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A Unified Kantorovich-type Convergence Analysis of Newton-like Methods for Solving

Generalized Equations under the Aubin Property

Samundra Regmi1, Ioannis K. Argyros2,∗, Santhosh George3, and Jefferey Warden2
1Department of Mathematics, University of Houston, Houston, TX, 77024, USA

sregmi5@uh.edu
2Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

iargyros@cameron.edu, jefferey.warden@cameron.edu
3Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka,

India-575 025
sgeorge@nitk.edu.in

∗Correspondence: iargyros@cameron.edu

Abstract. Numerous applications from diverse disciplines reduce to solving generalized equationsin a Banach space setting. These equations are solved mostly iteratively, when a sequence is gen-erated approximating a solution provided that certain conditions are valid on the starting point andthe operators appearing on the method. In particular, Newton-like methods are developed whosespecializations reduce to well known methods such as Newton, modified Newton, Secant, Kurchatovand Steffensen to mention a few. A unified semi-local analysis of these methods is presented usingthe contraction mapping principle under the Aubin property of a set valued operator, and generalizedcontinuity assumption on the operators on these methods.

1. Introduction
Let B1 and B2 stand for complete normed spaces; D be an open and convex subset of B1;operator F : D −→ B2 be continuous and G : B1 ⇒ B2 be a set-valued operator with closedgraph, which is a nonempty set [15].We are concerned with the problem of finding a solution x∗ ∈ B1 of the generalized equationiteratively in the form:

Find x ∈ B1 so that F (x) + G(x) 3 0. (1.1)
Many applications from diverse disciplines, especially in Mathematical programming can be for-mulated like the generalized equation (1.1) [1–15, 23–25, 34]. S. M. Robinson inaugurated the

Received: 26 Dec 2023.
Key words and phrases. Generalized equation; Newton-like methods; Aubin property; Banach space; Local-semi-local convergence; Newton’s method. 1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.3


Eur. J. Math. Anal. 10.28924/ada/ma.4.3 2study of generalized equations in [23–25]. A solution x∗ ∈ B1 in analytical form or closed form iscomputationally hard or impossible to find. Thus, researchers and practitioners generate iterativemethods approximating x∗ if certain conditions related to the starting point and the operators onthe methods are fulfilled. N, H. Josephy introduced the Newton method for solving the generalizedequation (1.1) in [18]. Later, numerous other authors worked on various other iterative methodsunder diverse convergence conditions ( see [1–3,7–15] and references there in).All these iterative methods are useful and provide insight in the solutions of generalized equa-tions. But as far as we know there is not a unified convergence analysis for the existing iterativemethods. That is very useful, since this way under the same set of conditions the convergence andcomparison of numerous iterative methods becomes possible. This is our motivation for the presentarticle. In particular, consider the Newton-like iterative method (NLM) for solving the generalizedequation in the following form
F (xn) + L(xn)(xn+1 − xn) + G(xn+1) 3 0, n = 0, 1, 2, . . . , (1.2)

where L(.) : B1 −→ L(B1, B2) which stands for the space of linear operators which are boundedmapping from B1 into B2. By specializing the linear operator L, many iterative methods can beobtained such as:
Newton’s Method [4,5,21,22]: Select L(x) = F ′(x), x ∈ B1 to obtain

F (xn) + F ′(xn)(xn+1 − xn) + G(xn+1) 3 0, n = 0, 1, 2, . . . ,

where F ′ denotes the derivative according to Fréchet of the operator F.
Modified Newton’s Method [4,5,20]: Set L(x) = F ′(x0), x ∈ B1 to obtain

F (xn) + F ′(x0)(xn+1 − xn) + G(xn+1) 3 0, n = 0, 1, 2, . . . .

Secant Method [20]: Let L(xn) = [xn−1, xn;F ], n = 0, 1, 2, . . . a divided difference of order one [20].Then, iterative method (1.1) becomes
F (xn) + [xn−1, xn;F ](xn+1 − xn) + G(xn+1).

Modified Secant Method [4,5]: Take L(xn) = [x−1, x0;F ], x−1, x0 ∈ B−1, n = 0, 1, 2, . . . to obtain
F (xn) + [x−1, x0;F ](xn+1 − xn) + G(xn+1).

