Sensitivity Analysis for Nonlinear Set-Valued Variational Equations in Banach Framework

Abstract

The main purpose of this paper is to study the sensitivity analysis for nonlinear set-valued variational equations based on (𝐴,𝜂)-resolvent operator technique. The obtained results encompass a broad model of results.

1. Introduction

The set-valued inclusions problem, which was initiated by Di Bella [1] and Huang et al. [2, 3], is a useful extension of the mathematical analysis and variational equation and is important context in the set-valued equations. Verma [4], Huang [5], Fang and Huang [6], Lan et al. [7], Khan and Salahuddin [8, 9], Li et al. [10], and Yen and Lee [11] introduced the concepts of 𝐴-monotone, 𝐻-monotone, (𝐴,𝜂)-monotone operator, (𝐴,𝜂)-accretive mappings, and resolvent operator associated with them, respectively. Dafermos [12] studied the sensitivity property of solutions of particular kinds of variational inequality on parameter which takes values on an open subset of Euclidean space 𝑅^𝐾. Tobin [13], Verma [14, 15], Lee and Salahuddin [3], Kyparisis [16], Moudafi [17], Noor [18], Robinson [19], Yen and Lee [11], and Hussain et al. [20] studied the sensitivity analysis of various types of variational inequalities.

In this paper, we present the sensitivity analysis for (𝐴,𝜂)-accretive variational equations based on the (𝐴,𝜂)-resolvent operator technique. The obtained results generalize a wide range of results on the sensitivity analysis for nonlinear set-valued variational equations in Banach spaces.

2. Preliminaries

Let 𝑋 be a real Banach space with dual space 𝑋^∗, and let ⟨⋅,⋅⟩ be the dual pair between 𝑋 and 𝑋^∗, 2^𝑋 denote the family of all the nonempty subset of 𝑋 and CB(𝑋) the family of all nonempty closed bounded subsets of 𝑋. The generalized duality mapping 𝐽_𝑞∶𝑋→2^𝑋∗ is defined by 𝐽_𝑞(𝑥)=𝑓^∗∈𝑋^∗∶𝑢,𝑓^∗=‖𝑢‖^𝑞,𝑓^∗=‖𝑢‖^𝑞−1,∀𝑢∈𝑋, (1) where 𝑞>1 is constant.

The modulus of smoothness of 𝑋 is the function 𝜌_𝑋∶[0,∞)→[0,∞) defined by 𝜌_𝑋(𝑡)1=sup2(‖𝑢+𝑣‖+‖𝑢−𝑣‖)−1∶‖𝑢‖≤1,‖𝑣‖<𝑡. (2) A Banach space 𝑋 is called uniformly smooth if lim_𝑡→0𝜌_𝑋(𝑡)𝑡=0. (3)𝑋 is called 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌_𝑋(𝑡)≤𝑐𝑡^𝑞𝑞>1. (4)

Let 𝜂∶𝑋×𝑋→𝑋 be a single-valued mapping. The map 𝜂 is called 𝜏-Lipschitz continuous if there exists a constant 𝜏>0 such that 𝜂(𝑢,𝑣)≤𝜏‖𝑢−𝑣‖,∀𝑢,𝑣∈𝑋. (5)

Definition 1. Let 𝜂∶𝑋×𝑋→𝑋 be a single-valued mapping, and let 𝑀∶𝑋→2^𝑋 be a mapping on 𝑋. A set-valued mapping 𝑀 is said to be(i)accretive if 𝑢^∗−𝑣^∗,𝐽_𝑞(𝑢−𝑣)≥0,∀𝑢,𝑣∈𝑋,𝑢^∗∈𝑀(𝑢),𝑣^∗∈𝑀(𝑣); (6)(ii)𝜂-accretive if 𝑢^∗−𝑣^∗,𝐽_𝑞𝜂(𝑢,𝑣)≥0,∀𝑢,𝑣∈𝑋,𝑢^∗∈𝑀(𝑢),𝑣^∗∈𝑀(𝑣); (7)(iii)(𝑟,𝜂)-strongly accretive if 𝑢^∗−𝑣^∗,𝐽_𝑞𝜂(𝑢,𝑣)≥𝑟‖𝑢−𝑣‖^𝑞,∀𝑢,𝑢^∗,𝑣,𝑣^∗∈Graph(𝑀); (8)(iv)𝜂-pseudomonotone if ⟨𝑣^∗,𝐽_𝑞𝜂(𝑢,𝑣)⟩≥0, (9) implying ⟨𝑢^∗,𝐽_𝑞𝜂(𝑢,𝑣)⟩≥0,∀𝑢,𝑢^∗,𝑣,𝑣^∗∈Graph(𝑀); (10)(v)(𝑚,𝜂)-relaxed accretive if there exists a positive constant 𝑚 such that 𝑢^∗−𝑣^∗,𝐽_𝑞𝜂(𝑢,𝑣)≥(−𝑚)‖𝑢−𝑣‖^𝑞,∀𝑢,𝑢^∗,𝑣,𝑣^∗∈Graph(𝑀). (11)

