System of Nonlinear Set-Valued Variational Inclusions Involving a Finite Family of 𝐻(⋅,⋅)-Accretive Operators in Banach Spaces

Abstract

We study a new system of nonlinear set-valued variational inclusions involving a finite family of 𝐻(⋅,⋅)-accretive operators in Banach spaces. By using the resolvent operator technique associated with a finite family of 𝐻(⋅,⋅)-accretive operators, we prove the existence of the solution for the system of nonlinear set-valued variational inclusions. Moreover, we introduce a new iterative scheme and prove a strong convergence theorem for finding solutions for this system.

1. Introduction

Variational inequality theory has become a very effective and powerful tool for studying a wide range of problems arising in pure and applied sciences which include work on differential equations, control problems, mechanics, general equilibrium problems in transportation and economics. In 1994, Hassouni and Moudafi [1] introduced and studied a class of variational inclusions and developed a perturbed algorithm for finding approximate solutions of the variational inclusions. In 1996, Adly [2] obtained some important extensions and generalizations of the results in [1] for nonlinear variational inclusions. Recently, Ding [3] introduced and studied a class of generalized quasivariational inclusions and Kazmi [4] introduced and studied another class of quasivariational inclusions in the same year. In [5, 6], Ansari et al. introduced the system of vector equilibrium problems and they proved the existence of solutions for such problems (see also in [7–9]). In 2004, Verma [10] studied nonlinear variational inclusion problems based on the generalized resolvent operator technique involving 𝐴-monotone mapping. For existence result and approximating solution of the system of set-valued variational inclusions and the class of nonlinear relaxed cocoercive variational inclusions, we refer the reader to Yan et al. [11], Plubtieng and Sriprad [12], Verma [13] and Cho et al. [14].

Very recently, Verma [15] introduced and studied approximation solvability of a general class of nonlinear variational inclusion problems based on (𝐴,𝜂)-resolvent operator technique in a Hilbert space. On the other hand, Zou and Huang [16] studied the Lipschitz continuity of resolvent operator for the 𝐻(⋅,⋅)-accretive operator in Banach spaces. Moreover, they also applied these new concepts to solve a variational-like inclusion problem. One year later, Zou and Huang [17] introduced and studied a new class of system of variational inclusions involving 𝐻(⋅,⋅)-accretive operator in Banach spaces. By using the resolvent operator technique associated with 𝐻(⋅,⋅)-accretive operator, they proved the existence of the solution for the system of inclusions. Moreover, they also develop a step-controlled iterative algorithm to approach the unique solution.

In this paper, we introduce a new system of nonlinear set-valued variational inclusions involving a finite family of 𝐻(⋅,⋅)-accretive operators in Banach spaces. By using the resolvent operators technique associated with a finite family of 𝐻(⋅,⋅)-accretive operator, we prove the existence of the solution for the system of nonlinear set-valued variational inclusions. Moreover, we introduce a new iterative scheme and prove a strong convergence theorem for finding solutions of this system.

2. Preliminaries

Let 𝑋 be a real Banach space with dual space 𝑋^∗,⟨⋅,⋅⟩ the dual pair between 𝑋 and 𝑋^∗ and 2^𝑋 and 𝐶(𝑋) denote the family of all the nonempty subsets of 𝑋 and the family of all closed subsets of 𝑋, respectively. The generalized duality mapping 𝐽_𝑞∶𝑋→2^𝑋∗ is defined by𝐽_𝑞(𝑋)=𝑓^∗∈𝑋^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖^𝑞,‖𝑓^∗‖=‖𝑥‖^𝑞−1,∀𝑥∈𝑋,(2.1) where 𝑞>1 is a constant. It is known that, in general, 𝐽_𝑞(𝑥)=‖𝑥‖^𝑞−1𝐽₂(𝑥) for all 𝑥≠0 and 𝐽_𝑞 is single-valued if 𝑋^∗ is strictly convex. In the sequel, we always assume that 𝑋 is a real Banach space such that 𝐽_𝑞 is single-valued.

The modulus of smoothness of 𝑋 is the function 𝜌_𝑋∶[0,∞)→[0,∞) defined by𝜌_𝑋(𝑡)=sup‖𝑥+𝑦‖+‖𝑥−𝑦‖2−1∶‖𝑥‖≤1,‖𝑦‖≤𝑡.(2.2) A Banach space 𝑋 is called uniformly smooth iflim_𝑡→0𝜌_𝑋(𝑡)𝑡=0.(2.3)𝑋 is called 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that𝜌_𝑋(𝑡)≤𝑐𝑡^𝑞,𝑞>1.(2.4)

Note that 𝐽_𝑞 is single valued if 𝑋 is uniformly smooth. In the study of characteristic inequalities in 𝑞-uniformly smooth Banach spaces, Xu [18] proved the following result.

Definition 2.1. Let 𝐻,𝜂∶𝑋×𝑋→𝑋 be two single-valued mappings and 𝐴,𝐵∶𝑋→𝑋 two single-valued mappings.(i)𝐴 is said to be accretive if𝐴𝑥−𝐴𝑦,𝐽_𝑞(𝑥−𝑦)≥0,∀𝑥,𝑦∈𝑋,(2.5)(ii)𝐴 is said to be strictly accretive if 𝐴 is accretive and 𝐴𝑥−𝐴𝑦,𝐽_𝑞(𝑥−𝑦)=0,∀𝑥,𝑦∈𝑋,(2.6) if and only if 𝑥=𝑦;(iii)𝐻(𝐴,⋅) is said to be 𝛼-strongly accretive with respect to 𝐴 if there exists a constant 𝛼>0 such that𝐻(𝐴𝑥,𝑢)−𝐻(𝐴𝑦,𝑢),𝐽_𝑞(𝑥−𝑦)≥𝛼‖𝑥−𝑦‖^𝑞,∀𝑥,𝑦,𝑢∈𝑋;(2.7)(iv)𝐻(⋅,𝐵) is said to be 𝛽-relaxed accretive with respect to 𝐵 if there exists a constant 𝛽>0 such that𝐻(𝑢,𝐵𝑥)−𝐻(𝑢,𝐵𝑦),𝐽_𝑞(𝑥−𝑦)≥−𝛽‖𝑥−𝑦‖^𝑞,∀𝑥,𝑦,𝑢∈𝑋;(2.8)(v)𝐻(⋅,⋅) is said to be 𝛾-Lipschitz continuous with respect to 𝐴 if there exists a constant 𝛾>0 such that‖𝐻(𝐴𝑥,𝑢)−𝐻(𝐴𝑦,𝑢)‖≤𝛾‖𝑥−𝑦‖^𝑞,∀𝑥,𝑦,𝑢∈𝑋;(2.9)(vi)𝐴 is said to be 𝜃-Lipschitz continuous if there exists a constant 𝜃>0 such that‖𝐴𝑥−𝐴𝑦‖≤𝜃‖𝑥−𝑦‖^𝑞,∀𝑥,𝑦∈𝑋;(2.10)(vii)𝜂(⋅,⋅) is said to be strongly accretive with respect to 𝐻(𝐴,𝐵) if there exists a constant 𝜌>0 such that𝜂(𝑥,𝑢)−𝜂(𝑦,𝑢),𝐽_𝑞(𝐻(𝐴𝑥,𝐵𝑥)−𝐻(𝐴𝑦,𝐵𝑦))≥𝜌‖𝑥−𝑦‖^𝑞,∀𝑥,𝑦,𝑢∈𝑋.(2.11)

