1 Introduction

Many authors interested in fractional differential equations for many decades. In [1], Hsiang and Chang proved the existence of mild and classical solutions of nonlocal Cauchy problems. Also, show that the linear part generates an analytic compact bounded semigroup, and that the nonlinear part is a Hölder continuous function with respect to the fractional power norm of the linear part. In [2], Mahdy gave a new numerical method for solving a linear system of fractional integrodifferential equations. He conceder the fractional derivative is considered in the Caputo sense to demonstrate the accuracy and applicability of the presented method. In [3], Aly Abdou proposed a new fractional sub-equation method with a fractional complex transform for constructing exact solutions of fractional partial differential equations arising in plasma physics in order to a modified Riemann-Liouville derivative. In [4], N’Guerekata discussed the existence and uniqueness of solutions to the Cauchy problem for the fractional differential equation with non-local conditions as.

In [5], Mophou et al. order to the existence solution of the Cauchy problem for the fractional differential equation with non local conditions in Rⁿ:

In [6], Pei-Luan and Chang-Jin investigated the boundary value problems of fractional order differential equations with non instantaneous impulse. By some fixed-point theorems, the existence solution of mild solution are established. Reference [7], Chang and Ponce established the existence of mild solutions to a multi-term fractional differential equation.

One of the important topics in ocean wave are the characteristics and mechanism of linear and nonlinear structures like solitary waves, cotidal waves, rogue waves, etc. in fluid mediums (including in seas and oceans plasma physics). The moving waves in ocean or shallow water or generally in fluid mediums is very important part in ocean and in science generally. Since the waves can cause disasters, during several decade the scientists study the behaviors of the waves in different aspects. Mathematicians present several linear and nonlinear differential and integrable equations that describe the nonlinear phenomena in shallow water and water in oceans.

Dabas et al. [8], considered a solution for a semilinear problems and proved its existence and uniqueness. Authors in [9, 10] under the assumption of the generator, our discussion focused on the uniqueness and existence of the solution to the following IDE on an analytic semigroup generator ζ, and the maps f, g which are nonlinear with the kernel q, Also, [11–22] introduced some property related to existence of mild solutions for fractional impulsive integro-differential equation. In addition, they focus in improving the techniques to reach the optimal solutions that can predict the phenomena of waves in water or ocean very well.

In, this paper we are order to solving the following impulsive fractional order integrodifferential equation denoted by (IDE) with nonlocal conditions via fractional operators in a Banach space denoted by (BS) X; (1) where We let −ζ Analyzes a semigroup G(z), z ≥ 0, on a BS X on Eq (1), the map q is locally integrable on [z₀, ∞) and real valued, we define the nonlinear maps f and g on [z₀, ∞) × X into X. The function φ is continuous and will be specified later.

The rest of this paper is organized as follows: In Sec. 2, we introduce some definitions and lemmas; In Sec. 3, a set of sufficient condition is required for the local existence of mild solutions for fractional impulsive integrodifferential equation; In Sec. 4, The proof of global existence of a mild solution are stated. In Sec. 5, an example is given to illustrate our main results.

2 Preliminaries

Let X be BS and let the closure of the interval [z₀, T), z₀ < T ≤ ∞ be denoted by I. Suppose that −ζ ∈ X be the infinitesimal-generator of an analytic-semigroup (denoted by IGAS) {G(z), z ≥ 0}. We note that as an analytic semigroup −ζ is the infinitesimal generator and −(ζ + αI) is invertible for α > 0, then −(ζ + αI) is analytic and bounded semigroup. In this way, we can reduce the case in which −ζ is an IGAS to the case in which is bounded semigroup and the generator is invertible. In a resolvent set, ρ(−ζ) is the set of all −ζ without loss of generality 0 ∈ ρ(−ζ). Furtermore, the fractional power ζ^α for 0 < α < 1, is define as a closed linear operator with inverse ζ^−α on its domain D(ζ^α). The following are the basic properties ζ^α. For semigroup G(z) there is an M ≥ 1 where ‖G(z)‖ ≤ M for all z₀ ≤ z ≤ T, the following properties will be used:

Theorem 2.1 [23]

1. X_α = D(ζ^α) is a BS with norm ‖x‖_α = ‖ζ^αx‖, for x ∈ D(ζ^α).
2. G(z):X → X_α, for each x > 0.
3. ζ^αG(z)x = G(z)ζ^αX, ∀x ∈ D(ζ^α) and z > 0.
4. For every z > 0, ζ^αG(z) is bounded on X and there exist C_α > 0 and δ > 0 where
5. ζ^−α is a linear bounded operator in X with D(ζ^α) = im(ζ^−α).
6. If 0 < α ≤ β, then .

