Abstract

Let E be a reflexive real Banach space with uniformly Gâteaux differentiable norm and F, K : 𝐸→𝐸 be Lipschitz accretive maps with 𝐷(𝐾)=𝑅(𝐹)=𝐸. Suppose that the Hammerstein equation 𝑢+𝐾𝐹𝑢=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 𝑆∶={𝑥∈𝐸∶‖𝑥‖=1}. 𝐸 is said to have a Gâteaux differentiable norm if the limit lim𝑡→0+‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(1.1) exists for each 𝑥,𝑦∈𝑆. The normed space 𝐸 is said to have a uniformly Gâteaux differentiable norm if for each 𝑦∈𝑆 the limit is attained uniformly for 𝑥∈𝑆. Furthermore, 𝐸 is said to be uniformly smooth if the limit exists uniformly for (𝑥,𝑦)∈𝑆×𝑆.

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,∞)→[0,∞) defined by 𝜌𝐸(𝑡)∶=sup‖𝑥+𝑦‖+‖𝑥−𝑦‖2−1∶‖𝑥‖=1,‖𝑦‖=𝑡.(1.2) In terms of the modulus of smoothness, the space 𝐸 is called uniformly smooth if and only if lim𝑡→0+(𝜌𝐸(𝑡)/𝑡)=0.𝐸 is called ğ‘ž-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌𝐸(𝑡)â‰¤ğ‘ğ‘¡ğ‘ž,𝑡>0. 𝐿𝑝 (and ℓ𝑝) spaces, 1<𝑝<+∞ are ğ‘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<ğ‘ž<+∞, every ğ‘ž-uniformly smooth real Banach space is uniformly smooth and thus has a uniformly Gâteaux differentiable norm.

Let 𝐸 be a real normed linear space with dual 𝐸∗. For ğ‘ž>1, we denote by ğ½ğ‘ž the generalized duality mapping from 𝐸 to 2𝐸∗ defined by ğ½ğ‘žî€½ğ‘“(𝑥)∶=∗∈𝐸∗∶⟨𝑥,𝑓∗⟩=‖𝑥‖⋅‖𝑓∗‖,‖𝑓∗‖=â€–ğ‘¥â€–ğ‘žâˆ’1,(1.3) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. 𝐽2 is denoted by 𝐽. If 𝐸∗ is strictly convex, then ğ½ğ‘ž is single-valued. We note that ğ½ğ‘ž(𝑥)=â€–ğ‘¥â€–ğ‘žâˆ’2𝐽(𝑥), for 𝑥≠0. It is known that if 𝐸 is a real Banach space with uniform Gâteaux 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 𝑘>0 such that for every 𝑥,𝑦∈𝐷(𝐺), there exists ğ‘—ğ‘ž(𝑥−𝑦)âˆˆğ½ğ‘ž(𝑥−𝑦) such that 𝐺𝑥−𝐺𝑦,ğ‘—ğ‘žî¬(𝑥−𝑦)â‰¥ğ‘˜â€–ğ‘¥âˆ’ğ‘¦â€–ğ‘ž.(1.4) If 𝑘=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 ğ‘¢î…ž(𝑡)+𝐴𝑢(𝑡)=0,𝑢(0)=𝑢0,(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 Schrödinger equations. If in (1.5), 𝑢(𝑡) is independent of 𝑡, then (1.5) reduces to 𝐴𝑢=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 𝑚-accretive that is, 𝑅(𝐼+𝐴)=𝐸. Clearly, a consequence of this is that the following equation 𝑢+𝐴𝑢=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: 𝑢(𝑥)+Ω𝜅(𝑥,𝑦)𝑓(𝑦,𝑢(𝑦))𝑑𝑦=ℎ(𝑥),(1.8) where 𝑑𝑦 is a ğœŽ-finite measure on the measure space Ω; the real kernel 𝜅 is defined on Ω×Ω,  𝑓 is a real-valued function defined on Ω×ℝ and is, in general, nonlinear, and ℎ is a given function on Ω. If we now define an operator 𝐾 by 𝐾𝑣(𝑥)∶=Ω𝜅(𝑥,𝑦)𝑣(𝑦)𝑑𝑦;𝑥∈Ω,(1.9) and the so-called superposition or Nemytskii operator by 𝐹𝑢(𝑦)∶=𝑓(𝑦,𝑢(𝑦)) then, the integral equation (1.8) can be put in operator theoretic form as follows: 𝑢+𝐾𝐹𝑢=0,(1.10) where without loss of generality, we have taken ℎ≡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(𝑡)𝑑𝑡2+ğ‘Ž2[],sin𝑣(𝑡)=𝑧(𝑡),𝑡∈0,1𝑣(0)=𝑣(1)=0,(1.11) where the driving force 𝑧 is odd. The constant ğ‘Ž, (ğ‘Žâ‰ 0) depends on the length of the pendulum and gravity. Since Green's function of the problem: ğ‘£î…žî…ž(𝑡)=0,𝑣(0)=𝑣(1)=0(1.12) is given by: 𝑘(𝑡,𝑠)∶=𝑡(1−𝑠),0≤𝑡≤𝑠≤1,𝑠(1−𝑡),0≤𝑠≤𝑡≤1,(1.13) it follows that problem (1.11) is equivalent to the nonlinear integral equation 𝑣(𝑡)=−10𝑘(𝑡,𝑠)𝑧(𝑠)âˆ’ğ‘Ž2[].sin𝑣(𝑠)𝑑𝑠,𝑡∈0,1(1.14) If we set 𝑔(𝑡)∶=10[],𝑘(𝑡,𝑠)𝑧(𝑠)𝑑𝑠,𝑢(𝑡)∶=𝑣(𝑡)+𝑔(𝑡),𝑡∈0,1(1.15) then 𝑣=𝑢−𝑔(1.16) and (1.14) can be written as 𝑢(𝑡)+10𝑘(𝑡,𝑠)ğ‘Ž2sin(𝑔(𝑠)−𝑢(𝑠))𝑑𝑠=0,(1.17) which is in the Hammerstein equation form: 𝑢(𝑡)+10𝑘(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠=0,(1.18) where 𝑓(𝑡,𝑠)=ğ‘Ž2sin(𝑔(𝑡)−𝑠),𝑡,𝑠∈[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., Brézis 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 𝐾−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, Brézis and Browder [7] proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.10) (see also Brézis and Browder [9]). In fact, they proved the following Theorem.