Definition 2.2. Let 𝜂∶𝑋×𝑋→𝑋 be single-valued mapping. Let 𝑀∶𝑋→2^𝑋 be a set-valued mapping.(i)𝜂 is said to be 𝒯-Lipschitz continuous if there exists a constant 𝒯>0 such that‖𝜂(𝑥,𝑦)‖≤𝒯‖𝑥−𝑦‖,∀𝑥,𝑦∈𝑋;(2.12)(ii) 𝑀 is said to be accretive if𝑢−𝑣,𝐽_𝑞(𝑥−𝑦)≥0,∀𝑥,𝑦∈𝑋,𝑢∈𝑀(𝑥),𝑣∈𝑀(𝑦);(2.13)(iii) 𝑀 is said to be 𝜂-accretive if𝑢−𝑣,𝐽_𝑞(𝜂(𝑥,𝑦))≥0,∀𝑥,𝑦∈𝑋,𝑢∈𝑀(𝑥),𝑣∈𝑀(𝑦);(2.14)(iv)𝑀 is said to be strictly 𝜂-accretive if 𝑀 is 𝜂-accretive and equality holds if and only if 𝑥=𝑦;(v) 𝑀 is said to be 𝛾-strongly 𝜂-accretive if there exists a positive constant 𝛾>0 such that𝑢−𝑣,𝐽_𝑞(𝜂(𝑥,𝑦))≥𝛾‖𝑥−𝑦‖^𝑞,∀𝑥,𝑦∈𝑋,𝑢∈𝑀(𝑥),𝑣∈𝑀(𝑦);(2.15)(vi) 𝑀 is said to be 𝛼-relaxed 𝜂-accretive if there exists a positive constant 𝛼>0 such that𝑢−𝑣,𝐽_𝑞(𝜂(𝑥,𝑦))≥−𝛼‖𝑥−𝑦‖^𝑞,∀𝑥,𝑦∈𝑋,𝑢∈𝑀(𝑥),𝑣∈𝑀(𝑦).(2.16)

Definition 2.3. Let 𝐴,𝐵∶𝑋→𝑋, 𝐻∶𝑋×𝑋→𝑋 be three single-valued mappings. Let 𝑀∶𝑋→2^𝑋 be a set-valued mapping. 𝑀 is said to be 𝐻(⋅,⋅)-accretive with respect to 𝐴 and 𝐵 (or simply 𝐻(⋅,⋅)-accretive in the sequel), if 𝑀 is accretive and (𝐻(𝐴,𝐵)+𝜆𝑀)(𝑋)=𝑋 for every 𝜆>0.

Lemma 2.4. Let 𝑋 be a real uniformly smooth Banach space. Then 𝑋 is 𝑞-uniformly smooth if and only if there exists a constant 𝑐_𝑞>0 such that for all 𝑥,𝑦∈𝑋‖𝑥+𝑦‖^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝐽_𝑞(𝑥)+𝑐_𝑞‖𝑦‖^𝑞.(2.17)

Lemma 2.5 (see[16]). Let 𝐻(𝐴,𝐵) be 𝛼-strongly accretive with respect to 𝐴, 𝛽-relaxed accretive with respect to 𝐵, and 𝛼>𝛽. Let 𝑀 be an 𝐻(⋅,⋅)-accretive operator with respect to 𝐴 and 𝐵. Then, the operator 𝐻((𝐴,𝐵)+𝜆𝑀)⁻¹ is single valued. Based on Lemma 2.4, one can define the resolvent operator 𝑅^{𝐻(⋅,⋅)}_𝑀,𝜆 as follows.

Definition 2.6. Let 𝐻,𝐴,𝐵,𝑀 be defined as in Definition 2.3. Let 𝐻(𝐴,𝐵) be 𝛼-strongly accretive with respect to 𝐴, 𝛽-relaxed accretive with respect to 𝐵, and 𝛼>𝛽. Let 𝑀 be an 𝐻(⋅,⋅)-accretive operator with respect to 𝐴 and 𝐵. The resolvent operator 𝑅^{𝐻(⋅,⋅)}_𝑀,𝜆∶𝑋→𝑋 is defined by 𝑅^{𝐻(⋅,⋅)}_𝑀,𝜆(𝑧)=(𝐻(𝐴,𝐵)+𝜆𝑀)⁻¹(𝑧),∀𝑧∈𝑋,(2.18) where 𝜆>0 is a constant.

Lemma 2.7 (see [16]). Let 𝐻,𝐴,𝐵,𝑀 be defined as in Definition 2.3. Let 𝐻(𝐴,𝐵) be 𝛼-strongly accretive with respect to 𝐴, 𝛽-relaxed accretive with respect to 𝐵, and 𝛼>𝛽. Suppose that 𝑀∶𝑋→2^𝑋 is an 𝐻(⋅,⋅)-accretive operator. Then resolvent operator 𝑅^{𝐻(⋅,⋅)}_𝑀,𝜆 defined by (2.18) is 1/(𝛼−𝛽) Lipschitz continuous. That is, ‖‖𝑅^{𝐻(⋅,⋅)}_𝑀,𝜆(𝑥)−𝑅^{𝐻(⋅,⋅)}_𝑀,𝜆(𝑦)‖‖≤1𝛼−𝛽‖𝑥−𝑦‖,∀𝑥,𝑦∈𝑋.(2.19)

We define a Hausdorff pseudometric 𝐷∶2^𝑋×2^𝑋→[0,+∞] by𝐷(𝑈,𝑉)=maxsup_𝑢∈𝑈inf_𝑣∈𝑉‖𝑢−𝑣‖,sup_𝑢∈𝑉inf_𝑣∈𝑈‖𝑢−𝑣‖(2.20) for any given 𝑈,𝑉∈2^𝑋. Note that if the domain of 𝐷 is restricted to closed bounded subsets, then 𝐷 is the Hausdorff metric.

