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ISRN Mathematical Analysis
VolumeΒ 2012Β (2012), Article IDΒ 169751, 12 pages
http://dx.doi.org/10.5402/2012/169751
Research Article

Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type

1Mathematics Institute, African University of Sciences and Technology, Abuja, Nigeria
2Mathematics Department, Gaston Berger University, Saint Louis, Senegal

Received 29 November 2011; Accepted 10 January 2012

Academic Editors: F.Β Arandiga and J.Β Cui

Copyright Β© 2012 C. E. Chidume and N. DjittΓ©. 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

Suppose that 𝐻 is a real Hilbert space and 𝐹,πΎβˆΆπ»β†’π» are bounded monotone maps with 𝐷(𝐾)=𝐷(𝐹)=𝐻. Let π‘’βˆ— denote a solution of the Hammerstein equation 𝑒+𝐾𝐹𝑒=0. An explicit iteration process is shown to converge strongly to π‘’βˆ—. No invertibility or continuity assumption is imposed on 𝐾 and the operator 𝐹 is not restricted to be angle-bounded. Our result is a significant improvement on the Galerkin method of BrΓ©zis and Browder.

1. Introduction

Let 𝑋 be a real normed linear space with dual π‘‹βˆ—. For π‘ž>1, we denote by π½π‘ž the generalized duality mapping from 𝑋 to 2π‘‹βˆ— defined byπ½π‘žξ€½π‘“(π‘₯)∢=βˆ—βˆˆπ‘‹βˆ—βˆΆβŸ¨π‘₯,π‘“βˆ—βŸ©=β€–π‘₯β€–β‹…β€–π‘“βˆ—β€–,β€–π‘“βˆ—β€–=β€–π‘₯β€–π‘žβˆ’1ξ€Ύ,(1.1) where βŸ¨β‹…,β‹…βŸ© denotes the generalized duality pairing. 𝐽2 is denoted by 𝐽. If π‘‹βˆ— is strictly convex, then π½π‘ž is single-valued. A map 𝐺 with domain 𝐷(𝐺) in a normed linear space 𝑋 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.(1.2) If π‘˜=0, 𝐺 is said to be accretive. If 𝑋 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 formπ‘₯ξ…ž(𝑑)+𝐴π‘₯(𝑑)=0,π‘₯(0)=π‘₯0,(1.3) 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 SchrΓΆdinger equations. If in (1.3), π‘₯(𝑑) is independent of 𝑑, then (1.3) reduces to𝐴𝑒=0,(1.4) whose solutions correspond to the equilibrium points of the system (1.3). Consequently, considerable research efforts have been devoted, especially within the past 30 years or so, to methods of finding approximate solutions (when they exist) of (1.4). An early fundamental result in the theory of accretive operators, due to Browder [1], states that the initial value problem (1.3) is solvable if 𝐴 is locally Lipschitzian and accretive on 𝑋. Utilizing the existence result for (1.3), Browder [1] proved that if 𝐴 is locally Lipschitzian and accretive on 𝑋, then 𝐴 is π‘š-accretive, that is, 𝑅(𝐼+𝐴)=𝑋, where 𝑅(𝐼+𝐴) denotes the range of (𝐼+𝐴). Clearly, a consequence of this is that the equation𝑒+𝐴𝑒=0(1.5) has a solution. One important generalization of (1.5) 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‘₯),(1.6) 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 byξ€œπΎπ‘£(π‘₯)∢=Ξ©πœ…(π‘₯,𝑦)𝑣(𝑦)𝑑𝑦,π‘₯∈Ω,(1.7) and the so-called superposition or Nemytskii operator by 𝐹𝑒(𝑦)∢=𝑓(𝑦,𝑒(𝑦)) then, the integral equation (1.6) can be put in operator theoretic form as follows:𝑒+𝐾𝐹𝑒=0,(1.8) where, without loss of generality, we have taken β„Žβ‰‘0.

Interest in (1.8) 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 (1.8) (see e.g., Pascali and Sburlan [5], Chapter IV). 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], Browder and Gupta [11], Cydotchepanovich [12], and De Figueiredo and Gupta [13]). For the iterative approximation of solutions of (1.4) and (1.5), 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 be monotone. In the special case in which the operators are defined on subsets 𝐷 of 𝑋 which are compact (or more generally, angle-bounded see e.g., Pascali and Sburlan [5] for definition), BrΓ©zis and Browder [7] have proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.8) (see also BrΓ©zis and Browder [9]).

It is our purpose in this paper to prove that an explicit coupled iteration process recently introduced by Chidume and Zegeye [17] which does not involve πΎβˆ’1 which is also required to be monotone converges strongly to a solution of (1.8) when 𝐾 and 𝐹 are bounded and monotone. Our new method of proof is also of independent interest.

2. Preliminaries

In the sequel, we will need the followings results.

