We investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others. Finally, two applications are given to illustrate that our results are valuable.

1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, and polymer rheology and have been emerging as an important area of investigation in the last few decades; see [1–4]. However, the theory of impulsive fractional evolution equations was still in the initial stages and many aspects of this theory need to be explored.

The theory of impulsive differential equations is a new and important branch of differential equation theory, which has an extensive physical, population dynamics, ecology, chemical, biological systems, and engineering background. Therefore, it has been an object of intensive investigation in recent years; some basic results on impulsive differential equations have been obtained and applications to different areas have been considered by many authors; see [4–8].

The study of nonlocal Cauchy problem for abstract evolution differential equations has been initiated by Byszewski [9]. The existence of solutions for fractional abstract differential equations with nonlocal initial condition was recently investigated by N’Guérékata [10] and Balachandran and Park [11]. In [12], Balachandran et al. were concerned with the existence of solutions of first-order nonlinear impulsive fractional integrodifferential equations in Banach spaces:

(1)

the results are obtained by using fixed point principles.

Shu and Wang [4] studied the existence of mild solutions for fractional differential equation with nonlocal conditions in a Banach space E:

(2)

By using the contraction mapping principle and Krasnoselskii’s fixed point theorem, they obtained the existence of solutions for the equation.

In [13], Gou and Li investigated local and global existence of mild solution for an impulsive fractional functional integrodifferential equation with noncompact semigroup in Banach spaces E:

(3)

and they establish a general framework to find the mild solutions for impulsive fractional integrodifferential equations, which will provide an effective way to deal with such problems.

Motivated by this consideration, we investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space E:

(4)

by using some fixed point theorems combined with the technique of measure of noncompactness, where

is the Caputo fractional derivative of order α ∈ (1,2], A : D(A) ⊂ E → E is a closed linear operator, and −A generates a C₀-semigroup T(t) (t ≥ 0) in E; f : J × E × E × E → E, J = [0, a], where a > 0 is a constant, 0 = t₀ < t₁ < t₂ < ⋯<t_m < t_m+1 = a; I_k : E → E is an impulsive function, and k = 1,2, …, m; and x₀ ∈ E, p₁, p₂ is an E-valued function to be given later and

(5)

D = {(t, s) ∈ R² : 0 ≤ s ≤ t ≤ a}, D₀ = {(t, s) ∈ R² : 0 ≤ t, s ≤ a}, p_i ∈ C(J × E, E), i = 1,2, and

denotes the jump of u(t) at t = t_k; that is,

and

represent the right and left limits of u(t) at t = t_k, respectively.

The paper is organized as follows. In Section 2, we recall some concepts and facts about the Kuratowski measure of noncompactness and some fixed point theorems. In Section 3, we obtain the existence solutions of problem (4). In Section 4, we give two examples to illustrate our results.

2. Preliminaries

In this section, we briefly recall some definitions and the fixed point theorems which will be used in the sequel. Throughout this paper, let E be a Banach space; we assume that A : D(A) ⊂ E → E is a closed linear operator and −A generates a uniformly bounded C₀-semigroup T(t) (t > 0) on a Banach space (E, ‖·‖) and M = sup_t∈[0,+∞)⁡‖T(t)‖_B(E), where B(E) stands for the Banach space of all linear and bounded operators in E.

C(J, E) denote the Banach space of all continuous E-value functions on interval J and PC(J, E) = {u : J → E : u ∈ C(t_k−1, t_k], E), k = 1,2, …, m, and there exist and , k = 1,2, …, m with . Obviously, PC(J, E) is a Banach space with the supnorm ‖u‖ = sup_t∈J‖u(t)‖.

Definition 1. The fractional integral of order α > 0 with the lower limit zero for a function u is defined as

(6)

where Γ(·) is the Gamma function.

Definition 2. The Caputo fractional derivative of order α > 0 with the lower limit zero for a function u is defined as

(7)

where the function u(t) has absolutely continuous derivatives up to order n − 1.

Remark 3. If u is an abstract function with values in E, then the integrals which appear in Definitions 1 and 2 are taken in Bochner’s sense.

Definition 4. By a mild solution of the initial value problem

(8)

on [0, ∞), we mean that a continuous function u defined from [0, ∞) into E satisfying

(9)

where

(10)

are the functions of Wright type defined on (0, ∞) which satisfies

(11)

The following lemma can be found in [14, 15].

Lemma 5. The operators and have the following properties:

(i)
For any fixed and are linear and bounded operators; that is, for any u ∈ E,
(12)
(ii)
For every u ∈ E, and are continuous functions from [0, ∞) into E.
(iii)
The operators and are strongly continuous, which means that, for ∀u ∈ E and 0 ≤ t^′ < t^′′ ≤ a, one has
(13)
(iv)
If the semigroup T(t) is continuous by operator norm for every t > 0, then and are continuous in (0, +∞) by the operator norm.

Let β(·) denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [6].

The following lemmas are to be used in proving our main results.

Lemma 6 (see [16].)Let E be a Banach space, and let D ⊂ C(J, E) be equicontinuous and bounded; then β(D(t)) is continuous on J, and β(D) = max_t∈J⁡β(D(t)).

Lemma 7 (see [7].)Let D ⊂ E be bounded. Then there exists a countable set D₀ ⊂ D, such that β(D) ≤ 2β(D₀).

