Skip to content

Advertisement

  • Research
  • Open Access

Coupled fixed point theorems for generalized Mizoguchi-Takahashi contractions with applications

  • 1,
  • 2,
  • 3 and
  • 3Email author
Fixed Point Theory and Applications20122012:51

https://doi.org/10.1186/1687-1812-2012-51

  • Received: 27 October 2011
  • Accepted: 26 March 2012
  • Published:

Abstract

We derive some new coupled fixed point theorems for nonlinear contractive maps that satisfied a generalized Mizoguchi-Takahashi's condition in the setting of ordered metric spaces. Presented theorems extends and generalize many well-known results in the literature. As an application, we give an existence and uniqueness theorem for the solution to a two-point boundary value problem.

2000 Mathematics Subject Classification: 54H25; 47H10.

Keywords

  • coupled fixed point
  • ordered metric space
  • mixed monotone mapping
  • generalized Mizoguchi-Takahashi's contraction
  • two-point boundary value problem

1 Introduction

Let (X, d) be a metric space. Denote by P(X) the set of all nonempty subsets of X and CB(X) the family of all nonempty closed and bounded subsets of X. A point x in X is a fixed point of a multivalued map T : XP(X), if x Tx. Nadler [1] extended the Banach contraction principle to multivalued mappings.

Theorem 1.1 (Nadler[1]) Let (X, d) be a complete metric space and let T : XCB(X) be a multivalued map. Assume that there exists r [0,1) such that
H ( T x , T y ) r d ( x , y )

for all x, y X, where H is the Hausdorff metric with respect to d. Then T has a fixed point.

Reich [2] proved the following generalization of Nadler's fixed point theorem.

Theorem 1.2 (Reich[2]) Let (X, d) be a complete metric space and T : XC(X) be a multi-valued map with non empty compact values. Assume that
H ( T x , T y ) φ ( d ( x , y ) ) d ( x , y )

for all x, y X, where φ is a function from [0, ∞) into [0,1) satisfying lim  sup s t + φ ( s ) < 1 for all t > 0. Then T has a fixed point.

Mizoguchi and Takahashi [3] proved the following generalization of Nadler's fixed point theorem for a weak contraction which is a partial answer of Problem 9 in Reich [4].

Theorem 1.3 (Mizoguchi and Takahashi[3]) Let (X, d) be a complete metric space and T : XCB(X) be a multivalued map. Assume that
H ( T x , T y ) φ ( d ( x , y ) ) d ( x , y )

for all x, y X, where φ is a function from [0, ∞) into [0,1) satisfying lim  sup s t + φ ( s ) < 1 for all t ≥ 0. Then T has a fixed point.

Suzuki [5] gave a very simple proof of Theorem 1.3.

Very recently, Amini-Harandi and O'Regan [6] obtained a nice generalization of Mizoguchi and Takahashi's fixed point theorem. Throughout the article, let Ψ be the family of all functions ψ : [0, ∞) → [0, ∞) satisfying the following conditions:
  1. (a)

    ψ(s) = 0 s = 0,

     
  2. (b)

    ψ is nondecreasing,

     
  3. (c)

    lim  sup s 0 + s ψ ( s ) < .

     

We denote by Φ the set of all functions φ : [0, ∞) → [0,1) satisfying lim sup r t + φ ( r ) < 1 for all t ≥ 0.

Theorem 1.4 (Amini-Harandi and O'Regan[6]) Let (X, d) be a complete metric space and T : XCB(X) be a multivalued map. Assume that
ψ ( H ( T x , T y ) ) φ ( ψ ( d ( x , y ) )) ψ ( d ( x , y ) )

for all x, y X, where ψ Ψ is lower semicontinuous and φ Φ. Then T has a fixed point.

The existence of fixed point in partially ordered sets has been investigated recently in [727] and references therein.

Du [13] proved some coupled fixed point results for weakly contractive single-valued maps that satisfy Mizoguchi-Takahashi's condition in the setting of quasiordered metric spaces. Before recalling the main results in [13], we need some definitions.

Definition 1.1 (Bhaskar and Lakshmikantham[11]) Let X be a nonempty set and A : X × XX be a given map. We call an element (x, y) X × X a coupled fixed point of A if x = A(x, y) and y = A(y, x).

Definition 1.2 (Bhaskar and Lakshmikantham[11]) Let (X, ) be a quasiordered set and A : XXX a map. We say that A has the mixed monotone property on X if A(x, y) is monotone nondecreasing in x X and is monotone nonincreasing in y X, that is, for any x, y X,
x 1 , x 2 X w i t h x 1 x 2 A ( x 1 , y ) A ( x 2 , y ) , y 1 , y 2 X w i t h y 1 y 2 A ( x , y 1 ) A ( x , y 2 ) .

