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

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 lim_𝑡→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 𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(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 lim_𝑡→0⁺(𝜌_𝐸(𝑡)/𝑡)=0.𝐸 is called 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌_𝐸(𝑡)≤𝑐𝑡^𝑞,𝑡>0. 𝐿_𝑝 (and ℓ_𝑝) spaces, 1<𝑝<+∞ are 𝑞-uniformly smooth. In particular, 𝐿_𝑝𝑖𝑠2-uniformly smooth if 2≤𝑝<+∞ and 𝑝-uniformly smooth if 1<𝑝<2. It is easy to see that for 1<𝑞<+∞, every 𝑞-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 𝑞>1, we denote by 𝐽_𝑞 the generalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽_𝑞(𝑥)∶=𝑓^∗∈𝐸^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖⋅‖𝑓^∗‖,‖𝑓^∗‖=‖𝑥‖^𝑞−1,(1.3) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. 𝐽₂ is denoted by 𝐽. If 𝐸^∗ is strictly convex, then 𝐽_𝑞 is single-valued. We note that 𝐽_𝑞(𝑥)=‖𝑥‖^𝑞−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 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝐺𝑥−𝐺𝑦,𝑗_𝑞(𝑥−𝑦)≥𝑘‖𝑥−𝑦‖^𝑞.(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 𝑢^(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 𝜎-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+𝑎²sin𝑣𝑣((0𝑡))==𝑣𝑧((𝑡1)),=𝑡0,∈[0,1],(1.11) where the driving force 𝑧 is odd. The constant 𝑎, (𝑎≠0) depends on the length of the pendulum and gravity. Since Green's function of the problem: 𝑣^(𝑡)=0,𝑣(0)=𝑣(1)=0(1.12) is given by: 𝑘(𝑡,𝑠)∶=𝑡𝑠((11−−𝑠𝑡)),,00≤≤𝑡𝑠≤≤𝑠𝑡≤≤11,,(1.13) it follows that problem (1.11) is equivalent to the nonlinear integral equation 𝑣(𝑡)=−¹⁰𝑘(𝑡,𝑠)𝑧(𝑠)−𝑎²sin𝑣(𝑠)𝑑𝑠,𝑡∈[0,1].(1.14) If we set 𝑔(𝑡)∶=¹⁰𝑘(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)𝑎²sin(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)=𝑎²sin(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+𝐾_𝑛𝐹_𝑛𝑢_𝑛=𝑃^∗𝑓,(1.20) where 𝐾_𝑛=𝑃^∗𝑛𝐾𝑃_𝑛∶𝐻→𝐶_𝑛 and 𝐹_𝑛=𝑃_𝑛𝐹𝑃^∗𝑛∶𝐶_𝑛→𝐻, where the symbols have their usual meanings (see, [5]). Then, for each 𝑛∈ℕ, the Galerkin approximation (1.20) admits a unique solution 𝑢_𝑛 in 𝐶_𝑛 and {𝑢_𝑛} 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 {𝑢_𝑛} and {𝑣_𝑛} be sequences in 𝐻 defined iteratively from arbitrary points 𝑢₁,𝑣₁∈𝐻 as follows: 𝑣𝑢_𝑛+1=𝑢_𝑛−𝜆_𝑛𝐹𝑢_𝑛−𝑣_𝑛−𝜆_𝑛𝜃_𝑛𝑢_𝑛−𝑢₁,𝑛≥1,_𝑛+1=𝑣_𝑛−𝜆_𝑛𝐾𝑣_𝑛+𝑢_𝑛−𝜆_𝑛𝜃_𝑛𝑣_𝑛−𝑣₁,𝑛≥1,(1.21) where {𝜆_𝑛} and {𝜃_𝑛} are sequences in (0,1) satisfying the following conditions: (1)lim𝜃_𝑛=0, (2)∑^∞𝑛=1𝜆_𝑛𝜃_𝑛=∞,𝜆_𝑛=𝑜(𝜃_𝑛),(3)lim_𝑛→∞(((𝜃_𝑛−1)/𝜃_𝑛)−1)/𝜆_𝑛𝜃_𝑛=0. Suppose that 𝑢+𝐾𝐹𝑢=0 has a solution in 𝐻. Then, there exists a constant 𝑑₀>0 such that if 𝜆_𝑛≤𝑑₀𝜃_𝑛 for all 𝑛≥𝑛₀ for some 𝑛₀≥1, then the sequence {𝑢_𝑛} converges to 𝑢^∗, a solution of 𝑢+𝐾𝐹𝑢=0.

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

It is our purpose in this paper to prove convergence theorems in reflexive real Banach spaces that include all 𝐿_𝑝 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 𝐿_𝑝 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 𝐿_𝑝 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 {𝜆_𝑛} be a sequence of nonnegative numbers and {𝛼_𝑛}⊆(0,1) a sequence such that ∑^∞𝑛=1𝛼_𝑛=∞. Let the recursive inequality 𝜆_𝑛+1≤𝜆_𝑛−2𝛼_𝑛𝜓𝜆_𝑛+1+𝜎_𝑛,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 𝜎_𝑛=𝑜(𝛼_𝑛). Then 𝜆_𝑛→0, as 𝑛→∞.

