Open Access

Fixed point theorems for decreasing operators in ordered Banach spaces with lattice structure and their applications

Fixed Point Theory and Applications20132013:18

DOI: 10.1186/1687-1812-2013-18

Received: 2 December 2012

Accepted: 14 January 2013

Published: 30 January 2013

Abstract

This paper presents some theorems of the fixed point for decreasing operators in Banach spaces with lattice structure. The results are applied to nonlinear second-order elliptic equations.

MSC:47H10, 34B15.

Keywords

decreasing operators lattice structure nonlinear elliptic equations

1 Introduction and preliminaries

The fixed point theory for monotone operators in ordered Banach spaces has been investigated extensively in the past 30 years [18]. Many new fixed point theorems have been proved under the nonlinear contractive condition by using the theorem of cone and monotone iterative technique. These results have been applied to study the ordinary differential equations, partial differential equations, and integral equations.

In this paper, we investigate decreasing operators in ordered Banach spaces with lattice structure. The theoretical results of fixed points are extended by using the famous Schauder fixed point theorem for the operators. We weaken the conditions of the Schauder fixed point theorem. The results of this paper have no need for the closed bounded and convex property of domains for the operators. To demonstrate the applicability of our results, we apply them to study a problem of nonlinear second-order elliptic equations in the final section of the paper, and the existence of solution is obtained.

Let E be a Banach space and P be a cone of E. We define a partial ordering ≤ with respect to P by x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq1_HTML.gif if only if y x P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq2_HTML.gif. A cone P E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq3_HTML.gif is called normal if there is a constant N > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq4_HTML.gif such that θ x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq5_HTML.gif implies x N y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq6_HTML.gif for all x , y E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq7_HTML.gif. The least positive constant N satisfying the above inequality is called the normal constant of P.

Let E be a partially ordered set. We call E a lattice in the partial ordering ≤. For arbitrary x , y E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq7_HTML.gif, sup { x , y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq8_HTML.gif and inf { x , y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq9_HTML.gif exist. One can see [7] for the definition and the properties of the lattice.

Let D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq10_HTML.gif, the operator A : D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq11_HTML.gif is said to be an increasing operator if x , y D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq12_HTML.gif, x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq1_HTML.gif, implies A x A y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq13_HTML.gif; the operator A : D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq11_HTML.gif is said to be a decreasing operator if x , y D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq12_HTML.gif, x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq1_HTML.gif, implies A y A x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq14_HTML.gif.

Lemma 1.1 [9]

Let E be a real Banach space, D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq10_HTML.gif be nonempty, closed bounded convex, and A : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq15_HTML.gif be condensing. Then A has a fixed point in D.

Lemma 1.2 [10]

Let E be a real Banach space, D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq10_HTML.gif be nonempty, closed bounded convex, and A : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq15_HTML.gif be completely continuous. Then A has a fixed point in D.

Lemma 1.3 [11]

Let E be a real Banach space, D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq10_HTML.gif be nonempty, closed bounded convex, and A : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq15_HTML.gif be strict-set-contraction mappings. Then A has a fixed point in D.

Remark 1 Lemma 1.1 is the famous Sadovskii fixed point theorem; Lemma 1.2 is the famous Schauder fixed point theorem; Lemma 1.3 is the famous Darbo fixed point theorem.

2 Main results

Theorem 2.1 Let E be an ordered Banach space with lattice structure, D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq10_HTML.gif be bounded, and A : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq16_HTML.gif be a decreasing and condensing operator. Then the operator A has a fixed point in D.

Proof For any x D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq17_HTML.gif, since A : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq18_HTML.gif, we have A x D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq19_HTML.gif.

Since E is a Banach space with lattice structure and D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq10_HTML.gif is bounded, there exists u 0 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq20_HTML.gif such that
inf { A x , x } = u 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equa_HTML.gif
That is,
u 0 A x , u 0 x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ1_HTML.gif
(2.1)
Since A is a decreasing operator, we have
A 2 x A u 0 , A x A u 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ2_HTML.gif
(2.2)
(2.1) and (2.2) show that
u 0 A u 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ3_HTML.gif
(2.3)
Similar to the proof of (2.3), there exists v 0 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq21_HTML.gif such that
sup { A x , x } = v 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equb_HTML.gif
That is,
A x v 0 , x v 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ4_HTML.gif
(2.4)
Since A is a decreasing operator, we have
A v 0 A 2 x , A v 0 A x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ5_HTML.gif
(2.5)
(2.4) and (2.5) show that
A v 0 v 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ6_HTML.gif
(2.6)
(2.3) and (2.6) together with u 0 v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq22_HTML.gif show that
u 0 A v 0 A u 0 v 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ7_HTML.gif
(2.7)
For any x [ u 0 , v 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq23_HTML.gif, since A is a decreasing operator, we have
A v 0 A x A u 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equc_HTML.gif
By (2.7), we have
A [ u 0 , v 0 ] [ u 0 , v 0 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equd_HTML.gif

