Theorem 14.If 0 < A+B+C+D< 1 and e‡ d, then the equilibrium point x given by (7) of Eq. (1) is global attractor.
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
![Theorem 14.If 0< A+B+ C+D<1 and e+ d, then
the equilibrium point x given by (7) of Eq.(1) is global
attractor.
Proof.We consider the following function
by
F(x, y, z, w) = Ax+By+Cz+ Dw+
(50)
(dy- ez)'
ez)?>bez and
where dy # ez, provided that B(dy -
C(dy- ez)? + bey>0. It is easy to verify the condition (i)
of Theorem 3. Let us now verify the condition (ii) of
Theorem 3 as follows:
[F(x, х, х, х) — х (х— X) —
b
K5x-
X
е —
- -
X
[Ks — 1] (е- d)
S x(e- d) [Ks – 1] – b \
2
e-d
1
(51)
[K5 – 1]
where K5 = (A+B+C+D). Since 0 < K5 < 1 and e+ d,
then we deduce from (51) that
[F(x, х, х, х) — х] (х— 3) <0.
(52)
According to Theorem 3, is global attractor. Thus, the
proof is now completed.O
On combining the two Theorems 4 and 14, we have the
result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e6fa233-173d-4c75-ac91-9f159b3c4543%2F96e4f06c-2544-4e6b-afd4-bffa669d8e72%2F1lyptew_processed.jpeg&w=3840&q=75)
![The objective of this article is to investigate some
qualitative behavior of the solutions of the nonlinear
difference equation
bxn-k
Xn+1 = Axn+ Bxŋ-k+Cxn-1+Dxn-o+
[dxn-k– exn-1]
(1)
n = 0,1,2, ...
where the coefficients A, B, C, D, b, d, e E (0,0), while
k, 1 and o are positive integers. The initial conditions
X-g,..., X_1,..., X_k ..., X_1, Xo are arbitrary positive real
numbers such that k <1< o. Note that the special cases
of Eq. (1) have been studied in [1] when B=C= D=0,
and k= 0,1= 1, b is replaced by – b and in [27] when
B=C= D=0, and k = 0, b is replaced by – b and in
[33] when B = C = D = 0, 1= 0 and in [32] when
A= C= D=0, 1=0, b is replaced by – b.
6.
•.•9
-
6.
|
(7)
[(A+B+C+ D) – 1] (e – d)
Let
us
now
introduce
a
continuous
function
F: (0, 0)* → (0, ) which is defined by
ai
bu
Theorem 3.[12]. Consider the difference equation (2).
Let xe I be an equilibrium point of Eq. (2). Suppose also
that
(i) F
is a nondecreasing function in each of its
arguments.
(ii) The function F satisfies the negative feedback
property
[F (x, X,х, х) — х (х- %) <0 for all х€1-{},
where I is an open interval of real numbers. Then x is
global attractor for all solutions of Eq. (2).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e6fa233-173d-4c75-ac91-9f159b3c4543%2F96e4f06c-2544-4e6b-afd4-bffa669d8e72%2Fapkvba7_processed.png&w=3840&q=75)
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