Lemma 2.2. There exists a map 𝜇_𝑛∶ℓ^∞(ℕ)→ℝ such that (1)𝜇_𝑛 is linear; that is, 𝜇_𝑛(𝑥+𝑦)=𝜇_𝑛(𝑥)+𝜇_𝑛(𝑦) and 𝜇_𝑛(𝑐⋅𝑥)=𝑐⋅𝜇_𝑛(𝑥),(2)𝜇_𝑛 is positive: 𝜇_𝑛(𝑥)≥0 for every 𝑥∈ℓ^∞(ℕ) with 𝑥_𝑛≥0,  for  all  𝑛,(3)𝜇_𝑛 is normalized: 𝜇_𝑛(𝑒)=1, where 𝑒=(1,1,…),(4)𝜇_𝑛 is shift invariant: 𝜇_𝑛(𝑆𝑥)=𝜇_𝑛(𝑥), for all 𝑥∈ℓ^∞(ℕ), where 𝑆∶ℓ^∞(ℕ)→ℓ^∞(ℕ) is the shift operator defined by 𝑆∶𝑥₁,𝑥₂,𝑥₃,…⟼𝑥₂,𝑥₃,…,(2.2) the above properties on the functional 𝜇_𝑛 imply the following: (5)𝜇_𝑛 has norm one; thus, |𝜇_𝑛(𝑥)|≤‖𝑥‖, for  all  𝑥∈ℓ^∞(ℕ), (6)𝜇_𝑛 extends lim on the subspace of the convergent sequences: lim_𝑛→∞𝑥_𝑛=𝑐⟹𝜇_𝑛(𝑥)=𝑐,where𝑥=𝑥_𝑛,(2.3)(7)for any 𝑥=(𝑥_𝑛)∈ℓ^∞(ℕ): liminf_𝑛→∞𝑥_𝑛≤𝜇_𝑛(𝑥)≤limsup_𝑛→∞𝑥_𝑛.(2.4)

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

Lemma 2.4 (see [24, 25]). Let 𝑥=(𝑥_𝑛)∈ℓ_∞ be such that 𝜇_𝑛(𝑥)≤0 for all Banach limits 𝜇_𝑛. If limsup_𝑛(𝑥_𝑛+1−𝑥_𝑛)≤0, then limsup_𝑛𝑥_𝑛≤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 {𝑢_𝑛} and {𝑣_𝑛} be sequences in 𝐸 defined iteratively from arbitrary points 𝑢₁,𝑣₁∈𝐸 as follows: 𝑣𝑢_𝑛+1=𝑢_𝑛−𝜆_𝑛𝛼_𝑛𝐹𝑢_𝑛−𝑣_𝑛−𝜆_𝑛𝜃_𝑛𝑢_𝑛−𝑢₁,𝑛≥1,_𝑛+1=𝑣_𝑛−𝜆_𝑛𝛼_𝑛𝐾𝑣_𝑛+𝑢_𝑛−𝜆_𝑛𝜃_𝑛𝑣_𝑛−𝑣₁,𝑛≥1,(3.1) where {𝜆_𝑛}, {𝛼_𝑛}, and {𝜃_𝑛} are sequences in (0,1) satisfying the following conditions: (1)lim𝜆_𝑛=0, lim𝛼_𝑛=0, (2)𝜆_𝑛=𝑜(𝜃_𝑛),𝛼_𝑛=𝑜(𝜃_𝑛). Suppose that 𝑢+𝐾𝐹𝑢=0 has a solution in 𝐸. Then the sequences {𝑢_𝑛} and {𝑣_𝑛} 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 {𝑤_𝑛} in 𝑋 by: 𝑤_𝑛∶=(𝑢_𝑛,𝑣_𝑛). 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 {𝑤_𝑛} is bounded in 𝑋.
For this, define 𝐿∶=max𝐿₁,𝐿₂,𝛾₀∶=√2−18(𝐿+1)(𝐿+2).(3.2) Since 𝜆_𝑛=𝑜(𝜃_𝑛) and 𝛼_𝑛=𝑜(𝜃_𝑛), there exists 𝑁∈ℕ such that 𝛼_𝑛/𝜃_𝑛<𝛾₀ and 𝜆_𝑛/𝜃_𝑛<𝛾₀ for all 𝑛≥𝑁. Let 𝑟>0 sufficiently large such that 𝑤₁∈𝐵(𝑤^∗,𝑟/2) and 𝑤_𝑁∈𝐵(𝑤^∗,𝑟). Define 𝐵∶=𝐵(𝑤^∗,𝑟).