It is easy to know that [ u 0 , v 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq24_HTML.gif is a closed convex set. Since D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq10_HTML.gif is bounded, we have [ u 0 , v 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq24_HTML.gif is bounded. Hence, [ u 0 , v 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq24_HTML.gif is a closed bounded convex set. Thus, Lemma 1.1 implies that the operator A has a fixed point in D. □

Theorem 2.2 Let E be an ordered Banach space with lattice structure, P E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq3_HTML.gif be a normal cone, and A : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq25_HTML.gif be a decreasing and condensing operator. Then the operator A has a fixed point in E.

Proof For any x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq26_HTML.gif, since A : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq25_HTML.gif, we have A x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq27_HTML.gif.

Since E is a Banach space with lattice structure, there exists u 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq28_HTML.gif such that
inf { A x , x } = u 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Eque_HTML.gif
That is,
u 0 A x , u 0 x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ8_HTML.gif
(2.8)
Since A is a decreasing operator, we have
A 2 x A u 0 , A x A u 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ9_HTML.gif
(2.9)
(2.8) and (2.9) show that
u 0 A u 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ10_HTML.gif
(2.10)
Similar to the proof of (2.10), there exist v 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq29_HTML.gif such that
sup { A x , x } = v 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equf_HTML.gif
That is,
A x v 0 , x v 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ11_HTML.gif
(2.11)
Since A is a decreasing operator, we have
A v 0 A 2 x , A v 0 A x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ12_HTML.gif
(2.12)
(2.11) and (2.12) show that
A v 0 v 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ13_HTML.gif
(2.13)
(2.10) and (2.13) together with u 0 v 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq22_HTML.gif show that
u 0 A v 0 A u 0 v 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ14_HTML.gif
(2.14)
For any x [ u 0 , v 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq23_HTML.gif, since A is a decreasing operator, we have
A v 0 A x A u 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equg_HTML.gif
By (2.14), we have
A [ u 0 , v 0 ] [ u 0 , v 0 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equh_HTML.gif

It is easy to know that [ u 0 , v 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq24_HTML.gif is a closed convex set. Since P is a normal cone of E, we have [ u 0 , v 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq24_HTML.gif is bounded. Hence, [ u 0 , v 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq24_HTML.gif is a closed bounded convex set. Thus, Lemma 1.1 implies that the operator A has a fixed point in D. □

3 Corollaries and relative results

Similar to the proof of Theorem 2.1, by Lemma 1.2 and Lemma 1.3, we can get the following corollaries and relative results.

Corollary 3.1 Let E be an ordered Banach space with lattice structure, D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq10_HTML.gif be bounded, and A : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq16_HTML.gif be a decreasing and completely continuous operator. Then the operator A has a fixed point in D.

Corollary 3.2 Let E be an ordered Banach space with lattice structure, P E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq3_HTML.gif be a normal cone, and A : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq25_HTML.gif be a decreasing and completely continuous operator. Then the operator A has a fixed point in E.

Corollary 3.3 Let E be an ordered Banach space with lattice structure, D E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq10_HTML.gif be bounded, and A : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq16_HTML.gif be a decreasing and strict-set-contraction mapping. Then the mapping A has a fixed point in D.

Corollary 3.4 Let E be an ordered Banach space with lattice structure, P E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq3_HTML.gif be a normal cone, and A : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq25_HTML.gif be a decreasing and strict-set-contraction mapping. Then the mapping A has a fixed point in E.

4 Applications

In this section, we use Theorem 2.1 to show the existence of a solution for the uniformly elliptic differential problem. Let Ω be a bounded convex domain in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq30_HTML.gif ( n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq31_HTML.gif) whose boundary Ω is assumed to be sufficiently smooth. Consider a uniformly elliptic differential operator on Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq32_HTML.gif
L u = i , j = 1 n a i j ( x ) 2 u x i x j + i , j = 1 n b i ( x ) u x i + c ( x ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equi_HTML.gif

i.e., there exists a positive constant μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq33_HTML.gif such that i , j = 1 n a i j ( x ) ξ i ξ j μ 0 | ξ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq34_HTML.gif for any x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq35_HTML.gif and ξ = ( ξ 1 , ξ 2 , , ξ n ) R n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq36_HTML.gif, where a i j ( x ) = a j i ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq37_HTML.gif, c ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq38_HTML.gif. For the sake of simplicity, we will assume that all functions a j i ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq39_HTML.gif, b i ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq40_HTML.gif, c ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq41_HTML.gif are sufficiently smooth.

