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A Modified Algorithms for New Krasnoselskii’s Type for Strongly Monotone and Lipschitz

Mappings

Furmose Mendy, John T Mendy∗
University of the Gambia, Gambia

furmosemendy111@gmail.com, jt.mendy@yahoo.com
∗Correspondence: jt.mendy@yahoo.com

Abstract. Let E be a 2 uniformly smooth and convex real Banach space and let a mapping A : E → E∗be lipschitz and strongly monotone such that A−1(0) 6= ∅. For an arbitrary ({x1}, {y1}) ∈ E, wedefine the sequences {xn} and {yn} by{
yn = xn − θnJ−1(Axn), n ≥ 1

xn+1 = yn − λnJ−1(Ayn), n ≥ 1where λn and θn are positive real number and J is the duality mapping of E. Letting (λn, θn) ∈ (0, 1), then xn and yn converges strongly to ρ∗, a unique solution of the equation Ax = 0. We also appliedour algorithm in convex minimization and also proved the convergence of it in Lp, `p or Wm,p . At theend we proposed the algorithm of it in Lp(Ω) and its inverse Lq(Ω).

1. Introduction
Definition 1.1. A map A : E → E∗ is called monotone if for each x, y ∈ E, the following inequalityholds:

〈Ax − Ay, x − y〉 ≥ 0A is called strongly monotone if there exists k ∈ (0, 1) such that for each x, y ∈ E, the followinginequality holds:
〈Ax − Ay, x − y〉 ≥ k‖x − y‖2A map A : E → E is called accretive if for each x, y ∈ E, there exists j(x − y) ∈ J(x − y) suchthat
〈Ax − Ay, j(x − y)〉 ≥ 0A is called strongly accretive if there exists k ∈ (0, 1) such that for each x, y ∈ E, there exists

j(x − y) ∈ J(x − y) such that
〈Ax − Ay, j(x − y)〉 ≥ k‖x − y‖2

Received: 23 Apr 2023.
Key words and phrases. Krasnoselskii-type algorithm; monotone operators; Lipschitz mappings.1

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https://doi.org/10.28924/ada/ma.3.18
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Eur. J. Math. Anal. 10.28924/ada/ma.3.18 2A map A : E → E∗ is called Lipschitzian, if for each constant L > 0 and for all x, y ∈ E, thefollowing inequality holds:
i):

〈Ax − Ay〉 ≤ L‖x − y‖

ii):
L2(d2 − 1) < k2

Many physical problems in applications can be modeled in the following form: find x ∈ H suchthat
0 ∈ Ax (1.1)where A is a monotone operator on a real Hilbert space H. Typical examples where monotone oper-ators occur and satisfy the inclusion 0 ∈ Ax include the equilibrium state of evolution equations andcritical points of some functionals and convex optimization, linear programing, monotone inclusionsand elliptic differential equations defined on Hilbert spaces (see e.g., Browder [2], Mustafa [19],Stephen [26], Sina [24], Mendy et al, [17] and Chidume [3]). For precisely, the classical convexoptimization problem: let h : H → R ∪ {+∞} be a proper convex function. The sub-differential of

h at x ∈ H; is defined by ∂h : H → 2H

∂(x) = {x∗ ∈ h : h(y)− h(x) ≥ 〈y − x, x∗〉,∀y ∈ h}. (1.2)
Clearly, ∂h : H → 2His monotone operator on H, and 0 ∈ ∂(x0) if and only if x0 is a minimizer of
h. In the case of setting ∂(x) ≡ A ; solving the inclusion 0 ∈ Ax is solving for a minimizer of h.There have been fruitful works on approximating zero point of A in Hilbert spaces (see e.g.,Takahashi and Ueda [31], Song and Chen [25], and Cho et al. [9]). The proximal point algorithm
(PPA) is recognized as a powerful and successful algorithm in finding a numerical solution ofmonotone operators equation 0 ∈ Ax which was introduced by Martinet [13] and studied furtherby Rockafellar [22] and a host of other authors. That is, given xk ∈ H;

xn+1 = Jλnxn. (1.3)
where Jλn = (I + λnA)−1 is the resolvent of operator A. Since Rockafellar [22] only obtained theweak convergence of the algorithm 1.3 as λn →∞ ; so he proposed two open questions for obtainingthe strong convergence of the proximal point algorithm: (1) Does the proximal point algorithmalways converge weakly? (2) Can the proximal point algorithm be modified to guarantee strongconvergence? In studying the strong convergence, many authors have modified the proximal pointalgorithm (PPA) to guarantee strong convergence under different settings, see e.g., Takahashi [29],Reich [20], Lehdili and Moudafi [12], Chidume et al. [6], and the references therein.Let E be a real normed space, E∗ its topological dual space. The map J : E → 2E

