) and the results of Samet [16] (Theorem 1.2) under the assumption that the metric has an additional monotonicity type property, called semi-monotonicity (see Definition 2.3). We investigate both the existence and the uniqueness of (coupled) fixed points for mixed monotone mappings in partially ordered metric spaces with semi-monotone metric by the method of successive approximations, while weakening contraction conditions (1), (2) and assumptions (a), (b) in Theorems 1.1 and 1.2. As an application, we prove the existence and uniqueness of the solution for a first-order differential equations with periodic boundary conditions and complement a similar result of Gnana Bhaskar and Lakshmikantham [3]. 2. Main results Before we formulate and prove the main fixed point theorems, we need to introduce and study some new notions. 2.1. The s-composition of bivariate mappings The following concept is motivated by the necessity of properly defining the iterates of bivariate mappings, in order to be able to apply the method of successive approximation in the study of (coupled) fixed points. We also refer to Gnana Bhaskar and Lakshmikantham [3, p. 1381] and Samet [16, p. 4510] for a direct approach on this matter. Definition 2.1. Let X , Y , Z be nonempty sets and A : X × X → Y , B : Y × Y → Z . We define the symmetric composition (or, the s-composition for short) of A and B by B ∗ A : X × X → Z,

(B ∗ A)(x, y) = B(A(x, y), A(y, x)) (x, y ∈ X ).

For each nonempty set X , denote by PX the projection mapping PX : X × X → X ,

P (x, y) = x (x, y ∈ X ).

The following basic properties of the s-composition follow in an elementary manner, hence we omit the proofs. Proposition 2.1 (Associativity). If A : X × X → Y , B : Y × Y → Z and C : Z × Z → W , then (C ∗ B) ∗ A = C ∗ (B ∗ A). Proposition 2.2 (Identity Element). If A : X × X → Y , then A ∗ PX = PY ∗ A = A. Proposition 2.3 (Mixed Monotonicity). If (X , ≼), (Y , ≼), (Z , ≼) are partially ordered sets and the mappings A : X × X → Y , B : Y × Y → Z are mixed monotone, then B ∗ A is mixed monotone. Remark 2.1. If A is a bivariate self-map of X , i.e., A : X × X → X , then a direct consequence of Propositions 2.1 and 2.2 is that one can define the functional powers (i.e., the iterates) of the mapping A with respect to the s-composition (see also [3, p. 1381], [16, p. 4510]) by An+1 = A ∗ An = An ∗ A (n = 0, 1, . . .), A0 = PX . Moreover, if (X , ≼) is a partially ordered set and A is mixed monotone, then An is mixed monotone for every n, by Proposition 2.3. 2.2. Symmetric Meir–Keeler type contractions In what follows, we define and study a new type of Meir–Keeler contraction condition for bivariate mappings. Definition 2.2. Let (X , ≼) be a partially ordered set and d a metric on X . We say that a mapping A : X × X → X is a symmetric Meir–Keeler type contraction if for each ε > 0 there exists δ = δ(ε) > 0 such that x, y ∈ X ,

x ≺ y,

ε ≤ d(x, y) < ε + δ ⇒ d(A(x, y), A(y, x)) < ε.

(3)

Remark 2.2. By letting u := y and v := x in (2), we obtain (3), hence any generalized Meir–Keeler type function (see Definition 1.2) is a symmetric Meir–Keeler type contraction. Remark 2.3. Using the s-composition, one may choose to write (d ∗ A)(x, y) in place of d(A(x, y), A(y, x)), for short.

M.-D. Rus / Nonlinear Analysis 74 (2011) 1804–1813

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Some properties of symmetric Meir–Keeler type contractions are discussed next. Proposition 2.4. Let (X , ≼) be a partially ordered set and d a metric on X . If a mapping A : X × X → X is a symmetric Meir–Keeler type contraction, then d(A(x, y), A(y, x)) < d(x, y) for all x, y ∈ X comparable, with x ̸= y

(4)

and lim d(An (x, y), An (y, x)) = 0 for all x, y ∈ X comparable.