Theorem BB (Brézis and Browder). Let 𝐻 be a separable real Hilbert space and 𝐶 be a closed subspace of 𝐻. Let 𝐾∶𝐻→𝐶 be a bounded continuous monotone operator and 𝐹∶𝐶→𝐻 be an angle-bounded and weakly compact mapping. For a given 𝑓∈𝐶, consider the Hammerstein equation (𝐼+𝐾𝐹)𝑢=𝑓,(1.19) and its 𝑛th Galerkin approximation given by 𝐼+𝐾𝑛𝐹𝑛𝑢𝑛=𝑃∗𝑓,(1.20) where 𝐾𝑛=𝑃∗𝑛𝐾𝑃𝑛∶𝐻→𝐶𝑛 and 𝐹𝑛=𝑃𝑛𝐹𝑃∗𝑛∶𝐶𝑛→𝐻, where the symbols have their usual meanings (see, [5]). Then, for each 𝑛∈ℕ, the Galerkin approximation (1.20) admits a unique solution 𝑢𝑛 in 𝐶𝑛 and {𝑢𝑛} converges strongly in 𝐻 to the unique solution 𝑢∈𝐶 of (1.19).

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

We observe that the Galerkin method of Brézis 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,𝐾∶𝐻→𝐻 be bounded and maximal monotone operators. Let {𝑢𝑛} and {𝑣𝑛} be sequences in 𝐻 defined iteratively from arbitrary points 𝑢1,𝑣1∈𝐻 as follows: 𝑢𝑛+1=𝑢𝑛−𝜆𝑛𝐹𝑢𝑛−𝑣𝑛−𝜆𝑛𝜃𝑛𝑢𝑛−𝑢1𝑣,𝑛≥1,𝑛+1=𝑣𝑛−𝜆𝑛𝐾𝑣𝑛+𝑢𝑛−𝜆𝑛𝜃𝑛𝑣𝑛−𝑣1,𝑛≥1,(1.21) where {𝜆𝑛} and {𝜃𝑛} are sequences in (0,1) satisfying the following conditions: (1)lim𝜃𝑛=0, (2)âˆ‘âˆžğ‘›=1𝜆𝑛𝜃𝑛=∞,𝜆𝑛=𝑜(𝜃𝑛),(3)limğ‘›â†’âˆž(((𝜃𝑛−1)/𝜃𝑛)−1)/𝜆𝑛𝜃𝑛=0. Suppose that 𝑢+𝐾𝐹𝑢=0 has a solution in 𝐻. Then, there exists a constant 𝑑0>0 such that if 𝜆𝑛≤𝑑0𝜃𝑛 for all 𝑛≥𝑛0 for some 𝑛0≥1, then the sequence {𝑢𝑛} converges to 𝑢∗, a solution of 𝑢+𝐾𝐹𝑢=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 Gâteaux 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 𝑢+𝐾𝐹𝑢=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 {𝜆𝑛} be a sequence of nonnegative numbers and {𝛼𝑛}⊆(0,1) a sequence such that âˆ‘âˆžğ‘›=1𝛼𝑛=∞. Let the recursive inequality 𝜆𝑛+1≤𝜆𝑛−2𝛼𝑛𝜓𝜆𝑛+1+ğœŽğ‘›,𝑛=1,2,…,(2.1) be given where 𝜓∶[0,∞)→[0,∞) is a strictly increasing function such that it is positive on (0,∞), 𝜓(0)=0. If ğœŽğ‘›=𝑜(𝛼𝑛). Then 𝜆𝑛→0, as ğ‘›â†’âˆž.

Lemma 2.2. There exists a map ğœ‡ğ‘›âˆ¶â„“âˆž(ℕ)→ℝ such that (1)𝜇𝑛 is linear; that is, 𝜇𝑛(𝑥+𝑦)=𝜇𝑛(𝑥)+𝜇𝑛(𝑦) and 𝜇𝑛(𝑐⋅𝑥)=𝑐⋅𝜇𝑛(𝑥),(2)𝜇𝑛 is positive: 𝜇𝑛(𝑥)≥0 for every ğ‘¥âˆˆâ„“âˆž(ℕ) with 𝑥𝑛≥0,  for  all  𝑛,(3)𝜇𝑛 is normalized: 𝜇𝑛(𝑒)=1, where 𝑒=(1,1,…),(4)𝜇𝑛 is shift invariant: 𝜇𝑛(𝑆𝑥)=𝜇𝑛(𝑥), for all ğ‘¥âˆˆâ„“âˆž(ℕ), where ğ‘†âˆ¶â„“âˆž(ℕ)→ℓ∞(ℕ) is the shift operator defined by 𝑥𝑆∶1,𝑥2,𝑥3⟼𝑥,…2,𝑥3,…,(2.2) the above properties on the functional 𝜇𝑛 imply the following: (5)𝜇𝑛 has norm one; thus, |𝜇𝑛(𝑥)|≤‖𝑥‖, for  allâ€‰â€‰ğ‘¥âˆˆâ„“âˆž(ℕ), (6)𝜇𝑛 extends lim on the subspace of the convergent sequences: limğ‘›â†’âˆžî€·ğ‘¥ğ‘›î€¸=𝑐⟹𝜇𝑛𝑥(𝑥)=𝑐,where𝑥=𝑛,(2.3)(7)for any 𝑥=(𝑥𝑛)∈ℓ∞(ℕ): liminfğ‘›â†’âˆžğ‘¥ğ‘›â‰¤ğœ‡ğ‘›(𝑥)≤limsupğ‘›â†’âˆžğ‘¥ğ‘›.(2.4)

