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ISRN Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 963802, 15 pages
http://dx.doi.org/10.5402/2012/963802
Research Article

Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type with Lipschitz and Bounded Nonlinear Operators

Section de MathΓ©matiques AppliquΓ©es, UniversitΓ© Gaston Berger, BP 234 Saint Louis, Senegal

Received 20 March 2012; Accepted 10 May 2012

Academic Editors: M.Β Idemen and J.Β Kou

Copyright Β© 2012 N. Djitte and M. Sene. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let E be a reflexive real Banach space with uniformly GΓ’teaux differentiable norm and F, K : 𝐸→𝐸 be Lipschitz accretive maps with 𝐷(𝐾)=𝑅(𝐹)=𝐸. Suppose that the Hammerstein equation 𝑒+𝐾𝐹𝑒=0 has a solution. An explicit iteration method is shown to converge strongly to a solution of the equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorems are significant improvements on important recent results (e.g., (Chiume and Djitte, 2012)).

1. Introduction

Let 𝐸 be a real normed space and let π‘†βˆΆ={π‘₯βˆˆπΈβˆΆβ€–π‘₯β€–=1}. 𝐸 is said to have a GΓ’teaux differentiable norm if the limit lim𝑑→0+β€–π‘₯+π‘‘π‘¦β€–βˆ’β€–π‘₯‖𝑑(1.1) exists for each π‘₯,π‘¦βˆˆπ‘†. The normed space 𝐸 is said to have a uniformly GΓ’teaux differentiable norm if for each π‘¦βˆˆπ‘† the limit is attained uniformly for π‘₯βˆˆπ‘†. Furthermore, 𝐸 is said to be uniformly smooth if the limit exists uniformly for (π‘₯,𝑦)βˆˆπ‘†Γ—π‘†.

Let 𝐸 be a normed linear space with dimension greater than or equal to 2. The modulus of smoothness of 𝐸 is the function 𝜌𝐸∢[0,∞)β†’[0,∞) defined by πœŒπΈξ‚»(𝑑)∢=supβ€–π‘₯+𝑦‖+β€–π‘₯βˆ’π‘¦β€–2ξ‚Όβˆ’1βˆΆβ€–π‘₯β€–=1,‖𝑦‖=𝑑.(1.2) In terms of the modulus of smoothness, the space 𝐸 is called uniformly smooth if and only if lim𝑑→0+(𝜌𝐸(𝑑)/𝑑)=0.𝐸 is called π‘ž-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌𝐸(𝑑)β‰€π‘π‘‘π‘ž,𝑑>0. 𝐿𝑝 (and ℓ𝑝) spaces, 1<𝑝<+∞ are π‘ž-uniformly smooth. In particular, 𝐿𝑝𝑖𝑠2-uniformly smooth if 2≀𝑝<+∞ and 𝑝-uniformly smooth if 1<𝑝<2. It is easy to see that for 1<π‘ž<+∞, every π‘ž-uniformly smooth real Banach space is uniformly smooth and thus has a uniformly GΓ’teaux differentiable norm.

Let 𝐸 be a real normed linear space with dual πΈβˆ—. For π‘ž>1, we denote by π½π‘ž the generalized duality mapping from 𝐸 to 2πΈβˆ— defined by π½π‘žξ€½π‘“(π‘₯)∢=βˆ—βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βˆ—βŸ©=β€–π‘₯β€–β‹…β€–π‘“βˆ—β€–,β€–π‘“βˆ—β€–=β€–π‘₯β€–π‘žβˆ’1ξ€Ύ,(1.3) where βŸ¨β‹…,β‹…βŸ© denotes the generalized duality pairing. 𝐽2 is denoted by 𝐽. If πΈβˆ— is strictly convex, then π½π‘ž is single-valued. We note that π½π‘ž(π‘₯)=β€–π‘₯β€–π‘žβˆ’2𝐽(π‘₯), for π‘₯β‰ 0. It is known that if 𝐸 is a real Banach space with uniform GΓ’teaux differentiable norm, then the duality map 𝐽 is norm-to-weak* uniformly continuous on bounded subsets of 𝐸.

A map 𝐺 with domain 𝐷(𝐺) in a normed linear space 𝐸 is said to be strongly accretive if there exists a constant π‘˜>0 such that for every π‘₯,π‘¦βˆˆπ·(𝐺), there exists π‘—π‘ž(π‘₯βˆ’π‘¦)βˆˆπ½π‘ž(π‘₯βˆ’π‘¦) such that 𝐺π‘₯βˆ’πΊπ‘¦,π‘—π‘žξ¬(π‘₯βˆ’π‘¦)β‰₯π‘˜β€–π‘₯βˆ’π‘¦β€–π‘ž.(1.4) If π‘˜=0, 𝐺 is said to be accretive. If 𝐸 is a Hilbert space, accretive operators are called monotone.

The accretive mappings were introduced independently in 1967 by Browder [1] and Kato [2]. Interest in such mappings stems mainly from their firm connection with equations of evolution. It is known (see, e.g., Zeidler [3]) that many physically significant problems can be modelled by initial-value problems of the form π‘’ξ…ž(𝑑)+𝐴𝑒(𝑑)=0,𝑒(0)=𝑒0,(1.5) where 𝐴 is an accretive operator in an appropriate Banach space. Typical examples where such evolution equations occur can be found in the heat, wave, or the SchrΓΆdinger equations. If in (1.5), 𝑒(𝑑) is independent of 𝑑, then (1.5) reduces to 𝐴𝑒=0,(1.6) whose solutions correspond to the equilibrium points of the system (1.5). Consequently, considerable research efforts have been devoted, especially within the past 40 years or so, to methods of finding approximate solutions (when they exist) of (1.6). An early fundamental result in the theory of accretive operators, due to Browder [1], states that the initial-value problem (1.5) is solvable if 𝐴 is locally Lipschitzian and accretive on 𝐸. Utilizing the existing result for (1.5), Browder [1] proved that if 𝐴 is locally Lipschitzian and accretive on 𝐸, then 𝐴 is π‘š-accretive that is, 𝑅(𝐼+𝐴)=𝐸. Clearly, a consequence of this is that the following equation 𝑒+𝐴𝑒=0(1.7) has a solution.

