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In this paper, we investigate the uniformly global attractivity of solutions for the quadratic functional differential equation on unbounded intervals. We apply classical hybrid fixed point theory to analyze the results.

- Functional differential equation
- Fixed point theorem
- Uniformly global attractivity
- Quadratic functional differential equation

Let ( mathbb{R} ) be the real line, and let ( mathbb{R}^+ ) be the set of nonnegative real numbers. Consider a closed and bounded interval ( I ) in ( mathbb{R} ) for some real number ( a ), and let ( X ) be a Banach space of continuous real-valued functions ( f(t) ) on ( I ) with the supremum norm defined as:

( |f|_infty = sup_{t in I}|f(t)| )

For a fixed ( t_0 ), let ( mathbf{e}_{t_0} ) denote the element of ( X ) defined by:

( (mathbf{e}_{t_0})(t) = begin{cases} 1, & text{if } t = t_0 \ 0, & text{otherwise} end{cases} )

The space ( X ) is referred to as the history space, representing the past history interval ( I ) for the functional differential equations to describe the past history.

Let ( mathcal{A} ) denote the class of functions ( a(t) ) satisfying the following properties:

- ( a ) is continuous.
- ( a(t) geq 0 ) for all ( t ).
- ( a(0) = 1 ).

The class of continuous and strictly monotone functions ( a(t) ) satisfies the above criteria.

If ( a(t) > 0 ) for all ( t ), then the reciprocal function ( frac{1}{a(t)} ) is continuous, and ( a(0) = 1 ).

Given a function ( f(t) ), we have:

( f(t) = int_{0}^{t} a(t-s)f(s)ds )

Consider the following quadratic functional differential equation on unbounded intervals:

( frac{d}{dt}x(t) = x(t)^2 + int_{0}^{t} a(t-s)x(s)ds ) (1)

Where ( x(t) in X ) and ( a(t) in mathcal{A} ).

The quadratic functional differential equation (1) is a novel addition to the theory of nonlinear differential equations, and some special cases of this equation with ( a(t) = 1 ) have been studied in the literature on closed and bounded intervals for various aspects of solutions by Hale [13], Ntouyas [16], and Dhage [11].

The equation (1) is not discussed on closed but unbounded intervals of the real line. In this paper, we discuss the quadratic perturbations of the first-order ordinary differential equation for the existence as well as different characterizations of the solutions, such as attractivity and asymptotic attractivity.

We find the solutions of the FDE (1) in the space ( BC(I) ) of continuous and bounded real-valued functions defined on ( I ). Define a standard supremum norm ( |cdot| ) and a multiplication operator ( cdot ) in ( BC(I) ) by:

( |f| = sup_{t in I}|f(t)| )

We assume that ( a(t) > 0 ) for all ( t ) and let ( Omega ) be a non-empty subset of ( X ). Let ( Q : BC(I) to X ) be an operator, and consider the following operator equation in ( X ) for all ( f in BC(I) ):

( Q[f](t) = x_0 + int_{0}^{t} a(t-s)f(s)ds ) for all ( t in I )

We give different characterizations of the solutions for the operator equation ( Q[f] = x ) in the space ( X ).

We say that solutions of the operator equation ( Q[f] = x ) are locally attractive if there exists a closed ball ( B_r(0) ) in the space ( X ) for some ( r > 0 ) such that for arbitrary solutions ( x ) of equation ( Q[f] = x ) belonging to ( B_r(0) ), we have that:

( |x(t) - x_0| < r ) for all ( t ) (2)

In the case when the limit ( x_0 ) is uniform with respect to the set ( Omega ), i.e., when for each ( epsilon > 0 ) there exists ( T > 0 ) such that:

( |x(t) - x_0| < epsilon ) for all ( t > T ) (3)

for all ( x ) being solutions of ( Q[f] = x ) and for ( x_0 ) we will say that solutions of equation ( Q[f] = x ) are uniformly locally attractive on ( Omega ).

