# Nontrivial solutions for nonlinear discrete boundary value

## Transcript Of Nontrivial solutions for nonlinear discrete boundary value

Nontrivial solutions for nonlinear discrete boundary value problems of the fourth order

By Danielle Layne

Lingju Kong Professor and Graduate Coordinator (Major Advisor)

Roger Nichols Associate Professor (Committee Member)

John R. Graef Professor (Committee Member)

Jin Wang Professor and UNUM Chair of Excellence in Applied Mathematics (Committee Member)

Nontrivial solutions for nonlinear discrete boundary value problems of the fourth order

By Danielle Layne

A Thesis Submitted to the University of Tennessee at Chattanooga in Partial

Fulﬁllment of the Requirements of the Degree of Master of Science: Mathematics

The University of Tennessee at Chattanooga Chattanooga, Tennessee May 2021

ii

ABSTRACT We study the existence of multiple nontrivial solutions for two nonlinear fourth order discrete boundary value problems. We ﬁrst establish criteria for the existence of at least two nontrivial solutions of the problems and obtain conditions to guarantee that the two solutions are sign-changing. Under some appropriate assumptions, we further prove that the problems have at least three nontrivial solutions, which are respectively positive, negative, and sign-changing. We include two examples to illustrate the applicability of our results. Our theorems are proved by employing variational approaches, combined with the classic mountain pass lemma and a result from the theory of invariant sets of descending ﬂow.

iii

TABLE OF CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 PRELIMINARY RESULTS . . . . . . . . . . . . . . . . . . . . . . 5 3 EXISTENCE OF TWO NONTRIVIAL SOLUTIONS . . . . . . . . . . . 16

3.1 MAIN RESULTS . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 PROOFS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 EXISTENCE OF THREE NONTRIVIAL SOLUTIONS. . . . . . . . . . . 23 4.1 MAIN RESULTS . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 PROOFS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

iv

CHAPTER 1 INTRODUCTION

It is well known that nonlinear difference equations of higher order appear naturally as discrete analogue and as numerical solutions of differential equations. The applications of such equations have been well documented in [16, 17]. In recent years, the existence of solutions of boundary value problems (BVPs) of difference equations, with various boundary conditions (BCs), has received increasing attention from many researchers. In this thesis, let [c, d]Z = {z ∈ Z | c ≤ z ≤ d}, where c, d ∈ Z with c ≤ d. We are concerned with the existence of multiple nontrivial solutions of the BVP for the discrete beam equation

∆4u(k − 2) − α∆2u(k − 1) + β u(k)

=

f (k, u(k)),

k

∈

[a + 1, b + 1]Z,

(1.1)

u(a)

=

∆2u(a − 1)

=

0,

u(b + 2)

=

∆2u(b + 1)

=

0,

where α, β ∈ [0, ∞), a, b ∈ Z with b ≥ a, f : [a + 1, b + 1]Z × R → R is continuous, and ∆ is the forward difference operator deﬁned by ∆u(k) = u(k + 1) − u(k) and ∆nu(k) = ∆(∆n−1u(k)). By a solution of BVP (1.1), we mean a function u : [a − 1, b + 3]Z → R such that u satisﬁes both the equation and the BCs in (1.1). If u(k) > 0 for all k ∈ [a + 1, b + 1]Z, then u is called a positive solution; if u(k) < 0 for all k ∈ [a + 1, b + 1]Z, then u is said to be a negative solution; and if u(k) changes signs on [a + 1, b + 1]Z, then u is called a sign-changing solution. We also obtain existence criteria for the BVP

∆4u(k − 2) − α∆2u(k − 1) + β u(k)

=

λ

f (k, u(k)),

k

∈

[a + 1, b + 1]Z,

(1.2)

u(a)

=

∆2u(a − 1)

=

0,

u(b + 2)

=

∆2u(b + 1)

=

0,

where λ is a positive parameter.