∗ defined by
Jx :

{
x∗ ∈ E∗ : 〈x, x∗〉 = ‖x‖.‖x∗‖ = ‖x‖2 = ‖x∗‖2

}
.

https://doi.org/10.28924/ada/ma.3.18


Eur. J. Math. Anal. 10.28924/ada/ma.3.18 3is called the normalized duality map on E. where 〈, 〉 denotes the generalized duality pairingbetween E and E∗.In a Hilbert space, the normalized duality map is the identity map. Hence, in Hilbert spaces,monotonicity and accretivity coincide. For an accretive-type operator A,solutions of the equation Ax = 0, in many cases, represent the equilibrium state of somedynamical system (see, for example, [29], page 116). To approximate a solution of Ax = 0, assumingexistence, where A : E → E is of accretive type, Browder [2] defined an operator T : E → E by
T := I − A, where I is the identity map on E. He called such an operator pseudo-contractive.It is trivial to observe that zeros of A correspond to fixed points of T . For Lipschitz stronglypseudo-contractive maps, Chidume [6] proved the following theorem.
Theorem 1.1. (Chidume, [7]. Let E = Lp, 2 ≤ p < 8, and K ⊂ E be nonempty closed convex
and bounded. Let T : K → K be a strongly pseudo-contractive and Lipschitz map. For arbitrary
x0 ∈ K, let a sequence {xn} be defined iteratively by xn+1 = (1 − λn)xn + λnTxn, n ≥ 0, where

{λn} ⊂ (0, 1) satisfies the following conditions: ,(i)
∞∑
n=1

λn =∞ , (i i)

∞∑
n=1

λ2n ≤ ∞. Then {xn}

converges strongly to the unique fixed point of T .

By setting T := I − A in Theorem 1.1, the following theorem for approximating a solution of
Ax = 0 where A is a strongly accretive and bounded operator can be proved.Unfortunately, the success achieved in using geometric properties developed from the mid-1980sto early 1990s in approximating zeros of accretive-type mappings has not carried over to approx-imating zeros of monotone-type operators in general Banach spaces. Part of the problem is thatsince A maps E to E∗, for xn ∈ E,Axn is in E∗. Consequently, a recursion formula containing xnand Axn may not be well defined. Attempts have been made to overcome this difficulty by introduc-ing the inverse of the normalized duality mapping in the recursion formulas for approximating zerosof monotone-type mappings.Examples Chidume [4], [5], Moudafi [18], Reich [21], Takahashi [30],Zegeye [37], Djitte [17], Mendy [ [15], [10]]Motivated by approximating zeros of monotone mappings, Chidume et al. [8] proposed aKrasnoselskii-type scheme and proved a strong convergence theorem in Lp, 2 ≤ p < ∞. In fact,they obtained the following result.
Theorem 1.2. (Chidume et al. [8]). Let X = Lp, 2 ≤ p < ∞, and A : X → X∗ be a Lipschitz map.
Assume that there exists a constant k ∈ (0, 1) such that A satisfies the condition

〈Ax − Ay, x − y〉 ≥ k‖x − y‖
p
p−1 (1.4)

and that A−1(0) 6= ∅. For arbitrary x1 ∈ X , define the sequence {xn} iteratively by

xn+1 = J−1(Jxn − λnAxn) n ≥ 0

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Eur. J. Math. Anal. 10.28924/ada/ma.3.18 4

where λn ∈ (0, δp) and δp is some positive constant. Then the sequence {xn} converges strongly
to the unique solution of the equation Ax = 0.

