International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 709715 | https://doi.org/10.5402/2011/709715

Sayyedeh Zahra Nazemi, " ( 𝐻 , πœ™ ) - πœ‚ -Accretive Mappings and a New System of Generalized Variational Inclusions with ( 𝐻 , πœ™ ) - πœ‚ -Accretive Mappings in Banach Spaces", International Scholarly Research Notices, vol. 2011, Article ID 709715, 20 pages, 2011. https://doi.org/10.5402/2011/709715

( 𝐻 , πœ™ ) - πœ‚ -Accretive Mappings and a New System of Generalized Variational Inclusions with ( 𝐻 , πœ™ ) - πœ‚ -Accretive Mappings in Banach Spaces

Academic Editor: Y.-K. Chang
Received05 Jun 2011
Accepted29 Jun 2011
Published02 Oct 2011

Abstract

We introduce a new class of generalized accretive mappings, named (𝐻,πœ™)-πœ‚-accretive mappings, in Banach spaces. We define a resolvent operator associated with (𝐻,πœ™)-πœ‚-accretive mappings and show its Lipschitz continuity. We also introduce and study a new system of generalized variational inclusions with (𝐻,πœ™)-πœ‚-accretive mappings in Banach spaces. By using the resolvent operator technique associated with (𝐻,πœ™)-πœ‚-accretive mappings, we construct a new iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We also prove the existence of solutions for the generalized variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.

1. Introduction

Variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, and engineering sciences, and so forth (see, e.g., [1–4]).

Recently Ding [5], Huang and Fang [6], Verma [7], Fang and Huang [8, 9], Huang and Fang [10], Fang et al. [11], Kazmi and Khan [12], and Lan et al. [13, 14] introduced the concepts of πœ‚-subdifferential operators, maximal πœ‚-monotone operators, A-monotone operators, and (𝐻,πœ‚)-monotone operators in Hilbert spaces, H-accretive operators, generalized m-accretive mappings, (𝐻,πœ‚)-accretive operators, P-πœ‚-accretive operators, and (𝐴,πœ‚)-accretive mappings in Banach spaces, and their resolvent operators, respectively. In [15], Luo and Huang introduced a new concept of (𝐻,πœ™)-πœ‚-monotone mappings in Banach spaces and defined the proximal mapping associated with (𝐻,πœ™)-πœ‚-monotone mappings.

Motivated and inspired by the research work going on this field, in this paper, we introduce a new concept of (𝐻,πœ™)-πœ‚-accretive mappings and give the definition of its resolvent operator in Banach spaces. We also introduce and study a new system of generalized variational inclusions with (𝐻,πœ™)-πœ‚-accretive mappings in Banach spaces, and we construct a new iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We also prove the existence of solutions for the generalized variational inclusions and the convergence of iterative sequences generated by algorithm. The results in this paper improve and extend some known results in the literature.

2. Preliminaries

Let 𝐸 be a real Banach space equipped with norm β€–β‹…β€–; let πΈβˆ— be the topological dual space of 𝐸; let βŸ¨β‹…,β‹…βŸ© be the pair between 𝐸 and πΈβˆ—; let 2𝐸 be the power set of 𝐸; let 𝐻(β‹…,β‹…) be the Hausdorff metric on CB(𝐸) defined by𝐻(𝐴,𝐡)=maxsupπ‘₯βˆˆπ΄π‘‘(π‘₯,𝐡),supπ‘¦βˆˆπ΅ξƒ°π‘‘(𝐴,𝑦),βˆ€π΄,𝐡∈CB(𝐸).(2.1)

Definition 2.1 (see [16, 17]). For π‘ž>1, a mapping π½π‘žβˆΆπΈβ†’2πΈβˆ— is said to be generalized duality mapping if it is defined by π½π‘žξ€½(π‘₯)=π‘“βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–π‘ž,β€–π‘₯β€–π‘žβˆ’1ξ€Ύ=‖𝑓‖,βˆ€π‘₯∈𝐸.(2.2)
In particular, 𝐽2 is the usual normalized duality mapping on 𝐸.
It is well known that π½π‘ž(π‘₯)=β€–π‘₯β€–π‘žβˆ’1𝐽2(π‘₯),βˆ€π‘₯(β‰ 0)∈𝐸.(2.3)
Note that if 𝐸=β„‹ is a real Hilbert space, then 𝐽2 becomes the identity mapping on β„‹.

Definition 2.2 (see [18]). A Banach space 𝐸 is called smooth, if for every π‘₯∈𝐸 with β€–π‘₯β€–=1, there exists a unique π‘“βˆˆπΈβˆ— such that ‖𝑓‖=𝑓(π‘₯)=1. The modulus of 𝐸 is the function 𝜌𝐸∢[0,∞)β†’[0,∞), defined by πœŒπΈξ‚»(𝜏)=supβ€–π‘₯+𝑦‖+β€–π‘₯βˆ’π‘¦β€–2ξ‚Ό.βˆ’1∢π‘₯,π‘¦βˆˆπΈ,β€–π‘₯β€–=1,‖𝑦‖=𝜏(2.4)

Definition 2.3 (see [17]). The Banach space 𝐸 is said to be(i)uniformly smooth, if limπœβ†’0𝜌𝐸(𝜏)𝜏=0;(2.5)(ii)q-uniformly smooth, for π‘ž>1, if there exists a constant 𝜏>0 such that 𝜌𝐸(𝜏)β‰€π‘πœπ‘ž[,𝜏∈0,∞).(2.6) It is well known (see [16]) thatπΏπ‘žξ€·orπ‘™π‘žξ€Έisξ‚»π‘ž-uniformlysmooth,if1<π‘žβ‰€2,2-uniformlysmooth,ifπ‘žβ‰₯2.(2.7)
Note that if 𝐸 is uniformly smooth, π½π‘ž becomes single-valued, and π½π‘ž is single-valued if πΈβˆ— is strictly convex. In the sequel, unless otherwise specified, we always suppose that 𝐸 is a real Banach space such that π½π‘ž is single-valued.

Lemma 2.4 (see [19]). Let π‘ž>1 be a real number and let 𝐸 be a smooth Banach space. Then 𝐸 is q-uniformly smooth if and only if there exists a constant π‘π‘ž>0 such that for every π‘₯,π‘¦βˆˆπΈ, β€–π‘₯+π‘¦β€–π‘žβ‰€β€–π‘₯β€–π‘žξ«+π‘žπ‘¦,π½π‘žξ¬(π‘₯)+π‘π‘žβ€–π‘¦β€–π‘ž.(2.8)

Definition 2.5 (see [9, 20]). Let π‘ƒβˆΆπΈβ†’πΈ be a single-valued mapping. 𝑃 is said to be(i)accretive if 𝑃(π‘₯)βˆ’π‘ƒ(𝑦),π½π‘žξ¬(π‘₯βˆ’π‘¦)β‰₯0,iffπ‘₯,π‘¦βˆˆπΈ;(2.9)(ii)strictly accretive if P is accretive and 𝑃(π‘₯)βˆ’π‘ƒ(𝑦),π½π‘žξ¬(π‘₯βˆ’π‘¦)=0ifπ‘₯=𝑦;(2.10)(iii)r-strongly accretive if there exists a constant π‘Ÿ>0 such that 𝑃(π‘₯)βˆ’π‘ƒ(𝑦),π½π‘žξ¬(π‘₯βˆ’π‘¦)β‰₯π‘Ÿβ€–π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,π‘¦βˆˆπΈ;(2.11)(iv)m-relaxed accretive if there exists a constant π‘š>0 such that 𝑃(π‘₯)βˆ’π‘ƒ(𝑦),π½π‘žξ¬(π‘₯βˆ’π‘¦)β‰₯βˆ’π‘šβ€–π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,π‘¦βˆˆπΈ;(2.12)(v)(𝛼,πœ‰)-relaxed cocoercive if there exist constants 𝛼,πœ‰>0 such that 𝑃(π‘₯)βˆ’π‘ƒ(𝑦),π½π‘žξ¬β€–(π‘₯βˆ’π‘¦)β‰₯βˆ’π›Όβ€–π‘ƒ(π‘₯)βˆ’π‘ƒ(𝑦)π‘ž+πœ‰β€–π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,π‘¦βˆˆπΈ;(2.13)(vi)s-Lipschitz continuous if there exists a constant 𝑠>0 such that ‖𝑃(π‘₯)βˆ’π‘ƒ(𝑦)‖≀𝑠‖π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΈ.(2.14)