Claim 1. 𝑤_𝑛 is in 𝐵 for all 𝑛≥𝑁. The proof is by induction. By construction 𝑤_𝑁∈𝐵. Suppose that 𝑤_𝑛∈𝐵 for some 𝑛≥𝑁. We prove that 𝑤_𝑛+1∈𝐵. Assume for contradiction that 𝑤_𝑛+1∉𝐵. Then we have ‖𝑤_𝑛+1−𝑤^∗‖_𝐸>𝑟. We compute as follows: ‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝑋=‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸+‖‖𝑣_𝑛+1−𝑣^∗‖‖^2𝐸.(3.3) Using Lemma 2.5, We have ‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸=‖‖𝑢_𝑛−𝑢^∗−𝜆_𝑛𝛼_𝑛𝐹𝑢_𝑛−𝑣_𝑛+𝜃_𝑛𝑢_𝑛−𝑢₁‖‖^2𝐸≤‖‖𝑢_𝑛−𝑢^∗‖‖^2𝐸−2𝜆_𝑛𝛼_𝑛𝐹𝑢_𝑛−𝑣_𝑛+𝜃_𝑛𝑢_𝑛−𝑢₁,𝑗𝑢_𝑛+1−𝑢^∗.(3.4) Observing that 𝛼_𝑛𝐹𝑢_𝑛−𝑣_𝑛,𝑗𝑢_𝑛+1−𝑢^∗=+𝛼_𝑛𝛼𝐹𝑢_𝑛+1−𝐹𝑢^∗,𝑗𝑢_𝑛+1−𝑢^∗_𝑛𝜃𝐹𝑢_𝑛−𝐹𝑢_𝑛+1+𝛼_𝑛𝑣^∗−𝑣_𝑛,𝑗𝑢_𝑛+1−𝑢^∗_𝑛,𝑢_𝑛−𝑢₁,𝑗𝑢_𝑛+1−𝑢^∗=+𝜃_𝑛𝜃𝑢_𝑛+1−𝑢^∗,𝑗𝑢_𝑛+1−𝑢^∗_𝑛𝑢_𝑛−𝑢_𝑛+1+𝜃_𝑛𝑢^∗−𝑢₁,𝑗𝑢_𝑛+1−𝑢^∗,(3.5) and using the fact that 𝐹 is accretive, we obtain the following estimate: ‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸≤‖‖𝑢_𝑛−𝑢^∗‖‖^2𝐸−2𝜆_𝑛𝜃_𝑛‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸++22𝜆𝜆_𝑛𝑛𝛼_𝑛𝐹𝑢_𝑛+1−𝐹𝑢_𝑛+𝛼_𝑛𝑣_𝑛−𝑣^∗,𝑗𝑢_𝑛+1−𝑢^∗𝜃_𝑛𝑢_𝑛+1−𝑢_𝑛+𝜃_𝑛𝑢₁−𝑢^∗,𝑗𝑢_𝑛+1−𝑢^∗.(3.6) Using Schwartz's inequality and the fact that 𝐹 is Lipschitz, it follows that: ‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸≤‖‖𝑢_𝑛−𝑢^∗‖‖^2𝐸−2𝜆_𝑛𝜃_𝑛‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸+2𝜆_𝑛𝛼_𝑛𝐿₁‖‖𝑢_𝑛+1−𝑢_𝑛‖‖_𝐸+𝛼_𝑛‖‖𝑣_𝑛−𝑣^∗‖‖_𝐸‖‖𝑢_𝑛+1−𝑢^∗‖‖_𝐸+2𝜆_𝑛𝜃_𝑛‖‖𝑢_𝑛+1−𝑢_𝑛‖‖_𝐸+𝜃_𝑛‖‖𝑢₁−𝑢^∗‖‖_𝐸‖‖𝑢_𝑛+1−𝑢^∗‖‖_𝐸.