In [8], the authors posed the following open problem. If E = Lp, 2 ≤ p <∞, attempts to obtainstrong convergence of the Krasnoselskii-type sequence defined for x0 ∈ E by
xn+1 = J−1(Jxn − λnAxn) n ≥ 0

to a solution of the equation Ax = 0, where A is strongly monotone and Lipschitz, have notyielded any positive result.Following the works of Chidume et al [8], and motivation of finding the zeros of the monotonetype mapping, several strong convergence results have been established by various authors (seee.g [17], [10], [15], [23], [16]).Following this great work, in 2023, Mendy [16] constructed the following two-step proximalalgorithm for the zero point of monotone mapping and proof a strong convergency of the sequences
{xn} and {yn} to a unique point x∗ ∈ A−1(0).{

yn+1 = J−1(Jxn − λnAxn), n ≥ 0

xn+1 = J−1(Jyn+1 − λn+1Ayn+1), n ≥ 0
(1.5)

In this paper, we study the two step size of the new Krasnoselskii-type algorithm introduced bySene et al. [23] and prove a strong convergence theorem to approximate the unique zero of aLipschitz and strongly monotone mapping 2−uniformly smooth and convex real Banach space for
p ≥ 2. This class of Banach spaces contains all Lp-spaces, 2 ≤ p < ∞ and Sobolev space. Thenwe apply our results to the convex minimization problem. Finally, our method of proof generalizedand extended various authors in this way of work.

2. Preliminaries

Let E be a normed linear space. E is said to be smooth if
lim
t→0

‖x + ty‖ − ‖x‖
t

(2.1)
exist for each x, y ∈ SE (Here SE := {x ∈ E : ||x || = 1} is the unit sphere of E). E is said to beuniformly smooth if it is smooth and the limit is attained uniformly for each x, y ∈ SE , and E isFréchet differentiable if it is smooth and the limit is attained uniformly for y ∈ SE .Let E be a real normed linear space of dimension ≥ 2. The modulus of smoothness of E , ρE , isdefined by:

ρE(τ) := sup

{
‖x + y‖+ ‖x − y‖

2
− 1 : ‖x‖ = 1, ‖y‖ = τ

}
; τ > 0.

A normed linear space E is called uniformly smooth if
lim
τ→0

ρE(τ)

τ
= 0.

https://doi.org/10.28924/ada/ma.3.18


Eur. J. Math. Anal. 10.28924/ada/ma.3.18 5If there exist a constant c > 0 and a real number q > 1 such that ρE(τ) ≤ cτq , then E is said tobe q-uniformly smooth.A normed linear space E is said to be strictly convex if:
‖x‖ = ‖y‖ = 1, x 6= y ⇒

∥∥∥x + y

2

∥∥∥ < 1.

The modulus of convexity of E is the function δE : (0, 2]→ [0, 1] defined by:
δE(ε) := inf

{
1−

1

2
‖x + y‖ : ‖x‖ = ‖y‖ = 1, ‖x − y‖ ≥ ε

}
.

E is uniformly convex if and only if δE(ε) > 0 for every ε ∈ (0, 2]. For p > 1, E is said to be
p-uniformly convex if there exists a constant c > 0 such that δE(ε) ≥ cεp for all ε ∈ (0, 2]. Observethat every p-uniformly convex space is uniformly convex.Typical examples of such spaces are the Lp , `p and Wm

p spaces for 1 < p <∞ where,
Lp (or lp) or Wm

p is { 2− uniformly smooth and p − uniformly convex if 2 ≤ p <∞;

2− uniformly convex and p − uniformly smooth if 1 < p < 2.

Remark 1. Note also that duality mapping exists in each Banach space.We recall from [11] someof the examples of this mapping in `p, Lp,Wm,p−spaces, 1 < p <∞

• `p : Jx = ‖x‖2−p`p
y ∈ `q, x = (x1, x2, ..., xn, ...), y = (x1|x1|p−2, x2|x2|p−2, ..., xn|xn|p−2, ...)