Definition 1.3 (Du [13]) Let (X, d) be a metric space with a quasi-order . A nonempty subset M of X is said to be

(i) sequentially ↑-complete if every -nondecreasing Cauchy sequence in M converges;

(ii) sequentially ↓-complete if every -nonincreasing Cauchy sequence in M converges;

(iii) sequentially ↕-complete if it is both ↑-complete and ↓-complete.

Theorem 1.5 (Du[13]) Let (X, d, ) be a sequentially ↕-complete metric space and A : X × XX be a continuous map having the mixed monotone property on X. Assume that there exists a function φ Φ such that
d ( A ( x , y ) , A ( u , v ) ) 1 2 φ ( d ( x , u ) + d ( y , v ) ) ( d ( x , u ) + d ( y , v ) )

for all x u and y v. If there exist x0, y0 X such that x 0 A (x 0 , y 0 ) and y 0 A(y 0 , x 0 ), then A has a coupled fixed point.

Theorem 1.6 (Du[13]) Let (X, d, ) be a sequentially ↕-complete metric space and A : X × XX be a map having the mixed monotone property on X. Assume that

(i) any -nondecreasing sequence (x n ) with x n x implies x n x for all n,

(ii) any -nonincreasing sequence (y n ) with y n y implies for all n.

Assume also that there exists a function φ Φ such that
d ( A ( x , y ) , A ( u , v ) ) 1 2 φ ( d ( x , u ) + d ( y , v ) ) ( d ( x , u ) + d ( y , v ) )

for all x u and y v. If there exist x0,y0 X such that x 0 A(x 0 , y 0 ) and y 0 A(y 0 , x 0 ), then A has a coupled fixed point.

Very recently, Gordji and Ramezani [14] established a new fixed point theorem for a self-map T : XX satisfying a generalized Mizoguchi-Takahashi's condition in the setting of ordered metric spaces. The main result in [14] is the following.

Theorem 1.7 (Gordji and Ramezani[14]) Let (X, d, ) be a complete ordered metric space and T : XX an increasing mapping such that there exists an element x0 X with x 0 Tx 0 . Suppose that there exists a lower semicontinuous function ψ Ψ and φ Φ such that
ψ ( d ( T x , T y ) ) φ ( ψ ( d ( x , y ) ) ψ ( d ( x , y ) )

for all x, y X such that x and y are comparable. Assume that either T is continuous or X is such that the following holds: any -nondecreasing sequence (x n ) with x n x implies x n x for all n. Then T has a fixed point.

In this article, we present new coupled fixed point theorems for mixed monotone mappings satisfying a generalized Mizoguchi-Takahashi's condition in the setting of ordered metric spaces. Presented theorems extend and generalize Du [[13], Theorems 2.8 and 2.10], Bhaskar and Lakshmikantham [[11], Theorems 2.1 and 2.2], Harjani et al. [[15], Theorems 2 and 3], and other existing results in the literature. Moreover, some applications to ordinary differential equations are presented.

2 Main results

Through this article, we will use the following notation: if (X, ) is an ordered set, we endow the product set X × X with the order given by
( x , y ) , ( u , v ) X × X , ( x , y ) ( u , v ) x u , y v .

Our first result is the following.

Theorem 2.1 Let (X, d, ) be a sequentially ↕-complete metric space and A : X × XX be a map having the mixed monotone property on X. Suppose that there exist ψ Ψ and φ Φ such that for any (x, y),(u, v) X×X with (u, v) (x, y),
ψ ( d ( A ( x , y ) , A ( u , v ) ) ) φ ( ψ ( max { d ( x , u ) , d ( y , v ) } ) ) ψ ( max { d ( x , u ) , d ( y , v ) } ) .
(1)

Suppose also that either A is continuous or (X, d, ) has the following properties:

(i) any -nondecreasing sequence (x n ) with x n x implies x n x for each n,

(ii) any -nonincreasing sequence (y n ) with y n y implies y n y for each n.

If there exist x0,y0 X such that x 0 A(x 0 , y 0 ) and y 0 A(y 0 , x 0 ), then there exist a, b X such that a = A(a, b) and b = A(b, a).

Proof. Define the sequences (x n ) and (y n ) in X by
x n + 1 = A ( x n , y n ) , y n + 1 = A ( y n , x n ) for all n 0 .

