We consider an impulsive neutral fractional integrodifferential equation with infinite delay in an arbitrary Banach space X. The existence of mild solution is established by using solution operator and Hausdorff measure of noncompactness.

1. Introduction

In recent years, fractional calculus has becomes an active area of research due to its demonstrated applications in widespread fields of science and engineering such as mechanics, electrical engineering, medicine, biology, ecology, and many others. The memory and hereditary properties of various materials and processes can be described by a differential equation with fractional order. The fractional differential equation also describes the efficiency of nonlinear oscillations of the earthquake. The details on the theory and its applications can be found in [1–4] and references given therein.

On the other hand, many real world processes and phenomena which are subjected during their development to short-term external influences can be modeled as impulsive differential equation with fractional order which have been used efficiently in modelling many practical problems. Their duration is negligible compared with the total duration of the entire process and phenomena. Such process is investigated in various fields such as biology, physics, control theory, population dynamics, economics, chemical technology, and medicine. In addition, the improvement of the hypothesis of the functional differential equation with infinite delay relies on a choice of phase space. There are various phase spaces which have been studied. Hale and Kato in [5] introduced a common phase space 𝒫. For more details on phase space, we refer to books by Hale and Kato [5], Hino et al. [6] and papers [7–10]. For the study of impulsive differential equation, we refer to papers [7, 8, 11–18] and references given therein.

The purpose of this work is to establish the existence of mild solution for impulsive fractional differential equation with infinite delay:

(1)

where 0 < T < ∞, 0 < q < 1, A : D(A) ⊂ X → X is a closed and densely defined linear operator and infinitesimal generator of a solution (resolvent) operator {S_q(t)} _t≥0 on Banach space X, and

denotes the fractional derivative in Caputo sense and

denotes the Riemann-Liouville fractional integral operator. The history u_t : (−∞, 0] → X defined by u_t(s) = u(t + s) for s ∈ (−∞, 0] belongs to some abstract phase space 𝒫 defined axiomatically and I_i ∈ C(X, X) (i = 1, …, n); 0 ≤ t₀ < t₁ < ⋯<t_n ≤ t_n+1 = T are fixed numbers and Δu(t) denotes the jump of the function u at the point t, given by Δu(t) = u(t⁺) − u(t⁻). The functions f, g : J × 𝒫 → X are appropriate functions and satisfy some conditions to be specified later.

In [13], authors have considered the following impulsive fractional differential equation in a Banach space of the form

(2)

where A : D(A) ⊂ X → X is the infinitesimal generator of a C₀-semigroup {T(t) : t ≥ 0} on a Banach space, f : J × X → X is continuous, and u₀, y_i are the element of X. Authors have established some existence and uniqueness results for system (2) under the different assumptions on initial conditions.

In this work, we adopt the idea of Wang et al. [13] and establish the existence of a mild solution for the problem (1) by using the measure of noncompactness and solution operator. The tool of measure of noncompactness has been used in linear operator theory, theory of differential and integral equations, the fixed point theory, and many others. For an initial study of theory of the measure of noncompactness, we refer to book of Banaś and Goebel [19] and Akhmerov et al. [20] and papers [9, 21–25] and references given therein.

This paper is organized as follows: In Section 2 we recall some basic definitions, lemmas, and theorems. We will prove the existence of a mild solution for the system (1) in Section 3. In the last section, we shall discuss an example to illustrate the application of the abstract results.

2. Preliminaries

Now we provide some basic definitions, notations, theorems, lemmas, and preliminary facts which will be used throughout this paper.

Let X be a Banach space and let C([0, T]; X) be the Banach space of continuous functions u(t) from [0, T] to X equipped with the norm ∥y∥_C = sup⁡_t∈[0,T]∥y(t)∥_X and L^p((0, T); X) denotes the Banach space of all Bochner-measurable functions from (0, T) to X with the norm

(3)

Assume that 0 ∈ ρ(A), that is, A is invertible. Then, this permits us to define the positive fractional power A^α as closed linear operator with domain D(A^α)⊆H for α ∈ (0,1]. Moreover, D(A^α) is dense in H with the norm

(4)

It is easy to see that D(A^α) which is dense in X is a Banach space. Henceforth, we use X_α as notation of D(A^α). Also, we have that X_κ↪X_α for 0 < α < κ and, therefore, the embedding is continuous. Then, we define X_−α = (X_α) ^*, for each α > 0. The space X_−α, standing for the dual space of X_α, is a Banach space with the norm ∥z∥_−α = ∥A^−αz∥ for z ∈ X_−α. For more details on the fractional powers of closed linear operators, we refer to book by Pazy [26].