• Lp : Ju = ‖u‖2−pLp
|u|p−2u ∈ Lq

• Wm,p : Ju = ‖u‖2−pWm,p

∑
|α≤m|

(−1)|α|Dα(|Dαu|p−2Dαu) ∈ W−m,p

In Lp, `p and Wm,p spaces for 1 < p <∞ are q−uniformly smooth real Banach spaces with
q, as

q = min{2, p} and dq ≥ 1 (2.2)
is given by

dq =

{
1+τq−1

(1+τ)q−1 , i f 1 < p < 2;
p − 1, i f 2 ≤ p <∞. (2.3)

and τ(0, 1) as the unique solution of the equation
(q − 2)tq−1 + (q − 1)tq−2 − 1 = 0

It is well known that
• E is smooth if and only if J is single-valued.
• If E is uniformly smooth then J is uniformly continuous on bounded subsets of E.
• If E is reflexive and strictly convex dual then J−1 is single-valued, one-to-one, surjective,uniformly continuous on bounded subsets and it is the duality mapping from E∗ into E and
J−1J = IE and JJ−1 = IE .
• J−1 is uniformly continuous if and only if it has a modulus of continuity.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.18 6

Lemma 2.1 (Xu [32]). . Let q > 1 be a real number and E be a Banach space. Then the following
assertion are equivalent

i): E is q−uniformly smooth
ii): There exists a constant dn > 0, such that for all x, y ∈ E, then the following holds

‖x + q‖q ≤ ‖x‖q + q〈y , Jq(x)〉+ dq‖y‖q. (2.4)
3. Main Result

We now prove the following result
Theorem 3.1. Let E be a 2 uniformly smooth and convex real Banach space and let a mapping A :

E → E∗ be lipschitz strongly monotone such that A−1(0) 6= ∅. For an arbitrary ({x1}, {y1}) ∈ E,
we define the sequences {xn} and {yn} by

{
yn = xn − θnJ−1(Axn), n ≥ 1

xn+1 = yn − λnJ−1(Ayn), n ≥ 1
(3.1)

where λn and θn are positive real number and J is the duality mapping of E. Letting (λn, θn) ∈
(0, 1) , then {xn} and {yn} converges strongly to ρ∗, a unique solution of the equation Ax = 0.

Proof. Letting ρ∗ = x∗ ∈ E be the unique solution of Ax = 0. From inequality 2.4 in lemma 2.1with 3.1, knowingly that ‖J−1w‖ = ‖w‖ for all w ∈ E∗, then we have the following estimates:
‖xn+1 − ρ∗‖2 = ‖yn − ρ∗ − λnJ−1(Ayn)‖2

= ‖λnJ−1(Ayn)‖2 − 2〈yn − ρ∗, J(λnJ
−1(Ayn))〉+ d2‖yn − ρ∗‖2

≤ λ2n‖(Ayn)‖2 − 2λn〈yn − ρ∗, Ayn)〉+ d2‖yn − ρ∗‖2

≤ λ2nL
2‖yn − ρ∗‖2 − 2λnk‖yn − ρ∗‖2 + d2‖yn − ρ∗‖2

=
(
λ2nL

2 − 2kλn + d2

)
‖yn − ρ∗‖2 (3.2)

For the fact that 0 <
(
λ2nL

2 − 2kλn + d2

)
< 1, we have the following

‖xn+1 − ρ∗‖2 ≤ δ(λ1)‖yn − ρ∗‖2 (3.3)
where δ(λ1) =

(
λ2nL

2 − 2kλn + d2

).Using 3.1 , Lipschitz property of A, with the same computational we have the following:

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Eur. J. Math. Anal. 10.28924/ada/ma.3.18 7

‖yn − ρ∗‖2 = ‖xn − ρ∗ − θnJ−1(Axn)‖2

= ‖θnJ−1(Axn)‖2 − 2〈xn − ρ∗, J(θnJ
−1(Axn))〉+ d2‖xn − ρ∗‖2

≤ θ2n‖(Axn)‖2 − 2θn〈xn − ρ∗, Axn)〉+ d2‖xn − ρ∗‖2

≤ θ2nL
2‖xn − ρ∗‖2 − 2θnk‖xn − ρ∗‖2 + d2‖xn − ρ∗‖2

=
(
θ2nL

2 − 2kθn + d2

)
‖xn − ρ∗‖2 (3.4)