Lemma 2.8 (see [19]). Let {𝑐_𝑛} and {𝑘_𝑛} be two real sequences of nonnegative numbers that satisfy the following conditions:(i)0<𝑘_𝑛<1 for 𝑛=0,1,2,…, and limsup_𝑛𝑘_𝑛<1;(ii)𝑐_𝑛+1≤𝑘_𝑛𝑐_𝑛 for 𝑛=0,1,2,….Then, 𝑐_𝑛 converges to 0 as 𝑛→∞.

3. Main Result

Let 𝑋 be 𝑞-uniformly smooth real Banach space and 𝐶(𝑋) a nonempty closed convex set. Let 𝑆_𝑖,𝐻_𝑖∶𝑋×𝑋→𝑋, 𝐴_𝑖,𝐵_𝑖∶𝑋→𝑋 be single-valued operators, for all 𝑖=1,2,…,𝑁. For any fix 𝑖∈{1,2,…,𝑁}, we let 𝑀_𝑖∶𝑋→2^𝑋, 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖)-accretive set-valued operator and 𝑈_𝑖∶𝑋→2^𝑋 a set-valued mapping which nonempty values. The system of nonlinear set-valued variational inclusions is to find 𝑎₁,…,𝑎_𝑁∈𝑋, 𝑢₁∈𝑈₁(𝑎_𝑁),…,𝑢_𝑁∈𝑈_𝑁(𝑎₁) such that0∈𝑆_𝑖𝑎_𝑖,𝑢_𝑖+𝑀_𝑖𝑎_𝑖,∀𝑖=1,2,…,𝑁.(3.1)

If 𝑁=2, then system of nonlinear set-valued variational inclusions (3.1) becomes following system of variational inclusions: finding 𝑎₁,𝑎₂∈𝑋, 𝑢₁∈𝑈₁(𝑎₂) and 𝑢₂∈𝑈₂(𝑎₁) such that0∈𝑆₁𝑎₁,𝑢₁+𝑀₁𝑎₁,0∈𝑆₂𝑎₂,𝑢₂+𝑀₂𝑎₂.(3.2)

If 𝑁=1, then system of nonlinear set-valued variational inclusions (3.1) becomes the following class of nonlinear set-valued variational inclusions see [15]: finding 𝑎∈𝑋, 𝑢∈𝑈(𝑎) such that0∈𝑆(𝑎,𝑢)+𝑀(𝑎).(3.3)

For solving the system of nonlinear set-valued variational inclusions involving a finite family of 𝐻(⋅,⋅)-accretive operators in Banach spaces, let us give the following assumptions.

For any 𝑖∈{1,2,…,𝑁}, we suppose that(A1)𝐻(𝐴_𝑖,𝐵_𝑖) is 𝛼_𝑖-strongly accretive with respect to 𝐴_𝑖, 𝛽_𝑖-relaxed accretive with respect to 𝐵_𝑖 and 𝛼_𝑖>𝛽_𝑖,(A2)𝑀_𝑖∶𝑋→2^𝑋 is an 𝐻_𝑖(⋅,⋅)-accretive single-valued mapping,(A3)𝑈_𝑖∶𝑋→𝐶(𝑋) is a contraction set-valued mapping with 0≤𝐿_𝑖<1 and nonempty values,(A4)𝐻_𝑖(𝐴_𝑖,𝐵_𝑖) is 𝑟_𝑖-Lipschitz continuous with respect to 𝐴_𝑖 and 𝑡_𝑖-Lipschitz continuous with respect to 𝐵_𝑖,(A5)𝑆_𝑖∶𝑋×𝑋→𝑋 is 𝑙_𝑖-Lipschitz continuous with respect to its first argument and 𝑚_𝑖-Lipschitz continuous with respect to its second argument,(A6)𝑆_𝑖(⋅,𝑢) is 𝑠_𝑖-strongly accretive with respect to 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖).

Theorem 3.1. For given 𝑎₁,…,𝑎_𝑁∈𝑋, 𝑢₁∈𝑈₁(𝑎_𝑁),…,𝑢_𝑁∈𝑈_𝑁(𝑎₁), it is a solution of problem (3.1) if and only if 𝑎_𝑖=𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎_𝑖,𝐵_𝑖𝑎_𝑖−𝜆_𝑖𝑆_𝑖𝑎_𝑖,𝑢_𝑖,(3.4) where 𝜆_𝑖>0 are constants.

Proof. We note from the Definition 2.6 that 𝑎₁,…,𝑎_𝑁∈𝑋, 𝑢₁∈𝑈₁(𝑎_𝑁),…,𝑢_𝑁∈𝑈_𝑁(𝑎₁) is a solution of (3.1) if and only if, for each 𝑖∈{1,2,…,𝑁}, we have 𝑎_𝑖=𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎_𝑖,𝐵_𝑖𝑎_𝑖−𝜆_𝑖𝑆_𝑖𝑎_𝑖,𝑢_𝑖⟺𝑎_𝑖=𝐻_𝑖𝐴_𝑖,𝐵_𝑖+𝜆_𝑖𝑀_𝑖⁻¹𝐻_𝑖𝐴_𝑖𝑎_𝑖,𝐵_𝑖𝑎_𝑖−𝜆_𝑖𝑆_𝑖𝑎_𝑖,𝑢_𝑖⟺𝐻_𝑖𝐴_𝑖𝑎_𝑖,𝐵_𝑖𝑎_𝑖−𝜆_𝑖𝑆_i𝑎_𝑖,𝑢_𝑖∈𝐻_𝑖𝐴_𝑖,𝐵_𝑖+𝜆_𝑖𝑀_𝑖𝑎_𝑖⟺−𝜆_𝑖𝑆_𝑖𝑎_𝑖,𝑢_𝑖∈𝜆_𝑖𝑀_𝑖𝑎_𝑖⟺0∈𝑆_𝑖𝑎_𝑖,𝑢_𝑖+𝑀_𝑖𝑎_𝑖.(3.5)