(3.7) Thus, we obtain ‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸≤‖‖𝑢_𝑛−𝑢^∗‖‖^2𝐸−2𝜆_𝑛𝜃_𝑛‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸+2𝜆_𝑛𝛼_𝑛𝐿₁+𝜃_𝑛‖‖𝑢_𝑛+1−𝑢_𝑛‖‖_𝐸‖‖𝑢_𝑛+1−𝑢^∗‖‖_𝐸+2𝜆_𝑛𝛼_𝑛‖‖𝑣_𝑛−𝑣^∗‖‖_𝐸+𝜃_𝑛‖‖𝑢₁−𝑢^∗‖‖_𝐸‖‖𝑢_𝑛+1−𝑢^∗‖‖_𝐸.(3.8) Using the fact that ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖_𝐸=‖‖𝜆_𝑛𝛼_𝑛𝐹𝑢_𝑛−𝑣_𝑛+𝜆_𝑛𝜃_𝑛𝑢_𝑛−𝑢₁‖‖_𝐸≤𝜆_𝑛𝛼_𝑛𝐿₁‖‖𝑢_𝑛−𝑢^∗‖‖_𝐸+𝜆_𝑛𝛼_𝑛‖‖𝑣_𝑛−𝑣^∗‖‖_𝐸+𝜆_𝑛𝜃_𝑛‖‖𝑢_𝑛−𝑢₁‖‖_𝐸=𝜆_𝑛𝛼_𝑛𝐿₁‖‖𝑢_𝑛−𝑢^∗‖‖_𝐸+𝛼_𝑛‖‖𝑣_𝑛−𝑣^∗‖‖_𝐸+𝜃_𝑛‖‖𝑢_𝑛−𝑢₁‖‖_𝐸,(3.9) we have ‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸≤‖‖𝑢_𝑛−𝑢^∗‖‖^2𝐸−2𝜆_𝑛𝜃_𝑛‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸+2𝜆_𝑛𝛼_𝑛𝐿₁+𝜃_𝑛𝜆_𝑛𝛼_𝑛𝐿₁‖‖𝑢_𝑛−𝑢^∗‖‖_𝐸‖‖𝑢_𝑛+1−𝑢^∗‖‖_𝐸+2𝜆_𝑛𝛼_𝑛𝐿₁+𝜃_𝑛𝜆_𝑛𝛼_𝑛‖‖𝑣_𝑛−𝑣^∗‖‖_𝐸‖‖𝑢_𝑛+1−𝑢^∗‖‖_𝐸+2𝜆_𝑛𝛼_𝑛𝐿₁+𝜃_𝑛𝜆_𝑛𝜃_𝑛‖‖𝑢_𝑛−𝑢₁‖‖_𝐸‖‖𝑢_𝑛+1−𝑢^∗‖‖_𝐸+2𝜆_𝑛𝛼_𝑛‖‖𝑣_𝑛−𝑣^∗‖‖_𝐸+𝜃_𝑛‖‖𝑢₁−𝑢^∗‖‖_𝐸‖‖𝑢_𝑛+1−𝑢^∗‖‖_𝐸.(3.10) Following the same argument, we also obtain ‖‖𝑣_𝑛+1−𝑣^∗‖‖^2𝐸≤‖‖𝑣_𝑛−𝑣^∗‖‖^2𝐸−2𝜆_𝑛𝜃_𝑛‖‖𝑣_𝑛+1−𝑣^∗‖‖^2𝐸+2𝜆_𝑛𝛼_𝑛𝐿₂+𝜃_𝑛𝜆_𝑛𝛼_𝑛𝐿₂‖‖𝑣_𝑛−𝑣^∗‖‖_𝐸‖‖𝑣_𝑛+1−𝑣^∗‖‖_𝐸+2𝜆_𝑛𝛼_𝑛𝐿₂+𝜃_𝑛𝜆_𝑛𝛼_𝑛‖‖𝑢_𝑛−𝑢^∗‖‖_𝐸‖‖𝑣_𝑛+1−𝑣^∗‖‖_𝐸+2𝜆_𝑛𝛼_𝑛𝐿₂+𝜃_𝑛𝜆_𝑛𝜃_𝑛‖‖𝑣_𝑛−𝑣₁‖‖_𝐸‖‖𝑣_𝑛+1−𝑣^∗‖‖_𝐸+2𝜆_𝑛𝛼_𝑛‖‖𝑢_𝑛−𝑢^∗‖‖_𝐸+𝜃_𝑛‖‖𝑣₁−𝑣^∗‖‖_𝐸‖‖𝑣_𝑛+1−𝑣^∗‖‖_𝐸.(3.11) Using the fact that ‖𝑢‖_𝐸+‖𝑣‖_𝐸≤√2‖𝑤‖_𝑋,∀𝑤=(𝑢,𝑣)∈𝑋,(3.12) we have ‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝑋≤‖‖𝑤_𝑛−𝑤^∗‖‖^2𝑋−2𝜆_𝑛𝜃_𝑛‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝑋+2𝜆_𝑛√2𝛼_𝑛𝐿+𝜃_𝑛𝜆_𝑛𝛼_𝑛𝐿‖‖𝑤_𝑛−𝑤^∗‖‖_𝑋‖‖𝑤_𝑛+1−𝑤^∗‖‖_𝑋+2𝜆_𝑛√2𝛼_𝑛𝐿+𝜃_𝑛𝜆_𝑛𝛼_𝑛‖‖𝑤_𝑛−𝑤^∗‖‖_𝑋‖‖𝑤_𝑛+1−𝑤^∗‖‖_𝑋+2𝜆_𝑛√2𝛼_𝑛𝐿+𝜃_𝑛𝜆_𝑛𝜃_𝑛‖‖𝑤_𝑛−𝑤₁‖‖_𝑋‖‖𝑤_𝑛+1−𝑤^∗‖‖_𝑋+2𝜆_𝑛√2𝛼_𝑛‖‖𝑤_𝑛−𝑤^∗‖‖_𝑋+√2𝜃_𝑛‖‖𝑤₁−𝑤^∗‖‖_𝑋‖‖𝑤^𝑛+1−𝑤^∗‖‖_𝑋,(3.13) which implies that ‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝑋≤‖‖𝑤_𝑛−𝑤^∗‖‖^2𝑋−2𝜆_𝑛𝜃_𝑛‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝑋+2𝜆_𝑛2√2𝛼_𝑛𝐿+𝜃_𝑛𝜆_𝑛𝛼_𝑛𝐿‖‖𝑤_𝑛−𝑤^∗‖‖_𝑋‖‖𝑤_𝑛+1−𝑤^∗‖‖_𝑋+2𝜆_𝑛√2𝛼_𝑛𝐿+𝜃_𝑛𝜆_𝑛𝜃_𝑛‖‖𝑤_𝑛−𝑤₁‖‖_𝑋‖‖𝑤_𝑛+1−𝑤^∗‖‖_𝑋+2𝜆_𝑛√2𝛼_𝑛‖‖𝑤_𝑛−𝑤^∗‖‖_𝑋+√2𝜃_𝑛‖‖𝑤₁−𝑤^∗‖‖_𝑋‖‖𝑤^𝑛+1−𝑤^∗‖‖_𝑋.(3.14) Since 𝑤_𝑛∈𝐵 and 𝑤₁∈𝐵(𝑤^∗,𝑟/2), we have ‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝑋≤‖‖𝑤_𝑛−𝑤^∗‖‖^2𝑋−2𝜆_𝑛𝜃_𝑛‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝑋+2𝜆_𝑛2√2𝛼_𝑛𝐿+𝜃_𝑛𝜆_𝑛𝛼_𝑛𝐿𝑟+2𝜃_𝑛𝑟‖‖𝑤_𝑛+1−𝑤^∗‖‖_𝑋+2𝜆_𝑛2√2𝛼_𝑛𝑟+√22𝜃^𝑛𝑟‖‖𝑤_𝑛+1−𝑤^∗‖‖_𝑋.(3.15) By assumption, ‖𝑤_𝑛+1−𝑤^∗‖^2𝑋>|𝑤_𝑛−𝑤^∗‖^2𝑋. So we have 𝜃_𝑛‖‖𝑤_𝑛+1−𝑤^∗‖‖_𝑋≤2√2𝛼_𝑛𝐿+𝜃_𝑛𝜆_𝑛𝛼_𝑛𝐿+2𝜃_𝑛𝑟+2√2𝛼_𝑛𝑟+√22𝜃^𝑛𝑟.(3.16) Hence ‖‖𝑤_𝑛+1−𝑤^∗‖‖_𝑋≤2√2𝜆^𝑛𝜃_𝑛(𝐿+1)(𝐿+2)𝑟+2√2𝛼^𝑛𝜃_𝑛𝑟+√22𝑟.(3.17) Therefore, using the fact that 𝜆_𝑛/𝜃_𝑛<𝛾₀ and 𝛼_𝑛/𝜃_𝑛<𝛾₀, it follows that ‖𝑤_𝑛+1−𝑤^∗‖_𝑋<𝑟. A contradiction. So 𝑤_𝑛+1∈𝐵. This prove the boundedness of the sequences {𝑢_𝑛} and {𝑣_𝑛}.

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 {𝑢_𝑛}_𝑛≥1 and {𝑣_𝑛}_𝑛≥1 be sequences in 𝐸 defined iteratively from arbitrary 𝑢₁,𝑣₁∈𝐸 by 𝑣𝑢_𝑛+1=𝑢_𝑛−𝜆_𝑛𝛼_𝑛𝐹𝑢_𝑛−𝑣_𝑛−𝜆_𝑛𝜃_𝑛𝑢_𝑛−𝑢₁,𝑛≥1,_𝑛+1=𝑣_𝑛−𝜆_𝑛𝛼_𝑛𝐾𝑣_𝑛+𝑢_𝑛−𝜆_𝑛𝜃_𝑛𝑣_𝑛−𝑣₁,𝑛≥1,(3.18) where {𝜆_𝑛}_𝑛≥1, {𝛼_𝑛}_𝑛≥1, and {𝜃_𝑛}_𝑛≥1 are real sequences in (0,1) such that 𝜆_𝑛=𝑜(𝜃_𝑛),𝛼_𝑛=𝑜(𝜃_𝑛), and ∑^∞𝑛=1𝜆_𝑛𝜃_𝑛=+∞. Suppose that the equation 𝑢+𝐾𝐹𝑢=0 has a solution 𝑢^∗. Then there exists a subset 𝐾_min of 𝐸×𝐸 such that if (𝑢^∗,𝑣^∗)∈𝐾_min with 𝑣^∗=𝐹𝑢^∗, then the sequence {𝑢_𝑛}_𝑛≥1 converges strongly to 𝑢^∗.