Again, with the fact that 0 <
(
θ2nL

2 − 2kθn + d2

)
< 1, we have the following

‖yn − ρ∗‖2 ≤ δ(λ2)‖xn − ρ∗‖2 (3.5)
where δ(λ2) =

(
θ2nL

2 − 2kθn + d2

)
Putting 3.5 in 3.3, we have the following

‖xn+1 − ρ∗‖2 ≤ δ(λ1)δ(λ2)‖xn − ρ∗‖2 (3.6)
‖xn+1 − ρ∗‖ ≤

√
δ(λ1)δ(λ2)‖xn − ρ∗‖ (3.7)

‖xn+1 − ρ∗‖ ≤ µ‖xn − ρ∗‖where µ =
√
δ(λ1)δ(λ2).Therefore the sequences {xn} and {yn} converges strongly to ρ∗. This complete the proof. �

Corollary 3.1. Let E = Lp, 2 ≤ p <∞, and A : E → E∗ be a Lipschitz strongly monotone mapping
such that A−1(0) 6= ∅. For arbitrary (x1, y1) ∈ E, define the sequence {xn} and {yn} iteratively by{

yn = xn − θnJ−1(Axn), n ≥ 1

xn+1 = yn − λnJ−1(Ayn), n ≥ 1
(3.8)

where λn and θn are positive real number and J is the duality mapping of E. Letting (λn, θn) ∈
(0, 1), then xn and yn converges strongly to ρ∗, a unique solution of the equation Ax = 0.

Proof. Since E = Lp spaces, 2 ≤ p <∞, are 2−uniformly smooth and convex real Banach spaces,then the proof follows from Theorem 3.1.
�

4. Convergence in Lp, `p or Wm,p, 2 ≤ p <∞

Theorem 4.1. Let E be a 2 uniformly smooth and convex real Banach space either
Lp, `p or W

m,p, 2 ≤ p <∞ with it dual E∗. Let a mapping A : E → E∗ be lipschitz and strongly
monotone such that A−1(0) 6= ∅. For an arbitrary ({x1}, {y1}) ∈ E, we define the sequences {xn}
and {yn} by 3.1 converges strongly to ρ∗, a unique solution of the equation Ax = 0.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.18 8

Proof. Since Lp, `p or Wm,p, 2 ≤ p < ∞ are 2− uniformly smooth Banach spaces, then with thesame computation in 3.1, the proof follows.
�

Corollary 4.1. Let E be a Banach space either Lp, `p or Wm,p, 2 ≤ p <∞ with it dual E∗. Let a
mapping A : E → E∗ be lipschitz and strongly monotone such that A−1(0) 6= ∅. For an arbitrary
({x1}, {y1}) ∈ E, we define the sequences {xn} and {yn} by 3.1 converges strongly to ρ∗, a unique
solution of the equation Ax = 0.

Proof. Since Lp, `p or Wm,p, 2 ≤ p < ∞ are 2− uniformly smooth Banach spaces, then fromTheorem 4.1 with the same computation in 3.1, the proof follows.
�

5. Application to Convex minimization problem

Now, we present a convex minimization problem for a convex function ∇ : E → R.The following results are well known.
Remark 2. Let ∆ : E → R be a differentiable convex function and ρ∗ ∈ E, then the point ρ∗ is aminimizer of ∇ on E if and only if d∇(ρ∗) = 0.
Definition 5.1. A function ∇ : E → R is said to be strongly convex if there exists γ > 0 such thatthe following condition holds:

∇(βx + (1− β)y) ≤ β∇x + (1− β)∇y − γ‖x − y‖2 (5.1)
for every x, y ∈ E with x 6= y and β ∈ (0, 1),
Lemma 5.2. Let E be normed linear space and∇ : E → R a convex differentiable function. Suppose
that ∇ is strongly convex. Then the differential map d∇ : E → E∗ is strongly monotone, i.e., there
exists k > 0 such that