Algorithm 3.2. For given 𝑎¹₀,…,𝑎^𝑁₀∈𝑋, 𝑢¹₀∈𝑈₁(𝑎^𝑁₀),…,𝑢^𝑁₀∈𝑈_𝑁(𝑎¹₀), we let 𝑎^𝑖₁=𝜎₀𝑎^𝑖₀+1−𝜎₀𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎^𝑖₀,𝐵_𝑖𝑎^𝑖₀−𝜆_𝑖𝑆_𝑖𝑎^𝑖₀,𝑢^𝑖₀,(3.6) for all 𝑖=1,2,…,𝑁, where 0<𝜎₀≤1. By Nadler theorem [20], there exists 𝑢¹₁∈𝑈₁(𝑎^𝑁₁),…,𝑢^𝑁₁∈𝑈_𝑁(𝑎¹₁) such that ‖‖𝑢^𝑖₁−𝑢^𝑖₀‖‖≤(1+1)𝐷𝑈_𝑖𝑎^{𝑁−(𝑖−1)}₁,𝑈_𝑖𝑎^{𝑁−(𝑖−1)}₀,∀𝑖=1,2,…,𝑁,(3.7) where 𝐷(⋅,⋅) is the Hausdorff pseudo metric on 2^𝑋. Continuing the above process inductively, we can obtain the sequences {𝑎^𝑖_𝑛} and {𝑢^𝑖_𝑛} such that 𝑎^𝑖_𝑛+1=𝜎_𝑛𝑎^𝑖_𝑛+1−𝜎_𝑛𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎ⁱ_𝑛,𝐵_𝑖𝑎^𝑖_𝑛−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛,(3.8) for all 𝑛=1,2,3,…,𝑖=1,2,…,𝑁, where 0<𝜎_𝑛≤1 with limsup_𝑛→∞𝜎_𝑛<1. Therefore, by Nadler theorem [20], there exists 𝑢¹_𝑛+1∈𝑈₁(𝑎^𝑁_𝑛+1),…,𝑢^𝑁_𝑛+1∈𝑈_𝑁(𝑎¹_𝑛+1) such that ‖‖𝑢^𝑖_𝑛+1−𝑢^𝑖_𝑛‖‖≤1+(1+𝑛)⁻¹𝐷𝑈_𝑖𝑎^{𝑁−(𝑖−1)}_𝑛+1,𝑈_𝑖𝑎^{𝑁−(𝑖−1)}_𝑛,∀𝑛=1,2,3,…,𝑖=1,2,…,𝑁.(3.9)
The idea of the proof of the next theorem is contained in the paper of Verma [15] and Zou and Huang [17].

Theorem 3.3. Let 𝑋 be q-uniformly smooth real Banach space. Let 𝐴_𝑖,𝐵_𝑖∶𝑋→𝑋 be single-valued operators, 𝐻_𝑖∶𝑋×𝑋→𝑋 a single-valued operator satisfy (A1) and 𝑀_𝑖, 𝑈_𝑖, 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖), 𝑆_𝑖, 𝑆_𝑖(⋅,𝑢) satisfy conditions (A2)–(A6), respectively. If there exists a constant 𝑐_𝑞,𝑖 such that ^𝑞𝑟_𝑖+𝑡_𝑖^𝑞−𝑞𝜆_𝑖𝑠_𝑖+𝑐_𝑞,𝑖𝜆^𝑞_𝑖𝑙^𝑞_𝑖𝛼_𝑖−𝛽_𝑖+𝜆_𝑖𝑚_𝑖𝛼_𝑖−𝛽_𝑖<1(3.10) for all 𝑖=1,2,…,𝑁, then problem (3.1) has a solution 𝑎₁,…,𝑎_𝑁, 𝑢₁∈𝑈₁(𝑎_𝑁),…,𝑢_𝑁∈𝑈_𝑁(𝑎₁).