Definition 2.6 (see [12]). A mapping πœ‚βˆΆπΈΓ—πΈβ†’πΈ is said to be 𝜏-Lipschitz continuous if there exists a constant 𝜏>0 such that β€–πœ‚(π‘₯,𝑦)β€–β‰€πœβ€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΈ.(2.15)

Definition 2.7 (see [20]). Let πœ‚βˆΆπΈΓ—πΈβ†’πΈ and π‘ƒβˆΆπΈβ†’πΈ be single-valued mappings. Then a multi-valued mapping π‘‡βˆΆπΈβ†’2𝐸 is said to be(i)πœ‚-accretive if ξ«π‘’βˆ’π‘£,π½π‘žξ¬(πœ‚(π‘₯,𝑦))β‰₯0,βˆ€π‘₯,π‘¦βˆˆπΈ,βˆ€π‘’βˆˆπ‘‡(π‘₯),π‘£βˆˆπ‘‡(𝑦);(2.16)(ii)strictly πœ‚-accretive if ξ«π‘’βˆ’π‘£,π½π‘žξ¬(πœ‚(π‘₯,𝑦))β‰₯0,βˆ€π‘₯,π‘¦βˆˆπΈ,βˆ€π‘’βˆˆπ‘‡(π‘₯),π‘£βˆˆπ‘‡(𝑦),(2.17) and equality holds if and only if π‘₯=𝑦;(iii)𝛾-strongly πœ‚-accretive if there exists a constant 𝛾>0 such that ξ«π‘’βˆ’π‘£,π½π‘žξ¬(πœ‚(π‘₯,𝑦))β‰₯𝛾‖π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,π‘¦βˆˆπΈ,βˆ€π‘’βˆˆπ‘‡(π‘₯),π‘£βˆˆπ‘‡(𝑦);(2.18)(iv)m-relaxed πœ‚-accretive if there exists a constant π‘š>0 such that ξ«π‘’βˆ’π‘£,π½π‘žξ¬(πœ‚(π‘₯,𝑦))β‰₯βˆ’π‘šβ€–π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,π‘¦βˆˆπΈ,βˆ€π‘’βˆˆπ‘‡(π‘₯),π‘£βˆˆπ‘‡(𝑦);(2.19)(v)πœ‚-π‘š-accretive if 𝑇 is πœ‚-accretive and (𝐼+πœŒπ‘‡)(𝐸)=𝐸 holds for all 𝜌>0;(vi)𝑃-πœ‚-accretive if 𝑇 is πœ‚-accretive and (𝑃+πœŒπ‘‡)(𝐸)=𝐸 holds for all 𝜌>0.

Definition 2.8. Let 𝐸1,𝐸2,…,𝐸𝑛 be real Banach spaces, and, for 𝑖=1,2,…,𝑛, let π‘π‘–βˆΆβˆπ‘›π‘—=1𝐸𝑗→𝐸𝑖 be a single-valued mapping. Then 𝑁𝑖 is said to be πœ‰π‘–π‘—-Lipschitz continuous in the 𝑗th argument, if there exists a constant πœ‰π‘–π‘—>0 such that ‖‖𝑁𝑖π‘₯1,…,π‘₯π‘—βˆ’1,𝑦𝑗1,π‘₯𝑗+1,…,π‘₯π‘›ξ€Έβˆ’π‘π‘–ξ€·π‘₯1,…,π‘₯π‘—βˆ’1,𝑦𝑗2,π‘₯𝑗+1,…,π‘₯π‘›ξ€Έβ€–β€–β‰€πœ‰π‘–π‘—β€–β€–π‘¦π‘—1βˆ’π‘¦π‘—2β€–β€–,βˆ€π‘¦π‘—1,𝑦𝑗2βˆˆπΈπ‘—,π‘₯π‘–βˆˆπΈπ‘–(π‘–βˆˆ{1,2,…,𝑛},𝑖≠𝑗).(2.20)

Definition 2.9. Let π΄βˆΆπΈβ†’CB(𝐸) be a set-valued mapping. 𝐴 is said to be 𝐻-Lipschitz continuous, if there exists a constant 𝑑>0 such that 𝐻(𝐴(π‘₯),𝐡(𝑦))≀𝑑‖π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΈ,(2.21) where 𝐻(β‹…,β‹…) denotes the Hausdorff metric on CB(𝐸).

Definition 2.10. Let β„‹ be a Hilbert space, and let π»βˆΆβ„‹β†’β„‹ be a single-valued mapping. 𝐻 is said to be(i)coercive if limβ€–π‘₯β€–β†’βˆžβŸ¨π»(π‘₯),π‘₯βŸ©β€–π‘₯β€–=+∞;(2.22)(ii)hemicontinuous if for any fixed π‘₯,𝑦,π‘§βˆˆβ„‹, the function π‘‘β†’βŸ¨π»(π‘₯+𝑑𝑦),π‘§βŸ© is continuous at 0+.

3. (𝐻,πœ™)-πœ‚-Accretive Mappings

In this section, we will introduce a new class of generalized accretive mappings, (𝐻,πœ™)-πœ‚-accretive mappings, and discuss some properties of (𝐻,πœ™)-πœ‚-accretive mappings.

Definition 3.1. Let 𝐸 be a Banach space and let 𝐻,πœ™βˆΆπΈβ†’πΈ, πœ‚βˆΆπΈΓ—πΈβ†’πΈ be single-valued mappings and π‘€βˆΆπΈβ†’2𝐸 a multi-valued mapping. The mapping 𝑀 is said to be a (𝐻,πœ™)-πœ‚-accretive mapping, if πœ™βˆ˜π‘€ is πœ‚-accretive and (𝐻+πœ™βˆ˜π‘€)(𝐸)=𝐸.

Remark 3.2. (i) If 𝑀 is πœ‚-accretive and πœ™(π‘₯)=πœ†π‘₯, for all π‘₯∈𝐸, πœ†>0, then (𝐻,πœ™)-πœ‚-accretive mapping reduces to the 𝑃-πœ‚-accretive mapping studied by Kazmi and Khan [12].
(ii) If πœ‚(π‘₯,𝑦)=π‘₯βˆ’π‘¦, for all π‘₯,π‘¦βˆˆπΈ, 𝑀 is πœ‚-accretive, and πœ™(π‘₯)=πœ†π‘₯, for all π‘₯∈𝐸, πœ†>0, then (𝐻,πœ™)-πœ‚-accretive mapping reduces to the 𝐻-accretive mapping studied by Fang and Huang [9].
(iii) If 𝑀 is π‘š-relaxed πœ‚-accretive and πœ™(π‘₯)=πœ†π‘₯, for all π‘₯∈𝐸, πœ†>0, then (𝐻,πœ™)-πœ‚-accretive mapping reduces to the (𝐴,πœ‚)-accretive mapping studied by Lan et al. [14].
Similarly, we give the following definition.

Definition 3.3. Let 𝐻,πœ™βˆΆπΈβ†’πΈ be single-valued mappings and π‘€βˆΆπΈβ†’2𝐸 a multi-valued mapping. The mapping 𝑀 is said to be a (𝐻,πœ™)-accretive mapping, if πœ™βˆ˜π‘€ is accretive and (𝐻+πœ™βˆ˜π‘€)(𝐸)=𝐸.

Example 3.4. Let β„‹ be a Hilbert space and for every π‘₯,π‘¦βˆˆβ„‹, πœ‚(x,𝑦)=π‘₯βˆ’π‘¦, πœ™(π‘₯)=πœ†π‘₯, where πœ†>0 is a constant. Let π‘€βˆΆβ„‹β†’2β„‹ be a maximal monotone mapping and π»βˆΆβ„‹β†’β„‹ a bounded, coercive, hemicontinuous, and 𝛼-strongly πœ‚-accretive mapping. Then it follows from Corollary  32.26 of [21] that 𝑀 is (𝐻,πœ™)-πœ‚-accretive mapping.

