Cyclic generalized contractions and fixed point results with applications to an integral equation

  • Hemant Kumar Nashine1,

    Affiliated with

    • Wutiphol Sintunavarat2 and

      Affiliated with

      • Poom Kumam2Email author

        Affiliated with

        Fixed Point Theory and Applications20122012:217

        DOI: 10.1186/1687-1812-2012-217

        Received: 13 June 2012

        Accepted: 12 November 2012

        Published: 28 November 2012

        Abstract

        We set up a new variant of cyclic generalized contractive mappings for a map in a metric space and present existence and uniqueness results of fixed points for such mappings. Our results generalize or improve many existing fixed point theorems in the literature. To illustrate our results, we give some examples. At the same time as applications of the presented theorems, we prove an existence theorem for solutions of a class of nonlinear integral equations.

        MSC: 47H10, 54H25.

        Keywords

        fixed point cyclic generalized http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq1_HTML.gif -contraction integral equation

        1 Introduction and preliminaries

        All the way through this paper, by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq2_HTML.gif , we designate the set of all real nonnegative numbers, while ℕ is the set of all natural numbers.

        The celebrated Banach’s [1] contraction mapping principle is one of the cornerstones in the development of nonlinear analysis. This principle has been extended and improved in many ways over the years (see, e.g., [25]). Fixed point theorems have applications not only in various branches of mathematics but also in economics, chemistry, biology, computer science, engineering, and other fields. In particular, such theorems are used to demonstrate the existence and uniqueness of a solution of differential equations, integral equations, functional equations, partial differential equations, and others. Owing to the magnitude, generalizations of the Banach fixed point theorem have been explored heavily by many authors. This celebrated theorem can be stated as follows.

        Theorem 1.1 ([1])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq3_HTML.gif be a complete metric space and T be a mapping of X into itself satisfying
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ1_HTML.gif
        (1)

        where k is a constant in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq4_HTML.gif . Then T has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq5_HTML.gif .

        Inequality (1) implies the continuity of T. A natural question is whether we can find contractive conditions which will imply the existence of a fixed point in a complete metric space but will not imply continuity.

        On the other hand, cyclic representations and cyclic contractions were introduced by Kirk et al.[6]. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq6_HTML.gif is called cyclic if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq8_HTML.gif , where A, B are nonempty subsets of a metric space http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq3_HTML.gif . Moreover, T is called a cyclic contraction if there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq9_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq10_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq12_HTML.gif . Notice that although a contraction is continuous, a cyclic contraction need not to be. This is one of the important gains of this theorem.

        Definition 1.1 (See [6, 7])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq3_HTML.gif be a metric space. Let p be a positive integer, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq13_HTML.gif be nonempty subsets of X, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq14_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq15_HTML.gif . Then Y is said to be a cyclic representation of Y with respect to T if

        (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq16_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq17_HTML.gif are nonempty closed sets, and

        (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq18_HTML.gif .

        Following the paper in [6], a number of fixed point theorems on a cyclic representation of Y with respect to a self-mapping T have appeared (see, e.g., [3, 715]).

        In this paper, we introduce a new class of cyclic generalized http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq1_HTML.gif -contractive mappings, and then investigate the existence and uniqueness of fixed points for such mappings. Our main result generalizes and improves many existing theorems in the literature. We supply appropriate examples to make obvious the validity of the propositions of our results. To end with, as applications of the presented theorems, we achieve fixed point results for a generalized contraction of integral type and we prove an existence theorem for solutions of a system of integral equations.

        2 Main results

        In this section, we introduce two new notions of a cyclic contraction and establish new results for such mappings.

        In the sequel, we fixed the set of functions by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq19_HTML.gif such that

        (i) ℱ is nondecreasing, continuous, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq20_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq21_HTML.gif ;

        (ii) ψ is nondecreasing, right continuous, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq22_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq23_HTML.gif .

        Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq25_HTML.gif .

        We state the notion of a cyclic generalized http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq26_HTML.gif -contraction as follows.

