From which we have (a₁ + a₂) D+ (a3 + α4 + α5) d−(1 − A) (B₁ + ß₂) D² = (1 − A) (B3 + ß4 + 35) Dd (5.38) and (a₁ + a₂)d + (a3 + α4 + α5) D−(1 − A) (B₁ + ß₂) ď² = (1 − A) (B3 +³4 + ß5) Dd (5.39) From (5.38) and (5.39), we obtain Cd-D) {[(a₁ + a₂) - (a3 + a₁ + α5)] − (1 − A) (B₁ + B₂) (d+ D)} = 0. - (5.40)
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Explain the determine green and the information is here
![Case 3. Let the function H(uo, ..., u5) is non-decreasing in uo,u1,u2 and
non-increasing in u3, U4, U5.
Suppose that (d, D) is a solution of the system
H(D,D, D, d, d, d)
аnd
d = H(d, d, d, D, D, D).
D =
Then we get
aD+a2D+azd + a4d + a5d
BịD+ B2D+ B3d + B4d + B5d
a1d + a2d + a3D+a4D+a5D
Bid + Bad + B3D+ B4D+ B;D
D = AD+
аnd d — Ad+
or
D (1 – A) =
(a1 + a2) D+ (a3 + a4 + a5) d
(В1 + В2) D + (Bз + Ва + Bs) d
(а1 + 02) d + (аз + а4 + as) D
(B1 + B2) d+ (B3 + B4 + B5) D
and d(1 – A) =
18
621
A. M. Alotaibi ET AL 604-627
COMPU
ONAL ANALYSIS AND APPLICATIONS, VOL. 26, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC
From which we have
(а1 + aэ) D + (aз + aд t as) d- (1— А) (31 + Bә) D" — (1 — А) (3з + Bа + В5) Dd
(5.38)
and
(ат + 09) d + (aз + ад + a5) D-(1 — А) (31 + B2) d? — (1 — А) (83 + Ba + B5) Dd
(5.39)
From (5.38) and (5.39), we obtain
(а — D) {(aл + a2) — (аз + од +as)] — (1 — А) (B1 + B) (d + D)} —D0.
(5.40)
Since A < 1 and (a3 + a4 + a5) > (a1 + a2), we deduce from (5.40) that
D = d. It follows by Theorem 2, that ỹ of Eq.(1.1) is a global attractor.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7410b99-86dc-4094-add7-8a5727a4f3de%2Feb807076-74ec-487e-81ce-7f2e934e9a4a%2F1w0n8lf_processed.jpeg&w=3840&q=75)
![The main focus of this article is to discuss some qualitative behavior of
the solutions of the nonlinear difference equation
a1Ym-1+a2Ym-2 + a3ym-3+ a4Ym-4 + a5Ym-5
Aym+
B1ym-1 + B2ym-2 + B3Ym-3 + B4Ym-4 + BsYm-5
т %3D 0, 1, 2, ...,
Ym+1 =
(1.1)
where the coefficients A, ai, Bi E (0, 00), i = 1, ..., 5, while the initial condi-
tions y-5,y-4,Y–3,Y-2, y-1, yo are arbitrary positive real numbers. Note that
the special case of Eq.(1.1) has been discussed in [4] when az =
B4
when a4 = B4 = a5 = B5 = 0 and Eq.(1.1) has been discussed in [5] in the
special case when az = B5 = 0.
B3 = a4 =
B5 = 0 and Eq.(1.1) has been studied in [8] in the special case
= a5 =
Theorem 2 ([6). Let H : [a, b]k+1 → [a, b] be a continuous function, where
k is a positive integer, and where [a, b] is an interval of real numbers. Con-
sider the difference equation (1.2). Suppose that H satisfies the following
conditions:
1. For each integer i with1 < i < k+ 1; the function H(z1, z2, ..., Zk+1)
is weakly monotonic in zi for fixed z1, z2, ..., Zi-1, Zi+1, ..., Zk+1•
2. If (d, D) is a solution of the system
d = H(d1, d2, ., de+1) and D= H(D1, D2, .., Dk+1),
then d = D, where for each i = 1, 2,
..., k +1, we set
d
di = {
if F is non – decreasing in zi
if F is non – increasing in zi
D
аnd
{
(D if F is non – decreasing in z;
Di =
if
F is non – increasing in zị.
Then there exists exuctly one equilibrium y of Eq.(1.2), and every solution
of Eq. (1.2) converges to y.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7410b99-86dc-4094-add7-8a5727a4f3de%2Feb807076-74ec-487e-81ce-7f2e934e9a4a%2Fdgsyskf_processed.png&w=3840&q=75)
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