Definition 2. A single-valued mapping 𝐴∶𝑋→𝑋 is said to be(i)accretive if 𝐴(𝑢)−𝐴(𝑣),𝐽_𝑞(𝑢−𝑣)≥0,∀𝑢,𝑣∈𝑋; (12)(ii)strictly accretive if 𝐴 is accretive and 𝐴(𝑢)−𝐴(𝑣),𝐽_𝑞(𝑢−𝑣)=0,iff𝑢=𝑣,∀𝑢,𝑣∈𝑋; (13)(iii)(𝑟,𝜂)-strongly accretive if there exists a constant 𝑟>0 such that 𝐴(𝑢)−𝐴(𝑣),𝐽_𝑞𝜂(𝑢,𝑣)≥𝑟‖𝑢−𝑣‖^𝑞,∀𝑢,𝑣∈𝑋; (14)(iv)𝛼-Lipschitz continuous if there exists a constant 𝛼>0 such that ‖𝐴(𝑢)−𝐴(𝑣)‖≥𝛼‖𝑢−𝑣‖,∀𝑢,𝑣∈𝑋. (15)

Definition 3. A single-valued mapping 𝑁∶𝑋×𝑋→𝑋 is said to be(i)(𝜇,𝜐)-Lipschitz continuous if there exist constants 𝜇,𝜐>0 such that 𝑁𝑥₁,𝑦₁−𝑁𝑥₂,𝑦₂≤𝜇𝑥₁−𝑥₂+𝜐𝑦₁−𝑦₂,∀𝑥_𝑖,𝑦_𝑖∈𝑋,𝑖=1,2; (16)(ii)(𝜑,𝜅)−𝑄-relaxed cocoercive with respect to 𝐴𝑄 in the second argument if there exists constant 𝜑,𝜅>0 and for all 𝑥_𝑖∈𝑋, 𝑦_𝑖∈𝑄(𝑢_𝑖), (𝑖=1,2) such that 𝑁𝑦₁,⋅−𝑁𝑦₂,⋅,𝐽_𝑞𝐴𝑦₁−𝐴𝑦₂≥𝜑𝑁𝑦₁,⋅−𝑁𝑦₂,⋅^𝑞+𝜅𝑥₁−𝑥₂^𝑞, (17) where 𝐴,𝑄∶𝑋→𝑋 are single-valued mappings.

Definition 4. A mapping 𝑀∶𝑋→2^𝑋 is said to be maximal (𝑚,𝜂)-relaxed accretive if(i)𝑀 is (𝑀,𝜂)-accretive; (ii)for (𝑢,𝑢^∗)∈𝑋×𝑋 and 𝑢^∗−𝑣^∗,𝐽_𝑞𝜂()≥(‖𝑢,𝑣−𝑚)‖𝑢−𝑣^𝑞,∀𝑣,𝑣^∗∈Graph(𝑀), (18) one has 𝑢^∗∈𝑀(𝑢).