(5)

n→∞

Proof. Let x, y ∈ X be comparable. If x = y, then (5) is obviously verified, hence we may assume that x ≺ y due to the symmetry of (4) and (5). In order to obtain (4), it is enough to use (3) with ε := d(x, y) > 0. Next, we prove (5). Let εn := d(An (x, y), An (y, x)) for every n. If there exists k such that Ak (x, y) = Ak (y, x), then clearly An (x, y) = An (y, x) (hence, εn = 0) for all n ≥ k, making the conclusion trivial. Therefore, we may assume that An (x, y) ̸= An (y, x) for all n and, by further using the mixed monotonicity of An (Remark 2.1), it follows that An (x, y) ≺ An (y, x) for all n. By applying (4), we obtain that 0 < εn+1 = d(An+1 (x, y), An+1 (y, x)) = d A An (x, y), An (y, x) , A An (y, x), An (x, y)

 







< d(An (x, y), An (y, x)) = εn for all n, proving that (εn ) is a decreasing sequence of positive numbers, hence convergent to some ε ≥ 0. We prove that ε = 0 by assuming the contrary, i.e., ε > 0. Let δ = δ(ε) > 0 such that (3) is satisfied and let k be such that εk ∈ [ε, ε + δ). Then, by (3),

ε ≤ εk = d(Ak (x, y), Ak (y, x)) < ε + δ ⇒ εk+1 = d(Ak+1 (x, y), Ak+1 (y, x)) < ε, which is clearly a contradiction and concludes the argument.



Corollary 2.1. Let (X , ≼) be a partially ordered set and d a metric on X . If a mapping A : X × X → X is a symmetric Meir–Keeler type contraction and (x, y) ∈ X × X is a coupled fixed point of A such that x, y are comparable, then x = y. Proof. This is a direct consequence of (4) in Proposition 2.4.



2.3. Partially ordered metric spaces with semi-monotone metric The following concept is a natural extension of the notion of semi-monotone norm from partially ordered normed linear spaces (see, e.g., [20, p. 24]) to the framework of partially ordered metric spaces. Definition 2.3. Let (X , ≼) be a partially ordered set and d a metric on X . We say that d is semi-monotone if there exists a constant c ≥ 1 such that x, y, u, v ∈ X ,

x ≼ u ≼ v ≼ y ⇒ d(u, v) ≤ c · d(x, y).

If c = 1, then d is said to be monotone. Next, we present some properties of partially ordered metric spaces with semi-monotone metric. Proposition 2.5. Let (X , ≼) be a partially ordered set and d a semi-monotone metric on X . Let (xn ), (yn ), (un ) be sequences in X and x ∈ X such that xn ≼ un ≼ yn

for all n

(xn ) → x,

(yn ) → x (as n → ∞).

(6)

and

Then (un ) → x (as n → ∞). Proof. By applying the triangle inequality, (6) and the semi-monotonicity of d, it follows that d(un , x) ≤ d(un , xn ) + d(xn , x) ≤ c · d(xn , yn ) + d(xn , x), which proves, by letting n → ∞, that (un ) → x.



In what follows, if (X , ≼) is a partially ordered set and x, y ∈ X are such that x ≼ y, then by [x, y] we mean the set of all elements z ∈ X such that x ≼ z ≼ y.

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M.-D. Rus / Nonlinear Analysis 74 (2011) 1804–1813

Proposition 2.6. Let (X , ≼) be a partially ordered set and d a semi-monotone metric on X . Let (xn ), (yn ) be sequences in X such that xn ≼ yn for all n. We assume either (a) n≥0 [xn , yn ] is nonempty, or (b) (xn ) is nondecreasing, (yn ) is nonincreasing and d is complete.



