International Journal of Analysis and Applications

Volume 17, Number 5 (2019), 850-863

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-850

FIXED POINTS OF NON-SMOOTH FUNCTIONS ON FINITE DIMENSIONAL

ORDERED BANACH SPACES VIA CLARKE GENERALIZED JACOBIAN

ZOHARI AND MARDANBEIGI∗

1Department of mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

∗Corresponding author: mrmardanbeigi@srbiau.ac.ir

Abstract. Considering Lipschitz functions which are not necessarily Fréchet differentiable, we obtain a

non-smooth version of Lakshmikantham’s theorem in finite dimensional ordered Banach spaces . We also

present an application of the obtained result in dynamical Coulomb friction problem.

1. Introduction

Ordered Banach spaces are very significant class of vector spaces which are studied widely in theory and

applications of mathematics. This class of vector spaces is considered in nonlinear integral equations [2],

nonlinear boundary value problems [4], optimal control theory [8], operator equations [20] and etc. On

the other hand, an important theory in mathematical analysis is fixed point theory. This theory and its

applications in orederd Banach spaces have been considered by many researchers. (see [2, 3, 5, 6, 11, 14, 16, 17]

and the references therein.)

Recently, Lakshmikantham et al. [14] have proved some fixed point theorems in ordered Banach space

X for a Fréchet differentiable mapping T : X → X. They showed the applications of their results in ODE

initial value problems and semilinear parabolic initial boundary value problems. Vijesh and Kumar [19] and

Mouhadjer and Benahmed [16] obtained some generalizations of Lakshmikantham’s fixed point theorems.

Received 2019-05-21; accepted 2019-07-04; published 2019-09-02.

2010 Mathematics Subject Classification. 47H10, 49J52, 46B40.

Key words and phrases. Fixed point; Ordered Banach spaces; Clarke generalized Jacobian;Coulomb friction.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

850

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-850


Int. J. Anal. Appl. 17 (5) (2019) 851

To obtain fixed points in a class of nonlinear operators in ordered Banach spaces, Mouhadjer and Benahmed

introduced a monotone Newton-like method in [15], by using Lakshmikantham’s fixed point theorems.

In this paper, we study these fixed point theorems for Lipschitz mappings on finite Bancah spaces which

are not necessary Fréchet differentiable. Our main tool is Clarke generalized Jacobian which was firstly

introduced by Clarke in [9] for a mapping between two finite dimensional vector spaces. Clarke generalized

Jacobian has heavy calculus rules and this paper, to the best of our knowledge, is the first work which deal

to non-smooth fixed point theorem using generalized gradient. Since every finite dimensional vector space

is isomorphic and homeomorphic to Rn for some n, we only focus on mappings F : Rn → Rn which are

not necessary differentiable and we prove some fixed point theorems for Lipschitzian ones. Finally, to show

one of the applications of the obtained results, we consider a non-smooth conic complementary problem and

then we investigate the relation between obtained fixed points and the solutions of the problem. This conic

complementary problem is arised in the linear discrete Coulomb friction problem which studied by Acary et

al. [1].

The paper is organized as follows. In Section 2 preliminaries are given. In Section 3 we introduce some

new definitions related to Clarke generalized Jacobian and then we investigate main results. Section 4 is

devoted to an application of the obtained results.

2. Preliminaries

Let S be a set. Denote the convex hull of S (the set of all finite convex combinations of members of S)

by co(S). Let C ⊆ Rn be nonempty. Then C is a closed pointed convex cone, when

i) C is a closed convex set,

ii) For every x ∈ C and every scalar λ ≥ 0, λx ∈ C,

iii) C ∩−C = {0}. We have C + C ⊆ C for every convex cone C. Using a closed pointed convex cone

C ⊆ Rn, we can define the following order relation on Rn:

x ≤ y ⇐⇒ y −x ∈ C.

This relation is reflexive, antisymmetric and transitive. If x ≤ y, then

i) x + z ≤ y + z, for each z ∈ Rn,

ii) αx ≤ αy, for all scalar α ≥ 0.

For x̄, ȳ ∈ Rn such that x̄ ≤ ȳ, the order interval [x̄, ȳ] is defined as

[x̄, ȳ] := {z ∈ Rn : x̄ ≤ z ≤ ȳ}.