〈d∇x − d∇y , x − y〉 ≥ k‖x − y‖2 ∀ x, y ∈ E. (5.2)
Now we present the following result.
Theorem 5.3. Let d∇ : E∗ → E be a L-Lipschitz continuous and strongly monotone mapping
such that d∇−1(0) 6= ∅. Let E = Lp, p ≥ 2 and ∇ : E → R be a differentiable, strongly convex
real-valued function. For given x1, y1 ∈ E, define the sequence {xn} and {yn} as follows:{

yn = xn − θnd∇xn), n ≥ 1

xn+1 = yn − λnd∇yn), n ≥ 1
(5.3)

where the sequences {λn} and {θn}, are in the interval [0, 1] . Then ∇ has a unique minimizer
ρ∗ ∈ E such that if

(
λn, θn

)
∈ [0, 1], the sequence {xn} and {yn} converges strongly to ρ∗.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.18 9

Proof. From Remark 2 it follows that ∇ has a unique minimizer ρ∗ and is obtained by d∇(ρ∗) = 0.From Lemma 5.2 and using the fact that the differential mapping d∇ : E → E∗ is Lipschitz,considering the result of Theorem 3.1, we can complete the proof. �

6. The proposed algorithm in Lp(Ω)

Now, From [14], the duality mapping J is known precisely in Lp(Ω) for 1 < p <∞ by
Jv = ‖v‖2−pLp

|v |p−2v ,∀v ∈ Lp(Ω)

and if Lp(Ω) is reflexive, smooth and strictly convex real Banach space, for 1 < p < ∞, then theduality mapping J is surjective, one-to-one and its inverse J−1 is given by
Ju = ‖‖2−qL |u|q−2u,∀u ∈ Lq(Ω)

with 1

p
+

1

q
= 1Now from 3.1, we defined x1, y1 ∈ Lq(Ω){

yn = xn − θn‖Axn‖2−qLq
|Axn|2−qLq

Axn, n ≥ 1

xn+1 = yn − λn‖Ayn‖2−qLq
|Ayn|2−qLq

Ayn, n ≥ 1
(6.1)

Conclusion
In this paper, we proposed and analyzed the strong convergence theorem of two step size of thenew Krasnoselskii-type algorithm introduced by Sene et al. [7] and prove a strong convergencetheorem to approximate the unique zero of a Lipschitz strongly monotone mapping 2−uniformlysmooth and p−uniformly convex real Banach space for p ≥ 2. This class of Banach spaces containsall Lp-spaces, 2 ≤ p <∞ and Sobolev space. Then we apply our results to the convex minimizationproblem. We also complemented and generalized previous worked been done under this setting.

References
[1] Ya. Alber, I. Ryazantseva, Nonlinear Ill Posed Problems of Monotone Type, Springer, London, UK, 2006.[2] F.E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc. 73(1967) 875-882. https://doi.org/10.1090/s0002-9904-1967-11823-8.[3] C.E. Chidume, M.O. Osilike, Iterative solution of nonlinear integral equations of Hammerstein-type, J. Niger. Math.Soc. Appl. Anal. 11 (1992) 9-18.[4] C.E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings, Proc. Amer.Math. Soc. 99 (1987) 283-283. https://doi.org/10.1090/s0002-9939-1987-0870786-4.[5] C.E. Chidume, M.O. Osilike, Iterative solutions of nonlinear accretive operator equations in arbitrary Banach spaces,Nonlinear Anal.: Theory Meth. Appl. 36 (1999) 863-872. https://doi.org/10.1016/s0362-546x(97)00611-1.[6] C.E. Chidume, A. Adamu, L.C. Okereke, A Krasnoselskii-type algorithm for approximating solutions of variationalinequality problems and convex feasibility problems. J. Nonlinear Var. Anal. 2 (2018) 203-218.[7] C.E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudo-contractive mappings, Proc.Amer. Math. Soc. 99 (1987) 283-288.

https://doi.org/10.28924/ada/ma.3.18
https://doi.org/10.1090/s0002-9904-1967-11823-8
https://doi.org/10.1090/s0002-9939-1987-0870786-4
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https://doi.org/10.1080/01630563.2019.1606825
https://doi.org/10.1186/1029-242x-2013-566

	1. Introduction
	2. Preliminaries
	3. Main Result
	4. Convergence in Lp,p   or  Wm,p, 2p < 
	5. Application to Convex minimization problem
	6. The proposed algorithm in Lp()
	Conclusion
	References

