The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

  • Zhiqun Xue1Email author,

    Affiliated with

    • Guiwen Lv1 and

      Affiliated with

      • BE Rhoades2

        Affiliated with

        Fixed Point Theory and Applications20122012:206

        DOI: 10.1186/1687-1812-2012-206

        Received: 11 May 2012

        Accepted: 29 October 2012

        Published: 22 November 2012

        Abstract

        Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq1_HTML.gif be a generalized Lipschitz Φ-hemi-contractive mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq2_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq6_HTML.gif be four real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq7_HTML.gif and satisfy the conditions (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq8_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq10_HTML.gif ; (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq11_HTML.gif . For some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq12_HTML.gif , let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq14_HTML.gif be any bounded sequences in D, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φ-quasi-accretive mapping.

        MSC: 47H10.

        Keywords

        generalized Lipschitz mapping Φ-hemi-contractive mapping Ishikawa iterative sequence with errors uniformly smooth real Banach space

        1 Introduction and preliminary

        Let E be a real Banach space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq16_HTML.gif be its dual space. The normalized duality mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq17_HTML.gif is defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equa_HTML.gif

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq18_HTML.gif denotes the generalized duality pairing. It is well known that

        1. (i)

          If E is a smooth Banach space, then the mapping J is single-valued;

           
        2. (ii)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq19_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq21_HTML.gif ;

           
        3. (iii)

          If E is a uniformly smooth Banach space, then the mapping J is uniformly continuous on any bounded subset of E. We denote the single-valued normalized duality mapping by j.

           

        Definition 1.1 ([1])

        Let D be a nonempty closed convex subset of E, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq1_HTML.gif be a mapping.

        (1) T is called strongly pseudocontractive if there is a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq22_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq23_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equb_HTML.gif
        (2) T is called ϕ-strongly pseudocontractive if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq23_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq24_HTML.gif and a strictly increasing continuous function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq25_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq26_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equc_HTML.gif
        (3) T is called Φ-pseudocontractive if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq23_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq24_HTML.gif and a strictly increasing continuous function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq27_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq28_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equd_HTML.gif

        It is obvious that Φ-pseudocontractive mappings not only include ϕ-strongly pseudocontractive mappings, but also strongly pseudocontractive mappings.

        Definition 1.2 ([1])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq1_HTML.gif be a mapping and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq29_HTML.gif .

        (1) T is called ϕ-strongly-hemi-pseudocontractive if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq31_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq32_HTML.gif and a strictly increasing continuous function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq25_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq26_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Eque_HTML.gif
        (2) T is called Φ-hemi-pseudocontractive if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq31_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq32_HTML.gif and the strictly increasing continuous function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq27_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq28_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equf_HTML.gif

        Closely related to the class of pseudocontractive-type mappings are those of accretive type.

        Definition 1.3 ([1])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq33_HTML.gif . The mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq34_HTML.gif is called strongly quasi-accretive if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq35_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq36_HTML.gif and a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq37_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq38_HTML.gif ; T is called ϕ-strongly quasi-accretive if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq35_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq36_HTML.gif and a strictly increasing continuous function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq25_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq26_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq39_HTML.gif ; T is called Φ-quasi-accretive if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq35_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq36_HTML.gif and a strictly increasing continuous function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq27_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq28_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq40_HTML.gif .

        Definition 1.4 ([2])

        For arbitrary given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq12_HTML.gif , the Ishikawa iterative process with errors http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq41_HTML.gif is defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ1_HTML.gif
        (1.1)
        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq14_HTML.gif are any bounded sequences in D; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq6_HTML.gif are four real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq7_HTML.gif and satisfy http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq43_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq44_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq45_HTML.gif , then the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ2_HTML.gif
        (1.2)

        is called the Mann iterative process with errors.

        Definition 1.5 ([3, 4])

        A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq1_HTML.gif is called generalized Lipschitz if there exists a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq46_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq48_HTML.gif .

        The aim of this paper is to prove the convergent results of the above Ishikawa and Mann iterations with errors for generalized Lipschitz Φ-hemi-contractive mappings in uniformly smooth real Banach spaces. For this, we need the following lemmas.

        Lemma 1.6 ([5])

        Let E be a uniformly smooth real Banach space, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq49_HTML.gif be a normalized duality mapping. Then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ3_HTML.gif
        (1.3)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq50_HTML.gif .

        Lemma 1.7 ([6])

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq51_HTML.gif be a nonnegative sequence which satisfies the following inequality:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ4_HTML.gif
        (1.4)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq52_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq53_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq54_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq55_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif .

        2 Main results

        Theorem 2.1 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq56_HTML.gif be a generalized Lipschitz Φ-hemi-contractive mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq2_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq6_HTML.gif be four real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq7_HTML.gif and satisfy the conditions (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq57_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq10_HTML.gif ; (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq11_HTML.gif . For some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq12_HTML.gif , let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq14_HTML.gif be any bounded sequences in D, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the unique fixed point q of T.

        Proof Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq1_HTML.gif is a generalized Lipschitz Φ-hemi-contractive mapping, there exists a strictly increasing continuous function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq27_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq28_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equg_HTML.gif
        i.e.,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equh_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equi_HTML.gif

        for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq31_HTML.gif .

        Step 1. There exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq59_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq60_HTML.gif (range of Φ). Indeed, if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq61_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq62_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq63_HTML.gif ; if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq64_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq65_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq66_HTML.gif , there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq67_HTML.gif in D such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq68_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq69_HTML.gif . Furthermore, we obtain that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq70_HTML.gif is bounded. Hence, there exists a natural number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq71_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq72_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq73_HTML.gif , then we redefine http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq75_HTML.gif .

        Step 2. For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif is bounded. Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq76_HTML.gif , then from Definition 1.2(2), we obtain that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq77_HTML.gif . Denote http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq78_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq79_HTML.gif . Since T is generalized Lipschitz, so T is bounded. We may define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq80_HTML.gif . Next, we want to prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq81_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq82_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq83_HTML.gif . Now, assume that it holds for some n, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq81_HTML.gif . We prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq84_HTML.gif . Suppose it is not the case, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq85_HTML.gif . Since J is uniformly continuous on a bounded subset of E, then for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq86_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq87_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq88_HTML.gif when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq89_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq90_HTML.gif . Now, denote
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equj_HTML.gif
        Owing to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq91_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif , without loss of generality, assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq92_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq44_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq10_HTML.gif , denote http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq93_HTML.gif . So, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ5_HTML.gif
        (2.1)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ6_HTML.gif
        (2.2)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ7_HTML.gif
        (2.3)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ8_HTML.gif
        (2.4)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ9_HTML.gif
        (2.5)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ10_HTML.gif
        (2.6)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ11_HTML.gif
        (2.7)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ12_HTML.gif
        (2.8)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ13_HTML.gif
        (2.9)
        Therefore,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equk_HTML.gif
        Using Lemma 1.6 and the above formulas, we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ14_HTML.gif
        (2.10)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ15_HTML.gif
        (2.11)
        Substitute (2.11) into (2.10)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ16_HTML.gif
        (2.12)

        this is a contradiction. Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq84_HTML.gif , i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq94_HTML.gif is a bounded sequence. So, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq95_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq96_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq97_HTML.gif are all bounded sequences.

        Step 3. We want to prove http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq98_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif . Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq99_HTML.gif .

        By (2.10), (2.11), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ17_HTML.gif
        (2.13)
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ18_HTML.gif
        (2.14)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq100_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq102_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif .

        Taking (2.14) into (2.13),
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ19_HTML.gif
        (2.15)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq103_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif .

        Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq104_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq105_HTML.gif . If it is not the case, we assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq106_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq107_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq108_HTML.gif , i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq109_HTML.gif . Thus, from (2.15) it follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ20_HTML.gif
        (2.16)
        This implies that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ21_HTML.gif
        (2.17)
        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq110_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq111_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq112_HTML.gif . Then we get that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equl_HTML.gif
        Applying Lemma 1.7, we get that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq55_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif . This is a contradiction and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq105_HTML.gif . Therefore, there exists an infinite subsequence such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq113_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq114_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq115_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq116_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq114_HTML.gif . In view of the strictly increasing continuity of Φ, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq117_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq114_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq118_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq114_HTML.gif . Next, we want to prove http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq98_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq119_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq120_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq121_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq123_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq124_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq125_HTML.gif . First, we want to prove http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq126_HTML.gif . Suppose it is not the case, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq127_HTML.gif . Using (1.1), we may get the following estimates:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ22_HTML.gif
        (2.18)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ23_HTML.gif
        (2.19)
        Since Φ is strictly increasing, then (2.19) leads to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq128_HTML.gif . From (2.15), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ24_HTML.gif
        (2.20)

        which is a contradiction. Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq126_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq129_HTML.gif holds. Repeating the above course, we can easily prove that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq130_HTML.gif holds. Therefore, for any m, we obtain that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq129_HTML.gif , which means http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq98_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif . This completes the proof. □

        Theorem 2.2 Let E be an arbitrary uniformly smooth real Banach space, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq34_HTML.gif be a generalized Lipschitz Φ-quasi-accretive mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq131_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq6_HTML.gif be four real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq7_HTML.gif and satisfy the conditions (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq57_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq10_HTML.gif ; (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq11_HTML.gif . For some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq12_HTML.gif , let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq14_HTML.gif be any bounded sequences in E, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif be an Ishikawa iterative sequence with errors defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ25_HTML.gif
        (2.21)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq132_HTML.gif is defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq133_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq20_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif converges strongly to the unique solution of the equation http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq134_HTML.gif (or the unique fixed point of S).