A cone, especially used for many applications, is standard cone in finite dimensional Euclidean spaces:

Rn+ := {(x1, . . . ,xn) ∈ R
n : xi ≥ 0, i = 1 . . . ,n}.



Int. J. Anal. Appl. 17 (5) (2019) 852

Definition 2.1. Let C ⊆ Rn be a closed pointed convex cone.

i) C is called normal if there exists a positive constant δ such that for every x,y ∈ Rn,

0 ≤ x ≤ y ⇒‖x‖≤ δ‖y‖,

ii) C is called regular if every increasing (decreasing) sequence {xn}n∈N which is bounded from above

(below) converges.

Since every finite-dimensional normed space is reflexive, then for closed pointed convex cone C ⊆ Rn,

normality and regularity are equivalent [12].

Definition 2.2. The mapping F : Rn → Rm is strictly differentiable at x, if there is an m×n-matrix M

such that

lim
y→x,u→0

F(y + u) −F(y) −M(u)
‖u‖

= 0.

In this case, M is called strict derivative of F at x.

Definition 2.3. The mapping F : Rn → Rm is Fréchet differentiable at x, if there is an m×n-matrix M

such that

lim
u→0

F(x + u) −F(x) −M(u)
‖u‖

= 0.

In this case, M is called Fréchet derivative of F at x.

Note that, strictly differentiable functions are Fréchet differentiable while the converse is not true in

general [13].

Let the vector-valued function F : Rn → Rm be locally Lipschitz at a given point x, that is, there exists

a neighborhood U of x and a positive k such that

‖F(x1) −F(x2)‖≤ k‖x1 −x2‖, ∀x1,x2 ∈ U.

By Rademacher’s theorem [9], F is differentiable almost everywhere (in the sense of Lebesgue measure) on

U.

Definition 2.4. [9, 13] Let the vector-valued function F : Rn → Rm be locally Lipschitz at a given point x.

The Clarke generalized Jacobian of F at x, denoted by ∂F(x) is defined as

∂F(x) := co
{

lim
i→∞

∇F(xi) : xi ∈ Ω,xi → x
}
,

where Ω is the set of points in U at which F is differentiable at them.

Note that ∂F(x) is a nonempty convex and compact subset of L(Rn,Rm), the space of real m×n-matrices.



Int. J. Anal. Appl. 17 (5) (2019) 853

3. Main Results

In [14], Lakshmikantham et al. considered a mapping T : E → E where E is an ordered Banach space

and they worked on an order interval [v0,w0] which satisfies some assumptions. Fréchet differentiability of

T on [v0,w0] is a vital assumption in [14]. One of the Lakshmikantham’s abstract fixed point theorems is as

follows.

Theorem 3.1. [14, Theorem 2.1] Let E be an ordered Banach space with regular order cone E+. Suppose

T : E → E satisfies the following hypotheses:

(i) There exist v0,w0 ∈ E such that v0 ≤ Tv0, Tw0 ≤ w0 and v0 ≤ w0.

(ii) The Fréchet derivative T ′(u) exists for every u ∈ [v0,w0], and the mapping u 7→ T ′(u)v is increasing

on [v0,w0] for all v ∈ E+.

(iii) [I −T ′(u)]−1 exists and is a bounded and positive operator for all u ∈ [v0,w0].

Then, for n ∈ N, relations

vn+1 = Tvn + T
′(vn)(vn+1 −vn), wn+1 = Twn + T ′(vn)(wn+1 −wn),

define an increasing sequence {vn}∞n=0 and a decreasing sequence {wn}∞n=0 which both converge to fixed points

of T . These fixed points are equal if

(iv) Tu1 −Tu0 < u1 −u0 whenever v0 ≤ u0 < u1 ≤ w0.

Assumptions (ii) and (iii) are deeply dependent on Fréchet differentiability of T on [v0,w0] , so even for

a simple function f(x) = |x| on R we cannot use this theorem (since f is not differentiable at x = 0).

In this section, we assume that, the mappings are Lipschitz and we impose strictly differentiability only in

v̄, the left bound of the order interval [v̄, w̄], and we do not need to assume differentiability in other members

of the interval.

Definition 3.1. Let F : Rn → Rn be locally Lipschitz on [v̄, w̄] and let C be a regular convex cone . We say

the set-valued u 7→ ∂F(u) is semi-increasing on [v̄, w̄] if for all v,w ∈ [v̄, w̄], with w −v ∈ C,

∂F(w) ◦z ⊆ ∂F(v) ◦z + C, ∀z ∈ C, (3.1)

where ◦ denotes the matrix multiplication.