One important generalization of (1.7) is the so-called equation of Hammerstein type (see e.g., Hammerstein [4]), where a nonlinear integral equation of Hammerstein type is one of the form: ξ€œπ‘’(π‘₯)+Ξ©πœ…(π‘₯,𝑦)𝑓(𝑦,𝑒(𝑦))𝑑𝑦=β„Ž(π‘₯),(1.8) where 𝑑𝑦 is a 𝜎-finite measure on the measure space Ξ©; the real kernel πœ… is defined on Ω×Ω,  𝑓 is a real-valued function defined on Ω×ℝ and is, in general, nonlinear, and β„Ž is a given function on Ξ©. If we now define an operator 𝐾 by ξ€œπΎπ‘£(π‘₯)∢=Ξ©πœ…(π‘₯,𝑦)𝑣(𝑦)𝑑𝑦;π‘₯∈Ω,(1.9) and the so-called superposition or Nemytskii operator by 𝐹𝑒(𝑦)∢=𝑓(𝑦,𝑒(𝑦)) then, the integral equation (1.8) can be put in operator theoretic form as follows: 𝑒+𝐾𝐹𝑒=0,(1.10) where without loss of generality, we have taken β„Žβ‰‘0.

Interest in (1.10) stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Greens functions can, as a rule, be transformed into the form (1.10). Among these, we mention the problem of the forced oscillations of finite amplitude of a pendulum (see, e.g., Pascali and Sburlan [5], Chapter IV).

Example 1.1. We consider the problem of the pendulum. 𝑑2𝑣(𝑑)𝑑𝑑2+π‘Ž2[],sin𝑣(𝑑)=𝑧(𝑑),π‘‘βˆˆ0,1𝑣(0)=𝑣(1)=0,(1.11) where the driving force 𝑧 is odd. The constant π‘Ž, (π‘Žβ‰ 0) depends on the length of the pendulum and gravity. Since Green's function of the problem: π‘£ξ…žξ…ž(𝑑)=0,𝑣(0)=𝑣(1)=0(1.12) is given by: ξ‚»π‘˜(𝑑,𝑠)∢=𝑑(1βˆ’π‘ ),0≀𝑑≀𝑠≀1,𝑠(1βˆ’π‘‘),0≀𝑠≀𝑑≀1,(1.13) it follows that problem (1.11) is equivalent to the nonlinear integral equation ξ€œπ‘£(𝑑)=βˆ’10ξ€Ίπ‘˜(𝑑,𝑠)𝑧(𝑠)βˆ’π‘Ž2ξ€»[].sin𝑣(𝑠)𝑑𝑠,π‘‘βˆˆ0,1(1.14) If we set ξ€œπ‘”(𝑑)∢=10[],π‘˜(𝑑,𝑠)𝑧(𝑠)𝑑𝑠,𝑒(𝑑)∢=𝑣(𝑑)+𝑔(𝑑),π‘‘βˆˆ0,1(1.15) then 𝑣=π‘’βˆ’π‘”(1.16) and (1.14) can be written as ξ€œπ‘’(𝑑)+10π‘˜(𝑑,𝑠)π‘Ž2sin(𝑔(𝑠)βˆ’π‘’(𝑠))𝑑𝑠=0,(1.17) which is in the Hammerstein equation form: ξ€œπ‘’(𝑑)+10π‘˜(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠=0,(1.18) where 𝑓(𝑑,𝑠)=π‘Ž2sin(𝑔(𝑑)βˆ’π‘ ),𝑑,π‘ βˆˆ[0,1].
Equations of Hammerstein type play a crucial role in the theory of optimal control systems and in automation and network theory (see, e.g., Dolezal [6]).
Several existence and uniqueness theorems have been proved for equations of the Hammerstein type (see, e.g., BrΓ©zis and Browder [7–9], Browder [1], Browder et al. [10], Bowder and Gupta [11], Chepanovich [12], and De Figueiredo and Gupta [13]).
For the iterative approximation of solutions of (1.6) and (1.7), the monotonicity/accretivity of 𝐴 is crucial. The Mann iteration scheme (see, e.g., Mann [14]) has successfully been employed (see, e.g., the recent monographs of Berinde [15] and Chidume [16]). The recurrence formulas used involved πΎβˆ’1 which is also assumed to be strongly monotone, and this, apart from limiting the class of mappings to which such iterative schemes are applicable, is also not convenient in applications. Part of the difficulty is the fact that the composition of two monotone operators need not to be monotone. In the special case in which the operators are defined on subsets 𝐷 of 𝐸 which are compact, BrΓ©zis and Browder [7] proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.10) (see also BrΓ©zis and Browder [9]). In fact, they proved the following Theorem.

Theorem BB (BrΓ©zis and Browder). Let 𝐻 be a separable real Hilbert space and 𝐢 be a closed subspace of 𝐻. Let πΎβˆΆπ»β†’πΆ be a bounded continuous monotone operator and πΉβˆΆπΆβ†’π» be an angle-bounded and weakly compact mapping. For a given π‘“βˆˆπΆ, consider the Hammerstein equation (𝐼+𝐾𝐹)𝑒=𝑓,(1.19) and its 𝑛th Galerkin approximation given by 𝐼+𝐾𝑛𝐹𝑛𝑒𝑛=π‘ƒβˆ—π‘“,(1.20) where 𝐾𝑛=π‘ƒβˆ—π‘›πΎπ‘ƒπ‘›βˆΆπ»β†’πΆπ‘› and 𝐹𝑛=π‘ƒπ‘›πΉπ‘ƒβˆ—π‘›βˆΆπΆπ‘›β†’π», where the symbols have their usual meanings (see, [5]). Then, for each π‘›βˆˆβ„•, the Galerkin approximation (1.20) admits a unique solution 𝑒𝑛 in 𝐢𝑛 and {𝑒𝑛} converges strongly in 𝐻 to the unique solution π‘’βˆˆπΆ of (1.19).

Theorem BB is a special case of the actual theorem of BrΓ©zis and Browder in which the Banach space is a separable real Hilbert space. The main theorem of BrΓ©zis and Browder is proved in an arbitrary separable real Banach space.

We observe that the Galerkin method of BrΓ©zis and Browder is not iterative. The first attempt to construct an iterative method for the approximation of a solution of a Hammerstein equation, as far as we know, was made by Chidume and Zegeye [17] who constructed a sequence in the cartesian product 𝐸×𝐸 and proved the convergence of the sequence to a solution of the Hammerstein equation. In subsequent papers, [18, 19], these authors were able to construct explicit coupled algorithms in the original space 𝐸 which converge strongly to a solution of the equation. Following this, Chidume and Djitte studied this explicit coupled algorithms and proved several strong convergence theorems (see, [20–22]).

Recently, Chidume and Djitte [20] introduced and studied a coupled explicit iterative process (see Theorem CD below) to approximate solutions of nonlinear equations of Hammerstein-type in real Hilbert spaces when the operators 𝐾 and 𝐹 are bounded and maximal monotone. They proved the following theorem.

