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The significance of functional differential equations with maxima lies in real-world problems related to automatic regulation in technical systems. These equations are a special class where the current state depends on the maximum value of an earlier state within a prior time interval. See Mangomedov[21, 22] and Myshkis [23] for more on this subject.

Consider the initial value problem of the first-order functional differential equation (FDE) with a maximum:

$$frac{d}{dt}x(t) = f(t, x_t), quad t in [a, b],$$

where (f) is a continuous function and (x_t) denotes the maximum value of (x) in the interval ([a, t]).

In this paper, unless and until mentioned, let (E) denote a partially ordered real normed linear space with an order relation and norm where usual addition and scalar multiplication by positive real numbers are preserved.

More detailed insights into a partially ordered normed linear space can be found in Heikkila and Lakshmikantham [18] and related references.

Two elements (a) and (b) in (E) are considered comparable if either the relation (a leq b) or (b leq a) holds.

A non-empty subset (C) of (E) is called a chain or totally ordered if all the elements of (C) are comparable.

The regularity of a set (E) signifies that a non-decreasing sequence in set (E) converges to a limit within (E). Conditions guaranteeing the regularity of set (E) can be found in Heikkila and Lakshmikantham [18].

**Definition 6.3.1:** A mapping is called isotone or nondecreasing if it preserves the order relation, i.e., if (x leq y) implies (f(x) leq f(y)).

**Definition 6.3.2:** An operator is called partially continuous at a point if for any given (epsilon > 0) there exists a (delta > 0) such that whenever (x) is comparable to (a) and (|x - a| < delta), then (|f(x) - f(a)| < epsilon).

It is called partially continuous on (E) if it is partially continuous at every point of (E).

**Definition 6.3.3:** A non-empty subset (S) of the partially ordered Banach space (E) is called partially bounded if every chain (C) in (S) is bounded. An operator is called partially bounded if every chain (C) in (T(E)) is bounded. (T(E)) is called uniformly partially bounded if all chains (C) in (T(E)) are bounded by a unique constant.

**Definition 6.3.4:** A non-empty subset (S) of the partially ordered Banach space (E) is called partially compact if every chain (C) in (S) is compact. An operator is called partially compact if every chain (C) in (T(E)) is a relatively compact subset of (E). (T(E)) is called uniformly partially compact if it is uniformly partially bounded and partially compact on (E).

**Definition 6.3.5:** The order relation and the metric (d) on a non-empty set (E) are said to be comparable if a monotonic sequence in (E) converges if and only if its subsequence converges to the same limit. Similarly, given a partially ordered normed linear space, the order relation and the norm are said to be comparable if the metric defined through the norm and the order relation are comparable.

**Definition 6.3.6:** An upper semi-continuous and nondecreasing function is called a (phi)-function provided (lim_{t to infty} phi(t) = infty). A mapping is called partially nonlinear (phi)-Lipschitz if there exists a (phi)-function such that (|f(x) - f(y)| leq phi(|x - y|)) for all comparable elements (x) and (y). If this holds with a Lipschitz constant (k), then it is called partially Lipschitz with a Lipschitz constant (k).

**Definition 6.3.7:** An operator is said to be positive if the range (f(E)) is a positive cone in (E).

**Lemma 6.3.1:** If (f) is a positive operator, then (f(E)) is a positive cone in (E).

**Theorem 6.3.1:** [Dhage[3,5]] Let (E) be a regular partially ordered complete normed linear algebra such that every compact chain of (E) is compatible Banach Space. Let (f), (g), and (h) be three nondecreasing operators satisfying certain conditions...

Furthermore, we introduce definitions and properties related to operators in a partially ordered normed linear space, such as partially bounded, partially compact, uniformly partially bounded, and partially continuous operators. These definitions are fundamental in establishing the conditions for solutions to the functional differential equations.

Additionally, various conditions and functions, like -functions and their properties, are defined and utilized in the subsequent theorems and proofs.

We consider the first-order functional differential equation (FDE) in the function space of continuous real-valued functions defined on (J). We define a norm and the order relation, establishing that (E) is a Banach algebra and partially ordered.

Let (E) be a partially ordered Banach space with a norm and order relation. Then, every partially compact subset (S) of (E) is Banach.

**Definitions:** Lower and upper solutions for the FDE are defined. A set of assumptions (A0)-(C1) are introduced.

Under assumptions (A0)-(C1), if a specific inequality (6.4.4) holds, the FDE (6.2.1) has a solution defined on ([a, b]). The sequence of successive approximations converges monotonically to the solution.

**Step I:** The operators (f) and (g) are shown to be nondecreasing on (E).

**Step II:** The operators (f) and (g) are proven to be partially bounded and partially -Lipschitz on (E).

**Step III:** The operator (h) is established as partially continuous on (E).

**Step IV:** The operator (h) is shown to be uniformly partially compact on (E).

**Step V:** The lower solution (u) satisfies the operator inequality.

**Step VI:** The (phi)-function of the operator satisfies the growth condition.

By satisfying all conditions of Theorem 6.3.1, it can be concluded that the operator equation has a solution, implying the existence of a positive solution to the integral equation and the FDE (6.2.1) defined on ([a, b]). Additionally, the sequence of successive approximations converges monotonically.

**Remark 6.4.2:** The conclusion of Theorem 6.4.1 also holds true with an upper solution instead of a lower solution.

**Remark 6.4.3:** If the FDE has both lower and upper solutions and certain conditions hold, then the corresponding solutions are the minimal and maximal solutions of the FDE in the vector segment of the Banach space.

These conclusions provide insights into the existence and properties of solutions for the given functional differential equation with maxima in the specified interval.

This paper has presented a comprehensive exploration of first-order functional differential equations (FDEs) with maxima in a partially ordered Banach algebra. By blending the concepts of hybrid differential equations and equations with maxima, the paper establishes the existence and numerical aspects of solutions for these specialized FDEs.

The main contributions of this work include:

- The formulation of a step-by-step algorithm for the solution of FDEs with maxima.
- Establishing conditions under which a sequence of successive approximations converges monotonically to the solution of related perturbed differential equations.
- Defining and proving key properties of operators in a partially ordered normed linear space, essential for establishing the existence and convergence of solutions.
- Demonstrating the relationship between FDEs with maxima and nonlinear functional integral equations.

By proving the existence and convergence of solutions, the paper offers valuable insights into the behavior and properties of systems governed by functional differential equations with maxima, contributing to the broader understanding of differential equations in hybrid systems and their real-world applications.

The findings of this paper open avenues for further research in the field of functional differential equations with hybrid conditions and may have implications in diverse areas including automatic regulation of technical systems and dynamic systems modeling.

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