1

BVP (1.1) can be regarded as a discrete analogue of the beam problem

u(4)(t) − αu

(t) + β u(t) =

f (t, u(t)),

t ∈ (0, 1),

(1.3)

u(0) = u

(0) = 0,

u(1) = u (1) = 0,

where the BCs correspond to both ends of the beam being hinged when there is no bending moment. The equation in BVP (1.3) is often referred to as the beam equation since it describes the deﬂection of a beam under a certain force. BVP (1.3) and a number of its variations have been investigated by many authors. A small sample of the work can be found in, for example, [2, 5, 27, 28] and the included references.

The existence of solutions of discrete BVPs, with various BCs, of the fourth order has been extensively studied in the literature. The reader is refered to [1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 24, 25, 26, 29, 31] for some work on this subject. In particular, paper [8] studied the existence of three solutions of the BVP

∆4u(k − 2) − α∆2u(k − 1) + β u(k)

=

λ

f (k, u(k)),

k

∈

[1, T ]Z,

(1.4)

u(0)

=

∆u(−1)

=

∆2u(T )

=

0,

∆3u(T

− 1) − α∆u(T )

=

µ g(u(T

+ 1)),

where T ≥ 2 is an integer, α, β , λ , µ ∈ R are parameters, f ∈ C([1, T ]Z × R, R), and g ∈ C(R, R); while the existence of inﬁnitely many solutions of BVP (1.4) was considered in [7]. The existence theorems in [7, 8] give the existence of solutions for the parameters λ and µ in different intervals. Paper [6] investigated the existence of two solutions of the BVP

∆4u(k − 2) − α∆2u(k − 1) + β u(k)

=

f (k, u(k)),

k

∈

[1, T ]Z,

u(−1)

=

∆u(−1)

=

0,

u(T

+ 1)

=

∆2u(T )

=

0.

Variational methods and critical point theory were used in [6, 7, 8] to prove the existence results. By using ﬁxed point theory, paper [1] obtained a number of criteria for the existence of positive

2

solutions of BVP (1.2) with β = 0, i.e., the problem

∆4u(k − 2) − α∆2u(k − 1)

=

λ

f (k, u(k)),

k

∈

[a + 1, b + 1]Z,

(1.5)

u(a)

=

∆2u(a − 1)

=

0,

u(b + 2)

=

∆2u(b + 1)

=

0.

Later, paper [9] studied the following slightly more general version of BVP (1.5)

∆4u(t − 2) − β ∆2u(t − 1)

=

λ

[ f (t, u(t), u(t)) + r(t, u(t))] ,

t

∈

[a + 1, b − 1]Z,

(1.6)

u(a)

=

∆2u(a − 1)

=

0,

u(b)

=

∆2u(b − 1)

=

0,

where f : [a + 1, b − 1]Z × [0, ∞) × (0, ∞) → [0, ∞) and r : [a + 1, b − 1]Z × [0, ∞) → [0, ∞) are continuous. By applying some results from mixed monotone operator theory, not only was the existence and uniqueness of positive solutions of BVP (1.6) obtained , but also the dependence of positive solutions on the parameter λ was discussed. Moreover, two sequences are constructed in such a way so that they converge uniformly to the unique positive solution of the problem. See [9, Theorems 3.1 and 3.2] for details. Paper [10] studied the existence of solutions of BVP (1.2) by using the critical point theory and monotone operator theory. We comment that none of these papers studied the existence of sign-changing solutions. One of the goals of this work is to study sign-changing solutions of BVPs (1.1) and (1.2).

In this thesis, we prove some new existence criteria for multiple nontrivial solutions of BVPs (1.1) and (1.2). We ﬁrst establish an equivalent variational structure for BVP (1.1). During the process, we derive a symmetric positive deﬁnite matrix M, deﬁned by (2.19) below, whose eigenvalues are exactly eigenvalues of a linear BVP. See Remark 2.0.3 in Chapter 2 for details. The smallest and largest eigenvalues of the matrix M are used in the statements and proofs of our theorems. Spectral properties of several BVPs for the linear discrete beam equation have been studied by Ji and Yang in [12, 13, 14]. In our ﬁrst existence result (Theorem 3.1.1) for BVP (1.1), we utilize variational approaches, combined with the classic mountain pass lemma, to show that BVP (1.1) has at least two nontrivial solutions. Then, by the positivity of the associated Green’s function (see Remark 2.0.5), we further establish sufﬁcient conditions to guarantee that the two nontrivial solutions are sign-changing. In our second existence result (Theorem 4.1.1) for BVP (1.1), we combine variational methods with the theory of