In order to make the proof more comprehensive we will divide it into several steps.

Step 1. x n xn+1and y n yn+1for all n ≥ 0.

We use mathematical induction.

As x 0 A(x 0 , y 0 ) = x1 and y 0 A(y 0 , x 0 ) = y1, our claim is satisfied for n = 0.

Suppose that our claim holds for some fixed n ≥ 0. Then, since x n xn+1and y n yn+1, and as A has the mixed monotone property, we get
x n + 1 = A ( x n , y n ) A ( x n + 1 , y n ) A ( x n + 1 , y n + 1 ) = x n + 2
and
y n + 1 = A ( y n , x n ) A ( y n + 1 , x n ) A ( y n + 1 , x n + 1 ) = y n + 2 .

This proves our claim.

Step 2. limn→∞ψ(max{d(x n+ 1,x n ),d(y n+ 1,y n )}) = 0.

Since x n x n+1 and y n y n+1 (Step 1), we have (x n , y n ) (x n+ 1,y n+ 1), and by (1), we have
ψ ( d ( A ( x n + 1 , y n + 1 ) , A ( x n , y n ) ) ) φ ( ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) ) ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) .
(2)
Similarly, since (y n+ 1,x n+ 1) (y n , x n ), by (1), we have
ψ ( d ( A ( y n , x n ) , A ( y n + 1 , x n + 1 ) ) ) φ ( ψ ( max { d ( y n , y n + 1 ) , d ( x n , x n + 1 ) } ) ) ψ ( max { d ( y n , y n + 1 ) , d ( x n , x n + 1 ) } ) ψ ( max { d ( y n , y n + 1 ) , d ( x n , x n + 1 ) } ) .
(3)
From (2) and (3), we get
max { ψ ( d ( x n + 2 , x n + 1 ) , ψ ( d ( y n + 2 , y n + 1 ) ) } ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) .
Since ψ is nondecreasing, this implies that
ψ max d ( x n + 2 , x n + 1 ) d ( y n + 2 , y n + 1 ) ψ max d ( x n + 1 , x n ) , d ( y n + 1 , y n )
(4)
for all n ≥ 0. Now, (4) means that (ψ(max{d(x n+ 1,x n ),d(y n+ 1,y n )})) is a non increasing sequence. On the other hand, this sequence is bounded below; thus there exists μ ≥ 0 such that
lim n ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) = μ .
(5)
Since φ Φ, we have lim  sup r μ + φ ( r ) < 1 and φ (μ) < 1. Then, there exist α [0,1) and ε > 0 such that φ(r) ≤ α for all r [μ,μ + ε). From (5), we can take n0 ≥ 0 such that μψ(max{d(x n+ 1,x n ),d(y n+ 1,y n )}) ≤ μ + ε for all nn0. Then, from (2), for all nn0, we have
ψ ( d ( x n + 1 , x n + 2 ) ) φ ( ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) ) ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) α ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) .
(6)
Similarly, from (3), for all n > n0, we have
ψ ( d ( y n + 1 , y n + 2 ) ) φ ( ψ ( max { d ( y n , y n + 1 ) , d ( x n , x n + 1 ) } ) ) ψ ( max { d ( y n , y n + 1 ) , d ( x n , x n + 1 ) } ) α ψ ( max { d ( y n , y n + 1 ) , d ( x n , x n + 1 ) } ) .
(7)
Now, from (6) and (7), we get
ψ max d ( x n + 1 , x n + 2 ) d ( y n + 1 , y n + 2 ) α ψ max d ( y n , y n + 1 ) , d ( x n , x n + 1 )
(8)
for all nn0. Letting n → ∞ in the above inequality and using (5), we obtain that
μ α μ .
Since α [0,1), this implies that μ = 0. Thus, we proved that
lim n ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) = 0 .
(9)
  • Step 3. limn→∞max{d(x n+ 1,x n ),d(y n+ 1,y n )} = 0.

Since (ψ(max{d(x n+ 1,x n ),d(y n+ 1,y n )})) is a decreasing sequence and ψ is nondecreasing, then (max{d(x n+ 1,x n ),d(y n+ 1,y n )}) is a decreasing sequence of positive numbers. This implies that there exists θ ≥ 0 such that
lim n max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) = θ + .
Since ψ is nondecreasing, we have
ψ ( max d ( x n + 1 , x n ) , d ( y n + 1 , y n ) ) ψ ( θ ) .
Letting n → ∞ in the above inequality, from (9), we obtain that 0 ≥ ψ(θ), which implies that θ = 0. Thus, we proved that
lim n max d ( x n + 1 , x n ) , d ( y n + 1 , y n ) = 0 .
(10)

Step 4. (x n ) and (y n ) are Cauchy sequences in (X, d).