Kurchatov Method [29,30]: Set L(xn) = [2xn − xn−1, xn−1;F ] to obtain
F (xn) + [2xn − xn−1, xn−1;F ](xn+1 − xn) + G(xn+1).

Modified Kurchatov Method [29,30]: Let L(x) = [2x0 − x−1, x−1;F ] to get
F (xn) + [2x0 − x−1, x−1;F ](xn+1 − xn) + G(xn+1).

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Eur. J. Math. Anal. 10.28924/ada/ma.4.3 3

Picard Method [20]: Pick L(x) = I, x ∈ B1 for B1 = B2 to obtain
F (xn) + xn+1 − xn + G(xn+1), n = 0, 1, 2, . . . .

Steffensen’s Method [4,5]: Define L(x) = [x + F (x), x − F (x);F ] for B1 = B2 to get
F (xn) + [xn + F (xn), xn − F (xn);F ](xn+1 − xn) + G(xn+1), n = 0, 1, 2, . . . .

Stirling’s Method [20,31]: Define L(x) = I − F ′(x), x ∈ B1 to obtain
F (xn) + (I − F ′(xn))(xn+1 − xn) + G(xn+1), n = 0, 1, 2, . . . .

Traub and Other Multi-point and Multi-step Methods [4,5,17,31,32]: Therefore, it is importantto develop unifying conditions for the convergence of (1.2). There are two popular convergenceapproaches in the literature. We develop, the semi-local analysis of convergence. In the localcase information is used to produce usually a ball centered at x∗, so that if one picks a pointinside of it the convergence of the iterative method is assured. Note that , in the semi-localcase the convergence ball is centered at the starting point x0. The convergence conditions usuallyinvolve Lipschitz [4, 5] and Hölder-type conditions [20]. The new convergence analysis in bothcases involves generalized continuity conditions, majorant functions, majorizing sequences (in thesemi-local case) in combination under the Aubin property of the set valued operator on the methodand the celebrated contraction mapping principle [15]. Upper error estimates on ‖x∗ − xn‖ for thesolution are developed which are computable.The rest of the article contains: The Mathematical background necessary to make this article asself contained as possible appears in Section 2; semi-local convergence results appear is Section3. The article ends with concluding remarks in Section 4.
2. Mathematical Bachground

Certain standard concepts are restated in order to make the article as self-contained as possible.More detailed information can be found in [15].The graph of a set-valued operator G : B1 ⇒ B2 is
gpG = {(v1, v2) ∈ B1 × B2 : v2 ∈ G(v1)}

the domain Dom(G) = {v ∈ B1 : G(v) 6= ∅}; the Rge(G) = {v2 ∈ B2 : for some v1 ∈ B1, v2 ∈
G(v1)}.Moreover, the inverse G−1 : B2 ⇒ B1 is

G−1(v2) = {v1 ∈ B − 1 : v2 ∈ G(v1)}.

Furthermore, for sets C1 and C2 in B1, define
d(v , C1) = inf

v1∈C1
d(v , v1) and e(C1, C2) = sup

v1∈C1
d(v , C2),

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Eur. J. Math. Anal. 10.28924/ada/ma.4.3 4where d, e are standard symbols for the distance from v to C2 and the excess of C1 to C2. Recallthat e(∅, ∅) = +∞, d(v , C2) = +∞, if C2 = ∅ and e(∅, C2) = 0, if C2 6= ∅ (by convention), where
∅ is the symbol for the empty set.Next, some more definitions and standard results are stated.
Definition 2.1. The inverse operator G−1 of G has the Aubin property for v1 ∈ B1 at v2 ∈ B2 of
modulus λ ≥ 0, if when v1 ∈ G−1(v2), there exist α > 0 and β > 0 so that

e(G−1(v4) ∩H[v1, α], G−1(v3)) ≤ λ‖v4 − v3‖ for each v3, v4 ∈ H[v2, β] (2.1)
and the inverse operator G−1 is locally closed at the pair (v2, v1), where H(v , α), H[v , α] denote
open and closed balls, respectively of center v ∈ B1 with radius α > 0.