3

invariant sets of descending ﬂow to show that, under some suitable conditions, BVP (1.1) has at least three nontrivial solutions consisting of one positive, one negative, and one sign-changing solutions. The theory of invariant sets of descending ﬂow was introduced by Liu and Sun in [22] in 2001 and has now become a useful tool in the study of existence theory for nonlinear problems. We refer the reader to [19, 20, 21, 23] for some recent applications of this theory. As applications of Theorems 3.1.1 and 4.1.1, we also obtain several criteria for the existence of multiple nontrivial solutions of BVP (1.2).

The rest of this thesis is organized as follows. Chapter 2 contains some preliminaries. Chapter 3 studies the existence of at least two nontrivial solutions of BVPs (1.1) and (1.2) and the existence of at least three nontrivial solutions of the problems are investigated in Chapter 4.

4

CHAPTER 2 PRELIMINARY RESULTS

In this chapter, we ﬁrst obtain the equivalent variational structure for BVP (1.1). Deﬁne a set X of functions by

X = u : [a − 1, b + 3]Z → R | u(a) = ∆2u(a − 1) = 0, u(b + 2) = ∆2u(b + 1) = 0 . (2.1)

Then, X is a vector space with au + bv = {au(k) + bv(k)} for any u, v ∈ X and a, b ∈ R. Moreover, X is a b − a + 1 dimensional Banach space equipped with the norm

u=

b+1

1/2

∑ (u(k))2

k=a+1

for any u ∈ X.

Deﬁne the functionals Φ, Ψ, I : X → R by

∑ ∑ ∑ Φ(u) = 1 b+2 (∆2u(k − 2))2 + 1 α b+2 (∆u(k − 1))2 + 1 β b+1 u2(k),

2 k=a+2

2 k=a+1

2 k=a+1

(2.2)

b+1

Ψ(u) = ∑ F(k, u(k)), k=a+1

and

I(u) = Φ(u) − Ψ(u),

(2.3)

where u ∈ X and

x

F(t, x) = f (k, s)ds.

(2.4)

0

5

Then, Φ, Ψ, I are well deﬁned and continuously differentiable whose derivatives are the linear functionals Φ (u), Ψ (u), and I (u) given by

b+2

b+2

b+1

∑ ∑ ∑ Φ (u)(v) =

∆2u(k − 2)∆2v(k − 2) + α

∆u(k − 1)∆v(k − 1) + β

u(k)v(k),

k=a+2

k=a+1

k=a+1

b+1

Ψ (u)(v) = ∑ f (k, u(k))v(k), k=a+1

and

b+2

b+2

∑ ∑ I (u)(v) =

∆2u(k − 2)∆2v(k − 2) + α

∆u(k − 1)∆v(k − 1)

k=a+2

k=a+1

b+1

b+1

∑ ∑ +β

u(k)v(k) −

f (k, u(k))v(k)

k=a+1

k=a+1

(2.5)

for any u, v ∈ X. Using the summation by parts formula

n

n

∑ ∑ fk∆gk = fn+1gn+1 − fmgm − gk+1∆ fk,

k=m

k=m

(2.6)

we can prove the following lemma. Lemma 2.0.1For any u, v ∈ X, we have

b+2

b+1

∑ ∑ ∆2u(k − 2)∆2v(k − 2) =

∆4u(k − 2)v(k)

k=a+2

k=a+1

(2.7)

and

b+2

b+1

∑ ∑ ∆u(k − 1)∆v(k − 1) = −

∆2u(k − 1)v(k).

k=a+1

k=a+1

Proof. For any u, v ∈ X, we have

(2.8)

∆2u(a − 1) = ∆2u(b + 1) = v(a) = v(b + 2) = 0.

