http://jipam.vu.edu.au/
Volume 2, Issue 3, Article 27, 2001
GENERALIZED AUXILIARY PROBLEM PRINCIPLE AND SOLVABILITY OF A CLASS OF NONLINEAR VARIATIONAL INEQUALITIES INVOLVING
COCOERCIVE AND CO-LIPSCHITZIAN MAPPINGS
RAM U. VERMA UNIVERSITY OFTOLEDO, DEPARTMENT OFMATHEMATICS,
TOLEDO, OHIO43606, USA.
Received — ; accepted 15 March, 2001 Communicated by D. Bainov
ABSTRACT. The approximation-solvability of the following class of nonlinear variational in- equality (NVI) problems, based on a new generalized auxiliary problem principle, is discussed.
Find an elementx∗∈Ksuch that
h(S−T) (x∗), x−x∗i+f(x)−f(x∗)≥0 for allx∈K,
whereS, T :K→H are mappings from a nonempty closed convex subsetKof a real Hilbert spaceH into H, and f : K → Ris a continuous convex functional on K.The generalized auxiliary problem principle is described as follows: for given iteratexk ∈Kand, for constants ρ >0andσ >0), findxk+1such that
ρ(S−T) yk
+h0 xk+1
−h0 yk
, x−xk+1
+ρ(f(x)−f(xk+1))≥0 for all x∈K, where
σ(S−T) xk
+h0 yk
−h0 xk
, x−yk
+σ(f(x)−f(yk)) ≥ 0 for all x ∈ K, wherehis a functional onKandh0the derivative ofh.
Key words and phrases: Generalized auxiliary variational inequality problem, Cocoercive mappings, Approximation- solvability, Approximate solutions, Partially relaxed monotone mappings.
2000Mathematics Subject Classification. 49J40.
1. INTRODUCTION
Recently, Zhu and Marcotte [23], based on the auxiliary problem principle introduced by Co- hen [3], investigated the approximation-solvability of a class of variational inequalities involv- ing the cocoercive and partially cocoercive mappings in the Rnspace. The auxiliary problem technique introduced by Cohen [3], is quite similar to that of the iterative algorithm character- ized as the auxiliary variational inequality studied by Marcotte and Wu [12], but the estimates
ISSN (electronic): 1443-5756
c 2001 Victoria University. All rights reserved.
025-01
for the approximate solutions seem to be significantly different, which makes a difference es- tablishing the convergence of the sequence of approximate solutions to a given solution of the original variational inequality under consideration. On the top of that, using the auxiliary prob- lem principle, one does not require any projection formula leading to a fixed point and eventu- ally the solution of the variational inequality, which has been the case following the variational inequality type algorithm adopted by Marcotte and Wu [12]. Recently Verma [21] introduced an iterative scheme characterized as an auxiliary variational inequality type of algorithm and ap- plied to the approximation-solvability of a class of nonlinear variational inequalities involving cocoercive as well as partially relaxed monotone mappings [18] in a Hilbert space setting. The partially relaxed monotone mappings seem to be weaker than cocoercive and strongly mono- tone mappings. In this paper, we first intend to introduce the generalized auxiliary problem principle, and then apply the generalized auxiliary problem principle, which includes the aux- iliary problem principle of Cohen [3] as a special case, to approximation-solvability of a class of nonlinear variational inequalities involving cocoercive mappings. The obtained results do complement the earlier works of Cohen [3], Zhu and Marcotte [23] and Verma [18] on the approximation- solvability of nonlinear variational inequalities in different space settings.
LetH be a real Hilbert space with the inner producth·,·iand normk·k. LetS,T : K →H be any mappings andK a closed convex subset of H. Letf : K →Rbe a continuous convex function. We consider a class of nonlinear variational inequality (abbreviated as NVI) problems:
find an elementx∗ ∈K such that
(1.1) h(S−T) (x∗), x−x∗i+f(x)−f(x∗)≥0 for all x∈K.
Now we need to recall the following auxiliary result, most commonly used in the context of the approximation-solvability of the nonlinear variational inequality problems based on the iterative procedures.
Lemma 1.1. An elementu∈K is a solution of the NVI problem (1.1) if h(S−T)(u), x−ui+f(x)−f(u)≥0 for all x∈K.
A mappingS:H →H is said to beα-cocoercive [19] if for allx, y ∈H, we have kx−yk2 ≥α2kS(x)−S(y)k2+kα(S(x)−S(y))−(x−y)k2, whereα >0is a constant.