Remark 2.3. Functions 𝜇𝑛 as above are called Banach Limits.

Lemma 2.4 (see [24, 25]). Let 𝑥=(𝑥𝑛)∈ℓ∞ be such that 𝜇𝑛(𝑥)≤0 for all Banach limits 𝜇𝑛. If limsup𝑛(𝑥𝑛+1−𝑥𝑛)≤0, then limsup𝑛𝑥𝑛≤0.

Lemma 2.5. Let 𝐸 be a real normed linear space. Then, the following inequality holds: ‖𝑥+𝑦‖2≤‖𝑥‖2+2⟨𝑦,𝑗(𝑥+𝑦)⟩∀𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦),∀𝑥,𝑦∈𝐸.(2.5)

3. Main Results

We now prove the following theorems.

Theorem 3.1. Let 𝐸 be a real Banach space and 𝐹,𝐾∶𝐸→𝐸 be Lipschitz and accretive maps with 𝐷(𝐾)=𝑅(𝐹)=𝐸 and Lipschitz constants 𝐿1 and 𝐿2, respectively. Let {𝑢𝑛} and {𝑣𝑛} be sequences in 𝐸 defined iteratively from arbitrary points 𝑢1,𝑣1∈𝐸 as follows: 𝑢𝑛+1=𝑢𝑛−𝜆𝑛𝛼𝑛𝐹𝑢𝑛−𝑣𝑛−𝜆𝑛𝜃𝑛𝑢𝑛−𝑢1𝑣,𝑛≥1,𝑛+1=𝑣𝑛−𝜆𝑛𝛼𝑛𝐾𝑣𝑛+𝑢𝑛−𝜆𝑛𝜃𝑛𝑣𝑛−𝑣1,𝑛≥1,(3.1) where {𝜆𝑛}, {𝛼𝑛}, and {𝜃𝑛} are sequences in (0,1) satisfying the following conditions: (1)lim𝜆𝑛=0, lim𝛼𝑛=0, (2)𝜆𝑛=𝑜(𝜃𝑛),𝛼𝑛=𝑜(𝜃𝑛). Suppose that 𝑢+𝐾𝐹𝑢=0 has a solution in 𝐸. Then the sequences {𝑢𝑛} and {𝑣𝑛} are bounded.