Definition 5. Let 𝐴∶𝑋→𝑋 and 𝜂∶𝑋×𝑋→𝑋 be two single-valued mappings; the mapping 𝑀∶𝑋→2^𝑋 is said to be (𝐴,𝜂)-accretive if(i)𝑀 is (𝑀,𝜂)-relaxed accretive; (ii)𝑅(𝐴+𝜌𝑀)=𝑋 for 𝜌>0. Note that alternatively, the mapping 𝑀∶𝑋→2^𝑋 is said to be (𝐴,𝜂)-accretive if (i)𝑀 is (𝑀,𝜂)-relaxed accretive; (ii)𝐴+𝜌𝑀 is 𝜂-pseudoaccretive for 𝜌>0.
The following propositions will be needed in the sequel. For more details one can refer to [7, 10, 15].

Proposition 6. Let 𝐴∶𝑋→𝑋 be an (𝑟,𝜂)-strongly accretive single-valued mapping, and let 𝑀∶𝑋→2^𝑋 be an (𝐴,𝜂)-accretive mapping. Let 𝜂∶𝑋×𝑋→𝑋 be 𝜏-Lipschitz continuous single-valued mapping. Then, 𝑀 is maximal (𝑚,𝜂)-relaxed accretive, and (𝐴+𝜌𝑀)𝑋=𝑋 for 0<𝜌<𝑟/𝑚.

Proposition 7. Let 𝐴∶𝑋→𝑋 be an (𝑟,𝜂)-strongly accretive single-valued mapping, and let 𝑀∶𝑋→2^𝑋 be an (𝐴,𝜂)-accretive mapping. In addition, let 𝜂∶𝑋×𝑋→𝑋 be 𝜏-Lipschitz continuous. Then (𝐴+𝜌𝑀) is maximal 𝜂-accretive for 0<𝜌<𝑟/𝑚.

Proposition 8. Let 𝐴∶𝑋→𝑋 be an (𝑟,𝜂)-strongly accretive mapping, and let 𝑀∶𝑋→2^𝑋 be an (𝐴,𝜂)-accretive mapping. If, in addition, 𝜂∶𝑋×𝑋→𝑋 is 𝜏-Lipschitz continuous, then, the operator (𝐴+𝜌𝑀)⁻¹ is single-valued for 0<𝜌<𝑟/𝑚.

Definition 9. Let 𝜂∶𝑋×𝑋→𝑋 be a single-valued mapping. Let 𝐴∶𝑋→𝑋 be an (𝑟,𝜂)-strongly accretive mapping, and let 𝑀∶𝑋→2^𝑋 be an (𝐴,𝜂)-accretive mapping. Then, the generalized resolvent operator 𝑅_𝑀^{𝜂,𝜌,𝐴}∶𝑋→𝑋 is defined by 𝑅_𝑀^{𝜂,𝜌,𝐴}(𝑢)=𝐴+𝜌𝑀⁻¹(𝑢), (19) where 𝜌>0 is a constant.

Definition 10. The mapping 𝑓∶𝑋×𝑋×𝐿→𝑋 is said to be 𝛽-strongly accretive with respect to 𝐴 in the first argument if there exists a positive constant 𝛽 such that 𝑓(𝑥,𝜆)−𝑓𝑦,𝜆,𝐽_𝑞𝑥−𝑦≥𝛽𝑥−𝑦^𝑞,∀𝑥,𝑦,𝑢,𝜆∈𝑋×𝑋×𝑋×𝐿. (20)

3. Sensitivity Analysis

Let 𝑄, let 𝑇, and let 𝑉∶𝑋×𝐿→2^𝑋 be three set-valued mappings, and let a nonlinear mapping 𝑁∶𝑋×𝑋×𝐿→𝑋 be an 𝐴𝑄-accretive mapping with respect to first argument, and let 𝑀∶𝑋×𝑋×𝐿→2^𝑋 be the mapping, where 𝐿 is a nonempty open subset of 𝑋. Let 𝑓,𝑔∶𝑋×𝐿→𝑋 be the single-valued mappings. Furthermore, let 𝜂∶𝑋×𝑋→𝑋 be a nonlinear mapping. Then, the problem of finding an element 𝑢∈𝑋, 𝑥∈𝑄(𝑢,𝜆), 𝑦∈𝑇(𝑢,𝜆), 𝑤∈𝑉(𝑢,𝜆) such that 𝑢∈𝑓𝑦,𝜆+𝑁𝑥,𝑔(𝑢,𝜆),𝜆+𝑀(𝑢,𝑤,𝜆), (21) where 𝐴∈𝐿 is the set-valued parameter is called a generalized set-valued variational inclusions.