Theorem 3.5. Let πœ™βˆΆπΈβ†’πΈ, πœ‚βˆΆπΈΓ—πΈβ†’πΈ be single-valued mappings, π»βˆΆπΈβ†’πΈ a strictly πœ‚-accretive mapping, and π‘€βˆΆπΈβ†’2𝐸 a (𝐻,πœ™)-πœ‚-accretive mapping. Then (𝐻+πœ™βˆ˜π‘€)βˆ’1 is a single-valued mapping.

Proof. For any given π‘§βˆˆπΈ, let π‘₯,π‘¦βˆˆ(𝐻+πœ™βˆ˜π‘€)βˆ’1(𝑧). It follows that π‘§βˆ’π»(π‘₯)βˆˆπœ™βˆ˜π‘€(π‘₯),π‘§βˆ’π»(𝑦)βˆˆπœ™βˆ˜π‘€(𝑦).(3.1) Then πœ‚-accretivity of πœ™βˆ˜π‘€ implies that ξ«π‘§βˆ’π»(π‘₯)βˆ’(π‘§βˆ’π»(𝑦)),π½π‘žξ¬=(πœ‚(π‘₯,𝑦))βˆ’π»(π‘₯)+𝐻(𝑦),π½π‘žξ¬(πœ‚(π‘₯,𝑦))β‰₯0.(3.2) This implies that π‘₯=𝑦 and so (𝐻+πœ™βˆ˜π‘€)βˆ’1 is a single-valued mapping. This completes the proof.

By Theorem 3.5, we can define the resolvent operator 𝐽𝐻,πœ‚π‘€,πœ™ associated with an (𝐻,πœ™)-πœ‚-accretive mapping 𝑀 as follows.

Definition 3.6. Let πœ™βˆΆπΈβ†’πΈ, πœ‚βˆΆπΈΓ—πΈβ†’πΈ be single-valued mappings, let π»βˆΆπΈβ†’πΈ be a strictly πœ‚-accretive mapping, and let π‘€βˆΆπΈβ†’2𝐸 be a (𝐻,πœ™)-πœ‚-accretive mapping. A resolvent operator 𝐽𝐻,πœ‚π‘€,πœ™βˆΆπΈβ†’πΈ is defined by 𝐽𝐻,πœ‚π‘€,πœ™(π‘₯)=(𝐻+πœ™βˆ˜π‘€)βˆ’1(π‘₯),βˆ€π‘₯∈𝐸.(3.3)

Remark 3.7. (i) If 𝑀 is πœ‚-accretive and πœ™(π‘₯)=πœ†π‘₯, for all π‘₯∈𝐸, πœ†>0, then the resolvent operator 𝐽𝐻,πœ‚π‘€,πœ™ reduces to the 𝑃-πœ‚-proximal point mapping π½π‘€πœ‚,𝜌 introduced by Kazmi and Khan [12].
(ii) If πœ‚(π‘₯,𝑦)=π‘₯βˆ’π‘¦, for all π‘₯,π‘¦βˆˆπΈ, 𝑀 is πœ‚-accretive, and πœ™(π‘₯)=πœ†π‘₯, for all π‘₯∈𝐸, πœ†>0, then the resolvent operator 𝐽𝐻,πœ‚π‘€,πœ™ reduces to the proximal-point mapping introduced by Fang and Huang [9].
(iii) If 𝑀 is π‘š-relaxed πœ‚-accretive and πœ™(π‘₯)=πœ†π‘₯, for all π‘₯∈𝐸, πœ†>0, then the resolvent operator 𝐽𝐻,πœ‚π‘€,πœ™ reduces to the resolvent operator π‘…πœŒ,π΄πœ‚,𝑀 introduced by Lan at el. [14].

Theorem 3.8. Let πœ™βˆΆπΈβ†’πΈ be a single-valued mapping, let πœ‚βˆΆπΈΓ—πΈβ†’πΈ be a 𝜏-Lipschitz continuous mapping, let π»βˆΆπΈβ†’πΈ be a 𝛾-strongly πœ‚-accretive mapping, and let π‘€βˆΆπΈβ†’2𝐸 be a (𝐻,πœ™)-πœ‚-accretive mapping. Then the resolvent operator 𝐽𝐻,πœ‚π‘€,πœ™ is Lipschitz continuous with constant πœπ‘žβˆ’1/𝛾, that is, ‖‖𝐽𝐻,πœ‚π‘€,πœ™(π‘₯)βˆ’π½π»,πœ‚π‘€,πœ™β€–β€–β‰€πœ(𝑦)π‘žβˆ’1𝛾‖π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΈ.(3.4)

Proof. Let π‘₯,𝑦 be any given points in 𝐸, it follows from Definition 3.6 that 𝐽𝐻,πœ‚π‘€,πœ™(π‘₯)=(𝐻+πœ™βˆ˜π‘€)βˆ’1(π‘₯),𝐽𝐻,πœ‚π‘€,πœ™(𝑦)=(𝐻+πœ™βˆ˜π‘€)βˆ’1(𝑦).(3.5) This implies that 𝐽π‘₯βˆ’π»π»,πœ‚π‘€,πœ™ξ‚ξ‚€π½(π‘₯)βˆˆπœ™βˆ˜π‘€π»,πœ‚π‘€,πœ™ξ‚,𝐽(π‘₯)π‘¦βˆ’π»π»,πœ‚π‘€,πœ™ξ‚ξ‚€π½(𝑦)βˆˆπœ™βˆ˜π‘€π»,πœ‚π‘€,πœ™ξ‚.(𝑦)(3.6) Since 𝑀 is (𝐻,πœ™)-πœ‚-accretive, we have 𝐽π‘₯βˆ’π»π»,πœ‚π‘€,πœ™ξ‚βˆ’ξ‚€ξ‚€π½(π‘₯)π‘¦βˆ’π»π»,πœ‚π‘€,πœ™(𝑦),π½π‘žξ‚€πœ‚ξ‚€π½π»,πœ‚π‘€,πœ™(π‘₯),𝐽𝐻,πœ‚π‘€,πœ™=𝐻𝐽(𝑦)π‘₯βˆ’π‘¦βˆ’π»,πœ‚π‘€,πœ™ξ‚ξ‚€π½(π‘₯)βˆ’π»π»,πœ‚π‘€,πœ™(𝑦),π½π‘žξ‚€πœ‚ξ‚€π½π»,πœ‚π‘€,πœ™(π‘₯),𝐽𝐻,πœ‚π‘€,πœ™(𝑦)β‰₯0.(3.7) The inequality above implies that β€–β€–πœ‚ξ‚€π½β€–π‘₯βˆ’π‘¦β€–π»,πœ‚π‘€,πœ™(π‘₯),𝐽𝐻,πœ‚π‘€,πœ™ξ‚β€–β€–(𝑦)π‘žβˆ’1=‖‖𝐽‖π‘₯βˆ’π‘¦β€–π‘žξ‚€πœ‚ξ‚€π½π»,πœ‚π‘€,πœ™(π‘₯),𝐽𝐻,πœ‚π‘€,πœ™β€–β€–β‰₯(𝑦)π‘₯βˆ’π‘¦,π½π‘žξ‚€πœ‚ξ‚€π½π»,πœ‚π‘€,πœ™(π‘₯),𝐽𝐻,πœ‚π‘€,πœ™β‰₯𝐻𝐽(𝑦)𝐻,πœ‚π‘€,πœ™ξ‚ξ‚€π½(π‘₯)βˆ’π»π»,πœ‚π‘€,πœ™ξ‚(𝑦),π½π‘žξ‚€πœ‚ξ‚€π½π»,πœ‚π‘€,πœ™(π‘₯),𝐽𝐻,πœ‚π‘€,πœ™β€–β€–π½(𝑦)β‰₯𝛾𝐻,πœ‚π‘€,πœ™(π‘₯)βˆ’π½π»,πœ‚π‘€,πœ™β€–β€–(𝑦)π‘ž.(3.8) Since πœ‚ is Lipschitz continuous with a constant 𝜏, we have β€–β€–πœ‚ξ‚€π½π»,πœ‚π‘€,πœ™(π‘₯),𝐽𝐻,πœ‚π‘€,πœ™ξ‚β€–β€–β€–β€–π½(𝑦)β‰€πœπ»,πœ‚π‘€,πœ™(π‘₯)βˆ’π½π»,πœ‚π‘€,πœ™β€–β€–.(𝑦)(3.9) It follows from (3.8) and (3.9) that β€–π‘₯βˆ’π‘¦β€–πœπ‘žβˆ’1‖‖𝐽𝐻,πœ‚π‘€,πœ™(π‘₯)βˆ’π½π»,πœ‚π‘€,πœ™β€–β€–(𝑦)π‘žβˆ’1‖‖𝐽β‰₯𝛾𝐻,πœ‚π‘€,πœ™(π‘₯)βˆ’π½π»,πœ‚π‘€,πœ™β€–β€–(𝑦)π‘ž.(3.10) Hence, we get ‖‖𝐽𝐻,πœ‚π‘€,πœ™(π‘₯)βˆ’π½π»,πœ‚π‘€,πœ™β€–β€–β‰€πœ(𝑦)π‘žβˆ’1𝛾‖π‘₯βˆ’π‘¦β€–.(3.11) This completes the proof.