Theorem CD (Chidume and Djitte, [20]). Let 𝐻 be a real Hilbert space and 𝐹,πΎβˆΆπ»β†’π» be bounded and maximal monotone operators. Let {𝑒𝑛} and {𝑣𝑛} be sequences in 𝐻 defined iteratively from arbitrary points 𝑒1,𝑣1∈𝐻 as follows: 𝑒𝑛+1=π‘’π‘›βˆ’πœ†π‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1𝑣,𝑛β‰₯1,𝑛+1=π‘£π‘›βˆ’πœ†π‘›ξ€·πΎπ‘£π‘›+π‘’π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘£π‘›βˆ’π‘£1ξ€Έ,𝑛β‰₯1,(1.21) where {πœ†π‘›} and {πœƒπ‘›} are sequences in (0,1) satisfying the following conditions: (1)limπœƒπ‘›=0, (2)βˆ‘βˆžπ‘›=1πœ†π‘›πœƒπ‘›=∞,πœ†π‘›=π‘œ(πœƒπ‘›),(3)limπ‘›β†’βˆž(((πœƒπ‘›βˆ’1)/πœƒπ‘›)βˆ’1)/πœ†π‘›πœƒπ‘›=0. Suppose that 𝑒+𝐾𝐹𝑒=0 has a solution in 𝐻. Then, there exists a constant 𝑑0>0 such that if πœ†π‘›β‰€π‘‘0πœƒπ‘› for all 𝑛β‰₯𝑛0 for some 𝑛0β‰₯1, then the sequence {𝑒𝑛} converges to π‘’βˆ—, a solution of 𝑒+𝐾𝐹𝑒=0.

It is Kown that 𝐿𝑝 spaces, 𝑝=2, are Hilbert and 𝐿𝑝 spaces, 1<𝑝<∞,𝑝≠2, are not. So, the results of Chidume and Djitte [20] do not cover 𝐿𝑝 spaces, 1<𝑝<∞,𝑝≠2.

It is our purpose in this paper to prove convergence theorems in reflexive real Banach spaces that include all 𝐿𝑝 spaces, 1<𝑝<∞. In fact, it is proved in this paper, that an iteration process converges strongly in reflexive real Banach spaces with uniformly GΓ’teaux differentiable norm, to a solution of the Hammerstein equation (assuming existence) when the operators 𝐾 and 𝐹 are Lipschitz and accretive. These spaces include 𝐿𝑝 spaces, 1<𝑝<∞. This complements the results of Chidume and Djitte [20] to provide iterative methods for the approximation of solutions of the Hammerstein equation 𝑒+𝐾𝐹𝑒=0 in all 𝐿𝑝 spaces, 1<𝑝<∞. Our method of proof is different and is also of independent interest.

2. Preliminaries

In the sequel, we will need the followings results.

Lemma 2.1 (see, e.g., [23]). Let {πœ†π‘›} be a sequence of nonnegative numbers and {𝛼𝑛}βŠ†(0,1) a sequence such that βˆ‘βˆžπ‘›=1𝛼𝑛=∞. Let the recursive inequality πœ†π‘›+1β‰€πœ†π‘›βˆ’2π›Όπ‘›πœ“ξ€·πœ†π‘›+1ξ€Έ+πœŽπ‘›,𝑛=1,2,…,(2.1) be given where πœ“βˆΆ[0,∞)β†’[0,∞) is a strictly increasing function such that it is positive on (0,∞), πœ“(0)=0. If πœŽπ‘›=π‘œ(𝛼𝑛). Then πœ†π‘›β†’0, as π‘›β†’βˆž.

Lemma 2.2. There exists a map πœ‡π‘›βˆΆβ„“βˆž(β„•)→ℝ such that (1)πœ‡π‘› is linear; that is, πœ‡π‘›(π‘₯+𝑦)=πœ‡π‘›(π‘₯)+πœ‡π‘›(𝑦) and πœ‡π‘›(𝑐⋅π‘₯)=π‘β‹…πœ‡π‘›(π‘₯),(2)πœ‡π‘› is positive: πœ‡π‘›(π‘₯)β‰₯0 for every π‘₯βˆˆβ„“βˆž(β„•) with π‘₯𝑛β‰₯0,  for  all  𝑛,(3)πœ‡π‘› is normalized: πœ‡π‘›(𝑒)=1, where 𝑒=(1,1,…),(4)πœ‡π‘› is shift invariant: πœ‡π‘›(𝑆π‘₯)=πœ‡π‘›(π‘₯), for all π‘₯βˆˆβ„“βˆž(β„•), where π‘†βˆΆβ„“βˆž(β„•)β†’β„“βˆž(β„•) is the shift operator defined by ξ€·π‘₯π‘†βˆΆ1,π‘₯2,π‘₯3ξ€ΈβŸΌξ€·π‘₯,…2,π‘₯3ξ€Έ,…,(2.2) the above properties on the functional πœ‡π‘› imply the following: (5)πœ‡π‘› has norm one; thus, |πœ‡π‘›(π‘₯)|≀‖π‘₯β€–, for  all  π‘₯βˆˆβ„“βˆž(β„•), (6)πœ‡π‘› extends lim on the subspace of the convergent sequences: limπ‘›β†’βˆžξ€·π‘₯𝑛=π‘βŸΉπœ‡π‘›ξ€·π‘₯(π‘₯)=𝑐,whereπ‘₯=𝑛,(2.3)(7)for any π‘₯=(π‘₯𝑛)βˆˆβ„“βˆž(β„•): liminfπ‘›β†’βˆžπ‘₯π‘›β‰€πœ‡π‘›(π‘₯)≀limsupπ‘›β†’βˆžπ‘₯𝑛.(2.4)

Remark 2.3. Functions πœ‡π‘› as above are called Banach Limits.

Lemma 2.4 (see [24, 25]). Let π‘₯=(π‘₯𝑛)βˆˆβ„“βˆž be such that πœ‡π‘›(π‘₯)≀0 for all Banach limits πœ‡π‘›. If limsup𝑛(π‘₯𝑛+1βˆ’π‘₯𝑛)≀0, then limsup𝑛π‘₯𝑛≀0.

Lemma 2.5. Let 𝐸 be a real normed linear space. Then, the following inequality holds: β€–π‘₯+𝑦‖2≀‖π‘₯β€–2+2βŸ¨π‘¦,𝑗(π‘₯+𝑦)βŸ©βˆ€π‘—(π‘₯+𝑦)∈𝐽(π‘₯+𝑦),βˆ€π‘₯,π‘¦βˆˆπΈ.(2.5)

3. Main Results

We now prove the following theorems.