Proof. For any 𝑖∈{1,2,…,𝑁} and 𝜆_𝑖>0, we define 𝐹_𝑖∶𝑋×𝑋→𝑋 by 𝐹_𝑖(𝑢,𝑣)=𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖(𝑢),𝐵_𝑖(𝑢)−𝜆_𝑖𝑆_𝑖(𝑢,𝑣),(3.11) for all 𝑢,𝑣∈𝑋. Let 𝐽_𝑖(𝑥,𝑦)=𝐻_𝑖(𝐴_𝑖(𝑥),𝐵_𝑖(𝑦)). For any (𝑢₁,𝑣₁),(𝑢₂,𝑣₂)∈𝑋×𝑋, we note by (3.11) and Lemma 2.7 that ‖‖𝐹_𝑖𝑢₁,𝑣₁−𝐹_𝑖𝑢₂,𝑣₂‖‖=‖‖𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑢₁,𝐵_𝑖𝑢₁−𝜆_𝑖𝑆_𝑖𝑢₁,𝑣₁−𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑢₂,𝐵_𝑖𝑢₂−𝜆_𝑖𝑆_𝑖𝑢₂,𝑣₂‖‖=‖‖𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐽_𝑖𝑢₁,𝑢₁−𝜆_𝑖𝑆_𝑖𝑢₁,𝑣₁−𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐽_𝑖𝑢₂,𝑢₂−𝜆_𝑖𝑆_𝑖𝑢₂,𝑣₂‖‖≤1𝛼_𝑖−𝛽_𝑖‖‖𝐽_𝑖𝑢₁,𝑢₁−𝜆_𝑖𝑆_𝑖𝑢₁,𝑣₁−𝐽_𝑖𝑢₂,𝑢₂−𝜆_𝑖𝑆_𝑖𝑢₂,𝑣₂‖‖=1𝛼_𝑖−𝛽_𝑖‖‖𝐽_𝑖𝑢₁,𝑢₁−𝐽_𝑖𝑢₂,𝑢₂−𝜆_𝑖𝑆_𝑖𝑢₁,𝑣₁−𝑆_𝑖𝑢₂,𝑣₂‖‖≤1𝛼_𝑖−𝛽_𝑖‖‖𝐽_𝑖𝑢₁,𝑢₁−𝐽_𝑖𝑢₂,𝑢₂−𝜆_𝑖𝑆_𝑖𝑢₁,𝑣₁−𝑆_𝑖𝑢₂,𝑣₁‖‖+𝜆_𝑖𝛼_𝑖−𝛽_𝑖‖‖𝑆_𝑖𝑢₂,𝑣₁−𝑆_𝑖𝑢₂,𝑣₂‖‖.(3.12) By Lemma 2.4, we have ‖‖𝐽_𝑖𝑢₁,𝑢₁−𝐽_𝑖𝑢₂,𝑢₂−𝜆_𝑖𝑆_𝑖𝑢₁,𝑣₁−𝑆_𝑖𝑢₂,𝑣₁‖‖^𝑞≤‖‖𝐽_𝑖𝑢₁,𝑢₁−𝐽_𝑖𝑢₂,𝑢₂‖‖^𝑞−𝑞𝜆_𝑖𝑆_𝑖𝑢₁,𝑣₁−𝑆_𝑖𝑢₂,𝑣₁,𝐽_𝑞𝐽_𝑖𝑢₁,𝑢₁−𝐽_𝑖𝑢₂,𝑢₂+𝑐_𝑞,𝑖𝜆^𝑞_𝑖‖‖𝑆_𝑖𝑢₁,𝑣₁−𝑆_𝑖𝑢₂,𝑣₁‖‖^𝑞.(3.13) Moreover, by (A4), we obtain ‖‖𝐽_𝑖𝑢₁,𝑢₁−𝐽_𝑖𝑢₂,𝑢₂‖‖≤‖‖𝐽_𝑖𝑢₁,𝑢₁−𝐽_𝑖𝑢₂,𝑢₁‖‖+‖‖𝐽_𝑖𝑢₂,𝑢₁−𝐽_𝑖𝑢₂,𝑢₂‖‖≤𝑟_𝑖‖‖𝑢₁−𝑢₂‖‖+𝑡_𝑖‖‖𝑢₁−𝑢₂‖‖≤𝑟_𝑖+𝑡_𝑖‖‖𝑢₁−𝑢₂‖‖.(3.14) From (A6), we have −𝑞𝜆_𝑖𝑆_𝑖𝑢₁,𝑣₁−𝑆_𝑖𝑢₂,𝑣₁,𝐽_𝑞𝐽_𝑖𝑢₁,𝑢₁−𝐽_𝑖𝑢₂,𝑢₂≤−𝑞𝜆_𝑖𝑠_𝑖‖‖𝑢₁−𝑢₂‖‖^𝑞.(3.15) Moreover, from (A5), we obtain ‖‖𝑆_𝑖𝑢₁,𝑣₁−𝑆_𝑖𝑢₂,𝑣₁‖‖≤𝑙_𝑖‖‖𝑢₁−𝑢₂‖‖,(3.16)‖‖𝑆_𝑖𝑢₂,𝑣₁−𝑆_𝑖𝑢₂,𝑣₂‖‖≤𝑚_𝑖‖‖𝑣₁−𝑣₂‖‖.(3.17) From (3.13)–(3.16), we have ‖‖𝐽_𝑖𝑢₁,𝑢₁−𝐽_𝑖(𝑢₂,𝑢₂)−𝜆_𝑖𝑆_𝑖𝑢₁,𝑣₁−𝑆_𝑖𝑢₂,𝑣₁‖‖^𝑞≤^𝑞𝑟_𝑖+𝑡_𝑖^𝑞−𝑞𝜆_𝑖𝑠_𝑖+𝑐_𝑞,𝑖𝜆^𝑞_𝑖𝑙^𝑞_𝑖‖‖𝑢₁−𝑢₂‖‖.