Lemma 2.1 (see Xu [18]). Let {π‘Žπ‘›} be a sequence of nonnegative real numbers satisfying the following relations:π‘Žπ‘›+1≀1βˆ’π›Όπ‘›ξ€Έπ‘Žπ‘›+π›Όπ‘›πœŽπ‘›+𝛾𝑛,𝑛β‰₯0,(2.1) where (i){𝛼𝑛}βŠ‚(0,1), βˆ‘π›Όπ‘›=∞; (ii)limsupπœŽπ‘›β‰€0; (iii)𝛾𝑛β‰₯0, (𝑛β‰₯0), βˆ‘π›Ύπ‘›<∞. Then, π‘Žπ‘›β†’0 as π‘›β†’βˆž.

Lemma 2.2 (see Chidume and Djitte, [19, Lemma  2.5]). Let 𝐻 be a real Hilbert space and π΄βˆΆπ»β†’π» be a map with 𝐷(𝐴)=𝐻. Suppose that 𝐴 is π‘š-accretive, that is, (i) for all𝑒,π‘£βˆˆπ», βŸ¨π΄π‘’βˆ’π΄π‘£,π‘’βˆ’π‘£βŸ©β‰₯0; (ii)𝑅(𝐼+𝑠0𝐴)=𝐻 for some 𝑠0>0. Then 𝐴 satisfies the range condition, that is, 𝑅(𝐼+𝑠𝐴)=𝐻 for all 𝑠>0.

We now prove the following result.

Lemma 2.3. Let 𝐻 be a real Hilbert space and 𝐹,πΎβˆΆπ»β†’π» be maps with 𝐷(𝐹)=𝐷(𝐾)=𝐻. Let 𝐸=𝐻×𝐻 and π‘‡βˆΆπΈβ†’πΈ be the map defined by: 𝑇𝑀=(πΉπ‘’βˆ’π‘£,𝐾𝑣+𝑒),βˆ€π‘€=(𝑒,𝑣)∈𝐸.(2.2) Assume that 𝐹 and 𝐾 are monotones and satisfy the range condition. Then, 𝑇 is monotone and also satisfies the range condition.

Proof. On 𝐸 we have the natural norm ‖⋅‖𝐸 and natural inner product βŸ¨β‹…,β‹…βŸ©πΈ given by: ‖𝑀‖𝐸=‖𝑒‖2𝐻+‖𝑣‖2𝐻1/2,for𝑀=(𝑒,𝑣)∈𝐸,βŸ¨π‘€1,𝑀2⟩𝐸=βŸ¨π‘’1,𝑒2⟩𝐻+βŸ¨π‘£1,𝑣2⟩𝐻,for𝑀1=𝑒1,𝑣1ξ€Έ,𝑀2=𝑒2,𝑣2ξ€ΈβˆˆπΈ.(2.3)Step 1. We prove that 𝑇 is monotone. Let 𝑀1=(𝑒1,𝑣1), 𝑀2=(𝑒2,𝑣2)∈𝐸. We have 𝑇𝑀1=(𝐹𝑒1βˆ’π‘£1,𝐾𝑣1+𝑒1) and 𝑇𝑀2=(𝐹𝑒2βˆ’π‘£2,𝐾𝑣2+𝑒2). So, 𝑇𝑀1βˆ’π‘‡π‘€2=(𝐹𝑒1βˆ’πΉπ‘’2+𝑣2βˆ’π‘£1,𝐾𝑣1βˆ’πΎπ‘£2+𝑒1βˆ’π‘’2). Therefore, using the fact that 𝐹 and 𝐾 are monotone, we obtain,βŸ¨π‘‡π‘€1βˆ’π‘‡π‘€2,𝑀1βˆ’π‘€2⟩𝐸=βŸ¨πΉπ‘’1βˆ’πΉπ‘’2+𝑣2βˆ’π‘£1,𝑒1βˆ’π‘’2⟩𝐻+βŸ¨πΎπ‘£1βˆ’πΎπ‘£2+u1βˆ’π‘’2,𝑣1βˆ’π‘£2⟩𝐻=βŸ¨πΉπ‘’1βˆ’πΉπ‘’2,𝑒1βˆ’π‘’2⟩𝐻+βŸ¨πΎπ‘£1βˆ’πΎπ‘£2,𝑣1βˆ’π‘£2⟩𝐻β‰₯0.(2.4) So, 𝑇 is monotone.Step 2. We show that β„›(𝐼𝐸+π‘Ÿπ‘‡)=𝐸 for all π‘Ÿ, 0<π‘Ÿ<1. In fact let π‘Ÿ0 such that 0<π‘Ÿ0<1. Since 𝐹 and 𝐾 are monotone and satisfy the range condition, then it is known that (𝐼+π‘Ÿ0𝐹) and (𝐼+π‘Ÿ0𝐾) are bijective and moreover, the resolvent π½πΉπ‘Ÿ0∢=(𝐼+π‘Ÿ0𝐹)βˆ’1 of 𝐹 and the resolvent π½πΎπ‘Ÿ0∢=(𝐼+π‘Ÿ0𝐾)βˆ’1 of 𝐾 are nonexpansive.
Let β„Ž=(β„Ž1,β„Ž2)∈𝐸. Define πΊβˆΆπΈβ†’πΈ by𝐽𝐺𝑀=πΉπ‘Ÿ0ξ€·β„Ž1+π‘Ÿ0𝑣,π½πΎπ‘Ÿ0ξ€·β„Ž2βˆ’π‘Ÿ0𝑒,βˆ€π‘€=(𝑒,𝑣)∈𝐸.(2.5) Using the fact that π½πΉπ‘Ÿ0 and π½πΎπ‘Ÿ0 are nonexpansive, we have, ‖‖𝐺𝑀1βˆ’πΊπ‘€2β€–β€–πΈβ‰€π‘Ÿ0‖‖𝑀1βˆ’π‘€2‖‖𝐸,βˆ€π‘€1,𝑀2∈𝐸.(2.6) Therefore 𝐺 is a contraction. So, by the Banach fixed point theorem, 𝐺 has a unique fixed point π‘€βˆ—=(π‘’βˆ—,π‘£βˆ—)∈𝐸, that is πΊπ‘€βˆ—=π‘€βˆ— or equivalently, π‘’βˆ—=π½πΉπ‘Ÿ0ξ€·β„Ž1+π‘Ÿ0π‘£βˆ—ξ€Έ,π‘£βˆ—=π½πΎπ‘Ÿ0ξ€·β„Ž2βˆ’π‘Ÿ0π‘’βˆ—ξ€Έ.(2.7) These imply (𝐼𝐸+π‘Ÿ0𝑇)π‘€βˆ—=β„Ž. Therefore, β„›(𝐼𝐸+π‘Ÿ0𝑇)=𝐸.
By Lemma 2.2, it follows that 𝑇 satisfies the range condition. This completes the proof.