If lim n→∞ d(xn , yn ) = 0, then (xn ), (yn ) are convergent and have the same limit, say, x. Moreover, considering assumption (a), then n≥0 [xn , yn ] = {x}. Proof. First, assume (a) and let x ∈ d(xn , x) ≤ c · d(xn , yn )



n≥0

[xn , yn ]. By the semi-monotonicity of d it follows that

and d(yn , x) ≤ c · d(xn , yn ) for all n,

which assures the conclusion. Moreover, n≥0 [xn , yn ] = {x} by the uniqueness of the limit. Now, assume (b). Since xn ≼ xn+p ≼ yn+p ≼ yn for all n, p, it follows by the semi-monotonicity of d that



d(xn , xn+p ) ≤ c · d(xn , yn ) and

d(yn , yn+p ) ≤ c · d(xn , yn ) for all n, p,

meaning that (xn ), (yn ) are Cauchy sequences, hence convergent by the completeness of d. Also, if (xn ) → x, (yn ) → y (as n → ∞), then d(x, y) = limn→∞ d(xn , yn ) = 0, hence x = y.  2.4. Fixed point and coupled fixed point theorems The main results of this paper are the following (coupled) fixed point theorems for symmetric Meir–Keeler type contractions with the mixed monotone property that extend and complement the results of Samet [16] (Theorem 1.2). As a direct consequence, we also establish a fixed point theorem that complements the result of Lakshmikantham and Ćirić [10] (Theorem 1.1). In our first theorem, we investigate the existence and local uniqueness of (coupled) fixed points. Theorem 2.1. Let (X , ≼) be a partially ordered set and d a semi-monotone and complete metric on X . Let A : X × X → X be a symmetric Meir–Keeler type contraction that has the mixed monotone property. Suppose either: (a) A has the following property: if (xn ) is a nondecreasing sequence in X and (yn ) is a nonincreasing sequence in X such that (xn ) → x, (yn ) → x (as n → ∞) and xn ≼ yn for all n, then (A(xn , yn )) → A(x, x) or (A(yn , xn )) → A(x, x) (as n → ∞), or (b) X has the following property: if (xn ) is a nondecreasing sequence in X and (yn ) is a nonincreasing sequence in X such that (xn ) → x, (yn ) → x (as n → ∞) and xn ≼ yn for all n, then xn ≼ x ≼ yn for all n. If there exist x0 , y0 ∈ X comparable such that x0 ≼ A(x0 , y0 ) and

y0 ≽ A(y0 , x0 ),

(7)

then x0 ≼ y0 , the mapping A has a fixed point x∗ ∈ X and the sequence (An (x, y)) converges to x∗ for each x, y ∈ [x0 , y0 ]. Furthermore, considering assumption (b), then x∗ ∈ [x0 , y0 ], (x∗ , x∗ ) is the unique coupled fixed point of A in [x0 , y0 ]×[x0 , y0 ] and x∗ is the unique fixed point of A in [x0 , y0 ]. Proof. Consider the sequences (xn ), (yn ) in X defined by xn = An (x0 , y0 ),

yn = An (y0 , x0 )

(n = 0, 1, . . .).

Equivalently, xn+1 = A(xn , yn ),

yn+1 = A(yn , xn )

(n = 0, 1, . . .).

(8)

Since An is mixed monotone for every n (Remark 2.1), it follows by (7) that xn = An (x0 , y0 ) ≼ An+1 (x0 , y0 ) = xn+1 ,

yn = An (y0 , x0 ) ≽ An+1 (y0 , x0 ) = yn+1

for all n,

hence

(xn ) is nondecreasing and (yn ) is nonincreasing.

(9)

Moreover, since x0 , y0 are comparable, it follows by Proposition 2.4 that lim d(xn , yn ) = 0.

n→∞

(10)

We next show that x0 ≼ y0 is the only possible situation. Indeed, by assuming the contrary, i.e., x0 ≻ y0 , (9) leads to yn ≼ y0 ≺ x0 ≼ xn for all n, hence 0 < d(x0 , y0 ) ≤ c · d(xn , yn ) for all n by the semi-monotonicity of d, which clearly contradicts (10).

