On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation

We discuss the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Well-known Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. +e simulation of an example is also given to show the applicability of our results.

Positive solution of fractional ODEs have already discussed in [13][14][15]. However, until now research on existence of positive solution for fractional functional differential equations (FFDEs) is rare. Research in different fields including engineering, physics, and biosciences have proved that numerous system structures explain more exactly with the help of FDEs [16][17][18][19][20][21] and similarly FDEs with delay [22][23][24][25][26][27] are more accurate to illustrate the real world problems compared to FDEs without delay. So, in [28], Miller and Ross mentioned that this field has been touched by almost all fields of engineering and science.
Idea of Ulam stability started in 1940, when during the talk at Wisconsin University Ulam posed a problem that "When can we assert that an approximate solution of a functional equation can be approximated by a solution of the corresponding equation?" (for more details, see [29]). After one year, Hyers first gave the answer to the Ulam's question [30] in case of Banach spaces. After that, stability of this type is called the Ulam-Hyers stability. Rassias [31] in 1978 provided an outstanding generalization of the Ulam-Hyers stability of mappings by considering variables.
Usually, the discussion of the existence and uniqueness of a solution for nonlinear FDEs normally fixed point theory has been used [16,32,33]. Motivated by these and [34][35][36], in this work, we have discussed existence and uniqueness of solution also after applying some sufficient conditions, obtained positive solution, and at the end established local generalized Ulam-Hyers stability and local generalized Ulam-Hyers-Rassias stability and presented stability results graphically for the class of nonlinear FDEs with delay given by c D

Preliminaries
Let Q be a cone in a real Banach space X and partial order ≤ introduced by Q in X is as Definition 1. Let u, v ∈ X be the order interval 〈u, v〉 defined as implies that for any (t, x) and (t, y) ∈ I × C.
Definition 3 (see [11]). e fractional integral of order c for a function g with lower limit 0 can be defined as where Γ is the gamma function and right-hand side of upper equality is defined pointwise on R + .
Definition 4 (see [11]). e left Caputo fractional derivative of order c is given by where n � [c] + 1 ([c] stands for the bracket function of c ).
Lemma 1 (Leray Schauder fixed point theorem, see [11]). Let S r be a nonempty, closed, bounded, and convex subset of Banach space X and Q: S r ⟶ S r is a compact and continuous map; then, Q has a fixed point in S r .
where Q is a cone of space of partial order X, F: U ⟶ X be nondecreasing. If there exists

Main Results
In this section, we have discussed existence and uniqueness of solution and some conditions for positive solution of equation (1). Consider X � C[− τ, T] is a Banach space endowed with the supremum norm and cone Q is defined as and we can easily get that So, x 0 � ψ and for every w ∈ C(I, R + ) with w(0) � 0, and we defined w as Let x(·) satisfy equation (9), and we can decompose z(.) as z(t) � w(t) + x(t), for 0 ≤ t ≤ T so that z t � w t + x t , for every 0 ≤ t ≤ T, where w(t) is such that |w(t)| < 1, ∀t ∈ I and the function w(·) satisfies Let the operator P: H ⟶ H be defined by Mathematical Problems in Engineering Before proving main results, we introduce the following conditions: (C1) Let us take f, g: where function g is nondecreasing and |g(t, z t )| ≤ L 1 and f and g satisfy Lipchitz condition that is, and there exists nonnegative constants c 1 and c 2 such that (C2) If G ⊂ H be bounded, f is nondecreasing and where Define the operator P by Hence, PG is bounded. Now, we show that PG is equicontinuous, and the proof is divided into three possible steps.
Step 1: for every w ∈ G and t 1 , t 2 ∈ I such that t 2 > t 1 , then

Mathematical Problems in Engineering 3
As t 1 ⟶ t 2 , the right-hand side of above inequality tends to zero.
Step 2: for every w ∈ G, Step 3: for every w ∈ G, ∀ ϵ > 0, and t 1 , Hence, PG is equicontinuous. So, PG is compact by Arzela-Ascoli eorem. □ Theorem 3. Suppose that C1 and C2 hold. Let f, g: Proof. Consider the operator P: H ⟶ H defined as We will show that P is a contraction. Let w 1 , w 2 ∈ C(I, R + ), then we obtain the following sequence of inequalities: erefore, So, P is a contraction, and hence by Banach's contraction principle P has a unique fixed point.
e function m ∈ X is said to be a lower solution of equation (1) and the function n ∈ X is said to be an upper solution of equation (1) If inequalities are strict, then m(t) and n(t) are strict lower and upper solutions. Proof. We only take fixed point operator P. By eorem 2, P is completely continuous and obviously by equation (16), u 0 and v 0 are lower and upper solutions of P, respectively. By (H1), w 1 , w 2 ∈ H, w 1 ≤ w 2 , we have So, the operator P is nondecreasing. Also, we have As m ∈ X is a lower solution, therefore we can say that Pm(0) ≥ m(0), similarly Pn(0) ≤ n(0) from the definition of upper solution of P.
Hence, P: 〈m 0 , n 0 〉 ⟶ 〈m 0 , n 0 〉 is a completely continuous operator. Since Q is a normal cone, eorem 1 implies that P has a fixed point w ∈ 〈m 0 , n 0 〉. Proof. Consider the equation Obviously, equation which is a lower solution of equation (1). Similarly, consider the equation We know that n(t) � n 0 − g(t 0 , ψ) + g(t, n t ) + (Mt c /Γ(c + 1)), t ∈ I is an upper solution of equation (1) and n(t) ≥ m(t), so the above definition proves that equation (1) has minimum one positive solution.
For getting another result for a positive solution of equation (1), now we take the more general case of (1) and find the existence of a positive solution for the equations: Mathematical Problems in Engineering 5 Let P: H ⟶ H be an operator defined as Since Here, δ < ((R − L 1 )Γ(c + 1)/N(](0) + R)) 1/c .