It is useful to recall that there is a relationship between the Aubin property and the metricregularity (see e.g. [15, Theorem 3.7]). In particular, G−1 : B2 ⇒ B1 has the Aubin property at
v1, v2) with modulus λ > 0 if and only if G : B1 ⇒ B2 is metrically regular at (v1, v2) with the sameconstant λ. Therefore, the results that follows are given equivalently in terms of metric regularity.The celebrated contraction Mapping Principle [15,21,22] plays a vital role in our investigations.
THEOREM 2.2. Let us consider a set-valued operator Ψ : B1 ⇒ B2 and v ∈ B1. Assume that
there exist constants γ0 > 0 and δ0 ∈ (0, 1) so that gphΨ ∩ (H[v , γ0]×H[v , δ0]) is a closed set:
(i) d(v ,Ψ(v)) ≤ γ0(1− δ0) and
(ii) e(Ψ(v1) ∩ H[v , γ0],Ψ(v2)) ≤ δ0m(v1, v2) for each v1, v2 ∈ H[v , γ0], where m is some metric.
Then, the operator Ψ admit a fixed point in the closed ball H[v , γ0].

Majorizing sequences play an important role in the study of iterative methods.
Definition 2.3. Let {sn} stand for a nonnegative sequence of numbers and let {zn} be a sequence
in a Banach space. Assume:

‖zn+1 − zn‖ ≤ sn+1 − sn for each n = 0, 1, 2 . . . .

Then, the sequence {sn} is said to be majorizing for the sequence {yn}. In the case of convergence
of the sequence {sn}, the sequence {zn} is Cauchy in the Banach space and as such it is convergent
to some z∗, i.e., limn−→∞ zn = z∗.

3. Convergence
Let T = [0,+∞). The following conditions are used in the semi-local convergence analysis ofthe NLM.Assume:(A1) There exists a continuous and nondecreasing function w0 : T −→ R such that the equation

w0(t)− 1 = 0 has a smallest solution ρ0 ∈ T − {0}. Set T0 = [0, ρ0).

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Eur. J. Math. Anal. 10.28924/ada/ma.4.3 5(A2) There exist CNF w : T0 −→ R, and w1 : T −→ R. Let λ > 0. Define the sequence {sn}for s0 = 0, some s1 ∈ [0, ρ0), and each n = 0, 1, 2, ... by
sn+1

= sn +
[
∫ 1
0 w((1− θ)(sn − sn−1))dθ + w0(sn−1) + w1(sn−1)](sn − sn−1)

1− w0(sn) (3.1)
It is shown in Theorem 3.1 that {sn} is a majorizing sequence for {xn}. But let us firstpresent a general convergence criterion for it.(A3) There exists a parameter ρ ∈ [0, ρ0) such that for each n = 0, 1, 2, ...

w0(sn) < 1 and sn ≤ ρ.It follows by (3.1) and (A3) that 0 ≤ sn ≤ sn+1 ≤ ρand there exists s ∈ [0, ρ) such that limn→+∞ sn = s .The functions "w" and sequence {sn} are connected to the operators on NLM.(A4) There exists a linear operator M such that
λ‖L(x)−M‖ ≤ w0(‖x − x0‖) for each x ∈ D.Set D0 = D ∩ S(x0, ρ0).(A5) λ‖F ′(x)− F ′(y)‖ ≤ w(‖x − y‖) for each x, y ∈ D0 and
λ‖L(x)−M‖ ≤ w1(‖x − x0‖) for each x ∈ D0.(A6) There exist x1 ∈ D generated by NLM so that ‖x1 − x0‖ ≤ s1, and the multi-operator
(F (x0) +M(.− x0) +G(.))−1 is Aubin continuous at (0, x1) with corresponding parameters
α and β.(A7) For ρ > s1

2ρ− s1 < α,
1

λ

[∫ 1
0

w0((1− θ)ρ)dθ + w0(ρ) +

∫ 1
0

w((1− θ)ρ)dθ + w1(ρ)

]
ρ ≤ β,

and w0(ρ) < 1.and(A8) S[x0, s] ⊂ D.Next, the semi-local convergence analysis of NLM is developed using the conditions (A1)− (A8).
THEOREM 3.1. Assume that the conditions (A1) − (A8) are valid. Then, the sequence {xn}
generated by NLM is well defined in S(x0, s), remains in S(x0, s) for each n = 0, 1, 2, .. and is
convergent to some x∗ ∈ S[x0, s] solving the generalized equation (1.1). Moreover, the following
error estimates hold for each n = 0, 1, 2, ...

‖x∗ − xn‖ ≤ s − sn. (3.2)

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Eur. J. Math. Anal. 10.28924/ada/ma.4.3 6

Proof. Mathematical induction is employed to show the assertion for each n = 0, 1, 2, ...