Theorem 2.4 (see Reich [20]). Let 𝐻 be a real Hilbert space. Let π΄βˆΆπ»β†’π» be monotone with 𝐷(𝐴)=𝐻 and suppose that 𝐴 satisfies the range condition: β„›(𝐼+π‘Ÿπ΄)=𝐻for all π‘Ÿ>0. Let 𝐽𝑑π‘₯∢=(𝐼+𝑑𝐴)βˆ’1π‘₯, 𝑑>0 be the resolvent of 𝐴, and assume that π΄βˆ’1(0) is nonempty. Then for each π‘₯∈𝐻, limπ‘‘β†’βˆžπ½π‘‘π‘₯βˆˆπ΄βˆ’1(0).

3. Main Results

Let 𝐻 be a real Hilbert space and 𝐹,πΎβˆΆπ»β†’π» be maps with 𝐷(𝐾)=𝐷(𝐹)=𝐻 such that the following conditions hold:(i)𝐹 is bounded and monotone, that is, βŸ¨πΉπ‘’1βˆ’πΉπ‘’2,𝑒1βˆ’π‘’2⟩β‰₯0,βˆ€π‘’1,𝑒2∈𝐻,(3.1)(ii)𝐾 is bounded and monotone, that is, βŸ¨πΎπ‘’1βˆ’πΎπ‘’2,𝑒1βˆ’π‘’2⟩β‰₯0,βˆ€π‘’1,𝑒2∈𝐻,(3.2)(iii)𝐹 and K satisfy the range condition.

With these assumptions, we prove the following theorem.

Theorem 3.1. Let 𝐻 be a real Hilbert space. 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,(3.3) 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.

