Interpretation on nonlocal neutral functional di ﬀ erential equations with delay

: This work deals with the existence and continuous dependence of an integral solution for neutral integro-di ﬀ erential equations with a nonlocal condition. This result is established by using an integrated resolvent operator under conditions of Lipschitz continuity and uniqueness via the Banach ﬁxed point technique. We also study the existence of a strict solution on reﬂexive and general Banach spaces. In the last section, an example is provided related to this theory.


Introduction
Neutral differential equations are studied by many authors with or without delay, to model many real situations in different fields like population studies, electronics, chemical kinetics and biological science. The below system is used to describe the heat conduction materials in [2].
−∞ e 1 (t 1 − ξ)z(ξ, x)dξ] = d∆z(t 1 ) + t 1 −∞ e 2 (t 1 − ξ)∆z(ξ, x) + f (t 1 , z(·, x)), t 1 ≥ 0, z(t 1 , x) = 0, for x ∈ ∂Ω, where Ω ⊂ R n is open, (t 1 , x) ∈ [0, ∞) × Ω and z(t 1 , x) denotes the heat in x at any time t 1 . Let d > 0; e i : R → R represents the internal energy of fading memory materials. In [7,8] Ezzinbi et al. proved the existence and regularity of solutions of neutral equations by using resolvent operator theory and fixed point theorems. In [3,15,21] the authors proved the existence solution of neutral integro-differential equations by using fractional powers of operators and the Schauder fixed point theorem. Also, in [1,28], the authors proved the existence of solutions of differential equations by using fractional powers of operators under the condition of Krasnoselskii's fixed point theorems. In [24] Murugesu and Suguna proved the existence solution for neutral functional integro-differential equations by using fractional powers of operators and Sadovskii's fixed point theorem. The existence result for integro-differential equations in [9,22,27] was proved by using resolvent operator theory and Monch-Krasnoselskii's and Sadovskii's fixed point theorems. In [4,23] the authors established the existence of a mild solution for neutral differential equations by using Schaefer fixed-point theorem.
The nonlocal initial conditions are more effective, realistic and accurate in the solutions and uniqueness than the classical one proved by many researchers see [6,20]. Recently published [18,19] proves the existence and uniqueness of solutions of functional integro-differential equations with nonlocal conditions; the authors also proved the existence of a strict solution by using an integrated resolvent operator. The main tool for proving the uniqueness and existence of solutions of differential equations by using the Banach fixed point theorem has been established in [12][13][14]. In [26] the authors proved that the mild solution, strong solution and classical solutions obtained by using the semigroup theory of evolution equations also explained the uniqueness of the solution. The semigroup and resolvent operator theories are important methods to find the solutions of integro-differential equations in Banach space(BS), and the authors established integrated semigroup theory in [16]. In recent years, many differential equations have been reformed as integral equations and scholars have proved that the existence of solutions can be obtained via appropriate fixed-point theorems, which is the common technique for proving the existence of solutions of the integral equations. In [11,17], proved the existence solutions of integro-differential equations through the use of resolvent operators with finite delay furthermore, the authors used the integrated resolvent operator in [11].
In the recently published article [29] the authors established the following system For this problem they proved the existence of the solution of nondensely defined neutral equations via the integrated resolvent operator technique. They also proved continuous dependence and differentiability. They assumed that A is a closed linear operator on X and its domain does not equals to X. Motivated by this above-mentioned article, we established the theory for the neutral integrodifferential equations with nonlocal and finite delay. This theory contains the integrated resolvent operator in the proof of the existence of the solution and assumptions of Lipschitz continuity; we also prove the uniqueness by applying Banach fixed-point theory and verified its differentiability.
Regarding this, we have to show the existence of the integral solution of the below system: for t ∈ [0, a] = I, (1.1) In this article, E denotes the BS and D is the closed linear operator on E; its domain D(D) E, which satisfies the Hille-Yosida theorem. Let H(t) be the set of bounded linear operators in E with D(D) ⊂ D(B(t)), t ≥ 0 from D(D) = Y into E. The functions q : I × C → E, h : I × I × C → E and ϕ : I × C × E → E are continuous as specified later. Let C = C([−r, 0]; X) be a set of continuous functions on [−r, 0] in E and φ, g be continuous functions defined on C.
Note that ω belongs to the continuous function C([−r, ∞); E), t ≥ 0; the function ω t ∈ C given that ω t (σ) = ω(t + σ) for σ ∈ [−r, 0]. The general form of (1.1) is an abstract formation of a large number of partial integro-differntial equations, particularly for applications such as electronic circuits, economics, biological sciences, medicine and more. In this article we use the Banach theorem to prove the existence of a solution to the nonlocal system given by Eq (1.1). The existence and uniqueness of the abstract form given by Eq (1.1) have been established in previous articles and by using different approaches this is particularly true for the existence of solutions and valid properties of differential equations which have been established by applying the resolvent operator technique in [5,7,10].
This paper is summarized as follows. In Section 2, we provide the preliminary results and definitions regarding integrated resolvent operator theory. In Section 3, we discuss the existence and uniqueness of the solution and continuous dependence. In Section 4, we prove the differentiability of the solution; in Section 5, we provide an example related to our basic results.