The solvability of problem (21) depends on the equivalence between (21) and the problem of finding the fixed point of the associated resolvent operator.

Note that if 𝑀 is (𝐴,𝜂)-accretive, then the corresponding resolvent operator defined by 𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑣,𝜆)}(𝑢)=𝐴+𝜌𝑀(⋅,𝑣,𝜆)⁻¹(𝑢),∀𝑢∈𝑋, (22) where 𝜌>0 and 𝐴 is an (𝑟,𝜂)-strongly accretive mapping.

Lemma 11 (see [7, 10, 15]). Let 𝑋 be a real Banach spaces, and let 𝜂∶𝑋×𝑋→𝑋 be a 𝜏-Lipschitz continuous nonlinear mapping. Let 𝐴∶𝑋→𝑋 be (𝑟,𝜂)-strongly accretive mapping, and let 𝑀∶𝑋×𝑋×𝐿→2^𝑋 be (𝐴,𝜂)-accretive in the first variable with 𝑚. Then, the resolvent operator associated with 𝑀(⋅,𝑣,𝜆) for a fixed 𝑣∈𝑋, defined by 𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑣,𝜆)}(𝑢)=𝐴+𝜌𝑀(⋅,𝑣,𝜆)⁻¹(𝑢),∀𝑢∈𝑋, (23) is (𝜏^𝑞−1/(𝑟−𝜌𝑚))-Lipschitz continuous, where 𝜌∈(0,𝑟/𝑚); that is, 𝑅_𝑀()^{𝜂,𝜌,𝐴}_{⋅,𝑤,𝜆}(𝑢)−𝑅_𝑀()^{𝜂,𝜌,𝐴}_{⋅,𝑤,𝜆}(𝑣≤𝜏_𝑞−1)‖‖𝑟−𝜌𝑚𝑢−𝑣,∀𝑢,𝑣,𝑤∈𝑋. (24)

Lemma 12 (see [7, 10, 15]). Let 𝑋 be a real uniformly smooth Banach spaces, let 𝐴∶𝑋→𝑋 be the (𝑟,𝜂)-strongly accretive mapping, and let 𝑀∶𝑋×𝑋×𝐿→2^𝑋 be an (𝐴,𝜂)-accretive in the first variable. Let 𝜂∶𝑋×𝑋→𝑋 be a 𝜏-Lipschitz continuous nonlinear mapping; then the following statements are mutually equivalent: (i)an element 𝑢∈𝑋, 𝑥∈𝑄(𝑢,𝜆),  𝑦∈𝑇(𝑢,𝜆), 𝑤∈𝑉(𝑢,𝜆) is a solution to (21);(ii)the mapping 𝐺∶𝑋×𝐿→𝑋 defined by 𝐺(𝑢,𝜆)=𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑤,𝜆)}𝐴(𝑢)−𝜌𝑁𝑥,𝑔(𝑢,𝜆),𝜆+𝜌𝑢−𝜌𝑓𝑦,𝜆 (25) has a fixed point.

Lemma 13 (see [21]). Let 𝑋 be a real uniformly smooth Banach spaces. Then, 𝑋 is 𝑞-uniformly smooth if and only if there exists a constant 𝑐_𝑞>0 such that for all 𝑢,𝑣∈𝑋‖𝑢+𝑣‖^𝑞≤‖𝑢‖^𝑞+𝑞𝑣,𝐽_𝑞(𝑢)+𝑐_𝑞‖𝑣‖^𝑞. (26)