Suppose that max{d(x m+ 1,x m ), d(y m+ 1,y m )} = 0 for some m ≥ 0. Then, we have d(x m +1,x m ) = d(y m+ 1,y m ) = 0, which implies that (x m , y m ) = (x m+ 1,y m+ 1), that is, x m = A(x m , y m ) and y m = A(y m , x m ). Then, (x m , y m ) is a coupled fixed point of A.

Now, suppose that max{d(x n +1,x n ), d(y n+ 1,y n )} 0 for all n ≥ 0.

Denote
a n = ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) for all n 0 .
From (8), we have
a n + 1 α a n for all n n 0 .
Then, we have
n = 0 a n n = 0 n 0 a n + n = n 0 + 1 α n - n 0 a n 0 < .
On the other hand, we have
lim  sup n max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ψ ( max { d ( x n + 1 , x n ) , d ( y n + 1 , y n ) } ) lim  sup s 0 + s ψ ( s ) < .

Then ∑ n max{d(x n , x n +1),d(y n , y n +1)} < ∞. Hence, (x n ) and (y n ) are Cauchy sequences in X.

Step 5. Existence of a coupled fixed point.

Since (X, d, ) is sequentially ↕-complete metric space and (x n ) is -nondecreasing Cauchy sequence, there exists a X such that
lim n x n = a .
(11)
Similarly, since (X, d, ) is sequentially ↕-complete metric space and (y n ) is -noincreasing Cauchy sequence, there exists b X such that
lim n y n = b .
(12)

Case 1. A is continuous.

From the continuity of A and using (11) and (12), we get
a = lim n x n + 1 = lim n A x n , y n = A ( lim n x n , lim n y n ) = A ( a , b )
and
b = lim n y n + 1 = lim n A y n , x n = A ( lim n y n , lim n x n ) = A ( b , a ) .

Case 2. (X, d, ) satisfies (i) and (ii).

Since (x n ) is -nondecreasing and limn→∞x n = a, then from (i), we have x n a for all n. Similarly, from (ii), since (y n ) is -nonincreasing and limn→∞y n = b, we have y n b for all n. Then, we have (x n , y n ) (a, b) for all n. Now, applying our contractive condition (1), we get
ψ ( d ( x n + 1 , A ( a , b ) ) ) = ψ ( d ( A ( x n , y n ) , A ( a , b ) ) ) φ ( ψ ( max { d ( x n , a ) , d ( y n , b ) } ) ) ψ ( max { d ( x n , a ) , d ( y n , b ) } ) ψ ( max { d ( x n , a ) , d ( y n , b ) } ) .
Since ψ is nondecreasing, this implies that
d ( x n + 1 , A ( a , b ) ) max { d ( x n , a ) , d ( y n , b ) } .

letting n → ∞ in the above inequality, we obtain that d(a, A(a, b)) ≤ 0, that is, a = A(a, b). Similarly, we can show that b = A(b, a). □

Remark 2.1 In our presented theorems we don't need the hypothesis: ψ is lower semicontinuous. Such hypothesis is considered in Theorem 1.7 of Gordji and Ramezani[14].

In what follows, we give a sufficient condition for the uniqueness of the coupled fixed point in Theorem 2.1. We consider the following hypothesis:

(H): For all (x, y), (u, v) X × X, there exists (w, z) X × X such that (x, y) (w, z) and (u, v) (w, z).

Theorem 2.2 Adding condition (H) to the hypotheses of Theorem 2.1, we obtain uniqueness of the coupled fixed point of A.

Proof. Suppose that (a, b) and (c, e) are coupled fixed points of A, that is, a = A(a, b), b = A(b, a), c = A(c, e) and e = A(e, c). From (H), there exists (f0, g0) X × X such that (a, b) (f0, g0) and (c, e) (f0, g0).

We construct the sequences (f n ) and (g n ) in X defined by
f n + 1 = A ( f n , g n ) , g n + 1 = A ( g n , f n ) for all n 0 .

We claim that (a, b) (f n , g n ) for all n ≥ 0.

In fact, we will use mathematical induction.

Since (a, b) (f0, g0), then our claim is satisfied for n = 0.