Basic results and definitions
Here, this section includes some basic results and definitions regarding integrated resolvent operators. Let E ba a BS and D be a closed linear operator; 0 ∈ ρ(D) then D −1 exists. Let Next we recall a few definitions and results about the integrated resolvent operators established in [25] for linear nondensely defined integro-differential equations.
Consider the below homogeneous linear integro-differential system: (2.1) Here, the operators D and H(·) are defined already in Eq (2.1). Then the integrated resolvent operator for Eq (2.1) is as follows: A set of operators (Q(t)) t≥0 in L(E) constitute an integrated resolvent operator for Eq (2.1) if it satisfies the following: The operator (Q(t)) t≥0 defined in the above definition is locally Lipschitz continuous (LLC) if ∀a > 0and ∃K a = K(a) > 0 implies the following Consider the following non homogeneous integro-differential system: We follow a previous article see [25] to write the integral solution and strict solution of Eq (2.2) as follows: [25]) Assume that (Q(t)) t≥0 is an LLC integrated resolvent operator of Eq (2.2) with ρ(D) ∅ then, we have the following: (i) If v 0 ∈ D(D) and f ∈ L 1 ([0, +∞); E) then ∃ a unique integral solution v(·) of problem Eq (2.2); then, Here C 1 > 0 is a constant.

Existence result
Here, we have to show the existence of the solution of Eq (1.1). Due to Lemma 2.4, the integral solution of Eq (1.1) with the nonlocal condition is as follows: is an integral solution of the system given by Eq (1.1) if it satisfies the following: To establish the solution of the existence of Eq (1.1), we need the support of the below assumptions: (H1) The function q : I × C → D(D) is Lipschitz continuous; there exists a constant L 1 > 0; then, and ϕ(t, ξ 1 , ν 1 ) ≤ L 2 ( ξ 1 C + ν 1 ) for every ξ 1 , ξ 2 ∈ C, ν 1 , ν 2 ∈ E.
(H3) The map g : C([0, a]; E) → C is Lipschitz continuous and ∃L 3 > 0; then, Theorem 3.3. Let ω 0 ∈ D(D), 0 ∈ ρ(D) satisfy (H1-H4) and ∀φ ∈ C; then, the system given by Eq (1.1) has at least one mild solution on [−r, +∞) provided that Proof. Let a > 0 and C([0, a]; E) is a set of continuous maps from [0, a] into E with the uniform norm topology. We prove this existence by using the Banach fixed point theorem.
We prove that this operator Γ has a fixed point in the closed ball B r = {ω ∈ C([−r, a]; E), ω ≤ r}. Before we prove that Γ is a map on B r , for each ω ∈ B r and t ∈ [−r, 0], we take Γ 1 = φ(t) + g(ω)(t); we have Using the hypotheses, we have It follows from the above two cases that Hence the operator Γ is well defined in B r ; next, we show that Γ is a contractive map on B r . The map Γ is defined on B r as The extensionω : [−r, 0] → E is as follows By using the hypotheses Thus from Eq (3.2), Hence Γ has a fixed point ω(·) and is a unique integral solution of Eq (1.1) on [0, a].
Next we consider the continuous dependence of the solution for Eq (1.1) in the sense of the below theorem: Theorem 3.4. Suppose that the axioms of Theorem 3.3 hold and let u(·), v(·) be solutions of Eq (1.1) with the initial conditions u 0 , v 0 ∈ D(D) respectively; then, the solution of Eq (1.1) has continuous dependence upon initial values, provided that Proof. Let u = u(·), v = v(·) be two solutions of Eq (1.1). For t ∈ [0, a], Thus, Hence by Gronwall's lemma, Thus from Eq (3.4), the integral solution of Eq (1.1) has continuous dependence on the initial conditions.