Proof. Let 𝑋∶=𝐸×𝐸 with the norm ‖𝑧‖𝑋=(‖𝑢‖2𝐸+‖𝑣‖2𝐸)1/2, where 𝑧=(𝑢,𝑣). Define the sequence {𝑤𝑛} in 𝑋 by: 𝑤𝑛∶=(𝑢𝑛,𝑣𝑛). Let 𝑢∗∈𝐸 be a solution of 𝑢+𝐾𝐹𝑢=0, set 𝑣∗∶=𝐹𝑢∗ and 𝑤∗∶=(𝑢∗,𝑣∗). We observe that 𝑢∗=−𝐾𝑣∗. We show that the sequence {𝑤𝑛} is bounded in 𝑋.
For this, define 𝐿𝐿∶=max1,𝐿2,𝛾0√∶=2−18.(𝐿+1)(𝐿+2)(3.2) Since 𝜆𝑛=𝑜(𝜃𝑛) and 𝛼𝑛=𝑜(𝜃𝑛), there exists 𝑁∈ℕ such that 𝛼𝑛/𝜃𝑛<𝛾0 and 𝜆𝑛/𝜃𝑛<𝛾0 for all 𝑛≥𝑁. Let 𝑟>0 sufficiently large such that 𝑤1∈𝐵(𝑤∗,𝑟/2) and 𝑤𝑁∈𝐵(𝑤∗,𝑟). Define 𝐵∶=𝐵(𝑤∗,𝑟).
Claim 1. 𝑤𝑛 is in 𝐵 for all 𝑛≥𝑁. The proof is by induction. By construction 𝑤𝑁∈𝐵. Suppose that 𝑤𝑛∈𝐵 for some 𝑛≥𝑁. We prove that 𝑤𝑛+1∈𝐵. Assume for contradiction that 𝑤𝑛+1∉𝐵. Then we have ‖𝑤𝑛+1−𝑤∗‖𝐸>𝑟. We compute as follows: ‖‖𝑤𝑛+1−𝑤∗‖‖2𝑋=‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸+‖‖𝑣𝑛+1−𝑣∗‖‖2𝐸.(3.3) Using Lemma 2.5, We have ‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸=‖‖𝑢𝑛−𝑢∗−𝜆𝑛𝛼𝑛𝐹𝑢𝑛−𝑣𝑛+𝜃𝑛𝑢𝑛−𝑢1‖‖2𝐸≤‖‖𝑢𝑛−𝑢∗‖‖2𝐸−2𝜆𝑛𝛼𝑛𝐹𝑢𝑛−𝑣𝑛+𝜃𝑛𝑢𝑛−𝑢1𝑢,𝑗𝑛+1−𝑢∗.(3.4) Observing that 𝛼𝑛𝐹𝑢𝑛−𝑣𝑛𝑢,𝑗𝑛+1−𝑢∗=𝛼𝑛𝐹𝑢𝑛+1−𝐹𝑢∗𝑢,𝑗𝑛+1−𝑢∗+𝛼𝑛𝐹𝑢𝑛−𝐹𝑢𝑛+1+𝛼𝑛𝑣∗−𝑣𝑛𝑢,𝑗𝑛+1−𝑢∗,𝜃𝑛𝑢𝑛−𝑢1𝑢,𝑗𝑛+1−𝑢∗=𝜃𝑛𝑢𝑛+1−𝑢∗𝑢,𝑗𝑛+1−𝑢∗+𝜃𝑛𝑢𝑛−𝑢𝑛+1+𝜃𝑛𝑢∗−𝑢1𝑢,𝑗𝑛+1−𝑢∗,(3.5) and using the fact that 𝐹 is accretive, we obtain the following estimate: ‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸≤‖‖𝑢𝑛−𝑢∗‖‖2𝐸−2𝜆𝑛𝜃𝑛‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸+2𝜆𝑛𝛼𝑛𝐹𝑢𝑛+1−𝐹𝑢𝑛+𝛼𝑛𝑣𝑛−𝑣∗𝑢,𝑗𝑛+1−𝑢∗+2𝜆𝑛𝜃𝑛𝑢𝑛+1−𝑢𝑛+𝜃𝑛𝑢1−𝑢∗𝑢,𝑗𝑛+1−𝑢∗.(3.6) Using Schwartz's inequality and the fact that 𝐹 is Lipschitz, it follows that: ‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸≤‖‖𝑢𝑛−𝑢∗‖‖2𝐸−2𝜆𝑛𝜃𝑛‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸+2𝜆𝑛𝛼𝑛𝐿1‖‖𝑢𝑛+1−𝑢𝑛‖‖𝐸+𝛼𝑛‖‖𝑣𝑛−𝑣∗‖‖𝐸‖‖𝑢𝑛+1−𝑢∗‖‖𝐸+2𝜆𝑛𝜃𝑛‖‖𝑢𝑛+1−𝑢𝑛‖‖𝐸+𝜃𝑛‖‖𝑢1−𝑢∗‖‖𝐸‖‖𝑢𝑛+1−𝑢∗‖‖𝐸.(3.7) Thus, we obtain ‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸≤‖‖𝑢𝑛−𝑢∗‖‖2𝐸−2𝜆𝑛𝜃𝑛‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸+2𝜆𝑛𝛼𝑛𝐿1+𝜃𝑛‖‖𝑢𝑛+1−𝑢𝑛‖‖𝐸‖‖𝑢𝑛+1−𝑢∗‖‖𝐸+2𝜆𝑛𝛼𝑛‖‖𝑣𝑛−𝑣∗‖‖𝐸+𝜃𝑛‖‖𝑢1−𝑢∗‖‖𝐸‖‖𝑢𝑛+1−𝑢∗‖‖𝐸.(3.8) Using the fact that ‖‖𝑢𝑛+1−𝑢𝑛‖‖𝐸=‖‖𝜆𝑛𝛼𝑛𝐹𝑢𝑛−𝑣𝑛+𝜆𝑛𝜃𝑛𝑢𝑛−𝑢1‖‖𝐸≤𝜆𝑛𝛼𝑛𝐿1‖‖𝑢𝑛−𝑢∗‖‖𝐸+𝜆𝑛𝛼𝑛‖‖𝑣𝑛−𝑣∗‖‖𝐸+𝜆𝑛𝜃𝑛‖‖𝑢𝑛−𝑢1‖‖𝐸=𝜆𝑛𝛼𝑛𝐿1‖‖𝑢𝑛−𝑢∗‖‖𝐸+𝛼𝑛‖‖𝑣𝑛−𝑣∗‖‖𝐸+𝜃𝑛‖‖𝑢𝑛−𝑢1‖‖𝐸,(3.9) we have ‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸≤‖‖𝑢𝑛−𝑢∗‖‖2𝐸−2𝜆𝑛𝜃𝑛‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸+2𝜆𝑛𝛼𝑛𝐿1+𝜃𝑛𝜆𝑛𝛼𝑛𝐿1‖‖𝑢𝑛−𝑢∗‖‖𝐸‖‖𝑢𝑛+1−𝑢∗‖‖𝐸+2𝜆𝑛𝛼𝑛𝐿1+𝜃𝑛𝜆𝑛𝛼𝑛‖‖𝑣𝑛−𝑣∗‖‖𝐸‖‖𝑢𝑛+1−𝑢∗‖‖𝐸+2𝜆𝑛𝛼𝑛𝐿1+𝜃𝑛𝜆𝑛𝜃𝑛‖‖𝑢𝑛−𝑢1‖‖𝐸‖‖𝑢𝑛+1−𝑢∗‖‖𝐸+2𝜆𝑛𝛼𝑛‖‖𝑣𝑛−𝑣∗‖‖𝐸+𝜃𝑛‖‖𝑢1−𝑢∗‖‖𝐸‖‖𝑢𝑛+1−𝑢∗‖‖𝐸.(3.10) Following the same argument, we also obtain ‖‖𝑣𝑛+1−𝑣∗‖‖2𝐸≤‖‖𝑣𝑛−𝑣∗‖‖2𝐸−2𝜆𝑛𝜃𝑛‖‖𝑣𝑛+1−𝑣∗‖‖2𝐸+2𝜆𝑛𝛼𝑛𝐿2+𝜃𝑛𝜆𝑛𝛼𝑛𝐿2‖‖𝑣𝑛−𝑣∗‖‖𝐸‖‖𝑣𝑛+1−𝑣∗‖‖𝐸+2𝜆𝑛𝛼𝑛𝐿2+𝜃𝑛𝜆𝑛𝛼𝑛‖‖𝑢𝑛−𝑢∗‖‖𝐸‖‖𝑣𝑛+1−𝑣∗‖‖𝐸+2𝜆𝑛𝛼𝑛𝐿2+𝜃𝑛𝜆𝑛𝜃𝑛‖‖𝑣𝑛−𝑣1‖‖𝐸‖‖𝑣𝑛+1−𝑣∗‖‖𝐸+2𝜆𝑛𝛼𝑛‖‖𝑢𝑛−𝑢∗‖‖𝐸+𝜃𝑛‖‖𝑣1−𝑣∗‖‖𝐸‖‖𝑣𝑛+1−𝑣∗‖‖𝐸.(3.11) Using the fact that ‖𝑢‖𝐸+‖𝑣‖𝐸≤√2‖𝑤‖𝑋,∀𝑤=(𝑢,𝑣)∈𝑋,(3.12) we have ‖‖𝑤𝑛+1−𝑤∗‖‖2𝑋≤‖‖𝑤𝑛−𝑤∗‖‖2𝑋−2𝜆𝑛𝜃𝑛‖‖𝑤𝑛+1−𝑤∗‖‖2𝑋+2𝜆𝑛√2𝛼𝑛𝐿+𝜃𝑛𝜆𝑛𝛼𝑛𝐿‖‖𝑤𝑛−𝑤∗‖‖𝑋‖‖𝑤𝑛+1−𝑤∗‖‖𝑋+2𝜆𝑛√2𝛼𝑛𝐿+𝜃𝑛𝜆𝑛𝛼𝑛‖‖𝑤𝑛−𝑤∗‖‖𝑋‖‖𝑤𝑛+1−𝑤∗‖‖𝑋+2𝜆𝑛√2𝛼𝑛𝐿+𝜃𝑛𝜆𝑛𝜃𝑛‖‖𝑤𝑛−𝑤1‖‖𝑋‖‖𝑤𝑛+1−𝑤∗‖‖𝑋+2𝜆𝑛√2𝛼𝑛‖‖𝑤𝑛−𝑤∗‖‖𝑋+√2𝜃𝑛‖‖𝑤1−𝑤∗‖‖𝑋‖‖𝑤𝑛+1−𝑤∗‖‖𝑋,(3.13) which implies that ‖‖𝑤𝑛+1−𝑤∗‖‖2𝑋≤‖‖𝑤𝑛−𝑤∗‖‖2𝑋−2𝜆𝑛𝜃𝑛‖‖𝑤𝑛+1−𝑤∗‖‖2𝑋+2𝜆𝑛2√2𝛼𝑛𝐿+𝜃𝑛𝜆𝑛𝛼𝑛𝐿‖‖𝑤𝑛−𝑤∗‖‖𝑋‖‖𝑤𝑛+1−𝑤∗‖‖𝑋+2𝜆𝑛√2𝛼𝑛𝐿+𝜃𝑛𝜆𝑛𝜃𝑛‖‖𝑤𝑛−𝑤1‖‖𝑋‖‖𝑤𝑛+1−𝑤∗‖‖𝑋+2𝜆𝑛√2𝛼𝑛‖‖𝑤𝑛−𝑤∗‖‖𝑋+√2𝜃𝑛‖‖𝑤1−𝑤∗‖‖𝑋‖‖𝑤𝑛+1−𝑤∗‖‖𝑋.(3.14) Since 𝑤𝑛∈𝐵 and 𝑤1∈𝐵(𝑤∗,𝑟/2), we have ‖‖𝑤𝑛+1−𝑤∗‖‖2𝑋≤‖‖𝑤𝑛−𝑤∗‖‖2𝑋−2𝜆𝑛𝜃𝑛‖‖𝑤𝑛+1−𝑤∗‖‖2𝑋+2𝜆𝑛2√2𝛼𝑛𝐿+𝜃𝑛𝜆𝑛𝛼𝑛𝐿𝑟+2𝜃𝑛𝑟‖‖𝑤𝑛+1−𝑤∗‖‖𝑋+2𝜆𝑛2√2𝛼𝑛√𝑟+22𝜃𝑛𝑟‖‖𝑤𝑛+1−𝑤∗‖‖𝑋.(3.15) By assumption, ‖𝑤𝑛+1−𝑤∗‖2𝑋>|𝑤𝑛−𝑤∗‖2𝑋. So we have 𝜃𝑛‖‖𝑤𝑛+1−𝑤∗‖‖𝑋√≤22𝛼𝑛𝐿+𝜃𝑛𝜆𝑛𝛼𝑛𝐿+2𝜃𝑛√𝑟+22𝛼𝑛√𝑟+22𝜃𝑛𝑟.(3.16) Hence ‖‖𝑤𝑛+1−𝑤∗‖‖𝑋√≤22𝜆𝑛𝜃𝑛√(𝐿+1)(𝐿+2)𝑟+22𝛼𝑛𝜃𝑛√𝑟+22𝑟.(3.17) Therefore, using the fact that 𝜆𝑛/𝜃𝑛<𝛾0 and 𝛼𝑛/𝜃𝑛<𝛾0, it follows that ‖𝑤𝑛+1−𝑤∗‖𝑋<𝑟. A contradiction. So 𝑤𝑛+1∈𝐵. This prove the boundedness of the sequences {𝑢𝑛} and {𝑣𝑛}.