Theorem 14. Let 𝑋 be a 𝑞-uniformly smooth Banach spaces, and let 𝜂∶𝑋×𝑋→𝑋 be a 𝜏-Lipschitz continuous nonlinear mapping. Let 𝐴∶𝑋→𝑋 be the (𝑟,𝜂)-strongly accretive, and let 𝑠-Lipschitz continuous mapping and 𝑀∶𝑋×𝑋×𝐿→2^𝑋 be the (𝐴,𝜂)-accretive in the first variable with 𝑚. Let 𝑇,𝑄,𝑉∶𝑋×𝐿→2^𝑋 be the set-valued mappings and ℋ-Lipschitz continuous with 𝛾, 𝜎, and 𝜁 constants, respectively. Let 𝑔,𝑓∶𝑋×𝐿→𝑋 be the 𝜋-Lipschitz continuous and 𝜒-Lipschitz continuous mappings. Let 𝑁∶𝑋×𝑋×𝐿→𝑋 be the 𝜇-Lipschitz continuous with first variable and 𝜐-Lipschitz continuous in the second variable. Let 𝑁 be the (𝜑,𝜅)−𝑄-relaxed cocoercive with respect to 𝐴𝑄 in the first argument. Let 𝑓 be the strongly accretive mapping. If the following condition holds: 𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑤,𝜆)}(𝑢)−𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑤,𝜆)}(𝑣≤𝜏)^𝑞−1‖‖,𝑅𝑟−𝜌𝑚𝑢−𝑣_{𝜂,𝜌,𝐴𝑀(⋅,𝑤1,𝜆)}(𝑢)−𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑤2,𝜆)}(𝑢)≤𝛿𝑤₁−𝑤₂𝜏_𝑞,𝑠^𝑞+𝑐_𝑞𝜌^𝑞𝜇^𝑞𝜎^𝑞+𝑞𝜌𝜓𝜇^𝑞𝜎^𝑞−𝑞𝜌𝜅^1/𝑞+𝜌𝜐𝜋+1+𝑐_𝑞𝜒^𝑞𝛾^𝑞−𝑞𝛽^1/𝑞<𝑟−𝜌𝑚(1−𝛿𝜁), (27)(28)(29) where 𝑐_𝑞>0 is the same as in Lemma 13 and 𝜌∈(0,𝑟/𝑚). Consequently, for each 𝜆∈𝐿, the mapping 𝐺(𝑢,𝜆) in the light (28) has a unique fixed point 𝑧(𝜆). Hence, in light of Lemma 12, 𝑧(𝜆) is a unique solution to (21). Thus, one has 𝐺(𝑧(𝜆),𝜆)=𝑧(𝜆). (30)