Proof. Since, by Theorem 3.1, the sequences {𝑢_𝑛}_𝑛≥1 and {𝑣_𝑛}_𝑛≥1 are bounded and 𝐹 is bounded, there exists an 𝑅>0 such that 𝑢_𝑛∈𝐵₁∶=𝐵(𝑢^∗,𝑅) and 𝑣_𝑛∈𝐵₂∶=𝐵(𝑣^∗,𝑅) for all 𝑛≥1. Furthermore, the sets 𝐵₁ and 𝐵₂ are nonempty closed convex and bounded subsets of 𝐸. Let 𝜇_𝑛 be a Banach limit. Define the maps 𝜑₁∶𝐸→ℝ and 𝜑₂∶𝐸→ℝ by 𝜑₁(𝑢)∶=𝜇_𝑛‖𝑢_𝑛−𝑢‖²,𝜑₂(𝑣)∶=𝜇_𝑛‖𝑣_𝑛−𝑣‖².(3.19) Then for 𝑖=1,2,𝜑_𝑖 is weakly lower semicontinuous. Since 𝐸 reflexive, then Argmin_𝑢∈𝐵1𝜑₁(𝑢)≠∅,Argmin_𝑣∈𝐵2𝜑₂(𝑣)≠∅.(3.20) Set 𝐾_min∶=Argmin_𝑢∈𝐵1𝜑₁(𝑢)×Argmin_𝑣∈𝐵2𝜑₂(𝑣).(3.21) Now assume that 𝑤^∗=(𝑢^∗,𝑣^∗)∈𝐾_min 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 ‖‖𝑢_𝑛−𝑢^∗−𝑡𝑢₁−𝑢^∗‖‖²≤‖‖𝑢_𝑛−𝑢^∗‖‖²−2𝑡𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗−𝑡𝑢₁−𝑢^∗∀𝑛∈ℕ.(3.22) Thus, taking Banach limits gives 𝜇_𝑛‖‖𝑢^𝑛−𝑢^∗−𝑡𝑢₁−𝑢^∗‖‖²≤𝜇^𝑛‖‖𝑢^𝑛−𝑢^∗‖‖²−2𝑡𝜇^𝑛𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗−𝑡𝑢₁−𝑢^∗.(3.23) That is, 𝜑₁(1−𝑡)𝑢^∗+𝑡𝑢₁≤𝜑₁𝑢^∗−2𝑡𝜇_𝑛𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗−𝑡𝑢₁−𝑢^∗.(3.24) This implies that 𝜇_𝑛(⟨𝑢₁−𝑢^∗,𝑗(𝑢_𝑛−𝑢^∗−𝑡(𝑢₁−𝑢^∗))⟩)≤0forall𝑛∈ℕ. Furthermore, the fact that 𝐸 has a uniformly Gâteaux differentiable norm gives, as 𝑡→0, that 𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗−𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗−𝑡𝑢₁−𝑢^∗⟶0.(3.25) Thus, for all 𝜖>0, there exists 𝛿_𝜖>0 such that for all 𝑡∈]0,𝛿_𝜖[ and for all 𝑛∈ℕ, 𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗−𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗−𝑡𝑢₁−𝑢^∗<𝜖.(3.26) Thus, 𝜇_𝑛𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗−𝜇_𝑛𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗−𝑡𝑢₁−𝑢^∗≤𝜖,(3.27) which implies that 𝜇_𝑛𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗≤0.(3.28) Using the same argument, we have 𝜇_𝑛𝑣₁−𝑣^∗,𝑗𝑣_𝑛−𝑣^∗≤0.(3.29) Moreover, since {𝑢_𝑛}_𝑛≥1,{𝑣_𝑛}_𝑛≥1, {𝐹𝑢_𝑛}_𝑛≥1, and {𝐾𝑣_𝑛}_𝑛≥1 are all bounded, then, from (3.18) and that there exist positive constants 𝑀₁ and 𝑀₂ such that ‖‖‖‖𝑣𝑢_𝑛+1𝑛+1−𝑢_𝑛‖‖≤𝜆_𝑛𝛼_𝑛‖‖𝐹𝑢_𝑛‖‖+‖‖𝑣_𝑛‖‖+𝜃_𝑛‖‖𝑢_𝑛−𝑢₁‖‖≤𝜆_𝑛𝑀₁,−𝑣_𝑛‖‖≤𝜆_𝑛𝛼_𝑛‖‖𝐾𝑣_𝑛‖‖+‖‖𝑢_𝑛‖‖+𝜃_𝑛‖‖𝑣_𝑛−𝑣₁‖‖≤𝜆_𝑛𝑀₂.(3.30) Thus, lim_𝑛→∞‖𝑢_n+1−𝑢_𝑛‖=0 and lim_𝑛→∞‖𝑣_𝑛+1−𝑣_𝑛‖=0.