Theorem 3.1. Let 𝐸 be a real Banach space and 𝐹,πΎβˆΆπΈβ†’πΈ be Lipschitz and accretive maps with 𝐷(𝐾)=𝑅(𝐹)=𝐸 and Lipschitz constants 𝐿1 and 𝐿2, respectively. Let {𝑒𝑛} and {𝑣𝑛} be sequences in 𝐸 defined iteratively from arbitrary points 𝑒1,𝑣1∈𝐸 as follows: 𝑒𝑛+1=π‘’π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1𝑣,𝑛β‰₯1,𝑛+1=π‘£π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΎπ‘£π‘›+π‘’π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘£π‘›βˆ’π‘£1ξ€Έ,𝑛β‰₯1,(3.1) where {πœ†π‘›}, {𝛼𝑛}, and {πœƒπ‘›} are sequences in (0,1) satisfying the following conditions: (1)limπœ†π‘›=0, lim𝛼𝑛=0, (2)πœ†π‘›=π‘œ(πœƒπ‘›),𝛼𝑛=π‘œ(πœƒπ‘›). Suppose that 𝑒+𝐾𝐹𝑒=0 has a solution in 𝐸. Then the sequences {𝑒𝑛} and {𝑣𝑛} are bounded.

Proof. Let π‘‹βˆΆ=𝐸×𝐸 with the norm ‖𝑧‖𝑋=(‖𝑒‖2𝐸+‖𝑣‖2𝐸)1/2, where 𝑧=(𝑒,𝑣). Define the sequence {𝑀𝑛} in 𝑋 by: π‘€π‘›βˆΆ=(𝑒𝑛,𝑣𝑛). Let π‘’βˆ—βˆˆπΈ be a solution of 𝑒+𝐾𝐹𝑒=0, set π‘£βˆ—βˆΆ=πΉπ‘’βˆ— and π‘€βˆ—βˆΆ=(π‘’βˆ—,π‘£βˆ—). We observe that π‘’βˆ—=βˆ’πΎπ‘£βˆ—. We show that the sequence {𝑀𝑛} is bounded in 𝑋.
For this, define ξ€½πΏπΏβˆΆ=max1,𝐿2ξ€Ύ,𝛾0√∢=2βˆ’18.(𝐿+1)(𝐿+2)(3.2) Since πœ†π‘›=π‘œ(πœƒπ‘›) and 𝛼𝑛=π‘œ(πœƒπ‘›), there exists π‘βˆˆβ„• such that 𝛼𝑛/πœƒπ‘›<𝛾0 and πœ†π‘›/πœƒπ‘›<𝛾0 for all 𝑛β‰₯𝑁. Let π‘Ÿ>0 sufficiently large such that 𝑀1∈𝐡(π‘€βˆ—,π‘Ÿ/2) and π‘€π‘βˆˆπ΅(π‘€βˆ—,π‘Ÿ). Define 𝐡∢=𝐡(π‘€βˆ—,π‘Ÿ).
Claim 1. 𝑀𝑛 is in 𝐡 for all 𝑛β‰₯𝑁. The proof is by induction. By construction π‘€π‘βˆˆπ΅. Suppose that π‘€π‘›βˆˆπ΅ for some 𝑛β‰₯𝑁. We prove that 𝑀𝑛+1∈𝐡. Assume for contradiction that 𝑀𝑛+1βˆ‰π΅. Then we have ‖𝑀𝑛+1βˆ’π‘€βˆ—β€–πΈ>π‘Ÿ. We compute as follows: ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–2𝑋=‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2𝐸+‖‖𝑣𝑛+1βˆ’π‘£βˆ—β€–β€–2𝐸.(3.3) Using Lemma 2.5, We have ‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2𝐸=β€–β€–π‘’π‘›βˆ’π‘’βˆ—βˆ’πœ†π‘›ξ€·π›Όπ‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έ+πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1β€–β€–ξ€Έξ€Έ2πΈβ‰€β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2πΈβˆ’2πœ†π‘›ξ«π›Όπ‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έ+πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1𝑒,𝑗𝑛+1βˆ’π‘’βˆ—.(3.4) Observing that ξ«π›Όπ‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—=𝛼𝑛𝐹𝑒𝑛+1βˆ’πΉπ‘’βˆ—ξ€Έξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—+ξ«π›Όξ€Έξ¬π‘›ξ€·πΉπ‘’π‘›βˆ’πΉπ‘’π‘›+1ξ€Έ+π›Όπ‘›ξ€·π‘£βˆ—βˆ’π‘£π‘›ξ€Έξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—,ξ«πœƒξ€Έξ¬π‘›ξ€·π‘’π‘›βˆ’π‘’1𝑒,𝑗𝑛+1βˆ’π‘’βˆ—=ξ«πœƒξ€Έξ¬π‘›ξ€·π‘’π‘›+1βˆ’π‘’βˆ—ξ€Έξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—+ξ«πœƒξ€Έξ¬π‘›ξ€·π‘’π‘›βˆ’π‘’π‘›+1ξ€Έ+πœƒπ‘›ξ€·π‘’βˆ—βˆ’π‘’1𝑒,𝑗𝑛+1βˆ’π‘’βˆ—,(3.5) and using the fact that 𝐹 is accretive, we obtain the following estimate: ‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2πΈβ‰€β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2πΈβˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–2𝐸+2πœ†π‘›ξ«π›Όπ‘›ξ€·πΉπ‘’π‘›+1βˆ’πΉπ‘’π‘›ξ€Έ+π›Όπ‘›ξ€·π‘£π‘›βˆ’π‘£βˆ—ξ€Έξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—ξ€Έξ¬+2πœ†π‘›ξ«πœƒπ‘›ξ€·π‘’π‘›+1βˆ’π‘’π‘›ξ€Έ+πœƒπ‘›ξ€·π‘’1βˆ’π‘’βˆ—ξ€Έξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—.(3.6) Using Schwartz's inequality and the fact that 𝐹 is Lipschitz, it follows that: ‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2πΈβ‰€β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2πΈβˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–2𝐸+2πœ†π‘›ξ€Ίπ›Όπ‘›πΏ1‖‖𝑒𝑛+1βˆ’π‘’π‘›β€–β€–πΈ+π›Όπ‘›β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–πΈξ€»β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–πΈ+2πœ†π‘›ξ€Ίπœƒπ‘›β€–β€–π‘’π‘›+1βˆ’π‘’π‘›β€–β€–πΈ+πœƒπ‘›β€–β€–π‘’1βˆ’π‘’βˆ—β€–β€–πΈξ€»β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–πΈ.(3.7) Thus, we obtain ‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2πΈβ‰€β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2πΈβˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–2𝐸+2πœ†π‘›π›Όξ€Ίξ€·π‘›πΏ1+πœƒπ‘›ξ€Έβ€–β€–π‘’π‘›+1βˆ’π‘’π‘›β€–β€–πΈξ€»β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–πΈ+2πœ†π‘›ξ€Ίπ›Όπ‘›β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–πΈ+πœƒπ‘›β€–β€–π‘’1βˆ’π‘’βˆ—β€–β€–πΈξ€»β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–πΈ.