Theorem 3.2. Let 𝐸 be reflexive real Banach space with uniformly Gâteaux differentiable norm and 𝐹,𝐾∶𝐸→𝐸 be Lipschitz accretive mappings. Let {𝑢𝑛}𝑛≥1 and {𝑣𝑛}𝑛≥1 be sequences in 𝐸 defined iteratively from arbitrary 𝑢1,𝑣1∈𝐸 by 𝑢𝑛+1=𝑢𝑛−𝜆𝑛𝛼𝑛𝐹𝑢𝑛−𝑣𝑛−𝜆𝑛𝜃𝑛𝑢𝑛−𝑢1𝑣,𝑛≥1,𝑛+1=𝑣𝑛−𝜆𝑛𝛼𝑛𝐾𝑣𝑛+𝑢𝑛−𝜆𝑛𝜃𝑛𝑣𝑛−𝑣1,𝑛≥1,(3.18) where {𝜆𝑛}𝑛≥1, {𝛼𝑛}𝑛≥1, and {𝜃𝑛}𝑛≥1 are real sequences in (0,1) such that 𝜆𝑛=𝑜(𝜃𝑛),𝛼𝑛=𝑜(𝜃𝑛), and âˆ‘âˆžğ‘›=1𝜆𝑛𝜃𝑛=+∞. Suppose that the equation 𝑢+𝐾𝐹𝑢=0 has a solution 𝑢∗. Then there exists a subset 𝐾min of 𝐸×𝐸 such that if (𝑢∗,𝑣∗)∈𝐾min with 𝑣∗=𝐹𝑢∗, then the sequence {𝑢𝑛}𝑛≥1 converges strongly to 𝑢∗.