Proof. For any elements (𝑢,𝑣,𝜆)∈𝑋×𝑋×𝐿 and 𝑥₁∈𝑄(𝑢,𝜆), 𝑥₂∈𝑄(𝑣,𝜆), 𝑦₁∈𝑇(𝑢,𝜆), 𝑦₂∈𝑇(𝑣,𝜆), 𝑤₁∈𝑉(𝑢,𝜆), and 𝑤₂∈𝑉(𝑣,𝜆), we have ‖𝐺()(=𝑅𝑢,𝜆−𝐺𝑣,𝜆)‖_{𝑀𝜂,𝜌,𝐴⋅,𝑤1,𝜆}𝐴(𝑢)−𝜌𝑁𝑥₁()𝑦,𝑔𝑢,𝜆,𝜆+𝜌𝑢−𝜌𝑓₁,𝜆−𝑅_𝑀^{𝜂,𝜌,𝐴}_{⋅,𝑤2,𝜆}𝐴(𝑣)−𝜌𝑁𝑥₂()𝑦,𝑔𝑣,𝜆,𝜆+𝜌𝑣−𝜌𝑓₂≤𝑅,𝜆_{𝑀𝜂,𝜌,𝐴⋅,𝑤1,𝜆}𝐴(𝑢)−𝜌𝑁𝑥₁()𝑦,𝑔𝑢,𝜆,𝜆+𝜌𝑢−𝜌𝑓₁,𝜆−𝑅_𝑀^{𝜂,𝜌,𝐴}_{⋅,𝑤1,𝜆}𝐴(𝑣)−𝜌𝑁𝑥₂()𝑦,𝑔𝑣,𝜆,𝜆+𝜌𝑣−𝜌𝑓₂+𝑅,𝜆_{𝑀𝜂,𝜌,𝐴⋅,𝑤1,𝜆}×𝐴(𝑣)−𝜌𝑁𝑥₂,𝑔(𝑣,𝜆),𝜆+𝜌𝑣−𝜌𝑓𝑦₂,𝜆−𝑅_𝑀^{𝜂,𝜌,𝐴}_{⋅,𝑤2,𝜆}×𝐴(𝑣)−𝜌𝑁𝑥₂,𝑔(𝑣,𝜆),𝜆+𝜌𝑣−𝜌𝑓𝑦₂≤𝜏,𝜆_𝑞−1𝐴(𝑢)(𝑣)𝑁𝑥𝑟−𝜌𝑚−𝐴−𝜌₁,𝑔(𝑢,𝜆),𝜆−𝑁𝑥₂()𝑓𝑦,𝑔𝑣,𝜆,𝜆+𝜌𝑢−𝑣−₁,𝜆−𝑓𝑦₂𝑤,𝜆+𝛿₁−𝑤₂≤𝜏^𝑞−1𝐴(𝑢)(𝑣)𝑁𝑥𝑟−𝜌𝑚−𝐴−𝜌₁,𝑔(𝑢,𝜆),𝜆−𝑁𝑥₂()𝑁𝑥,𝑔𝑢,𝜆,𝜆+𝜌₂,𝑔(𝑢,𝜆),𝜆−𝑁𝑥₂()𝑓𝑦,𝑔𝑣,𝜆,𝜆+𝜌𝑢−𝑣−₁,𝜆−𝑓𝑦₂𝑤,𝜆+𝛿₁−𝑤₂. (31)
Since 𝑇, 𝑉, and 𝑄 are ℋ-Lipschitz continuous and 𝑁 is Lipschitz continuous with respect to first and second arguments, 𝑓, 𝑔 are Lipschitz continuous, we have 𝑁𝑥₁,𝑔(𝑢,𝜆),𝜆−𝑁𝑥₂()𝑥,𝑔𝑢,𝜆,𝜆≤𝜇₁−𝑥₂(𝑄()(‖‖,𝑁𝑥≤𝜇ℋ𝑢,𝜆,𝑄𝑣,𝜆))≤𝜇𝜎𝑢−𝑣₂,𝑔(𝑢,𝜆),𝜆−𝑁𝑥₂()𝑔()(‖‖,𝑓𝑦,𝑔𝑣,𝜆,𝜆≤𝜐𝑢,𝜆−𝑔𝑣,𝜆)≤𝜐𝜋𝑢−𝑣₁,𝜆−𝑓𝑦₂,𝜆≤𝜒𝑦₁−𝑦₂(𝑇()(‖‖,𝑤≤𝜒ℋ𝑢,𝜆,𝑇𝑣,𝜆))≤𝜒𝛾𝑢−𝑣₁−𝑤₂≤ℋ(𝑉(𝑢,𝜆),𝑉(𝑣,𝜆))≤𝜁‖𝑢−𝑣‖. (32)
Since 𝑁(⋅,⋅) is (𝜑,𝜅)−𝑄-relaxed cocoercive with respect to 𝐴𝑄 in the first argument and 𝑄 is ℋ-Lipschitz continuous mapping, so we obtain 𝐴(𝑢)(𝑣)𝑁𝑥−𝐴−𝜌₁,𝑔(𝑢,𝜆),𝜆−𝑁𝑥₂,𝑔(𝑢,𝜆),𝜆^𝑞≤‖𝐴(𝑢)−𝐴(𝑣)‖^𝑞−𝑞𝜌𝑁𝑥₁()𝑥,𝑔𝑢,𝜆,𝜆−𝑁₂,𝑔(𝑢,𝜆),𝜆,𝐽_𝑞(𝐴(𝑢)−𝐴(𝑣))+𝑐_𝑞𝜌^𝑞𝑁𝑥₁,𝑔(𝑢,𝜆),𝜆−𝑁𝑥₂,𝑔(𝑢,𝜆),𝜆^𝑞≤𝑠^𝑞−𝑞𝜌𝜅+𝑞𝜌𝜓𝜇^𝑞𝜎^𝑞+𝑐_𝑞𝜌^𝑞𝜇^𝑞𝜎^𝑞‖𝑢−𝑣‖^𝑞. (33)
Since 𝑓 is 𝛽-strongly accretive and 𝑇 is the ℋ-Lipschitz continuous, we have 𝑢−𝑣−𝑓𝑦₁,𝜆−𝑓𝑦₂,𝜆^𝑞≤‖𝑢−𝑣‖^𝑞−𝑞𝑓𝑦₁,𝜆−𝑓𝑦₂,𝜆,𝐽_𝑞(𝑢−𝑣)+𝑐_𝑞𝑓𝑦₁,𝜆−𝑓𝑦₂,𝜆^𝑞≤1+𝑐_𝑞𝜒^𝑞𝛾^𝑞−𝑞𝛽‖𝑢−𝑣‖^𝑞. (34)
Combining (31)–(34), we can get ‖𝐺(𝑢,𝜆)−𝐺(𝑣,𝜆)‖≤𝜃‖𝑢−𝑣‖, (35) where 𝜏𝜃=^𝑞−1𝑠𝑟−𝜌𝑚^𝑞−𝑞𝜌𝜅+𝑐_𝑞𝜌^𝑞𝜇^𝑞𝜎^𝑞+𝑞𝜌𝜓𝜇^𝑞𝜎^𝑞1/𝑞+𝜌𝜐𝜋+𝜌1+𝑐_𝑞𝜒^𝑞𝛾^𝑞−𝑞𝛽^1/𝑞+𝛿𝜁. (36)
It follows from (29) such that 𝜃<1; it concludes the proof.