Considering the Dirichlet problem
L u = f ( x , u ) , u | Ω = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equ15_HTML.gif
(4.1)

we have the following conclusions.

Theorem 4.1 Suppose that f ( x , u ) C ( Ω ¯ × [ 0 , ) , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq42_HTML.gif, which is decreasing on u, then the problem (4.1) has a positive solution.

Proof It is easy to know that E = C ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq43_HTML.gif is a Banach space with a maximum norm https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq44_HTML.gif and it is also a lattice. Let P = { u E u ( t ) 0 , t I } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq45_HTML.gif and P be a normal cone in E. It is well known (see [1, 10]) that the solution of the Dirichlet problem (4.1) is equivalent to the fixed point of the integral operator A
A u ( x ) = Ω ¯ G ( x , y ) f ( y , u ( y ) ) d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equj_HTML.gif
where G ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq46_HTML.gif denotes the Green function of a differential operator L with boundary condition u | Ω = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq47_HTML.gif. It is also well known that G ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq46_HTML.gif satisfies the following inequality:
0 < G ( x , y ) < { K 0 | x y | 2 n , n > 2 , K 0 | ln | x y | | , n = 2 ( x , y Ω , x y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equk_HTML.gif
Hence, the linear integral operator
B v ( x ) = Ω ¯ G ( x , y ) v ( y ) d y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equl_HTML.gif

is a completely continuous operator from E into E. Clearly, the superposition operator F ϕ ( x ) = f ( x , ϕ ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq48_HTML.gif that maps P into P is continuous and bounded. Therefore, the operator A = B F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq49_HTML.gif that maps P into P is completely continuous, and thus A is condensing.

Moreover, the mapping A is decreasing in u. In fact, by hypotheses, for u v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_IEq50_HTML.gif,
f ( t , u ( x ) ) f ( t , v ( x ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equm_HTML.gif
implies that
( A u ) ( x ) = Ω ¯ G ( x , y ) f ( y , u ( y ) ) d y Ω ¯ G ( x , y ) f ( y , v ( y ) ) d y = ( A v ) ( x ) , x Ω ¯ , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_Article_349_Equn_HTML.gif

so A is decreasing.

So, the condition of Theorem 2.1 holds, Theorem 4.1 is proved. □

Declarations

Acknowledgements

The first author was supported financially by the NSFC (71240007, 11001151), NSFSP (ZR2010AM005).

Authors’ Affiliations

(1)
Center for Economic Research, Harbin University of Commerce

References

  1. Guo D: Positive fixed points and eigenvectors of noncompact decreasing operators with applications to nonlinear integral equations. Chin. Ann. Math., Ser. B 1993, 4: 419–426.
  2. Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. 2007, 23(12):2203–2212.View Article
  3. O’Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026MathSciNetView Article
  4. Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces. Proc. Am. Math. Soc. 2007, 135: 2505–2517. 10.1090/S0002-9939-07-08729-1View Article
  5. Sadarangani K, Caballero J, Harjani J: Contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations. Fixed Point Theory Appl. 2010., 2010: Article ID 916064
  6. Nieto JJ: An abstract monotone iterative technique. Nonlinear Anal. 1997, 28: 1923–1933. 10.1016/S0362-546X(97)89710-6MathSciNetView Article
  7. Wu Y: New fixed point theorems and applications of mixed monotone operator. J. Math. Anal. Appl. 2008, 341: 883–893. 10.1016/j.jmaa.2007.10.063MathSciNetView Article
  8. Gnana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. TMA 2006, 65(7):1379–1393. 10.1016/j.na.2005.10.017MathSciNetView Article
  9. Sadovskii BN: A fixed point principle. Funct. Anal. Appl. 1967, 1: 151–153.View Article
  10. Gnana Bhaskar T, Bose RK: Some Topics in Nonlinear Functional Analysis. Wiley, New Delhi; 1985.
  11. Darbo G: Punti uniti in trasformazioni a condominio non compatto. Rend. Semin. Mat. Univ. Padova 1955, 24: 84–92.MathSciNet

Copyright

© Li and Wang; licensee Springer 2013

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.