‖xn+1 − xn‖ ≤ sn+1 − sn < s (3.3)
The assertion (3.3) holds for n = 0 by (A2) and the definition of s1 in (A6). Let us assume thatthere exist x1, ..., xm generated by NLM satisfying for all integers m = 0, 1, 2, ..., n − 1

‖xm − xm−1‖ ≤ sm − sm−1.

Then,
‖xm − x0‖ ≤ ‖xm − xm−1‖+ ‖xm−1 − xm−2‖+ ...+ ‖x1 − x0‖

≤ sm − sm−1 + sm−1 − sm−2 + ...+ s1 − s0 = sm < s,

and
‖xm − x1‖ ≤ ‖xm − x0‖+ ‖x0 − x1‖ ≤ sm − s1 ≤ s − s1.

Pic x ∈ U(xm, ‖xm − x0‖) to be arbitrary.
Define the operator

Q = F (x0) +M(x − x0) + G(x),

and the multi-operator
ψm(x) = Q−1[F (x0) +M(x − x0)− F (xm)− L(xm)(x − xm)].

The conditions of the Theorem 2.2 are validated in turn next. By applying the conditions (A5) and(A7), we get
‖F (x0) +M(x − x0)− F (xm)− L(xm)(x − xm)‖

≤ ‖F (x)− F (x0)−M(x − x0)‖

+‖F (x)− F (xm)− F ′(xm)(x − xm)‖

+‖F ′(xm)−M‖‖x − xm‖+ ‖M − L(xm)‖‖x − xm‖
1

λ

[∫ 1
0

w0((1− θ)‖x − x0‖)dθ‖x − x0‖

+

∫ 1
0

w((1− θ)‖x − xm‖)dθ‖x − xm‖

+w0(‖xm − x0‖)‖x − xm‖+ w1)‖xm − x0‖)‖x − xm‖]

≤
1

λ

[∫ 1
0

w0((1− θ)ρ)dθ + w0(ρ) +

∫ 1
0

w((1− θ)ρ)dθ + w1(ρ)

]
ρ ≤ β.

Notice that xm ∈ Q−1[F (x0) +M(xm − x0)− F (xm−1 − L(xm−1(xm − xm−1)].By Aubin property of Q−1(.) at (0, x1) with modulus λ and parameters α, β we have in turn

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Eur. J. Math. Anal. 10.28924/ada/ma.4.3 7

d(xm, ψm(xm))

≤ e{Q−1[F (x0) +M(xm − x0)− F (xm−1)− L(xm−1)(xm − xm−1)]

∩S(x1, α), ψm(xm)}

≤ λ‖F (xm)− F (xm−1)− L(xm−1)(xm − xm−1)‖

≤ λ‖F (xm)− F (xm−1)− F ′(xm−1)(xm − xm−1)‖

+λ‖(L(xm−1)− F ′(xm−1))(xm − xm−1)‖

≤ λ‖F (xm)− F (xm−1)− F ′(xm−1)(xm − xm−1)‖

+λ‖(L(xm−1)−M)(xm − xm−1)‖

+λ‖(F ′(xm−1)−M)(xm − xm−1)‖

≤
[∫ 1
0

w((1− θ)‖xm − xm−1‖)dθ‖xm − xm−1‖

+ w0(‖xm−1 − x0‖)‖xm − xm−1‖+ w1(‖xm−1 − x0‖)‖xm − xm−1‖]

= γ(1− w0(‖xm − x0‖)),

where
γ =

[∫ 1
0 w((1− θ)‖xm − xm−1‖)dθ + w0(‖xm−1 − x0‖) + w1(‖xm−1 − x0‖)

]
1− w0(‖xm − x0‖)

×‖xm − xm−1‖ (3.4)
Pick v1, v2 ∈ S(xm, ‖xm − x0‖). Then, we get

e{ψm(v1) ∩ S(xm, ‖xm − x0‖), ψm(v2)}

≤ e{ψm(v1) ∩ S(x1, α), ψm(v2)}

≤ λ‖M − L(xm)‖‖v1 − v2‖

≤ w0(‖xm − x0‖)‖v1 − v2‖ ≤ w0(ρ)‖v1 − v2‖,

where w0(ρ) < 1, by the definition of ρ. Thus, the Theorem 2.2 is applicable if we take ψ =