(2.9)

6

By Danielle Layne

Lingju Kong Professor and Graduate Coordinator (Major Advisor)

Roger Nichols Associate Professor (Committee Member)

John R. Graef Professor (Committee Member)

Jin Wang Professor and UNUM Chair of Excellence in Applied Mathematics (Committee Member)

Nontrivial solutions for nonlinear discrete boundary value problems of the fourth order

By Danielle Layne

A Thesis Submitted to the University of Tennessee at Chattanooga in Partial

Fulﬁllment of the Requirements of the Degree of Master of Science: Mathematics

The University of Tennessee at Chattanooga Chattanooga, Tennessee May 2021

ii

ABSTRACT We study the existence of multiple nontrivial solutions for two nonlinear fourth order discrete boundary value problems. We ﬁrst establish criteria for the existence of at least two nontrivial solutions of the problems and obtain conditions to guarantee that the two solutions are sign-changing. Under some appropriate assumptions, we further prove that the problems have at least three nontrivial solutions, which are respectively positive, negative, and sign-changing. We include two examples to illustrate the applicability of our results. Our theorems are proved by employing variational approaches, combined with the classic mountain pass lemma and a result from the theory of invariant sets of descending ﬂow.

iii

TABLE OF CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 PRELIMINARY RESULTS . . . . . . . . . . . . . . . . . . . . . . 5 3 EXISTENCE OF TWO NONTRIVIAL SOLUTIONS . . . . . . . . . . . 16

3.1 MAIN RESULTS . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 PROOFS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 EXISTENCE OF THREE NONTRIVIAL SOLUTIONS. . . . . . . . . . . 23 4.1 MAIN RESULTS . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 PROOFS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

iv

CHAPTER 1 INTRODUCTION

It is well known that nonlinear difference equations of higher order appear naturally as discrete analogue and as numerical solutions of differential equations. The applications of such equations have been well documented in [16, 17]. In recent years, the existence of solutions of boundary value problems (BVPs) of difference equations, with various boundary conditions (BCs), has received increasing attention from many researchers. In this thesis, let [c, d]Z = {z ∈ Z | c ≤ z ≤ d}, where c, d ∈ Z with c ≤ d. We are concerned with the existence of multiple nontrivial solutions of the BVP for the discrete beam equation

∆4u(k − 2) − α∆2u(k − 1) + β u(k)

=

f (k, u(k)),

k

∈

[a + 1, b + 1]Z,

(1.1)

u(a)

=

∆2u(a − 1)

=

0,

u(b + 2)

=

∆2u(b + 1)

=

0,

where α, β ∈ [0, ∞), a, b ∈ Z with b ≥ a, f : [a + 1, b + 1]Z × R → R is continuous, and ∆ is the forward difference operator deﬁned by ∆u(k) = u(k + 1) − u(k) and ∆nu(k) = ∆(∆n−1u(k)). By a solution of BVP (1.1), we mean a function u : [a − 1, b + 3]Z → R such that u satisﬁes both the equation and the BCs in (1.1). If u(k) > 0 for all k ∈ [a + 1, b + 1]Z, then u is called a positive solution; if u(k) < 0 for all k ∈ [a + 1, b + 1]Z, then u is said to be a negative solution; and if u(k) changes signs on [a + 1, b + 1]Z, then u is called a sign-changing solution. We also obtain existence criteria for the BVP

∆4u(k − 2) − α∆2u(k − 1) + β u(k)

=

λ

f (k, u(k)),

k

∈

[a + 1, b + 1]Z,

(1.2)

u(a)

=

∆2u(a − 1)

=

0,

u(b + 2)

=

∆2u(b + 1)

=

0,

where λ is a positive parameter.

