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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 560248, 15 pages
http://dx.doi.org/10.1155/2012/560248
Research Article

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

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 15 February 2012; Accepted 27 March 2012

Academic Editor: GiuseppeΒ Marino

Copyright Β© 2012 Prapairat Junlouchai and Somyot Plubtieng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a new system of nonlinear set-valued variational inclusions involving a finite family of 𝐻(β‹…,β‹…)-accretive operators in Banach spaces. By using the resolvent operator technique associated with a finite family of 𝐻(β‹…,β‹…)-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,πœ‚)-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(β‹…,β‹…)-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(β‹…,β‹…)-accretive operator in Banach spaces. By using the resolvent operator technique associated with 𝐻(β‹…,β‹…)-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(β‹…,β‹…)-accretive operators in Banach spaces. By using the resolvent operators technique associated with a finite family of 𝐻(β‹…,β‹…)-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 π‘‹βˆ—,βŸ¨β‹…,β‹…βŸ© the dual pair between 𝑋 and π‘‹βˆ— and 2𝑋 and 𝐢(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 π½π‘žβˆΆπ‘‹β†’2π‘‹βˆ— is defined byπ½π‘ž(𝑋)=ξ€½π‘“βˆ—βˆˆπ‘‹βˆ—βˆΆβŸ¨π‘₯,π‘“βˆ—βŸ©=β€–π‘₯β€–π‘ž,β€–π‘“βˆ—β€–=β€–π‘₯β€–π‘žβˆ’1ξ€Ύ,βˆ€π‘₯βˆˆπ‘‹,(2.1) where π‘ž>1 is a constant. It is known that, in general, π½π‘ž(π‘₯)=β€–π‘₯β€–π‘žβˆ’1𝐽2(π‘₯) for all π‘₯β‰ 0 and π½π‘ž is single-valued if π‘‹βˆ— is strictly convex. In the sequel, we always assume that 𝑋 is a real Banach space such that π½π‘ž is single-valued.

The modulus of smoothness of 𝑋 is the function πœŒπ‘‹βˆΆ[0,∞)β†’[0,∞) defined byπœŒπ‘‹(𝑑)=supξ‚»β€–π‘₯+𝑦‖+β€–π‘₯βˆ’π‘¦β€–2βˆ’1βˆΆβ€–π‘₯‖≀1,‖𝑦‖≀𝑑.(2.2) A Banach space 𝑋 is called uniformly smooth iflim𝑑→0πœŒπ‘‹(𝑑)𝑑=0.(2.3)𝑋 is called π‘ž-uniformly smooth if there exists a constant 𝑐>0 such thatπœŒπ‘‹(𝑑)β‰€π‘π‘‘π‘ž,π‘ž>1.(2.4)

Note that π½π‘ž is single valued if 𝑋 is uniformly smooth. In the study of characteristic inequalities in π‘ž-uniformly smooth Banach spaces, Xu [18] proved the following result.

Definition 2.1. Let 𝐻,πœ‚βˆΆπ‘‹Γ—π‘‹β†’π‘‹ be two single-valued mappings and 𝐴,π΅βˆΆπ‘‹β†’π‘‹ two single-valued mappings.(i)𝐴 is said to be accretive if𝐴π‘₯βˆ’π΄π‘¦,π½π‘ž(π‘₯βˆ’π‘¦)β‰₯0,βˆ€π‘₯,π‘¦βˆˆπ‘‹,(2.5)(ii)𝐴 is said to be strictly accretive if 𝐴 is accretive and 𝐴π‘₯βˆ’π΄π‘¦,π½π‘ž(π‘₯βˆ’π‘¦)=0,βˆ€π‘₯,π‘¦βˆˆπ‘‹,(2.6) if and only if π‘₯=𝑦;(iii)𝐻(𝐴,β‹…) is said to be 𝛼-strongly accretive with respect to 𝐴 if there exists a constant 𝛼>0 such that𝐻(𝐴π‘₯,𝑒)βˆ’π»(𝐴𝑦,𝑒),π½π‘ž(π‘₯βˆ’π‘¦)β‰₯𝛼‖π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,𝑦,π‘’βˆˆπ‘‹;(2.7)(iv)𝐻(β‹…,𝐡) is said to be 𝛽-relaxed accretive with respect to 𝐡 if there exists a constant 𝛽>0 such that𝐻(𝑒,𝐡π‘₯)βˆ’π»(𝑒,𝐡𝑦),π½π‘ž(π‘₯βˆ’π‘¦)β‰₯βˆ’π›½β€–π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,𝑦,π‘’βˆˆπ‘‹;(2.8)(v)𝐻(β‹…,β‹…) is said to be 𝛾-Lipschitz continuous with respect to 𝐴 if there exists a constant 𝛾>0 such that‖𝐻(𝐴π‘₯,𝑒)βˆ’π»(𝐴𝑦,𝑒)‖≀𝛾‖π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,𝑦,π‘’βˆˆπ‘‹;(2.9)(vi)𝐴 is said to be πœƒ-Lipschitz continuous if there exists a constant πœƒ>0 such that‖𝐴π‘₯βˆ’π΄π‘¦β€–β‰€πœƒβ€–π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,π‘¦βˆˆπ‘‹;(2.10)(vii)πœ‚(β‹…,β‹…) is said to be strongly accretive with respect to 𝐻(𝐴,𝐡) if there exists a constant 𝜌>0 such thatξ«πœ‚(π‘₯,𝑒)βˆ’πœ‚(𝑦,𝑒),π½π‘ž(𝐻(𝐴π‘₯,𝐡π‘₯)βˆ’π»(𝐴𝑦,𝐡𝑦))β‰₯πœŒβ€–π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,𝑦,π‘’βˆˆπ‘‹.(2.11)

Definition 2.2. Let πœ‚βˆΆπ‘‹Γ—π‘‹β†’π‘‹ be single-valued mapping. Let π‘€βˆΆπ‘‹β†’2𝑋 be a set-valued mapping.(i)πœ‚ is said to be 𝒯-Lipschitz continuous if there exists a constant 𝒯>0 such thatβ€–πœ‚(π‘₯,𝑦)‖≀𝒯‖π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπ‘‹;(2.12)(ii) 𝑀 is said to be accretive ifξ«π‘’βˆ’π‘£,π½π‘ž(π‘₯βˆ’π‘¦)β‰₯0,βˆ€π‘₯,π‘¦βˆˆπ‘‹,π‘’βˆˆπ‘€(π‘₯),π‘£βˆˆπ‘€(𝑦);(2.13)(iii) 𝑀 is said to be πœ‚-accretive ifξ«π‘’βˆ’π‘£,π½π‘ž(πœ‚(π‘₯,𝑦))β‰₯0,βˆ€π‘₯,π‘¦βˆˆπ‘‹,π‘’βˆˆπ‘€(π‘₯),π‘£βˆˆπ‘€(𝑦);(2.14)(iv)𝑀 is said to be strictly πœ‚-accretive if 𝑀 is πœ‚-accretive and equality holds if and only if π‘₯=𝑦;(v) 𝑀 is said to be 𝛾-strongly πœ‚-accretive if there exists a positive constant 𝛾>0 such thatξ«π‘’βˆ’π‘£,π½π‘ž(πœ‚(π‘₯,𝑦))β‰₯𝛾‖π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,π‘¦βˆˆπ‘‹,π‘’βˆˆπ‘€(π‘₯),π‘£βˆˆπ‘€(𝑦);(2.15)(vi) 𝑀 is said to be 𝛼-relaxed πœ‚-accretive if there exists a positive constant 𝛼>0 such thatξ«π‘’βˆ’π‘£,π½π‘ž(πœ‚(π‘₯,𝑦))β‰₯βˆ’π›Όβ€–π‘₯βˆ’π‘¦β€–π‘ž,βˆ€π‘₯,π‘¦βˆˆπ‘‹,π‘’βˆˆπ‘€(π‘₯),π‘£βˆˆπ‘€(𝑦).(2.16)

Definition 2.3. Let 𝐴,π΅βˆΆπ‘‹β†’π‘‹, π»βˆΆπ‘‹Γ—π‘‹β†’π‘‹ be three single-valued mappings. Let π‘€βˆΆπ‘‹β†’2𝑋 be a set-valued mapping. 𝑀 is said to be 𝐻(β‹…,β‹…)-accretive with respect to 𝐴 and 𝐡 (or simply 𝐻(β‹…,β‹…)-accretive in the sequel), if 𝑀 is accretive and (𝐻(𝐴,𝐡)+πœ†π‘€)(𝑋)=𝑋 for every πœ†>0.

Lemma 2.4. Let 𝑋 be a real uniformly smooth Banach space. Then 𝑋 is π‘ž-uniformly smooth if and only if there exists a constant π‘π‘ž>0 such that for all π‘₯,π‘¦βˆˆπ‘‹β€–π‘₯+π‘¦β€–π‘žβ‰€β€–π‘₯β€–π‘ž+π‘žξ«π‘¦,π½π‘ž(π‘₯)+π‘π‘žβ€–π‘¦β€–π‘ž.(2.17)

Lemma 2.5 (see[16]). Let 𝐻(𝐴,𝐡) be 𝛼-strongly accretive with respect to 𝐴, 𝛽-relaxed accretive with respect to 𝐡, and 𝛼>𝛽. Let 𝑀 be an 𝐻(β‹…,β‹…)-accretive operator with respect to 𝐴 and 𝐡. Then, the operator 𝐻((𝐴,𝐡)+πœ†π‘€)βˆ’1 is single valued. Based on Lemma 2.4, one can define the resolvent operator 𝑅𝐻(β‹…,β‹…)𝑀,πœ† as follows.

Definition 2.6. Let 𝐻,𝐴,𝐡,𝑀 be defined as in Definition 2.3. Let 𝐻(𝐴,𝐡) be 𝛼-strongly accretive with respect to 𝐴, 𝛽-relaxed accretive with respect to 𝐡, and 𝛼>𝛽. Let 𝑀 be an 𝐻(β‹…,β‹…)-accretive operator with respect to 𝐴 and 𝐡. The resolvent operator 𝑅𝐻(β‹…,β‹…)𝑀,πœ†βˆΆπ‘‹β†’π‘‹ is defined by 𝑅𝐻(β‹…,β‹…)𝑀,πœ†(𝑧)=(𝐻(𝐴,𝐡)+πœ†π‘€)βˆ’1(𝑧),βˆ€π‘§βˆˆπ‘‹,(2.18) where πœ†>0 is a constant.

Lemma 2.7 (see [16]). Let 𝐻,𝐴,𝐡,𝑀 be defined as in Definition 2.3. Let 𝐻(𝐴,𝐡) be 𝛼-strongly accretive with respect to 𝐴, 𝛽-relaxed accretive with respect to 𝐡, and 𝛼>𝛽. Suppose that π‘€βˆΆπ‘‹β†’2𝑋 is an 𝐻(β‹…,β‹…)-accretive operator. Then resolvent operator 𝑅𝐻(β‹…,β‹…)𝑀,πœ† defined by (2.18) is 1/(π›Όβˆ’π›½) Lipschitz continuous. That is, ‖‖𝑅𝐻(β‹…,β‹…)𝑀,πœ†(π‘₯)βˆ’π‘…π»(β‹…,β‹…)𝑀,πœ†(𝑦)‖‖≀1π›Όβˆ’π›½β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπ‘‹.(2.19)

We define a Hausdorff pseudometric 𝐷∢2𝑋×2𝑋→[0,+∞] by𝐷(π‘ˆ,𝑉)=maxξ‚»supπ‘’βˆˆπ‘ˆinfπ‘£βˆˆπ‘‰β€–π‘’βˆ’π‘£β€–,supπ‘’βˆˆπ‘‰infπ‘£βˆˆπ‘ˆβ€–π‘’βˆ’π‘£β€–ξ‚Ό(2.20) for any given π‘ˆ,π‘‰βˆˆ2𝑋. 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 {𝑐𝑛} and {π‘˜π‘›} be two real sequences of nonnegative numbers that satisfy the following conditions:(i)0<π‘˜π‘›<1 for 𝑛=0,1,2,…, and limsupπ‘›π‘˜π‘›<1;(ii)𝑐𝑛+1β‰€π‘˜π‘›π‘π‘› for 𝑛=0,1,2,….Then, 𝑐𝑛 converges to 0 as π‘›β†’βˆž.

3. Main Result

Let 𝑋 be π‘ž-uniformly smooth real Banach space and 𝐢(𝑋) a nonempty closed convex set. Let 𝑆𝑖,π»π‘–βˆΆπ‘‹Γ—π‘‹β†’π‘‹, 𝐴𝑖,π΅π‘–βˆΆπ‘‹β†’π‘‹ be single-valued operators, for all 𝑖=1,2,…,𝑁. For any fix π‘–βˆˆ{1,2,…,𝑁}, we let π‘€π‘–βˆΆπ‘‹β†’2𝑋, 𝐻𝑖(𝐴𝑖,𝐡𝑖)-accretive set-valued operator and π‘ˆπ‘–βˆΆπ‘‹β†’2𝑋 a set-valued mapping which nonempty values. The system of nonlinear set-valued variational inclusions is to find π‘Ž1,…,π‘Žπ‘βˆˆπ‘‹, 𝑒1βˆˆπ‘ˆ1(π‘Žπ‘),…,π‘’π‘βˆˆπ‘ˆπ‘(π‘Ž1) such that0βˆˆπ‘†π‘–ξ€·π‘Žπ‘–,𝑒𝑖+π‘€π‘–ξ€·π‘Žπ‘–ξ€Έ,βˆ€π‘–=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 π‘Ž1,π‘Ž2βˆˆπ‘‹, 𝑒1βˆˆπ‘ˆ1(π‘Ž2) and 𝑒2βˆˆπ‘ˆ2(π‘Ž1) such that0βˆˆπ‘†1ξ€·π‘Ž1,𝑒1ξ€Έ+𝑀1ξ€·π‘Ž1ξ€Έ,0βˆˆπ‘†2ξ€·π‘Ž2,𝑒2ξ€Έ+𝑀2ξ€·π‘Ž2ξ€Έ.(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 π‘Žβˆˆπ‘‹, π‘’βˆˆπ‘ˆ(π‘Ž) such that0βˆˆπ‘†(π‘Ž,𝑒)+𝑀(π‘Ž).(3.3)

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

For any π‘–βˆˆ{1,2,…,𝑁}, we suppose that(A1)𝐻(𝐴𝑖,𝐡𝑖) is 𝛼𝑖-strongly accretive with respect to 𝐴𝑖, 𝛽𝑖-relaxed accretive with respect to 𝐡𝑖 and 𝛼𝑖>𝛽𝑖,(A2)π‘€π‘–βˆΆπ‘‹β†’2𝑋 is an 𝐻𝑖(β‹…,β‹…)-accretive single-valued mapping,(A3)π‘ˆπ‘–βˆΆπ‘‹β†’πΆ(𝑋) is a contraction set-valued mapping with 0≀𝐿𝑖<1 and nonempty values,(A4)𝐻𝑖(𝐴𝑖,𝐡𝑖) is π‘Ÿπ‘–-Lipschitz continuous with respect to 𝐴𝑖 and 𝑑𝑖-Lipschitz continuous with respect to 𝐡𝑖,(A5)π‘†π‘–βˆΆπ‘‹Γ—π‘‹β†’π‘‹ is 𝑙𝑖-Lipschitz continuous with respect to its first argument and π‘šπ‘–-Lipschitz continuous with respect to its second argument,(A6)𝑆𝑖(β‹…,𝑒) is 𝑠𝑖-strongly accretive with respect to 𝐻𝑖(𝐴𝑖,𝐡𝑖).

Theorem 3.1. For given π‘Ž1,…,π‘Žπ‘βˆˆπ‘‹, 𝑒1βˆˆπ‘ˆ1(π‘Žπ‘),…,π‘’π‘βˆˆπ‘ˆπ‘(π‘Ž1), it is a solution of problem (3.1) if and only if π‘Žπ‘–=𝑅𝐻𝑖(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–,𝑒𝑖,(3.4) where πœ†π‘–>0 are constants.

Proof. We note from the Definition 2.6 that π‘Ž1,…,π‘Žπ‘βˆˆπ‘‹, 𝑒1βˆˆπ‘ˆ1(π‘Žπ‘),…,π‘’π‘βˆˆπ‘ˆπ‘(π‘Ž1) is a solution of (3.1) if and only if, for each π‘–βˆˆ{1,2,…,𝑁}, we have π‘Žπ‘–=𝑅𝐻𝑖(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–,π‘’π‘–ξ€Έξ€»βŸΊπ‘Žπ‘–=𝐻𝑖𝐴𝑖,𝐡𝑖+πœ†π‘–π‘€π‘–ξ€»βˆ’1ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–,π‘’π‘–ξ€Έξ€»βŸΊξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–ξ€Έξ€Έβˆ’πœ†π‘–π‘†iξ€·π‘Žπ‘–,π‘’π‘–ξ€Έξ€»βˆˆξ€Ίπ»π‘–ξ€·π΄π‘–,𝐡𝑖+πœ†π‘–π‘€π‘–ξ€»ξ€·π‘Žπ‘–ξ€ΈβŸΊβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–,π‘’π‘–ξ€Έβˆˆπœ†π‘–π‘€π‘–ξ€·π‘Žπ‘–ξ€ΈβŸΊ0βˆˆπ‘†π‘–ξ€·π‘Žπ‘–,𝑒𝑖+π‘€π‘–ξ€·π‘Žπ‘–ξ€Έ.(3.5)

Algorithm 3.2. For given π‘Ž10,…,π‘Žπ‘0βˆˆπ‘‹, 𝑒10βˆˆπ‘ˆ1(π‘Žπ‘0),…,𝑒𝑁0βˆˆπ‘ˆπ‘(π‘Ž10), we let π‘Žπ‘–1=𝜎0π‘Žπ‘–0+ξ€·1βˆ’πœŽ0𝑅𝐻𝑖(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–0ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–0ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–0,𝑒𝑖0ξ€Έξ€»,(3.6) for all 𝑖=1,2,…,𝑁, where 0<𝜎0≀1. By Nadler theorem [20], there exists 𝑒11βˆˆπ‘ˆ1(π‘Žπ‘1),…,𝑒𝑁1βˆˆπ‘ˆπ‘(π‘Ž11) such that ‖‖𝑒𝑖1βˆ’π‘’π‘–0‖‖≀(1+1)π·ξ‚€π‘ˆπ‘–ξ‚€π‘Žπ‘βˆ’(π‘–βˆ’1)1,π‘ˆπ‘–ξ‚€π‘Žπ‘βˆ’(π‘–βˆ’1)0,βˆ€π‘–=1,2,…,𝑁,(3.7) where 𝐷(β‹…,β‹…) is the Hausdorff pseudo metric on 2𝑋. Continuing the above process inductively, we can obtain the sequences {π‘Žπ‘–π‘›} and {𝑒𝑖𝑛} such that π‘Žπ‘–π‘›+1=πœŽπ‘›π‘Žπ‘–π‘›+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Ži𝑛,π΅π‘–ξ€·π‘Žπ‘–π‘›ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–π‘›,𝑒𝑖𝑛,(3.8) for all 𝑛=1,2,3,…,𝑖=1,2,…,𝑁, where 0<πœŽπ‘›β‰€1 with limsupπ‘›β†’βˆžπœŽπ‘›<1. Therefore, by Nadler theorem [20], there exists 𝑒1𝑛+1βˆˆπ‘ˆ1(π‘Žπ‘π‘›+1),…,𝑒𝑁𝑛+1βˆˆπ‘ˆπ‘(π‘Ž1𝑛+1) such that ‖‖𝑒𝑖𝑛+1βˆ’π‘’π‘–π‘›β€–β€–β‰€ξ€·1+(1+𝑛)βˆ’1ξ€Έπ·ξ‚€π‘ˆπ‘–ξ‚€π‘Žπ‘βˆ’(π‘–βˆ’1)𝑛+1,π‘ˆπ‘–ξ‚€π‘Žπ‘βˆ’(π‘–βˆ’1)𝑛,βˆ€π‘›=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 𝐴𝑖,π΅π‘–βˆΆπ‘‹β†’π‘‹ be single-valued operators, π»π‘–βˆΆπ‘‹Γ—π‘‹β†’π‘‹ a single-valued operator satisfy (A1) and 𝑀𝑖, π‘ˆπ‘–, 𝐻𝑖(𝐴𝑖,𝐡𝑖), 𝑆𝑖, 𝑆𝑖(β‹…,𝑒) satisfy conditions (A2)–(A6), respectively. If there exists a constant π‘π‘ž,𝑖 such that π‘žξ”ξ€·π‘Ÿπ‘–+π‘‘π‘–ξ€Έπ‘žβˆ’π‘žπœ†π‘–π‘ π‘–+π‘π‘ž,π‘–πœ†π‘žπ‘–π‘™π‘žπ‘–π›Όπ‘–βˆ’π›½π‘–+πœ†π‘–π‘šπ‘–π›Όπ‘–βˆ’π›½π‘–<1(3.10) for all 𝑖=1,2,…,𝑁, then problem (3.1) has a solution π‘Ž1,…,π‘Žπ‘, 𝑒1βˆˆπ‘ˆ1(π‘Žπ‘),…,π‘’π‘βˆˆπ‘ˆπ‘(π‘Ž1).

Proof. For any π‘–βˆˆ{1,2,…,𝑁} and πœ†π‘–>0, we define πΉπ‘–βˆΆπ‘‹Γ—π‘‹β†’π‘‹ by 𝐹𝑖(𝑒,𝑣)=𝑅𝐻𝑖(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–(𝑒),𝐡𝑖(𝑒)ξ€Έβˆ’πœ†π‘–π‘†π‘–(𝑒,𝑣)ξ€»,(3.11) for all 𝑒,π‘£βˆˆπ‘‹. Let 𝐽𝑖(π‘₯,𝑦)=𝐻𝑖(𝐴𝑖(π‘₯),𝐡𝑖(𝑦)). For any (𝑒1,𝑣1),(𝑒2,𝑣2)βˆˆπ‘‹Γ—π‘‹, we note by (3.11) and Lemma 2.7 that ‖‖𝐹𝑖𝑒1,𝑣1ξ€Έβˆ’πΉπ‘–ξ€·π‘’2,𝑣2ξ€Έβ€–β€–=‖‖𝑅𝐻𝑖(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘’1ξ€Έ,𝐡𝑖𝑒1ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘’1,𝑣1ξ€Έξ€»βˆ’π‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘’2ξ€Έ,𝐡𝑖𝑒2ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘’2,𝑣2ξ€Έξ€»β€–β€–=‖‖𝑅𝐻𝑖(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ½π‘–ξ€·π‘’1,𝑒1ξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘’1,𝑣1ξ€Έξ€»βˆ’π‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ½π‘–ξ€·π‘’2,𝑒2ξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘’2,𝑣2‖‖≀1π›Όπ‘–βˆ’π›½π‘–β€–β€–ξ€Ίπ½π‘–ξ€·π‘’1,𝑒1ξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘’1,𝑣1ξ€Έξ€»βˆ’ξ€Ίπ½π‘–ξ€·π‘’2,𝑒2ξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘’2,𝑣2ξ€Έξ€»β€–β€–=1π›Όπ‘–βˆ’π›½π‘–β€–β€–ξ€Ίπ½π‘–ξ€·π‘’1,𝑒1ξ€Έβˆ’π½π‘–ξ€·π‘’2,𝑒2ξ€Έξ€»βˆ’πœ†π‘–ξ€Ίπ‘†π‘–ξ€·π‘’1,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣2‖‖≀1π›Όπ‘–βˆ’π›½π‘–β€–β€–ξ€Ίπ½π‘–ξ€·π‘’1,𝑒1ξ€Έβˆ’π½π‘–ξ€·π‘’2,𝑒2ξ€Έξ€»βˆ’πœ†π‘–ξ€Ίπ‘†π‘–ξ€·π‘’1,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣1ξ€Έξ€»β€–β€–+πœ†π‘–π›Όπ‘–βˆ’π›½π‘–β€–β€–ξ€Ίπ‘†π‘–ξ€·π‘’2,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣2ξ€Έξ€»β€–β€–.(3.12) By Lemma 2.4, we have ‖‖𝐽𝑖𝑒1,𝑒1ξ€Έβˆ’π½π‘–ξ€·π‘’2,𝑒2ξ€Έβˆ’πœ†π‘–ξ€Ίπ‘†π‘–ξ€·π‘’1,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣1ξ€Έξ€»β€–β€–π‘žβ‰€β€–β€–π½π‘–ξ€·π‘’1,𝑒1ξ€Έβˆ’π½π‘–ξ€·π‘’2,𝑒2ξ€Έβ€–β€–π‘žβˆ’π‘žπœ†π‘–ξ«π‘†π‘–ξ€·π‘’1,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣1ξ€Έ,π½π‘žξ€·π½π‘–ξ€·π‘’1,𝑒1ξ€Έβˆ’π½π‘–ξ€·π‘’2,𝑒2+π‘π‘ž,π‘–πœ†π‘žπ‘–β€–β€–π‘†π‘–ξ€·π‘’1,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣1ξ€Έβ€–β€–π‘ž.(3.13) Moreover, by (A4), we obtain ‖‖𝐽𝑖𝑒1,𝑒1ξ€Έβˆ’π½π‘–ξ€·π‘’2,𝑒2‖‖≀‖‖𝐽𝑖𝑒1,𝑒1ξ€Έβˆ’π½π‘–ξ€·π‘’2,𝑒1ξ€Έβ€–β€–+‖‖𝐽𝑖𝑒2,𝑒1ξ€Έβˆ’π½π‘–ξ€·π‘’2,𝑒2ξ€Έβ€–β€–β‰€π‘Ÿπ‘–β€–β€–π‘’1βˆ’π‘’2β€–β€–+𝑑𝑖‖‖𝑒1βˆ’π‘’2β€–β€–β‰€ξ€·π‘Ÿπ‘–+𝑑𝑖‖‖𝑒1βˆ’π‘’2β€–β€–.(3.14) From (A6), we have βˆ’π‘žπœ†π‘–ξ«π‘†π‘–ξ€·π‘’1,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣1ξ€Έ,π½π‘žξ€·π½π‘–ξ€·π‘’1,𝑒1ξ€Έβˆ’π½π‘–ξ€·π‘’2,𝑒2ξ€Έξ€Έξ¬β‰€βˆ’π‘žπœ†π‘–π‘ π‘–β€–β€–π‘’1βˆ’π‘’2β€–β€–π‘ž.(3.15) Moreover, from (A5), we obtain ‖‖𝑆𝑖𝑒1,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣1‖‖≀𝑙𝑖‖‖𝑒1βˆ’π‘’2β€–β€–,(3.16)‖‖𝑆𝑖𝑒2,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣2ξ€Έβ€–β€–β‰€π‘šπ‘–β€–β€–π‘£1βˆ’π‘£2β€–β€–.(3.17) From (3.13)–(3.16), we have ‖‖𝐽𝑖𝑒1,𝑒1ξ€Έβˆ’π½π‘–(𝑒2,𝑒2)βˆ’πœ†π‘–ξ€Ίπ‘†π‘–ξ€·π‘’1,𝑣1ξ€Έβˆ’π‘†π‘–ξ€·π‘’2,𝑣1ξ€Έξ€»β€–β€–π‘žβ‰€π‘žξ”ξ€·π‘Ÿπ‘–+π‘‘π‘–ξ€Έπ‘žβˆ’π‘žπœ†π‘–π‘ π‘–+π‘π‘ž,π‘–πœ†π‘žπ‘–π‘™π‘žπ‘–β€–β€–π‘’1βˆ’π‘’2β€–β€–.(3.18) It follows from (3.12), (3.17), and (3.18) that ‖‖𝐹𝑖𝑒1,𝑣1ξ€Έβˆ’πΉπ‘–ξ€·π‘’2,𝑣2ξ€Έβ€–β€–β‰€π‘žξ”ξ€·π‘Ÿπ‘–+π‘‘π‘–ξ€Έπ‘žβˆ’π‘žπœ†π‘–π‘ π‘–+π‘π‘ž,π‘–πœ†π‘žπ‘–π‘™π‘žπ‘–π›Όπ‘–βˆ’π›½π‘–β€–β€–π‘’1βˆ’π‘’2β€–β€–+πœ†π‘–π‘šπ‘–π›Όπ‘–βˆ’π›½π‘–β€–β€–π‘£1βˆ’π‘£2β€–β€–.(3.19) Put πœƒπ‘–1=π‘žξ”ξ€·π‘Ÿπ‘–+π‘‘π‘–ξ€Έπ‘žβˆ’π‘žπœ†π‘–π‘ π‘–+π‘π‘ž,π‘–πœ†π‘žπ‘–π‘™π‘žπ‘–π›Όπ‘–βˆ’π›½π‘–,πœƒπ‘–2=πœ†π‘–π‘šπ‘–π›Όπ‘–βˆ’π›½π‘–.(3.20) Define β€–β‹…β€– on π‘‹Γ—β‹―Γ—π‘‹ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘βˆ’times by β€–(π‘₯1,…,π‘₯𝑁)β€–=β€–π‘₯1β€–+β‹―β€–π‘₯𝑁‖ for all (π‘₯1,…,π‘₯𝑁)βˆˆπ‘‹Γ—β‹―Γ—π‘‹ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘βˆ’times. It is easy to see that (π‘‹Γ—β‹―Γ—π‘‹ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘βˆ’times,β€–β‹…β€–) is a Banach space. For any given π‘₯1,…,π‘₯π‘βˆˆπ‘‹, we choose a finite sequence 𝑀1βˆˆπ‘ˆ1(π‘₯𝑁),…,π‘€π‘βˆˆπ‘ˆπ‘(π‘₯1). Define π‘„βˆΆπ‘‹Γ—β‹―Γ—π‘‹ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘βˆ’timesβ†’π‘‹Γ—β‹―Γ—π‘‹ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘βˆ’times by 𝑄(π‘₯1,…,π‘₯𝑁)=(𝐹1(π‘₯1,𝑀1),…,𝐹𝑁(π‘₯𝑁,𝑀𝑁)). Set π‘˜=max{(πœƒ11+πœƒπ‘2𝐿𝑁),…,(πœƒ12𝐿1+πœƒπ‘1)}, where 𝐿1,…,𝐿𝑁 are contraction constants of π‘ˆ1,…,π‘ˆπ‘, respectively. We note that πœƒπ‘–1+πœƒπ‘–2𝐿𝑖<πœƒπ‘–1+πœƒπ‘–2<1, for all 𝑖=1,2,…,𝑁, and so π‘˜<1. Let π‘₯1,…,π‘₯π‘βˆˆπ‘‹, 𝑀1βˆˆπ‘ˆ1(π‘₯𝑁),…,π‘€π‘βˆˆπ‘ˆπ‘(π‘₯1) and 𝑦1,…,π‘¦π‘βˆˆπ‘‹, 𝑧1βˆˆπ‘ˆ1(𝑦𝑁),…,π‘§π‘βˆˆπ‘ˆπ‘(𝑦1). By (A3), we get ‖‖𝑄π‘₯1,…,π‘₯π‘ξ€Έβˆ’π‘„ξ€·π‘¦1,…,𝑦𝑁‖‖=‖‖𝐹1ξ€·π‘₯1,𝑀1ξ€Έ,…,𝐹𝑁π‘₯𝑁,π‘€π‘ξ€Έξ€Έβˆ’ξ€·πΉ1𝑦1,𝑧1ξ€Έ,…,𝐹𝑁𝑦𝑁,𝑧𝑁‖‖=‖‖𝐹1ξ€·π‘₯1,𝑀1ξ€Έβˆ’πΉ1𝑦1,𝑧1ξ€Έβ€–β€–+β‹―+‖‖𝐹𝑁π‘₯𝑁,π‘€π‘ξ€Έβˆ’πΉπ‘ξ€·π‘¦π‘,π‘§π‘ξ€Έβ€–β€–β‰€ξ€·πœƒ11β€–β€–π‘₯1βˆ’π‘¦1β€–β€–+πœƒ12‖‖𝑀1βˆ’π‘§1β€–β€–ξ€Έ+β‹―+ξ€·πœƒπ‘1β€–β€–π‘₯π‘βˆ’π‘¦π‘β€–β€–+πœƒπ‘2β€–β€–π‘€π‘βˆ’π‘§π‘β€–β€–ξ€Έβ‰€ξ€·πœƒ11β€–β€–π‘₯1βˆ’π‘¦1β€–β€–+πœƒ12𝐿1β€–β€–π‘₯π‘βˆ’π‘¦π‘β€–β€–ξ€Έ+β‹―+ξ€·πœƒπ‘1β€–β€–π‘₯π‘βˆ’π‘¦π‘β€–β€–+πœƒπ‘2𝐿𝑁‖‖π‘₯1βˆ’π‘¦1β€–β€–ξ€Έ=ξ€·πœƒ11+πœƒπ‘2𝐿𝑁‖‖π‘₯1βˆ’π‘¦1β€–β€–+β‹―+ξ€·πœƒπ‘1+πœƒ12𝐿1ξ€Έβ€–β€–π‘₯π‘βˆ’π‘¦π‘β€–β€–β‰€π‘˜β€–β€–π‘₯1βˆ’π‘¦1β€–β€–+β‹―+π‘˜β€–β€–π‘₯π‘βˆ’π‘¦π‘β€–β€–=π‘˜ξ€·β€–β€–π‘₯1βˆ’π‘¦1β€–β€–+β‹―+β€–β€–π‘₯π‘βˆ’π‘¦π‘β€–β€–ξ€Έ=π‘˜β€–β€–ξ€·π‘₯1,…,π‘₯π‘ξ€Έβˆ’ξ€·π‘¦1,…,𝑦𝑁‖‖,(3.21) and so 𝑄 is a contraction on π‘‹Γ—β‹―Γ—π‘‹ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘βˆ’times. Hence there exists π‘Ž1,…,π‘Žπ‘βˆˆπ‘‹,𝑒1βˆˆπ‘ˆ1(π‘Žπ‘),…,π‘’π‘βˆˆπ‘ˆπ‘(π‘Ž1) such that π‘Ž1=𝐹1(π‘Ž1,𝑒1),…,π‘Žπ‘=𝐹𝑁(π‘Žπ‘,𝑒𝑁). From Theorem 3.1, π‘Ž1,…,π‘Žπ‘βˆˆπ‘‹, 𝑒1βˆˆπ‘ˆ1(π‘Žπ‘),…,π‘’π‘βˆˆπ‘ˆπ‘(π‘Ž1) is the solution of the problem (3.1).

Theorem 3.4. Let 𝑋 be π‘ž-uniformly smooth real Banach space. For 𝑖=1,2,…,𝑁. Let 𝐴𝑖,π΅π‘–βˆΆπ‘‹β†’π‘‹ be single-valued operators, π»π‘–βˆΆπ‘‹Γ—π‘‹β†’π‘‹ single-valued operator satisfy (A1) and suppose that 𝑀𝑖, π‘ˆπ‘–, 𝐻𝑖(𝐴𝑖,𝐡𝑖), 𝑆𝑖, 𝑆𝑖(β‹…,𝑒) satisfy conditions (A2)–(A6), respectively. Then, for any π‘–βˆˆ{1,2,…,𝑁}, the sequences {π‘Žπ‘–π‘›}βˆžπ‘›=1 and {𝑒𝑖𝑛}βˆžπ‘›=1 generated by Algorithm 3.2 converge strongly to π‘Žπ‘–, π‘’π‘–βˆˆπ‘ˆπ‘–(π‘Žπ‘βˆ’(π‘–βˆ’1)), respectively.

Proof. By Theorem 3.3, the problem (3.1) has a solution π‘Ž1,…,π‘Žπ‘βˆˆπ‘‹, 𝑒1βˆˆπ‘ˆ1(π‘Žπ‘),…,π‘’π‘βˆˆπ‘ˆπ‘(π‘Ž1). From Theorem 3.1, we note that π‘Žπ‘–=πœŽπ‘›π‘Žπ‘–+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–ξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–,𝑒𝑖,(3.22) for all 𝑖=1,2,…,𝑁. Hence, by (3.8) and (3.22), we have β€–β€–π‘Žπ‘–π‘›+1βˆ’π‘Žπ‘–π‘›β€–β€–=β€–β€–πœŽπ‘›π‘Žπ‘–π‘›+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έξ€»βˆ’ξ‚ƒπœŽπ‘›π‘Žπ‘–π‘›βˆ’1+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›βˆ’1ξ€Έξ€»ξ‚„β€–β€–β‰€πœŽπ‘›β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–+ξ€·1βˆ’πœŽπ‘›ξ€Έβ€–β€–π‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έξ€»βˆ’π‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›βˆ’1ξ€Έξ€»β€–β€–=πœŽπ‘›β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–+ξ€·1βˆ’πœŽπ‘›ξ€Έβ€–β€–π‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ½π‘–ξ€·π‘Žπ‘–π‘›,π‘Žπ‘–π‘›ξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έξ€»βˆ’π‘…π»π‘–(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1ξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›βˆ’1ξ€Έξ€»β€–β€–β‰€πœŽπ‘›β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–+ξ€·1βˆ’πœŽπ‘›ξ€Έ1π›Όπ‘–βˆ’π›½π‘–β€–β€–ξ€Ίπ½π‘–ξ€·π‘Žπ‘–π‘›,π‘Žπ‘–π‘›ξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έξ€»βˆ’ξ€Ίπ½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1ξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›βˆ’1ξ€Έξ€»β€–β€–=πœŽπ‘›β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–+ξ€·1βˆ’πœŽπ‘›ξ€Έ1π›Όπ‘–βˆ’π›½π‘–β€–β€–ξ€Ίπ½π‘–ξ€·π‘Žπ‘–π‘›,π‘Žπ‘–π‘›ξ€Έβˆ’π½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1ξ€Έξ€»βˆ’πœ†π‘–ξ€Ίπ‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›βˆ’1ξ€Έξ€»β€–β€–β‰€πœŽπ‘›β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–+ξ€·1βˆ’πœŽπ‘›ξ€Έ1π›Όπ‘–βˆ’π›½π‘–β€–β€–ξ€Ίπ½π‘–ξ€·π‘Žπ‘–π‘›,π‘Žπ‘–π‘›ξ€Έβˆ’π½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1ξ€Έξ€»βˆ’πœ†π‘–ξ€Ίπ‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,𝑒𝑖𝑛‖‖+ξ€·1βˆ’πœŽπ‘›ξ€Έ1π›Όπ‘–βˆ’π›½π‘–β€–β€–π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›βˆ’1ξ€Έβ€–β€–.(3.23) By Lemma 2.4, we obtain β€–β€–π½π‘–ξ€·π‘Žπ‘–π‘›,π‘Žπ‘–π‘›ξ€Έβˆ’π½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1ξ€Έβˆ’πœ†π‘–ξ€Ίπ‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›ξ€Έξ€»β€–β€–π‘žβ‰€β€–β€–π½π‘–ξ€·π‘Žπ‘–π‘›,π‘Žπ‘–π‘›ξ€Έβˆ’π½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1ξ€Έβ€–β€–π‘žβˆ’π‘žπœ†π‘–ξ«π‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,𝑒𝑖𝑛,π½π‘ž,π‘–ξ€·π½π‘–ξ€·π‘Žπ‘–π‘›,π‘Žπ‘–π‘›ξ€Έβˆ’π½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1+π‘π‘ž,π‘–πœ†π‘žπ‘–β€–β€–π‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›ξ€Έβ€–β€–π‘ž.(3.24) From (A4), we note that β€–β€–π½π‘–ξ€·π‘Žπ‘–π‘›,π‘Žπ‘–π‘›ξ€Έβˆ’π½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1ξ€Έβ€–β€–=β€–β€–π»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›ξ€Έξ€Έβˆ’π»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έξ€Έβ€–β€–β‰€β€–β€–π»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›ξ€Έξ€Έβˆ’π»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›ξ€Έξ€Έβ€–β€–+β€–β€–π»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›ξ€Έξ€Έβˆ’π»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–π‘›βˆ’1ξ€Έξ€Έβ€–β€–β‰€ξ€·π‘Ÿi+π‘‘π‘–ξ€Έβ€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–.(3.25) From (3.24) and (A6), it follows that βˆ’π‘žπœ†π‘–ξ«π‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,𝑒𝑖𝑛,π½π‘ž,1ξ€·π½π‘–ξ€·π‘Žπ‘–π‘›,π‘Žπ‘–π‘›ξ€Έβˆ’π½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1ξ€Έξ€Έξ¬β‰€βˆ’π‘žπœ†π‘–π‘ π‘–β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–π‘ž.(3.26) By (3.23), (3.24), and (A5), we have β€–β€–π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›βˆ’1ξ€Έβ€–β€–β‰€π‘šπ‘–β€–β€–π‘’π‘–π‘›βˆ’π‘’π‘–π‘›βˆ’1β€–β€–β‰€π‘šπ‘–π‘‘π‘–ξ€·1+π‘›βˆ’1ξ€Έβ€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–,(3.27)β€–β€–π‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›ξ€Έβ€–β€–β‰€π‘™π‘–β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–.(3.28) From (3.23)–(3.28), we obtain β€–β€–π½π‘–ξ€·π‘Ži𝑛,π‘Žπ‘–π‘›ξ€Έβˆ’π½π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘Žπ‘–π‘›βˆ’1ξ€Έβˆ’πœ†π‘–ξ€Ίπ‘†π‘–ξ€·π‘Žπ‘–π‘›,π‘’π‘–π‘›ξ€Έβˆ’π‘†π‘–ξ€·π‘Žπ‘–π‘›βˆ’1,π‘’π‘–π‘›ξ€Έξ€»β€–β€–π‘žβ‰€π‘žξ”ξ€·π‘Ÿπ‘–+π‘‘π‘–ξ€Έπ‘žβˆ’π‘žπœ†π‘–π‘ π‘–+π‘π‘ž,π‘–πœ†π‘žπ‘–π‘™π‘žπ‘–π›Όπ‘–βˆ’π›½π‘–β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–+πœ†π‘–π‘šπ‘–π›Όπ‘–βˆ’π›½π‘–π‘‘π‘–ξ€·1+π‘›βˆ’1ξ€Έβ€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–.(3.29) Hence, by (3.23), (3.28) and (3.29), we have β€–β€–π‘Žπ‘–π‘›+1βˆ’π‘Žπ‘–π‘›β€–β€–β‰€πœŽπ‘›β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘žξ”ξ€·π‘Ÿπ‘–+π‘‘π‘–ξ€Έπ‘žβˆ’π‘žπœ†π‘–π‘ π‘–+π‘π‘ž,π‘–πœ†π‘žπ‘–π‘™π‘žπ‘–π›Όπ‘–βˆ’π›½π‘–β€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–+ξ€·1βˆ’πœŽπ‘›ξ€Έπœ†π‘–π‘šπ‘–π›Όπ‘–βˆ’π›½π‘–π‘‘π‘–ξ€·1+π‘›βˆ’1ξ€Έβ€–β€–π‘Žπ‘–π‘›βˆ’π‘Žπ‘–π‘›βˆ’1β€–β€–.(3.30) Put π‘˜=max{πœ‹1…,πœ‹π‘}, where πœ‹π‘–=π‘žξ”ξ€·π‘Ÿπ‘–+π‘‘π‘–ξ€Έπ‘žβˆ’π‘žπœ†π‘–π‘ π‘–+π‘π‘ž,π‘–πœ†π‘žπ‘–π‘™π‘žπ‘–π›Όπ‘–βˆ’π›½π‘–+πœ†π‘–π‘šπ‘–π‘‘π‘–ξ€·1+π‘›βˆ’1ξ€Έπ›Όπ‘–βˆ’π›½π‘–.(3.31) It follows from (3.30) that β€–β€–π‘Ž1𝑛+1βˆ’π‘Ž1𝑛‖‖+β‹―+β€–β€–π‘Žπ‘π‘›+1βˆ’π‘Žπ‘π‘›β€–β€–β‰€πœŽπ‘›β€–β€–π‘Ž1π‘›βˆ’π‘Ž1π‘›βˆ’1β€–β€–+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘˜β€–β€–π‘Ž1π‘›βˆ’π‘Ž1π‘›βˆ’1β€–β€–+β‹―+πœŽπ‘›β€–β€–π‘Žπ‘π‘›βˆ’π‘Žπ‘π‘›βˆ’1β€–β€–+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘˜β€–β€–π‘Žπ‘π‘›βˆ’π‘Žπ‘π‘›βˆ’1β€–β€–.(3.32) Set 𝑐𝑛=β€–π‘Ž1π‘›βˆ’π‘Ž1π‘›βˆ’1β€–+β‹―+β€–π‘Žπ‘π‘›βˆ’π‘Žπ‘π‘›βˆ’1β€– and π‘˜π‘›=π‘˜+(1βˆ’π‘˜)πœŽπ‘›. From (3.32), we obtain 𝑐𝑛+1β‰€π‘˜π‘›π‘π‘›,βˆ€π‘›=0,1,2,….(3.33) Since limsupπ‘›β†’βˆžπœŽπ‘›<1, we have limsupπ‘›β†’βˆžπ‘˜π‘›<1. Thus, it follows from Lemma 2.8 that 𝑐𝑛+1β†’0 and hence limπ‘›β†’βˆžβ€–π‘Žπ‘–π‘›+1βˆ’π‘Žπ‘–π‘›β€–=0. Therefore, {π‘Žπ‘–π‘›} is a Cauchy sequence and hence there exists π‘Žπ‘–βˆˆπ‘‹ such that π‘Žπ‘–π‘›β†’π‘Žπ‘– as π‘›β†’βˆž, for all 𝑖=1,2,…,𝑁. Next, we will show that 𝑒1𝑛→𝑒1βˆˆπ‘ˆ1(π‘Žπ‘) as π‘›β†’βˆž. Hence, it follows from (3.9) that {𝑒1𝑛} is also a Cauchy sequence. Thus there exists 𝑒1βˆˆπ‘‹ such that 𝑒1𝑛→𝑒1 as π‘›β†’βˆž. Consider 𝑑𝑒1,π‘ˆ1ξ€·π‘Žπ‘ξ€Έξ€Έ=inf‖‖𝑒1βˆ’π‘žβ€–β€–βˆΆπ‘žβˆˆπ‘ˆ1ξ€·π‘Žπ‘ξ€Έξ€Ύβ‰€β€–β€–π‘’1βˆ’π‘’1𝑛‖‖+𝑑𝑒1𝑛,π‘ˆ1ξ€·π‘Žπ‘ξ€Έξ€Έβ‰€β€–β€–π‘’1βˆ’π‘’1𝑛‖‖+π·ξ€·π‘ˆ1ξ€·π‘Žπ‘π‘›ξ€Έ,π‘ˆ1ξ€·π‘Žπ‘ξ€Έξ€Έβ‰€β€–β€–π‘’1βˆ’π‘’1𝑛‖‖+𝑑1β€–β€–π‘Žπ‘π‘›βˆ’π‘Žπ‘β€–β€–βŸΆ0(3.34) as π‘›β†’βˆž. Since π‘ˆ1(π‘Žπ‘) is a closed set and 𝑑(𝑒1,π‘ˆ1(π‘Žπ‘))=0, we have 𝑒1βˆˆπ‘ˆ1(π‘Žπ‘). By continuing the above process, there exist 𝑒2βˆˆπ‘ˆ2(π‘Žπ‘βˆ’1),…,π‘’π‘βˆˆπ‘ˆπ‘(π‘Ž1) such that 𝑒2𝑛→𝑒2,…,𝑒𝑁𝑛→𝑒𝑁 as π‘›β†’βˆž. Hence, by (3.8), we obtain π‘Žπ‘–=𝑅𝐻𝑖(β‹…,β‹…)𝑀𝑖,πœ†π‘–ξ€Ίπ»π‘–ξ€·π΄π‘–ξ€·π‘Žπ‘–ξ€Έ,π΅π‘–ξ€·π‘Žπ‘–ξ€Έξ€Έβˆ’πœ†π‘–π‘†π‘–ξ€·π‘Žπ‘–,𝑒𝑖.(3.35) Therefore, it follows from Theorem 3.1 that π‘Ž1,…,π‘Žπ‘ 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 π‘ž-uniformly smooth real Banach spaces. Let 𝐴𝑖,π΅π‘–βˆΆπ‘‹β†’π‘‹ be singled valued operators, π»π‘–βˆΆπ‘‹Γ—π‘‹β†’π‘‹ a single-valued operator such that 𝐻(𝐴𝑖,𝐡𝑖) is 𝛼𝑖-strongly accretive with respect to 𝐴𝑖, 𝛽𝑖-relaxed accretive with respect to 𝐡𝑖 and 𝛼𝑖>𝛽𝑖 and suppose that π‘€π‘–βˆΆπ‘‹β†’2𝑋 is an 𝐻𝑖(β‹…,β‹…)-accretive set-valued mapping and π‘ˆπ‘–βˆΆπ‘‹β†’πΆ(𝑋) contraction set-valued mapping with 0≀𝐿𝑖<1 and nonempty values, for all 𝑖=1,2. Assume that 𝐻𝑖(𝐴𝑖,𝐡𝑖) is π‘Ÿπ‘–-Lipschitz continuous with respect to 𝐴𝑖 and 𝑑𝑖-Lipschitz continuous with respect to 𝐡𝑖, π‘†π‘–βˆΆπ‘‹Γ—π‘‹β†’π‘‹ is 𝑙𝑖-Lipschitz continuous with respect to its first argument and π‘šπ‘–-Lipschitz continuous with respect to its second argument, 𝑆1(β‹…,𝑦) is 𝑠1-strongly accretive with respect to 𝐻1(𝐴1,𝐡1), and S2(π‘₯,β‹…) is 𝑠2-strongly accretive with respect to 𝐻2(𝐴2,𝐡2), for all 𝑖=1,2. If π‘žξ”ξ€·π‘Ÿπ‘–+π‘‘π‘–ξ€Έπ‘žβˆ’π‘žπœ†π‘–π‘ π‘–+π‘π‘ž,π‘–πœ†π‘žπ‘–π‘™π‘žπ‘–π›Όπ‘–βˆ’π›½π‘–+πœ†π‘–π‘šπ‘–π›Όπ‘–βˆ’π›½π‘–<1,(3.36) for all π‘–βˆˆ{1,2}, then problem (3.2) has a solution π‘Ž1,π‘Ž2βˆˆπ‘‹, 𝑒1βˆˆπ‘ˆ1(π‘Ž2),𝑒2βˆˆπ‘ˆ2(π‘Ž1).

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

Corollary 3.6. Let 𝑋 be π‘ž-uniformly smooth real Banach spaces. Let 𝐴,π΅βˆΆπ‘‹β†’π‘‹ be two singled valued operators, π»βˆΆπ‘‹Γ—π‘‹β†’π‘‹ a single-valued operator such that 𝐻(𝐴,𝐡) is 𝛼-strongly accretive with respect to 𝐴, 𝛽-relaxed accretive with respect to 𝐡, and 𝛼>𝛽 and suppose that π‘€βˆΆπ‘‹β†’2𝑋 is an 𝐻(β‹…,β‹…)-accretive set-valued mapping, π‘ˆβˆΆπ‘‹β†’πΆ(𝑋) is contraction set-valued mapping with 0≀𝐿<1 and nonempty values. Assume that 𝐻(𝐴,𝐡) is π‘Ÿ-Lipschitz continuous with respect to 𝐴 and 𝑑-Lipschitz continuous with respect to 𝐡, π‘†βˆΆπ‘‹Γ—π‘‹β†’π‘‹ is 𝑙-Lipschitz continuous with respect to its first argument and π‘š-Lipschitz continuous with respect to its second argument, 𝑆(β‹…,𝑦) is 𝑠-strongly accretive with respect to 𝐻(𝐴,𝐡). If π‘žβˆš(π‘Ÿ+𝑑)π‘žβˆ’π‘žπœ†π‘ +π‘π‘ž,πœ†π‘žπ‘™π‘žπ›Όβˆ’π›½+πœ†π‘šπ›Όβˆ’π›½<1,(3.37) then problem (3.3) has a solution π‘Žβˆˆπ‘‹ and π‘’βˆˆπ‘ˆ(π‘Ž).

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.

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