Again, using the fact that 𝐸 has a uniformly Gâteaux differentiable norm, we obtain that lliimm_{𝑛→∞𝑛→∞}𝑢₁−𝑢^∗,𝑗𝑢_𝑛+1−𝑢^∗−𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗=0,𝑣₁−𝑣^∗,𝑗𝑣_𝑛+1−𝑣^∗−𝑣₁−𝑣^∗,𝑗𝑣_𝑛−𝑣^∗=0.(3.31) Therefore, the sequences {⟨𝑢₁−𝑢^∗,𝑗(𝑢_𝑛−𝑢^∗)⟩}_𝑛≥1 and {⟨𝑣₁−𝑣^∗,𝑗(𝑣_𝑛−𝑣^∗)⟩}_𝑛≥1 satisfy the conditions of Lemma 2.4.
Hence, lliimmssuup_𝑛→∞p_𝑛→∞𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗≤0,𝑣₁−𝑣^∗,𝑗𝑣_𝑛−𝑣^∗≤0.(3.32) Define 𝜉𝜎_𝑛∶=max𝑢₁−𝑢^∗,𝑗𝑢_𝑛+1−𝑢^∗,0_𝑛,∶=max𝑣₁−𝑣^∗,𝑗𝑣_𝑛+1−𝑣^∗,0,(3.33) then lim_𝑛→∞𝜎_𝑛=0=lim_𝑛→∞𝜉_𝑛. Moreover, 𝑣₁𝑢₁−𝑢^∗,𝑗𝑢_𝑛−𝑢^∗≤𝜎_𝑛,−𝑣^∗,𝑗𝑣_𝑛−𝑣^∗≤𝜉_𝑛,∀𝑛∈ℕ.(3.34) Using Lemma 2.5, and (3.18), we have: ‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸≤‖‖𝑢_𝑛−𝑢^∗‖‖^2𝐸−2𝜆_𝑛𝜃_𝑛‖‖𝑢_𝑛+1−𝑢^∗‖‖^2𝐸++22𝜆𝜆_𝑛𝑛𝛼_𝑛𝐹𝑢_𝑛+1−𝐹𝑢_𝑛+𝛼_𝑛𝑣_𝑛−𝑣^∗,𝑗𝑢_𝑛+1−𝑢^∗‖‖𝑣𝜃_𝑛𝑢_𝑛+1−𝑢_𝑛+𝜃_𝑛𝑢₁−𝑢^∗,𝑗𝑢_𝑛+1−𝑢^∗_𝑛+1,−𝑣^∗‖‖^2𝐸≤‖‖𝑣_𝑛−𝑣^∗‖‖^2𝐸−2𝜆_𝑛𝜃_𝑛‖‖𝑣_𝑛+1−𝑣^∗‖‖^2𝐸++22𝜆𝜆_𝑛𝑛𝛼_𝑛𝐾𝑣_𝑛+1−𝐾𝑣_𝑛+𝛼_𝑛𝑣^∗−𝑣_𝑛,𝑗𝑣_𝑛+1−𝑣^∗𝜃_𝑛𝑣_𝑛+1−𝑣_𝑛+𝜃_𝑛𝑣₁−𝑣^∗,𝑗𝑣_𝑛+1−𝑣^∗.(3.35) Using the fact that 𝐹 and 𝐾 are Lipschitz and the sequences {𝑢_𝑛} and {𝑣_𝑛} are bounded, we obtain the following estimate: ‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝐸≤‖‖𝑤_𝑛−𝑤^∗‖‖^2𝐸−2𝜆_𝑛𝜃_𝑛‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝐸=‖‖+𝑤2_𝑛𝜆_𝑛𝛼_𝑛+𝜆^2𝑛𝑐+2𝜆_𝑛𝜃_𝑛𝜎_𝑛+𝜉_𝑛−𝑤^∗‖‖^2𝐸−2𝜆_𝑛𝜃_𝑛‖‖𝑤_𝑛+1−𝑤^∗‖‖^2𝐸+𝛾_𝑛,(3.36) where 𝛾_𝑛=2(𝜆_𝑛𝛼_𝑛+𝜆^2𝑛)𝑐+2𝜆_𝑛𝜃_𝑛(𝜎_𝑛+𝜉_𝑛)=𝑜(𝜆_𝑛𝜃_𝑛) for some 𝑐>0. Hence, by Lemma 2.1, 𝑤_𝑛→𝑤^∗ as 𝑛→∞. But 𝑤_𝑛=(𝑢_𝑛,𝑣_𝑛) and 𝑤^∗=(𝑢^∗,𝑣^∗). This implies that 𝑢_𝑛→𝑢^∗ as 𝑛→∞. This completes the proof.