(3.8) Using the fact that ‖‖𝑒𝑛+1βˆ’π‘’π‘›β€–β€–πΈ=β€–β€–πœ†π‘›π›Όπ‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έ+πœ†π‘›πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1ξ€Έβ€–β€–πΈβ‰€πœ†π‘›π›Όπ‘›πΏ1β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–πΈ+πœ†π‘›π›Όπ‘›β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–πΈ+πœ†π‘›πœƒπ‘›β€–β€–π‘’π‘›βˆ’π‘’1‖‖𝐸=πœ†π‘›ξ€·π›Όπ‘›πΏ1β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–πΈ+π›Όπ‘›β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–πΈ+πœƒπ‘›β€–β€–π‘’π‘›βˆ’π‘’1‖‖𝐸,(3.9) we have ‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2πΈβ‰€β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2πΈβˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–2𝐸+2πœ†π‘›π›Όξ€Ίξ€·π‘›πΏ1+πœƒπ‘›πœ†ξ€Έξ€·π‘›π›Όπ‘›πΏ1β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–πΈβ€–β€–π‘’ξ€Έξ€»π‘›+1βˆ’π‘’βˆ—β€–β€–πΈ+2πœ†π‘›π›Όξ€Ίξ€·π‘›πΏ1+πœƒπ‘›πœ†ξ€Έξ€·π‘›π›Όπ‘›β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–πΈβ€–β€–π‘’ξ€Έξ€»π‘›+1βˆ’π‘’βˆ—β€–β€–πΈ+2πœ†π‘›π›Όξ€Ίξ€·π‘›πΏ1+πœƒπ‘›πœ†ξ€Έξ€·π‘›πœƒπ‘›β€–β€–π‘’π‘›βˆ’π‘’1‖‖𝐸‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–πΈ+2πœ†π‘›ξ€Ίπ›Όπ‘›β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–πΈ+πœƒπ‘›β€–β€–π‘’1βˆ’π‘’βˆ—β€–β€–πΈξ€»β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–πΈ.(3.10) Following the same argument, we also obtain ‖‖𝑣𝑛+1βˆ’π‘£βˆ—β€–β€–2πΈβ‰€β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–2πΈβˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘£π‘›+1βˆ’π‘£βˆ—β€–β€–2𝐸+2πœ†π‘›π›Όξ€Ίξ€·π‘›πΏ2+πœƒπ‘›πœ†ξ€Έξ€·π‘›π›Όπ‘›πΏ2β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–πΈβ€–β€–π‘£ξ€Έξ€»π‘›+1βˆ’π‘£βˆ—β€–β€–πΈ+2πœ†π‘›π›Όξ€Ίξ€·π‘›πΏ2+πœƒπ‘›πœ†ξ€Έξ€·π‘›π›Όπ‘›β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–πΈβ€–β€–π‘£ξ€Έξ€»π‘›+1βˆ’π‘£βˆ—β€–β€–πΈ+2πœ†π‘›π›Όξ€Ίξ€·π‘›πΏ2+πœƒπ‘›πœ†ξ€Έξ€·π‘›πœƒπ‘›β€–β€–π‘£π‘›βˆ’π‘£1‖‖𝐸‖‖𝑣𝑛+1βˆ’π‘£βˆ—β€–β€–πΈ+2πœ†π‘›ξ€Ίπ›Όπ‘›β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–πΈ+πœƒπ‘›β€–β€–π‘£1βˆ’π‘£βˆ—β€–β€–πΈξ€»β€–β€–π‘£π‘›+1βˆ’π‘£βˆ—β€–β€–πΈ.(3.11) Using the fact that ‖𝑒‖𝐸+β€–π‘£β€–πΈβ‰€βˆš2‖𝑀‖𝑋,βˆ€π‘€=(𝑒,𝑣)βˆˆπ‘‹,(3.12) we have ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–2π‘‹β‰€β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–2π‘‹βˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–2𝑋+2πœ†π‘›ξ‚ƒβˆš2𝛼𝑛𝐿+πœƒπ‘›πœ†ξ€Έξ€·π‘›π›Όπ‘›πΏβ€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–π‘‹ξ€Έξ‚„β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–π‘‹+2πœ†π‘›ξ‚ƒβˆš2𝛼𝑛𝐿+πœƒπ‘›πœ†ξ€Έξ€·π‘›π›Όπ‘›β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–π‘‹ξ€Έξ‚„β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–π‘‹+2πœ†π‘›ξ‚ƒβˆš2𝛼𝑛𝐿+πœƒπ‘›πœ†ξ€Έξ€·π‘›πœƒπ‘›β€–β€–π‘€π‘›βˆ’π‘€1‖‖𝑋‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–π‘‹+2πœ†π‘›ξ‚ƒβˆš2π›Όπ‘›β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–π‘‹+√2πœƒπ‘›β€–β€–π‘€1βˆ’π‘€βˆ—β€–β€–π‘‹ξ‚„β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–π‘‹,(3.13) which implies that ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–2π‘‹β‰€β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–2π‘‹βˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–2𝑋+2πœ†π‘›ξ‚ƒ2√2𝛼𝑛𝐿+πœƒπ‘›πœ†ξ€Έξ€·π‘›π›Όπ‘›πΏβ€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–π‘‹ξ€Έξ‚„β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–π‘‹+2πœ†π‘›ξ‚ƒβˆš2𝛼𝑛𝐿+πœƒπ‘›πœ†ξ€Έξ€·π‘›πœƒπ‘›β€–β€–π‘€π‘›βˆ’π‘€1‖‖𝑋‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–π‘‹+2πœ†π‘›ξ‚ƒβˆš2π›Όπ‘›β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–π‘‹+√2πœƒπ‘›β€–β€–π‘€1βˆ’π‘€βˆ—β€–β€–π‘‹ξ‚„β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–π‘‹.(3.14) Since π‘€π‘›βˆˆπ΅ and 𝑀1∈𝐡(π‘€βˆ—,π‘Ÿ/2), we have ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–2π‘‹β‰€β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–2π‘‹βˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–2𝑋+2πœ†π‘›ξ‚ƒ2√2𝛼𝑛𝐿+πœƒπ‘›ξ€Έπœ†π‘›ξ€·π›Όπ‘›πΏπ‘Ÿ+2πœƒπ‘›π‘Ÿξ€Έξ‚„β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–π‘‹+2πœ†π‘›ξƒ¬2√2π›Όπ‘›βˆšπ‘Ÿ+22πœƒπ‘›π‘Ÿξƒ­β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–π‘‹.(3.15) By assumption, ‖𝑀𝑛+1βˆ’π‘€βˆ—β€–2𝑋>|π‘€π‘›βˆ’π‘€βˆ—β€–2𝑋. So we have πœƒπ‘›β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–π‘‹βˆšβ‰€22𝛼𝑛𝐿+πœƒπ‘›ξ€Έπœ†π‘›ξ€·π›Όπ‘›πΏ+2πœƒπ‘›ξ€Έβˆšπ‘Ÿ+22π›Όπ‘›βˆšπ‘Ÿ+22πœƒπ‘›π‘Ÿ.(3.16) Hence ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–π‘‹βˆšβ‰€22πœ†π‘›πœƒπ‘›βˆš(𝐿+1)(𝐿+2)π‘Ÿ+22π›Όπ‘›πœƒπ‘›βˆšπ‘Ÿ+22π‘Ÿ.(3.17) Therefore, using the fact that πœ†π‘›/πœƒπ‘›<𝛾0 and 𝛼𝑛/πœƒπ‘›<𝛾0, it follows that ‖𝑀𝑛+1βˆ’π‘€βˆ—β€–π‘‹<π‘Ÿ. A contradiction. So 𝑀𝑛+1∈𝐡. This prove the boundedness of the sequences {𝑒𝑛} and {𝑣𝑛}.