Remark 2.1 In [1] and by Theorem 2.1 (2), (3), we can see that the restriction G_α(z) of G(z) to X_α is exactly the part of G(z) in X_α. Let x ∈ X_α. Since and as z decreases to 0 for all x ∈ X_α it follows that (G(z)_{z ≥ 0}) is a family of strongly continuous semigroup on X_α and ‖G_α(z)‖ ≤ ‖G(z)‖ for all z ≥ 0.

We define a mild solution to (1) on I a function w which is continuous where w: I → X satisfying the following integral equation If there exists a T₀, z₀ ≤ T₀ ≤ T and a continuous w which defined as w: I₀ = [z₀, T₀]→X, then we say that (1) has a local mild solution w to (1) on I₀.

To start the analysis, we need to know some basic properties and definitions of fractional calculus.

Definition 2.1 [24] A space C_μ, and if p > μ, such that f(x) = x^pf₁(x), where f₁ ∈ C[0, ∞) and f(x), x ≥ 0 be a real function. The space C_μ is called to be if and only if f^(m) ∈ C_μ, .

Definition 2.2 [24] The fractional integral operator Riemann-Liouville of order 0 ≤ α, of a function f ∈ C_μ, μ ≥ −1 is defined as

A Riemann-Liouville derivative has certain disadvantages when modeled with fractional differential equations to describe some real-world phenomena. Therefore, we will introduce a modified fractional method proposed by M. Caputo.

Definition 2.3 [25] The Caputo fractional derivative of f(x) is defined as

3 Local existence of mild solutions

For problem (1), in this section we show its existence. Let us list the hypothesis for some α ∈ (0, 1). To determine whether a unique mild solution exists (1), we require the following assumption on the maps f and g.

(H1) Let W be an open subset of [0, ∞)×X_α. The map f and g is called to be satisfy (H1) if for every (z, x)∈W, there exist a neighborhood V ⊆ W of (z, x) and constants L₀, such that (H2) It is possible to assume without losing generality that IGAS by −ζ is bounded and that −ζ is invertible. In addition, suppose that 0 < T < ∞ to establish local existence.

(H3) I_k ∈ C(X_α, X_α), k = 1, 2, …, m, and Ψ_k ∈ C(I, R₊), k = 1, 2, …, m is nondecreasing maps and There exists ρ > 0 and (H3′) there exists ρ₁ > 0 and The notations will be continued considered in the previous section.

An equivalent integral equation for z ≥ z₀ is associated with the existence of a solution to (1) as the following: The following theorem follows from these simplifications.

Theorem 3.1 Let −ζ be the operator generates the analytic-semigroup G(z) where 0 ∈ ρ(−ζ). If the maps f and g satisfy (H1) and q be the real valued map where it integrable in I, then the Eq (1) has a unique local mild solution for every w₀ ∈ X_α. Therefore, if the kernel q satisfied the condition blow

(H4) There exist positive constants C₀ > 0 and 0 < β ≤ 1 such that for all z, s ∈ I. Hence the mild solutionis also Hölder continuous.

Proof. We will use the concepts and notations introduced in the previous section. Let (z₀, w₀) be a point in an open subset U of [0, ∞) × X_α and choose such that (H1) satisfied, with some positive number L₀ > 0 holds for the set of maps f and g Let and Choose τ > tz₀ such that and (2) where L(r) = (L₀r + B₁) + a_T(L₀r + B₂) and C_α is a positive constant depending on α satisfying (3) Let endow Y = C([z₀, τ];X) with the supremum norm Thus Y is a BS. Then Fy be a map on Y and is given by It clear by hypothesis (H1) on the functions f and g, and condition (3) that F: Y → Y and ∀y ∈ Y, Fy(z₀) = ζ^α(w₀−φ(y)).

Let Ω be the non-empty bounded and closed set given by

Then for y ∈ Ω, we have where the last two inequalities follows from above assumption. Hence, we have F: Ω → Ω. We can see that F is a strict contraction in Ω. There will be a unique continuous function satisfying this requirement (1). For y ∈ Ω and θ ∈ Ω, then Assuming (H1) on g and f and condition (3) we get In the last inequality we use (2). Then the map F is a strict contraction where F: Ω → Ω and hence we have a unique fixed point y of F in Ω, i.e. there is a unique y ∈ Ω such that Fy = y, by using the Banach contraction principle.

Suppose that w = ζ^−αy. Thus for z ∈ [z₀, τ], we have This follow that w is a unique local mild solution to (1).