        Definition 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq3_HTML.gif be a metric space. Let p be a positive integer, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq13_HTML.gif be nonempty subsets of X and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq14_HTML.gif . An operator http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq27_HTML.gif is said to be a cyclic generalized http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq1_HTML.gif -contraction for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq30_HTML.gif if

        (a) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

        (b) for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq17_HTML.gif (with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq33_HTML.gif ),
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equa_HTML.gif
        where
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equb_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equc_HTML.gif

        Our first main result is the following.

        Theorem 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq34_HTML.gif be a complete metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq36_HTML.gif be nonempty closed subsets of X, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq37_HTML.gif . Suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq38_HTML.gif is a cyclic generalized http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq1_HTML.gif -contraction mapping for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif . Then T has a unique fixed point. Moreover, the fixed point of T belongs to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq40_HTML.gif .

        Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq41_HTML.gif (such a point exists since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq42_HTML.gif ). Define the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq43_HTML.gif in X by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equd_HTML.gif
        We shall prove that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ2_HTML.gif
        (2)
        If, for some k, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq44_HTML.gif , then (2) follows immediately. So, we can suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq45_HTML.gif for all n. From the condition (a), we observe that for all n, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq46_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq47_HTML.gif . Then, from the condition (b), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ3_HTML.gif
        (3)
        On the other hand, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Eque_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equf_HTML.gif
        Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq48_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq49_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq50_HTML.gif , so
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equg_HTML.gif
        a contradiction. Hence,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equh_HTML.gif
        and thus
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ4_HTML.gif
        (4)
        Similarly, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ5_HTML.gif
        (5)
        Thus, from (4) and (5), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equi_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq51_HTML.gif . Now, from
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equj_HTML.gif

        and the property of ψ, we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq52_HTML.gif , and consequently (2) holds.

        Now, we shall prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq43_HTML.gif is a Cauchy sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq3_HTML.gif . Suppose, on the contrary, that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq43_HTML.gif is not a Cauchy sequence. Then there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq53_HTML.gif for which we can find two sequences of positive integers http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq55_HTML.gif such that for all positive integers k,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ6_HTML.gif
        (6)
        Further, corresponding to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq56_HTML.gif , we can choose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq57_HTML.gif in such a way that it is the smallest integer with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq58_HTML.gif satisfying (6). Then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ7_HTML.gif
        (7)
        Using (6), (7), and the triangular inequality, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equk_HTML.gif
        Thus, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equl_HTML.gif
        Passing to the limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq59_HTML.gif in the above inequality and using (2), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ8_HTML.gif
        (8)
        On the other hand, for all k, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq60_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq61_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq62_HTML.gif (for k large enough, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq63_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq64_HTML.gif lie in different adjacently labeled sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq66_HTML.gif for certain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq67_HTML.gif . Using (b), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ9_HTML.gif
        (9)
        for all k. Now, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ10_HTML.gif
        (10)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ11_HTML.gif
        (11)
        for all k. Using the triangular inequality, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equm_HTML.gif
        which implies from (8) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ12_HTML.gif
        (12)
        Using (2), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ13_HTML.gif
        (13)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ14_HTML.gif
        (14)
        Again, using the triangular inequality, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equn_HTML.gif
        Passing to the limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq68_HTML.gif in the above inequality, using (14) and (12), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ15_HTML.gif
        (15)
        Similarly, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equo_HTML.gif
        Passing to the limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq68_HTML.gif , using (2) and (12), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ16_HTML.gif
        (16)
        Similarly, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ17_HTML.gif
        (17)
        Now, it follows from (12)-(16) and the continuity of φ that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ18_HTML.gif
        (18)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ19_HTML.gif
        (19)
        Passing to the limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq68_HTML.gif in (9), using (17), (18), (19), and the condition (ii), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equp_HTML.gif

        which is a contradiction. Thus, we proved that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq43_HTML.gif is a Cauchy sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq3_HTML.gif .

        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq3_HTML.gif is complete, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq5_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ20_HTML.gif
        (20)
        We shall prove that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ21_HTML.gif
        (21)

        From the condition (a), and since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq41_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq69_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq70_HTML.gif is closed, from (20), we get that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq71_HTML.gif . Again, from the condition (a), we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq72_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq73_HTML.gif is closed, from (20), we get that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq74_HTML.gif . Continuing this process, we obtain (21).