1

BVP (1.1) can be regarded as a discrete analogue of the beam problem

u(4)(t) − αu

(t) + β u(t) =

f (t, u(t)),

t ∈ (0, 1),

(1.3)

u(0) = u

(0) = 0,

u(1) = u (1) = 0,

where the BCs correspond to both ends of the beam being hinged when there is no bending moment. The equation in BVP (1.3) is often referred to as the beam equation since it describes the deﬂection of a beam under a certain force. BVP (1.3) and a number of its variations have been investigated by many authors. A small sample of the work can be found in, for example, [2, 5, 27, 28] and the included references.

The existence of solutions of discrete BVPs, with various BCs, of the fourth order has been extensively studied in the literature. The reader is refered to [1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 24, 25, 26, 29, 31] for some work on this subject. In particular, paper [8] studied the existence of three solutions of the BVP

∆4u(k − 2) − α∆2u(k − 1) + β u(k)

=

λ

f (k, u(k)),

k

∈

[1, T ]Z,

(1.4)

u(0)

=

∆u(−1)

=

∆2u(T )

=

0,

∆3u(T

− 1) − α∆u(T )

=

µ g(u(T

+ 1)),

where T ≥ 2 is an integer, α, β , λ , µ ∈ R are parameters, f ∈ C([1, T ]Z × R, R), and g ∈ C(R, R); while the existence of inﬁnitely many solutions of BVP (1.4) was considered in [7]. The existence theorems in [7, 8] give the existence of solutions for the parameters λ and µ in different intervals. Paper [6] investigated the existence of two solutions of the BVP

∆4u(k − 2) − α∆2u(k − 1) + β u(k)

=

f (k, u(k)),

k

∈

[1, T ]Z,

u(−1)

=

∆u(−1)

=

0,

u(T

+ 1)

=

∆2u(T )

=

0.

Variational methods and critical point theory were used in [6, 7, 8] to prove the existence results. By using ﬁxed point theory, paper [1] obtained a number of criteria for the existence of positive

2

solutions of BVP (1.2) with β = 0, i.e., the problem

∆4u(k − 2) − α∆2u(k − 1)

=

λ

f (k, u(k)),

k

∈

[a + 1, b + 1]Z,

(1.5)

u(a)

=

∆2u(a − 1)

=

0,

u(b + 2)

=

∆2u(b + 1)

=

0.

Later, paper [9] studied the following slightly more general version of BVP (1.5)

∆4u(t − 2) − β ∆2u(t − 1)

=

λ

[ f (t, u(t), u(t)) + r(t, u(t))] ,

t

∈

[a + 1, b − 1]Z,

(1.6)

u(a)

=

∆2u(a − 1)

=

0,

u(b)

=

∆2u(b − 1)

=

0,

where f : [a + 1, b − 1]Z × [0, ∞) × (0, ∞) → [0, ∞) and r : [a + 1, b − 1]Z × [0, ∞) → [0, ∞) are continuous. By applying some results from mixed monotone operator theory, not only was the existence and uniqueness of positive solutions of BVP (1.6) obtained , but also the dependence of positive solutions on the parameter λ was discussed. Moreover, two sequences are constructed in such a way so that they converge uniformly to the unique positive solution of the problem. See [9, Theorems 3.1 and 3.2] for details. Paper [10] studied the existence of solutions of BVP (1.2) by using the critical point theory and monotone operator theory. We comment that none of these papers studied the existence of sign-changing solutions. One of the goals of this work is to study sign-changing solutions of BVPs (1.1) and (1.2).

In this thesis, we prove some new existence criteria for multiple nontrivial solutions of BVPs (1.1) and (1.2). We ﬁrst establish an equivalent variational structure for BVP (1.1). During the process, we derive a symmetric positive deﬁnite matrix M, deﬁned by (2.19) below, whose eigenvalues are exactly eigenvalues of a linear BVP. See Remark 2.0.3 in Chapter 2 for details. The smallest and largest eigenvalues of the matrix M are used in the statements and proofs of our theorems. Spectral properties of several BVPs for the linear discrete beam equation have been studied by Ji and Yang in [12, 13, 14]. In our ﬁrst existence result (Theorem 3.1.1) for BVP (1.1), we utilize variational approaches, combined with the classic mountain pass lemma, to show that BVP (1.1) has at least two nontrivial solutions. Then, by the positivity of the associated Green’s function (see Remark 2.0.5), we further establish sufﬁcient conditions to guarantee that the two nontrivial solutions are sign-changing. In our second existence result (Theorem 4.1.1) for BVP (1.1), we combine variational methods with the theory of

