2.5 Review on some optimization problems
2.5.7 Some important lemmas
1. Chang et al. [70] obtained a weak convergence result and under some compactness assumption, they obtained some strong convergence results.
Another effective iterative method for finding solutions of MPs is the viscosity implicit method (VIM). Many authors have obtained convergence results using the VIM in more general spaces (see [137, 264] and other references therein). This study has also been extended to the Hadamard space (see [265]) and like every other type of iterative method, the PPA type of VIM has also been studied in this setting. In 2015, Xu et. al. [263] applied the VIM to the implict midpoint rule (IMR) for nonexpansive mappings in Hilbert spaces and proposed a viscosity implicit midpoint rule (VIMR). They proved that the sequence generated converges strongly under suitable conditions to a fixed point of the nonexpansive mapping which is also a solution of the variational inequality.
Recently, Ahmad and Ahmad [4] proposed a VIM of IMR in Hadamard space and they defined it as follows;
Algorithm 2.5.31.
wn = xn⊕2xn+1,
yn =αn(wn)⊕βng(wn)⊕τnT(wn), xn+1 =T(yn),
(2.5.36)
where {αn},{βn} and {τn} are sequences in (0,1), g is a contraction with a coefficient θ ∈[0,1) andT is a nonexpansive mapping onD.They also obtained a strong convergence of (2.5.36) to a fixed point of the nonexpansive mapping.
Remark 2.5.6. Besides the above mentioned works which mainly motivated our study of MPs in Hadamard spaces, there are very few other results on MPs in Hadamard spaces.
Thus, it is very important to further develop and generalize this study in Hadamard spaces.
Lemma 2.5.10. [261, Theorem 4.1] Assume that the solution set V I(A,C)of the classical VIP (1.2.1) is nonempty and C is defined as C :={x∈ H |c(x)≤0}, where c:H →R is a continuously differential convex function. Let p ∈ C. Then, p∈ V I(A,C) if and only if either
(i) Ap= 0, or
(ii) p∈∂C and there existsρ >0such thatAp=−ρc′(p),where∂C denotes the boundary of C.
Following Attouch and Cabot [29, pages 5, 10], we note that if xn+1 =xn+θn(xn−xn−1), then for all n ≥1, we have that
xn+1−xn =
n
Y
j=1
θj
!
(x1−x0), which implies that
xn =x1 +
n−1
X
j=1 l
Y
j=1
θj
!
(x1−x0).
Thus, {xn} converges if and only ifx1 =x0 or if
∞
P
l=1 l
Q
j=1
θj <∞.
Therefore, we assume henceforth that
∞
X
l=i l
Y
j=i
θj
!
<∞ ∀i≥1. (2.5.37)
Hence, we can define the sequence {ti} in R by ti :=
∞
X
l=i−1 l
Y
j=i
θj
!
= 1 +
∞
X
l=i l
Y
j=i
θj
!
, (2.5.38)
with the convention
i−1
Q
j=i
θj = 1 ∀i≥1.
Remark 2.5.11. (See also [29]).
Assumption (2.5.37) ensures that the sequence {ti} given by (2.5.38) is well-defined, and ti = 1 +θiti+1, ∀i≥1. (2.5.39) The following proposition provides a criterion for ensuring assumption (2.5.37).
Proposition 2.5.32. [29, Proposition 3.1] Let {θn} be a sequence such that θn ∈ [0,1) for every n≥1. Assume that
n→∞lim
1
1−θn+1 − 1 1−θn
=c, for some c∈[0,1). Then,
(i) assumption (2.5.37) holds, and tn+1 ∼ (1−c)(1−θ1
n) as n → ∞,
(ii) the equivalence 1−θn ∼ 1− θn+1 holds true as n → ∞. Hence, tn+1 ∼ tn+2 as n → ∞.
Remark 2.5.12. Using Proposition 2.5.32, we can see thatθn= 1−nθ¯, θ >¯ 1, is a typical example of a sequence satisfying assumption (2.5.37).
Indeed, we have that
n→∞lim
1 1−θn+1
− 1 1−θn
= lim
n→∞
1
θ¯(n+ 1)−1 θ¯n
= 1
θ¯∈[0,1),
which satisfies the assumption of Proposition 2.5.32. Hence by Proposition 2.5.32(i), assumption (2.5.37) holds.
It is worthy of note that the exampleθn= 1−θn¯,θ >¯ 1, falls within the setting of Nesterov’s extrapolation methods. In fact, many practical choices for θn satisfy assumption (2.5.37) (for instance, see [29, 40, 68,187]).
The corresponding finite sum expression for {ti} is defined for i, n≥1, by
ti,n:=
n−1
P
l=i−1 l
Q
j=i
θj
!