M.-D. Rus / Nonlinear Analysis 74 (2011) 1804–1813

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Next, x0 ≼ y0 and the mixed monotonicity of An leads to xn = An (x0 , y0 ) ≼ An (y0 , x0 ) = yn

for all n.

All the conditions in Proposition 2.6(b) are now satisfied, therefore (xn ), (yn ) are convergent and have the same limit, say, x∗ . We prove that x∗ is a fixed point of A. Considering assumption (a), then xn+1 = A(xn , yn ) → A(x∗ , x∗ ) or yn+1 = A(yn , xn ) → A(x∗ , x∗ )

(as n → ∞),

hence A(x , x ) = x . Considering assumption (b), then ∗

∗

∗

xn ≼ x∗ ≼ yn

for all n,

(11)

hence x ∈ [x0 , y0 ] and ∗



[xn , yn ] = {x∗ }

(12)

n ≥0

by Proposition 2.6(a). Using (9), the mixed monotonicity of A and (11), we have x0 ≼ xn+1 = A(xn , yn ) ≼ A(x∗ , x∗ ) ≼ A(yn , xn ) = yn+1 ≼ y0

for all n,

hence A(x , x ) ∈ n≥0 [xn , yn ], which proves by (12) that A(x , x ) = x∗ . Now, let x, y ∈ [x0 , y0 ]. Since An is mixed monotone, we obtain that ∗

∗

∗



xn = An (x0 , y0 ) ≼ An (x, y) ≼ An (y0 , x0 ) = yn

∗

for all n,

hence (A (x, y)) → x (as n → ∞), by Proposition 2.5. In particular, if (x, y) is a coupled fixed point of A in [x0 , y0 ]×[x0 , y0 ], then ∗

n

x = A n ( x, y ) → x∗ ,

y = An (y, x) → x∗

∗

hence x = y = x and the proof is complete.

(as n → ∞),



Next, we study the problem of global uniqueness of (coupled) fixed points. Theorem 2.2. Let (X , ≼) be a partially ordered set and d a semi-monotone metric on X . Let A : X × X → X be a symmetric Meir–Keeler type contraction that has the mixed monotone property. Assume that (X × X , .) is quasi-directed and that A has a coupled fixed point (x∗ , y∗ ). Then x∗ = y∗ , (x∗ , x∗ ) is the unique coupled fixed point of A in X × X , x∗ is the unique fixed point of A in X and the sequence n (A (x, y)) converges to x∗ for each x, y ∈ X . Proof. First, we show that each nonempty finite subset of X has a lower bound and an upper bound. Clearly, it is enough to show that for each x, y ∈ X , there exist u, v ∈ X such that x, y ∈ [u, v]. Let x, y ∈ X . Since (X × X , .) is quasi-directed, {(x, y), (y, x)} has a lower bound or an upper bound. If (u, v) is a lower bound for {(x, y), (y, x)}, then (u, v) . (x, y) and (u, v) . (y, x), i.e., u ≼ x,

v ≽ y and u ≼ y,

v ≽ x,

hence x, y ∈ [u, v]. Similarly, if (u, v) is an upper bound for {(x, y), (y, x)}, then x, y ∈ [v, u]. Next, we prove that lim d(An (x, y), An (u, v)) = 0

n→∞

for all x, y, u, v ∈ X .

(13)

Let x, y, u, v ∈ X and let s, t ∈ X be a lower bound and an upper bound, respectively, for {x, y, u, v}, i.e., x, y, u, v ∈ [s, t ]. This assures, by the mixed monotonicity of An (Remark 2.1), that An (s, t ) ≼ An (x, y) ≼ An (t , s) and An (s, t ) ≼ An (u, v) ≼ An (t , s) for all n. By further applying the triangle inequality and the semi-monotonicity of d, it follows that d(An (x, y), An (u, v)) ≤ d(An (x, y), An (s, t )) + d(An (s, t ), An (u, v))