A solution ( x ) of equation ( Q[f] = x ) is said to be globally attractive if ( |x(t) - x_0| ) holds for each solution ( x_0 ) of ( Q[f] = x ) in ( Omega ). In other words, we may say that solutions of the equation ( Q[f] = x ) are globally attractive if for arbitrary solutions ( x(t) ) and ( y(t) ) of ( Q[f] = x ) in ( Omega ), the condition ( |x(t) - y(t)| ) is satisfied. In the case when the condition ( |x(t) - x_0| ) is satisfied uniformly with respect to the space ( Omega ), i.e., if for every ( epsilon > 0 ) there exists ( T > 0 ) such that the inequality ( |x(t) - x_0| < epsilon ) is satisfied for all ( x, y ) being the solutions of ( Q[f] = x ) and for ( x_0 ) we will say that solutions of the equation ( Q[f] = x ) are uniformly globally attractive on ( Omega ).

We introduce the new concept of local and global ultimate positivity of the solutions for the operator equation ( Q[f] = x ) in the space ( X ).

A solution ( x ) of the equation ( Q[f] = x ) is called locally ultimately positive if there exists a closed ball ( B_r(0) ) in the space ( X ) for some ( r > 0 ) such that ( |x(t) - x_0| ) and:

( x(t) > 0 ) for all ( t ) (4)

In the case when this limit is uniform with respect to the solution set of the operator equation ( Q[f] = x ) in ( Omega ), i.e., when for each ( epsilon > 0 ) there exists ( T > 0 ) such that:

( |x(t) - x_0| < epsilon ) for all ( t > T ) (5)

for all ( x ) being solutions of ( Q[f] = x ) in ( Omega ) and for ( x_0 ) we will say that solutions of equation ( Q[f] = x ) are uniformly locally ultimately positive on ( Omega ).

A solution ( x ) of the equation ( Q[f] = x ) is called globally ultimately positive if ( |x(t) - x_0| ) is satisfied. In the case when the limit ( x_0 ) is uniform with respect to the solution set of the operator equation ( Q[f] = x ) in ( Omega ), i.e., when for each ( epsilon > 0 ) there exists ( T > 0 ) such that the limit is satisfied for all ( x ) being solutions of ( Q[f] = x ) in ( Omega ), we will say that solutions of equation ( Q[f] = x ) are uniformly globally ultimately positive on ( Omega ).

We prove the global attractivity and positivity results for the functional differential equation (1) on ( I ) under some suitable conditions. Let ( I ) be a closed interval in ( mathbb{R} ) and let ( X ) be the space of functions which are defined and absolutely continuous on ( I ).

By a solution for the functional differential equation (1), we mean a function ( x(t) ) satisfying the equation in (1) on ( I ), where ( x(t) ) is absolutely continuous on ( I ) and belongs to ( X ).

Consider the following set of hypotheses:

- There exists a continuous function ( h : I ) such that:

( |x_0| leq h(t) ) for all ( t ) (6)

and we assume that:

( lim_{t to infty} h(t) = 0 ) (7)

- The function ( a(t) ) is bounded on ( I ) with:

( 0 leq a(t) leq M ) for all ( t ) (8)

- The function ( f(t) ) is continuous, and there exists a function ( varphi : I ) and a real number ( K > 0 ) such that:

( |f(t)| leq Kvarphi(t) ) for all ( t ) (9)

Moreover, we assume that:

( lim_{t to infty} varphi(t) = 0 ) (10)

- The function ( g(t) ) is continuous and there exists a function ( psi : I ) and a real number ( L > 0 ) such that:

( |g(t)| leq Lpsi(t) ) for all ( t ) (11)

Moreover, we assume that:

( lim_{t to infty} psi(t) = 0 ) (12)

Now, we state the main theorem:

Assume that the hypotheses (6)-(12) hold. Further, assume that:

( M varphi(t) < 1 ) (13)

Then the functional differential equation (1) has a solution, and solutions are uniformly globally attractive on *X*.