Suppose that our claim holds for some fixed n ≥ 0. Then, we have (a, b) (f n , g n ), that is, a f n and b g n . Using the mixed monotone property of A, we get
f n + 1 = A ( f n , g n ) A ( a , g n ) A ( a , b ) = a
and
g n + 1 = A ( g n , f n ) A ( b , f n ) A ( b , a ) = b .

This proves that (a, b) (f n+1 , g n+1 ). Then, our claim holds.

Now, we can apply (1) with (u, v) = (a, b) and (x, y) = (f n , g n ). We get
ψ ( d ( f n + 1 , a ) ) = ψ ( d ( A ( f n , g n ) , A ( a , b ) ) ) φ ( ψ ( max { d ( f n , a ) , d ( g n , b ) } ) ) ψ ( max { d ( f n , a ) d ( g n , b ) } ) .
(13)
Similarly, we have
ψ ( d ( g n + 1 , b ) ) = ψ ( d ( A ( g n , f n ) , A ( b , a ) ) ) φ ( ψ ( max { d ( f n , a ) , d ( g n , b ) } ) ) ψ ( max { d ( f n , a ) d ( g n , b ) } ) .
(14)
Combining (13) with (14), we obtain
ψ ( max { d ( f n + 1 , a ) , d ( g n + 1 , b ) } ) φ ( ψ ( max { d ( f n , a ) , d ( g n , b ) } ) ) ψ ( max { d ( f n , a ) , d ( g n , b ) } ) ψ ( max { d ( f n , a ) , d ( g n , b ) } ) .
(15)

Consequently, (ψ(max{d(f n , a), d(g n , b)})) is a nonnegative decreasing sequence and hence possesses a limit γ ≥ 0. Following the same strategy used in the proof of Theorem 2.1, one can show that γ = 0 and limn→∞max{d(f n , a),d(g n , b)} = 0.

Analogously, it can be proved that limn→∞max{d(f n , c),d(g n , e)} = 0.

Now, we have
d ( a , c ) d ( a , f n ) + d ( f n , c ) max { d ( f n , a ) , d ( g n , b ) } + max { d ( f n , c ) , d ( g n , e ) }
and
d ( b , e ) d ( b , g n ) + d ( g n , e ) max { d ( f n , a ) , d ( g n , b ) } + max { d ( f n , c ) , d ( g n , e ) } .
Then, we have
max { d ( a , c ) , d ( b , e ) } max { d ( f n , a ) , d ( g n , b ) } + max { d ( f n , c ) , d ( g n , e ) } .
Letting n → ∞ in the above inequality, we get
max { d ( a , c ) , d ( b , e ) } = 0 ,

which implies that d(a, c) = d(b, e) = 0. Then, (a, b) = (c, e). □

Theorem 2.3 Under the assumptions of Theorem 2.1, suppose that x0and y 0 are comparable, then the coupled fixed point (a, b) X × X satisfies a = b.

Proof. Assume that x 0 y 0 (the same strategy can be used if y 0 x 0 ). Using the mixed monotone property of A, it is easy to show that x n y n for all n ≥ 0.

Now, using the contractive condition, as x n y n , we have
ψ ( d ( y n + 1 , x n + 1 ) ) = ψ ( d ( A ( y n , x n ) , A ( x n , y n ) ) ) φ ( ψ ( d ( x n , y n ) ) ψ ( d ( x n , y n ) ) ψ ( d ( x n , y n ) ) .
(16)

Thus limn→∞ψ(d(x n , y n )) = θ for certain θ ≥ 0. Since φ Φ, we have lim  sup r θ + φ ( r ) < 1 and φ(θ) < 1. Then, there exist α [0,1) and ε > 0 such that φ(r) <α for all r [θ,θ + ε).

Now, we take n0 ≥ 0 such that θψ(d(x n , y n )) ≤ θ + ε for all n ≥ n0. Then, from (16), for all nn0, we have
ψ ( d ( y n + 1 , x n + 1 ) ) α ψ ( d ( x n , y n ) ) .
Letting n → ∞ in the above inequality, we obtain that
θ α θ .
Since α [0,1), this implies that θ = 0. Thus, we proved that
lim n ψ ( d ( x n , y n ) ) = 0 ,
which implies that limn→ ∞d(x n , y n ) = 0. Now, we have
0 = lim n d ( x n , y n ) = d ( lim n x n , lim n y n ) = d ( a , b )

and thus a = b. This finishes the proof. □

Now, we present some consequences of our theorems.