Proof. Let 𝐸∢=𝐻×𝐻 with the norm ‖𝑧‖𝐸=(‖𝑒‖2𝐻+‖𝑣‖2𝐻)1/2, where 𝑧=(𝑒,𝑣). Define the sequence {𝑀𝑛} in 𝐸 by: π‘€π‘›βˆΆ=(𝑒𝑛,𝑣𝑛). Let π‘’βˆ—βˆˆπ» be a solution of 𝑒+𝐾𝐹𝑒=0, π‘£βˆ—βˆΆ=πΉπ‘’βˆ— and π‘€βˆ—βˆΆ=(π‘’βˆ—,π‘£βˆ—). We observe that π‘’βˆ—=βˆ’πΎπ‘£βˆ—. It suffices to show that {𝑀𝑛} converges to π‘€βˆ— in 𝐸.
For this, let 𝑛0βˆˆβ„•, there exists π‘Ÿ>0 sufficiently large such that 𝑀1∈𝐡(π‘€βˆ—,π‘Ÿ/2), 𝑀𝑛0∈𝐡(π‘€βˆ—,π‘Ÿ), where 𝐡(π‘€βˆ—,π‘Ÿ) denotes the ball of center π‘€βˆ— and radius π‘Ÿ. Define 𝐡∢=𝐡(π‘€βˆ—,π‘Ÿ). Since 𝐹 and 𝐾 are bounded, we set 𝑀1∢=sup{‖𝐹π‘₯βˆ’π‘¦β€–2𝐻+π‘Ÿ2∢(π‘₯,𝑦)∈𝐡}<∞ and 𝑀2∢=sup{‖𝐾𝑦+π‘₯‖𝐻+π‘Ÿ2∢(π‘₯,𝑦)∈𝐡}<∞. Let π‘€βˆΆ=𝑀1+𝑀2. We split the proof in three steps.
Step 1. We first prove that the sequence {𝑀𝑛} is bounded in 𝐸. Indeed, it suffices to show that 𝑀𝑛 is in 𝐡 for all 𝑛β‰₯𝑛0. The proof is by induction. By construction, 𝑀𝑛0∈𝐡. Suppose that π‘€π‘›βˆˆπ΅ for 𝑛β‰₯𝑛0. 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€–2=‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2𝐻+‖‖𝑣𝑛+1βˆ’π‘£βˆ—β€–β€–2𝐻.(3.4) We have ‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2𝐻=β€–β€–π‘’π‘›βˆ’π‘’βˆ—βˆ’πœ†π‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›ξ€Έβˆ’πœ†π‘›πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1ξ€Έβ€–β€–2=β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2π»βˆ’2πœ†π‘›ξ«πΉπ‘’π‘›βˆ’π‘£π‘›+πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1ξ€Έ,π‘’π‘›βˆ’π‘’βˆ—ξ¬+πœ†2π‘›β€–β€–πΉπ‘’π‘›βˆ’π‘£π‘›+πœƒπ‘›(π‘’π‘›βˆ’π‘’1)β€–β€–2π»β‰€β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2π»βˆ’2πœ†π‘›βŸ¨πΉπ‘’π‘›βˆ’π‘£π‘›+πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1ξ€Έ,π‘’π‘›βˆ’π‘’βˆ—βŸ©+πœ†2𝑛𝑀1.(3.5) Observing that ξ«πΉπ‘’π‘›βˆ’π‘£π‘›+πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1ξ€Έ,π‘’π‘›βˆ’π‘’βˆ—ξ¬=βŸ¨πΉπ‘’π‘›βˆ’πΉπ‘’βˆ—,π‘’π‘›βˆ’π‘’βˆ—βŸ©βˆ’βŸ¨π‘£π‘›βˆ’π‘£βˆ—,π‘’π‘›βˆ’π‘’βˆ—βŸ©+πœƒπ‘›β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2𝐻+πœƒπ‘›βŸ¨π‘’βˆ—βˆ’π‘’1,π‘’π‘›βˆ’π‘’βˆ—βŸ©,(3.6) and using (3.1), we obtain the following estimate: ‖‖𝑒𝑛+1βˆ’π‘’βˆ—β€–β€–2𝐻≀1βˆ’2πœ†π‘›πœƒπ‘›ξ€»β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2𝐻+πœ†2𝑛𝑀1+2πœ†π‘›βŸ¨π‘£π‘›βˆ’π‘£βˆ—,π‘’π‘›βˆ’π‘’βˆ—βŸ©βˆ’2πœ†π‘›πœƒπ‘›βŸ¨π‘’βˆ—βˆ’π‘’1,π‘’π‘›βˆ’π‘’βˆ—βŸ©.(3.7) Following the same argument, we also obtain ‖‖𝑣𝑛+1βˆ’π‘£βˆ—β€–β€–2𝐻≀1βˆ’2πœ†π‘›πœƒπ‘›ξ€»β€–β€–π‘£nβˆ’π‘£βˆ—β€–β€–2𝐻+πœ†2𝑛𝑀2βˆ’2πœ†π‘›βŸ¨π‘’π‘›βˆ’π‘’βˆ—,π‘£π‘›βˆ’π‘£βˆ—βŸ©βˆ’2πœ†π‘›πœƒπ‘›βŸ¨π‘£βˆ—βˆ’π‘£1,π‘£π‘›βˆ’π‘£βˆ—βŸ©.(3.8) Thus, we obtain ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–2𝐸≀1βˆ’2πœ†π‘›πœƒπ‘›ξ€»β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–2𝐸+π‘€πœ†2π‘›βˆ’2πœ†π‘›πœƒπ‘›βŸ¨π‘’βˆ—βˆ’π‘’1,π‘’π‘›βˆ’π‘’βˆ—βŸ©βˆ’2πœ†π‘›πœƒπ‘›βŸ¨π‘£βˆ—βˆ’π‘£1,π‘£π‘›βˆ’π‘£βˆ—βŸ©.