Remark 3.1. If F : Rn → Rn is strictly differentiable on [v̄, w̄], then for every x ∈ [v̄, w̄], ∂F(·) is a

singleton and Lipschitz on [v̄, w̄] [9]. We denote this derivative by F ′(x). If F is semi-increasing on [v̄, w̄]

and C = Rn+. Then, for all v,w ∈ [v̄, w̄], with w −v ∈ Rn+ we have,

F ′(w) ◦z ⊆ F ′(v) ◦z + Rn+, ∀z ∈ R
n
+.



Int. J. Anal. Appl. 17 (5) (2019) 854

Thus, for some r ∈ Rn+, F ′(w)◦z = F ′(v)◦z+r. This shows that for v ≤ w, F ′(v)◦z ≤ F ′(w)◦z. Therefore,

u 7→ T ′(u)z is increasing on [v̄, w̄] for all z ∈ Rn+. Thus, semi-increasing map notion is a generalization of

increasing map notion.

Example 3.1. Consider the function F : R2 → R2, defined by F(x,y) = (|x|, |y|) on [(0, 0)T , (1, 1)T ] and

C = R2+. Where T denotes Matrix Transposition. Then,

∂F(0, 0) =




α 0

0 β


 : α,β ∈ [−1, 1]




∂F(x, 0) =




1 0

0 β


 : β ∈ [−1, 1]


 0 < x ≤ 1

∂F(0,y) =




α 0

0 1


 : α ∈ [−1, 1]


 0 < y ≤ 1

∂F(x,y) =




1 0

0 1




 0 < x,y ≤ 1.

If v = (0, 0)T , v ≤ w and z = (z1,z2)T is arbitrary, then Figure 1 shows that ∂F(w)◦z ⊆ ∂F(0, 0)◦z+R2+. It

is easy to check that in all other cases, (3.1) holds too. Thus, F is a semi-increasing map on [(0, 0)T , (1, 1)T ].

Figure 1. semi-increasing map in Example 3.1

Lemma 3.1. Let F : Rn → Rn be locally Lipschitz on [v̄, w̄]. If the set-valued u 7→ ∂F(u) is semi-increasing

on [v̄, w̄] and v,w ∈ [v̄, w̄], with w −v ∈ C, then

∂F(w) ◦z ⊆ ∂F(v) ◦z −C, ∀z ∈−C.



Int. J. Anal. Appl. 17 (5) (2019) 855

Proof. Let z ∈ −C and ζz ∈ ∂F(w) ◦ z be arbitrary. Since −z ∈ C and ζ(−z) ∈ ∂F(w) ◦ (−z) we have

ζ(−z) ∈ ∂F(v) ◦ (−z) + C. Thus, ζ(−z) = η(−z) + c, for some η ∈ ∂F(v) and c ∈ C. On the other hand,

∂F(u) ⊆ L(Rn) for every u ∈ Rn, thus, ζ(−z) = −ζz and η(−z) = −ηz and we have ζz = ηz − c. Hence,

ζz ∈ ∂F(v) ◦z −C, which proves the lemma. �

The following lemma plays an important role in the reminder of this paper.

Lemma 3.2. Let F : Rn → Rn be Lipschitz and semi-increasing on an order interval [v̄, w̄]. Then for

v̄ ≤ v ≤ w ≤ w̄ we have,

F(v) −F(w) ∈ ∂F(v) ◦ (v −w) −C.

Proof. According to the mean value theorem [9, Proposition 2.6.5],

F(v) −F(w) ∈ co(∂F(co{v,w})) ◦ (v −w).

The right-hand side above denotes the convex hull of all points of the form η(v −w), where η ∈ ∂F(u) for

some point u ∈ co{v,w}. Assume that u = v + t(w −v) where 0 ≤ t ≤ 1. There exist λ1, . . . ,λm ∈ R such

that λi ≥ 0 for i ∈{1, . . . ,m} and
∑m
i=1 λi = 1 and we have,

F(v) −F(w) =

(
m∑
i=1

λiζi

)
(v −w), (3.2)

where ζi ∈ ∂F(u). On the other hand, since v −w ∈−C and u−v ∈ C, by Lemma 3.1 we have,

∂F(u) ◦ (v −w) ⊆ ∂F(v) ◦ (v −w) −C. (3.3)

Now, considering the convexity of ∂F(u), also (3.2) and (3.3), we have

F(v) −F(w) ∈ ∂F(v) ◦ (v −w) −C.