≤ 2c · d(An (s, t ), An (t , s)) for all n. Since limn→∞ d(An (s, t ), An (t , s)) = 0 by (5) in Proposition 2.4, the proof of (13) is complete. Now, let (x∗ , y∗ ) ∈ X × X be a coupled fixed point of A. By (13), d(x∗ , y∗ ) = d(An (x∗ , y∗ ), An (y∗ , x∗ )) → 0

(as n → ∞),

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M.-D. Rus / Nonlinear Analysis 74 (2011) 1804–1813

hence x∗ = y∗ . Moreover, if x, y ∈ X , then d(An (x, y), x∗ ) = d(An (x, y), An (x∗ , x∗ )) → 0 (as n → ∞) by (13), hence (An (x, y)) converges to x∗ . In particular, if (x, y) is a coupled fixed point of A, then x = An (x, y) → x∗ ,

y = An (y, x) → x∗

∗

hence x = y = x and the proof is complete.

(as n → ∞),



The following result is a consequence of Theorems 2.1 and 2.2. Theorem 2.3. In addition to the hypotheses of Theorem 2.1, suppose that (X × X , .) is quasi-directed. Then (x∗ , x∗ ) is the unique coupled fixed point of A in X × X , x∗ is the unique fixed point of A in X and the sequence (An (x, y)) converges to x∗ for each x, y ∈ X . Proof. By Theorem 2.1, x∗ is a fixed point of A (i.e., (x∗ , x∗ ) is a coupled fixed point of A). The conclusion now follows by Theorem 2.2.  Remark 2.4. By comparing our results (Theorems 2.1–2.3) with the results of Samet [16] (Theorem 1.2 in this paper), we remark that in the case of partially ordered metric spaces with semi-monotone metric, our results yield better conclusions on the existence, uniqueness and effective construction of fixed points by the method of successive approximation. Remark 2.5. From the proofs of Theorems 2.1–2.3, we can see that the contraction condition on A of symmetric Meir–Keeler type has been used exclusively for proving (10) and (13) by means of Proposition 2.4, hence can be replaced with the weaker assumption (see (5) in Proposition 2.4): lim d(An (x, y), An (y, x)) = 0

n→∞

for all x, y ∈ X with x ≺ y.

(14)

We conclude this section with a consequence of Theorems 2.1 and 2.3. Our result is an improvement of Theorem 1.1 [10] when the metric is semi-monotone. Theorem 2.4. Let (X , ≼) be a partially ordered set and d a semi-monotone and complete metric on X . Let A : X × X → X be a mixed monotone mapping and assume there exists a function φ : (0, +∞) → [0, +∞) with

φ(t ) < t and

lim sup φ(s) < t s→t +

for all t > 0,

(15)

such that d(A(x, y), A(y, x)) ≤ φ (d(x, y))

for all x, y ∈ X with x ≺ y.

(16)

Suppose that either assumptions (a), (b) in Theorem 2.1 takes place. If there exist x0 , y0 ∈ X comparable such that x0 ≼ A(x0 , y0 ) and

y0 ≽ A(y0 , x0 ),

then x0 ≼ y0 , the mapping A has a fixed point x∗ ∈ X and the sequence (An (x, y)) converges to x∗ for each x, y ∈ [x0 , y0 ]. Furthermore, 1. considering assumption (b) of Theorem 2.1, then x∗ ∈ [x0 , y0 ], (x∗ , x∗ ) is the unique coupled fixed point of A in [x0 , y0 ] × [x0 , y0 ] and x∗ is the unique fixed point of A in [x0 , y0 ]; 2. if (X × X , .) is quasi-directed, then (x∗ , x∗ ) is the unique coupled fixed point of A in X × X , x∗ is the unique fixed point of A in X and the sequence (An (x, y)) converges to x∗ for each x, y ∈ X . Proof. By Theorems 2.1 and 2.3, it is enough to prove that A is a symmetric Meir–Keeler type contraction. Let ε > 0. By (15) it follows that