Corollary 2.1 Let (X, d, ) be a sequentially ↕-complete metric space and A : X × XX be a map having the mixed monotone property on X. Suppose that there exist ψ Ψ and φ ̃ : [ 0 , ) [ 0 , ) with lim inf s t + ( φ ̃ ( s ) / ψ ( s ) ) > 0 for all t ≥ 0 such that for any (x, y),(u, v) X × X with(u, v) (x, y),
ψ ( d ( A ( x , y ) , A ( u , v ) ) ) ψ ( max { d ( x , u ) , d ( y , v ) } ) - φ ̃ ( ψ ( max { d ( x , u ) , d ( y , v ) } ) ) .

Suppose also that either A is continuous or (X, d, ) has the following properties:

(i) any -nondecreasing sequence (x n ) with x n x implies x n x for each n,

(ii) any -nonincreasing sequence (y n ) with y n y implies y n y for each n.

If there exist x0, y0 X such that x 0 A (x 0 , y 0 ) and y 0 A(y 0 , x 0 ), then there exist a, b X such that a = A(a, b) and b = A(b, a).

Proof. It follows immediately from Theorem 2.1 by considering φ ( s ) = 1 - φ ̃ ( s ) / ψ ( s ) . □

Remark 2.2 Corollary 2.1 is an extension of Harjani et al. [[15], Theorems 2 and 3].

Corollary 2.2 Let (X, d, ) be a sequentially ↕-complete metric space and A : X × XX be a map having the mixed monotone property on X. Suppose that there exists a nondecreasing function φ : [0, ∞) → [0, 1) such that for any (x, y), (u, v) X × X with (u, v) (x, y),
d ( A ( x , y ) , A ( u , v ) ) φ ( 2 max { d ( x , u ) , d ( y , v ) } ) max { d ( x , u ) , d ( y , v ) } .

Suppose also that either A is continuous or (X, d, ) has the following properties:

(i) any -nondecreasing sequence (x n ) with x n x implies xn x for each n,

(ii) any -nonincreasing sequence (y n ) with y n y implies yn y for each n.

If there exist x0, y0 X such that x 0 A(x 0 , y 0 ) and y 0 A(y 0 , x 0 ), then there exist a, b X such that a = A(a, b) and b = A(b, a).

Proof. It follows from Theorem 2.1 by considering ψ(s) = 2s

Remark 2.3 If φ is nondecreasing, Corollary 2.2 generalizes Du [[13], Theorems 1.5 and 1.6].

Corollary 2.3 Let (X, d, ) be a sequentially ↕-complete metric space and A : X × XX be a map having the mixed monotone property on X. Suppose that there exists k [0, 1) such that for any (x, y),(u, v) X × X with (u, v) (x, y),
d ( A ( x , y ) , A ( u , v ) ) k max { d ( x , u ) , d ( y , v ) } .

Suppose also that either A is continuous or (X, d, ) has the following properties:

(i) any -nondecreasing sequence (x n ) with x n x implies x n x for each n,

(ii) any -nonincreasing sequence (y n ) with y n y implies y n y for each n.

If there exist x0, y0 X such that x 0 A(x 0 , y 0 ) and y 0 A(y 0 , x 0 ), then there exist a, b X such that a = A(a, b) and b = A(b, a).

Proof. It follows immediately from Corollary 2.2 by considering φ(s) = k

Remark 2.4 Corollary 2.3 is a generalization of Bhaskar and Lakshmikantham [[11], Theorems 2.1 and 2.2].

3 An application

In this section, we apply our main results to study the existence and uniqueness of solution to the two-point boundary value problem
- d 2 x d t 2 ( t ) = f ( t , x ( t ) , x ( t ) ) , t [ 0 , 1 ] x ( 0 ) = x ( 1 ) = 0 ,
(17)

where f : [0, 1] × × is a continuous function.

Previously we considered the space X = C(I, )(I = [0, 1]) of continuous functions defined on I. Obviously, this space with the metric given by
d ( x , y ) = max { | x ( t ) - y ( t ) | : t I } for x , y I
is a complete metric space. The space X can also be equipped with a partial order given by
x , y I , x y x ( t ) y ( t ) for all t I .

Obviously, (X, ) satisfies condition (H) since for x, y X the functions max{x, y} and min{x, y} are least upper and greatest lower bounds of x and y, respectively. Moreover, in [21] it is proved that (X, d, ) satisfies conditions (i) and (ii) of Theorem 2.1.

Now, we consider the following assumptions:
  1. (a)

    f : [0, 1] × × is continuous.