�

In the following, we prove our main results.

Theorem 3.2. Let Rn be ordered with regular cone C. Assume that the mapping F : Rn → Rn satisfies the

following hypotheses.

(i) There exist v̄, w̄ ∈ Rn with w̄ − v̄ ∈ C, such that F is Lipschitz function on order interval [v̄, w̄],

v̄ ≤ F(v̄) and F(w̄) ≤ w̄.

(ii) The set-valued u 7→ ∂F(u) is semi-increasing on [v̄, w̄].

(iii) F is strictly differentiable at v̄ with ∂F(v̄) = {ζv̄} such that [I−ζv̄]−1 exists and is a bounded positive

operator, that is, [I − ζv̄]−1(C) ⊆ C.

Then for n ∈ N, relations

vn+1 = F(vn) + ζv̄(vn+1 −vn), wn+1 = F(wn) + ζv̄(wn+1 −wn), (3.4)



Int. J. Anal. Appl. 17 (5) (2019) 856

with v0 := v̄, define an increasing sequence {vn}n∈N and a decreasing sequence {wn}n∈N which both converge

to fixed points of F . These fixed points are equal if

(iv) Fu1 −Fu0 < u1 −u0 whenever v̄ ≤ u0 < u1 ≤ w0.

Proof. We recall that since F is strictly differentiable at v̄, ∂F(v̄) is a singleton set (see [13, P. 15]). Consider

v1 = F(v̄) + ζv̄(v1 − v̄). Thus, [I − ζv̄]v1 = F(v̄) − ζv̄v̄. This implies v1 = [I − ζv̄]−1(F(v̄) − ζv̄v̄). Since

[I − ζv̄]−1 is bounded, v1 is well-defined. We show that v̄ ≤ v1 ≤ w̄. The assumption (1) implies:

v̄ −v1 ≤ F(v̄) − [F(v̄) + ζv̄(v1 − v̄)] = ζv̄(v̄ −v1).

Therefore, [I − ζv̄](v̄ −v1) ≤ 0. Since, [I − ζv̄]−1 is a positive operator, (v̄ −v1) ≤ [I − ζv̄]−1(0) = 0 and we

have v̄ ≤ v1. Now, by (i)

v1 − w̄ ∈ F(v̄) + ζv̄(v1 − v̄) −F(w̄) −C.

and using Lemma 3.2 we have,

v1 − w̄ ∈ ∂F(v̄) ◦ (v̄ − w̄) + ζv̄(v1 − v̄) −C.

Therefore, for some c ∈ C we have,

v1 − w̄ = ζv̄(v̄ − w̄) + ζv̄(v1 − v̄) − c. (3.5)

Then (3.5) implies:

v1 − w̄ ≤ ζv̄(v1 − w̄).

Thus [I − ζv̄](v1 − w̄) ≤ 0 and (v1 − w̄) ≤ [I − ζv̄]−1(0) = 0 which implies v1 ≤ w̄. In a similar way we

can show that there exists a point w1 such that w1 = F(w̄) + ζv̄(w1 − w̄) and v̄ ≤ w1 ≤ w̄. We claim that

v1 ≤ w1. To prove this claim, note that:

v1 −w1 = F(v̄) + ζv̄(v1 − v̄) − [F(w̄) + ζv̄(w1 − w̄)].

Thus, by Lemma 3.2 we have,

v1 −w1 ∈ ∂F(v̄) ◦ (v̄ − w̄) + ζv̄(v1 − v̄) − ζv̄(w1 − w̄) −C,

similarly, v1 −w1 ≤ ζv̄(v1 −w1) and we have v1 ≤ w1.

Up to now, we showed that v̄ ≤ v1 ≤ w1 ≤ w̄. We claim for every j ∈ N, v̄ ≤ vj ≤ vj+1 ≤ wj+1 ≤ wj ≤ w̄.

