An intermediate value theorem for monotone operators in ordered Banach spaces

  • Vadim Kostrykin1Email author and

    Affiliated with

    • Anna Oleynik1, 2

      Affiliated with

      Fixed Point Theory and Applications20122012:211

      DOI: 10.1186/1687-1812-2012-211

      Received: 5 June 2012

      Accepted: 5 November 2012

      Published: 22 November 2012

      Abstract

      We consider a monotone increasing operator in an ordered Banach space having http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq2_HTML.gif as a strong super- and subsolution, respectively. In contrast with the well-studied case http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq3_HTML.gif , we suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq4_HTML.gif . Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the order interval http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq5_HTML.gif .

      MSC: 47H05, 47H10, 46B40.

      Keywords

      fixed point theorems in ordered Banach spaces
      It is an elementary consequence of the intermediate value theorem for continuous real-valued functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq6_HTML.gif that if either
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equ1_HTML.gif
      (1)
      or
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equ2_HTML.gif
      (2)

      then f has a fixed point in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq7_HTML.gif . It is a natural question whether this result can be extended to the case of ordered Banach spaces. A number of fixed point theorems with assumptions of type (1) are well known; see, e.g., [[1], Section 2.1]. However, to the best of our knowledge, fixed point theorems with assumptions of type (2) have not been known so far. In the present note, we prove the following fixed point theorem of this type.

      Theorem 1 Let X be a real Banach space with an order cone K satisfying

      1. (a)

        K has a nonempty interior,

         
      2. (b)

        K is normal and minihedral.

         
      Assume that there are two points in X, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq8_HTML.gif , and a monotone increasing compact continuous operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq9_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq1_HTML.gif is a strong supersolution of T and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq2_HTML.gif is a strong subsolution, that is,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equa_HTML.gif

      then T has a fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq10_HTML.gif .

      Here http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq5_HTML.gif denotes the order interval http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq11_HTML.gif .

      Theorem 1 generalizes an idea developed by the present authors in [2], where the existence of solutions to a certain nonlinear integral equation of Hammerstein type has been shown.

      Before we present the proof, we recall some notions. We write http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq12_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq14_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq15_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq16_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq17_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq18_HTML.gif is the interior of the cone K.

      A cone K is called minihedral if for any pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq20_HTML.gif , bounded above in order there exists the least upper bound http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq21_HTML.gif , that is, an element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq22_HTML.gif such that

      1. (1)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq24_HTML.gif ,

         
      2. (2)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq26_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq27_HTML.gif .

         

      Obviously, a cone K is minihedral if and only if for any pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq20_HTML.gif , bounded below in order there exists the greatest lower bound http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq28_HTML.gif . If a minihedral cone has a nonempty interior, then any pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq20_HTML.gif is bounded above in order. Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq21_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq28_HTML.gif exist for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq20_HTML.gif .

      A cone K is called normal if there exists a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq29_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq31_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq32_HTML.gif .

      By the Kakutani-Krein brothers theorem [[3], Theorem 6.6] a real Banach space X with an order cone K satisfying assumptions (a) and (b) of Theorem 1 is isomorphic to the Banach space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq33_HTML.gif of continuous functions on a compact Hausdorff space Q. The image of K under this isomorphism is the cone of nonnegative continuous functions on Q.

      An operator T acting in the Banach space X is called monotone increasing if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq34_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq35_HTML.gif .

      Consider the operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq36_HTML.gif defined by
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equ3_HTML.gif
      (3)

      Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq37_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq38_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq2_HTML.gif is a fixed point of the operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq39_HTML.gif . Similarly, one shows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq1_HTML.gif is also a fixed point.

      Lemma 2 The operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq39_HTML.gif is continuous, monotone increasing, compact and maps the order interval http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq5_HTML.gif into itself.

      Proof For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq40_HTML.gif , the maps http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq42_HTML.gif are continuous; see, e.g., Corollary 3.1.1 in [4]. Due to the continuity of T, it follows immediately that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq39_HTML.gif is continuous as well. The operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq39_HTML.gif is monotone increasing since inf and sup are monotone increasing with respect to each argument. Therefore, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq43_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equb_HTML.gif

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq44_HTML.gif be an arbitrary sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq5_HTML.gif . Since T is compact, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq45_HTML.gif has a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq46_HTML.gif converging to some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq47_HTML.gif . From the continuity of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq39_HTML.gif , it follows that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq48_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq49_HTML.gif , thus, proving that the range of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq39_HTML.gif is relatively compact. □

      Lemma 3 There exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq50_HTML.gif with
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equc_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equd_HTML.gif

      Proof Due to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq51_HTML.gif , there is a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq52_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq53_HTML.gif . The preimage of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq54_HTML.gif under the continuous mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq55_HTML.gif contains a ball http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq56_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq57_HTML.gif holds for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq58_HTML.gif . By the same argument, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq59_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq60_HTML.gif . Choosing http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq61_HTML.gif sufficiently small, we can achieve that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq62_HTML.gif .

      Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq63_HTML.gif . We choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq64_HTML.gif so small that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq66_HTML.gif so close to 1 that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq67_HTML.gif . Then we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq68_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Eque_HTML.gif
      Due to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq70_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq71_HTML.gif . Further, we obtain
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equf_HTML.gif

      From http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq70_HTML.gif it follows that there is an element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq72_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq73_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq74_HTML.gif . Then we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq75_HTML.gif . However, in view of the Kakutani-Krein brothers theorem, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq76_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq77_HTML.gif . Thus, it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq78_HTML.gif and, therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq79_HTML.gif . Similarly one shows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq80_HTML.gif . □

      The main tool for the proof of Theorem 1 is Amann’s theorem on three fixed points (see, e.g., [[5], Theorem 7.F and Corollary 7.40]):

      Theorem 4 Let X be a real Banach space with an order cone having a nonempty interior. Assume there are four points in X,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equg_HTML.gif
      and a monotone increasing image compact operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq81_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_Equh_HTML.gif

      Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq39_HTML.gif has a third fixed point p satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq82_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq83_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq84_HTML.gif .

      Recall that the operator is called image compact if it is continuous and its image is a relatively compact set.

      We choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq85_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq86_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq87_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq88_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq89_HTML.gif is as in Lemma 3. Since the cone K is normal, by Theorem 1.1.1 in [1], http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq5_HTML.gif is norm bounded. Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq39_HTML.gif is image compact.

      Theorem 4 yields the existence of a fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq90_HTML.gif of the operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq39_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq91_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_277_IEq90_HTML.gif is a fixed point of the operator T as well. This observation completes the proof of Theorem 1.

      Declarations

      Acknowledgements

      The authors thank H.-P. Heinz for useful comments. This work has been supported in part by the Deutsche Forschungsgemeinschaft, Grant KO 2936/4-1.

      Authors’ Affiliations

      (1)
      FB 08 - Institut für Mathematik, Johannes Gutenberg-Universität Mainz
      (2)
      Department of Mathematics, University of Uppsala

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