4. A New System of Generalized Variational Inclusions

In this section, we will introduce a new system of generalized variational inclusions with (𝐻,πœ™)-πœ‚-accretive mappings and construct a new iterative algorithm for solving this system of generalized variational inclusions. In what follows, for each 𝑖=1,2,3, suppose that 𝐸𝑖 is a Banach space, 𝐻𝑖,𝑓𝑖,πœ™π‘–,π‘π‘–βˆΆπΈπ‘–β†’πΈπ‘–, πœ‚π‘–βˆΆπΈπ‘–Γ—πΈπ‘–β†’πΈπ‘–, 𝐹𝑖,π‘π‘–βˆΆβˆ3π‘˜=1πΈπ‘˜β†’πΈπ‘– are single-valued mappings, π‘‡π‘˜π‘–βˆΆπΈπ‘˜β†’CB(𝐸𝑖) is a set-valued mapping, and π‘€π‘–βˆΆπΈπ‘–Γ—πΈπ‘–β†’2𝐸𝑖 is a (𝐻𝑖,πœ™π‘–)-πœ‚π‘–-accretive mapping in the second argument. Assume that 𝑓𝑖(𝐸𝑖)∩dom𝑀(𝑀𝑖,β‹…)β‰ βˆ…, for each π‘€π‘–βˆˆπΈi. We consider the following system of generalized variational inclusions. Find (π‘₯1,π‘₯2,π‘₯3,𝑒11,𝑒12,𝑒13,𝑒21,𝑒22,𝑒23,𝑒31,𝑒32,𝑒33) such that for each 𝑖=1,2,3, π‘₯π‘–βˆˆπΈπ‘–, 𝑒1π‘–βˆˆπ‘‡1𝑖(π‘₯1), 𝑒2π‘–βˆˆπ‘‡2𝑖(π‘₯2), 𝑒3π‘–βˆˆπ‘‡3𝑖(π‘₯3), and0βˆˆπΉπ‘–ξ€·π‘₯1βˆ’π‘1ξ€·π‘₯1ξ€Έ,π‘₯2βˆ’π‘2ξ€·π‘₯2ξ€Έ,π‘₯3βˆ’π‘3ξ€·π‘₯3ξ€Έξ€Έ+𝑁𝑖𝑒𝑖1,𝑒𝑖2,𝑒𝑖3ξ€Έ+𝑀𝑖π‘₯𝑖,𝑓𝑖π‘₯𝑖.ξ€Έξ€Έ(4.1) The following are some special cases of problem (4.1).

(i) If 𝐸1=β„‹1, 𝐸2=β„‹2, and 𝐸3=β„‹3 are three Hilbert spaces, and, for each 𝑖=1,2,3, 𝑝𝑖=0, 𝑁𝑖=0, 𝑀1(π‘₯,𝑓1(π‘₯))=πœ•πœ‚1πœ‘1(π‘₯), for all π‘₯∈𝐸1, 𝑀2(𝑦,𝑓2(𝑦))=πœ•πœ‚2πœ‘2(𝑦), for all π‘¦βˆˆπΈ2, and 𝑀3(𝑧,𝑓3(𝑧))=πœ•πœ‚3πœ‘3(𝑧), for all π‘§βˆˆπΈ3, where πœ‘π‘–βˆΆπΈπ‘–β†’π‘…βˆͺ{∞} is a proper lower semicontinuous and πœ‚π‘–-subdifferential function, πœ•πœ‚1πœ‘1(π‘₯) is the πœ‚1-subdifferential of πœ‘1 at π‘₯, πœ•πœ‚2πœ‘2(𝑦) is the πœ‚2-subdifferential of πœ‘2 at 𝑦, and πœ•πœ‚3πœ‘3(𝑧) is the πœ‚3-subdifferential of πœ‘3 at 𝑧, then SGVIP (4.1) reduces to the following system of variational inequalities, which is to find (π‘₯1,π‘₯2,π‘₯3)∈𝐸1×𝐸2×𝐸3 such that𝐹1ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ,πœ‚1ξ€·π‘Ž,π‘₯1+πœ‘1(π‘Ž)βˆ’πœ‘1ξ€·π‘₯1ξ€Έβ‰₯0,βˆ€π‘ŽβˆˆπΈ1,𝐹2ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ,πœ‚2𝑏,π‘₯2+πœ‘2(𝑏)βˆ’πœ‘2ξ€·π‘₯2ξ€Έβ‰₯0,βˆ€π‘βˆˆπΈ2,𝐹3ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ,πœ‚3𝑐,π‘₯3+πœ‘3(𝑐)βˆ’πœ‘3ξ€·π‘₯3ξ€Έβ‰₯0,βˆ€π‘βˆˆπΈ3.(4.2)

If πœ‚1(π‘Ž,π‘₯)=π‘Žβˆ’π‘₯, for all π‘Ž,π‘₯∈𝐸1, πœ‚2(𝑏,𝑦)=π‘βˆ’π‘¦, for all 𝑏,π‘¦βˆˆπΈ2, πœ‚3(𝑐,𝑧)=π‘βˆ’π‘§, for all 𝑐,π‘§βˆˆπΈ3, and 𝑀1(π‘₯,𝑓1(π‘₯))=πœ•πœ‘1(π‘₯) is the subdifferential of πœ‘1 at π‘₯, 𝑀2(𝑦,𝑓2(𝑦))=πœ•πœ‘2(𝑦) is the subdifferential of πœ‘2 at 𝑦, and 𝑀3(𝑧,𝑓3(𝑧))=πœ•πœ‘3(𝑧) is the subdifferential of πœ‘3 at 𝑧, then problem (4.2) reduces to the following system of variational inequalities, which is to find (π‘₯1,π‘₯2,π‘₯3)∈𝐸1×𝐸2×𝐸3 such that𝐹1ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ,π‘Žβˆ’π‘₯1+πœ‘1(π‘Ž)βˆ’πœ‘1ξ€·π‘₯1ξ€Έβ‰₯0,βˆ€π‘ŽβˆˆπΈ1,𝐹2ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ,π‘βˆ’π‘₯2+πœ‘2(𝑏)βˆ’πœ‘2ξ€·π‘₯2ξ€Έβ‰₯0,βˆ€π‘βˆˆπΈ2,𝐹3ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ,π‘βˆ’π‘₯3+πœ‘3(𝑐)βˆ’πœ‘3ξ€·π‘₯3ξ€Έβ‰₯0,βˆ€π‘βˆˆπΈ3.(4.3)

If 𝑀1(π‘₯,𝑓1(π‘₯))=πœ•π›Ώπ‘˜1(π‘₯), 𝑀2(𝑦,𝑓2(𝑦))=πœ•π›Ώπ‘˜2(𝑦), and 𝑀3(𝑧,𝑓3(𝑧))=πœ•π›Ώπ‘˜3(𝑧), for all π‘₯∈𝐾1, π‘¦βˆˆπΎ2, and π‘§βˆˆπΎ3, where 𝐾1βŠ‚πΈ1, 𝐾2βŠ‚πΈ2, and 𝐾3βŠ‚πΈ3 are three nonempty, closed, and convex subsets, π›Ώπ‘˜1, π›Ώπ‘˜2, and π›Ώπ‘˜3 denote the indicator functions of 𝐾1, 𝐾2, and 𝐾3, respectively, then problem (4.3) reduces to the following system of variational inequalities, which is to find (π‘₯1,π‘₯2,π‘₯3)∈𝐸1×𝐸2×𝐸3 such that𝐹1ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ,π‘Žβˆ’π‘₯1β‰₯0,βˆ€π‘ŽβˆˆπΎ1,𝐹2ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ,π‘βˆ’π‘₯2β‰₯0,βˆ€π‘βˆˆπΎ2,𝐹3ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ,π‘βˆ’π‘₯3β‰₯0,βˆ€π‘βˆˆπΎ3.(4.4)