Since v̄ ≤ v1 and w1 ≤ w̄, we only need to prove vj ≤ vj+1 ≤ wj and vj+1 ≤ wj+1 ≤ wj. We just prove

vj ≤ vj+1 and other inequalities are obtained similarly. By Lemma 3.2

vj −vj+1 = F(vj−1) + ζv̄(vj −vj−1) − [F(vj) + ζv̄(vj+1 −vj)]

∈ ∂F(vj−1) ◦ (vj−1 −vj) + ζv̄(vj −vj−1) − ζv̄(vj+1 −vj) −C



Int. J. Anal. Appl. 17 (5) (2019) 857

Since vj−1 − v̄ ∈ C and vj−1 −vj ∈−C, by Lemma 3.1 we have,

vj −vj+1 ∈ ∂F(v̄) ◦ (vj−1 −vj) + ζv̄(vj −vj−1) − ζv̄(vj+1 −vj) −C.

Therefore,

vj −vj+1 ≤ ζv̄(vj −vj+1),

so we have vj −vj+1 ≤ [I −ζv̄]−1(0) = 0 which proves our claim. We found an increasing sequence {vn}n∈N

and a decreasing sequence {wn}n∈N such that

v̄ ≤ v1 ≤ . . . ≤ vn ≤ wn ≤ . . . ≤ w1 ≤ w̄.

Since C is a regular cone, {vn}n∈N and {wn}n∈N are convergent. Suppose vn → v and wn → w. We will

prove vn+1 → F(v) which shows v is a fixed point of F. We have

vn+1 −F(v) = F(vn) + ζv̄(vn+1 −vn) −F(v)

Since F is continuous and vn → v, vn+1 −vn → 0 and F(vn) −F(v) → 0, thus vn+1 → F(v), so F(v) = v.

Similarly one can show that F(w) = w.

If (iv) holds and v < w, then w−v = Fw−Fv < w−v which is a contradiction. And the uniqueness of

the fixed point is proved. �

Note that under some conditions, the sequences in Theorem 3.2, converge quadratically as the following

theorem shows.

Theorem 3.3. Let Rn be ordered with regular cone C. Let the mapping F : Rn → Rn satisfies the hypotheses

of Theorem 3.2 and

‖ζv − ζv̄‖≤ L‖u−v‖, ∀ζv ∈ ∂F(v),

whenever v̄ ≤ v ≤ u ≤ w̄. Then, the sequences {vn}∞n=0 and {wn}∞n=0 converge quadratically to the same

fixed point of F .

Proof. Note that, by Theorem 3.2, both the sequences {vn}∞n=0 and {wn}∞n=0 converge to the same fixed

point u of F. We prove that these sequences converge quadratically. For sequence {vn}∞n=0 we have,

vn+1 −u = F(vn) −F(u) + ζv̄(vn+1 −vn)

⊆ ∂F(vn) ◦ (vn −u) + ζv̄(vn+1 −vn) −C.

Thus, for some ηvn ∈ ∂F(vn) and c ∈ C we have,

vn+1 −u = ζvn ◦ (vn −u) + ζv̄(vn+1 −vn) − c

= ζvn ◦ (vn −u) + ζv̄(vn+1 −u) − ζv̄(vn −u) − c.



Int. J. Anal. Appl. 17 (5) (2019) 858

Hence,

u−vn+1 = ζvn ◦ (u−vn) − ζv̄(vn+1 −u) + ζv̄(vn −u) + c.

This implies

[I − ζv̄](u−vn+1) = (ζvn − ζv̄) ◦ (u−vn) + c.

Since, [I − ζv̄]−1 is a positive operator, we have

0 ≤ u−vn+1 ≤ [I − ζv̄]−1(ζvn − ζv̄) ◦ (u−vn) + c̄,

where c̄ = [I −ζv̄]−1(c). On the other hand, the cone C is regular and therefore normal, thus, there exists a

positive constant δ such that

‖u−vn+1‖≤ δ‖[I − ζv̄]−1‖‖(ζvn − ζv̄)‖‖u−vn‖ + c̄

≤ δL‖[I − ζv̄]−1‖‖u−vn‖2 + c̄.

For sequence {wn}∞n=0 the claim can be proved by a similar manner. �

4. Application in the Dynamical Coulomb Friction Problem

The dynamical Coulomb friction problem in finite dimension with discretized time is associated to the

problem of simulating the dynamics of mechanical systems which involve unilateral contact between their

parts or with external objects. Acary et al. [1] presented a new formulation of this problem. They capture

and treat directly the friction model as a parametric quadratic optimization problem with second-order cone

constraints coupled with a fixed point equation. In this section, we just use the resulted fixed point theorems

obtained in this paper, to investigate problem. Firstly, we recall some preliminaries.

