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

Authors: A. Farajzadeh; Salahuddin
Journal: Journal of Function Spaces and Applications
 (2013)

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## 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.


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## Body

## 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 RK. 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

LetX be a real Banach space with dual space X*, and let 〈·,·〉 be the dual pair between X and X*, 2X 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 Jq:X→2X* is defined by
(1)Jq(x)={f*∈X*:〈u,f*〉=∥u∥q,∥f*∥=∥u∥q-1},∀u∈X,
where q>1 is constant.The modulus of smoothness ofX is the function ρX:[0,∞)→[0,∞) defined by
(2)ρX(t)=sup{12(∥u+v∥+∥u-v∥)-1:∥u∥≤1,∥v∥<t}.
A Banach space X is called uniformly smooth if
(3)limt→0ρX(t)t=0.X is called q-uniformly smooth if there exists a constant c>0 such that
(4)ρX(t)≤ctq(q>1).Letη: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
(5)∥η(u,v)∥≤τ∥u-v∥,∀u,v∈X.Definition 1.
Letη:X×X→X be a single-valued mapping, and let M:X→2X be a mapping on X. A set-valued mapping M is said to be(i)
accretive if(6)〈u*-v*,Jq(u-v)〉≥0,∀u,v∈X,u*∈M(u),v*∈M(v);(ii)
η-accretive if
(7)〈u*-v*,Jq(η(u,v))〉≥0,∀u,v∈X,u*∈M(u),v*∈M(v);(iii)
(r,η)-strongly accretive if
(8)〈u*-v*,Jq(η(u,v))〉≥r∥u-v∥q,∀(u,u*),(v,v*)∈Graph(M);(iv)
η-pseudomonotone if
(9)〈v*,Jq(η(u,v))〉≥0,
implying
(10)〈u*,Jq(η(u,v))〉≥0,∀(u,u*),(v,v*)∈Graph(M);(v)
(m,η)-relaxed accretive if there exists a positive constant m such that
(11)〈u*-v*,Jq(η(u,v))〉≥(-m)∥u-v∥q,∀(u,u*),(v,v*)∈Graph(M).Definition 2.
A single-valued mappingA:X→X is said to be(i)
accretive if(12)〈A(u)-A(v),Jq(u-v)〉≥0,∀u,v∈X;(ii)
strictly accretive ifA is accretive and
(13)〈A(u)-A(v),Jq(u-v)〉=0,iffu=v,∀u,v∈X;(iii)
(r,η)-strongly accretive if there exists a constant r>0 such that
(14)〈A(u)-A(v),Jq(η(u,v))〉≥r∥u-v∥q,∀u,v∈X;(iv)
α-Lipschitz continuous if there exists a constant α>0 such that
(15)∥A(u)-A(v)∥≥α∥u-v∥,∀u,v∈X.Definition 3.
A single-valued mappingN:X×X→X is said to be(i)
(μ,υ)-Lipschitz continuous if there exist constants μ,υ>0 such that
(16)∥N(x1,y1)-N(x2,y2)∥≤μ∥x1-x2∥+υ∥y1-y2∥,∀xi,yi∈X,i=1,2;(ii)
(φ,κ)-Q-relaxed cocoercive with respect to AQ in the second argument if there exists constant φ,κ>0 and for all xi∈X, yi∈Q(ui), (i=1,2) such that
(17)〈N(y1,·)-N(y2,·),Jq(A(y1)-A(y2))〉≥φ∥N(y1,·)-N(y2,·)∥q+κ∥x1-x2∥q,
where A,Q:X→X are single-valued mappings.Definition 4.
A mappingM:X→2X is said to be maximal (m,η)-relaxed accretive if(i)
M is (M,η)-accretive;(ii)
for(u,u*)∈X×X and
(18)〈u*-v*,Jq(η(u,v))〉≥(-m)∥u-v∥q,∀(v,v*)∈Graph(M),
one has u*∈M(u).Definition 5.
LetA:X→X and η:X×X→X be two single-valued mappings; the mapping M:X→2X 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→2X 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.
LetA:X→X be an (r,η)-strongly accretive single-valued mapping, and let M:X→2X 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.
LetA:X→X be an (r,η)-strongly accretive single-valued mapping, and let M:X→2X 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.
LetA:X→X be an (r,η)-strongly accretive mapping, and let M:X→2X 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.
Letη:X×X→X be a single-valued mapping. Let A:X→X be an (r,η)-strongly accretive mapping, and let M:X→2X be an (A,η)-accretive mapping. Then, the generalized resolvent operator RMη,ρ,A:X→X is defined by
(19)RMη,ρ,A(u)=(A+ρM)-1(u),
where ρ>0 is a constant.Definition 10.
The mappingf: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
(20)〈f(x,λ)-f(y,λ),Jq(x-y)〉≥β∥x-y∥q,∀(x,y,u,λ)∈X×X×X×L.