To consider the mild solution for the impulsive problem, we propose that the set 𝒫𝒞([0, T]; X) = {u : [0, T] → X : u is continuous at t ≠ t_i and left continuous at t =

. Clearly, 𝒫𝒞([0, T]; X) is a Banach space endowing the norm ∥u∥_𝒫𝒞 = sup⁡_t∈[0,T]∥u(s)∥. For a function u ∈ 𝒫𝒞([0, T]; X) and i ∈ {0,1, …, m}, we define the function

such that

(5)

For W ⊂ 𝒫𝒞([0, T]; X) and i ∈ {0,1, …, m}, we have

and following Accoli-Arzelà type criteria.

Lemma 1. A set B ⊂ 𝒫𝒞([0, T]; X) is relatively compact in 𝒫𝒞([0, T]; X) if and only if each set is relatively compact in C([t_j, t_j+1], X) (j = 0,1, …, m).

For the differential equation with infinite delay, Hale and Kato [5] proposed the phase space 𝒫 satisfying certain fundamental axioms.

Definition 2 (see [6].)A phase space 𝒫 is a linear space which contains all the functions mapping (−∞, 0] into Banach space X with a seminorm ∥·∥_𝒫. The fundamental axioms assumed on 𝒫 are the following,

(A)
If u : (−∞, a + T] → X, T > 0 is a continuous function on [a, a + T] such that u_a ∈ 𝒫 and u|_[a,a+T] ∈ 𝒫 ∈ 𝒫𝒞([a, a + T]; X), then for every t ∈ [a, a + T), the following conditions hold:
- (i)
  u_t ∈ 𝒫,
- (ii)
  ,
- (iii)
  .
Where H is a positive constant, N, K : [0, ∞)→[0, ∞), N is a locally bounded, K is continuous, and H, N, K are independent of u(·).
(A1)
For the function u in (A1), u_t is a 𝒫-valued continuous function for t ∈ [a, a + T].
(B)
The space 𝒫 is complete.

Now, we state some basic definitions and properties of fractional calculus.

Mittag-Leffler. The definition of one parameter Mittag-Leffler function is given as

(6)

and two-parameter Mittag-Leffler function is defined as

(7)

where C is a contour which starts and ends at −∞ and encircles the disc |μ | ≤|z|^1/2 counter clockwise. The Laplace transform of the Mittag-leffler is defined as

(8)

For more details we refer to [1].

Laplace transform of integer order derivatives is defined as

(9)

Definition 3. The Riemann-Liouville fractional integral operator ℐ is defined as

(10)

where F ∈ L¹((0, T); X) and q > 0 is the order of the fractional integration.

Definition 4. The Riemann-Liouville fractional derivative is given as

(11)

where

, F ∈ L¹((0, T); X), and

. Here, the notation W^n,1((0, T); X) stands for the Sobolev space defined as

(12)

Note that z(t) = yⁿ(t) and d_k = y^k(0).

Definition 5. The Caputo fractional derivative is given as

(13)

where F ∈ Cⁿ⁻¹((0, T); X)∩L¹((0, T); X) and the following holds

(14)

Definition 6 (see [27].)A family {S_q(t)} _t≥0 ⊂ ℒ(X) of bounded linear operators in X is called a resolvent (or solution operator) generating by A if the following conditions are fulfilled:

(1)
S_q(t) is strongly continuous on ℝ⁺ and S_q(0) = I;
(2)
for x ∈ D(A) and t ≥ 0, S_q(t)D(A) ⊂ D(A) and AS_q(t)x = S_q(t)Ax;
(3)
S_q(t)x is the solution of the equation
(15)

where ℒ(X) denotes the space of all bounded linear operators from X into X endowed with the norm of operators.

Also, the solution operator S_q(t) for (15) is defined as (see [27])

(16)

where ω ≥ 0 and {λ^q : Reλ > ω} ⊂ ρ(A).