Corollary 3.3. Let 𝐸 be a 𝑞-uniformly real Banach spaces, 𝑞>1 and 𝐹,𝐾∶𝐸→𝐸 be Lipschitz accretive mappings. Let {𝑢_𝑛}_𝑛≥1 and {𝑣_𝑛}_𝑛≥1 be sequences in 𝐸 defined iteratively from arbitrary 𝑢₁,𝑣₁∈𝐸 by 𝑣𝑢_𝑛+1=𝑢_𝑛−𝜆_𝑛𝛼_𝑛𝐹𝑢_𝑛−𝑣_𝑛−𝜆_𝑛𝜃_𝑛𝑢_𝑛−𝑢₁,𝑛≥1,_𝑛+1=𝑣_𝑛−𝜆_𝑛𝛼_𝑛𝐾𝑣_𝑛+𝑢_𝑛−𝜆_𝑛𝜃_𝑛𝑣_𝑛−𝑣₁,𝑛≥1,(3.37) where {𝜆_𝑛}_𝑛≥1, {𝛼_𝑛}_𝑛≥1, and {𝜃_𝑛}_𝑛≥1 are real sequences in (0,1) such that 𝜆_𝑛=𝑜(𝜃_𝑛),𝛼_𝑛=𝑜(𝜃_𝑛), and ∑^∞𝑛=1𝜆_𝑛𝜃_𝑛=+∞. Suppose that the equation 𝑢+𝐾𝐹𝑢=0 has a solution 𝑢^∗. Then there exists a set 𝐾_min in 𝐸×𝐸 such that if (𝑢^∗,𝑣^∗)∈𝐾_min with 𝑣^∗=𝐹𝑢^∗, then the sequence {𝑢_𝑛}_𝑛≥1 converges strongly to 𝑢^∗.

Corollary 3.4. Let 𝐸=𝐿_𝑝 (or ℓ_𝑝) space (1<𝑝<+∞) and 𝐹,𝐾∶𝐸→𝐸 be Lipschitz accretive mappings. Let {𝑢_𝑛}_𝑛≥1 and {𝑣_𝑛}_𝑛≥1 be sequences in 𝐸 defined iteratively from arbitrary 𝑢₁,𝑣₁∈𝐸 by 𝑣𝑢_𝑛+1=𝑢_𝑛−𝜆_𝑛𝛼_𝑛𝐹𝑢_𝑛−𝑣_𝑛−𝜆_𝑛𝜃_𝑛𝑢_𝑛−𝑢₁,𝑛≥1,_𝑛+1=𝑣_𝑛−𝜆_𝑛𝛼_𝑛𝐾𝑣_𝑛+𝑢_𝑛−𝜆_𝑛𝜃_𝑛𝑣_𝑛−𝑣₁,𝑛≥1,(3.38) where {𝜆_𝑛}_𝑛≥1, {𝛼_𝑛}_𝑛≥1, and {𝜃_𝑛}_𝑛≥1 are real sequences in (0,1) such that 𝜆_𝑛=𝑜(𝜃_𝑛),𝛼_𝑛=𝑜(𝜃_𝑛), and ∑^∞𝑛=1𝜆_𝑛𝜃_𝑛=+∞. Suppose that the equation 𝑢+𝐾𝐹𝑢=0 has a solution 𝑢^∗. Then there exists a subset 𝐾_min of 𝐸×𝐸 such that if (𝑢^∗,𝑣^∗)∈𝐾_min with 𝑣^∗=𝐹𝑢^∗, then the sequence {𝑢_𝑛}_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 {𝑢_𝑛}_𝑛≥1 and {𝑣_𝑛}_𝑛≥1 be sequences in 𝐻 defined iteratively from arbitrary 𝑢₁,𝑣₁∈𝐻 by 𝑣𝑢_𝑛+1=𝑢_𝑛−𝜆_𝑛𝛼_𝑛𝐹𝑢_𝑛−𝑣_𝑛−𝜆_𝑛𝜃_𝑛𝑢_𝑛−𝑢₁,𝑛≥1,_𝑛+1=𝑣_𝑛−𝜆_𝑛𝛼_𝑛𝐾𝑣_𝑛+𝑢_𝑛−𝜆_𝑛𝜃_𝑛𝑣_𝑛−𝑣₁,𝑛≥1,(3.39) where {𝜆_𝑛}_𝑛≥1, {𝛼_𝑛}_𝑛≥1, and {𝜃_𝑛}_𝑛≥1 are real sequences in (0,1) such that 𝜆_𝑛=𝑜(𝜃_𝑛),𝛼_𝑛=𝑜(𝜃_𝑛), and ∑^∞𝑛=1𝜆_𝑛𝜃_𝑛=+∞. Suppose that the equation 𝑢+𝐾𝐹u=0 has a solution 𝑢^∗. Then there exists a subset 𝐾_min of 𝐸×𝐸 such that if (𝑢^∗,𝑣^∗)∈𝐾_min with 𝑣^∗=𝐹𝑢^∗, then the sequence {𝑢_𝑛}_𝑛≥1 converges strongly to 𝑢^∗.

Remark 3.6. Real sequences that satisfy the hypotheses of Theorems 3.1 are 𝜆_𝑛=𝛼_𝑛=𝑛^−𝑎 and 𝜃_𝑛=𝑛^−𝑏, 𝑛≥1 with 0<𝑏<𝑎 and 𝑎+𝑏<1.