(3.9) Using ‖‖𝑒0β‰€βˆ—βˆ’π‘’1+ξ€·π‘’π‘›βˆ’π‘’βˆ—ξ€Έβ€–β€–2𝐻=β€–β€–π‘’βˆ—βˆ’π‘’1β€–β€–2𝐻+2βŸ¨π‘’βˆ—βˆ’π‘’1,π‘’π‘›βˆ’π‘’βˆ—β€–β€–π‘’βŸ©+π‘›βˆ’π‘’βˆ—β€–β€–2𝐻,‖‖𝑣0β‰€βˆ—βˆ’π‘£1+ξ€·π‘£π‘›βˆ’π‘£βˆ—ξ€Έβ€–β€–2𝐻=β€–β€–π‘£βˆ—βˆ’π‘£1β€–β€–2𝐻+2βŸ¨π‘£βˆ—βˆ’π‘£1,π‘£π‘›βˆ’π‘£βˆ—β€–β€–π‘£βŸ©+π‘›βˆ’π‘£βˆ—β€–β€–2𝐻,(3.10) we have βˆ’2βŸ¨π‘’βˆ—βˆ’π‘’1,π‘’π‘›βˆ’π‘’βˆ—β€–β€–π‘’βŸ©β‰€βˆ—βˆ’π‘’1β€–β€–2𝐻+β€–β€–π‘’π‘›βˆ’π‘’βˆ—β€–β€–2𝐻,βˆ’2βŸ¨π‘£βˆ—βˆ’π‘£1,π‘£π‘›βˆ’π‘£βˆ—β€–β€–π‘£βŸ©β‰€βˆ—βˆ’π‘£1β€–β€–2𝐻+β€–β€–π‘£π‘›βˆ’π‘£βˆ—β€–β€–2𝐻.(3.11) Therefore ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–2𝐸≀1βˆ’2πœ†π‘›πœƒπ‘›ξ€»β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–2𝐸+π‘€πœ†2𝑛+πœ†π‘›πœƒπ‘›β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–2𝐸+πœ†π‘›πœƒπ‘›β€–β€–π‘€βˆ—βˆ’π‘€1β€–β€–2𝐸.(3.12) So we obtain the following estimate: ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–2𝐸≀1βˆ’πœ†π‘›πœƒπ‘›ξ€»β€–β€–π‘€π‘›βˆ’π‘€βˆ—β€–β€–2𝐸+πœ†π‘›πœƒπ‘›β€–β€–π‘€βˆ—βˆ’π‘€1β€–β€–2𝐸+π‘€πœ†2𝑛.(3.13) Let 𝑑0=π‘Ÿ2/4𝑀. Then using the induction assumptions, the fact that 𝑀1∈𝐡(π‘€βˆ—,π‘Ÿ/2) and πœ†π‘›β‰€π‘‘0πœƒπ‘›, we obtain ‖‖𝑀𝑛+1βˆ’π‘€βˆ—β€–β€–2π»β‰€ξ‚Έπœ†1βˆ’π‘›πœƒπ‘›4ξ‚Ήπ‘Ÿ2<π‘Ÿ2,(3.14) a contradiction. Therefore, 𝑀𝑛+1∈𝐡. Thus by induction, {𝑀𝑛} is bounded and so are {𝑒𝑛} and {𝑣𝑛}.Step 2. We show that there exists a unique sequence 𝑧𝑛=(π‘₯𝑛,𝑦𝑛)∈𝐸 such that πœƒπ‘›ξ€·π‘₯π‘›βˆ’π‘’1ξ€Έ+𝐹π‘₯π‘›βˆ’π‘¦π‘›πœƒ=0,(3.15)π‘›ξ€·π‘¦π‘›βˆ’π‘£1ξ€Έ+𝐾𝑦𝑛+π‘₯𝑛=0,(3.16) and π‘₯𝑛→π‘₯βˆ—, π‘¦π‘›β†’π‘¦βˆ—, with π‘₯βˆ—+𝐾𝐹π‘₯βˆ—=0 and π‘¦βˆ—=𝐹π‘₯βˆ—.
In fact, let π‘‡βˆΆπΈβ†’πΈ be defined by 𝑇(𝑒,𝑣)=(πΉπ‘’βˆ’π‘£,𝐾𝑣+𝑒),for all (𝑒,𝑣)∈𝐸. Using the fact that 𝐹 and 𝐾 are monotone and satisfy the range condition, it follows from Lemma 2.3 that 𝑇 is monotone and also satisfies the range condition.
Applying Theorem 2.4, with 𝑑=1/πœƒπ‘› and π‘₯=(𝑒1,𝑣1), we obtain that lim𝑑→+βˆžπ½π‘‘π‘₯βˆˆπ‘‡βˆ’1(0) implies that lim𝑛→+βˆžξ‚΅1𝐼+πœƒπ‘›π‘‡ξ‚Άβˆ’1𝑒1,𝑣1ξ€Έβˆˆπ‘‡βˆ’1(0).(3.17) Set 𝑧𝑛=(π‘₯𝑛,𝑦𝑛)∢=(𝐼+(1/πœƒπ‘›)𝑇)βˆ’1(𝑒1,𝑣1). Then (𝐼+(1/πœƒπ‘›)𝑇)(π‘₯𝑛,𝑦𝑛)=(𝑒1,𝑣1), for all 𝑛β‰₯1. So we have, π‘₯𝑛+1πœƒπ‘›ξ€·πΉπ‘₯π‘›βˆ’π‘¦π‘›ξ€Έ=𝑒1,𝑦𝑛+1πœƒπ‘›ξ€·πΎπ‘¦π‘›+π‘₯𝑛=𝑣1.(3.18) Therefore, πœƒπ‘›ξ€·π‘₯π‘›βˆ’π‘’1ξ€Έ+𝐹π‘₯π‘›βˆ’π‘¦π‘›πœƒ=0,π‘›ξ€·π‘¦π‘›βˆ’π‘£1ξ€Έ+𝐾π‘₯𝑛+π‘₯𝑛=0.(3.19) Since 𝑇 is monotone and satisfies the range condition, then it is known that (𝐼+π‘Ÿπ‘‡) is bijective for every π‘Ÿ>0. So, the sequence {𝑧𝑛} is unique. Using (3.17) and Theorem 2.4, we have, limπ‘§π‘›βˆˆπ‘‡βˆ’1(0). Let π‘₯𝑛→π‘₯βˆ— and π‘¦π‘›β†’π‘¦βˆ—. Then (π‘₯βˆ—,π‘¦βˆ—)βˆˆπ‘‡βˆ’1(0). So, 𝑇(π‘₯βˆ—,π‘¦βˆ—)=0, that is, 𝐹π‘₯βˆ—βˆ’π‘¦βˆ—=0,πΎπ‘¦βˆ—+π‘₯βˆ—=0.(3.20) Therefore, π‘¦βˆ—=𝐹π‘₯βˆ— and π‘₯βˆ—+𝐾𝐹π‘₯βˆ—=0.
