A best proximity point theorem for Geraghty-contractions

  • J Caballero1,

    Affiliated with

    • J Harjani1 and

      Affiliated with

      • K Sadarangani1Email author

        Affiliated with

        Fixed Point Theory and Applications20122012:231

        DOI: 10.1186/1687-1812-2012-231

        Received: 16 May 2012

        Accepted: 10 December 2012

        Published: 27 December 2012

        Abstract

        The purpose of this paper is to provide sufficient conditions for the existence of a unique best proximity point for Geraghty-contractions.

        Our paper provides an extension of a result due to Geraghty (Proc. Am. Math. Soc. 40:604-608, 1973).

        Keywords

        fixed point Geraghty-contraction P-property best proximity point

        1 Introduction

        Let A and B be nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq1_HTML.gif .

        An operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq2_HTML.gif is said to be a k-contraction if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq3_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq4_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq5_HTML.gif . Banach’s contraction principle states that when A is a complete subset of X and T is a k-contraction which maps A into itself, then T has a unique fixed point in A.

        A huge number of generalizations of this principle appear in the literature. Particularly, the following generalization of Banach’s contraction principle is due to Geraghty [1].

        First, we introduce the class ℱ of those functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq6_HTML.gif satisfying the following condition:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equa_HTML.gif

        Theorem 1.1 ([1])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif be a complete metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq8_HTML.gif be an operator. Suppose that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq9_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq10_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equ1_HTML.gif
        (1)

        Then T has a unique fixed point.

        Since the constant functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq11_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq3_HTML.gif , belong to ℱ, Theorem 1.1 extends Banach’s contraction principle.

        Remark 1.1 Since the functions belonging to ℱ are strictly smaller than one, condition (1) implies that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equb_HTML.gif

        Therefore, any operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq8_HTML.gif satisfying (1) is a continuous operator.

        The aim of this paper is to give a generalization of Theorem 1.1 by considering a non-self map T.

        First, we present a brief discussion about a best proximity point.

        Let A be a nonempty subset of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq12_HTML.gif be a mapping. The solutions of the equation http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq13_HTML.gif are fixed points of T. Consequently, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq14_HTML.gif is a necessary condition for the existence of a fixed point for the operator T. If this necessary condition does not hold, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq15_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq16_HTML.gif and the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq17_HTML.gif does not have any fixed point. In this setting, our aim is to find an element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq16_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq18_HTML.gif is minimum in some sense. The best approximation theory and best proximity point analysis have been developed in this direction.

        In our context, we consider two nonempty subsets A and B of a complete metric space and a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq2_HTML.gif .

        A natural question is whether one can find an element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq19_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq20_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq21_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq16_HTML.gif , the optimal solution to this problem will be the one for which the value http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq22_HTML.gif is attained by the real valued function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq23_HTML.gif given by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq24_HTML.gif .

        Some results about best proximity points can be found in [29].

        2 Notations and basic facts

        Let A and B be two nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif .

        We denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq26_HTML.gif the following sets:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equc_HTML.gif

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq27_HTML.gif .

        In [8], the authors present sufficient conditions which determine when the sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq26_HTML.gif are nonempty.

        Now, we present the following definition.

        Definition 2.1 Let A, B be two nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif . A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq2_HTML.gif is said to be a Geraghty-contraction if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq9_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equd_HTML.gif
        Notice that since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq28_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Eque_HTML.gif

        Therefore, every Geraghty-contraction is a contractive mapping.

        In [10], the author introduces the following definition.

        Definition 2.2 ([10])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq29_HTML.gif be a pair of nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq30_HTML.gif . Then the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq29_HTML.gif is said to have the P-property if and only if for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq32_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equf_HTML.gif

        It is easily seen that for any nonempty subset A of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif , the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq33_HTML.gif has the P-property.

        In [10], the author proves that any pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq29_HTML.gif of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property.

        3 Main results

        We start this section presenting our main result.