= 1 +
n−1
P
l=i l
Q
j=i
θj
!
, i≤n,
0, otherwise.
(2.5.40)
In the same manner, we have that {ti,n} is well-defined and
ti,n= 1 +θiti+1,n ∀i≥1, n≥i+ 1. (2.5.41) The sequences {ti} and {ti,n} are very crucial to our convergence analysis. In fact, their effect can be seen in the following lemma which also plays a crucial role in establishing our convergence results.
Lemma 2.5.13. [29, page 42, Lemma B.1]. Let {an},{θn} and {bn} be sequences of real numbers satisfying
an+1 ≤θnan+bn for every n ≥1.
Assume that θn≥0 for every n≥1.
(a) For every n ≥1, we have
n
X
i=1
ai ≤t1,na1+
n−1
X
i=1
ti+1,nbi, where the double sequence {ti,n} is defined by (2.5.40).
(b) Under (2.5.37), assume that the sequence{ti}defined by (2.5.38)satisfies
∞
P
i=1
ti+1[bi]+<
∞. Then, the series P
i≥1
[ai]+ is convergent, and
∞
X
i=1
[ai]+ ≤t1[a1]++
∞
X
i=1
ti+1[bi]+ , where [t]+:= max{t,0} for any t ∈R.
Lemma 2.5.14. [29, page 7, Lemma 2.1]. Let {xn} be a sequence in H and {θn} be a sequence of real numbers. Given z ∈ H, define the sequence {Γn} by Γn := 12∥xn−z∥2. Then
Γn+1−Γn−θn(Γn−Γn−1) = 1
2(θn+θn2)∥xn−xn−1∥2+⟨xn+1−wn, wn−z⟩
+1
2∥xn+1−wn∥2, (2.5.42)
where wn =xn+θn(xn−xn−1).
The following lemmas are well-known.
Lemma 2.5.15. Let {xn} be any sequence in H such that xn⇀ x. Then, lim inf
n→∞ ∥xn−x∥<lim inf
n→∞ ∥xn−y∥,∀y̸=x.
Lemma 2.5.16. [122] Let C be a closed convex set in H, f be a real-valued function on H and define K :={x ∈ C : h(x) ≤0}. If K is nonempty and h is Lipschitz continuous on C with modulus L >0, then
dist(x, K)≥L−1max {f(x),0}, ∀x∈ C, where dist(x, K) denote the distance function from x to K.
Lemma 2.5.17. [264] Let H be a real Hilbert space and S : H → H be a nonexpansive mapping withF(S)̸=∅.If{xn}is a sequence inHconverging weakly tox∗ and{(I−S)xn} converges strongly to y, then (I −S)x∗ =y.
Lemma 2.5.18. [221] LetC ⊆ Hbe a nonempty, closed and convex subset of a real Hilbert space H. Let u∈ H be arbitrarily given, z :=PCu, and Ω :={x∈ H:⟨x−u, x−z⟩ ≤0}.
Then Ω∩ C ={z}.
Lemma 2.5.19. Let X be a smooth, strictly convex and reflexive Banach space. Let C be a nonempty, closed and convex subset of X, and let x1 ∈ X and z ∈ C. Then, z =PCx1 if and only if
⟨z−y, JX(x1−z)⟩ ≥0, ∀y∈ C.
Lemma 2.5.20. [164] Let X be a real Banach space. Let B : X → 2X be a maximal monotone operator and A : X → X be a k-inverse strongly monotone mapping on X. Define Tλ = (I+λB)−1(I −λA), λ >0. Then we have
(i) F(Tλ) = (A+B)−1(0);
(ii) for 0< s≤λ and x∈ X, ∥x−Tsx∥ ≤2∥x−Tλx∥.
Lemma 2.5.21. [128] LetX be a 2-uniformly convex Banach space andX∗ the dual space of X. Suppose A :X → X∗ is uniformly continuous on bounded subsets of X and B is a bounded subset of X. Then A(B) is bounded.
Lemma 2.5.22. [20] Suppose X is a 2- uniformly convex Banach space. Then there exists µ≥1 such that
ϕ(x, y)≥ 1
µ∥x−y∥2, ∀ x, y ∈ X,
where µ is the 2-uniform convexity constant of X. If X is a Hilbert space, then µ= 1.
Lemma 2.5.23. [95] Let x∈ X and ψ ≥σ >0. The following inequality holds;
∥x−ΠCJ−1(x−ψAx)∥
ψ ≤ ∥x−ΠCJ−1(x−σAx)∥
σ .