        Proof Since T is a generalized Lipschitz and Φ-quasi-accretive mapping, it follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equm_HTML.gif
        i.e.,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equn_HTML.gif
        i.e.,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equo_HTML.gif

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq135_HTML.gif . The rest of the proof is the same as that of Theorem 2.1. □

        Corollary 2.3 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq56_HTML.gif be a generalized Lipschitz Φ-hemi-contractive mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq2_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq5_HTML.gif be two real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq7_HTML.gif and satisfy the conditions (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq136_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq10_HTML.gif ; (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq11_HTML.gif . For some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq12_HTML.gif , let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq13_HTML.gif be any bounded sequence in D, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif be the Mann iterative sequence with errors defined by (1.2). Then (1.2) converges strongly to the unique fixed point q of T.

        Corollary 2.4 Let E be an arbitrary uniformly smooth real Banach space, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq34_HTML.gif be a generalized Lipschitz Φ-quasi-accretive mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq131_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq6_HTML.gif be two real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq7_HTML.gif and satisfy the conditions (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq136_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq10_HTML.gif ; (ii)  http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq11_HTML.gif . For some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq12_HTML.gif , let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq13_HTML.gif be any bounded sequence in E, and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif be the Mann iterative sequence with errors defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ26_HTML.gif
        (2.22)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq132_HTML.gif is defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq133_HTML.gif for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq20_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif converges strongly to the unique solution of the equation http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq134_HTML.gif (or the unique fixed point of S).

        Remark 2.5 It is mentioned that in 2006, Chidume and Chidume [1] proved the approximative theorem for zeros of generalized Lipschitz generalized Φ-quasi-accretive operators. This result provided significant improvements of some recent important results. Their result is as follows.

        Theorem CC ([[1], Theorem 3.1])

        Let E be a uniformly smooth real Banach space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq137_HTML.gif be a mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq138_HTML.gif . Suppose A is a generalized Lipschitz Φ-quasi-accretive mapping. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq5_HTML.gif be real sequences in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq7_HTML.gif satisfying the following conditions: (i) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq139_HTML.gif ; (ii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq140_HTML.gif ; (iii) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq141_HTML.gif ; (iv) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq142_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif be generated iteratively from arbitrary http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq143_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_Equ27_HTML.gif
        (2.23)

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq132_HTML.gif is defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq144_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq13_HTML.gif is an arbitrary bounded sequence in E. Then, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq146_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq148_HTML.gif , the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq15_HTML.gif converges strongly to the unique solution of the equation http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq149_HTML.gif .

        However, there exists a gap in the proof process of above Theorem CC. Here, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq150_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq151_HTML.gif ) does not hold in line 14 of Claim 2 on page 248, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq152_HTML.gif is a wrong case. For instance, set the iteration parameters: http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq153_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq155_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq156_HTML.gif ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq157_HTML.gif  . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq158_HTML.gif , but http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_322_IEq159_HTML.gif . Therefore, the proof of above Theorem CC is not reasonable. Up to now, we do not know the validity of Theorem CC. This will be an open question left for the readers!

        Declarations

        Authors’ Affiliations

        (1)
        Department of Mathematics and Physics, Shijiazhuang Tiedao University
        (2)
        Department of Mathematics, Indiana University

        References

        1. Chidume CE, Chidume CO: Convergence theorem for zeros of generalized Lipschitz generalized Φ-quasi-accretive operators. Proc. Am. Math. Soc. 2006,134(1):243–251.MathSciNetMATHView Article
        2. Xu YG: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 1998, 224:91–101.MathSciNetMATHView Article
        3. Zhou HY, Chen DQ: Iterative approximations of fixed points for nonlinear mappings of ϕ -hemicontractive type in normed linear spaces. Math. Appl. 1998,11(3):118–121.MathSciNetMATH
        4. Xue ZQ, Zhou HY, Cho YJ: Iterative solutions of nonlinear equations for m -accretive operators in Banach space. J. Nonlinear Convex Anal. 2003,1(3):313–320.MathSciNet
        5. Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1980.
        6. Weng X: Fixed point iteration for local strictly pseudocontractive mapping. Proc. Am. Math. Soc. 1991, 113:727–731.MATHView Article

        Copyright

        © Xue 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.