Let

(17)

Definition 7 (see [27].)A solution operator S_q(t) is said to be analytic if S_q(·) : ℝ⁺ → ℒ(X) admits analytic extension to a sector ∑ (0, θ₀) for some 0 < θ₀ ≤ π/2. Furthermore, An analytic resolvent S_q(t) is said to be of analyticity type (ω₀, θ₀) if, for θ₀ > θ and ω₀ < ω, there exists M = M(ω, θ) such that ∥S_q(t)∥ ≤ Me^ωRez for z ∈ ∑ (0, θ); here Rez means the real part of z.

In this work, we assume that solution operator {S_q(t)} _t≥0 is analytic; that is, {S_q(t)} _t≥0 satisfy the following property.

(HA) The map t ↦ S_q(t) is continuous from [0, T] to ℒ(X) endowed with the uniform operator norm ∥·∥_ℒ(X).

Without loss of generality, we have that there exist a positive constant M such that ∥S_q(t)∥ ≤ M, for t ≥ 0.

Definition 8 (see [19].)The Hausdorff measure of noncompactness χ_Z is defined as

(18)

for bounded set F ⊂ Z, where Z is a Banach space.

Lemma 9 (see [19].)For any bounded set U, V ⊂ Y, where Y is a Banach space. Then, the following properties are fulfilled:

(i)
χ_Y(U) = 0 if and only if U is pre-compact;
(ii)
, where conv U and denotes the convex hull and closure of U, respectively;
(iii)
χ_Y(U) ⊂ χ_Y(V), when U ⊂ V;
(iv)
χ_Y(U + V) ≤ χ_Y(U) + χ_Y(V), where U + V = {u + v : u ∈ U, v ∈ V};
(v)
χ_Y(U ∪ V) ≤ max⁡{χ_Y(U), χ_Y(V)};
(vi)
χ_Y(λU) = λ · χ_Y(U), for any λ ∈ ℝ;
(vii)
if the map P : D(P) ⊂ Y → 𝒵 is continuous and satisfies the Lipschitsz condition with constant κ, then we have that χ_𝒵(PU) ≤ κχ_Y(U) for any bounded subset U ⊂ D(P), where Y and 𝒵 are Banach space.

The details on the measure of noncompactness and its applications can be found in a book by Banaś and Goebel [19] and papers [9, 10, 21, 23, 24].

Lemma 10 (see [19].)A bounded and continuous map Q : D ⊂ Z → Z is a χ_Z-contraction if there exists a constant 0 < κ < 1 such that χ_Z(Q(U)) ≤ κχ_Z(U), for any bounded closed subset U ⊂ D, where Z is a Banach space.

Lemma 11 (see [28].)Let D ⊂ Z be closed and convex with 0 ∈ D and let the continuous map Q : D → D be a χ_Z-contraction. If the set {u ∈ D : u = λQu, for 0 < λ < 1} is bounded, then the map Q has a fixed point in D.

Lemma 12 ((Darbo-Sadovskii) [19]). Let D ⊂ Z be bounded, closed, and convex. If the continuous map Q : D → D is a χ_Z-contraction, then the map Q has a fixed point in D.

In this work, we consider that χ denotes the Hausdorff measure of noncompactness in X, χ_C denotes the Hausdorff measure in noncompactness of C([0, T]; X) and χ_𝒫𝒞 denotes the Hausdorff measure of noncompactness in 𝒫𝒞([0, T]; X).

Lemma 13 (see [19], [21].)If U is bounded subset of C([0, T]; X). Then, one has that χ(U(t)) ≤ χ_C(U), for all t ∈ [0, T], where U(t) = {u(t); u ∈ U}⊆X. Furthermore, if U is equicontinuous on [0, T], then χ(U(t)) is continuous on the interval [0, T] and

(19)

Lemma 14 (see [19].)If U ⊂ C([0, T]; X) is bounded and equicontinuous set, then χ(U(t)) is continuous and

(20)

where

Lemma 15 (see [29].)(1) If U ⊂ 𝒫𝒞([0, T]; X) is bounded, then χ(U(t)) ≤ χ_𝒫𝒞(U), for all t ∈ [0, T], where U(t) = {u(t) : u ∈ U} ⊂ X.