If 𝐸1=𝐸2=𝐸3=β„‹ is a Hilbert space, 𝐾1=𝐾2=𝐾3=𝐾 is a nonempty, closed, and convex subset, 𝐹1(π‘₯,𝑦,𝑧)=𝑠𝑇1(𝑦,𝑧,π‘₯)+π‘₯βˆ’π‘¦, 𝐹2(π‘₯,𝑦,𝑧)=𝑑𝑇2(𝑧,π‘₯,𝑦)+π‘¦βˆ’π‘§, and 𝐹3(π‘₯,𝑦,𝑧)=π‘Ÿπ‘‡3(π‘₯,𝑦,𝑧)+π‘§βˆ’π‘₯, for all π‘₯,𝑦,π‘§βˆˆπΎ, where 𝑇1,𝑇2,𝑇3βˆΆπΎΓ—πΎΓ—πΎβ†’β„‹ are mappings on 𝐾×𝐾×𝐾, π‘Ÿ,𝑠,𝑑>0 are three numbers, then problem (4.4) reduces to the following system of variational inequalities, which is to find π‘₯1,π‘₯2,π‘₯3∈𝐾 such that𝑠𝑇1ξ€·π‘₯2,π‘₯3,π‘₯1ξ€Έ+π‘₯1βˆ’π‘₯2,π‘Žβˆ’π‘₯1β‰₯0,βˆ€π‘ŽβˆˆπΎ,𝑑𝑇2ξ€·π‘₯3,π‘₯1,π‘₯2ξ€Έ+π‘₯2βˆ’π‘₯3,π‘Žβˆ’π‘₯2β‰₯0,βˆ€π‘ŽβˆˆπΎ,π‘Ÿπ‘‡3ξ€·π‘₯1,π‘₯2,π‘₯3ξ€Έ+π‘₯3βˆ’π‘₯1,π‘Žβˆ’π‘₯3β‰₯0,βˆ€π‘ŽβˆˆπΎ.(4.5) Problem (4.5) was introduced and studied by Cho and Qin [22].

(ii) If 𝐸2=𝐸3, 𝐹1(β‹…,β‹…,β‹…)=𝐹1(β‹…,β‹…), 𝐹2(β‹…,β‹…,β‹…)=𝐹2(β‹…,β‹…), and 𝐹2=𝐹3 and for each 𝑖=1,2,3, 𝑝𝑖=0, 𝑁𝑖=0, 𝑀2(𝑦,𝑓2(𝑦))=𝑀3(𝑧,𝑓3(𝑧)), for all (𝑦,𝑧)∈𝐸2×𝐸3, then SGVIP (4.1) reduces to the following system of variational inclusions, which is to find (π‘₯1,π‘₯2)∈𝐸1×𝐸2 such that0∈𝐹1ξ€·π‘₯1,π‘₯2ξ€Έ+𝑀1ξ€·π‘₯1,𝑓1ξ€·π‘₯1,ξ€Έξ€Έ0∈𝐹2ξ€·π‘₯1,π‘₯2ξ€Έ+𝑀2ξ€·π‘₯2,𝑓2ξ€·π‘₯2.ξ€Έξ€Έ(4.6)

If 𝐸1=β„‹1 and 𝐸2=β„‹2 are two Hilbert spaces and 𝑀1(π‘₯,𝑓1(π‘₯))=πœ•πœ‚1πœ‘1(π‘₯), for all π‘₯∈𝐸1, 𝑀2(𝑦,𝑓2(𝑦))=πœ•πœ‚2πœ‘2(𝑦), for all π‘¦βˆˆπΈ2, then the problem (4.6) reduces to the following system of variational inequalities, which is to find (π‘₯1,π‘₯2)∈𝐸1×𝐸2 such that𝐹1ξ€·π‘₯1,π‘₯2ξ€Έ,πœ‚1ξ€·π‘Ž,π‘₯1+πœ‘1(π‘Ž)βˆ’πœ‘1ξ€·π‘₯1ξ€Έβ‰₯0,βˆ€π‘ŽβˆˆπΈ1,𝐹2ξ€·π‘₯1,π‘₯2ξ€Έ,πœ‚2𝑏,π‘₯2+πœ‘2(𝑏)βˆ’πœ‘2ξ€·π‘₯2ξ€Έβ‰₯0,βˆ€π‘βˆˆπΈ2.(4.7)

If πœ‚1(π‘Ž,π‘₯)=π‘Žβˆ’π‘₯, for all π‘Ž,π‘₯∈𝐸1, πœ‚2(𝑏,𝑦)=π‘βˆ’π‘¦, for all 𝑏,π‘¦βˆˆπΈ2, 𝑀1(π‘₯,𝑓1(π‘₯))=πœ•πœ‘1(π‘₯) is the subdifferential of πœ‘1 at π‘₯, and 𝑀2(𝑦,𝑓2(𝑦))=πœ•πœ‘2(𝑦) is the subdifferential of πœ‘2 at 𝑦, then the problem (4.7) reduces to the following system of variational inequalities, which is to find (π‘₯1,π‘₯2)∈𝐸1×𝐸2 such that𝐹1ξ€·π‘₯1,π‘₯2ξ€Έ,π‘Žβˆ’π‘₯1+πœ‘1(π‘Ž)βˆ’πœ‘1ξ€·π‘₯1ξ€Έβ‰₯0,βˆ€π‘ŽβˆˆπΈ1,𝐹2ξ€·π‘₯1,π‘₯2ξ€Έ,π‘βˆ’π‘₯2+πœ‘2(𝑏)βˆ’πœ‘2ξ€·π‘₯2ξ€Έβ‰₯0,βˆ€π‘βˆˆπΈ2.(4.8) Problem (4.8) was introduced and studied by Cho et al. [23].

If 𝑀1(π‘₯,𝑓1(π‘₯))=πœ•π›Ώπ‘˜1(π‘₯), 𝑀2(𝑦,𝑓2(𝑦))=πœ•π›Ώπ‘˜2(𝑦), for all π‘₯∈𝐾1, π‘¦βˆˆπΎ2, where 𝐾1βŠ‚πΈ1, 𝐾2βŠ‚πΈ2 are two nonempty, closed, and convex subsets and πœ•π›Ώπ‘˜1 and πœ•π›Ώπ‘˜2 denote the indicator functions of 𝐾1, 𝐾2, respectively, then problem (4.8) reduces to the following system of variational inequalities, which is to find (π‘₯1,π‘₯2)∈𝐸1×𝐸2 such that𝐹1ξ€·π‘₯1,π‘₯2ξ€Έ,π‘Žβˆ’π‘₯1β‰₯0,βˆ€π‘ŽβˆˆπΎ1,𝐹2ξ€·π‘₯1,π‘₯2ξ€Έ,π‘βˆ’π‘₯2β‰₯0,βˆ€π‘βˆˆπΎ2.(4.9) Problem (4.9) is just the problem [24] with 𝐹1 and 𝐹2 being single-valued.

If 𝐸1=𝐸2=β„‹ is a Hilbert space and 𝐾1=𝐾2=𝐾 is a nonempty, closed, and convex subset, 𝐹1(π‘₯,𝑦)=πœŒπ‘‡(𝑦,π‘₯)+π‘₯βˆ’π‘¦ and 𝐹2(π‘₯,𝑦)=πœ†π‘‡(π‘₯,𝑦)+π‘¦βˆ’π‘₯, for all π‘₯,π‘¦βˆˆπΎ, where π‘‡βˆΆπΎΓ—πΎβ†’β„‹ is a mapping on 𝐾×𝐾, 𝜌,πœ†>0 are two numbers, then problem (4.9) reduces to the following problem, which is to find π‘₯1,π‘₯2∈𝐾 such thatπ‘₯πœŒπ‘‡2,π‘₯1ξ€Έ+π‘₯1βˆ’π‘₯2,π‘Žβˆ’π‘₯1π‘₯β‰₯0,βˆ€π‘ŽβˆˆπΎ,πœ†π‘‡1,π‘₯2ξ€Έ+π‘₯2βˆ’π‘₯1,π‘Žβˆ’π‘₯2β‰₯0,βˆ€π‘ŽβˆˆπΎ.(4.10) Problem (4.10) was introduced and studied by Verma [25].