(3.18) It follows from (3.12), (3.17), and (3.18) that ‖‖𝐹_𝑖𝑢₁,𝑣₁−𝐹_𝑖𝑢₂,𝑣₂‖‖≤^𝑞𝑟_𝑖+𝑡_𝑖^𝑞−𝑞𝜆_𝑖𝑠_𝑖+𝑐_𝑞,𝑖𝜆^𝑞_𝑖𝑙^𝑞_𝑖𝛼_𝑖−𝛽_𝑖‖‖𝑢₁−𝑢₂‖‖+𝜆_𝑖𝑚_𝑖𝛼_𝑖−𝛽_𝑖‖‖𝑣₁−𝑣₂‖‖.(3.19) Put 𝜃^𝑖₁=^𝑞𝑟_𝑖+𝑡_𝑖^𝑞−𝑞𝜆_𝑖𝑠_𝑖+𝑐_𝑞,𝑖𝜆^𝑞_𝑖𝑙^𝑞_𝑖𝛼_𝑖−𝛽_𝑖,𝜃^𝑖₂=𝜆_𝑖𝑚_𝑖𝛼_𝑖−𝛽_𝑖.(3.20) Define ‖⋅‖ on 𝑋×⋯×𝑋_{𝑁−times} by ‖(𝑥₁,…,𝑥_𝑁)‖=‖𝑥₁‖+⋯‖𝑥_𝑁‖ for all (𝑥₁,…,𝑥_𝑁)∈𝑋×⋯×𝑋_{𝑁−times}. It is easy to see that (𝑋×⋯×𝑋_{𝑁−times},‖⋅‖) is a Banach space. For any given 𝑥₁,…,𝑥_𝑁∈𝑋, we choose a finite sequence 𝑤₁∈𝑈₁(𝑥_𝑁),…,𝑤_𝑁∈𝑈_𝑁(𝑥₁). Define 𝑄∶𝑋×⋯×𝑋_{𝑁−times}→𝑋×⋯×𝑋_{𝑁−times} by 𝑄(𝑥₁,…,𝑥_𝑁)=(𝐹₁(𝑥₁,𝑤₁),…,𝐹_𝑁(𝑥_𝑁,𝑤_𝑁)). Set 𝑘=max{(𝜃¹₁+𝜃^𝑁₂𝐿_𝑁),…,(𝜃¹₂𝐿₁+𝜃^𝑁₁)}, where 𝐿₁,…,𝐿_𝑁 are contraction constants of 𝑈₁,…,𝑈_𝑁, respectively. We note that 𝜃^𝑖₁+𝜃^𝑖₂𝐿_𝑖<𝜃^𝑖₁+𝜃^𝑖₂<1, for all 𝑖=1,2,…,𝑁, and so 𝑘<1. Let 𝑥₁,…,𝑥_𝑁∈𝑋, 𝑤₁∈𝑈₁(𝑥_𝑁),…,𝑤_𝑁∈𝑈_𝑁(𝑥₁) and 𝑦₁,…,𝑦_𝑁∈𝑋, 𝑧₁∈𝑈₁(𝑦_𝑁),…,𝑧_𝑁∈𝑈_𝑁(𝑦₁). By (A3), we get ‖‖𝑄𝑥₁,…,𝑥_𝑁−𝑄𝑦₁,…,𝑦_𝑁‖‖=‖‖𝐹₁𝑥₁,𝑤₁,…,𝐹_𝑁𝑥_𝑁,𝑤_𝑁−𝐹₁𝑦₁,𝑧₁,…,𝐹_𝑁𝑦_𝑁,𝑧_𝑁‖‖=‖‖𝐹₁𝑥₁,𝑤₁−𝐹₁𝑦₁,𝑧₁‖‖+⋯+‖‖𝐹_𝑁𝑥_𝑁,𝑤_𝑁−𝐹_𝑁𝑦_𝑁,𝑧_𝑁‖‖≤𝜃¹₁‖‖𝑥₁−𝑦₁‖‖+𝜃¹₂‖‖𝑤₁−𝑧₁‖‖+⋯+𝜃^𝑁₁‖‖𝑥_𝑁−𝑦_𝑁‖‖+𝜃^𝑁₂‖‖𝑤_𝑁−𝑧_𝑁‖‖≤𝜃¹₁‖‖𝑥₁−𝑦₁‖‖+𝜃¹₂𝐿₁‖‖𝑥_𝑁−𝑦_𝑁‖‖+⋯+𝜃^𝑁₁‖‖𝑥_𝑁−𝑦_𝑁‖‖+𝜃^𝑁₂𝐿_𝑁‖‖𝑥₁−𝑦₁‖‖=𝜃¹₁+𝜃^𝑁₂𝐿_𝑁‖‖𝑥₁−𝑦₁‖‖+⋯+𝜃^𝑁₁+𝜃¹₂𝐿₁‖‖𝑥_𝑁−𝑦_𝑁‖‖≤𝑘‖‖𝑥₁−𝑦₁‖‖+⋯+𝑘‖‖𝑥_𝑁−𝑦_𝑁‖‖=𝑘‖‖𝑥₁−𝑦₁‖‖+⋯+‖‖𝑥_𝑁−𝑦_𝑁‖‖=𝑘‖‖𝑥₁,…,𝑥_𝑁−𝑦₁,…,𝑦_𝑁‖‖,(3.21) and so 𝑄 is a contraction on 𝑋×⋯×𝑋_{𝑁−times}. Hence there exists 𝑎₁,…,𝑎_𝑁∈𝑋,𝑢₁∈𝑈₁(𝑎_𝑁),…,𝑢_𝑁∈𝑈_𝑁(𝑎₁) such that 𝑎₁=𝐹₁(𝑎₁,𝑢₁),…,𝑎_𝑁=𝐹_𝑁(𝑎_𝑁,𝑢_𝑁). From Theorem 3.1, 𝑎₁,…,𝑎_𝑁∈𝑋, 𝑢₁∈𝑈₁(𝑎_𝑁),…,𝑢_𝑁∈𝑈_𝑁(𝑎₁) is the solution of the problem (3.1).