(2) If U is piecewise equicontinuous on [0, T], then χ(U(t)) is piecewise continuous for t ∈ [0, T] and

(21)

(3) If U ⊂ 𝒫𝒞([0, T]; X) is bounded and equicontinuous, then χ(U(t)) is piecewise continuous for t ∈ [0, T] and

(22)

where

3. Main Results

In this section, we will establish the existence results of solution for (1) by using solution operator and Hausdorff′s measure of noncompactness.

From [13], we adopt the following concept of solution for impulsive differential problem (1).

Definition 16. A piecewise continuous function u : (−∞, T] → X is said to be a mild solution for impulsive problem (1) if u₀ = φ, u(·)|_J ∈ 𝒫𝒞 and

(23)

where

(24)

Now we list the following assumptions which are required to establish main results.

(Hf)
The function f : J × 𝒫 → X satisfies the following conditions:
- (1)
  the function f(·, u) : J → X is strongly measurable for every u ∈ 𝒫 and u₀ ∈ 𝒫, u|_J ∈ 𝒫𝒞;
- (2)
  f(t, ·) : 𝒫 → X is a continuous function for each t ∈ J;
- (3)
  there exists an integrable function m_f : [0, T]→[0, ∞) and a nondecreasing continuous function Ω : [0, ∞)→(0, ∞) such that
  (25)
- (4)
  there exists an integrable function η : [0, T]→[0, ∞) such that, for any bounded set B ⊂ 𝒫, we have
  (26)
for almost everywhere τ ∈ J, where B(θ) = {u(θ) : u ∈ B}.
(Hg)
(1) For 0 < β < 1, A^βg(·, ·) is Lipschitz continuous function for all (t, v) ∈ J × 𝒫 and there exist positive constants C₁ and C₂ such that
(27)
- (2)
  there exists a constant L_g > 0 such that
  (28)
- for all u, v ∈ 𝒫.
(HI)
The functions I_i : X → X, (i = 1, …, n) are continuous functions and satisfy the following conditions:
- (1)
  There is a constant L_I > 0 such that
  (29)
- (2)
  There exist positive constants L_j (j = 1,2) such that
  (30)
- for all u ∈ 𝒫.
(H1)

(31)
where
(32)
(H2)

(33)

Now, let z : (−∞, T] → X be a function given by z₀ = φ and z(t) = S_q(t)φ(0) on J. It is easy to see that , where K_T = sup⁡_0≤t≤TK(t), N_T = sup⁡_0≤t≤TN(t).

Theorem 17. Suppose that hypotheses (HA), (Hf), (Hg), (HI), and (H1)−(H2) are satisfied. Then, there exists a mild solution for the impulsive problem (1).

Proof. Consider the space S(T) = {u : (−∞, T] → X : u₀ = 0, u|_J ∈ 𝒫} endowed with supremum norm ∥·∥_𝒫. Define the operator Q : S(T) → S(T) by

(34)

and we have that

(35)

where ∥u∥_t = sup⁡_0≤s≤t∥u(s)∥. Thus, Q is well defined and with the values in S(T) by our assumptions. By Lebesgue dominated convergence theorem, axioms of phase space and assumptions (Hf), (Hg), and (HI), it is clear that Q is continuous map. Furthermore, by uniformly continuity of the map t ↦ S_q(t) on (0, T], we obtain that set Q(S(T)) is equicontinuous. We prove the result in following steps.

Step 1. The set {u ∈ 𝒫𝒞 : u = λQu, for 0 < λ < 1} is bounded.

Let u_λ be a solution of u = λQu for 0 < λ < 1. Therefore, we have that

(36)

Take

, for t ∈ J. Then, we get that for t ∈ [0, t₁]

(37)

For t ∈ (t₁, t₂], we have

(38)

For t ∈ (t_n, T],

(39)

Therefore, for all t ∈ [0, T] = J, we have

(40)

From

, it implies that

(41)

and consequently,

(42)

where

(43)

Let

. Thus ξ_λ(0) = d. Therefore, we get

(44)

Integrating above inequality we have that

(45)

It gives that the functions ξ_λ(t) are bounded on interval J. Therefore, the functions v_λ(t) are bounded and u_λ(·) are also bounded on J.

Step 2. The map Q is a χ-contraction.