Let x ∈ Rd. The normal and tangential components of x with respect to a unit vector e ∈ Rd are defined

respectively as

xN := x
Te ∈ R, xT := x−xNe ∈ Rd.

For the unit vector e ∈ Rd and parameter µ ∈ (0, +∞), the second-order cone Ke,µ is defined by

Ke,µ := {x ∈ Rd : ‖xT‖≤ µxN}.

If µ = 0 or µ = +∞, we define

Ke,0 := {x ∈ Rd : xT = 0,xN ≥ 0}, Ke,∞ := {x ∈ Rd : xN ≥ 0}.

The dual cone of the second-order cone Ke,µ with µ ∈ (0,∞) is defined as

K∗e,µ := {s ∈ R
d : xTs ≥ 0, ∀x ∈ Ke,µ} = Ke, 1

µ
.



Int. J. Anal. Appl. 17 (5) (2019) 859

Also with the convention 1/0 = ∞ and 1/∞ = 0, (Ke,0)∗ = Ke,∞ and (Ke,∞)∗ = Ke,0. Consider e1, . . . ,en ∈

Rd and µ1, . . . ,µn ∈ [0, +∞]. The associated product cone and its dual cone are respectively

L = Ke1,µ1 ×···×Ken,µn ⊆ Rnd

, L∗ = K∗e1,µ1 ×···×K
∗
en,µn = Ke1, 1

µ1
×···×Ken, 1

µn
.

Set

I :=
{
i ∈{1, · · · ,n} : µi 6= 0

}
, nI := Card I.

Consider the problem:

Mv + f = HTr

ũ = Hv + w + Es

L∗ 3 ũ ⊥ r ∈ L

si = ‖ũiT‖ for i ∈ I (4.1)

with respect to the variable (v,r, ũ,s) ∈ Rm × Rnd × Rnd × RnI where ũ := (ũ1, . . . , ũn) ∈ Rnd, r :=

(r1, . . . ,rn) ∈ Rnd. Moreover, the data of the problem are

ei ∈ Rd, µi ∈ [0, +∞], M ∈ Rm×m, f ∈ Rm, H ∈ Rnd×m, w ∈ Rnd, E ∈ Rnd×nI,

where matrix M is definite positive and matrix E is constructed by concatenating nI columns Ei ∈ Rnd,

where Ei is itself the concatenation of n vectors of Rd, all zeros except for the i-th which is µiei.

Define

C(s) := {v ∈ Rm : Hv + w + Es ∈ L∗},

v(s) := arg min
v∈C(s)

{
1

2
vTMv + fTv

}
∈ Rm,

ũ(s) := Hv(s) + w + Es ∈ Rnd,

and

F(s) :=
(
‖ũiT (s)‖

)
i∈I ∈ R

nI

The following theorem shows the relation between fixed points of F and the solutions of (4.1).

Theorem 4.1. [1, Theorem 3.3] Let (v∗,r∗, ũ∗,s∗) solves the problem (4.1), then v∗ = v(s∗) and F(s∗) = s∗.

In dimension two , the inverse is also hold.

Theorem 4.2. [1, Theorem 3.7] Let the dimension d = 2. Then (v∗,r∗, ũ∗,s∗) solves the problem (4.1) if

and only if v∗ = v(s∗) and F(s∗) = s∗.



Int. J. Anal. Appl. 17 (5) (2019) 860

Using these theorems, we can compute a fixed point of F and then check whether the computed fixed

point is a solution of the problem (4.1). Note that F is a non-smooth function and to compute its fixed

points we need to use the non-smooth version fixed point theorem presented in this paper. In the following

Examples we show the efficiency of the resulted fixed point theorem.

Example 4.1. Let d = 2, n = 3, e1 = e2 = e3 =


0

1


, µ1 = 1, µ2 = 0, µ3 = 2 and m = 2. Then

E =




0 0

1 0

0 0

0 0

0 0

0 2



.