Theorem 3.2. Let 𝐸 be reflexive real Banach space with uniformly GΓ’teaux differentiable norm and 𝐹,πΎβˆΆπΈβ†’πΈ be Lipschitz accretive mappings. Let {𝑒𝑛}𝑛β‰₯1 and {𝑣𝑛}𝑛β‰₯1 be sequences in 𝐸 defined iteratively from arbitrary 𝑒1,𝑣1∈𝐸 by 𝑒𝑛+1=π‘’π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1𝑣,𝑛β‰₯1,𝑛+1=π‘£π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΎπ‘£π‘›+π‘’π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘£π‘›βˆ’π‘£1ξ€Έ,𝑛β‰₯1,(3.18) where {πœ†π‘›}𝑛β‰₯1, {𝛼𝑛}𝑛β‰₯1, and {πœƒπ‘›}𝑛β‰₯1 are real sequences in (0,1) such that πœ†π‘›=π‘œ(πœƒπ‘›),𝛼𝑛=π‘œ(πœƒπ‘›), and βˆ‘βˆžπ‘›=1πœ†π‘›πœƒπ‘›=+∞. Suppose that the equation 𝑒+𝐾𝐹𝑒=0 has a solution π‘’βˆ—. Then there exists a subset 𝐾min of 𝐸×𝐸 such that if (π‘’βˆ—,π‘£βˆ—)∈𝐾min with π‘£βˆ—=πΉπ‘’βˆ—, then the sequence {𝑒𝑛}𝑛β‰₯1 converges strongly to π‘’βˆ—.