ε > lim sup φ(s) = lim

sup

δ→0+ s∈(ε,ε+δ)

s→ε +

φ (s) = inf

sup

δ>0 s∈(ε,ε+δ)

φ(s),

hence there exists δ = δ(ε) > 0 such that sups∈(ε,ε+δ) φ(s) < ε . This leads to φ(s) < ε for each s ∈ (ε, ε + δ), and since φ(ε) < ε , we find that

ε ≤ s < ε + δ ⇒ φ(s) < ε. Now, let x, y ∈ X such that x ≺ y and ε ≤ d(x, y) < ε + δ . Then, by (16) and (17), d(A(x, y), A(y, x)) ≤ φ (d(x, y)) < ε, concluding the proof.



(17)

M.-D. Rus / Nonlinear Analysis 74 (2011) 1804–1813

1811

Remark 2.6. The proof of Theorem 2.2 shows that if (X , ≼) is an ordered set such that (X × X , .) is quasi-directed, then each pair of elements from X has both a lower bound and an upper bound (we will say that (X , ≼) is bi-directed). Conversely, an elementary check shows that if (X , ≼) is bi-directed, then (X × X , .) is also bi-directed, hence quasi-directed. Therefore, the condition ‘‘(X × X , .) is quasi-directed’’ in Theorems 1.2 and 2.2–2.4 can be replaced with the equivalent (and easier to check) condition ‘‘(X , ≼) is bi-directed’’. 3. Application Inspired by the works of Nieto and Rodríguez-López [2,4], Gnana Bhaskar and Lakshmikantham [3], we study the existence and uniqueness of the solution to a periodic boundary value problem, as an application to the fixed point theorems in Section 2.4. Consider the periodic boundary value problem (PBVP)



x′ (t ) = f (t , x(t )) + g (t , x(t )), x(0) = x(T ),

t ∈ I = [0, T ]

(18)

where T > 0 and f , g : I × R → R are continuous functions that satisfy the following conditions: Assumption 3.1. There exist λ1 , λ2 , λ3 ∈ R with λ1 > λ2 and λ3 > 2λ2 such that for all t ∈ I and u, v ∈ R with u ≼ v , f (t , v) − f (t , u) + λ1 (v − u) ≥ 0 g (t , v) − g (t , u) − λ2 (v − u) ≤ 0 f (t , v) − f (t , u) + λ3 (v − u) ≤ g (t , v) − g (t , u). We associate to problem (18) the following periodic system: x′ (t ) = f (t , x(t )) + g (t , y(t )) + λ2 (x(t ) − y(t )), y′ (t ) = f (t , y(t )) + g (t , x(t )) + λ2 (y(t ) − x(t )), x(0) = x(T ), y(0) = y(T ).



t ∈I t ∈I

(19)

Clearly, x is a solution of the PBVP (18) if and only if (x, x) is a solution of the periodic system (19). Also, a pair (x, y) is called a coupled lower and upper solution of the system (19) if x, y ∈ C 1 (I ) and

 ′ x (t ) ≤ f (t , x(t )) + g (t , y(t )) + λ2 (x(t ) − y(t )),   ′ y (t ) ≥ f (t , y(t )) + g (t , x(t )) + λ2 (y(t ) − x(t )), y(0) ≥ y(T )  x(0) ≤ x(T ), x(t ) ≤ y(t ) for all t ∈ I .

t ∈I t ∈I

It is a standard result that if x ∈ C 1 (I ) satisfies



x′ (t ) = h(t ), x(0) = x(T ),

t ∈I

(20)

with h ∈ C (I ), then for each λ ̸= 0, x verifies x(t ) =

T

∫

Gλ (t , s)(h(s) + λx(s))ds

for all t ∈ I ,

(21)

0

where

 λ(T +s−t ) e   ,  λT e −1 Gλ (t , s) = λ(s−t )   e , λ e T −1

0≤s
Conversely, if x ∈ C (I ) satisfies (21) for some λ ̸= 0, then x ∈ C 1 (I ) and verifies (20). By letting λ := λ1 − λ2 > 0, it follows from here that system (19) is equivalent to the coupled fixed point problem for the mapping A : C (I ) × C (I ) → C (I ) given by A(x, y)(t ) =

T

∫

Gλ (t , s) [f (s, x(s)) + g (s, y(s)) + λ1 x(s) − λ2 y(s)] ds,

x, y ∈ C (I ), t ∈ I .