Lemma 4.1. Let, for 𝑖=1,2,3, πœ™π‘–βˆΆπΈπ‘–β†’πΈπ‘– be a single-valued mapping satisfying πœ™π‘–(π‘₯+𝑦)=πœ™π‘–(π‘₯)+πœ™π‘–(𝑦) and kerπœ™π‘–={0}, πœ‚π‘–βˆΆπΈπ‘–Γ—πΈπ‘–β†’πΈπ‘– a single-valued mapping, π»π‘–βˆΆπΈπ‘–β†’πΈπ‘– a strictly πœ‚π‘–-accretive mapping and π‘€π‘–βˆΆπΈπ‘–Γ—πΈπ‘–β†’2𝐸𝑖 a (𝐻𝑖,πœ™π‘–)-πœ‚π‘–-accretive mapping in the second argument. Then (π‘₯1,π‘₯2,π‘₯3,𝑒11,𝑒12,𝑒13,𝑒21,𝑒22,𝑒23,𝑒31,𝑒32,𝑒33) in which π‘₯π‘–βˆˆπΈπ‘–, 𝑒1π‘–βˆˆπ‘‡1𝑖(π‘₯1), 𝑒2π‘–βˆˆπ‘‡2𝑖(π‘₯2), 𝑒3π‘–βˆˆπ‘‡3𝑖(π‘₯3)(𝑖=1,2,3) is a solution of the problem (4.1) if and only if 𝑓𝑖π‘₯𝑖=𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–ξ€·π‘₯𝑖,β‹…,πœ™π‘–ξ€Ίπ»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘–ξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯1βˆ’π‘1ξ€·π‘₯1ξ€Έ,π‘₯2βˆ’π‘2ξ€·π‘₯2ξ€Έ,π‘₯3βˆ’π‘3ξ€·π‘₯3ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’π‘–1,𝑒𝑖2,𝑒𝑖3,ξ€Έξ€»(4.11) where 𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–(π‘₯𝑖,β‹…),πœ™π‘–=(𝐻𝑖+πœ™π‘–βˆ˜π‘€π‘–(π‘₯𝑖,β‹…))βˆ’1.

Proof. The fact directly follows from Definition 3.6.

Algorithm 4.2. For any π‘₯0π‘–βˆˆπΈπ‘–, take 𝑒01π‘–βˆˆπ‘‡1𝑖(π‘₯01), 𝑒02π‘–βˆˆπ‘‡2𝑖(π‘₯02), and 𝑒03π‘–βˆˆπ‘‡3𝑖(π‘₯03)(𝑖=1,2,3). For 𝑖=1,2,3, let π‘₯1𝑖=π‘₯0π‘–βˆ’π‘“π‘–ξ€·π‘₯0𝑖+𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–ξ€·π‘₯0𝑖,β‹…,πœ™π‘–ξ€Ίπ»π‘–βˆ˜π‘“π‘–ξ€·π‘₯0π‘–ξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯01βˆ’π‘1ξ€·π‘₯01ξ€Έ,π‘₯02βˆ’π‘2ξ€·π‘₯02ξ€Έ,π‘₯03βˆ’π‘3ξ€·π‘₯03ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’0𝑖1,𝑒0𝑖2,𝑒0𝑖3.ξ€Έξ€»(4.12) Since 𝑒01π‘–βˆˆπ‘‡1𝑖(π‘₯01), 𝑒02π‘–βˆˆπ‘‡2𝑖(π‘₯02), 𝑒03π‘–βˆˆπ‘‡3𝑖(π‘₯03)(𝑖=1,2,3), by Nodler's theorem [26], there exist 𝑒11π‘–βˆˆπ‘‡1𝑖(π‘₯11), 𝑒12π‘–βˆˆπ‘‡2𝑖(π‘₯12), 𝑒13π‘–βˆˆπ‘‡3𝑖(π‘₯13)(𝑖=1,2,3), such that, for each 𝑖=1,2,3, ‖‖𝑒11π‘–βˆ’π‘’01𝑖‖‖≀𝐻𝑇(1+1)1𝑖π‘₯11ξ€Έ,𝑇1𝑖π‘₯01,‖‖𝑒12π‘–βˆ’π‘’02𝑖‖‖𝐻𝑇≀(1+1)2𝑖π‘₯12ξ€Έ,𝑇2𝑖π‘₯02,‖‖𝑒13π‘–βˆ’π‘’03𝑖‖‖≀𝐻𝑇(1+1)3𝑖π‘₯13ξ€Έ,𝑇3𝑖π‘₯03.ξ€Έξ€Έ(4.13) For 𝑖=1,2,3, let π‘₯2𝑖=π‘₯1π‘–βˆ’π‘“π‘–ξ€·π‘₯1𝑖+𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–ξ€·π‘₯1𝑖,β‹…,πœ™π‘–ξ€Ίπ»π‘–βˆ˜π‘“π‘–ξ€·π‘₯1π‘–ξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯11βˆ’π‘1ξ€·π‘₯11ξ€Έ,π‘₯12βˆ’π‘2ξ€·π‘₯12ξ€Έ,π‘₯13βˆ’π‘3ξ€·π‘₯13ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’1𝑖1,𝑒1𝑖2,𝑒1𝑖3.ξ€Έξ€»(4.14) Again by Nodler's theorem [26], there exist 𝑒21π‘–βˆˆπ‘‡1𝑖(π‘₯21), 𝑒22π‘–βˆˆπ‘‡2𝑖(π‘₯22), 𝑒23π‘–βˆˆπ‘‡3𝑖(π‘₯23)(𝑖=1,2,3), such that, for each 𝑖=1,2,3, ‖‖𝑒21π‘–βˆ’π‘’11𝑖‖‖≀11+2𝐻𝑇1𝑖π‘₯21ξ€Έ,𝑇1𝑖π‘₯11,‖‖𝑒22π‘–βˆ’π‘’12𝑖‖‖≀11+2𝐻𝑇2𝑖π‘₯22ξ€Έ,𝑇2𝑖π‘₯12,‖‖𝑒23π‘–βˆ’π‘’13𝑖‖‖≀11+2𝐻𝑇3𝑖π‘₯23ξ€Έ,𝑇3𝑖π‘₯13.ξ€Έξ€Έ(4.15) By induction, we can compute the sequences π‘₯𝑛𝑖, 𝑒𝑛1𝑖, 𝑒𝑛2𝑖, 𝑒𝑛3𝑖(𝑖=1,2,3) by the following iterative schemes such that, for each 𝑖=1,2,3, π‘₯𝑖𝑛+1=π‘₯π‘›π‘–βˆ’π‘“π‘–ξ€·π‘₯𝑛𝑖+𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–ξ€·π‘₯𝑛𝑖,β‹…,πœ™π‘–ξ€Ίπ»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯𝑛1βˆ’π‘1ξ€·π‘₯𝑛1ξ€Έ,π‘₯𝑛2βˆ’π‘2ξ€·π‘₯𝑛2ξ€Έ,π‘₯𝑛3βˆ’π‘3ξ€·π‘₯𝑛3ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’π‘›π‘–1,𝑒𝑛𝑖2,𝑒𝑛𝑖3,𝑒𝑛1π‘–βˆˆπ‘‡1𝑖π‘₯𝑛1ξ€Έ,‖‖𝑒𝑛1π‘–βˆ’π‘’π‘›βˆ’11𝑖‖‖≀11+𝑛𝐻𝑇1𝑖π‘₯𝑛1ξ€Έ,𝑇1𝑖π‘₯1π‘›βˆ’1,𝑒𝑛2π‘–βˆˆπ‘‡2𝑖π‘₯𝑛2ξ€Έ,‖‖𝑒𝑛2π‘–βˆ’π‘’π‘›βˆ’12𝑖‖‖≀11+𝑛𝐻𝑇2𝑖π‘₯𝑛2ξ€Έ,𝑇2𝑖π‘₯2π‘›βˆ’1,𝑒𝑛3π‘–βˆˆπ‘‡3𝑖π‘₯𝑛3ξ€Έ,‖‖𝑒𝑛3π‘–βˆ’π‘’π‘›βˆ’13𝑖‖‖≀11+𝑛𝐻𝑇3𝑖π‘₯𝑛3ξ€Έ,𝑇3𝑖π‘₯3π‘›βˆ’1,ξ€Έξ€Έ(4.16) for all 𝑛=0,1,2,….