Existence of a strict solution
Here, we study the strict solution of the problem given by Eq (1.1), by using the integrated resolvent operator theory and under some considerations.
Here L * > 0 is a constant and we prove that Γ has a fixed point on B r 0 . By Theorem 3.3, Γ(B r 0 ) ⊂ B r 0 ; it suffices to show that The extension of the operator solution Γ(ω(t)) is defined by where , Now take I 1 : From I 3 , we note that h(s, ξ,ω ξ )dξ ds, From the estimates of I 1 , I 2 , I 3 , we have where C * ∈ R, and different from L * , ω ∈ B r 0 ; thus, from (H6) where we assume that L * is large enough ≥ Thus Γ has a unique fixed point ω(·) and is an integral solution of Eq (1.1). Further ω(t) is Lipschitzcontinuous on [0, a]; moreover, is also Lipschitz continuous on [0, a] and E is a reflexive BS; hence, by the Radon-Nikodym property, By Lemma 2.4, we have that ω(t)−q(t, ω t ) is differentiable on [0, a] and also a strict solution of Eq (1.1) on [0, a]. Next we consider that E is a general BS; further, we assume the following: (H7) The function q ∈ C 1 (R + × E; Y) and the partial derivatives D 1 q(·, ·), D 2 q(·, ·) are Lipschitzcontinuous with respect to the second variable; ∃L i q > 0; then, (H8) The function ϕ ∈ C 1 (R + × C × E; E) and the partial derivatives D 1 ϕ(·, ·, ·), D 2 ϕ(·, ·, ·) are Lipschitzcontinuous function with respect to second variable; ∃L i ϕ > 0; then, Proof. Let ω(·) be an integral solution of Eq (1.1); see the following system  We shall prove that ω(·) = z(·) on [0, a].   Further, we obtain  Since ω 0 = z 0 putting Eqs (4.5)-(4.7) in Eq (4.4), we have By the values of J 1 , J 2 , J 3 we have Then by the Gronwall lemma, it follows that ω t = z t for all t ∈ [−r, a], which shows that ω(t) is continuously differentiable on [−r, a]; consequently, ω(·) is a strict solution of Eq (1.1).
Further let ω(t)( where L q 1 > 0 is a Lipschitz constant of q that is Lipschitz continuous on its domain C 2 ; also clearly, ϕ is satisfies (H2) with L 2 = L c . On the other hand, under the condition of (A3), for ω 1 , ω 2 ∈ C([−r, 0]; E), which shows that g satisfies (H3). According to Theorems 3.3 and 3.4, we state that the following: Moreover c ∈ C 2 ([0, 1] × R; R); then, it is clear that (H7) and (H8) are hold. If M 1 L q 1 < 1 then by Theorem 4.3, the integral solution of Eq (5.1) becomes a strict solution.

Conclusions
In this work, we obtained the existence results for the system of neutral integro-differential equations given by (1.1) with the nonlocal condition in finite delay situations by using the Banach fixed point theorem. Also, we verified that the integral solution of the system given by (1.1) has continuous dependence with respect to the initial data, and we proved the existence of a strict solution by using integrated resolvent operator theory and Gronwall's lemma. We considered most of the functions in Eq (1.1) to be Lipschitz continuous and then obtained the results. The future work will consider the partial neutral functional integro-differential equations with the initial conditions and we will apply the integrated resolvent operator technique to this system.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.