Lemma 2.5.24. [139] Let X be a smooth and uniformly convex real Banach space. Let {xn}and{yn}be two sequences inX.If either{xn}or{yn} is bounded andϕ(xn, yn)→0, as n→ ∞, then ∥xn−yn∥ →0, as n→ ∞.
Lemma 2.5.25. [196] Let X be an Hadamard space and f : X → (−∞,∞] be a proper convex and lower semi-continuous function. Then, d2(Jλfx, x)≤d2(Jµfx, x) for 0< λ < µ and x∈X.
Lemma 2.5.26. [196] Let X be an Hadamard space and fj : X → (−∞,∞], j = 1,2,· · ·, m be a finite family of proper, convex and lower semicontinuous functions. If 0< λ < µ and
∩mj=1F Jµ(j)
̸=∅. Then, F Πmj=1Jµ(j)
⊆
∩mj=1F Jλ(j)
, where, Πmj=1Jµ(j) =Jµ(1)◦Jµ(2)◦ · · · ◦Jµ(m).
Lemma 2.5.27. [158] Let X be an Hadamard space and f :X →(−∞,∞] be a proper, convex and lower semicontinuous function. Then, for all x, y ∈X and λ >0, we have
1
2λd2(Jλfx, y)− 1
2λd2(x, y) + 1
2λd2(x, Jλfx) +f(Jλfx)≤f(y).
Lemma 2.5.28. [93] LetX be a CAT(0) space,{v1, v2,· · · , vN} ⊂X and{λ1, λ2,· · · , λN} ⊂ (0,1) with
N
P
i=1
λi = 1. Then,
d
N
X
i=1
⊕λivi, x
!
≤
N
X
i=1
λid(vi, x) for each x∈X.
Remark 2.5.29. [93] For a CAT(0) space X, if {xi, i = 1,2, . . . , N} ⊂ X, and αi ∈ (0,1), i= 1,2, . . . , N. Then by induction, we can write
N
X
i=1
⊕αixi := (1−αN)
N−1
X
i=1
⊕ αi
1−αNxi⊕αNxN. (2.5.43) Lemma 2.5.30. Let X be a CAT(0) space, {v1, v2,· · ·, vN} ⊂X and{λ1, λ2,· · · , λN} ⊂ (0,1) with
N
P
i=1
λi = 1. Then,
d2
N
X
i=1
⊕λivi, x
!
≤
N
X
i=1
λid2(vi, x)−
N
X
i,j=1,i̸=j
λiλjd2(vi, vj).
Lemma 2.5.31. [194] Every bounded sequence in an Hadamard space has a△-convergence subsequence.
Lemma 2.5.32. [94] LetC be a nonempty convex subset of a CAT(0) space X andx∈X.
Then, u=PCx if and only if ⟨−ux,→ −uy⟩ ≤→ 0 ∀y∈C, where PC is the metric projection of X ontoC.
Lemma 2.5.33. [255] Let X be an Hadamard space and T :X → X be a nonexpansive mapping. Then T is a △- demiclosed.
Lemma 2.5.34. [242] Suppose {λn} and {θn} are two nonnegative real sequences such that
λn+1 ≤λn+ϕn, ∀n ≥1.
If P∞
n=1ϕn<∞, then lim
n→∞λn exists.
Lemma 2.5.35. [166] Let {an} be a sequence of nonnegative real numbers satisfying the following
an+1 ≤(1−βn)an+τn+γn, n≥1,
where {βn} is a sequence in (0,1) and {τn} is a real sequence. Suppose that
∞
P
n=1
γn < ∞ and τn≤βnM for some M >0. Then, {an} is a bounded sequence.
Lemma 2.5.36. [9] Let {an} be a sequence of non-negative real numbers, {γn} be a se- quence of real numbers in(0,1)with conditions
∞
P
n=1
γn=∞ and{dn}be a sequence of real numbers. Assume that
an+1 ≤(1−γn)an+γndn, n≥1.
If lim sup
k→∞
dnk ≤0 for every subsequence {ank} of {an} satisfying lim inf
k→∞ (ank+1−ank)≥0, then lim
n→∞an= 0.
CHAPTER 3
Results on Variational Inequality and Fixed Point Problems
3.1 Introduction
In this chapter, we present our results on monotone VIP and FPP of an infinite family of strict pseudo-contractive mappings in the real Hilbert space and apply these results to solve other nonlinear problems. Furthermore, using the projection technique, we present our results on VIP for a quasimonotone and Lipschitz operator in a real Hilbert space.
Also, we present our result on quasimonotone VIP and FPP of a quasi pseudocontractive mapping in the real Hilbert space. We provide some numerical experiments of our proposed methods in comparison to other existing methods in the literature.