Now we prove the existence of solution of the SGVIP (4.1) and the convergence of Algorithm 4.2.

Theorem 4.3. Let, for 𝑖=1,2,3, 𝐸𝑖 be a q-uniformly smooth Banach space and let πœ™π‘–βˆΆπΈπ‘–β†’πΈπ‘– be a πœƒπ‘–-Lipschitz continuous mapping satisfying πœ™π‘–(π‘₯+𝑦)=πœ™π‘–(π‘₯)+πœ™π‘–(𝑦) and kerπœ™π‘–={0}. Let πœ‚π‘–βˆΆπΈπ‘–Γ—πΈπ‘–β†’πΈπ‘– be πœπ‘–-Lipschitz continuous, let π»π‘–βˆΆπΈπ‘–β†’πΈπ‘– be a 𝛾𝑖-strongly πœ‚π‘–-accretive and 𝑠𝑖-Lipschitz continuous mapping, π‘“π‘–βˆΆπΈπ‘–β†’πΈπ‘– be (𝛼𝑖,πœ‡π‘–)-relaxed cocoercive and πœ‰π‘–-Lipschitz continuous, and let π‘π‘–βˆΆπΈπ‘–β†’πΈπ‘– be a strongly accretive mapping with constant 𝛿𝑝𝑖 and Lipschitz continuous with constant πœ†π‘π‘–. Suppose that πΉπ‘–βˆΆβˆ3π‘˜=1πΈπ‘˜β†’πΈπ‘– is π›½π‘–π‘˜-Lipschitz continuous in the kth argument and π‘π‘–βˆΆβˆ3π‘˜=1πΈπ‘˜β†’πΈπ‘– be πœπ‘–π‘˜-Lipschitz continuous in the kth argument for π‘˜=1,2,3, π‘€π‘–βˆΆπΈπ‘–Γ—πΈπ‘–β†’2𝐸𝑖 is a (𝐻𝑖,πœ™π‘–)-πœ‚π‘–-accretive mapping in the second argument, and set-valued mappings 𝑇1π‘–βˆΆπΈ1→𝐢𝐡(𝐸𝑖), 𝑇2π‘–βˆΆπΈ2→𝐢𝐡(𝐸𝑖), 𝑇3π‘–βˆΆπΈ3→𝐢𝐡(𝐸𝑖) are 𝐻-Lipschitz continuous with constants 𝑑1𝑖>0, 𝑑2𝑖>0, 𝑑3𝑖>0, respectively. In addition if for all 𝑖=1,2,3, one has ‖‖‖𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–ξ€·π‘₯𝑖1ξ€Έ,β‹…,πœ™π‘–ξ€·π‘§π‘–ξ€Έβˆ’π½π»π‘–,πœ‚π‘–π‘€π‘–ξ€·π‘₯𝑖2ξ€Έ,β‹…,πœ™π‘–ξ€·π‘§π‘–ξ€Έβ€–β€–β€–β‰€π›Ώπ‘–β€–β€–π‘₯𝑖1βˆ’π‘₯𝑖2β€–β€–,βˆ€π‘₯𝑖1,π‘₯𝑖2,π‘§π‘–βˆˆπΈπ‘–,(4.17)ξ€·0<1+π‘žπ›Ό1πœ‰π‘ž1βˆ’π‘žπœ‡1+π‘π‘žπœ‰π‘ž1ξ€Έ1/π‘ž+𝛿1+𝜏1π‘žβˆ’1𝛾1Γ—ξƒ©ξ‚€π‘ π‘ž1πœ‰π‘ž1+π‘žπœƒ1𝛽11𝑠1π‘žβˆ’1πœ‰1π‘žβˆ’1ξ€·1βˆ’π‘žπ›Ώπ‘1+π‘π‘žπœ†π‘žπ‘1ξ€Έ1/π‘ž+π‘π‘žπœƒπ‘ž1π›½π‘ž11ξ‚€1βˆ’π‘žπ›Ώπ‘1+π‘π‘žπœ†π‘žπ‘ƒ11/π‘ž+3𝑗=1πœƒ1𝜁1𝑗𝑑1𝑗ξƒͺ+ξƒ©πœƒ2𝛽21𝜏2π‘žβˆ’1𝛾2+πœƒ3𝛽31𝜏3π‘žβˆ’1𝛾3ξƒͺξ‚€1βˆ’π‘žπ›Ώπ‘1+π‘π‘žπœ†π‘žπ‘ƒ11/π‘žξ€·<1,0<1+π‘žπ›Ό2πœ‰π‘ž2βˆ’π‘žπœ‡2+π‘π‘žπœ‰π‘ž2ξ€Έ1/π‘ž+𝛿2+𝜏2π‘žβˆ’1𝛾2Γ—ξƒ©ξ‚€π‘ π‘ž2πœ‰π‘ž2+π‘žπœƒ2𝛽22𝑠2π‘žβˆ’1πœ‰2π‘žβˆ’1ξ€·1βˆ’π‘žπ›Ώπ‘2+π‘π‘žπœ†π‘žπ‘2ξ€Έ1/π‘ž+π‘π‘žπœƒπ‘ž2π›½π‘ž22ξ‚€1βˆ’π‘žπ›Ώπ‘2+π‘π‘žπœ†π‘žπ‘ƒ21/π‘ž+3𝑗=1πœƒ2𝜁2𝑗𝑑2𝑗ξƒͺ+ξƒ©πœƒ1𝛽12𝜏1π‘žβˆ’1𝛾1+πœƒ3𝛽32𝜏3π‘žβˆ’1𝛾3ξƒͺξ‚€1βˆ’π‘žπ›Ώπ‘2+π‘π‘žπœ†π‘žπ‘ƒ21/π‘žξ€·<1,0<1+π‘žπ›Ό3πœ‰π‘ž3βˆ’π‘žπœ‡3+π‘π‘žπœ‰π‘ž3ξ€Έ1/π‘ž+𝛿3+𝜏3π‘žβˆ’1𝛾3Γ—ξƒ©ξ‚€π‘ π‘ž3πœ‰π‘ž3+π‘žπœƒ3𝛽33𝑠3π‘žβˆ’1πœ‰3π‘žβˆ’1ξ€·1βˆ’π‘žπ›Ώπ‘3+π‘π‘žπœ†π‘žπ‘3ξ€Έ1/π‘ž+π‘π‘žπœƒπ‘ž3π›½π‘ž33ξ‚€1βˆ’π‘žπ›Ώπ‘3+π‘π‘žπœ†π‘žπ‘ƒ31/π‘ž+3𝑗=1πœƒ3𝜁3𝑗𝑑3𝑗ξƒͺ+ξƒ©πœƒ1𝛽13𝜏1π‘žβˆ’1𝛾1+πœƒ2𝛽23𝜏2π‘žβˆ’1𝛾2ξƒͺξ‚€1βˆ’π‘žπ›Ώπ‘3+π‘π‘žπœ†π‘žπ‘ƒ31/π‘ž<1.(4.18) Then the problem (4.1) admits a solution (π‘₯1,π‘₯2,π‘₯3,𝑒11,𝑒12,𝑒13,𝑒21,𝑒22,𝑒23,𝑒31,𝑒32,𝑒33) and sequences π‘₯𝑛1,π‘₯𝑛2,π‘₯𝑛3,𝑒𝑛11,𝑒𝑛12,𝑒𝑛13,𝑒𝑛21,𝑒𝑛22,𝑒𝑛23,𝑒𝑛31,𝑒𝑛32,𝑒𝑛33 converge to π‘₯1,π‘₯2,π‘₯3, 𝑒11, 𝑒12, 𝑒13, 𝑒21, 𝑒22, 𝑒23, 𝑒31, 𝑒32, 𝑒33, respectively, where π‘₯𝑛𝑖,𝑒𝑛1𝑖,𝑒𝑛2𝑖,𝑒𝑛3𝑖(𝑖=1,2,3) are sequences generated by Algorithm 4.2.

Proof. For 𝑖=1,2,3, let 𝐷𝑛𝑖=π»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯𝑛1βˆ’π‘1ξ€·π‘₯𝑛1ξ€Έ,π‘₯𝑛2βˆ’π‘2ξ€·π‘₯𝑛2ξ€Έ,π‘₯𝑛3βˆ’π‘3ξ€·π‘₯𝑛3ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’π‘›π‘–1,𝑒𝑛𝑖2,𝑒𝑛𝑖3ξ€Έ.(4.19) By Algorithm 4.2 and (4.17), we have β€–β€–π‘₯𝑖𝑛+1βˆ’π‘₯𝑛𝑖‖‖=β€–β€–β€–π‘₯π‘›π‘–βˆ’π‘“π‘–ξ€·π‘₯𝑛𝑖+𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–ξ€·π‘₯𝑛𝑖,β‹…,πœ™π‘–ξ€Ίπ»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯𝑛1βˆ’π‘1ξ€·π‘₯𝑛1ξ€Έ,π‘₯𝑛2βˆ’π‘2ξ€·π‘₯𝑛2ξ€Έ,π‘₯𝑛3βˆ’π‘3ξ€·π‘₯𝑛3ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’π‘›π‘–1,𝑒𝑛𝑖2,𝑒𝑛𝑖3βˆ’ξ€·π‘₯ξ€Έξ€»π‘–π‘›βˆ’1βˆ’π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έ+𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έ,β‹…,πœ™π‘–ξ€Ίπ»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯1π‘›βˆ’1βˆ’π‘1ξ€·π‘₯1π‘›βˆ’1ξ€Έ,π‘₯2π‘›βˆ’1βˆ’π‘2ξ€·π‘₯2π‘›βˆ’1ξ€Έ,π‘₯3π‘›βˆ’1βˆ’π‘3ξ€·π‘₯3π‘›βˆ’1ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’π‘›βˆ’1𝑖1,π‘’π‘›βˆ’1𝑖2,π‘’π‘›βˆ’1𝑖3‖‖‖≀‖‖π‘₯ξ€Έξ€»ξ€Έπ‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1βˆ’ξ€Ίπ‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1β€–β€–+‖‖‖𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–ξ€·π‘₯𝑛𝑖,β‹…,πœ™π‘–ξ€·π·π‘›π‘–ξ€Έβˆ’π½π»π‘–,πœ‚π‘–π‘€π‘–ξ€·π‘₯𝑛𝑖,β‹…,πœ™π‘–ξ€·π·π‘–π‘›βˆ’1ξ€Έβ€–β€–β€–+‖‖‖𝐽𝐻𝑖,πœ‚π‘–π‘€π‘–(π‘₯𝑛𝑖,β‹…),πœ™π‘–ξ€·π·π‘–π‘›βˆ’1ξ€Έβˆ’π½π»π‘–,πœ‚π‘–π‘€π‘–(π‘₯π‘–π‘›βˆ’1,β‹…),πœ™π‘–ξ€·π·π‘–π‘›βˆ’1‖‖‖≀‖‖π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1βˆ’ξ€Ίπ‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1β€–β€–+πœξ€Έξ€»π‘–π‘žβˆ’1π›Ύπ‘–β€–β€–π·π‘›π‘–βˆ’π·π‘–π‘›βˆ’1β€–β€–+𝛿𝑖‖‖π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1β€–β€–.(4.20) Since π‘“π‘–βˆΆπΈπ‘–β†’πΈπ‘– is (𝛼𝑖,πœ‡π‘–)-relaxed cocoercive and πœ‰π‘–-Lipschitz continuous and π‘π‘–βˆΆπΈπ‘–β†’πΈπ‘– is 𝛿𝑝𝑖-strongly accretive and πœ†π‘π‘–-Lipschitz continuous, we have β€–β€–π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1βˆ’ξ€Ίπ‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1β€–β€–ξ€Έξ€»π‘žβ‰€β€–β€–π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1β€–β€–π‘žξ«π‘“βˆ’π‘žπ‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έ,π½π‘žξ€·π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1+π‘π‘žβ€–β€–π‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έβ€–β€–π‘žβ‰€ξ€·1+π‘žπ›Όπ‘–πœ‰π‘žπ‘–βˆ’π‘žπœ‡π‘–+π‘π‘žπœ‰π‘žπ‘–ξ€Έβ€–β€–π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1β€–β€–π‘ž,β€–β€–π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1βˆ’ξ€Ίπ‘π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π‘π‘–ξ€·π‘₯π‘–π‘›βˆ’1β€–β€–ξ€Έξ€»π‘žβ‰€β€–β€–π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1β€–β€–π‘žξ«π‘βˆ’π‘žπ‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π‘π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έ,π½π‘žξ€·π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1+π‘π‘žβ€–β€–π‘π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π‘π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έβ€–β€–π‘žβ‰€ξ€·1βˆ’π‘žπ›Ώπ‘π‘–+π‘π‘žπœ†π‘žπ‘π‘–ξ€Έβ€–β€–π‘₯π‘›π‘–βˆ’π‘₯π‘–π‘›βˆ’1β€–β€–π‘ž.(4.21) From (4.19), we have β€–β€–π·π‘›π‘–βˆ’π·π‘–π‘›βˆ’1β€–β€–=β€–β€–π»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯𝑛1βˆ’π‘1ξ€·π‘₯𝑛1ξ€Έ,π‘₯𝑛2βˆ’π‘2ξ€·π‘₯𝑛2ξ€Έ,π‘₯𝑛3βˆ’π‘3ξ€·π‘₯𝑛3ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’π‘›π‘–1,𝑒𝑛𝑖2,𝑒𝑛𝑖3ξ€Έβˆ’ξ€·π»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯1π‘›βˆ’1βˆ’π‘1ξ€·π‘₯1π‘›βˆ’1ξ€Έ,π‘₯2π‘›βˆ’1βˆ’π‘2ξ€·π‘₯2π‘›βˆ’1ξ€Έ,π‘₯3π‘›βˆ’1βˆ’π‘3ξ€·π‘₯3π‘›βˆ’1ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’π‘›βˆ’1𝑖1,π‘’π‘›βˆ’1𝑖2,π‘’π‘›βˆ’1𝑖3β€–β€–β‰€β€–β€–π»ξ€Έξ€Έπ‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έ-ξ€·πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯𝑛1βˆ’π‘1ξ€·π‘₯𝑛1ξ€Έ,π‘₯𝑛2βˆ’π‘2ξ€·π‘₯𝑛2ξ€Έ,π‘₯𝑛3βˆ’π‘3ξ€·π‘₯𝑛3ξ€Έξ€Έβˆ’πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯1π‘›βˆ’1βˆ’π‘1ξ€·π‘₯1π‘›βˆ’1ξ€Έ,π‘₯2π‘›βˆ’1βˆ’π‘2ξ€·π‘₯2π‘›βˆ’1ξ€Έ,π‘₯3π‘›βˆ’1βˆ’π‘3ξ€·π‘₯3π‘›βˆ’1β€–β€–+β€–β€–πœ™ξ€Έξ€Έξ€Έπ‘–βˆ˜π‘π‘–ξ€·π‘’π‘›π‘–1,𝑒𝑛𝑖2,𝑒𝑛𝑖3ξ€Έβˆ’πœ™π‘–βˆ˜π‘π‘–ξ€·π‘’π‘›βˆ’1𝑖1,π‘’π‘›βˆ’1𝑖2,π‘’π‘›βˆ’1𝑖3ξ€Έβ€–β€–.(4.22) Since πΉπ‘–βˆΆβˆ3π‘˜=1πΈπ‘˜β†’πΈπ‘– is π›½π‘–π‘˜-Lipschitz continuous in the π‘˜th argument and by continuity of H𝑖, 𝑓𝑖, πœ™π‘–(𝑖=1,2,3), we have β€–β€–π»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘›π‘–ξ€Έβˆ’π»π‘–βˆ˜π‘“π‘–ξ€·π‘₯π‘–π‘›βˆ’1ξ€Έβˆ’ξ€·πœ™π‘–βˆ˜πΉπ‘–ξ€·π‘₯𝑛1βˆ’π‘1ξ€·π‘₯𝑛1ξ€Έ,π‘₯𝑛2βˆ’π‘2ξ€·π‘₯𝑛2ξ€Έ,π‘₯𝑛3βˆ’π‘3ξ€·