Firstly, we introduce decomposition of Q into Q = Q₁ + Q₂, for t ≥ 0 such that

(46)

To prove the result, we firstly show that Q₁ is Lipschitz continuous. For x₁, x₂ ∈ S(T) and t ∈ [0, t₁], we have that

(47)

For t ∈ (t_m, t_m+1], m = 1,2, …, n we have

(48)

Thus for t ∈ [0, T] we have

(49)

Taking supremum on [0, T], we get

(50)

Hence, it implies that Q₁ satisfies the Lipschitz condition with Lipschitz constant L, where L = K_T[∥A^−β∥L_g + nML_I + MnL_g(2 + L_I)].

Therefore, from Lemma 9 (vii), we have that for any bounded set B ∈ 𝒫𝒞

(51)

Next, we show that Q₂ is a χ-contraction. Let B be an arbitrary bounded subset S(T). Since S_q(t) is equicontinuous solution operator, therefore S_q(t − s)f(s, u_s + z_s) is piecewise continuous. From Lemma 9, we have that for any bounded set B ∈ 𝒫𝒞,

(52)

Thus for any bounded set B ∈ 𝒫𝒞

(53)

By the assumption (H2), we obtain that χ_𝒫𝒞(QB) < χ_𝒫𝒞(B); that is, Q is a contraction. Therefore, Q has at least one fixed point in B by Darbo fixed point theorem. Let u be a fixed point of Q on S(T), then y = u + z is a mild solution for (1).

Theorem 18. Suppose that (Hf), (Hg), and (HI) are satisfied and

(54)

Then, there exists a mild solution for the impulsive problem (1).

Proof. Proceeding as in the proof of Theorem 17, we infer that Q defined by (34) is continuous from S(T) into S(T). Next we indicate that there exists r > 0 such that Q(B_r) ⊂ B_r, where B is defined by B_r = {u ∈ S(T) : ∥u∥_T ≤ r}. To this end, let us assume that assertion is false, then for any r > 0 there exists u_r ∈ B_r and t_r ∈ J such that r < ∥Qu_r(t_r)∥. Therefore, for t_r ∈ [0, t₁] and u_r ∈ B_r,

(55)

For t_r ∈ (t₁, t₂], we have

(56)

For t_r ∈ (t_n, T], we have

(57)

which implies that r < max⁡{r₀, r₁, …, r_n}. Therefore, we conclude that

(58)

We divide both the sides of (58) by r and letting r → ∞, we obtain that

(59)

This contradicts the inequality (54). Hence, there exists a positive constant r > 0 such that Q(B_r) ⊂ B_r. Moreover, by uniform continuity of the map t ↦ S_q(t) on (0, T], we have that set Q(B_r) is equicontinuous. As the proof of Theorem 17, we infer that system (1) has a mild solution.

4. Example

In this section, we consider an example to illustrate the application of the theory. Here we take the space C₀ × L²(h, X) as the phase space 𝒫 (see, [7]).

Now we study the following fractional differential equation with infinite delay:

(60)

where φ ∈ C₀ × L²(h; X) and 0 < t₁ < ⋯<t_n < t_n+1 = T are prefixed numbers. Let X = L²[0, π] and consider the operator A : D(A) ⊂ X → X as Av = v^′′ with the domain

(61)

Then,

(62)

Here

, n ∈ ℕ is the orthogonal set of eigenvectors of A. It is clear that A is the infinitesimal generator of an analytic semigroup {S(t)} _t≥0 in X which is given by

(63)

By Theorem 3.1 in [27], we get that A is the infinitesimal generator of solution operator {S_q(t)} _t≥0 and there exists a positive constant M such that

for t ∈ [0, T]. The functions b, f, a_k satisfy the following conditions:

(1)
the functions b(s, μ, x), (∂b/∂x)(s, μ, x) are measurable such that b(s, μ, π) = b(s, μ, 0) = 0 and
(64)
(2)
f : ℝ⁴ → ℝ is continuous and there exist a η : C(ℝ², ℝ) such that
(65)
(3)
the functions a_k : [0, ∞) → ℝ, k = 1,2, …, n are continuous and
(66)

The impulsive system (60) might be reformulated as the abstract impulsive Cauchy problem (1) where

(67)

It may be verified that g, F satisfy the assumptions (Hf), (Hg), and (HI); that is, g(t, ·) and I_k, k = 1, …, n are bounded linear operators and the range of g(·) is contained in X_1/2, ∥A^1/2g(t, ·)∥ ≤ L_g, ∥I_k∥ ≤ L_I, k = 1, …, n, and ∥F(t, ϕ)∥ ≤ m_F(t)∥ϕ∥_𝒫 for every t ∈ [0, T], where m_F(t) is defined as

. Applying Theorem 17, we obtain that problem (60) has a mild solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the UGC (University Grants Commission, India) under Grant no. (6405-11-061).nt.