3

invariant sets of descending ﬂow to show that, under some suitable conditions, BVP (1.1) has at least three nontrivial solutions consisting of one positive, one negative, and one sign-changing solutions. The theory of invariant sets of descending ﬂow was introduced by Liu and Sun in [22] in 2001 and has now become a useful tool in the study of existence theory for nonlinear problems. We refer the reader to [19, 20, 21, 23] for some recent applications of this theory. As applications of Theorems 3.1.1 and 4.1.1, we also obtain several criteria for the existence of multiple nontrivial solutions of BVP (1.2).

The rest of this thesis is organized as follows. Chapter 2 contains some preliminaries. Chapter 3 studies the existence of at least two nontrivial solutions of BVPs (1.1) and (1.2) and the existence of at least three nontrivial solutions of the problems are investigated in Chapter 4.

4

CHAPTER 2 PRELIMINARY RESULTS

In this chapter, we ﬁrst obtain the equivalent variational structure for BVP (1.1). Deﬁne a set X of functions by

X = u : [a − 1, b + 3]Z → R | u(a) = ∆2u(a − 1) = 0, u(b + 2) = ∆2u(b + 1) = 0 . (2.1)

Then, X is a vector space with au + bv = {au(k) + bv(k)} for any u, v ∈ X and a, b ∈ R. Moreover, X is a b − a + 1 dimensional Banach space equipped with the norm

u=

b+1

1/2

∑ (u(k))2

k=a+1

for any u ∈ X.

Deﬁne the functionals Φ, Ψ, I : X → R by

∑ ∑ ∑ Φ(u) = 1 b+2 (∆2u(k − 2))2 + 1 α b+2 (∆u(k − 1))2 + 1 β b+1 u2(k),

2 k=a+2

2 k=a+1

2 k=a+1

(2.2)

b+1

Ψ(u) = ∑ F(k, u(k)), k=a+1

and

I(u) = Φ(u) − Ψ(u),

(2.3)

where u ∈ X and

x

F(t, x) = f (k, s)ds.

(2.4)

0

5

Then, Φ, Ψ, I are well deﬁned and continuously differentiable whose derivatives are the linear functionals Φ (u), Ψ (u), and I (u) given by

b+2

b+2

b+1

∑ ∑ ∑ Φ (u)(v) =

∆2u(k − 2)∆2v(k − 2) + α

∆u(k − 1)∆v(k − 1) + β

u(k)v(k),

k=a+2

k=a+1

k=a+1

b+1

Ψ (u)(v) = ∑ f (k, u(k))v(k), k=a+1

and

b+2

b+2

∑ ∑ I (u)(v) =

∆2u(k − 2)∆2v(k − 2) + α

∆u(k − 1)∆v(k − 1)

k=a+2

k=a+1

b+1

b+1

∑ ∑ +β

u(k)v(k) −

f (k, u(k))v(k)

k=a+1

k=a+1

(2.5)

for any u, v ∈ X. Using the summation by parts formula

n

n

∑ ∑ fk∆gk = fn+1gn+1 − fmgm − gk+1∆ fk,

k=m

k=m

(2.6)

we can prove the following lemma. Lemma 2.0.1For any u, v ∈ X, we have

b+2

b+1

∑ ∑ ∆2u(k − 2)∆2v(k − 2) =

∆4u(k − 2)v(k)

k=a+2

k=a+1

(2.7)

and

b+2

b+1

∑ ∑ ∆u(k − 1)∆v(k − 1) = −

∆2u(k − 1)v(k).

k=a+1

k=a+1

Proof. For any u, v ∈ X, we have

(2.8)

∆2u(a − 1) = ∆2u(b + 1) = v(a) = v(b + 2) = 0.

(2.9)

6