(22)

0

In particular, (18) is equivalent to the fixed point problem for the mapping A. Note that Gλ is a positive function, since λ > 0.

T 0

Gλ (t , s)ds = λ1 for all t ∈ I and

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M.-D. Rus / Nonlinear Analysis 74 (2011) 1804–1813

It is also a known fact that if x ∈ C 1 (I ) is a lower solution for (20), i.e.,



x′ (t ) ≤ h(t ), x(0) ≤ x(T ),

t ∈I

then x is a lower solution for (21), i.e., x(t ) ≤

T

∫

Gλ (t , s)(h(s) + λx(s))ds

for all t ∈ I .

0

Similarly, if x is an upper solution for (20), i.e.,



x′ (t ) ≥ h(t ), x(0) ≥ x(T ),

t ∈I

then x is an upper solution for (21), i.e., x(t ) ≥

T

∫

Gλ (t , s)(h(s) + λx(s))ds

for all t ∈ I .

0

In consequence, if (x, y) is a coupled lower and upper solution of system (19), then x(t ) ≤ A(x, y)(t )

and y(t ) ≥ A(y, x)(t )

for all t ∈ I .

Now, we are able to formulate and prove the main result in this section. Theorem 3.1. Let f , g ∈ C (I ) satisfy Assumption 3.1. If system (19) has a coupled lower and upper solution, then there exists a unique solution of the PBVP (18). Proof. It is sufficient to verify the conditions in Theorem 2.4 for X = C (I ) partially ordered by x, y ∈ X ,

x ≼ y ⇔ x(t ) ≤ y(t ) for all t ∈ I ,

the complete metric d induced by the sup-norm on X , i.e., d(x, y) = sup |x(t ) − y(t )| ,

x, y ∈ X

t ∈I

and the mapping A : X × X → X defined by (22). It is know that (X , ≼) is a lattice, hence (X , ≼) is bi-directed (equivalently, (X × X , .) is quasi-directed by Remark 2.6). Also, it is elementary to check that d is monotone and that X satisfies condition (b) in Theorem 2.1. We now check whether A is mixed monotone. Indeed, if x1 , x2 , y ∈ X are such that x1 ≼ x2 , then A(x2 , y)(t ) − A(x1 , y)(t ) =

T

∫

Gλ (t , s) [f (s, x2 (s)) − f (s, x1 (s)) + λ1 (x2 (s) − x1 (s))] ds 0

≥ 0 for all t ∈ I . Similarly, if x, y1 , y2 ∈ X such that y1 ≼ y2 , then A(x, y2 )(t ) − A(x, y1 )(t ) =

T

∫

Gλ (t , s) [g (s, y2 (s)) − g (s, y1 (s)) − λ2 (y2 (s) − y1 (s))] ds 0

≤ 0 for all t ∈ I . Now, let x, y ∈ X such that x ≼ y. Then d(A(x, y), A(y, x)) = sup |A(x, y)(t ) − A(y, x)(t )| t ∈I T

∫

Gλ (t , s)[f (s, y(s)) − f (s, x(s)) − (g (s, y(s)) − g (s, x(s)))

= sup t ∈I

0

+ (λ1 + λ2 ) (y(s) − x(s))]ds ∫ T ≤ sup Gλ (t , s)(λ1 + λ2 − λ3 ) (y(s) − x(s)) ds t ∈I

0

≤ (λ1 + λ2 − λ3 ) · d(x, y) · sup

T

∫

t ∈I

λ1 +λ2 −λ3 λ1 −λ2

Gλ (t , s) = 0

λ1 + λ2 − λ3 · d(x, y), λ1 − λ2

hence d(A(x, y), A(y, x)) ≤ α · d(x, y), with α := < 1. This proves (16) with φ(t ) = α t. Finally, we conclude that all the conditions in Theorem 2.4 are satisfied, hence A has a unique fixed point, which means the PVBP (18) has a unique solution. 