## 3. Sensitivity Analysis

LetQ, let T, and let V:X×L→2X 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→2X 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
(21)u∈f(y,λ)+N(x,g(u,λ),λ)+M(u,w,λ),
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 ifM is (A,η)-accretive, then the corresponding resolvent operator defined by
(22)RM(·,v,λ)η,ρ,A(u)=(A+ρM(·,v,λ))-1(u),∀u∈X,
where ρ>0 and A is an (r,η)-strongly accretive mapping.Lemma 11 (see [7, 10, 15]).
LetX 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→2X 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
(23)RM(·,v,λ)η,ρ,A(u)=(A+ρM(·,v,λ))-1(u),∀u∈X,
is (τq-1/(r-ρm))-Lipschitz continuous, where ρ∈(0,r/m); that is,
(24)∥RM(·,w,λ)η,ρ,A(u)-RM(·,w,λ)η,ρ,A(v)∥≤τq-1r-ρm∥u-v∥,∀u,v,w∈X.Lemma 12 (see [7, 10, 15]).
LetX be a real uniformly smooth Banach spaces, let A:X→X be the (r,η)-strongly accretive mapping, and let M:X×X×L→2X 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 elementu∈X, x∈Q(u,λ),  y∈T(u,λ), w∈V(u,λ) is a solution to (21);(ii)
the mappingG:X×L→X defined by
(25)G(u,λ)=RM(·,w,λ)η,ρ,A(A(u)-ρN(x,g(u,λ),λ)+ρu-ρf(y,λ))
has a fixed point.Lemma 13 (see [21]).
LetX be a real uniformly smooth Banach spaces. Then, X is q-uniformly smooth if and only if there exists a constant cq>0 such that for all u,v∈X(26)∥u+v∥q≤∥u∥q+q〈v,Jq(u)〉+cq∥v∥q.Theorem 14.
LetX 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→2X be the (A,η)-accretive in the first variable with m. Let T,Q,V:X×L→2X 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:
(27)∥RM(·,w,λ)η,ρ,A(u)-RM(·,w,λ)η,ρ,A(v)∥≤τq-1r-ρm∥u-v∥,(28)∥RM(·,w1,λ)η,ρ,A(u)-RM(·,w2,λ)η,ρ,A(u)∥≤δ∥w1-w2∥,(29)τq[(sq+cqρqμqσq+qρψμqσq-qρκ)1/q+ρ(υπ+(1+cqχqγq-qβ)1/q)]<(r-ρm)(1-δζ),
where cq>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
(30)G(z(λ),λ)=z(λ).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