Theorem 3.4. Let 𝑋 be 𝑞-uniformly smooth real Banach space. For 𝑖=1,2,…,𝑁. Let 𝐴_𝑖,𝐵_𝑖∶𝑋→𝑋 be single-valued operators, 𝐻_𝑖∶𝑋×𝑋→𝑋 single-valued operator satisfy (A1) and suppose that 𝑀_𝑖, 𝑈_𝑖, 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖), 𝑆_𝑖, 𝑆_𝑖(⋅,𝑢) satisfy conditions (A2)–(A6), respectively. Then, for any 𝑖∈{1,2,…,𝑁}, the sequences {𝑎^𝑖_𝑛}^∞_𝑛=1 and {𝑢^𝑖_𝑛}^∞_𝑛=1 generated by Algorithm 3.2 converge strongly to 𝑎_𝑖, 𝑢_𝑖∈𝑈_𝑖(𝑎_{𝑁−(𝑖−1)}), respectively.

Proof. By Theorem 3.3, the problem (3.1) has a solution 𝑎₁,…,𝑎_𝑁∈𝑋, 𝑢₁∈𝑈₁(𝑎_𝑁),…,𝑢_𝑁∈𝑈_𝑁(𝑎₁). From Theorem 3.1, we note that 𝑎_𝑖=𝜎_𝑛𝑎_𝑖+1−𝜎_𝑛𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎_𝑖,𝐵_𝑖𝑎_𝑖−𝜆_𝑖𝑆_𝑖𝑎_𝑖,𝑢_𝑖,(3.22) for all 𝑖=1,2,…,𝑁. Hence, by (3.8) and (3.22), we have ‖‖𝑎^𝑖_𝑛+1−𝑎^𝑖_𝑛‖‖=‖‖𝜎_𝑛𝑎^𝑖_𝑛+1−𝜎_𝑛𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛,𝐵_𝑖𝑎^𝑖_𝑛−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝜎_𝑛𝑎^𝑖_𝑛−1+1−𝜎_𝑛𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛−1,𝐵_𝑖𝑎^𝑖_𝑛−1−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛−1‖‖≤𝜎_𝑛‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖+1−𝜎_𝑛‖‖𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛,𝐵_𝑖𝑎^𝑖_𝑛−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛−1,𝐵_𝑖𝑎^𝑖_𝑛−1−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛−1‖‖=𝜎_𝑛‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖+1−𝜎_𝑛‖‖𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐽_𝑖𝑎^𝑖_𝑛,𝑎^𝑖_𝑛−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛−1‖‖≤𝜎_𝑛‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖+1−𝜎_𝑛1𝛼_𝑖−𝛽_𝑖‖‖𝐽_𝑖𝑎^𝑖_𝑛,𝑎^𝑖_𝑛−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛−1‖‖=𝜎_𝑛‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖+1−𝜎_𝑛1𝛼_𝑖−𝛽_𝑖‖‖𝐽_𝑖𝑎^𝑖_𝑛,𝑎^𝑖_𝑛−𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛−1‖‖≤𝜎_𝑛‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖+1−𝜎_𝑛1𝛼_𝑖−𝛽_𝑖‖‖𝐽_𝑖𝑎^𝑖_𝑛,𝑎^𝑖_𝑛−𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛‖‖+1−𝜎_𝑛1𝛼_𝑖−𝛽_𝑖‖‖𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛−1‖‖.(3.23) By Lemma 2.4, we obtain ‖‖𝐽_𝑖𝑎^𝑖_𝑛,𝑎^𝑖_𝑛−𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛‖‖^𝑞≤‖‖𝐽_𝑖𝑎^𝑖_𝑛,𝑎^𝑖_𝑛−𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1‖‖^𝑞−𝑞𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛,𝐽_𝑞,𝑖𝐽_𝑖𝑎^𝑖_𝑛,𝑎^𝑖_𝑛−𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1+𝑐_𝑞,𝑖𝜆^𝑞_𝑖‖‖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛‖‖^𝑞.(3.24) From (A4), we note that ‖‖𝐽_𝑖𝑎^𝑖_𝑛,𝑎^𝑖_𝑛−𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1‖‖=‖‖𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛,𝐵_𝑖𝑎^𝑖_𝑛−𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛−1,𝐵_𝑖𝑎^𝑖_𝑛−1‖‖≤‖‖𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛,𝐵_𝑖𝑎^𝑖_𝑛−𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛−1,𝐵_𝑖𝑎^𝑖_𝑛‖‖+‖‖𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛−1,𝐵_𝑖𝑎^𝑖_𝑛−𝐻_𝑖𝐴_𝑖𝑎^𝑖_𝑛−1,𝐵_𝑖𝑎^𝑖_𝑛−1‖‖≤𝑟_i+𝑡_𝑖‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖.(3.25) From (3.24) and (A6), it follows that −𝑞𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛,𝐽_𝑞,1𝐽_𝑖𝑎^𝑖_𝑛,𝑎^𝑖_𝑛−𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1≤−𝑞𝜆_𝑖𝑠_𝑖‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖^𝑞.(3.26) By (3.23), (3.24), and (A5), we have ‖‖𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛−1‖‖≤𝑚_𝑖‖‖𝑢^𝑖_𝑛−𝑢^𝑖_𝑛−1‖‖≤𝑚_𝑖𝑑_𝑖1+𝑛⁻¹‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖,(3.27)‖‖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛‖‖≤𝑙_𝑖‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖.(3.28) From (3.23)–(3.28), we obtain ‖‖𝐽_𝑖𝑎ⁱ_𝑛,𝑎^𝑖_𝑛−𝐽_𝑖𝑎^𝑖_𝑛−1,𝑎^𝑖_𝑛−1−𝜆_𝑖𝑆_𝑖𝑎^𝑖_𝑛,𝑢^𝑖_𝑛−𝑆_𝑖𝑎^𝑖_𝑛−1,𝑢^𝑖_𝑛‖‖^𝑞≤^𝑞𝑟_𝑖+𝑡_𝑖^𝑞−𝑞𝜆_𝑖𝑠_𝑖+𝑐_𝑞,𝑖𝜆^𝑞_𝑖𝑙^𝑞_𝑖𝛼_𝑖−𝛽_𝑖‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖+𝜆_𝑖𝑚_𝑖𝛼_𝑖−𝛽_𝑖𝑑_𝑖1+𝑛⁻¹‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖.(3.29) Hence, by (3.23), (3.28) and (3.29), we have ‖‖𝑎^𝑖_𝑛+1−𝑎^𝑖_𝑛‖‖≤𝜎_𝑛‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖+1−𝜎_𝑛^𝑞𝑟_𝑖+𝑡_𝑖^𝑞−𝑞𝜆_𝑖𝑠_𝑖+𝑐_𝑞,𝑖𝜆^𝑞_𝑖𝑙^𝑞_𝑖𝛼_𝑖−𝛽_𝑖‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖+1−𝜎_𝑛𝜆_𝑖𝑚_𝑖𝛼_𝑖−𝛽_𝑖𝑑_𝑖1+𝑛⁻¹‖‖𝑎^𝑖_𝑛−𝑎^𝑖_𝑛−1‖‖.(3.30) Put 𝑘=max{𝜋₁…,𝜋_𝑁}, where 𝜋_𝑖=^𝑞𝑟_𝑖+𝑡_𝑖^𝑞−𝑞𝜆_𝑖𝑠_𝑖+𝑐_𝑞,𝑖𝜆^𝑞_𝑖𝑙^𝑞_𝑖𝛼_𝑖−𝛽_𝑖+𝜆_𝑖𝑚_𝑖𝑑_𝑖1+𝑛⁻¹𝛼_𝑖−𝛽_𝑖.(3.31) It follows from (3.30) that ‖‖𝑎¹_𝑛+1−𝑎¹_𝑛‖‖+⋯+‖‖𝑎^𝑁_𝑛+1−𝑎^𝑁_𝑛‖‖≤𝜎_𝑛‖‖𝑎¹_𝑛−𝑎¹_𝑛−1‖‖+1−𝜎_𝑛𝑘‖‖𝑎¹_𝑛−𝑎¹_𝑛−1‖‖+⋯+𝜎_𝑛‖‖𝑎^𝑁_𝑛−𝑎^𝑁_𝑛−1‖‖+1−𝜎_𝑛𝑘‖‖𝑎^𝑁_𝑛−𝑎^𝑁_𝑛−1‖‖.(3.32) Set 𝑐_𝑛=‖𝑎¹_𝑛−𝑎¹_𝑛−1‖+⋯+‖𝑎^𝑁_𝑛−𝑎^𝑁_𝑛−1‖ and 𝑘_𝑛=𝑘+(1−𝑘)𝜎_𝑛. From (3.32), we obtain 𝑐_𝑛+1≤𝑘_𝑛𝑐_𝑛,∀𝑛=0,1,2,….(3.33) Since limsup_𝑛→∞𝜎_𝑛<1, we have limsup_𝑛→∞𝑘_𝑛<1. Thus, it follows from Lemma 2.8 that 𝑐_𝑛+1→0 and hence lim_𝑛→∞‖𝑎^𝑖_𝑛+1−𝑎^𝑖_𝑛‖=0. Therefore, {𝑎^𝑖_𝑛} is a Cauchy sequence and hence there exists 𝑎_𝑖∈𝑋 such that 𝑎^𝑖_𝑛→𝑎_𝑖 as 𝑛→∞, for all 𝑖=1,2,…,𝑁. Next, we will show that 𝑢¹_𝑛→𝑢₁∈𝑈₁(𝑎_𝑁) as 𝑛→∞. Hence, it follows from (3.9) that {𝑢¹_𝑛} is also a Cauchy sequence. Thus there exists 𝑢₁∈𝑋 such that 𝑢¹_𝑛→𝑢₁ as 𝑛→∞. Consider 𝑑𝑢₁,𝑈₁𝑎_𝑁=inf‖‖𝑢₁−𝑞‖‖∶𝑞∈𝑈₁𝑎_𝑁≤‖‖𝑢₁−𝑢¹_𝑛‖‖+𝑑𝑢¹_𝑛,𝑈₁𝑎_𝑁≤‖‖𝑢₁−𝑢¹_𝑛‖‖+𝐷𝑈₁𝑎^𝑁_𝑛,𝑈₁𝑎_𝑁≤‖‖𝑢₁−𝑢¹_𝑛‖‖+𝑑₁‖‖𝑎^𝑁_𝑛−𝑎_𝑁‖‖⟶0(3.34) as 𝑛→∞. Since 𝑈₁(𝑎_𝑁) is a closed set and 𝑑(𝑢₁,𝑈₁(𝑎_𝑁))=0, we have 𝑢₁∈𝑈₁(𝑎_𝑁). By continuing the above process, there exist 𝑢₂∈𝑈₂(𝑎_𝑁−1),…,𝑢_𝑁∈𝑈_𝑁(𝑎₁) such that 𝑢²_𝑛→𝑢₂,…,𝑢^𝑁_𝑛→𝑢_𝑁 as 𝑛→∞. Hence, by (3.8), we obtain 𝑎_𝑖=𝑅^{𝐻𝑖(⋅,⋅)}_{𝑀𝑖,𝜆𝑖}𝐻_𝑖𝐴_𝑖𝑎_𝑖,𝐵_𝑖𝑎_𝑖−𝜆_𝑖𝑆_𝑖𝑎_𝑖,𝑢_𝑖.(3.35) Therefore, it follows from Theorem 3.1 that 𝑎₁,…,𝑎_𝑁 is a solution of problem (3.1).
Setting 𝑁=2 in Theorem 3.3, we have the following result.