M.-D. Rus / Nonlinear Analysis 74 (2011) 1804–1813

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Remark 3.1. If x∗ is the unique solution of the PVBP (18), in the conditions of Theorem 3.1, then (An (x, y)) → x∗ (as n → ∞) for all x, y ∈ C (I ). In particular, if (x, y) is a coupled lower and upper solution of system (19), then (An (x, y)) is nondecreasing, (An (y, x)) is nonincreasing and An (x, y) ≼ x∗ ≼ An (y, x) for all n. Remark 3.2. Our theorem complements a result of Gnana Bhaskar and Lakshmikantham [3, Theorem 3.7] which studied the PBVP (18) under fairly similar assumptions, but with λ2 > λ1 (see Lemma 3.2 in [3]). References [1] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (5) (2004) 1435–1443. doi:10.1090/S0002-9939-03-07220-4. [2] J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (3) (2005) 223–239. doi:10.1007/s11083-005-9018-5. [3] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379–1393. doi:10.1016/j.na.2005.10.017. [4] J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (12) (2007) 2205–2212. doi:10.1007/s10114-005-0769-0. [5] Z. Drici, F. McRae, J.V. Devi, Fixed-point theorems in partially ordered metric spaces for operators with PPF dependence, Nonlinear Anal. 67 (2) (2007) 641–647. doi:10.1016/j.na.2006.06.022. [6] Z. Drici, F. McRae, J.V. Devi, Fixed point theorems for mixed monotone operators with PPF dependence, Nonlinear Anal. 69 (2) (2008) 632–636. doi:10.1016/j.na.2007.05.044. [7] R.P. Agarwal, M.A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87 (1) (2008) 109–116. doi:10.1080/00036810701556151. [8] D. O’Regan, A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2) (2008) 1241–1252. doi:10.1016/j.jmaa.2007.11.026. [9] L. Ćirić, N. Cakić, M. Rajović, J.S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. 2008 (2008) doi:10.1155/2008/131294. Art. ID 131294, 11 pages. [10] V. Lakshmikantham, L. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (12) (2009) 4341–4349. doi:10.1016/j.na.2008.09.020. [11] J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71 (7–8) (2009) 3403–3410. doi:10.1016/j.na.2009.01.240. [12] I. Altun, H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl. 2010 (2010) doi:10.1155/2010/621469. Art. ID 621469, 17 pages. [13] B.S. Choudhury, A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal. 73 (8) (2010) 2524–2531. doi:10.1016/j.na.2010.06.025. [14] J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (3–4) (2010) 1188–1197. doi:10.1016/j.na.2009.08.003. [15] S. Hong, Fixed points of multivalued operators in ordered metric spaces with applications, Nonlinear Anal. 72 (11) (2010) 3929–3942. doi:10.1016/j.na.2010.01.013. [16] B. Samet, Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (12) (2010) 4508–4517. doi:10.1016/j.na.2010.02.026. [17] B. Samet, H. Yazidi, Coupled fixed point theorems in partially ordered ϵ -chainable metric spaces, TJMCS 1 (3) (2010) 142–151. [18] X. Zhang, Fixed point theorems of multivalued monotone mappings in ordered metric spaces, Appl. Math. Lett. 23 (3) (2010) 235–240. doi:10.1016/j.aml.2009.06.011. [19] D.J. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11 (5) (1987) 623–632. doi:10.1016/0362546X(87)90077-0. [20] M.A. Krasnosel’ski˘ı, in: Leo F. Boron (Ed.), Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, 1964, Translated from the Russian by Richard E. Flaherty.