Stability
In this section, we will discuss local generalized Ulam-Hyers stability and local generalized Ulam-Hyers-Rassias stability for a class of fractional neutral differential equations. For the case I � [0, T] with f, g: I × C ⟶ R + are continuous functions on a closed interval or more generally compact sets, then they are bounded so we can replace the supremum by the maximum. In this case, norm is also called the maximum norm. Let X � x ∈ C[− τ, 1] { } be a Banach space with ‖x‖ � max t∈I |x(t)|, where τ is a nonnegative real number and z t � z(t + θ); − τ ≤ θ ≤ 0. We consider the following differential equation: where c D c 0 is a Caputo derivative with c ∈ (0, 1) and ψ ∈ C([− τ, 0], R + ). We focus on the following inequalities: Definition 6 (see [36]). Equation (36) Definition 7 (see [36]). Equation (36)

Remark 1. A solution of differential equation is stable (asymptotically stable) if it attracts all other solutions with sufficiently close initial values. On the contrary, in
Hyers-Ulam stability, we compare solution of given differential equation with the solution of differential inequality. We say solution of differential equation is stable if it stays close to solution of differential inequality. Hyers-Ulam stability may not imply the asymptotic stability. (38) if and only if there exists h ∈ C(I, R) such that , T], R) be a solution of inequality (37); then, v is a solution of the following inequality: 6 Mathematical Problems in Engineering
By using the same process as in Section 3, i.e., let x(·): [− τ, T] ⟶ R + be the function defined by erefore, x 0 � ψ and for every v ∈ C(I, Let x(·) satisfies equation (42). We can decompose u(.) By following the conditions in Section 3, we get from If v ∈ C([− τ, T], R) is a solution of inequality (38) then v is a solution of the following inequality: Before stating stability results, let us take condition (H4) as (H4) c 2 + (T c /Γ(c + 1))(c 1 + L) < 1
(b) f and g satisfy Lipchitz conditions: where c 1 and c 2 are nonnegative real numbers holds, then

Mathematical Problems in Engineering
We can see that ‖v(t) − w(t)‖ � 0, for t ∈ [− τ, 0]. For Proof. Following the same steps as in eorem 7, we can find the result, i.e., here we will obtain erefore,

Example
In this section, we present an example to explain the applicability of main results.

⎧ ⎨ ⎩
(60) One can easily prove that this problem has a unique fixed point and that it has a positive solution, i.e., Define the operator P as

Mathematical Problems in Engineering
For stability, we take (60) as and the inequalities To prove that equation (63) Applying I 1/2 0 + on both sides, we obtain Here, we take unique solution x(t) of equation (63) as By further simplification, we obtain Hence, (63) is local generalized Ulam-Hyers stable.
Remark 4. If we replace equation (66) by the inequality, By repeating the same process as in the above example, one can easily verify the main results of eorem 8.

Graphical Presentation
For graphical representation of the solution of the problem given in equation (60), we adopted Adams-Bashforth-Moulton scheme [38] to obtain the numerical solution for this fractional differential equation. To analyze the effect and contribution of time delayed factor, the modified predictor-corrector scheme [39] is incorporated for simulation. To check and demonstrate the stability of the consider model, graphical representation of the solution with different variations of the time delay factor along with other parameters is made. From the numerical results, we could note the UH-stability of the system with varying orders and delays. With the higher orders, the system will achieve the Ulam-Hyers stability more rapidly. It also holds

Conclusions
We present some new results about the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Leray-Schauder theorem, Arzela-Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We also establish local generalized Ulam-Hyers stability and local generalized Ulam-Hyers-Rassias stability for the same class of nonlinear fractional neutral differential equations. e simulation of an example is also given to show the applicability of our results. e current concepts have significant applications since it means that if we are studying local generalized Hyers-Ulam-Rassias stable (or Hyers-Ulam stable) system then one does not have to reach the exact solution. We just need to get a function which satisfies a suitable approximation inequality. In other words, local generalized Hyers-Ulam-Rassias stability (or Hyers-Ulam stability) guarantees that there exists a close exact solution. is is altogether useful in many applications where finding the exact solution is quite difficult such as optimization, numerical analysis, biology, and economics. It also helps, if the stochastic effects are small, to use deterministic model to approximate a stochastic one.

Data Availability
e data used to support the findings of this study are included within the article.