Also, I = {1, 3} and nI = 2. For x =
(
x1,x2

)T
and ei (i = 1, 2, 3), xN = x2 and xT =

(
x1, 0

)T
. Moreover,

Ke1,µ1 =
{
x ∈ R2 : |x1| ≤ x2

}
, K∗e1,µ1 = Ke1,µ1,

Ke2,µ2 =
{
x ∈ R2 : x1 = 0,x2 ≥ 0

}
, K∗e2,µ2 =

{
x ∈ R2 : x2 ≥ 0

}
Ke3,µ3 =

{
x ∈ R2 : |x1| ≤ 2x2

}
K∗e3,µ3 =

{
x ∈ R2 : |x1| ≤

1

2
x2

}
.

These cones and their dual cones are showed in Figure 2.

Figure 2. Cones and their dual cones in Example 4.1

Note that:

L =
{

(α,β,γ)T : α ∈ Ke1,µ1,β ∈ Ke2,µ2,γ ∈ Ke3,µ3
}
,

L∗ =
{

(α∗,β∗,γ∗):α∗ ∈ K∗e1,µ1,β
∗ ∈ K∗e2,µ2,γ

∗ ∈ K∗e3,µ3
}
.



Int. J. Anal. Appl. 17 (5) (2019) 861

Assume,

M =


1 0

0 1


 ,H =




1 0

0 1

1 0

1 0

0 1

1 0



,w =




1

0

0

1

0

0



,f =


1

0


 ,v =


v1
v2


 ,s =


s1
s3




r = (α1,α2,β1,β2,γ1,γ2)
T ∈ L, ũ = (α∗1,α

∗
2,β
∗
1,β
∗
2,γ
∗
1,γ
∗
2 )
T ∈ L∗.

Let v ∈ C(s). Thus, 


α∗1

α∗2

β∗1

β∗2

γ∗1

γ∗2



∈ L∗ ⇒




1 0

0 1

1 0

1 0

0 1

1 0



·


v1
v2


 +




1

0

0

1

0

0




+




0 0

1 0

0 0

0 0

0 0

0 2



·


s1
s3


 ∈ L∗.

Therefore, 


v1 + 1

v2 + s
1

v1

v1 + 1

v2

v1 + 2s
3



∈ L∗ ⇒


v1 + 1
v2 + s

1


 ∈ K∗e1,µ1,


 v1
v1 + 1


 ∈ K∗e2,µ2,


 v2
v1 + 2s

3


 ∈ K∗e3,µ3.

Hence,

|v1 + 1| ≤ v2 + s1,v1 + 1 ≥ 0, |v2| ≤
1

2
(v1 + 2s

3).

So, we have

C(s) =

{
(v1,v2)

T ∈ R2 : v1 + 1 ≤ v2 + s1, |v2| ≤
1

2
(v1 + 2s

3)

}
.

Note that C(s) is a closed convex set and J(v) := 1
2
vTMv+fTv is a convex function. Therefore, if C(s) 6= ∅,

then J(v) has a unique solution. In this example, if s1 ≥ −2s3 + 1, then C(s) 6= ∅. Now we must consider

the following convex optimization problem:

min
1

2
vTMv + fTv =

1

2
(v21 + v

2
2 ) + v1

subject to g1(v) := v1 −v2 −s1 + 1 ≤ 0

g2(v) := |v2|−
1

2
v1 −s3 ≤ 0



Int. J. Anal. Appl. 17 (5) (2019) 862

According to [10, Page 150], if v∗ = (v∗1,v
∗
2 )
T , then there exist λ1,λ2 ≥ 0 such that

0
0


 =


v∗1 + 1

v∗2


 + λ1


 1
−1


 + λ2


−12
α


 , λigi(v∗) = 0, i = 1, 2 (4.2)

where 


α = 1 if v∗2 > 0,

−1 ≤ α ≤ 1 if v∗2 = 0,

α = −1 if v∗2 < 0,

then v∗ is the optimal solution. The system (4.2) has a solution if s3 ≤ 1/2. With these condition, v∗1 = 0

and v∗2 = −2s3. Thus, v(s) = (0,−2s3). We also have ,

ũ(s) = Hv(s) + w + Es =




1 0

0 1

1 0

1 0

0 1

1 0





 0
−2s3


 +




1

0

0

1

0

0




+




0 0

1 0

0 0

0 0

0 0

0 2



·


s1
s3


 =




1

−2s3 + s1

0

1

−2s3

2s3




and

F(s) = (1,−2s3), where s3 ≤
1

2
.

Note that from Example 3.1, F is a semi-increasing map with C = R2+. Set v̄ = (−1,−1)T , w̄ = (2, 0)T .