References

1 Podlubny I., Fractional Differential Equations, 1993, Academic Press, New York, NY, USA, MR1658022.
Google Scholar
2 Miller K. S. and Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, 1993, John Wiley & Sons, New York, NY, USA.
Google Scholar
3 Samko S. G., Kilbas A. A., and Marichev O. I., Fractional Integrals and Derivatives: Theory and Applications, 1993, Gordon and Breach Science Publishers, Yverdon, Switzerland.
Google Scholar
4 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, 2006, Elsevier, Amsterdam, The Netherlands.
Google Scholar
5 Hale J. K. and Kato J., Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj. (1978) 21, no. 1, 11–41, MR0492721.
Google Scholar
6 Hino Y., Murakami S., and Naito T., Functional-Differential Equations with Infinite Delay, 1991, 1473, Springer, Berlin, Germany, Lecture Notes in Mathematics, MR1122588.
Google Scholar
7 Hernández E. M., Rabello M., and Henríquez H. R., Existence of solutions for impulsive partial neutral functional differential equations, Journal of Mathematical Analysis and Applications. (2007) 331, no. 2, 1135–1158, 2-s2.0-33947310146, https://doi.org/10.1016/j.jmaa.2006.09.043, MR2313705, ZBL1123.34062.
Web of Science® Google Scholar
8 Hernández E. and Henríquez H. R., Existence results for partial neutral functional-differential equations with unbounded delay, Journal of Mathematical Analysis and Applications. (1998) 221, no. 2, 452–475, 2-s2.0-0000308613, https://doi.org/10.1006/jmaa.1997.5875, MR1621730, ZBL0915.35110.
Google Scholar
9 Li F., An existence result for fractional differential equations of neutral type with infinite delay, Electronic Journal of Qualitative Theory of Differential Equations. (2011) 52, 1–15, 2-s2.0-79960470180, MR2825137.
10.1155/2011/354016
Google Scholar
10 Li F. and Zhang J., Existence of mild solutions to fractional integrodifferential equations of neutral type with infinite delay, Advances in Difference Equations. (2011) 2011, 15, 2-s2.0-79955042520, https://doi.org/10.1155/2011/963463, 963463, MR2774252, ZBL1213.45008.
Google Scholar
11 Agarwal R. P., Benchohra M., and Hamani S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Applicandae Mathematicae. (2010) 109, no. 3, 973–1033, 2-s2.0-77949264980, https://doi.org/10.1007/s10440-008-9356-6, MR2596185, ZBL1198.26004.
Google Scholar
12 Dabas J. and Chauhan A., Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Mathematical and Computer Modelling. (2013) 57, no. 3-4, 754–763, https://doi.org/10.1016/j.mcm.2012.09.001, MR3011198.
Google Scholar
13 Wang J., Fečkan M., and Zhou Y., On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynamics of Partial Differential Equations. (2011) 8, no. 4, 345–361, 2-s2.0-84255168779, https://doi.org/10.4310/DPDE.2011.v8.n4.a3, MR2901608, ZBL1264.34014.
Google Scholar
14 Benchohra M., Henderson J., and Ntouyas S., Impulsive Differential Equations and Inclusions, 2006, 2, Hindawi, New York, NY, USA, Contemporary Mathematics and Its Applications, https://doi.org/10.1155/9789775945501, MR2322133.
Google Scholar
15 Fečkan M., Zhou Y., and Wang J., On the concept and existence of solution for impulsive fractional differential equations, Communications in Nonlinear Science and Numerical Simulation. (2012) 17, no. 7, 3050–3060, 2-s2.0-84856291615, https://doi.org/10.1016/j.cnsns.2011.11.017, MR2880474, ZBL1252.35277.
Google Scholar
16 Lakshmikantham V., Baĭnov D. D., and Simeonov P. S., Theory of Impulsive Differential Equations, 1989, World Scientific Publishing, Teaneck, NJ, USA, Series in Modern Applied Mathematics, MR1082551.
Google Scholar