A mappingS :H →His calledα-cocoercive [12] if there exists a constantα >0such that hS(x)−S(y), x−yi ≥αkS(x)−S(y)k2 for all x, y ∈H.
Sis calledr-strongly monotone if for eachx, y ∈H, we have
hS(x)−S(y), x−yi ≥rkx−yk2 for a constant r >0.
This implies that
kS(x)−S(y)k ≥rkx−yk,
that is,Sisr-expanding, and whenr= 1, it is expanding. The mappingSis calledβ−Lipschitz continuous (orβ−Lipschitzian) if there exists a constantβ ≥0such that
kS(x)−S(y)k ≤βkx−yk for all x, y ∈H.
We note that ifS isα-cocoercive and expanding, thenS isα-strongly monotone. On the top of that, if S is α-strongly monotone and β−Lipschitz continuous, then S is
α β2
cocoercive forβ >0. Clearly everyα-cocoercive mappingSis α1
−Lipschitz continuous.
Proposition 1.2. [21]. LetS :H →H be a mapping from a Hilbert spaceHinto itself. Then the following statements are equivalent:
(i) For eachx, y ∈Hand for a constantα >0, we have
kx−yk2 ≥α2kS(x)−S(y)k2+kα(S(x)−S(y))−(x−y)k2. (ii) For eachx, y ∈H, we have
hS(x)−S(y), x−yi ≥αkS(x)−S(y)k2, whereα >0is a constant.
Lemma 1.3. For all elementsv, w∈H, we have kvk2+hv, wi ≥ −1
4kwk2.
A mappingS :H →His said to beγ−partially relaxed monotone[18] if for allx, y, z ∈H, we have
hS(x)−S(y), z−yi ≥ −γkz−xk2 for γ >0.
Proposition 1.4. [18]. LetS :H → H be anα-cocoercive mapping onH. ThenS is 4α1
− partially relaxed monotone.
Proof. We include the proof for the sake of the completeness. SinceSisα-cocoercive, it implies by Lemma 1.1, for allx, y, z ∈H, that
hS(x)−S(y), z−yi = hS(x)−S(y), x−yi+hS(x)−S(y), z−xi
≥ αkS(x)−S(y)k2+hS(x)−S(y), z−xi
= α
kS(x)−S(y)k2+ 1
α
hS(x)−S(y), z−xi
≥ − 1
4α
kz−xk2, that is,S is 4α1
−partially relaxed monotone.
A mappingT :H →His said to beµ-co-Lipschitz continuous if for eachx, y ∈H and for a constantµ >0, we have
kx−yk ≤µkT(x)−T(y)k. This clearly implies that
hT(x)−T(y), x−yi ≤µkT(x)−T(y)k2. Clearly, everyµ-co-Lipschitz continuous mappingT is
1 µ
−expanding.
2. GENERALIZEDAUXILIARY PROBLEM PRINCIPLE
This section deals with the approximation-solvability of the NVI problem (1.1), based on the generalized auxiliary nonlinear variational inequality problem principle by Verma [18], which includes the auxiliary problem principle introduced by Cohen [3] and later applied and studied by others, including Zhu and Marcotte [22]. This generalized auxiliary nonlinear variational inequality (GANVI) problem is as follows: for a given iteratexk, determine an xk+1 such that (fork ≥0):
(2.1)
ρ(S−T) yk
+h0 xk+1
−h0 yk
, x−xk+1
+ρ f(x)−f xk+1
≥0 for all x∈K,
where (2.2)
σ(S−T) xk
+h0 yk
−h0 xk
, x−yk
+σ f(x)−f yk
≥0 for all x∈K, and for a strongly convex functionhonK(whereh0 denotes the derivative ofh).
Whenσ =ρin the GANVI problem (2.1)-(2.2), we have GANVI problem as follows: for a given iteratexk, determine anxk+1 such that (fork≥0):
(2.3)
ρ(S−T) yk
+h0 xk+1
−h0 yk
, x−xk+1
+ρ f(x)−f xk+1
≥0 for all x∈K, where
(2.4)
ρ(S−T) xk
+h0 yk
−h0 xk
, x−yk
+ρ f(x)−f yk
≥0 for all x∈K.
Forσ = 0 andyk = xk, the GANVI problem (2.1)-(2.2) reduces to: for a given iterate xk , determine anxk+1such that (fork ≥0):
(2.5)
ρ(S−T) xk
+h0 xk+1
−h0 xk
, x−xk+1
+ρ f(x)−f xk+1
≥0 for all x∈K.