Proof. Since, by Theorem 3.1, the sequences {𝑒𝑛}𝑛β‰₯1 and {𝑣𝑛}𝑛β‰₯1 are bounded and 𝐹 is bounded, there exists an 𝑅>0 such that π‘’π‘›βˆˆπ΅1∢=𝐡(π‘’βˆ—,𝑅) and π‘£π‘›βˆˆπ΅2∢=𝐡(π‘£βˆ—,𝑅) for all 𝑛β‰₯1. Furthermore, the sets 𝐡1 and 𝐡2 are nonempty closed convex and bounded subsets of 𝐸. Let πœ‡π‘› be a Banach limit. Define the maps πœ‘1βˆΆπΈβ†’β„ and πœ‘2βˆΆπΈβ†’β„ by πœ‘1(𝑒)∢=πœ‡π‘›β€–π‘’π‘›βˆ’π‘’β€–2,πœ‘2(𝑣)∢=πœ‡π‘›β€–π‘£π‘›βˆ’π‘£β€–2.(3.19) Then for 𝑖=1,2,πœ‘π‘– is weakly lower semicontinuous. Since 𝐸 reflexive, then Argminπ‘’βˆˆπ΅1πœ‘1(𝑒)β‰ βˆ…,Argminπ‘£βˆˆπ΅2πœ‘2(𝑣)β‰ βˆ….(3.20) Set 𝐾min∢=Argminπ‘’βˆˆπ΅1πœ‘1(𝑒)Γ—Argminπ‘£βˆˆπ΅2πœ‘2(𝑣).(3.21) Now assume that π‘€βˆ—=(π‘’βˆ—,π‘£βˆ—)∈𝐾min and let π‘‘βˆˆ(0,1). Then, by convexity of 𝐡1, we have that (1βˆ’π‘‘)π‘’βˆ—+𝑑𝑒1∈𝐡1. Thus, πœ‘1(π‘’βˆ—)β‰€πœ‘1((1βˆ’π‘‘)π‘’βˆ—+𝑑𝑒1). It follows from Lemma 2.5 that β€–β€–π‘’π‘›βˆ’π‘’βˆ—ξ€·π‘’βˆ’π‘‘1βˆ’π‘’βˆ—ξ€Έβ€–β€–2β‰€β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2ξ«π‘’βˆ’2𝑑1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€·π‘’βˆ’π‘‘1βˆ’π‘’βˆ—ξ€Έξ€Έξ¬βˆ€π‘›βˆˆβ„•.(3.22) Thus, taking Banach limits gives πœ‡π‘›ξ‚€β€–β€–π‘’π‘›βˆ’π‘’βˆ—ξ€·π‘’βˆ’π‘‘1βˆ’π‘’βˆ—ξ€Έβ€–β€–2ξ‚β‰€πœ‡π‘›ξ‚€β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2ξ‚βˆ’2π‘‘πœ‡π‘›π‘’ξ€·ξ«1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€·π‘’βˆ’π‘‘1βˆ’π‘’βˆ—ξ€Έξ€Έξ¬ξ€Έ.(3.23) That is, πœ‘1ξ€·(1βˆ’π‘‘)π‘’βˆ—+𝑑𝑒1ξ€Έβ‰€πœ‘1ξ€·π‘’βˆ—ξ€Έβˆ’2π‘‘πœ‡π‘›π‘’ξ€·ξ«1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€·π‘’βˆ’π‘‘1βˆ’π‘’βˆ—ξ€Έξ€Έξ¬ξ€Έ.(3.24) This implies that πœ‡π‘›(βŸ¨π‘’1βˆ’π‘’βˆ—,𝑗(π‘’π‘›βˆ’π‘’βˆ—βˆ’π‘‘(𝑒1βˆ’π‘’βˆ—))⟩)≀0forallπ‘›βˆˆβ„•. Furthermore, the fact that 𝐸 has a uniformly GΓ’teaux differentiable norm gives, as 𝑑→0, that 𝑒1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—βˆ’ξ«π‘’ξ€Έξ¬1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€·π‘’βˆ’π‘‘1βˆ’π‘’βˆ—ξ€Έξ€Έξ¬βŸΆ0.(3.25) Thus, for all πœ–>0, there exists π›Ώπœ–>0 such that for all π‘‘βˆˆ]0,π›Ώπœ–[ and for all π‘›βˆˆβ„•, 𝑒1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—βˆ’ξ«π‘’ξ€Έξ¬1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€·π‘’βˆ’π‘‘1βˆ’π‘’βˆ—ξ€Έξ€Έξ¬<πœ–.(3.26) Thus, πœ‡π‘›π‘’ξ€·ξ«1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€Έξ¬ξ€Έβˆ’πœ‡π‘›π‘’ξ€·ξ«1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€·π‘’βˆ’π‘‘1βˆ’π‘’βˆ—ξ€Έξ€Έξ¬ξ€Έβ‰€πœ–,(3.27) which implies that πœ‡π‘›π‘’ξ€·ξ«1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€Έξ¬ξ€Έβ‰€0.(3.28) Using the same argument, we have πœ‡π‘›π‘£ξ€·ξ«1βˆ’π‘£βˆ—ξ€·π‘£,π‘—π‘›βˆ’π‘£βˆ—ξ€Έξ¬ξ€Έβ‰€0.(3.29) Moreover, since {𝑒𝑛}𝑛β‰₯1,{𝑣𝑛}𝑛β‰₯1, {𝐹𝑒𝑛}𝑛β‰₯1, and {𝐾𝑣𝑛}𝑛β‰₯1 are all bounded, then, from (3.18) and that there exist positive constants 𝑀1 and 𝑀2 such that ‖‖𝑒𝑛+1βˆ’π‘’π‘›β€–β€–β‰€πœ†π‘›ξ€Ίπ›Όπ‘›ξ€·β€–β€–πΉπ‘’π‘›β€–β€–+‖‖𝑣𝑛‖‖+ξ€·πœƒπ‘›β€–β€–π‘’π‘›βˆ’π‘’1β€–β€–ξ€Έξ€»β‰€πœ†π‘›π‘€1,‖‖𝑣𝑛+1βˆ’π‘£π‘›β€–β€–β‰€πœ†π‘›ξ€Ίπ›Όπ‘›ξ€·β€–β€–πΎπ‘£π‘›β€–β€–+‖‖𝑒𝑛‖‖+ξ€·πœƒπ‘›β€–β€–π‘£π‘›βˆ’π‘£1β€–β€–ξ€Έξ€»β‰€πœ†π‘›π‘€2.(3.30) Thus, limπ‘›β†’βˆžβ€–π‘’n+1βˆ’π‘’π‘›β€–=0 and limπ‘›β†’βˆžβ€–π‘£π‘›+1βˆ’π‘£π‘›β€–=0.
Again, using the fact that 𝐸 has a uniformly GΓ’teaux differentiable norm, we obtain that limπ‘›β†’βˆžπ‘’ξ€·ξ«1βˆ’π‘’βˆ—ξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—βˆ’ξ«π‘’ξ€Έξ¬1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€Έξ¬ξ€Έ=0,limπ‘›β†’βˆžπ‘£ξ€·ξ«1βˆ’π‘£βˆ—ξ€·π‘£,𝑗𝑛+1βˆ’π‘£βˆ—βˆ’ξ«π‘£ξ€Έξ¬1βˆ’π‘£βˆ—ξ€·π‘£,π‘—π‘›βˆ’π‘£βˆ—ξ€Έξ¬ξ€Έ=0.(3.31) Therefore, the sequences {βŸ¨π‘’1βˆ’π‘’βˆ—,𝑗(π‘’π‘›βˆ’π‘’βˆ—)⟩}𝑛β‰₯1 and {βŸ¨π‘£1βˆ’π‘£βˆ—,𝑗(π‘£π‘›βˆ’π‘£βˆ—)⟩}𝑛β‰₯1 satisfy the conditions of Lemma 2.4.
Hence, limsupπ‘›β†’βˆžξ«π‘’1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€Έξ¬β‰€0,limsupπ‘›β†’βˆžξ«π‘£1βˆ’π‘£βˆ—ξ€·π‘£,π‘—π‘›βˆ’π‘£βˆ—ξ€Έξ¬β‰€0.(3.32) Define πœŽπ‘›π‘’βˆΆ=max1βˆ’π‘’βˆ—ξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—ξ€Ύ,πœ‰ξ€Έξ¬,0π‘›π‘£βˆΆ=max1βˆ’π‘£βˆ—ξ€·π‘£,𝑗𝑛+1βˆ’π‘£βˆ—ξ€Ύ,,0(3.33) then limπ‘›β†’βˆžπœŽπ‘›=0=limπ‘›β†’βˆžπœ‰π‘›. Moreover, 𝑒1βˆ’π‘’βˆ—ξ€·π‘’,π‘—π‘›βˆ’π‘’βˆ—ξ€Έξ¬β‰€πœŽπ‘›,𝑣1βˆ’π‘£βˆ—ξ€·π‘£,π‘—π‘›βˆ’π‘£βˆ—ξ€Έξ¬β‰€πœ‰π‘›,βˆ€π‘›βˆˆβ„•.(3.34) Using Lemma 2.5, and (3.18), we have: ‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2πΈβ‰€β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2πΈβˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘’π‘›+1βˆ’π‘’βˆ—β€–β€–2𝐸+2πœ†π‘›ξ«π›Όπ‘›ξ€·πΉπ‘’π‘›+1βˆ’πΉπ‘’π‘›ξ€Έ+π›Όπ‘›ξ€·π‘£π‘›βˆ’π‘£βˆ—ξ€Έξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—ξ€Έξ¬+2πœ†π‘›ξ«πœƒπ‘›ξ€·π‘’π‘›+1βˆ’π‘’π‘›ξ€Έ+πœƒπ‘›ξ€·π‘’1βˆ’π‘’βˆ—ξ€Έξ€·π‘’,𝑗𝑛+1βˆ’π‘’βˆ—,‖‖𝑣𝑛+1βˆ’π‘£βˆ—β€–β€–2πΈβ‰€β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–2πΈβˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘£π‘›+1βˆ’π‘£βˆ—β€–β€–2𝐸+2πœ†π‘›ξ«π›Όπ‘›ξ€·πΎπ‘£π‘›+1βˆ’πΎπ‘£π‘›ξ€Έ+π›Όπ‘›ξ€·π‘£βˆ—βˆ’π‘£π‘›ξ€Έξ€·π‘£,𝑗𝑛+1βˆ’π‘£βˆ—ξ€Έξ¬+2πœ†π‘›ξ«πœƒπ‘›ξ€·π‘£π‘›+1βˆ’π‘£π‘›ξ€Έ+πœƒπ‘›ξ€·π‘£1βˆ’π‘£βˆ—ξ€Έξ€·π‘£,𝑗𝑛+1βˆ’π‘£βˆ—.(3.35) Using the fact that 𝐹 and 𝐾 are Lipschitz and the sequences {𝑒𝑛} and {𝑣𝑛} are bounded, we obtain the following estimate: ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–2πΈβ‰€β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–2πΈβˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–2πΈξ€·πœ†+2𝑛𝛼𝑛+πœ†2𝑛𝑐+2πœ†π‘›πœƒπ‘›ξ€·πœŽπ‘›+πœ‰π‘›ξ€Έ=β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–2πΈβˆ’2πœ†π‘›πœƒπ‘›β€–β€–π‘€π‘›+1βˆ’π‘€βˆ—β€–β€–2𝐸+𝛾𝑛,(3.36) where 𝛾𝑛=2(πœ†π‘›π›Όπ‘›+πœ†2𝑛)𝑐+2πœ†π‘›πœƒπ‘›(πœŽπ‘›+πœ‰π‘›)=π‘œ(πœ†π‘›πœƒπ‘›) for some 𝑐>0. Hence, by Lemma 2.1, π‘€π‘›β†’π‘€βˆ— as π‘›β†’βˆž. But 𝑀𝑛=(𝑒𝑛,𝑣𝑛) and π‘€βˆ—=(π‘’βˆ—,π‘£βˆ—). This implies that π‘’π‘›β†’π‘’βˆ— as π‘›β†’βˆž. This completes the proof.