Step 3. We show that {𝑀𝑛}β†’(π‘’βˆ—,π‘£βˆ—), where π‘’βˆ—+πΎπΉπ‘’βˆ—=0 and π‘£βˆ—=πΉπ‘’βˆ—.Claim 1. 𝑀𝑛+1βˆ’π‘§π‘›β†’0 as π‘›β†’βˆž. We compute as follows: ‖‖𝑀𝑛+1βˆ’π‘§π‘›β€–β€–2𝐸=‖‖𝑒𝑛+1βˆ’π‘₯𝑛‖‖2𝐻+‖‖𝑣𝑛+1βˆ’π‘¦π‘›β€–β€–2𝐻.(3.21) We have ‖‖𝑒𝑛+1βˆ’π‘₯𝑛‖‖2𝐻=β€–β€–π‘’π‘›βˆ’π‘₯π‘›βˆ’πœ†π‘›ξ€·πΉπ‘’π‘›βˆ’π‘£π‘›+πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1β€–β€–ξ€Έξ€Έ2𝐻=β€–β€–π‘’π‘›βˆ’π‘₯𝑛‖‖2π»βˆ’2πœ†π‘›ξ«πΉπ‘’π‘›βˆ’π‘£π‘›+πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1ξ€Έ,π‘’π‘›βˆ’π‘₯𝑛+πœ†2π‘›β€–β€–πΉπ‘’π‘›βˆ’π‘£π‘›+πœƒπ‘›ξ€·π‘’π‘›βˆ’π‘’1ξ€Έβ€–β€–2𝐻.(3.22) From the boundness of {𝑒𝑛}, {𝑣𝑛}, and 𝐹, there exists 𝑀3>0 such that β€–πΉπ‘’π‘›βˆ’π‘£π‘›+πœƒπ‘›(π‘’π‘›βˆ’π‘’1)β€–2𝐻≀𝑀3. Using (3.15) and the fact that 𝐹 is monotone, we obtain ‖‖𝑒𝑛+1βˆ’π‘₯𝑛‖‖2𝐻≀1βˆ’πœ†π‘›πœƒπ‘›ξ€Έβ€–β€–π‘’π‘›βˆ’π‘₯𝑛‖‖2π»βˆ’2πœ†π‘›βŸ¨πΉπ‘₯π‘›βˆ’π‘£π‘›,π‘’π‘›βˆ’π‘₯π‘›βŸ©βˆ’2πœ†π‘›βŸ¨π‘¦π‘›βˆ’πΉπ‘₯𝑛,π‘’π‘›βˆ’π‘₯π‘›βŸ©+𝑀3πœ†2𝑛=ξ€·1βˆ’πœ†π‘›πœƒπ‘›ξ€Έβ€–β€–π‘’π‘›βˆ’π‘₯𝑛‖‖2π»βˆ’2πœ†π‘›βŸ¨π‘¦π‘›βˆ’π‘£π‘›,π‘’π‘›βˆ’π‘₯π‘›βŸ©+𝑀3πœ†2𝑛,(3.23) for some constant 𝑀3>0. Using (3.16) and similar arguments, we obtain: ‖‖𝑣𝑛+1βˆ’π‘£π‘›β€–β€–2𝐻≀1βˆ’πœ†π‘›πœƒπ‘›ξ€Έβ€–β€–π‘£π‘›βˆ’π‘¦π‘›β€–β€–2π»βˆ’2πœ†π‘›βŸ¨πΎπ‘¦π‘›+𝑒𝑛,π‘£π‘›βˆ’π‘¦π‘›βŸ©+2πœ†π‘›βŸ¨π‘₯𝑛+𝐾𝑦𝑛,π‘£π‘›βˆ’π‘¦π‘›βŸ©+𝑀4πœ†2𝑛=ξ€·1βˆ’πœ†π‘›πœƒπ‘›ξ€Έβ€–β€–π‘£π‘›βˆ’π‘¦π‘›β€–β€–2𝐻+2πœ†π‘›βŸ¨π‘₯π‘›βˆ’π‘’π‘›,π‘£π‘›βˆ’π‘¦π‘›βŸ©+𝑀4πœ†2𝑛,(3.24) for some constant 𝑀4>0. Therefore, we have the following estimate: ‖‖𝑀𝑛+1βˆ’z𝑛‖‖2𝐸≀1βˆ’πœ†π‘›πœƒπ‘›ξ€Έβ€–β€–π‘€π‘›βˆ’π‘§π‘›β€–β€–2𝐸+π‘€β€²πœ†2𝑛,where𝑀′=𝑀3+𝑀4.(3.25) On the other hand, using the monotonicity of 𝐹 and 𝐾 we have β€–β€–π‘§π‘›βˆ’1βˆ’π‘§π‘›β€–β€–2𝐸≀‖‖π‘₯π‘›βˆ’1βˆ’π‘₯𝑛+πœƒπ‘›βˆ’1𝐹π‘₯π‘›βˆ’1βˆ’π‘¦π‘›βˆ’1βˆ’πΉπ‘₯𝑛+𝑦𝑛‖‖2𝐻+β€–β€–π‘¦π‘›βˆ’1βˆ’π‘¦π‘›+πœƒπ‘›βˆ’1ξ€·πΎπ‘¦π‘›βˆ’1+π‘₯π‘›βˆ’1βˆ’πΎπ‘¦π‘›βˆ’π‘₯𝑛‖‖2𝐻.(3.26) Using (3.15) and (3.16), we observe that π‘₯π‘›βˆ’1βˆ’π‘₯𝑛+1πœƒπ‘›ξ€·πΉπ‘₯π‘›βˆ’1βˆ’π‘¦π‘›βˆ’1βˆ’πΉπ‘₯𝑛+𝑦𝑛=πœƒπ‘›βˆ’πœƒπ‘›βˆ’1πœƒπ‘›ξ€·π‘₯π‘›βˆ’1βˆ’π‘’1ξ€Έ,π‘¦π‘›βˆ’1βˆ’π‘¦π‘›+1πœƒπ‘›ξ€·πΎπ‘¦π‘›βˆ’1+π‘₯π‘›βˆ’1βˆ’πΎπ‘¦π‘›βˆ’π‘₯𝑛=πœƒπ‘›βˆ’πœƒπ‘›βˆ’1πœƒπ‘›ξ€·π‘¦π‘›βˆ’1βˆ’π‘£1ξ€Έ.(3.27) Therefore, β€–β€–π‘§π‘›βˆ’1βˆ’π‘§π‘›β€–β€–πΈβ‰€πœƒnβˆ’1βˆ’πœƒπ‘›πœƒπ‘›β€–β€–π‘§π‘›βˆ’1βˆ’π‘€1‖‖𝐸.(3.28) Using (3.25) and the boundness of {π‘₯𝑛} and {𝑦𝑛}, we obtain that there exists 𝐢>0 such that: ‖‖𝑀𝑛+1βˆ’π‘§π‘›β€–β€–2𝐸≀1βˆ’πœ†π‘›πœƒπ‘›ξ€Έβ€–β€–π‘€π‘›βˆ’π‘§π‘›βˆ’1β€–β€–2π»ξ‚΅πœƒ+πΆπ‘›βˆ’1βˆ’πœƒπ‘›πœƒπ‘›ξ‚Ά+π‘€ξ…žπœ†2𝑛.(3.29) Thus, by Lemma 2.1, 𝑀𝑛+1βˆ’π‘§π‘›β†’0. Since 𝑧𝑛→(π‘₯βˆ—,π‘¦βˆ—), we obtain that 𝑀𝑛→(π‘₯βˆ—,π‘¦βˆ—). But since 𝑀𝑛=(𝑒𝑛,𝑣𝑛), this implies that π‘’π‘›β†’π‘’βˆ— and π‘£π‘›β†’π‘£βˆ—. This completes the proof.