ψm, γ0 = γ and δ0 = δ = w0(‖xm − x0‖). So, there exists xm+1 ∈ S[x∗, ρ] satisfying
xm+1 ∈ Q−1[F (x0) +M(xm+1 − x0)− F (xm)− L(xm)(xm+1 − xm)]

leading to
‖xm+1 − xm‖ ≤

[∫ 1
0 w((1− θ)(sm − sm−1))dθ + w0(‖sm−1‖) + w1(sm−1)

]
1− w0(sm)

×(sm − sm−1) (3.5)
≤ sm+1 − sm,

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Eur. J. Math. Anal. 10.28924/ada/ma.4.3 8where we used (A2), (3.4) and the induction hypothesis (3.3).By (A3) and (3.5), we obtain
∞∑
m=m0

‖xm+1 − xm‖ ≤
∞∑
m=m0

(sm+1 − sm)

≤ s − sm0 < +∞.

if follows that the sequence {xm} is complete in a Banach space B1, and as such it is convergentto some x∗ ∈ S[x0, s]. Then, by (3.3), we can write
‖xm+j − xn‖ ≤ ‖xm+j − xm+j−1‖+ ‖xm+j−1 − xm+j−2‖+ ‖xm+1 − xn‖

≤ sm+j − sm+j−1 + sm+j−1 − sm+j−2 + ...+ sm+1 − sn

= sm+j − sm. (3.6)
By letting j −→ +∞ in (3.6), we conclude that (3.2) is valid. In view of the definition the sequence
{xm}, 0 ∈ F (xm) +L(xm)(xm+1−xm) +G(xn+1) for each m = 0, 1, ... Then, by letting m −→ +∞,we deduce that 0 ∈ F (x∗) + G(x∗). �

REMARK 3.2. A popular choice for M = F ′(x0). But this is not necessarily the most flexible
choice. The two conditions in (A3) are very general. By specializing the functions w0, w and
w1, we can provide other stronger conditions that imply the ones in (A3). Let us consider the
interesting Lipchitz case, i.e. when w0(t) = l0t, w(t) = l t and w1(t) = l1t . Then, the sequence
{sn} in (A2) reduces for l2 = l0 + l1 to

sn+1 = sn +

(
l
2(sn − sn−1) + sn−1

)
(sn − sn−1)

1− l0(sn)
(3.7)

Such sequences appear as majorant of Newton-like methods for solving nonlinear equations (i.e.when G = {0}). The Kantorovich-type convergence conditions in such studies imply the ones in
(A3) but not necessarily vice versa [20,35]. Our approach for the study of majorizing sequence {sn}has provided even weaker convergence conditions than the Kantorovich-type [4,5,6].

4. Conclusion
A very general theory for studying the convergence of Newton-like methods is developed forgenerating sequences approximating a solution of a generalized equation involving set-valued op-erators. The semi-local analysis of convergence depend on the Aubin property and the conceptof generalized continuity. The error analysis includes, computable upper error bounds on thenorms ‖xn+1 − xn‖ and ‖x∗ − xn‖. In particular, the semi-local analysis of convergence is basedon majorizing sequences for {xn} generated by NLM. It is shown that even specializations of theoperators involved lead to better results when compared to existing ones (see Remark 3.2). The

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Eur. J. Math. Anal. 10.28924/ada/ma.4.3 9future direction of our research involves the application of the developed theory on other meth-ods [1–3,9, 14,17,18,21–35].
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https://doi.org/10.28924/ada/ma.4.3
https://www.jstor.org/stable/3689393
https://www.jstor.org/stable/3689393
https://doi.org/10.1007/978-3-642-68874-4_14
https://doi.org/10.1007/978-3-642-68874-4_14
https://doi.org/10.1007/BF01404880
https://eudml.org/doc/208686
https://www.jstor.org/stable/43660724
https://doi.org/10.1007/s11075-012-9585-7
https://doi.org/10.1016/j.amc.2003.12.025
https://doi.org/10.1007/s10957-021-01974-0
https://doi.org/10.1137/1.9781611970791
https://doi.org/10.1137/1.9781611970791
https://doi.org/10.1080/01630568708816254

	1. Introduction
	2. Mathematical Bachground
	3. Convergence 
	4. Conclusion
	References