Proof. Since, by Theorem 3.1, the sequences {𝑢𝑛}𝑛≥1 and {𝑣𝑛}𝑛≥1 are bounded and 𝐹 is bounded, there exists an 𝑅>0 such that 𝑢𝑛∈𝐵1∶=𝐵(𝑢∗,𝑅) and 𝑣𝑛∈𝐵2∶=𝐵(𝑣∗,𝑅) for all 𝑛≥1. Furthermore, the sets 𝐵1 and 𝐵2 are nonempty closed convex and bounded subsets of 𝐸. Let 𝜇𝑛 be a Banach limit. Define the maps 𝜑1∶𝐸→ℝ and 𝜑2∶𝐸→ℝ by 𝜑1(𝑢)∶=𝜇𝑛‖𝑢𝑛−𝑢‖2,𝜑2(𝑣)∶=𝜇𝑛‖𝑣𝑛−𝑣‖2.(3.19) Then for 𝑖=1,2,𝜑𝑖 is weakly lower semicontinuous. Since 𝐸 reflexive, then Argmin𝑢∈𝐵1𝜑1(𝑢)≠∅,Argmin𝑣∈𝐵2𝜑2(𝑣)≠∅.(3.20) Set 𝐾min∶=Argmin𝑢∈𝐵1𝜑1(𝑢)×Argmin𝑣∈𝐵2𝜑2(𝑣).(3.21) Now assume that 𝑤∗=(𝑢∗,𝑣∗)∈𝐾min and let 𝑡∈(0,1). Then, by convexity of 𝐵1, we have that (1−𝑡)𝑢∗+𝑡𝑢1∈𝐵1. Thus, 𝜑1(𝑢∗)≤𝜑1((1−𝑡)𝑢∗+𝑡𝑢1). It follows from Lemma 2.5 that ‖‖𝑢𝑛−𝑢∗𝑢−𝑡1−𝑢∗‖‖2≤‖‖𝑢𝑛−𝑢∗‖‖2𝑢−2𝑡1−𝑢∗𝑢,𝑗𝑛−𝑢∗𝑢−𝑡1−𝑢∗∀𝑛∈ℕ.(3.22) Thus, taking Banach limits gives 𝜇𝑛‖‖𝑢𝑛−𝑢∗𝑢−𝑡1−𝑢∗‖‖2≤𝜇𝑛‖‖𝑢𝑛−𝑢∗‖‖2−2𝑡𝜇𝑛𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗𝑢−𝑡1−𝑢∗.(3.23) That is, 𝜑1(1−𝑡)𝑢∗+𝑡𝑢1≤𝜑1𝑢∗−2𝑡𝜇𝑛𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗𝑢−𝑡1−𝑢∗.(3.24) This implies that 𝜇𝑛(⟨𝑢1−𝑢∗,𝑗(𝑢𝑛−𝑢∗−𝑡(𝑢1−𝑢∗))⟩)≤0forall𝑛∈ℕ. Furthermore, the fact that 𝐸 has a uniformly Gâteaux differentiable norm gives, as 𝑡→0, that 𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗−𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗𝑢−𝑡1−𝑢∗⟶0.(3.25) Thus, for all 𝜖>0, there exists 𝛿𝜖>0 such that for all 𝑡∈]0,𝛿𝜖[ and for all 𝑛∈ℕ, 𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗−𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗𝑢−𝑡1−𝑢∗<𝜖.(3.26) Thus, 𝜇𝑛𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗−𝜇𝑛𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗𝑢−𝑡1−𝑢∗≤𝜖,(3.27) which implies that 𝜇𝑛𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗≤0.(3.28) Using the same argument, we have 𝜇𝑛𝑣1−𝑣∗𝑣,𝑗𝑛−𝑣∗≤0.(3.29) Moreover, since {𝑢𝑛}𝑛≥1,{𝑣𝑛}𝑛≥1, {𝐹𝑢𝑛}𝑛≥1, and {𝐾𝑣𝑛}𝑛≥1 are all bounded, then, from (3.18) and that there exist positive constants 𝑀1 and 𝑀2 such that ‖‖𝑢𝑛+1−𝑢𝑛‖‖≤𝜆𝑛𝛼𝑛‖‖𝐹𝑢𝑛‖‖+‖‖𝑣𝑛‖‖+𝜃𝑛‖‖𝑢𝑛−𝑢1‖‖≤𝜆𝑛𝑀1,‖‖𝑣𝑛+1−𝑣𝑛‖‖≤𝜆𝑛𝛼𝑛‖‖𝐾𝑣𝑛‖‖+‖‖𝑢𝑛‖‖+𝜃𝑛‖‖𝑣𝑛−𝑣1‖‖≤𝜆𝑛𝑀2.(3.30) Thus, limğ‘›â†’âˆžâ€–ğ‘¢n+1−𝑢𝑛‖=0 and limğ‘›â†’âˆžâ€–ğ‘£ğ‘›+1−𝑣𝑛‖=0.
Again, using the fact that 𝐸 has a uniformly Gâteaux differentiable norm, we obtain that limğ‘›â†’âˆžğ‘¢î€·î«1−𝑢∗𝑢,𝑗𝑛+1−𝑢∗−𝑢1−𝑢∗𝑢,𝑗𝑛−𝑢∗=0,limğ‘›â†’âˆžğ‘£î€·î«1−𝑣∗𝑣,𝑗𝑛+1−𝑣∗−𝑣1−𝑣∗𝑣,𝑗𝑛−𝑣∗=0.(3.31) Therefore, the sequences {⟨𝑢1−𝑢∗,𝑗(𝑢𝑛−𝑢∗)⟩}𝑛≥1 and {⟨𝑣1−𝑣∗,𝑗(𝑣𝑛−𝑣∗)⟩}𝑛≥1 satisfy the conditions of Lemma 2.4.
Hence, limsupğ‘›â†’âˆžî«ğ‘¢1−𝑢∗𝑢,𝑗𝑛−𝑢∗≤0,limsupğ‘›â†’âˆžî«ğ‘£1−𝑣∗𝑣,𝑗𝑛−𝑣∗≤0.(3.32) Define ğœŽğ‘›ğ‘¢âˆ¶=max1−𝑢∗𝑢,𝑗𝑛+1−𝑢∗,𝜉,0𝑛𝑣∶=max1−𝑣∗𝑣,𝑗𝑛+1−𝑣∗,,0(3.33) then limğ‘›â†’âˆžğœŽğ‘›=0=limğ‘›â†’âˆžğœ‰ğ‘›. Moreover, 𝑢1−𝑢∗𝑢,ğ‘—ğ‘›âˆ’ğ‘¢âˆ—î€¸î¬â‰¤ğœŽğ‘›,𝑣1−𝑣∗𝑣,𝑗𝑛−𝑣∗≤𝜉𝑛,∀𝑛∈ℕ.(3.34) Using Lemma 2.5, and (3.18), we have: ‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸≤‖‖𝑢𝑛−𝑢∗‖‖2𝐸−2𝜆𝑛𝜃𝑛‖‖𝑢𝑛+1−𝑢∗‖‖2𝐸+2𝜆𝑛𝛼𝑛𝐹𝑢𝑛+1−𝐹𝑢𝑛+𝛼𝑛𝑣𝑛−𝑣∗𝑢,𝑗𝑛+1−𝑢∗+2𝜆𝑛𝜃𝑛𝑢𝑛+1−𝑢𝑛+𝜃𝑛𝑢1−𝑢∗𝑢,𝑗𝑛+1−𝑢∗,‖‖𝑣𝑛+1−𝑣∗‖‖2𝐸≤‖‖𝑣𝑛−𝑣∗‖‖2𝐸−2𝜆𝑛𝜃𝑛‖‖𝑣𝑛+1−𝑣∗‖‖2𝐸+2𝜆𝑛𝛼𝑛𝐾𝑣𝑛+1−𝐾𝑣𝑛+𝛼𝑛𝑣∗−𝑣𝑛𝑣,𝑗𝑛+1−𝑣∗+2𝜆𝑛𝜃𝑛𝑣𝑛+1−𝑣𝑛+𝜃𝑛𝑣1−𝑣∗𝑣,𝑗𝑛+1−𝑣∗.(3.35) Using the fact that 𝐹 and 𝐾 are Lipschitz and the sequences {𝑢𝑛} and {𝑣𝑛} are bounded, we obtain the following estimate: ‖‖𝑤𝑛+1−𝑤∗‖‖2𝐸≤‖‖𝑤𝑛−𝑤∗‖‖2𝐸−2𝜆𝑛𝜃𝑛‖‖𝑤𝑛+1−𝑤∗‖‖2𝐸𝜆+2𝑛𝛼𝑛+𝜆2𝑛𝑐+2ğœ†ğ‘›ğœƒğ‘›î€·ğœŽğ‘›+𝜉𝑛=‖‖𝑤𝑛−𝑤∗‖‖2𝐸−2𝜆𝑛𝜃𝑛‖‖𝑤𝑛+1−𝑤∗‖‖2𝐸+𝛾𝑛,(3.36) where 𝛾𝑛=2(𝜆𝑛𝛼𝑛+𝜆2𝑛)𝑐+2𝜆𝑛𝜃𝑛(ğœŽğ‘›+𝜉𝑛)=𝑜(𝜆𝑛𝜃𝑛) for some 𝑐>0. Hence, by Lemma 2.1, 𝑤𝑛→𝑤∗ as ğ‘›â†’âˆž. But 𝑤𝑛=(𝑢𝑛,𝑣𝑛) and 𝑤∗=(𝑢∗,𝑣∗). This implies that 𝑢𝑛→𝑢∗ as ğ‘›â†’âˆž. This completes the proof.