        Theorem 3.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq29_HTML.gif be a pair of nonempty closed subsets of a complete metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq25_HTML.gif is nonempty. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq2_HTML.gif be a Geraghty-contraction satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq34_HTML.gif . Suppose that the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq29_HTML.gif has the P-property. Then there exists a unique http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq35_HTML.gif in A such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq36_HTML.gif .

        Proof Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq25_HTML.gif is nonempty, we take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq19_HTML.gif .

        As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq37_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq38_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq39_HTML.gif . Similarly, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq40_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq41_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq42_HTML.gif . Repeating this process, we can get a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq43_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq25_HTML.gif satisfying
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equg_HTML.gif
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq29_HTML.gif has the P-property, we have that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equh_HTML.gif
        Taking into account that T is a Geraghty-contraction, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq44_HTML.gif , we have that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equ2_HTML.gif
        (2)

        Suppose that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq45_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq46_HTML.gif .

        In this case,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equi_HTML.gif

        and consequently, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq47_HTML.gif .

        Therefore,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equj_HTML.gif

        and this is the desired result.

        In the contrary case, suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq48_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq44_HTML.gif .

        By (2), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq49_HTML.gif is a decreasing sequence of nonnegative real numbers, and hence there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq50_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equk_HTML.gif

        In the sequel, we prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq51_HTML.gif .

        Assume http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq52_HTML.gif , then from (2) we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equl_HTML.gif

        The last inequality implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq53_HTML.gif and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq9_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq51_HTML.gif and this contradicts our assumption.

        Therefore,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equ3_HTML.gif
        (3)

        Notice that since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq54_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq55_HTML.gif , for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq56_HTML.gif fixed, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq57_HTML.gif , and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq29_HTML.gif satisfies the P-property, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq58_HTML.gif .

        In what follows, we prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq43_HTML.gif is a Cauchy sequence.

        In the contrary case, we have that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equm_HTML.gif
        By using the triangular inequality,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equn_HTML.gif
        By (2) and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq59_HTML.gif , by the above mentioned comment, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equo_HTML.gif
        which gives us
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equp_HTML.gif
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq60_HTML.gif and by (3), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq61_HTML.gif , from the last inequality it follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equq_HTML.gif

        Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq62_HTML.gif .

        Taking into account that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq9_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq63_HTML.gif and this contradicts our assumption.

        Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq43_HTML.gif is a Cauchy sequence.

        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq64_HTML.gif and A is a closed subset of the complete metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq65_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq66_HTML.gif .

        Since any Geraghty-contraction is a contractive mapping and hence continuous, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq67_HTML.gif .

        This implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq68_HTML.gif .

        Taking into account that the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq69_HTML.gif is a constant sequence with value http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq22_HTML.gif , we deduce
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equr_HTML.gif

        This means that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq35_HTML.gif is a best proximity point of T.

        This proves the part of existence of our theorem.

        For the uniqueness, suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq71_HTML.gif are two best proximity points of T with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq72_HTML.gif .

        This means that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equs_HTML.gif
        Using the P-property, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equt_HTML.gif
        Using the fact that T is a Geraghty-contraction, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equu_HTML.gif

        which is a contradiction.

        Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq73_HTML.gif .

        This finishes the proof. □

        4 Examples

        In order to illustrate our results, we present some examples.

        Example 4.1 Consider http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq74_HTML.gif with the usual metric.

        Let A and B be the subsets of X defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equv_HTML.gif

        Obviously, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq75_HTML.gif and A, B are nonempty closed subsets of X.

        Moreover, it is easily seen that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq77_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq2_HTML.gif be the mapping defined as
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equw_HTML.gif

        In the sequel, we check that T is a Geraghty-contraction.

        In fact, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq78_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq79_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equ4_HTML.gif
        (4)
        Now, we prove that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equ5_HTML.gif
        (5)

        Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq80_HTML.gif (the same reasoning works for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq81_HTML.gif ).

        Then, since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq82_HTML.gif is strictly increasing in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq83_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equx_HTML.gif

        This proves (5).

        Taking into account (4) and (5), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equ6_HTML.gif
        (6)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq82_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq84_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq85_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq87_HTML.gif .

        Obviously, when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq88_HTML.gif , the inequality (6) is satisfied.