        Now, we shall prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq75_HTML.gif is a fixed point of T. Indeed, from (21), since for all n there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq76_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq77_HTML.gif , applying (b) with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq79_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ22_HTML.gif
        (22)
        for all n. On the other hand, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equq_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equr_HTML.gif
        Passing to the limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq80_HTML.gif in the above inequality and using (20), we obtain that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ23_HTML.gif
        (23)
        Passing to the limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq80_HTML.gif in (22), using (23) and (20), we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equs_HTML.gif
        Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq81_HTML.gif . In this case, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equt_HTML.gif
        which implies that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equu_HTML.gif

        a contradiction. Then we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq82_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq75_HTML.gif is a fixed point of T.

        Finally, we prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq75_HTML.gif is the unique fixed point of T. Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq83_HTML.gif is another fixed point of T, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq84_HTML.gif . From the condition (a), this implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq85_HTML.gif . Then we can apply (b) for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq86_HTML.gif . We obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equv_HTML.gif
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq75_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq83_HTML.gif are fixed points of T, we can show easily that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq88_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq89_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equw_HTML.gif

        a contradiction. Then we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq90_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq91_HTML.gif . Thus, we proved the uniqueness of the fixed point. □

        In the following, we deduce some fixed point theorems from our main result given by Theorem 2.1.

        If we take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq93_HTML.gif in Theorem 2.1, then we get immediately the following fixed point theorem.

        Corollary 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq3_HTML.gif be a complete metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq94_HTML.gif satisfy the following condition: there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq30_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equx_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq95_HTML.gif . Then T has a unique fixed point.

        Remark 2.1 Corollary 2.1 extends and generalizes many existing fixed point theorems in the literature [1, 1621].

        Corollary 2.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq34_HTML.gif be a complete metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq36_HTML.gif be nonempty closed subsets of X, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq37_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq15_HTML.gif . Suppose that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif such that

        (a′) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

        (b′) for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq17_HTML.gif (with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq33_HTML.gif ),
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equy_HTML.gif

        Then T has a unique fixed point. Moreover, the fixed point of T belongs to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq40_HTML.gif .

        Remark 2.2 Corollary 2.2 is similar to Theorem 2.1 in [7].

        Remark 2.3 Taking in Corollary 2.2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq96_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq9_HTML.gif , we obtain a generalized version of Theorem 1.3 in [6].

        Corollary 2.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq34_HTML.gif be a complete metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq36_HTML.gif be nonempty closed subsets of X, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq37_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq15_HTML.gif . Suppose that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif such that

        (a′) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

        (b′) for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq17_HTML.gif (with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq33_HTML.gif ),
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equz_HTML.gif

        Then T has a unique fixed point. Moreover, the fixed point of T belongs to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq40_HTML.gif .

        Remark 2.4 Taking in Corollary 2.3 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq96_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq9_HTML.gif , we obtain a generalized version of Theorem 3 in [13].

        Corollary 2.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq34_HTML.gif be a complete metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq36_HTML.gif be nonempty closed subsets of X, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq37_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq15_HTML.gif . Suppose that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif such that

        (a′) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

        (b′) for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq17_HTML.gif (with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq33_HTML.gif ),
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equaa_HTML.gif

        Then T has a unique fixed point. Moreover, the fixed point of T belongs to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq40_HTML.gif .

        Remark 2.5 Taking in Corollary 2.4 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq96_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq9_HTML.gif , we obtain a generalized version of Theorem 5 in [13].

        Corollary 2.5 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq34_HTML.gif be a complete metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq36_HTML.gif be nonempty closed subsets of X, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq37_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq15_HTML.gif . Suppose that there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif such that

        (a) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq31_HTML.gif is a cyclic representation of Y with respect to T;

        (b) for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq17_HTML.gif (with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq33_HTML.gif ),
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equab_HTML.gif

        Then T has a unique fixed point. Moreover, the fixed point of T belongs to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq40_HTML.gif .

        We provide some examples to illustrate our obtained Theorem 2.1.

        Example 2.1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq97_HTML.gif with the usual metric. Suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq98_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq100_HTML.gif . Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq101_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq102_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq103_HTML.gif . It is clear that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq104_HTML.gif is a cyclic representation of Y with respect to T. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq28_HTML.gif be defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq105_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif of the form http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq106_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq107_HTML.gif . For all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq30_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equac_HTML.gif

        So, T is a cyclic generalized http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq1_HTML.gif -contraction for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq30_HTML.gif . Therefore, all conditions of Theorem 2.1 are satisfied ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq109_HTML.gif ), and so T has a unique fixed point (which is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq110_HTML.gif ).