(31)∥G(u,λ)-G(v,λ)∥=∥RM(·,w1,λ)η,ρ,A(f(y1,λ)A(u)-ρN(x1,g(u,λ),λ)+ρu-ρf(y1,λ))-RM(·,w2,λ)η,ρ,A(ρf(y2,λ)A(v)-ρN(x2,g(v,λ),λ)+ρv-ρf(y2,λ))RM(·,w1,λ)η,ρ,A∥≤∥RM(·,w1,λ)η,ρ,A[ρf(y2,λ)A(u)-ρN(x1,g(u,λ),λ)+ρu-ρf(y1,λ)]-RM(·,w1,λ)η,ρ,A[ρf(y2,λ)A(v)-ρN(x2,g(v,λ),λ)+ρv-ρf(y2,λ)]RM(·,w1,λ)η,ρ,A∥+∥RM(·,w1,λ)η,ρ,A×[A(v)-ρN(x2,g(v,λ),λ)+ρv-ρf(y2,λ)]-RM(·,w2,λ)η,ρ,A×[A(v)-ρN(x2,g(v,λ),λ)+ρv-ρf(y2,λ)]RM(·,w1,λ)η,ρ,A∥≤τq-1r-ρm[∥RM(·,w1,λ)η,ρ,AA(u)-A(v)-ρ(N(x1,g(u,λ),λ)-N(x2,g(v,λ),λ))+ρ(u-v-(f(y1,λ)-f(y2,λ)))RM(·,w1,λ)η,ρ,A∥]+δ∥w1-w2∥≤τq-1r-ρm[∥∥u-v-(f(y1,λ)-f(y2,λ)∥A(u)-A(v)-ρ(N(x1,g(u,λ),λ)-N(x2,g(u,λ),λ))∥+ρ∥N(x2,g(u,λ),λ)-N(x2,g(v,λ),λ)∥+ρ∥u-v-(f(y1,λ)-f(y2,λ)∥]+δ∥w1-w2∥.
SinceT, 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
(32)∥N(x1,g(u,λ),λ)-N(x2,g(u,λ),λ)∥≤μ∥x1-x2∥≤μℋ(Q(u,λ),Q(v,λ))≤μσ∥u-v∥,∥N(x2,g(u,λ),λ)-N(x2,g(v,λ),λ)∥≤υ∥g(u,λ)-g(v,λ)∥≤υπ∥u-v∥,∥f(y1,λ)-f(y2,λ)∥≤χ∥y1-y2∥≤χℋ(T(u,λ),T(v,λ))≤χγ∥u-v∥,∥w1-w2∥≤ℋ(V(u,λ),V(v,λ))≤ζ∥u-v∥.
SinceN(·,·) is (φ,κ)-Q-relaxed cocoercive with respect to AQ in the first argument and Q is ℋ-Lipschitz continuous mapping, so we obtain
(33)∥(N(x1,g(u,λ),λ)-N(x2,g(u,λ),λ))A(u)-A(v)-ρ(N(x1,g(u,λ),λ)-N(x2,g(u,λ),λ))∥q≤∥A(u)-A(v)∥q-qρ〈(x2,g(u,λ),λ),Jq(A(u)-A(v))N(x1,g(u,λ),λ)-N(x2,g(u,λ),λ),Jq(A(u)-A(v))〉+cqρq∥N(x1,g(u,λ),λ)-N(x2,g(u,λ),λ)∥q≤[sq-qρκ+qρψμqσq+cqρqμqσq]∥u-v∥q.
Sincef is β-strongly accretive and T is the ℋ-Lipschitz continuous, we have
(34)∥u-v-(f(y1,λ)-f(y2,λ))∥q≤∥u-v∥q-q〈f(y1,λ)-f(y2,λ),Jq(u-v)〉+cq∥f(y1,λ)-f(y2,λ)∥q≤[1+cqχqγq-qβ]∥u-v∥q.
Combining (31)–(34), we can get
(35)∥G(u,λ)-G(v,λ)∥≤θ∥u-v∥,
where
(36)θ=τq-1(r-ρm)[(sq-qρκ+cqρqμqσq+qρψμqσq)1/q+ρυπ+ρ(1+cqχqγq-qβ)1/q]+δζ.
It follows from (29) such that θ<1; it concludes the proof.Theorem 15.
LetX 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→2X be (A,η)-accretive in the first variable with m. Let T, Q, and V:X×L→2X 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:
(37)∥RM(·,w,λ)η,ρ,A(u)-RM(·,w,λ)η,ρ,A(v)∥≤τq-1r-ρm∥u-v∥,∥RM(·,w1,λ)η,ρ,A(u)-RM(·,w2,λ)η,ρ,A(u)∥≤δ∥w1-w2∥,τq[(sq+cqρqμqσq+qρψμqσq-qρκ)1/q+ρ(υπ+(1+cqχqγq-qβ)1/q)]<(r-ρm)(1-δζ),