Corollary 3.2. Let 𝐻 be a real Hilbert space and 𝐹,πΎβˆΆπ»β†’π» be maps with 𝐷(𝐾)=𝐷(𝐹)=𝐻 such that the following conditions hold:(i)𝐹 and 𝐾 are Lipschitz and monotone,(ii)𝐹 and 𝐾 satisfy the range condition. 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,(3.30) 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.

Let 𝑋 be a real Banach space with dual space π‘‹βˆ— and let π΄βˆΆπ‘‹β†’π‘‹βˆ— be a monotone linear operator. The mapping 𝐴 is said to be angle-bounded with constant 𝛼β‰₯0 if||||(𝐴π‘₯,𝑦)βˆ’(𝐴𝑦,π‘₯)≀2𝛼(𝐴π‘₯,π‘₯)1/2(A𝑦,𝑦)1/2,βˆ€π‘₯,π‘¦βˆˆπ·(𝐴),(3.31) where (β‹…,β‹…) denotes the duality pairing between elements of π‘‹βˆ— and those of 𝑋. The class of angle-bounded operators is a subclass of the class of monotone operators. The angle-boundness of 𝐴 with 𝛼=0 corresponds to the symmetry of 𝐴, that is,(𝐴π‘₯,𝑦)=(𝐴𝑦,π‘₯),βˆ€π‘₯,π‘¦βˆˆπ·(𝐴).(3.32) (See Pascali and Sburlan [5, Chapter IV, page 189]).