     
  2. (b)
    For all t I, z ≥ h, w ≤ r,
    0 f ( t , z , w ) - f ( t , h , r ) 4 [ ln ( z - h + 1 ) + ln ( r - w + 1 ) ] .
     
  3. (c)
    There exists (α, β) C 2 (I, ) × C 2 (I, ) solution to
    - d 2 α d t 2 ( t ) f ( t , α ( t ) , β ( t ) ) , t [ 0 , 1 ] - d 2 β d t 2 ( t ) f ( t , β ( t ) , α ( t ) ) , t [ 0 , 1 ] α ( 0 ) = α ( 1 ) = β ( 0 ) = β ( 1 ) = 0 .
    (18)
     
  4. (d)

    α β or β α.

     

Theorem 3.1 Under the assumptions (a)-(d), problem (17) has one and only one solution x* C2(I, ).

Proof. It is well known that the solution (in C2 (I, )) of problem (17) is equivalent to the solution (in C(I, )) of the following Hammerstein integral equation:
x ( t ) = 0 1 G ( t , s ) f ( x ( s ) ) d s , t I ,
where G(t, s) is the Green function of differential operator -d2/dt2 with Dirichlet boundary condition x(0) = x(1) = 0, i.e.,
G ( t , s ) = s ( 1 - t ) , 0 s t 1 , t ( 1 - s ) , 0 t s 1 .
Define A : X × XX by
A ( x , y ) ( t ) = 0 1 G ( t , s ) f ( s , x ( s ) , y ( s ) ) d s , t I ,

for all x, y X.

From (b), it is clear that A has the mixed monotone property with respect to the partial order in X.

Let x, y, u, v X such that x u and y υ. From (b), we have
d ( A ( x , y ) , A ( u , v ) ) = sup t I | A ( x , y ) ( t ) - A ( u , v ) ( t ) | = sup t I 0 1 G ( t , s ) f ( s , x ( s ) , y ( s ) ) - f ( s , u ( s ) , v ( s ) ) d s sup t I 0 1 4 G ( t , s ) [ ln ( x ( s ) - u ( s ) + 1 ) + ln ( v ( s ) - y ( s ) + 1 ) ] d s ( ln ( d ( x , u ) + 1 ) + ln ( d ( y , v ) + 1 ) ) sup t I 0 1 4 G ( t , s ) d s sup t I 0 1 8 G ( t , s ) d s ln ( max { d ( x , u ) , d ( y , v ) } + 1 )
On the other hand, for all t I, we have
0 1 G ( t , s ) d s = 1 2 t ( 1 - t ) ,
which implies that
sup t I 0 1 G ( t , s ) d s = 1 8 .
Then, we get
d ( A ( x , y ) , A ( u , v ) ) ln ( max { d ( x , u ) , d ( y , v ) } + 1 ) .
This implies that
ln d ( A ( x , y ) , A ( u , v ) ) + 1 ln ln max { d ( x , u ) , d ( y , v ) } + 1 + 1 = ln ln max { d ( x , u ) , d ( y , v ) } + 1 + 1 ln max { d ( x , u ) , d ( y , v ) } + 1 ln max { d ( x , u ) , d ( y , v ) } + 1 .

Thus, the contractive condition (1) of Theorem (2.1) is satisfied with ψ(t) = ln(t + 1) and φ(t) = ψ(t)/t.

Now, let (α, β) C2 (I, ) × C2 (I, ) be a solution to (18). We will show that α A(α, β) and β A(β, α). Indeed,
- α ( s ) f ( s , α ( s ) , β ( s ) ) , s [ 0 , 1 ] .
Multiplying by G(t, s), we get
0 1 - α ( s ) G ( t , s ) d s A ( α , β ) ( t ) , t [ 0 , 1 ] .
Then, for all t [0, 1], we have
- ( 1 - t ) 0 1 s α ( s ) d s - t t 1 ( 1 - s ) α ( s ) d s A ( α , β ) ( t ) .
Using an integration by parts, and since α(0) = α(1) = 0, for all t [0, 1], we get
- ( 1 - t ) ( t α ( t ) - α ( t ) ) - t ( - ( 1 - t ) α ( t ) - α ( t ) ) A ( α , β ) ( t ) .
Thus, we have
α ( t ) A ( α , β ) ( t ) , t [ 0 , 1 ] .

This implies that α A(α, β). Similarly, one can show that β A(β, α).