Next, we recall some auxiliary results crucial to the approximation-solvability of the NVI problem (1.1).
Lemma 2.1. [23]. Leth :K → Rbe continuously differentiable on a convex subsetK ofH.
Then we have the following conclusions:
(i) Ifhisb-strongly convex, then
h(x)−h(y)≥ hh0(y), x−yi+ b
2
kx−yk2 for all x, y ∈K.
(ii) If the gradienth0 isp−Lipschitz continuous, then
h(x)−h(y)≤ hh0(y), x−yi+ b
2
kx−yk2 for all x, y ∈K.
We are just about ready to present, based on the GANVI problem (2.1) – (2.2), the approximation- solvability of the NVI problem (1.1) involvingγ−cocoercive mappings in a Hilbert space set- ting.
Theorem 2.2. LetH be a real Hilbert space andS : K → H aγ−cocoercive mapping from a nonempty closed convex subset K of H into H. Let T : K → H be a µ-co-Lipschitz continuous mapping. Suppose that h : K → R is continuously differentiable and b-strongly convex, andh0, the derivative of h, isp−Lipschitz continuous. Thenxk+1 is a unique solution of (2.1) – (2.2). If in addition, ifx∗ ∈Kis any fixed solution of the NVI problem (1.1), thenxk is bounded and converges tox∗ for0< ρ < 2bγ,ρ+σ < band
xk+1−xk, xk−yk
≥0.
Proof. Before we can show that the sequences
xk converges to x∗, a solution of the NVI problem (1.1), we need to compute the estimates. Sincehisb−strongly convex, it ensures the
uniqueness of solutionxk+1 of the GANVI problem (2.1) – (2.2). Let us define a function Λ∗ by
Λ∗(x) := h(x∗)−h(x)− hh0(x), x∗ −xi
≥ b
2
kx∗−xk2 for x∈K,
wherex∗ is any fixed solution of the NVI problem (1.1). It follows foryk ∈K that Λ∗ yk
= h(x∗)−h yk
−
h0 yk
, x∗−yk
= h(x∗)−h yk
−
h0 yk
, x∗−xk+1+xk+1−yk . Similarly, we can have
Λ∗ xk+1
=h(x∗)−h xk+1
−
h0 xk+1
, x∗−xk+1 . Now we can write
Λ∗ yk
−Λ∗ xk+1 (2.6)
=h xk+1
−h yk
−
h0 yk
, xk+1−yk +
h0 xk+1
−h0 yk
, x∗−xk+1
≥ b
2
xk+1−yk
2+
h0 xk+1
−h0 yk
, x∗−xk+1
≥ b
2
xk+1−yk
2+ρ
(S−T) yk
, xk+1−x∗
+ρ f xk+1
−f(x∗) , forx=x∗ in (2.1).
If we replacexbyxk+1in (1.1) and combine with (2.6), we obtain Λ∗ yk
−Λ∗ xk+1
≥ b
2
xk+1−yk
2+ρ
(S−T) yk
, xk+1−x∗
−ρ
(S−T) (x∗), xk+1−x∗
= b
2
xk+1−yk
2+ρ
(S−T) yk
−(S−T) (x∗), xk+1−x∗
= b
2
xk+1−yk
2+ρ
(S−T) yk
−(S−T) (x∗), xk+1−yk+yk−x∗
= b
2
xk+1−yk
2+ρ
(S−T) yk
−(S−T) (x∗), yk−x∗ +ρ
(S−T) yk
−(S−T) (x∗), xk+1−yk .
SinceS isγ−cocoercive andT isµ-co-Lipschitz continuous, it implies that Λ∗ yk
−Λ∗ xk+1 (2.7)
≥ b
2
xk+1−yk
2+ργ
(S−T) yk
−(S−T) (x∗)
2
+ρ
(S−T) yk
−(S−T) (x∗), xk+1−yk
= b
2
xk+1−yk
2+ργ
(S−T) yk
−(S−T) (x∗) 2
+ 1
γ
(S−T) yk
−(S−T) (x∗), xk+1−yk
≥ b
2
xk+1−yk
2 −
ρ 4(γ−µ)
xk+1−yk
2 (by Lemma 1.3)
= 1 2
b−
ρ 2(γ−µ)
xk+1−yk
2 for γ−µ >0.