Theorem 15. Let 𝑋 be a real 𝑞-uniformly smooth Banach spaces, and let 𝜂∶𝑋×𝑋→𝑋 be a 𝜏-Lipschitz continuous nonlinear mapping. Let 𝐴∶𝑋→𝑋 be a (𝑟,𝜂)-strongly accretive and 𝑠-Lipschitz continuous mapping, and let 𝑀∶𝑋×𝑋×𝐿→2^𝑋 be (𝐴,𝜂)-accretive in the first variable with 𝑚. Let 𝑇, 𝑄, and 𝑉∶𝑋×𝐿→2^𝑋 be the set-valued mappings and ℋ-Lipschitz continuous with respect to 𝛾, 𝜎, and 𝜁 constants, respectively. Let 𝑔,𝑓∶𝑋×𝐿→𝑋 be the 𝜋-Lipschitz continuous and 𝜒-Lipschitz continuous mappings, respectively. Let 𝑁∶𝑋×𝑋×𝐿→𝑋 be the 𝜇-Lipschitz continuous with first variable and 𝜐-Lipschitz continuous with second variable. Let 𝑁 be the (𝜑,𝜅)−𝑄-relaxed cocoercive with respect to 𝐴𝑄 in the first variable. Let 𝑓 be the 𝛽-strongly accretive mapping if the following condition holds: 𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑤,𝜆)}(𝑢)−𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑤,𝜆)}(𝑣≤𝜏)^𝑞−1‖‖,𝑅𝑟−𝜌𝑚𝑢−𝑣_{𝜂,𝜌,𝐴𝑀(⋅,𝑤1,𝜆)}(𝑢)−𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑤2,𝜆)}(𝑢)≤𝛿𝑤₁−𝑤₂,𝜏^𝑞𝑠^𝑞+𝑐_𝑞𝜌^𝑞𝜇^𝑞𝜎^𝑞+𝑞𝜌𝜓𝜇^𝑞𝜎^𝑞−𝑞𝜌𝜅^1/𝑞+𝜌𝜐𝜋+1+𝑐_𝑞𝜒^𝑞𝛾^𝑞−𝑞𝛽^1/𝑞<𝑟−𝜌𝑚(1−𝛿𝜁), (37) where 𝑐_𝑞>0 is the same as in Lemma 13 and 𝜌∈(0,𝑟/𝑚).
If the mapping 𝜆→𝑁(𝑥₁,𝑔(𝑢,𝜆),𝜆), 𝜆→𝑓(𝑦,𝜆), and 𝜆→𝑅^{𝜂,𝜌,𝐴}_{𝑀(⋅,𝑤,𝜆)}, 𝜆→𝑇(𝑢,𝜆), 𝜆→𝑉(𝑢,𝜆), 𝜆→𝑔(𝑢,𝜆), and 𝜆→𝑄(𝑢,𝜆) are continuous (or Lipschitz continuous) from 𝐿 to 𝑋, then the solution 𝑧(𝜆) of (21) is continuous (or Lipschitz continuous) from 𝐿 to 𝑋.