Corollary 3.5. Let 𝑋 be 𝑞-uniformly smooth real Banach spaces. Let 𝐴_𝑖,𝐵_𝑖∶𝑋→𝑋 be singled valued operators, 𝐻_𝑖∶𝑋×𝑋→𝑋 a single-valued operator such that 𝐻(𝐴_𝑖,𝐵_𝑖) is 𝛼_𝑖-strongly accretive with respect to 𝐴_𝑖, 𝛽_𝑖-relaxed accretive with respect to 𝐵_𝑖 and 𝛼_𝑖>𝛽_𝑖 and suppose that 𝑀_𝑖∶𝑋→2^𝑋 is an 𝐻_𝑖(⋅,⋅)-accretive set-valued mapping and 𝑈_𝑖∶𝑋→𝐶(𝑋) contraction set-valued mapping with 0≤𝐿_𝑖<1 and nonempty values, for all 𝑖=1,2. Assume that 𝐻_𝑖(𝐴_𝑖,𝐵_𝑖) is 𝑟_𝑖-Lipschitz continuous with respect to 𝐴_𝑖 and 𝑡_𝑖-Lipschitz continuous with respect to 𝐵_𝑖, 𝑆_𝑖∶𝑋×𝑋→𝑋 is 𝑙_𝑖-Lipschitz continuous with respect to its first argument and 𝑚_𝑖-Lipschitz continuous with respect to its second argument, 𝑆₁(⋅,𝑦) is 𝑠₁-strongly accretive with respect to 𝐻₁(𝐴₁,𝐵₁), and S₂(𝑥,⋅) is 𝑠₂-strongly accretive with respect to 𝐻₂(𝐴₂,𝐵₂), for all 𝑖=1,2. If ^𝑞𝑟_𝑖+𝑡_𝑖^𝑞−𝑞𝜆_𝑖𝑠_𝑖+𝑐_𝑞,𝑖𝜆^𝑞_𝑖𝑙^𝑞_𝑖𝛼_𝑖−𝛽_𝑖+𝜆_𝑖𝑚_𝑖𝛼_𝑖−𝛽_𝑖<1,(3.36) for all 𝑖∈{1,2}, then problem (3.2) has a solution 𝑎₁,𝑎₂∈𝑋, 𝑢₁∈𝑈₁(𝑎₂),𝑢₂∈𝑈₂(𝑎₁).

Setting 𝑁=1 in Theorem 3.3, we have the following result.

Corollary 3.6. Let 𝑋 be 𝑞-uniformly smooth real Banach spaces. Let 𝐴,𝐵∶𝑋→𝑋 be two singled valued operators, 𝐻∶𝑋×𝑋→𝑋 a single-valued operator such that 𝐻(𝐴,𝐵) is 𝛼-strongly accretive with respect to 𝐴, 𝛽-relaxed accretive with respect to 𝐵, and 𝛼>𝛽 and suppose that 𝑀∶𝑋→2^𝑋 is an 𝐻(⋅,⋅)-accretive set-valued mapping, 𝑈∶𝑋→𝐶(𝑋) is contraction set-valued mapping with 0≤𝐿<1 and nonempty values. Assume that 𝐻(𝐴,𝐵) is 𝑟-Lipschitz continuous with respect to 𝐴 and 𝑡-Lipschitz continuous with respect to 𝐵, 𝑆∶𝑋×𝑋→𝑋 is 𝑙-Lipschitz continuous with respect to its first argument and 𝑚-Lipschitz continuous with respect to its second argument, 𝑆(⋅,𝑦) is 𝑠-strongly accretive with respect to 𝐻(𝐴,𝐵). If √^𝑞(𝑟+𝑡)^𝑞−𝑞𝜆𝑠+𝑐_𝑞,𝜆^𝑞𝑙^𝑞𝛼−𝛽+𝜆𝑚𝛼−𝛽<1,(3.37) then problem (3.3) has a solution 𝑎∈𝑋 and 𝑢∈𝑈(𝑎).

Acknowledgments

The first author would like to thank the Office of the Higher Education Commission, Thailand, financial support under Grant CHE-Ph.D-THA-SUP/191/2551, Thailand. Moreover, the second author would like to thank the Thailand Research Fund for financial support under Grant BRG5280016.