Corollary 3.3. Let 𝐸 be a π‘ž-uniformly real Banach spaces, π‘ž>1 and 𝐹,πΎβˆΆπΈβ†’πΈ be Lipschitz accretive mappings. Let {𝑒𝑛}𝑛β‰₯1 and {𝑣𝑛}𝑛β‰₯1 be sequences in 𝐸 defined iteratively from arbitrary 𝑒1,𝑣1∈𝐸 by 𝑒𝑛+1=π‘’π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1𝑣,𝑛β‰₯1,𝑛+1=π‘£π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΎπ‘£π‘›+π‘’π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘£π‘›βˆ’π‘£1ξ€Έ,𝑛β‰₯1,(3.37) where {πœ†π‘›}𝑛β‰₯1, {𝛼𝑛}𝑛β‰₯1, and {πœƒπ‘›}𝑛β‰₯1 are real sequences in (0,1) such that πœ†π‘›=π‘œ(πœƒπ‘›),𝛼𝑛=π‘œ(πœƒπ‘›), and βˆ‘βˆžπ‘›=1πœ†π‘›πœƒπ‘›=+∞. Suppose that the equation 𝑒+𝐾𝐹𝑒=0 has a solution π‘’βˆ—. Then there exists a set 𝐾min in 𝐸×𝐸 such that if (π‘’βˆ—,π‘£βˆ—)∈𝐾min with π‘£βˆ—=πΉπ‘’βˆ—, then the sequence {𝑒𝑛}𝑛β‰₯1 converges strongly to π‘’βˆ—.

Corollary 3.4. Let 𝐸=𝐿𝑝 (or ℓ𝑝) space (1<𝑝<+∞) and 𝐹,πΎβˆΆπΈβ†’πΈ be Lipschitz accretive mappings. Let {𝑒𝑛}𝑛β‰₯1 and {𝑣𝑛}𝑛β‰₯1 be sequences in 𝐸 defined iteratively from arbitrary 𝑒1,𝑣1∈𝐸 by 𝑒𝑛+1=π‘’π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1𝑣,𝑛β‰₯1,𝑛+1=π‘£π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΎπ‘£π‘›+π‘’π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘£π‘›βˆ’π‘£1ξ€Έ,𝑛β‰₯1,(3.38) where {πœ†π‘›}𝑛β‰₯1, {𝛼𝑛}𝑛β‰₯1, and {πœƒπ‘›}𝑛β‰₯1 are real sequences in (0,1) such that πœ†π‘›=π‘œ(πœƒπ‘›),𝛼𝑛=π‘œ(πœƒπ‘›), and βˆ‘βˆžπ‘›=1πœ†π‘›πœƒπ‘›=+∞. Suppose that the equation 𝑒+𝐾𝐹𝑒=0 has a solution π‘’βˆ—. Then there exists a subset 𝐾min of 𝐸×𝐸 such that if (π‘’βˆ—,π‘£βˆ—)∈𝐾min with π‘£βˆ—=πΉπ‘’βˆ—, then the sequence {𝑒𝑛}𝑛β‰₯1 converges strongly to π‘’βˆ—.

Corollary 3.5. Let H be a real Hilbert space and 𝐹,πΎβˆΆπ»β†’π» be Lipschitz monotone mappings. Let {𝑒𝑛}𝑛β‰₯1 and {𝑣𝑛}𝑛β‰₯1 be sequences in 𝐻 defined iteratively from arbitrary 𝑒1,𝑣1∈𝐻 by 𝑒𝑛+1=π‘’π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1𝑣,𝑛β‰₯1,𝑛+1=π‘£π‘›βˆ’πœ†π‘›π›Όπ‘›ξ€·πΎπ‘£π‘›+π‘’π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘£π‘›βˆ’π‘£1ξ€Έ,𝑛β‰₯1,(3.39) where {πœ†π‘›}𝑛β‰₯1, {𝛼𝑛}𝑛β‰₯1, and {πœƒπ‘›}𝑛β‰₯1 are real sequences in (0,1) such that πœ†π‘›=π‘œ(πœƒπ‘›),𝛼𝑛=π‘œ(πœƒπ‘›), and βˆ‘βˆžπ‘›=1πœ†π‘›πœƒπ‘›=+∞. Suppose that the equation 𝑒+𝐾𝐹u=0 has a solution π‘’βˆ—. Then there exists a subset 𝐾min of 𝐸×𝐸 such that if (π‘’βˆ—,π‘£βˆ—)∈𝐾min with π‘£βˆ—=πΉπ‘’βˆ—, then the sequence {𝑒𝑛}𝑛β‰₯1 converges strongly to π‘’βˆ—.

Remark 3.6. Real sequences that satisfy the hypotheses of Theorems 3.1 are πœ†π‘›=𝛼𝑛=π‘›βˆ’π‘Ž and πœƒπ‘›=π‘›βˆ’π‘, 𝑛β‰₯1 with 0<𝑏<π‘Ž and π‘Ž+𝑏<1.

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