        It is easily seen that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq89_HTML.gif by using elemental calculus.

        Therefore, T is a Geraghty-contraction.

        Notice that the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq29_HTML.gif satisfies the P-property.

        Indeed, if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equy_HTML.gif
        then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq90_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq91_HTML.gif and consequently,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equz_HTML.gif

        By Theorem 3.1, T has a unique best proximity point.

        Obviously, this point is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq92_HTML.gif .

        The condition A and B are nonempty closed subsets of the metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif is not a necessary condition for the existence of a unique best proximity point for a Geraghty-contraction http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq2_HTML.gif as it is proved with the following example.

        Example 4.2 Consider http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq74_HTML.gif with the usual metric and the subsets of X given by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equaa_HTML.gif

        Obviously, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq75_HTML.gif and B is not a closed subset of X.

        Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq77_HTML.gif .

        We consider the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq2_HTML.gif defined as
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equab_HTML.gif

        Now, we check that T is a Geraghty-contraction.

        In fact, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq78_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq79_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equ7_HTML.gif
        (7)
        In what follows, we need to prove that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equ8_HTML.gif
        (8)

        In fact, suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq80_HTML.gif (the same argument works for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq81_HTML.gif ).

        Put http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq95_HTML.gif (notice that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq96_HTML.gif since the function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq97_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq84_HTML.gif is strictly increasing).

        Taking into account that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equac_HTML.gif
        and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq98_HTML.gif , we have that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq99_HTML.gif , and consequently, from the last inequality it follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equad_HTML.gif
        Applying ϕ (notice that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq97_HTML.gif ) to the last inequality and taking into account the increasing character of ϕ, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equae_HTML.gif
        or equivalently,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equaf_HTML.gif

        and this proves (8).

        By (7) and (8), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equ9_HTML.gif
        (9)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq100_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq87_HTML.gif . Obviously, the inequality (9) is satisfied for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq101_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq88_HTML.gif .

        Now, we prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq9_HTML.gif .

        In fact, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq102_HTML.gif , then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq103_HTML.gif is a bounded sequence since in the contrary case, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq104_HTML.gif and thus http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq105_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq106_HTML.gif . This means that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq107_HTML.gif such that, for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq55_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq108_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq109_HTML.gif . The bounded character of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq103_HTML.gif gives us the existence of a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq110_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq111_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq110_HTML.gif convergent. Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq112_HTML.gif . From http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq113_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq114_HTML.gif and, as the unique solution of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq115_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq116_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq117_HTML.gif .

        Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq118_HTML.gif and this contradicts the fact that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq119_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq55_HTML.gif .

        Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq120_HTML.gif and this proves that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq9_HTML.gif .

        A similar argument to the one used in Example 4.1 proves that the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq29_HTML.gif has the P-property.

        On the other hand, the point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq92_HTML.gif is a best proximity point for T since
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equag_HTML.gif

        Moreover, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq121_HTML.gif is the unique best proximity point for T.

        Indeed, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq122_HTML.gif is a best proximity point for T, then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equah_HTML.gif
        and this gives us
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_Equai_HTML.gif

        Taking into account that the unique solution of this equation is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq123_HTML.gif , we have proved that T has a unique best proximity point which is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq121_HTML.gif .

        Notice that in this case B is not closed.

        Since for any nonempty subset A of X, the pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq33_HTML.gif satisfies the P-property, we have the following corollary.

        Corollary 4.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq7_HTML.gif be a complete metric space and A be a nonempty closed subset of X. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq124_HTML.gif be a Geraghty-contraction. Then T has a unique fixed point.

        Proof Using Theorem 3.1 when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq125_HTML.gif , the desired result follows. □

        Notice that when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_332_IEq126_HTML.gif , Corollary 4.1 is Theorem 1.1 due to Gerahty [1].

        Declarations

        Acknowledgements

        This research was partially supported by ‘Universidad de Las Palmas de Gran Canaria’, Project ULPGC 2010-006.

        Authors’ Affiliations

        (1)
        Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria

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        Copyright

        © Caballero 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.