where cq>0 is the same as in Lemma 13 and ρ∈(0,r/m).
If the mappingλ→N(x1,g(u,λ),λ), λ→f(y,λ), and λ→RM(·,w,λ)η,ρ,A, 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 anyλ,λ*∈L, we have
(38)∥z(λ)-z(λ*)∥=∥G(z(λ),λ)-G(z(λ*),λ*)∥≤∥G(z(λ),λ)-G(z(λ*),λ)∥+∥G(z(λ*),λ)-G(z(λ*),λ*)∥≤θ∥z(λ)-z(λ*)∥+∥G(z(λ*),λ)-G(z(λ*),λ*)∥.
It follows that(39)∥G(z(λ*),λ)-G(z(λ*),λ*)∥=∥RM(·,w(z(λ*),λ),λ)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ)),g((z(λ*),λ),λ)+ρz(λ*)-ρf(y(z(λ*),λ),λ)]-RM(·,w(z(λ*),λ*),λ*)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ*),g(z(λ*),λ*),λ*)+ρz(λ*)-ρf(y(z(λ*),λ*),λ*)]RM(·,w1,λ)η,ρ,A∥≤∥RM(·,w(z(λ*),λ),λ)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ),g(z(λ*),λ),λ)+ρz(λ*)-ρf(y(z(λ*),λ),λ)]-RM(·,w(z(λ*),λ),λ)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ*),g(z(λ*),λ*),λ*)+ρz(λ*)-ρf(y(z(λ*),λ*),λ*)]RM(·,w1,λ)η,ρ,A∥+∥RM(·,w(z(λ*),λ),λ)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ*),g(z(λ*),λ*),λ*)+ρz(λ*)-ρf(y(z(λ*),λ*),λ*)]-RM(·,w(z(λ*),λ*),λ*)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ*),g(z(λ*),λ*),λ*)+ρz(λ*)-ρf(y(z(λ*),λ*),λ*)]RM(·,w1,λ)η,ρ,A∥≤ρτq-1r-ρm[∥N(x(z(λ*),λ),g(z(λ*),λ),λ)-N(x(z(λ*),λ*),g(z(λ*),λ*),λ*)-(f(y(z(λ*),λ),λ)-f(y(z(λ*),λ*),λ*))∥]+∥RM(·,w(z(λ*),λ),λ)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ*),g(z(λ*),λ*),λ*)+ρz(λ*)-ρf(y(z(λ*),λ*),λ*)]-RM(·,w(z(λ*),λ*),λ*)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ*),g(z(λ*),λ*),λ*)+ρz(λ*)-ρf(y(z(λ*),λ*),λ*)]RM(·,w1,λ)η,ρ,A∥.
Hence, we get(40)∥z(λ)-z(λ*)∥≤ρτq-1(r-ρm)(1-θ)×[∥N(x(z(λ*),λ),g(z(λ*),λ),λ)-N(x(z(λ*),λ*),g(z(λ*),λ*),λ*)-(f(y(z(λ*),λ),λ))-f(y(z(λ*),λ*),λ*)∥]+11-θ∥RM(·,w(z(λ*),λ),λ)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ*),g(z(λ*),λ*),λ*)+ρz(λ*)-ρf(y(z(λ*),λ*),λ*)]-RM(·,w(z(λ*),λ*),λ*)η,ρ,A×[A(z(λ*))-ρN(x(z(λ*),λ*),g(z(λ*),λ*),λ*)+ρz(λ*)-ρf(y(z(λ*),λ*),λ*)A(z(λ*))]RM(·,w(z(λ*),λ),λ)η,ρ,A∥.
This completes the proof.



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