Thus for the kernel q satisfies the condition (H4), we show the Hölder continuity of the solution. Since there exist T₀, z₀ < T₀ < T and a map w where w is a mild unique solution to (1) on I₀ = (z₀, T₀) defined by where For z₀ < τ₁ < τ₂ < T and let v(z) = ζ^αw(z). Then, For simplification, set Then the above equation can be rewritten as Since the maps f and g satisfy assumption (H1) and w(z) is continuous on I₀, it clear that and are continuous. Hence bounded on I₀ and let Where, and we have, from which we deduce Hence Hence Therefore, the continuity of the function z → ‖G(z)‖ for z ∈ (0, T) allows us to conclude that For k = 1, …, m, we can write Hence Thus from the above inequality Hölder the continuity of v(z) follows.

4 Global existence

We need the following lemma to find the global existence of a mild solution to (1).

Lemma 4.1 [23] Let φ(z, s) = w₀(z, s) + φ(w(z, s)) ≥ 0 be continuous function on 0 ≤ s ≤ z ≤ T < ∞. If there exist two positive constant L, M and α where then there is a constant N and φ(z, s)≤N, for 0 ≤ s ≤ z ≤ T.

The following theorem prove that there is a global existence of a mild solution to (1).

Theorem 4.1 Let 0 ∈ D(ζ) and let −ζ be IGAS G(t) satisfying ‖G(z)‖ ≤ M for z ≥ z₀. Let f, g: [z₀, ∞) × X_α → X satisfy assumption H1. If there is continuous discreasing function k₁ and k₂ from [z₀, ∞) into [0, ∞) such that then the impulsive nonlocal problem (1) has a unique mild solution w on [z₀, ∞) for every w₀−φ(w)∈X_α.

Proof. By Theorem 3.1 we have a T₀, z₀ < T₀ and w on I₀ = [z₀, T₀] be a unique solution. For some positive constant C, if ‖w(z)‖_α ≤ C for z ∈ I₀, then on the right of T₀ the solution w(z) may be continued on it. Therefor, it suffices to show that if on [z₀, T], z₀ < T < ∞ there is a mild solution w to (1), then ‖w(z)‖_α is bounded as z ↑ T. Since w(z) is a mild solution. Hence we have By the fact that G(z) commutes with ζ and that ‖G(z)‖ ≤ M, ‖ζ^αG(z)‖ ≤ C_αz^−α for z ≥ z₀ in above equation taking norms of both sides and applying ζ^α, we have For the last equation we have the estimate For the equation incorporating the estimate, we have After a slight modification in the equation, we have The estimate in the equation is (4) Where C₁ and C₂ some positive constants depending on η, α, Ψ_k and T. When the equation is integrating over (z₀, z), we have. Now for integration changing the order We rewrite the equation as The estimate above equation (5) for some nonnegative C₃ and C₄ depending on η, α, and T only. Adding (4) and (5), we have for some nonnegative C₅ and C₆ depending on η, α and T only. Using Lemma 4.1 to above we have ‖w(z)‖ ≤ C on [z₀, T]. Which is the proof.

5 Application

Let ∂Ω be smooth boundary where be a bounded. Let the partial differential linear operator where for each multi-index α we defined a real or complex valued map a_α(x) on . Suppose that A(x, D) is strongly elliptic. i.e. there is a positive constant c > 0 where for every and . Consider the parabolic IDE. (6) Where D^j stands by any jth order derivative. Let f and g are continuous differentiable functions of all their variable, expect possibly in x.

By following abstract IDE in X = L^p(Ω), we can be reformulated the parabolic IDE (6): (7) And ζ_p: D(ζ_p) ⊂ X → X given by also F, G: [0, T) × D(A_p) → X are the Nemyckii operators is (8) (9) We can defined the maps f and g for the Nemyckii operators in (8) and (9) under the assumption sufficient Caratheodory and growth conditions. Now for λ is large so that ζ_p in invertible. We have that −ζ_p is IGAS on X. And from imbedding theorems we have that X_α is continuous imbedded in for 1 − 1/2m < α < 1 and p large value. We verified that F and G is satisfied by assumption. So a suitable assumptions on the kernel q, the existence of a unique global mild solution to (7) is ensures by Theorem 4.1 for p large. Hence, the existence of a unique global mild solution to (6) is guarantees.

6 Conclusion

In this manuscript, we have successfully established the sufficient conditions for the existence of solutions to fractional equations abstract integro-differential equation with impulsive. Further, we used the Banach fixed point theorem for existence of a unique solution. In the end, example are given to demonstrate the effectiveness of the obtained analytical results. Motivated by this manuscript, we can consider and investigate the existence of fractional order integro-differential equation with impulsive condition on time scales.

Acknowledgments

The authors wishes to thank the referees for useful comments and suggestions.