We have ∂F(v̄) =


0 0

0 −2


. With these informations, conditions (i), (ii), (iii) of Theorem 3.2 hold. Set

v0 = (−1,−1)T and

vn+1 = (1, 2‖v2n‖)
T +


0 0

0 −2


 (vn+1 −vn).

We have vn = (1, 0)
T for every n ∈ N. Thus (1, 0)T is a fixed point of F . It is easy to check that the

combination of these resulted quantities with r = (0, 0, 0, 0, 0, 1)T , is a solution of system (4.1).

References

[1] V. Acary, F. Cadoux, C. Lemaréchal, J. Malick, A formulation of the linear discrete Coulomb friction problem via convex

optimization. ZAMM, Z. Angew. Math. Mech. 91 (2011), 155-175.

[2] P.R. Agarwal, N. Hussain, M.A. Taoudi, Fixed point theorems in ordered banach spaces and applications to nonlinear

integral equations, Abstr. Appl. Anal. 2012 (2012), 245872.

[3] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM rev. 18 (1976),

620-709.

[4] H. Amann, Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems,

In: Nonlinear Operators and the Calculus of Variations 1976, Springer-Verlag Berlin Heidelberg , pp. 1-55.



Int. J. Anal. Appl. 17 (5) (2019) 863

[5] H. Andrei, P. Radu, Nonnegative solutions of nonlinear integral equations in ordered Banach spaces. Fixed Point Theory,

1 (2004), 65-70.

[6] M. Berzig, B. Samet, Positive fixed points for a class of nonlinear operatoes and applications. Positivity, 17 (2013), 235-255.

[7] S. Bonettini, I. Loris, F. Porta, M. Prato, Variable metric inexact line-search-based methods for non-smooth optimization.

SIAM J. Optim. 26 (2016), 891-921.

[8] J. Blot, N. Hayek, Infinite-Horizon Optimal Control in the Discrete-Time Framework. Springer-Verlag, New York, 2014.

[9] F.H. Clarke, Optimization and non-smooth Analysis. Society for Industrial and Applied Mathematics, 1990.

[10] A. Dhara, J. Dutta, Optimality conditions in convex optimization: a finite-dimensional view. CRC Press, 2011.

[11] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications. Non-

linear Anal. TMA, 65 (2006), 1379-1393.

[12] D. Guo, Y.J. Cho, J, Zhu, Partial ordering methods in nonlinear problems. Nova Science Publishers, New York, 2004.

[13] V. Jeyakumar, D.T. Luc, non-smooth Vector Functions and Continuous Optimization. Springer, New York, 2008.

[14] V. Lakshmikantham, S. Carl, S. Heikkilä, Fixed point theorems in ordered Banach spaces via quasilinearization. Nonlinear

Anal. TMA, 71 (2009), 3448-3458.

[15] L. Mouhadjer, B. Benahmed, A Monotone Newton-Like Method for the Computation of Fixed Points, In: Le Thi H, Pham

Dinh T, Nguyen N, editors. Modelling, Computation and Optimization in Information Systems and Management Sciences.

Advances in Intelligent Systems and Computing, vol 359. Springer, Cham, 2015, pp. 345-356.

[16] L. Mouhadjer, B. Benahmed, Fixed point theorem in ordered Banach spaces and applications to matrix equations. Positivity,

20 (2016), 981-998.

[17] J.J. Nieto, R. Rodŕıguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to

ordinary diferential equations. Acta Math. Sin. 3 (2007), 2203-2212.

[18] W. Rudin, Functional Analysis. McGraw-Hill, Inc. 1991.

[19] V.A. Vijesh, K.H. Kumar, Wavelet based quasilinearization method for semi-linear parabolic initial boundary value prob-

lems. Appl. Math. Comput. 266 (2015), 1163-1176.

[20] C.B. Zhai, C. Yang, C.M. Guo, Positive solutions of operator equations on ordered Banach spaces and applications.

Comput. Math. Appl. 56 (2008), 3150-3156.

[21] P. Zhou, J. Du, Z. Lü, Topology optimization of freely vibrating continuum structures based on non-smooth optimization.

Struct. Multidiscip. Optim. 56 (2017), 603-618.


	1. Introduction
	2. Preliminaries
	3. Main Results
	4. Application in the Dynamical Coulomb Friction Problem
	References