Let 𝐻 be a separable real Hilbert space and 𝐢 be a closed subspace of 𝐻. For a given π‘“βˆˆπΆ, consider the Hammerstein equation:(𝐼+𝐾𝐹)𝑒=𝑓,(3.33) and its 𝑛th Galerkin approximation given by𝐼+𝐾𝑛𝐹𝑛𝑒𝑛=π‘ƒβˆ—π‘“,(3.34) 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] for the meaning of the symbols). Under this setting, BrΓ©zis and Browder (see [9]) proved the following approximation theorem.

Theorem BB. Let 𝐻 be a separable real Hilbert space. 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. Then, for each π‘›βˆˆβ„•, the Galerkin approximation (3.34) admits a unique solution 𝑒𝑛 in 𝐢𝑛 and {𝑒𝑛} converges strongly in 𝐻 to the unique solution π‘’βˆˆπΆ of the (3.33).

Remark 3.3. Theorem BB is the 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 Banach space.

Remark 3.4. The class of mappings considered in our theorem (Theorem 3.1) is larger than that considered in Theorem BB. In particular, in Theorem BB, in addition to assuming that the operator 𝐾 is bounded and monotone, the authors also required 𝐾 to be continuous. Furthermore, the operator 𝐹 is restricted to the class of angle-bounded operators (a subclass of the monotone operators) and is also assumed to be weakly compact. In Theorem 3.1, the operators 𝐾 and 𝐹 are only assumed to be bounded and monotone and satisfy the range condition. We remark that continuity of the monotone map 𝐾 implies that 𝐾 is π‘š-accretive (see Martin [21]) and it is known that π‘š-accretive implies range condition.

Remark 3.5. Theorem BB guarantees the existence of a sequence {𝑒𝑛} which converges strongly to a solution of the Hammerstein equation (3.33). Our theorem provides an iterative sequence which converges strongly to a solution of (3.33).

Remark 3.6. Real sequences that satisfy the hypotheses of Theorem 3.1 are πœ†π‘›=(𝑛+1)βˆ’π‘Ž and πœƒπ‘›=(𝑛+1)βˆ’π‘ with 0<𝑏<π‘Ž and π‘Ž+𝑏<1.

We verify that these choices satisfy, in particular, condition (3) of Theorem 3.1. In fact, using the fact that (1+π‘₯)𝑝≀1+𝑝π‘₯, for π‘₯>βˆ’1 and 0<𝑝<1, we haveπœƒ0β‰€ξ€·ξ€·π‘›βˆ’1/πœƒπ‘›ξ€Έξ€Έβˆ’1πœ†π‘›πœƒπ‘›=ξ‚Έξ‚€11+π‘›ξ‚π‘ξ‚Ήβˆ’1β‹…(𝑛+1)π‘Ž+𝑏≀𝑏⋅(𝑛+1)π‘Ž+𝑏𝑛=𝑏⋅𝑛+1𝑛⋅1(𝑛+1)1βˆ’(π‘Ž+𝑏)⟢0,(3.35) as π‘›β†’βˆž.

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