Corollary 3.3. Let 𝐸 be a ğ‘ž-uniformly real Banach spaces, ğ‘ž>1 and 𝐹,𝐾∶𝐸→𝐸 be Lipschitz accretive mappings. Let {𝑢𝑛}𝑛≥1 and {𝑣𝑛}𝑛≥1 be sequences in 𝐸 defined iteratively from arbitrary 𝑢1,𝑣1∈𝐸 by 𝑢𝑛+1=𝑢𝑛−𝜆𝑛𝛼𝑛𝐹𝑢𝑛−𝑣𝑛−𝜆𝑛𝜃𝑛𝑢𝑛−𝑢1𝑣,𝑛≥1,𝑛+1=𝑣𝑛−𝜆𝑛𝛼𝑛𝐾𝑣𝑛+𝑢𝑛−𝜆𝑛𝜃𝑛𝑣𝑛−𝑣1,𝑛≥1,(3.37) where {𝜆𝑛}𝑛≥1, {𝛼𝑛}𝑛≥1, and {𝜃𝑛}𝑛≥1 are real sequences in (0,1) such that 𝜆𝑛=𝑜(𝜃𝑛),𝛼𝑛=𝑜(𝜃𝑛), and âˆ‘âˆžğ‘›=1𝜆𝑛𝜃𝑛=+∞. Suppose that the equation 𝑢+𝐾𝐹𝑢=0 has a solution 𝑢∗. Then there exists a set 𝐾min in 𝐸×𝐸 such that if (𝑢∗,𝑣∗)∈𝐾min with 𝑣∗=𝐹𝑢∗, then the sequence {𝑢𝑛}𝑛≥1 converges strongly to 𝑢∗.

