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

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.

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 xy if only if yxP. A cone PE is called normal if there is a constant N>0 such that θxy implies xNy for all x,yE. 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,yE, sup{x,y} and inf{x,y} exist. One can see [7] for the definition and the properties of the lattice.

Let DE, the operator A:DE is said to be an increasing operator if x,yD, xy, implies AxAy; the operator A:DE is said to be a decreasing operator if x,yD, xy, implies AyAx.

Lemma 1.1 [9]

Let E be a real Banach space, DE be nonempty, closed bounded convex, and A:DD be condensing. Then A has a fixed point in D.

Lemma 1.2 [10]

Let E be a real Banach space, DE be nonempty, closed bounded convex, and A:DD be completely continuous. Then A has a fixed point in D.

Lemma 1.3 [11]

Let E be a real Banach space, DE be nonempty, closed bounded convex, and A:DD 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, DE be bounded, and A:DD be a decreasing and condensing operator. Then the operator A has a fixed point in D.

Proof For any xD, since A:DD, we have AxD.

Since E is a Banach space with lattice structure and DE is bounded, there exists u 0 D such that

inf{Ax,x}= u 0 .

That is,

u 0 Ax, u 0 x.
(2.1)

Since A is a decreasing operator, we have

A 2 xA u 0 ,AxA u 0 .
(2.2)

(2.1) and (2.2) show that

u 0 A u 0 .
(2.3)

Similar to the proof of (2.3), there exists v 0 D such that

sup{Ax,x}= v 0 .

That is,

Ax v 0 ,x v 0 .
(2.4)

Since A is a decreasing operator, we have

A v 0 A 2 x,A v 0 Ax.
(2.5)

(2.4) and (2.5) show that

A v 0 v 0 .
(2.6)

(2.3) and (2.6) together with u 0 v 0 show that

u 0 A v 0 A u 0 v 0 .
(2.7)

For any x[ u 0 , v 0 ], since A is a decreasing operator, we have

A v 0 AxA u 0 .

By (2.7), we have

A[ u 0 , v 0 ][ u 0 , v 0 ].

It is easy to know that [ u 0 , v 0 ] is a closed convex set. Since DE is bounded, we have [ u 0 , v 0 ] is bounded. Hence, [ u 0 , v 0 ] 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, PE be a normal cone, and A:EE be a decreasing and condensing operator. Then the operator A has a fixed point in E.

Proof For any xE, since A:EE, we have AxE.

Since E is a Banach space with lattice structure, there exists u 0 E such that

inf{Ax,x}= u 0 .

That is,

u 0 Ax, u 0 x.
(2.8)

Since A is a decreasing operator, we have

A 2 xA u 0 ,AxA u 0 .
(2.9)

(2.8) and (2.9) show that

u 0 A u 0 .
(2.10)

Similar to the proof of (2.10), there exist v 0 E such that

sup{Ax,x}= v 0 .

That is,

Ax v 0 ,x v 0 .
(2.11)

Since A is a decreasing operator, we have

A v 0 A 2 x,A v 0 Ax.
(2.12)

(2.11) and (2.12) show that

A v 0 v 0 .
(2.13)

(2.10) and (2.13) together with u 0 v 0 show that

u 0 A v 0 A u 0 v 0 .
(2.14)

For any x[ u 0 , v 0 ], since A is a decreasing operator, we have

A v 0 AxA u 0 .

By (2.14), we have

A[ u 0 , v 0 ][ u 0 , v 0 ].

It is easy to know that [ u 0 , v 0 ] is a closed convex set. Since P is a normal cone of E, we have [ u 0 , v 0 ] is bounded. Hence, [ u 0 , v 0 ] 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, DE be bounded, and A:DD 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, PE be a normal cone, and A:EE 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, DE be bounded, and A:DD 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, PE be a normal cone, and A:EE 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 (n2) whose boundary Ω is assumed to be sufficiently smooth. Consider a uniformly elliptic differential operator on Ω ¯

Lu= 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

i.e., there exists a positive constant μ 0 such that i , j = 1 n a i j (x) ξ i ξj μ 0 | ξ | 2 for any x Ω ¯ and ξ=( ξ 1 , ξ 2 ,, ξ n ) R n , where a i j (x)= a j i (x), c(x)0. For the sake of simplicity, we will assume that all functions a j i (x), b i (x), c(x) are sufficiently smooth.

Considering the Dirichlet problem

Lu=f(x,u),u | Ω =0,
(4.1)

we have the following conclusions.

Theorem 4.1 Suppose that f(x,u)C( Ω ¯ ×[0,),[0,)), which is decreasing on u, then the problem (4.1) has a positive solution.

Proof It is easy to know that E=C( Ω ¯ ) is a Banach space with a maximum norm and it is also a lattice. Let P={uEu(t)0,tI} 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

Au(x)= Ω ¯ G(x,y)f ( y , u ( y ) ) dy,

where G(x,y) denotes the Green function of a differential operator L with boundary condition u | Ω =0. It is also well known that G(x,y) satisfies the following inequality:

0<G(x,y)<{ K 0 | x y | 2 n , n > 2 , K 0 | ln | x y | | , n = 2 (x,yΩ,xy).

Hence, the linear integral operator

Bv(x)= Ω ¯ G(x,y)v(y)dy

is a completely continuous operator from E into E. Clearly, the superposition operator Fϕ(x)=f(x,ϕ(x)) that maps P into P is continuous and bounded. Therefore, the operator A=BF 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 uv,

f ( t , u ( x ) ) f ( t , v ( x ) ) ,

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 ¯ ,

so A is decreasing.

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

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.

    Google Scholar 

  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.

    Article  Google Scholar 

  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.2007.11.026

    Article  MathSciNet  Google Scholar 

  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/S0002-9939-07-08729-1

    Article  Google Scholar 

  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

    Google Scholar 

  6. Nieto JJ: An abstract monotone iterative technique. Nonlinear Anal. 1997, 28: 1923–1933. 10.1016/S0362-546X(97)89710-6

    Article  MathSciNet  Google Scholar 

  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.2007.10.063

    Article  MathSciNet  Google Scholar 

  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.2005.10.017

    Article  MathSciNet  Google Scholar 

  9. Sadovskii BN: A fixed point principle. Funct. Anal. Appl. 1967, 1: 151–153.

    Article  Google Scholar 

  10. Gnana Bhaskar T, Bose RK: Some Topics in Nonlinear Functional Analysis. Wiley, New Delhi; 1985.

    Google Scholar 

  11. Darbo G: Punti uniti in trasformazioni a condominio non compatto. Rend. Semin. Mat. Univ. Padova 1955, 24: 84–92.

    MathSciNet  Google Scholar 

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Acknowledgements

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

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Correspondence to Xingchang Li.

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The authors declare that they have no competing interests.

Authors’ contributions

XL carried out the the main theorem and the main conclusion. ZW carried out the application of the main theorem. All authors read and approved the final manuscript.

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Li, X., Wang, Z. Fixed point theorems for decreasing operators in ordered Banach spaces with lattice structure and their applications. Fixed Point Theory Appl 2013, 18 (2013). https://doi.org/10.1186/1687-1812-2013-18

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Keywords

  • decreasing operators
  • lattice structure
  • nonlinear
  • elliptic equations