Similarly, we can have Λ∗ xk
−Λ∗ yk (2.8)
=h yk
−h xk
−
h0 xk
, yk−xk +
h0 yk
−h0 xk
, x∗−yk
≥ b
2
yk−xk
2+
h0 yk
−h0 xk
, x∗−yk
≥ b
2
yk−xk
2+σ T xk
, yk−x∗
+σ f yk
−f(x∗) , forx=x∗ in (2.2).
Again, if we replacexbyykin (1.1) and combine with (2.8), we obtain Λ∗ xk
−Λ∗ yk (2.9)
≥ b
2
yk−xk
2+σ
(S−T) xk
, yk−x∗
−σ
(S−T) (x∗), yk−x∗
= b
2
yk−xk
2+σ
(S−T) xk
−(S−T) (x∗), yk−x∗+xk−xk
= b
2
yk−xk
2+σ
(S−T) xk
−(S−T) (x∗), xk−x∗ +σ
(S−T) xk
−(S−T) (x∗), yk−xk
≥ b
2
yk−xk
2−
σ 4(γ−µ)
yk−xk
2
= 1
2 b−
σ 2(γ−µ)
yk−xk
2. Finally, we move toward finding the required estimate
Λ∗ xk
−Λ∗ xk+1 (2.10)
= Λ∗ xk
−Λ∗ yk
+ Λ∗ yk
−Λ∗ xk+1
≥ 1
2 b−
σ 2(γ−µ)
yk−xk
2+ 1
2 b−
ρ 2(γ−µ)
xk+1−yk
2
= 1
2 b−
σ 2(γ−µ)
yk−xk
2+ 1
2 b−
ρ 2(γ−µ)
×
xk+1−xk 2+
xk−yk
2+ 2
xk+1−xk, xk−yk
= 1
2 b−
ρ 2(γ−µ)
xk+1−xk 2+
b−
σ+ρ 4(γ−µ)
yk−xk
2
+
b−
ρ 2(γ−µ)
xk+1−xk, xk−yk
≥ 1
2 b−
ρ 2(γ−µ)
xk+1−xk
2 forb− ρ
2(γ−µ) >0, b− 4(γ−µ)σ+ρ >0and
xk+1−xk, xk−yk
≥0.
It follows from (2.10) that forxk+1 =yk =xkthatxkis a solution of the variational inequal- ity. If not, the conditionsb− 2(γ−µ)ρ >0,b− 4(γ−µ)σ+ρ >0and
xk+1−xk, xk−yk
≥0ensure that the sequence
Λ∗(xk)−Λ∗(xk+1) is nonnegative and, as a result, we have
k→∞lim
xk+1−xk = 0.
On the top of that,
x∗−xk
2 ≤ 2b
Λ∗ xk
and the sequence
Λ∗ xk is decreasing , that means
xk is a bounded sequence. Assume that x0 is a cluster point of
xk . Then as k → ∞ in (2.1) – (2.2), x0 is a solution of the variational inequality because there is no loss generality ifx∗is replaced byx0. If we associatex0toΛ0and defineΛ0 by
Λ0 xk
= h(x0)−h xk
−
h0 xk
, x0−xk
≤ p 2
x0−xk
2 (by Lemma 2.1), then we have
Λ0 xk
≤p 2
x0−xk
2. Since the sequence
Λ0 xk is strictly decreasing, it follows thatΛ0 xk
→ 0. On the other hand, we already have
Λ0 xk
≥ b
2
x0 −xk
2. Thus, we can conclude that the entire sequence
xk converges tox0, and this completes the proof. Forσ =ρin Theorem 2.2, we find:
Theorem 2.3. LetH be a real Hilbert space andT :K → Haγ−cocoercive mapping from a nonempty closed convex subsetK ofH intoH. Leth:K →Rbe continuously differentiable andb−strongly convex, andh0, the derivative ofh, isp−Lipschitz continuous. Thenxk+1 is a unique solution of (2.3) – (2.4).
If in addition,x∗ ∈K is any fixed solution of the NVI problem (1.1), then
xk is bounded and converges tox∗for0< ρ < 2bγ and
xk+1−xk, xk−yk
≥0.
Whenσ = 0andyk=xk, Theorem 2. reduces to:
Theorem 2.4. [23]. LetH be a real Hilbert space and T : K → H aγ−cocoercive mapping from a nonempty closed convex subset K of H into H. Let h : K → R be continuously differentiable and b−strongly convex, and h0, the derivative of h, is p−Lipschitz continuous.
Thenxk+1 is a unique solution of (2.5).
If in addition,x∗ ∈K is any fixed solution of the NVI problem (1.1), then
xk is bounded and converges tox∗for0< ρ < 2bγ.
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