Corollary 3.4. Let 𝐸=𝐿𝑝 (or ℓ𝑝) space (1<𝑝<+∞) and 𝐹,𝐾∶𝐸→𝐸 be Lipschitz accretive mappings. Let {𝑢𝑛}𝑛≥1 and {𝑣𝑛}𝑛≥1 be sequences in 𝐸 defined iteratively from arbitrary 𝑢1,𝑣1∈𝐸 by 𝑢𝑛+1=𝑢𝑛−𝜆𝑛𝛼𝑛𝐹𝑢𝑛−𝑣𝑛−𝜆𝑛𝜃𝑛𝑢𝑛−𝑢1𝑣,𝑛≥1,𝑛+1=𝑣𝑛−𝜆𝑛𝛼𝑛𝐾𝑣𝑛+𝑢𝑛−𝜆𝑛𝜃𝑛𝑣𝑛−𝑣1,𝑛≥1,(3.38) where {𝜆𝑛}𝑛≥1, {𝛼𝑛}𝑛≥1, and {𝜃𝑛}𝑛≥1 are real sequences in (0,1) such that 𝜆𝑛=𝑜(𝜃𝑛),𝛼𝑛=𝑜(𝜃𝑛), and âˆ‘âˆžğ‘›=1𝜆𝑛𝜃𝑛=+∞. Suppose that the equation 𝑢+𝐾𝐹𝑢=0 has a solution 𝑢∗. Then there exists a subset 𝐾min of 𝐸×𝐸 such that if (𝑢∗,𝑣∗)∈𝐾min with 𝑣∗=𝐹𝑢∗, then the sequence {𝑢𝑛}𝑛≥1 converges strongly to 𝑢∗.

Corollary 3.5. Let H be a real Hilbert space and 𝐹,𝐾∶𝐻→𝐻 be Lipschitz monotone mappings. Let {𝑢𝑛}𝑛≥1 and {𝑣𝑛}𝑛≥1 be sequences in 𝐻 defined iteratively from arbitrary 𝑢1,𝑣1∈𝐻 by 𝑢𝑛+1=𝑢𝑛−𝜆𝑛𝛼𝑛𝐹𝑢𝑛−𝑣𝑛−𝜆𝑛𝜃𝑛𝑢𝑛−𝑢1𝑣,𝑛≥1,𝑛+1=𝑣𝑛−𝜆𝑛𝛼𝑛𝐾𝑣𝑛+𝑢𝑛−𝜆𝑛𝜃𝑛𝑣𝑛−𝑣1,𝑛≥1,(3.39) where {𝜆𝑛}𝑛≥1, {𝛼𝑛}𝑛≥1, and {𝜃𝑛}𝑛≥1 are real sequences in (0,1) such that 𝜆𝑛=𝑜(𝜃𝑛),𝛼𝑛=𝑜(𝜃𝑛), and âˆ‘âˆžğ‘›=1𝜆𝑛𝜃𝑛=+∞. Suppose that the equation 𝑢+𝐾𝐹u=0 has a solution 𝑢∗. Then there exists a subset 𝐾min of 𝐸×𝐸 such that if (𝑢∗,𝑣∗)∈𝐾min with 𝑣∗=𝐹𝑢∗, then the sequence {𝑢𝑛}𝑛≥1 converges strongly to 𝑢∗.

Remark 3.6. Real sequences that satisfy the hypotheses of Theorems 3.1 are 𝜆𝑛=𝛼𝑛=ğ‘›âˆ’ğ‘Ž and 𝜃𝑛=𝑛−𝑏, 𝑛≥1 with 0<𝑏<ğ‘Ž and ğ‘Ž+𝑏<1.