17 Chang Y.-K., Anguraj A., and Arjunan M. M., Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Analysis: Hybrid Systems. (2008) 2, no. 1, 209–218, 2-s2.0-37049008497, https://doi.org/10.1016/j.nahs.2007.10.001, MR2382006, ZBL1170.35467.
Web of Science® Google Scholar
18 Chang Y.-K., Kavitha V., and Arjunan M. M., Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators, Nonlinear Analysis: Hybrid Systems. (2010) 4, no. 1, 32–43, 2-s2.0-70350564213, https://doi.org/10.1016/j.nahs.2009.07.004, MR2570180, ZBL1200.34095.
Web of Science® Google Scholar
19 Banaś J. and Goebel K., Measures of Noncompactness in Banach Spaces, 1980, Marcel Dekker, New York, NY, USA, Lecture Notes in Pure and Applied Mathematics, MR591679.
Google Scholar
20 Akhmerov R. R., Kamenskiĭ M. I., Potapov A. S., Rodkina A. E., and Sadovskiĭ B. N., Measures of Noncompactness and Condensing Operators, 1992, Birkhäuser, Berlin, Germany, MR1153247.
Google Scholar
21 Ji S. and Li G., A unified approach to nonlocal impulsive differential equations with the measure of noncompactness, Advances in Difference Equations. (2012) 2012, article 182, 14, https://doi.org/10.1186/1687-1847-2012-182, MR3017169.
Google Scholar
22 Chen P. and Li Y., Nonlocal problem for fractional evolution equations of mixed type with the measure of noncompactness, Abstract and Applied Analysis. (2013) 2013, 12, 784816, https://doi.org/10.1155/2013/784816, MR3045079, ZBL1274.35400.
10.1155/2013/784816
Google Scholar
23 Agarwal R. P., Benchohra M., and Seba D., On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results in Mathematics. (2009) 55, no. 3-4, 221–230, 2-s2.0-71149084117, https://doi.org/10.1007/s00025-009-0434-5, MR2571191, ZBL1196.26009.
Google Scholar
24 Xue X., Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces, Nonlinear Analysis: Theory, Methods and Applications. (2009) 70, no. 7, 2593–2601, 2-s2.0-59849128649, https://doi.org/10.1016/j.na.2008.03.046, MR2499726, ZBL1176.34071.
Google Scholar
25 Fan Z. and Mophou G., Nonlocal problems for fractional differential equations via resolvent operators, International Journal of Differential Equations. (2013) 2013, 9, 490673, https://doi.org/10.1155/2013/490673, MR3066280, ZBL1269.34007.
10.1155/2013/490673
Google Scholar
26 Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, 1983, Springer, New York, NY, USA, https://doi.org/10.1007/978-1-4612-5561-1, MR710486.
Google Scholar
27 Bazhlekova E., Fractional evolution equations in Banach spaces [Ph.D. thesis], 2001, Eindhoven University of Technology.
Google Scholar
28 Agarwal R. P., Meehan M., and O′Regan D., Fixed Point Theory and Applications, 2001, Cambridge University Press, New York, NY, USA, Cambridge Tracts in Mathematics, https://doi.org/10.1017/CBO9780511543005, MR1825411.
Google Scholar
29 Ye R., Existence of solutions for impulsive partial neutral functional differential equation with infinite delay, Nonlinear Analysis: Theory, Methods and Applications. (2010) 73, no. 1, 155–162, 2-s2.0-77955429215, https://doi.org/10.1016/j.na.2010.03.008, MR2645840, ZBL1198.34177.
Google Scholar

Citing Literature

All articles

Existence Results for an Impulsive Neutral Fractional Integrodifferential Equation with Infinite Delay

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Example

Conflict of Interests

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Existence Results for an Impulsive Neutral Fractional Integrodifferential Equation with Infinite Delay

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Example

Conflict of Interests

Acknowledgments

References

Citing Literature

References

Related

Information