Now, applying our Theorems 2.1 and 2.2, we deduce the existence of a unique (x, y) X2 solution to x = A(x, y) and y = A(y, x). Moreover, from (d), and using Theorem 2.3, we get x = y. Thus, we proved that x* = x = y C2([0, 1], ) is the unique solution to (17). □

Declarations

Acknowledgements

L. Ćirić and B. Damjanović were supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.

M. Jleli extends his appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group project No RGP-VPP-087.

Authors’ Affiliations

(1)
Faculty of Mechanical Engineering, University in Belgrade, Kraljice Marije 16, 11 000 Belgrade, Serbia
(2)
Faculty of Agriculture, University in Belgrade, Nemanjina 6, Belgrade, Serbia
(3)
Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

References

  1. Nadler SB Jr: Multi-valued contraction mappings. Pacific J Math 1969, 30: 475–488.MATHMathSciNetView ArticleGoogle Scholar
  2. Reich S: Fixed points of contractive functions. Boll Un Mat Ital 1972, 5: 26–42.MATHMathSciNetGoogle Scholar
  3. Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J Math Anal Appl 1989, 141: 177–188.MATHMathSciNetView ArticleGoogle Scholar
  4. Reich S: Some problems and results in fixed point theory, in Topological Methods in Nonlinear Functional Analysis (Toronto, Ont., 1982). In Contemp Math. Volume 21. American Mathematical Society, Providence, RI, USA; 1983:179–187.Google Scholar
  5. Suzuki T: Mizoguchi-Takahashi's fixed point theorem is a real generalization of Nadler's. J Math Anal Appl 2008, 340(1):752–755.MATHMathSciNetView ArticleGoogle Scholar
  6. Amini-Harandi A, O'Regan D: Fixed point theorems for set-valued contraction type maps in metric spaces. Fixed Point Theory Appl 2010, 2010: 7. Article ID 390183MathSciNetView ArticleGoogle Scholar
  7. Agarwal RP, El-Gebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Appl Anal 2008, 87(1):109–116.MATHMathSciNetView ArticleGoogle Scholar
  8. Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory Appl 2010, 2010: 17. Article ID 621492MathSciNetGoogle Scholar
  9. Beg I, Butt AR: Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal 2009, 71: 3699–3704.MATHMathSciNetView ArticleGoogle Scholar
  10. Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal 2011, 74: 4889–4897.MATHMathSciNetView ArticleGoogle Scholar
  11. Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal 2006, 65(7):1379–1393.MATHMathSciNetView ArticleGoogle Scholar
  12. Ćirić LjB, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl 2008, 2008: 11. Article ID 131294Google Scholar
  13. Du W-S: Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi's condition in quasiordered metric spaces. Fixed Point Theory Appl 2010, 2010: 9. Article ID 876372Google Scholar
  14. Gordji ME, Ramezani M: A generalization of Mizoguchi and Takahashi's theorem for single-valued mappings in partially ordered metric spaces. Nonlinear Anal 2011, 74: 4544–4549.MATHMathSciNetView ArticleGoogle Scholar
  15. Harjani J, López B, Sadarangani K: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal 2011, 74: 1749–1760.MATHMathSciNetView ArticleGoogle Scholar
  16. Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal 2010, 72(3–4):1188–1197.MATHMathSciNetView ArticleGoogle Scholar
  17. Jachymski J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal 2011, 74: 768–774.MATHMathSciNetView ArticleGoogle Scholar
  18. Lakshmikantham V, Ćirić LjB: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal 2009, 70: 4341–4349.MATHMathSciNetView ArticleGoogle Scholar
  19. Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal 2011, 74: 983–992.MATHMathSciNetView ArticleGoogle Scholar
  20. Nashine HK, Samet B, Kim JK: Fixed point results for contractions involving generalized altering distances in ordered metric spaces. Fixed Point Theory Appl 2011, 2011: 5.MathSciNetView ArticleGoogle Scholar
  21. Nieto JJ, López RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239.MATHMathSciNetView ArticleGoogle Scholar
  22. O'Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J Math Anal Appl 2008, 341: 1241–1252.MATHMathSciNetView ArticleGoogle Scholar
  23. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc Am Math Soc 2004, 132: 1435–1443.MATHMathSciNetView ArticleGoogle Scholar
  24. Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal 2010, 72: 4508–4517.MATHMathSciNetView ArticleGoogle Scholar
  25. Samet B, Vetro C: Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces. Nonlinear Anal 2011, 74: 4260–4268.MATHMathSciNetView ArticleGoogle Scholar
  26. Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal 2012, 75: 2154–2165.MATHMathSciNetView ArticleGoogle Scholar
  27. Turinici M: Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J Math Anal Appl 1986, 117: 100–127.MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Ćirić et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement