Sensitivity Analysis for a System of Generalized Nonlinear Mixed Quasi Variational Inclusions with H-Monotone Operators

Abstract

The existence of the solution for a new system of generalized nonlinear mixed quasi variational inclusions with H-monotone operators is proved by using implicit resolvent technique, and the sensitivity analysis of solution in Hilbert spaces is given. Our results improve and generalize some results of the recent ones.

1. Introduction

Sensitivity analysis of solution for variational inequalities and variational inclusions has been studied by many authors via quite different technique. (See [1–7] and the reference therein).

In 2004, Agarwal et al. [1] introduced and studied the following problem which is called the system of parametric generalized nonlinear mixed quasi variational inclusions.

Let ℋ be a real Hilbert space endowed with the product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Let Ω and Λ be two nonempty open subsets of ℋ in which the parametric 𝜔 and 𝜆 take values. Let 𝑀∶ℋ×Ω→2^ℋ and 𝑁∶ℋ×Λ→2^ℋ be two maximal monotone mappings with respect to the first argument. 𝐻₁,𝑆∶ℋ×Ω→ℋ and 𝐻₂,𝑇∶ℋ×Λ→ℋ be nonlinear single-valued mappings. The system of parametric generalized nonlinear mixed quasi variational inclusions problem [1] is to find (𝑥,𝑦)∈ℋ×ℋ such that 00∈∈𝑦𝑥−−𝑥𝑦++𝛾𝜌𝐻𝐻₁₂(𝑦,𝜔),𝑆(𝑦,𝜔)+𝜌𝑀(𝑥,𝜔),(𝑥,𝜆)+𝑇(𝑥,𝜆)+𝛾𝑁(𝑦,𝜆),(1.1) where 𝜌>0 and 𝛾>0 are two constants.

In this paper, we introduce a new system of parametric generalized nonlinear mixed quasi variational inclusions problem.

For each 𝜔∈Ω, 𝜆∈Λ, find 𝑥=𝑥(𝜔,𝜆), 𝑦=𝑦(𝜔,𝜆) such that 00∈∈𝑓𝑔((𝑥𝑥,,𝑦𝑦,,𝜆𝜔))++𝜌𝜌₁₂(𝐹(𝑥,𝑦,𝜔)+𝑀(𝑥,𝜔)),(𝐺(𝑥,𝑦,𝜆)+𝑁(𝑦,𝜆)),(1.2) where 𝑓,𝐹∶ℋ×ℋ×Ω→ℋ, 𝑔,𝐺∶ℋ×ℋ×Λ→ℋ are nonlinear single-valued mappings, 𝑀∶ℋ×Ω→2^ℋ, 𝑁∶ℋ×Λ→2^ℋ are multivalued mappings, 𝜌₁>0, 𝜌₂>0 are constants. By using implicit resolvent equations technique of 𝐻-monotone operator, the existence of solution is proved and the sensitivity analysis of solution for the problem (1.2) is given. Our results improve and generalize the known results of [1, 2, 8, 9].

2. Preliminaries

Let ℋ be a real Hilbert space, Ω, Λ be two nonempty open subsets of ℋ.

Definition 2.1 (see [1]). (i)A mapping 𝑇∶ℋ×Ω→ℋ is said to be monotone with respect to the first argument if ⟨𝑇(𝑥,𝜔)−𝑇(𝑦,𝜔),𝑥−𝑦⟩≥0,∀(𝑥,𝜔),(𝑦,𝜔)∈ℋ×Ω.(2.1)(ii)𝑇 is said to be 𝜅-strongly monotone with respect to the first argument if there exists a constant 𝜅>0 such that ⟨𝑇(𝑥,𝜔)−𝑇(𝑦,𝜔),𝑥−𝑦⟩≥𝜅‖𝑥−𝑦‖²,∀(𝑥,𝜔),(𝑦,𝜔)∈ℋ×Ω.(2.2)(iii)𝑇 is said to be (𝜉,𝜂)-Lipschitz continuous if there exist 𝜉>0,𝜂>0 such that ‖‖𝑇𝑥,𝜔₁−𝑇𝑦,𝜔₂‖‖≤𝜉‖𝑥−𝑦‖+𝜂‖‖𝜔₁−𝜔₂‖‖,∀𝑥,𝜔₁,𝑦,𝜔₂∈ℋ×Ω.(2.3)

Definition 2.2 (see [1]). A mapping 𝐹∶ℋ×ℋ×Ω→ℋ is said to be (𝜉,𝜂,𝜁)-Lipschitz continuous if there exist 𝜉>0,𝜂>0,𝜁>0 such that ‖‖≤𝐹𝜉‖‖𝑥𝑥₁₁,𝑦₁,𝜔₁−𝐹𝑥₂,𝑦₂,𝜔₂‖‖−𝑥₂‖‖+𝜂‖‖𝑦₁−𝑦₂‖‖+𝜁‖‖𝜔₁−𝜔₂‖‖,∀𝑥₁,𝑦₁,𝜔₁,𝑥₂,𝑦₂,𝜔₂∈ℋ×ℋ×Ω.(2.4)

Definition 2.3 (see [1]). A mapping 𝐹∶ℋ×ℋ×Ω→ℋ is said to be 𝛼-strongly monotone with respect to 𝐻 in the first argument if there exists 𝛼>0 such that 𝐹𝑥₁,𝑦,𝜔−𝐹𝑥₂,𝑦,𝜔,𝐻𝑥₁−𝐻𝑥₂≥𝛼‖‖𝑥₁−𝑥₂‖‖²,∀𝑥₁,𝑦,𝜔,𝑥₂,𝑦,𝜔∈ℋ×ℋ×Ω.(2.5) In a similar way, we can define the strong monotonicity of 𝐹 with respect to 𝐻 in the second argument.

Definition 2.4 (see [9]). Let 𝐻∶ℋ×Ω→ℋ be a single-valued mapping and 𝑀∶ℋ×Ω→2^ℋ be a multi-valued mapping. 𝑀 is said to be 𝐻-monotone if 𝑀 is monotone with respect to the first argument and (𝐻(⋅,𝜔)+𝜌𝑀(⋅,𝜔))(ℋ)=ℋholds for all 𝜌>0 and 𝜔∈Ω.

Definition 2.5 (see [9]). Let 𝐻∶ℋ×Ω→ℋ be a strictly monotone mapping and 𝑀∶ℋ×Ω→2^ℋ be an 𝐻-monotone mapping. The resolvent operator 𝐽^{𝐻𝑀(⋅,𝜔),𝜌}∶ℋ→ℋ is defined by 𝐽^{𝐻𝑀(⋅,𝜔),𝜌}(𝑢)=(𝐻(⋅,𝜔)+𝜌𝑀(⋅,𝜔))⁻¹(𝑢),∀𝑢∈ℋ.(2.6)

Lemma 2.6 (see [9]). Let 𝐻₁∶ℋ×Ω→ℋ be 𝛾₁-strongly monotone with respect to the first argument, 𝐻₂∶ℋ×Λ→ℋ be 𝛾₂-strongly monotone with respect to the first argument, and 𝑀∶ℋ×Ω→2^ℋ be 𝐻₁-monotone, and𝑁∶ℋ×Λ→2^ℋ be 𝐻₂-monotone. Then for any fixed 𝜔∈Ω,𝜆∈Λ, the resolvent operator 𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1} and 𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2} are Lipschitz continuous: ‖‖𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1}(𝑢)−𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1}(𝑣)‖‖≤1𝛾₁‖‖𝐽‖𝑢−𝑣‖,∀𝑢,𝑣∈ℋ,_{𝐻2𝑁(⋅,𝜆),𝜌2}(𝑢)−𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}(𝑣)‖‖≤1𝛾₂‖𝑢−𝑣‖,∀𝑢,𝑣∈ℋ,(2.7) where 𝛾₁>0, 𝛾₂>0 are constants.

Lemma 2.7. Let 𝑀∶ℋ×Ω→2^ℋ be 𝐻₁-monotone and 𝑁∶ℋ×Λ→2^ℋ be 𝐻₂-monotone. For any fixed (𝜔,𝜆)∈Ω×Λ. (𝑥(𝜔,𝜆),𝑦(𝜔,𝜆)) is a solution of (1.2) if and only if 𝑥(𝜔,𝜆)=≜𝐽𝑇(𝑥,𝑦,𝜔^𝐻1_{𝑀(⋅,𝜔),𝜌1},𝐻𝜆)₁,(𝑥(𝜔,𝜆),𝜔)−𝑓(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜔)−𝜌₁𝐹(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜔)(2.8₁𝑦(𝜔,𝜆)=𝐽_{𝐻2𝑁(⋅,𝜆),𝜌2})≜𝑆(𝑥,𝑦,𝜔,𝐻𝜆)₂,(𝑦(𝜔,𝜆),𝜆)−𝑔(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜆)−𝜌₂𝐺(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜆)(2.8₂) here 𝑥=𝑥(𝜔,𝜆), 𝑦=𝑦(𝜔,𝜆).

Proof. Assume that (𝑥(𝜔,𝜆),𝑦(𝜔,𝜆)) satisfies relations 2.8₁ and 2.8₂, since 𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1}=(𝐻₁(⋅,𝜔)+𝜌₁𝑀(⋅,𝜔))⁻¹, 𝐽^𝐻2_{𝑀(⋅,𝜔),𝜌2}=(𝐻₂(⋅,𝜔)+𝜌₂𝑁(⋅,𝜔))⁻¹, then 2.8₁ and 2.8₂ holds if and only if 00∈∈𝑔𝑓((𝑥𝑥((𝜔𝜔,,𝜆𝜆)),,𝑦𝑦((𝜔𝜔,,𝜆𝜆)),,𝜆𝜔))++𝜌𝜌₁₂𝐹(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜔)+𝜌₁𝑀(𝑥(𝜔,𝜆),𝜔),𝐺(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜆)+𝜌₂𝑁(𝑦(𝜔,𝜆),𝜆),(2.9) and hence (𝑥(𝜔,𝜆)),𝑦(𝜔,𝜆) is a solution of 2.8₁ and 2.8₂.

3. Main Results

Lemma 3.1. Let 𝑇∶ℋ×ℋ→ℋ, 𝑆∶ℋ×ℋ→ℋ be two continuous mappings. If there exist Θ₁, Θ₂, 0<Θ₁,Θ₂<1, such that ‖‖𝑇≤𝑥Θ₁₁,𝑦₁−𝑇𝑥₂,𝑦₂‖‖+‖‖𝑆𝑥₁,𝑦₁−𝑆𝑥₂,𝑦₂‖‖‖‖𝑥₁−𝑥₂‖‖+Θ₂‖‖𝑦₁−𝑦₂‖‖,∀𝑥₁,𝑥₂,𝑦₁,𝑦₂∈ℋ.(3.1) then there exist 𝑥^∗,𝑦^∗∈ℋ such that 𝑥^∗=𝑇𝑥^∗,𝑦^∗,𝑦^∗=𝑆𝑥^∗,𝑦^∗.(3.2)

Proof. For any 𝑥₀,𝑦₀∈ℋ, let 𝑥_𝑛+1=𝑇(𝑥_𝑥,𝑦_𝑛), 𝑦_𝑛+1=𝑆(𝑥_𝑛,𝑦_𝑛),𝑛=0,1,2,…, then by (3.1),forall𝑥₁,𝑥₂,…,𝑥_𝑛,𝑥_𝑛+1,𝑦₁,𝑦₂,…,𝑦_𝑛,𝑦_𝑛+1∈ℋ,we have ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤≤ΘΘ₁‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛−1‖‖+Θ₂‖‖𝑦_𝑛−𝑦_𝑛−1‖‖−𝑥_𝑛−1‖‖+‖‖𝑦_𝑛−𝑦_𝑛−1‖‖,(3.3) where Θ=max{Θ₁,Θ₂}. Let 𝑎=‖𝑥₁−𝑥₀‖+‖𝑦₁−𝑦₀‖, then ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤≤ΘΘ₂‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+‖‖𝑦_𝑛−𝑦_𝑛−1‖‖≤⋯≤‖‖𝑥Θ_𝑛−1𝑛−𝑥_𝑛−2‖‖+‖‖𝑦_𝑛−1−𝑦_𝑛−2‖‖‖‖𝑥₁−𝑥₀‖‖+‖‖𝑦₁−𝑦₀‖‖=Θ^𝑛𝑎,(3.4) and hence, 00≤≤‖‖‖‖𝑥𝑦_𝑛+1𝑛+1−𝑥_𝑛‖‖≤Θ^𝑛𝑎,−𝑦_𝑛‖‖≤Θ^𝑛𝑎.(3.5) Since 0<Θ=max{Θ₁,Θ₂}<1, (3.5) implies that {𝑥_𝑛} and {𝑦_𝑛} are both cauchy sequences. Therefore, there exist 𝑥^∗,𝑦^∗∈ℋ such that 𝑥_𝑛→𝑥^∗,𝑦_𝑛→𝑦^∗(𝑛→∞). By continuity of 𝑇 and 𝑆, 𝑥^∗=𝑇(𝑥^∗,𝑦^∗),𝑦^∗=𝑆(𝑥^∗,𝑦^∗).

Theorem 3.2. Let 𝐻₁∶ℋ×Ω→ℋ be (𝜉₁,𝜂₁)-Lipschitz continuous and 𝐻₂∶ℋ×Λ→ℋ be (𝜉₂,𝜂₂)-Lipschitz continuous. Let 𝑓∶ℋ×ℋ×Ω→ℋ be (𝜉_𝑓,𝜂_𝑓,𝜁_𝑓)-Lipschitz continuous and 𝛾_𝑓-strongly monotone with respect to 𝐻 in the first argument, 𝐹∶ℋ×ℋ×Ω→ℋ be (𝜉_𝐹,𝜂_𝐹,𝜁_𝐹)-Lipschitz continuous,𝑔∶ℋ×ℋ×Λ→ℋ be (𝜉_𝑔,𝜂_𝑔,𝜁_𝑔)-Lipschitz continuous and 𝛾_𝑔-strongly monotone with respect to 𝐻₂ in the second argument,𝐺∶ℋ×ℋ×Λ→ℋ be (𝜉_𝐺,𝜂_𝐺,𝜁_𝐺)-Lipschitz continuous. Suppose that 𝑀∶ℋ×Ω→2^ℋ is 𝐻₁-monotone and 𝑁∶ℋ×Λ→2^ℋ is 𝐻₂-monotone If Θ₁=1𝛾₁𝜉²¹−2𝛾_𝑓+𝜉^2𝑓+𝜌₁𝜉_𝐹+𝜉^𝑔+𝜌₂𝜉_𝐺𝛾₂Θ<1,₂=1𝛾₂𝜉²²−2𝛾_𝑔+𝜂^2𝑔+𝜌₂𝜂_𝐺+𝜂^𝑓+𝜌₁𝜂_𝐹𝛾₁<1,(3.6) then, for each 𝜔,𝜆∈Ω×Λ, the problem (1.2) has an unique solution (𝑥^∗(𝜔,𝜆),𝑦^∗(𝜔,𝜆)).

Proof. Let 𝑥₁=𝑥₁(𝜔,𝜆),𝑥₂=𝑥₂(𝜔,𝜆),𝑦₁=𝑦₁(𝜔,𝜆),𝑦₂=𝑦₂(𝜔,𝜆). By defining 2.8₁ of 𝑇 and Lemma 2.6, we have ‖‖𝑇𝑥₁,𝑦₁,𝜔,𝜆−𝑇𝑥₂,𝑦₂,𝜔,𝜆‖‖≤1𝛾₁‖‖+𝐻𝑓₁𝑥𝑥₁₂,𝜔−𝑓𝑥₁,𝑦₁,𝜔−𝜌₁𝐹𝑥₁,𝑦₁,𝜔−𝐻₁𝑥₂,𝜔≤1,𝑦₂,𝜔+𝜌₁𝐹𝑥₂,𝑦₂,𝜔‖‖𝛾₁+1‖‖𝐻₁𝑥₁,𝜔−𝐻₁𝑥₂,𝜔−𝑓𝑥₁,𝑦₁,𝜔+𝑓𝑥₂,𝑦₂,𝜔‖‖𝛾₁𝜌₁‖‖𝐹𝑥₁,𝑦₁,𝜔−𝐹𝑥₂,𝑦₂,𝜔‖‖.(3.7) Since 𝑓 is (𝜉_𝑓,𝜂_𝑓,𝜁_𝑓)-Lipschitz continuous and 𝛾_𝑓-strongly monotone with respect to 𝐻₁ in the first argument, 𝐹 is (𝜉_𝐹,𝜂_𝐹,𝜁_𝐹)-Lipschitz continuous, we have ‖‖𝐻≤₁‖‖𝑥𝐻₁,𝜔−𝐻₁𝑥₂,𝜔−𝑓𝑥₁,𝑦₁,𝜔+𝑓𝑥₂,𝑦₂,𝜔₁‖‖≤𝑥₁,𝜔−𝐻₁𝑥₂,𝜔−𝑓𝑥₁,𝑦₁,𝜔+𝑓𝑥₂,𝑦₁,𝜔‖‖+‖‖𝑓𝑥₂,𝑦₁,𝜔−𝑓𝑥₂,𝑦₂,𝜔‖‖𝜉²¹−2𝛾_𝑓+𝜉^2𝑓‖‖𝑥₁−𝑥₂‖‖+𝜂_𝑓‖‖𝑦₁−𝑦₂‖‖,(3.8)‖‖𝐹𝑥₁,𝑦₁,𝜔−𝐹𝑥₂,𝑦₂,𝜔‖‖≤𝜉_𝐹‖‖𝑥₁−𝑥₂‖‖+𝜂_𝐹‖‖𝑦₁−𝑦₂‖‖.(3.9) Combining (3.7)–(3.9), we have ‖‖𝑇𝑥₁,𝑦₁,𝜔,𝜆−𝑇𝑥₂,𝑦₂,𝜔,𝜆‖‖≤1𝛾₁𝜉²¹−2𝛾_𝑓+𝜉^2𝑓+𝜌₁𝜉_𝐹‖‖𝑥¹+𝜂_𝑓−𝑥₂‖‖+𝜌₁𝜂_𝐹𝛾₁‖‖𝑦₁−𝑦₂‖‖.(3.10) By defining 2.8₂ of 𝑆 and Lemma 2.6, we have ‖‖𝑆𝑥₁,𝑦₁,𝜔,𝜆−𝑆𝑥₂,𝑦₂,𝜔,𝜆‖‖≤1𝛾₂‖‖−𝐻𝐻₂₂𝑦₁,𝜆−𝑔𝑥₁,𝑦₁,𝜆−𝜌₂𝐺𝑥₁,𝑦₁,𝜆≤1𝑦₂,𝜆+𝑔𝑥₂,𝑦₂,𝜆+𝜌₂𝐺𝑥₂,𝑦₂,𝜆‖‖𝛾₂+𝜌‖‖𝐻₂𝑦₁,𝜆−𝐻₂𝑦₂,𝜆−𝑔𝑥₁,𝑦₁,𝜆+𝑔𝑥₂,𝑔₂,𝜆₂‖‖𝛾₂‖‖𝐺𝑥₁,𝑦₁,𝜆−𝐺𝑥₂,𝑦₂,𝜆‖‖.(3.11) Since 𝑔 is (𝜉_𝑔,𝜂_𝑔,𝜁_𝑔)-Lipschitz continuous and 𝛾_𝑔-strongly monotone with respect to 𝐻₂ in the second argument, 𝐺 is (𝜉_𝐺,𝜂_𝐺,𝜁_𝐺)-Lipschitz continuous. We have ‖‖𝐻≤₂‖‖𝑦𝐻₁,𝜆−𝐻₂𝑦₂,𝜆−𝑔𝑥₁,𝑦₁,𝜆+𝑔𝑥₂,𝑦₂,𝜆₂‖‖≤𝑦₁,𝜆−𝐻₂𝑦₂,𝜆−𝑔𝑥₁,𝑦₁,𝜆+𝑔𝑥₁,𝑦₂,𝜆‖‖+‖‖𝑔𝑥₁,𝑦₂,𝜆−𝑔𝑥₂,𝑦₂,𝜆‖‖𝜉²²−2𝛾_𝑔+𝜂^2𝑔‖‖𝑦₁−𝑦₂‖‖+𝜉_𝑔‖‖𝑥₁−𝑥₂‖‖,(3.12)‖‖𝐺𝑥₁,𝑦₁,𝜆−𝐺𝑥₂,𝑦₂,𝜆‖‖≤𝜉_𝐺‖‖𝑥₁−𝑥₂‖‖+𝜂_𝐺‖‖𝑦₁−𝑦₂‖‖.(3.13) Combining (3.11)–(3.13), we have ‖‖𝑆𝑥₁,𝑦₁,𝜔,𝜆−𝑆𝑥₂,𝑦₂,𝜔,𝜆‖‖≤1𝛾₂𝜉²²−2𝛾_𝑔+𝜂^2𝑔+𝜌₂𝜂_𝐺‖‖𝑦¹+𝜉_𝑔−𝑦₂‖‖+𝜌₂𝜉_𝐺𝛾₂‖‖𝑥₁−𝑥₂‖‖.(3.14) By (3.10) and (3.14), we have ‖‖𝑇≤𝑥₁,𝑦1₁,𝜔,𝜆−𝑇𝑥₂,𝑦₂,𝜔,𝜆‖‖+‖‖𝑆𝑥₁,𝑦₁,𝜔,𝜆−𝑆𝑥₂,𝑦₂,𝜔,𝜆‖‖𝛾₁𝜉²¹−2𝛾_𝑓+𝜉^2𝑓+𝜌₁𝜉_𝐹+𝜉^𝑔+𝜌₂𝜉_𝐺𝛾₂‖‖𝑥¹+1−𝑥₂‖‖𝛾₂𝜉²²−2𝛾_𝑔+𝜂^2𝑔+𝜌₂𝜂_𝐺+𝜂^𝑓+𝜌₁𝜂_𝐹𝛾₁‖‖𝑦¹=Θ₁−𝑦₂‖‖‖‖𝑥₁−𝑥₂‖‖+Θ₂‖‖𝑦₁−𝑦₂‖‖.(3.15) By (3.6) and Lemma 3.1, there exist 𝑥^∗=𝑥^∗(𝜔,𝜆),𝑦^∗=𝑦^∗(𝜔,𝜆).(𝑥^∗,𝑦^∗) such that 𝑥^∗=𝑇(𝑥^∗,𝑦^∗,𝜔,𝜆),𝑦^∗=𝑆(𝑥^∗,𝑦^∗,𝜔,𝜆). By Lemma 2.7, (𝑥^∗,𝑦^∗) is a solution of (1.2). From (3.15), we easily see that the solution of (1.2) is unique.

Assumption 1. For implicit resolvent operator 𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1},𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}, there are two constants 𝜉>0 and 𝜂>0 such that ‖‖‖𝐽‖‖‖𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1𝐻2𝑁(⋅,𝜆),𝜌2}(𝑢)−𝐽^𝐻1_{𝑀⋅,𝜔,𝜌1}(𝑢)‖‖‖≤𝜉‖‖𝜔−𝜔‖‖,(𝑣)−𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}(𝑣)‖‖‖≤𝜂‖‖𝜆−𝜆‖‖,∀𝑢,𝑣∈ℋ.(3.16)

Theorem 3.3. Suppose that the mappings 𝐻₁,𝑓,𝐹,𝑀,𝐻₂,𝑔,𝐺, and 𝑁 are the same as in Theorem 3.2, and for any fixed 𝑥,𝑦∈ℋ, the mappings 𝜔→𝐻₁(𝑥,𝜔),𝜔→𝑓(𝑥,𝑦,𝜔),𝜔→𝐹(𝑥,𝑦,𝜔),𝜆→𝐻₂(𝑦,𝜆),𝜆→𝑔(𝑥,𝑦,𝜆),𝜆→𝐺(𝑥,𝑦,𝜆) are continuous (or Lipschitz continuous.) If Assumption 1 and condition (3.6) hold, then the solution (𝑥(𝜔,𝜆),𝑦(𝜔,𝜆)) for the problem (1.2) is continuous (or Lipschitz continuous).

Proof. Suppose that 𝜔,𝜔∈Ω, 𝜆,𝜆∈Λ such that 𝜔→𝜔,𝜆→𝜆. From Theorem 3.2, we know that the problem (1.2) has solution (𝑥(𝜔,𝜆),𝑦(𝜔,𝜆)),𝑥(𝜔,𝜆),𝑦(𝜔,𝜆)), (𝑥(𝜔,𝜆),𝑦(𝜔,𝜆)) and (𝑥(𝜔,𝜆),𝑦(𝜔,𝜆)).
(A) Estimate ‖𝑥(𝜔,𝜆)−𝑥(𝜔,𝜆)‖ and ‖𝑦(𝜔,𝜆)−𝑦(𝜔,𝜆)‖. By Lemma 2.7,we have 𝑥(𝜔,𝜆)=≜𝐽𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1𝐻1𝑀(⋅,𝜔),𝜌1}𝐻₁(𝑥(𝜔,𝜆),𝜔)−𝑓(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜔)−𝜌₁𝐹(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜔)𝑥(𝑝),𝜔,𝜆=≜𝐽𝐽^𝐻1_{𝑀⋅,𝜔,𝜌1𝐻1𝑀⋅,𝜔,𝜌1}𝐻₁𝑥𝜔,𝜆,𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝜌₁𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔(𝑞).(3.17) It follows from Assumption 1, Lemmas 2.6 and 2.7 that ‖‖𝑥≤(𝜔‖‖,𝐽𝜆)−𝑥𝜔,𝜆‖‖=‖‖‖𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1}(𝑝)−𝐽^𝐻1_{𝑀⋅,𝜔,𝜌1}(𝑞)_{𝐻1𝑀(⋅,𝜔),𝜌1}‖‖‖(𝑝)−𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1}(𝑞)‖‖+‖‖‖𝐽^𝐻1_{𝑀(⋅,𝜔),𝜌1}(𝑞)−𝐽^𝐻1_{𝑀⋅,𝜔,𝜌1}(𝑞)‖‖‖≤1𝛾₁‖𝑝−𝑞‖+𝜉‖‖𝜔−𝜔‖‖.(3.18) Since 𝑓 is (𝜉_𝑓,𝜂_𝑓,𝜁_𝑓)-Lipschitz continuous, 𝛾_𝑓 -strongly monotone with respect to 𝐻₁ in the first argument and 𝐹 is (𝜉_𝐹,𝜂_𝐹,𝜁_𝐹)-Lipschitz continuous, we conclude ‖𝑝−𝑞‖=‖‖𝐻−𝐻₁(₁𝑥(𝜔,𝜆),𝜔)−𝑓(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜔)−𝜌₁𝐹(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜔)≤‖‖𝐻₁𝑥𝜔,𝜆,𝜔+𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔+𝜌₁𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖+‖‖𝐻(𝑥(𝜔,𝜆),𝜔)−𝐻₁𝑥𝜔,𝜆,𝜔−𝑓(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜔)+𝑓𝑥𝜔,𝜆,𝑦(𝜔,𝜆),𝜔₁‖‖+‖‖𝑓𝑥𝑥𝜔,𝜆,𝜔−𝐻₁𝑥𝜔,𝜆,𝜔‖‖+‖‖𝑓𝑥𝜔,𝜆,𝑦(𝜔,𝜆),𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖+𝜌₁𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖+𝜌₁‖‖𝐹(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜔)−𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖≤‖‖𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖+𝜂𝜉²¹_𝑓−2𝛾_𝑓+𝜉^2𝑓‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖+‖‖𝐻₁𝑥𝜔,𝜆,𝜔−𝐻₁𝑥𝜔,𝜆,𝜔‖‖+𝜌₁‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖+‖‖𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖+𝜌₁𝜉_𝐹‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖+𝜌₁𝜂_𝐹‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖‖‖𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖.(3.19) It follows from (3.18) and (3.19) that ‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖≤𝜃₁‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖+1𝛾₁𝜂_𝑓+1+𝜌₁𝜂_𝐹‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖𝛾₁+1‖‖𝐻₁𝑥𝜔,𝜆,𝜔−𝐻₁𝑥𝜔,𝜆,𝜔‖‖𝛾₁+𝜌₁‖‖𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖𝛾₁‖‖𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖+𝜉‖‖𝜔−𝜔‖‖,(3.20) that is, ‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖≤𝜂^𝑓+𝜌₁𝜂_𝐹1−𝜃₁𝛾₁+1‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖1−𝜃₁𝛾1₁+1‖‖𝐻₁𝑥𝜔,𝜆,𝜔−𝐻₁𝑥𝜔,𝜆,𝜔‖‖𝛾₁+𝜌₁‖‖𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖𝛾₁+𝜉‖‖‖‖𝜔𝐹−𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖𝜔‖‖,(3.21) where 𝜃₁=1𝛾₁𝜉²¹−2𝛾_𝑓+𝜉^2𝑓+𝜌₁𝜉_𝐹,(∗) By Lemma 2.7, we have 𝑦(𝜔,𝜆)≜=𝐽𝐽^𝐻2_{𝑁(⋅,𝜆)𝜌2𝐻2𝑁(⋅,𝜆),𝜌2}𝐻₂(𝑦(𝜔,𝜆),𝜆)−𝑔(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜆)−𝜌₂𝐺(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜆)𝑦(𝑠),𝜔,𝜆≜=𝐽𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2𝐻2𝑁(⋅,𝜆),𝜌2}𝐻₂𝑦𝜔,𝜆,𝜆−𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝜌₂𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆(𝑡).(3.22) It follows from Assumption 1, Lemmas 2.6 and 2.7 that ‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖=‖‖𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}(𝑠)−𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}(𝑡)‖‖≤1𝛾₂‖𝑠−𝑡‖.(3.23) Since 𝑔 is (𝜉_𝑔,𝜂_𝑔,𝜁_𝑔)-Lipschitz continuous and 𝛾_𝑔-strongly monotone with respect to 𝐻₂ in the second argument and 𝐺 is (𝜉_𝐺,𝜂_𝐺,𝜁_𝐺)-Lipschitz continuous,we conclude ‖𝑠−𝑡‖=‖‖𝐻−₂𝐻(𝑦₂(𝜔,𝜆),𝜆)−𝑔(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜆)−𝜌₂𝐺(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜆)≤‖‖𝐻𝑦𝜔,𝜆,𝜆+𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆+𝜌₂𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆₂‖‖+‖‖𝑔(𝑥𝑦((𝜔𝜔,,𝜆𝜆)),,𝑦𝜆)−𝐻₂𝑦𝜔,𝜆,𝜆−𝑔(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜆)+𝑔𝑥(𝜔,𝜆),𝑦𝜔,𝜆,𝜆‖‖+𝜌₂𝜔,𝜆,𝜆−𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖≤‖‖𝐺(𝑥(𝜔,𝜆),𝑦(𝜔,𝜆),𝜆)−𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖+𝜌𝜉²²₂−2𝛾_𝑔+𝜂^2𝑔‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖+𝜉_𝑔‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖𝜉_𝐺‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖+𝜌₂𝜂_𝐺‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖.(3.24) It follows from (3.23) and (3.24) that ‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖≤1𝛾₂𝜉²²−2𝛾_𝑔+𝜂^2𝑔+𝜌₂𝜂_𝐺‖‖𝑦(𝜔,𝜆)−𝑦+1𝜔,𝜆‖‖𝛾₂𝜉_𝑔+𝜌₂𝜉_𝐺‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖.(3.25) That is, ‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖≤11−𝜃₂1𝛾₂𝜉_𝑔+𝜌₂𝜉_𝐺‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖,(3.26) where 𝜃₂=1𝛾₂𝜉²²−2𝛾_𝑔+𝜂^2𝑔+𝜌₂𝜂_𝐺.(∗∗) By combining (3.21) and (3.26), we derive ‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖≤𝜂^𝑓+𝜌₁𝜂_𝐹𝜉_𝑔+𝜌₂𝜉_𝐺1−𝜃₁1−𝜃₂𝛾₁𝛾₂+1‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖1−𝜃₁𝛾1₁+1‖‖𝐻₁𝑥𝜔,𝜆,𝜔−𝐻₁𝑥𝜔,𝜆,𝜔‖‖𝛾₁+𝜌₁‖‖𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖𝛾₁+𝜉‖‖𝜔‖‖𝐹−𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖𝜔‖‖𝑠.(3.27) That is, ‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖≤11−𝜅1−1𝜃₁𝛾1₁+1‖‖𝐻₁𝑥𝜔,𝜆,𝜔−𝐻₁𝑥𝜔,𝜆,𝜔‖‖𝛾₁+𝜌₁‖‖𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖𝛾₁+𝜉‖‖𝜔‖‖𝐹−𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖‖‖𝑦(𝜔,𝜆)−𝑦𝜔‖‖,(3.28)𝜔,𝜆‖‖≤𝜉^𝑔+𝜌₂𝜉_𝐺1−𝜃₂𝛾₂11−𝜅1−1𝜃₁𝛾1₁+1‖‖𝐻₁𝑥𝜔,𝜆,𝜔−𝐻₁𝑥𝜔,𝜆,𝜔‖‖𝛾₁−𝑓𝑥‖‖𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔+𝜌₁𝜔,𝜆,𝑦𝜔,𝜆,𝜔‖‖𝛾₁−𝐹𝑥‖‖𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔+‖‖𝜉𝜔𝜔−,𝜆,𝑦𝜔,𝜆,𝜔‖‖𝜔‖‖,(3.29) where 𝜅=𝜂^𝑓+𝜌₁𝜂_𝐹𝜉_𝑔+𝜌₂𝜉_𝐺1−𝜃₁1−𝜃₂𝛾₁𝛾₂.(∗∗∗) By (3.6), 1−𝜃₁>1𝛾₂𝜉_𝑔+𝜌₂𝜉_𝐺,1−𝜃₂>1𝛾₁𝜂_𝑓+𝜌₁𝜂_𝐹,(3.30) and hence 0<𝜉^𝑔+𝜌₂𝜉_𝐺𝜂_𝑓+𝜌₁𝜂_𝐹1−𝜃₁1−𝜃₂𝛾₁𝛾₂<1,thatis,0<𝜅<1.(3.31)
(B) Estimate ‖𝑦(𝜔,𝜆)−𝑦(𝜔,𝜆)‖ and ‖𝑥(𝜔,𝜆)−𝑥(𝜔,𝜆)‖.
By Lemma 2.7, we have 𝑦𝜔,𝜆==𝐽𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2𝐻2𝑁(⋅,𝜆),𝜌2}𝐻²𝑦𝜔,𝜆,𝜆−𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝜌₂𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆𝑦(𝑚),𝜔,𝜆==𝐽𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2𝐻2𝑁(⋅,𝜆),𝜌2}𝐻₂𝑦𝜔,𝜆,𝜆−𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝜌₂𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆(𝑛).(3.32) It follows from Assumption 1, Lemmas 2.6 and 2.7 that ‖‖𝑦𝜔,𝜆−𝑦𝜔,𝜆‖‖=≤‖‖‖𝐽‖‖‖𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}(𝑚)−𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}(𝑛)_{𝐻2𝑁(⋅,𝜆),𝜌2}‖‖‖≤1(𝑚)−𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}(𝑛)‖‖‖+‖‖‖𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}(𝑛)−𝐽^𝐻2_{𝑁(⋅,𝜆),𝜌2}(𝑛)‖‖‖𝛾₂‖𝑚−𝑛‖+𝜂‖‖𝜆−𝜆‖‖.(3.33) Since 𝑔 is (𝜉_𝑔,𝜂_𝑔,𝜁_𝑔)-Lipschitz continuous and 𝛾_𝑔-strongly monotone with respect to 𝐻₂ in the second argument and 𝐺 is (𝜉_𝐺,𝜂_𝐺,𝜁_𝐺)-Lipschitz continuous,we conclude ‖𝑚−𝑛‖=‖‖𝐻−₂𝐻₂𝑦𝜔,𝜆,𝜆−𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝜌₂𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆≤‖‖𝐻𝑦𝜔,𝜆,𝜆+𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆+𝜌₂𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆₂‖‖−𝑔𝑦𝑥𝜔,𝜆,𝜆−𝐻₂𝑦𝜔,𝜆,𝜆+‖‖𝐻₂𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖+‖‖𝑔𝑥𝑦𝜔,𝜆,𝜆−𝐻₂𝑦𝜔,𝜆,𝜆‖‖+‖‖𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖+𝜌₂𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖+𝜌₂‖‖𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖≤‖‖𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖+𝜌𝜉²²₂−2𝛾_𝑔+𝜂^2𝑔‖‖𝑦𝜔,𝜆−𝑦𝜔,𝜆‖‖+𝜉_𝑔‖‖𝑥𝜔,𝜆−𝑥𝜔,𝜆‖‖𝜉^𝐺+‖‖𝑔𝑥‖‖𝑥𝜔,𝜆−𝑥𝜔,𝜆‖‖+𝜂_𝐺‖‖𝑦𝜔,𝜆−𝑦𝜔,𝜆‖‖+‖‖𝐻𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆₂‖‖+𝜌₂𝑦𝜔,𝜆,𝜆−𝐻₂𝑦𝜔,𝜆,𝜆‖‖‖‖𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆−𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖.(3.34) It follows from (3.33) and (3.34) that ‖‖𝑦𝜔,𝜆−𝑦𝜔,𝜆‖‖≤𝜉^𝑔+𝜌₂𝜉_𝐺1−𝜃₂𝛾₂+1‖‖𝑥𝜔,𝜆−𝑥𝜔,𝜆‖‖1−𝜃₂𝛾1₂‖‖𝐻₂+1𝑦𝜔,𝜆,𝜆−𝐻₂𝑦𝜔,𝜆,𝜆‖‖𝛾₂−‖‖𝑔𝑔𝑥𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆+𝜌𝜔,𝜆,𝑦𝜔,𝜆,𝜆₂‖‖𝛾₂−‖‖𝐺𝐺𝑥𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆+𝜂‖‖𝜆−𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖𝜆‖‖,(3.35) where 𝜃₂ defined by ∗∗.
By Lemma 2.7, we have 𝑥𝜔𝑥,𝜆=𝐽^𝐻1_{𝑀⋅,𝜔,𝜌1}𝐻¹𝑥𝜔,𝜆,𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝜌₁𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔,𝜔,𝜆=𝐽^𝐻1_{𝑀⋅,𝜔,𝜌1}𝐻₁𝑥𝜔,𝜆,𝜔−𝑓𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔−𝜌₁𝐹𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜔.(3.36) As the proof of (3.25),we have ‖‖𝑥𝜔,𝜆−𝑥𝜔,𝜆‖‖≤1𝛾₁𝜉²¹−2𝛾_𝑓+𝜉^2𝑓+𝜌₁𝜉_𝐹‖‖𝑥+1𝜔,𝜆−𝑥𝜔,𝜆‖‖𝛾₁𝜂_𝑓+𝜌₁𝜂_𝐹‖‖𝑦𝜔,𝜆−𝑦𝜔,𝜆‖‖,(3.37) that is, ‖‖𝑥𝜔,𝜆−𝑥𝜔,𝜆‖‖≤𝜂^𝑓+𝜌₁𝜂_𝐹1−𝜃₁𝛾₁‖‖𝑦𝜔,𝜆−𝑦𝜔,𝜆‖‖,(3.38) where 𝜃₁ is defined by ∗.
Therefore ‖‖𝑦𝜔,𝜆−𝑦𝜔,𝜆‖‖≤11−𝜅1−1𝜃₂𝛾1₂‖‖𝐻₂+1𝑦𝜔,𝜆,𝜆−𝐻₂𝑦𝜔,𝜆,𝜆‖‖𝛾₂‖‖𝑔−𝑔𝑥𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆+𝜌𝜔,𝜆,𝑦𝜔,𝜆,𝜆₂‖‖𝛾₂‖‖𝐺−𝐺𝑥𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆+𝜂‖‖𝜆−𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖‖‖𝑥𝜆‖‖,(3.39)𝜔,𝜆−𝑥𝜔,𝜆‖‖≤𝜂^𝑓+𝜌₁𝜂_𝐹1−𝜃₁𝛾₁11−𝜅1−1𝜃₂𝛾1₂‖‖𝐻₂+1𝑦𝜔,𝜆,𝜆−𝐻₂𝑦𝜔,𝜆,𝜆‖‖𝛾₂−𝑔‖‖𝑥𝑔𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆+𝜌𝜔,𝜆,𝑦𝜔,𝜆,𝜆₂‖‖𝛾₂−𝐺‖‖𝑥𝐺𝑥𝜔,𝜆,𝑦𝜔,𝜆,𝜆+𝜂‖‖𝜆−𝜔,𝜆,𝑦𝜔,𝜆,𝜆‖‖𝜆‖‖,(3.40) where 𝜅 is defined by ∗∗∗.
(C) Prove that the conclusion of Theorem 3.3.
From (3.28) and (3.40), by the assumptions for 𝐻₁,𝑓,𝐹,𝐻₂,𝑔, and 𝐺 in Theorem 3.3 and relation ‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖≤‖‖𝑥(𝜔,𝜆)−𝑥𝜔,𝜆‖‖+‖‖𝑥𝜔,𝜆−𝑥𝜔,𝜆‖‖,(3.41) we know that 𝑥(𝜔,𝜆) is continuous (or Lipschitz continuous).
From (3.29) and (3.39), by the assumptions for 𝐻₁,𝑓,𝐹,𝐻₂,𝑔 and 𝐺 in Theorem 3.3 and relation ‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖≤‖‖𝑦(𝜔,𝜆)−𝑦𝜔,𝜆‖‖+‖‖𝑦𝜔,𝜆−𝑦𝜔,𝜆‖‖,(3.42) we know that 𝑦(𝜔,𝜆) is continuous (or Lipschitz continuous).

Acknowledgment

The author would like to thank the anonymous referee for reading this paper carefully, providing valuable suggestions and comments. The work was supported by the National Science Foundation of China (no. 10561007) and Science and Technology Research Project of Education Department in Jiangxi province (GJJ10269).