Proof. From the hypothesis of theorem, for any 𝜆,𝜆^∗∈𝐿, we have 𝑧(𝜆)−𝑧𝜆^∗=𝐺(𝑧(𝜆),𝜆)−𝐺𝑧𝜆^∗,𝜆^∗≤𝐺(𝑧(𝜆),𝜆)−𝐺𝑧𝜆^∗+𝐺𝑧𝜆,𝜆_∗,𝜆−𝐺𝑧𝜆^∗,𝜆^∗≤𝜃𝑧(𝜆)−𝑧𝜆^∗+𝐺𝑧𝜆^∗,𝜆−𝐺𝑧𝜆^∗,𝜆^∗. (38)
It follows that𝐺𝑧𝜆^∗,𝜆−𝐺𝑧𝜆^∗,𝜆^∗=𝑅_𝑀𝑧𝜆^{𝜂,𝜌,𝐴}_{⋅,𝑤∗,𝜆,𝜆}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝑔𝑧𝜆,𝜆_∗,𝜆,𝜆+𝜌𝑧𝜆^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆,𝜆−𝑅_𝑀𝑧𝜆^{𝜂,𝜌,𝐴}_{⋅,𝑤∗,𝜆∗,𝜆∗}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝜌𝑧𝜆^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆^∗,𝜆^∗≤𝑅_𝑀𝑧𝜆^{𝜂,𝜌,𝐴}_{⋅,𝑤∗,𝜆,𝜆}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝜆,𝑔𝑧𝜆^∗𝜆,𝜆,𝜆+𝜌𝑧^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆,𝜆−𝑅_𝑀𝑧𝜆^{𝜂,𝜌,𝐴}_{⋅,𝑤∗,𝜆,𝜆}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝜌𝑧𝜆^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝑅_𝑀𝑧𝜆^{𝜂,𝜌,𝐴}_{⋅,𝑤∗,𝜆,𝜆}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝜌𝑧𝜆^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆^∗,𝜆^∗−𝑅_𝑀𝑧𝜆^{𝜂,𝜌,𝐴}_{⋅,𝑤∗,𝜆∗,𝜆∗}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝜌𝑧𝜆^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆^∗,𝜆^∗≤𝜌𝜏^𝑞−1𝑁𝑥𝑧𝜆𝑟−𝜌𝑚^∗,𝜆,𝑔𝑧𝜆^∗𝑥𝑧𝜆,𝜆,𝜆−𝑁^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗−𝑓𝑦𝑧𝜆^∗,𝜆,𝜆−𝑓𝑦𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝑅_𝑀𝑧𝜆^{𝜂,𝜌,𝐴}_{⋅,𝑤∗,𝜆,𝜆}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝜌𝑧𝜆^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆^∗,𝜆^∗−𝑅_𝑀𝑧𝜆^{𝜂,𝜌,𝐴}_{⋅,𝑤∗,𝜆∗,𝜆∗}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝜌𝑧𝜆^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆^∗,𝜆^∗. (39)
Hence, we get𝑧(𝜆)−𝑧𝜆^∗≤𝜌𝜏^𝑞−1()×𝑁𝑥𝑧𝜆𝑟−𝜌𝑚1−𝜃_∗,𝜆,𝑔𝑧𝜆^∗𝑥𝑧𝜆,𝜆,𝜆−𝑁^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗−𝑓𝑦𝑧𝜆^∗𝑦𝑧𝜆,𝜆,𝜆−𝑓^∗,𝜆^∗,𝜆^∗+1𝑅1−𝜃^{𝑀𝑧𝜆𝜂,𝜌,𝐴⋅,𝑤∗,𝜆,𝜆}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝜌𝑧𝜆^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆^∗,𝜆^∗−𝑅_𝑀𝑧𝜆^{𝜂,𝜌,𝐴}_{⋅,𝑤∗,𝜆∗,𝜆∗}×𝐴𝑧𝜆^∗−𝜌𝑁𝑥𝑧𝜆^∗,𝜆^∗,𝑔𝑧𝜆^∗,𝜆^∗,𝜆^∗+𝜌𝑧𝜆^∗−𝜌𝑓𝑦𝑧𝜆^∗,𝜆^∗,𝜆^∗. (40)
This completes the proof.