        Example 2.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq97_HTML.gif with the usual metric. Suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq111_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq100_HTML.gif . Define the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq113_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equad_HTML.gif

        Clearly, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq115_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq73_HTML.gif are nonempty closed subsets of X. Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq104_HTML.gif is a cyclic representation of Y with respect to T.

        Now, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq116_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq118_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equae_HTML.gif
        On the other hand, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equaf_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equag_HTML.gif
        Then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equah_HTML.gif
        Define the function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq119_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq120_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif of the form http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq106_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq30_HTML.gif . Then we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ24_HTML.gif
        (24)

        Moreover, we can show that (24) holds if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq121_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq122_HTML.gif . Similarly, we also get (24) holds for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq123_HTML.gif .

        Now, all the conditions of Theorem 2.1 are satisfied (with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq124_HTML.gif ), we deduce that T has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq125_HTML.gif .

        3 An application to an integral equation

        In this section, we apply the result given by Theorem 2.1 to study the existence and uniqueness of solutions to a class of nonlinear integral equations.

        We consider the nonlinear integral equation
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ25_HTML.gif
        (25)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq127_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq128_HTML.gif are continuous functions.

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq129_HTML.gif be the set of real continuous functions on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq130_HTML.gif . We endow X with the standard metric
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equai_HTML.gif

        It is well known that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq131_HTML.gif is a complete metric space.

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq132_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq133_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ26_HTML.gif
        (26)
        We suppose that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq134_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ27_HTML.gif
        (27)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ28_HTML.gif
        (28)
        We suppose that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq136_HTML.gif is a decreasing function, that is,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ29_HTML.gif
        (29)
        We suppose that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ30_HTML.gif
        (30)
        Finally, we suppose that, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq135_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq137_HTML.gif with ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq139_HTML.gif ) or ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq140_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq141_HTML.gif ),
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ31_HTML.gif
        (31)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq119_HTML.gif is a nondecreasing function that belongs to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq30_HTML.gif .

        Now, define the set
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equaj_HTML.gif

        We have the following result.

        Theorem 3.1 Under the assumptions (26)-(31), problem (25) has one and only one solution http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq143_HTML.gif .

        Proof Define the closed subsets of X, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq73_HTML.gif , by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equak_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equal_HTML.gif
        Define the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq144_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equam_HTML.gif
        We shall prove that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equ32_HTML.gif
        (32)
        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq145_HTML.gif , that is,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equan_HTML.gif
        Using condition (29), since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq146_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq147_HTML.gif , we obtain that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equao_HTML.gif
        The above inequality with condition (27) implies that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equap_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq134_HTML.gif . Then we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq148_HTML.gif .

        Similarly, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq149_HTML.gif , that is,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equaq_HTML.gif
        Using condition (29), since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq146_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq147_HTML.gif , we obtain that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equar_HTML.gif
        The above inequality with condition (28) implies that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equas_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq134_HTML.gif . Then we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq150_HTML.gif . Finally, we deduce that (32) holds.

        Now, let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq151_HTML.gif , that is, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq134_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equat_HTML.gif
        This implies, from condition (26), that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq134_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equau_HTML.gif
        Now, using conditions (30) and (31), we can write that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq134_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equav_HTML.gif
        This implies that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_Equaw_HTML.gif

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq29_HTML.gif of the form http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq152_HTML.gif . Using the same technique, we can show that the above inequality holds also if we take http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq153_HTML.gif .

        Now, all the conditions of Theorem 2.1 are satisfied (with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq124_HTML.gif ), we deduce that T has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq154_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-217/MediaObjects/13663_2012_325_IEq143_HTML.gif is the unique solution to (25). □

        Declarations

        Acknowledgements

        The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). Moreover, the third author was supported by the Commission on Higher Education (CHE), the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) (Grant No. MRG5580213).

        Authors’ Affiliations

        (1)
        Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg
        (2)
        Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)

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        Copyright

        © Nashine 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.