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

  • Xingchang Li1Email author and

    Affiliated with

    • Zhihao Wang1

      Affiliated with

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq1_HTML.gif if only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq2_HTML.gif . A cone http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq3_HTML.gif is called normal if there is a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq4_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq5_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq6_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq9_HTML.gif exist. One can see [7] for the definition and the properties of the lattice.

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq10_HTML.gif , the operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq11_HTML.gif is said to be an increasing operator if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq1_HTML.gif , implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq13_HTML.gif ; the operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq11_HTML.gif is said to be a decreasing operator if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq1_HTML.gif , implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq14_HTML.gif .

      Lemma 1.1[9]

      Let E be a real Banach space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq10_HTML.gif be nonempty, closed bounded convex, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq10_HTML.gif be nonempty, closed bounded convex, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq10_HTML.gif be nonempty, closed bounded convex, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq10_HTML.gif be bounded, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq16_HTML.gif be a decreasing and condensing operator. Then the operator A has a fixed point in D.

      Proof For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq17_HTML.gif , since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq18_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq19_HTML.gif .

      Since E is a Banach space with lattice structure and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq10_HTML.gif is bounded, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq20_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equa_HTML.gif
      That is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ1_HTML.gif
      (2.1)
      Since A is a decreasing operator, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ2_HTML.gif
      (2.2)
      (2.1) and (2.2) show that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ3_HTML.gif
      (2.3)
      Similar to the proof of (2.3), there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq21_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equb_HTML.gif
      That is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ4_HTML.gif
      (2.4)
      Since A is a decreasing operator, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ5_HTML.gif
      (2.5)
      (2.4) and (2.5) show that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ6_HTML.gif
      (2.6)
      (2.3) and (2.6) together with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq22_HTML.gif show that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ7_HTML.gif
      (2.7)
      For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq23_HTML.gif , since A is a decreasing operator, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equc_HTML.gif
      By (2.7), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equd_HTML.gif

      It is easy to know that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq24_HTML.gif is a closed convex set. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq10_HTML.gif is bounded, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq24_HTML.gif is bounded. Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq3_HTML.gif be a normal cone, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq25_HTML.gif be a decreasing and condensing operator. Then the operator A has a fixed point in E.

      Proof For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq26_HTML.gif , since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq25_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq27_HTML.gif .

      Since E is a Banach space with lattice structure, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq28_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Eque_HTML.gif
      That is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ8_HTML.gif
      (2.8)
      Since A is a decreasing operator, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ9_HTML.gif
      (2.9)
      (2.8) and (2.9) show that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ10_HTML.gif
      (2.10)
      Similar to the proof of (2.10), there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq29_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equf_HTML.gif
      That is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ11_HTML.gif
      (2.11)
      Since A is a decreasing operator, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ12_HTML.gif
      (2.12)
      (2.11) and (2.12) show that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ13_HTML.gif
      (2.13)
      (2.10) and (2.13) together with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq22_HTML.gif show that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ14_HTML.gif
      (2.14)
      For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq23_HTML.gif , since A is a decreasing operator, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equg_HTML.gif
      By (2.14), we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equh_HTML.gif

      It is easy to know that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq24_HTML.gif is a closed convex set. Since P is a normal cone of E, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq24_HTML.gif is bounded. Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq10_HTML.gif be bounded, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq3_HTML.gif be a normal cone, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq10_HTML.gif be bounded, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq3_HTML.gif be a normal cone, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq30_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq31_HTML.gif ) whose boundary Ω is assumed to be sufficiently smooth. Consider a uniformly elliptic differential operator on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq32_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equi_HTML.gif

      i.e., there exists a positive constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq33_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq34_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq36_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq38_HTML.gif . For the sake of simplicity, we will assume that all functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq41_HTML.gif are sufficiently smooth.

      Considering the Dirichlet problem
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equ15_HTML.gif
      (4.1)

      we have the following conclusions.

      Theorem 4.1 Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq43_HTML.gif is a Banach space with a maximum norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq44_HTML.gif and it is also a lattice. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equj_HTML.gif
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq46_HTML.gif denotes the Green function of a differential operator L with boundary condition http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq47_HTML.gif . It is also well known that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq46_HTML.gif satisfies the following inequality:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equk_HTML.gif
      Hence, the linear integral operator
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equl_HTML.gif

      is a completely continuous operator from E into E. Clearly, the superposition operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq48_HTML.gif that maps P into P is continuous and bounded. Therefore, the operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_IEq50_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_349_Equm_HTML.gif
      implies that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-18/MediaObjects/13663_2012_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

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