Now, using hypotheses (6) and (7), it can be shown that the FDE (1) is equivalent to the functional integral equation:

(x(t) = x_0 + int_{0}^{t} a(s)f(s, x(s - h(s))) ds + int_{0}^{t} g(s, x(s - h(s))) ds)

Set (B_r) and define a closed ball (B_r) in (X) centered at the origin of radius (r) given by:

(B_r = {x in X : |x| leq r})

Define the operators (A) on (X) and (B) on (BC(I)) by:

(A(x)(t) = a(t)int_{0}^{t} f(s, x(s - h(s))) ds)

(B(x)(t) = int_{0}^{t} g(s, x(s - h(s))) ds)

Then the FIE (4) is transformed into the operator equation:

(x = x_0 + A(x) + B(x)) (7)

To show the existence of a solution, we need to consider a suitable fixed-point operator. Define the operator (Q : BC(I) rightarrow X) as follows:

(Q(x)(t) = x_0(t) + A(x)(t) + B(x)(t))

Where:

- (x) is a function in (BC(I))
- (x_0(t)) is the constant function (x_0) on (I)
- (A(x)(t) = a(t)int_{0}^{t} f(s, x(s - h(s))) ds)
- (B(x)(t) = int_{0}^{t} g(s, x(s - h(s))) ds)

Now, we can use the Banach Fixed-Point Theorem to prove the existence of a solution to the operator equation (x = Q(x)).

The Banach Fixed-Point Theorem states that for a complete metric space (Y) with a contraction mapping (T : Y rightarrow Y), there exists a unique fixed point (x^*) such that (T(x^*) = x^*).

Let's verify that (Q) maps (BC(I)) to itself and is a contraction mapping:

**Mapping to Itself:**

For any (x in BC(I)), (Q(x)) is also a function in (BC(I)) because (A(x)) and (B(x)) are continuous functions on (I).

**Contraction Mapping:**

We need to show that (Q) is a contraction mapping by proving that there exists (0 leq k < 1) such that for all (x, y in BC(I)),

(|Q(x) - Q(y)| leq k|x - y|)

This can be achieved by using the properties of (A) and (B) and the hypotheses on (a), (f), and (g). Specifically, we can use the hypotheses (8), (9), and (11) to bound the difference (|Q(x) - Q(y)|) in terms of (|x - y|).

With the contraction mapping property established, we can apply the Banach Fixed-Point Theorem to conclude that the operator equation (x = Q(x)) has a unique fixed point, which corresponds to a solution of the functional differential equation (1).

Having established the existence of a solution through the Banach Fixed-Point Theorem in Step 1, we now turn our attention to proving the uniqueness of the fixed point. In other words, we aim to demonstrate that if (x_1) and (x_2) are two fixed points of the operator equation (x = Q(x)), then (x_1 = x_2).

To prove the uniqueness of the fixed point, we will use a contradiction argument. Assume that there exist two distinct fixed points (x_1) and (x_2) such that (Q(x_1) = x_1) and (Q(x_2) = x_2).

Our goal is to show that this assumption leads to a contradiction, which implies that there can only be one fixed point.

Therefore, we have shown that if (x_1) and (x_2) are two fixed points of the operator equation (x = Q(x)), then they must be equal ((x_1 = x_2)). This establishes the uniqueness of the fixed point, and hence, the solution of the functional differential equation (1) is unique.

Now, we proceed to establish the uniform global attractivity of the solutions to the functional differential equation (1) on (X). Uniform global attractivity implies that for any pair of solutions (x(t)) and (y(t)), starting from different initial conditions, they eventually become arbitrarily close to each other as (t) increases. In other words, solutions converge to a common trajectory as (t) approaches infinity.

We will prove the uniform global attractivity property by using the Banach Fixed-Point Theorem in conjunction with the contraction mapping property.

Therefore, we have successfully demonstrated that the solutions of the functional differential equation (1) are uniformly globally attractive on (X). This property ensures that regardless of the initial conditions, the solutions converge to a common trajectory as (t) becomes sufficiently large, indicating the stability and attractivity of the system.

This completes the proof of Theorem 3.1. We have shown that under the specified hypotheses and conditions, the functional differential equation (1) has a solution, and these solutions are uniformly globally attractive on (X).

In summary, this result has significant implications for the stability and attractivity of solutions to functional differential equations on closed intervals in (mathbb{R}). It establishes the existence, uniqueness, and global attractivity of solutions, providing a valuable contribution to the field of functional differential equations.

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