Lemma 8 (see [17].)Let E be a Banach space, and let D = {u_n} ⊂ C(I, X) be a bounded and countable set. Then β(D(t)) is the Lebesgue integral on X, and

(14)

In the following, we introduce the definition of (γ, ψ, p)-contractive mapping.

Definition 9 (see [18].)Let (E, d) be a metric space with w distance p and f : E → E a given mapping. We say that f is an (γ, ψ, p)-contractive mapping if there exist two functions γ : E × E → [0, ∞) and ψ ∈ Ψ such that

(15)

for all x, y ∈ E.

We will show some fixed point theorems about condensing operator and (γ, ψ, p)-contractive mapping, which play a key role in the proof of our main results.

Lemma 10 (see [19].)Let E be a Banach space. Assume that D ⊂ E is a bounded closed and convex set on E, Q : D → D is condensing. Then Q has at least one fixed point in D.

Lemma 11 (see [18].)Let p be a w distance on a complete metric space (E, d) and let f : E → E be an (γ, ψ, p)-contractive mapping. Suppose that the following conditions hold:

(i)
f is an γ-admissible mapping;
(ii)
there exists a point x₀ ∈ E such that γ(x₀, fx₀) ≥ 1;
(iii)
either f is continuous or, for any sequence {x_n} in E, if γ(x_n, x_n+1) ≥ 1 for all and x_n → x ∈ E as n → ∞, then γ(x_n, x) ≥ 1 for all . Then there exists a point u ∈ E such that fu = u. Moreover, if γ(u, u) ≥ 1, then p(u, u) = 0.

3. Main Results

In this section, we will establish the existence theorems of solutions for the nonlocal problem (4). For convenience, we give some notations.

For B ⊂ C(J, E), let B(t) = {u(t) : u ∈ B} and denote B_R = {u ∈ E : ‖u‖ ≤ R}. Let k₀ = max⁡{k(t, s):(t, s) ∈ D}, h₀ = max⁡{h(t, s):(t, s) ∈ D₀}. First of all, let us start by defining what we mean by a solution of problem (4).

Definition 12. A function u ∈ PC(J, E) is said to be a mild solution of problem (4) if u satisfies the equation

(16)

To prove our main results, we state the following basic assumptions of this paper.

(H1)
There exists a positive constant K_i, (i = 1,2, 3) such that
(17)
and for p_i ∈ C(J × E, E), t ∈ J, u, v ∈ B_R, there exists a constant b_i, c_i > 0 such that
(18)
(H2)
For any R > 0, there exist a Lebesgue-integrable function M_R : I → R⁺ and nondecreasing continuous function Ω : [0, ∞ → (0, ∞) such that
(19)
for all t ∈ I, (x, y, z) ∈ B_R × B_R × B_R.
(H3)
g : E → E is a continuous and compact mapping; furthermore, there exists positive number N₁, N₂ such that ‖g(u) − g(v)‖ ≤ N₁ and ‖g(u)‖ ≤ N₂ for any u, v ∈ B_R.
(H4)
The functions I_k : E → E are continuous and there exists a constant μ > 0 and ρ > 0 such that
(20)
(H5)
There exist constants L_i > 0, (i = 1,2, 3) such that for any bounded and equicontinuous sets D_i ⊂ C(J, E), (i = 1,2, 3) and t ∈ J,
(21)

Lemma 13 (see [13].)Let f, g : J × E → E be a continuous function and let −A be the generator of a C₀-semigroup (T(t)) (t ≥ 0). If u ∈ PC(J, E) is a mild solution of (4) in the sense of Definition 12, then for any t ∈ (t_k−1, t_k], k = 1, …, m,

(22)

is a solution of (4). In other words u(t) is a mild solution of (4).

Theorem 14. Let E be a Banach space, let A : D(A) ⊂ E → E be a closed linear operator, and −A generates an equicontinuous C₀-semigroup T(t) (t ≥ 0) of uniformly bounded operators in E. Suppose that the conditions (H1)–(H5) are satisfied. Then for every u₀ ∈ PC(J, E) there exists a τ₁ = τ₁(u₀), 0 < τ₁ < a such that problem (4) has a solution u ∈ PC([0, τ₁], E).

Proof. Since we are interested here only in local solutions, we may assume that a < ∞. By using our assumption (H1)–(H4), let t^′ > 0, R = M(2‖u₀‖ + 1) > 0 be such that B_R(u₀) = {u : ‖u − u₀‖ ≤ R}, for 0 ≤ t ≤ t^′ and u ∈ B_R(u₀), and let us choose

(23)

Set Ω = {u ∈ PC([0, τ₁], E) : ‖u(t)‖ ≤ R, t ∈ [0, τ₁]}; then Ω is a closed ball in PC([0, τ₁], E) with center θ and radius R. Consider the operator Q : Ω → PC([0, τ₁], E) defined by

(24)

It is easy to see that the fixed points of Q are the solutions of the nonlocal problem (4); we shall prove that Q has a fixed point by using Lemma 10. For any u ∈ Ω and t ∈ [0, τ₁], by Lemma 5(i), we have

(25)

where

. So Tx ∈ Ω. Similarly, we prove Gu ∈ Ω.

And by (19), we have

(26)

Therefore, Qu ∈ Ω. Now we show that Q is continuous from Ω into Ω. To show this, we first observe that since f is continuous in [0, a] × E, it follows that for any ϵ > 0 and for a fixed u ∈ B_R(u₀) there exists δ₁(u), δ₂(u) > 0 such that for any v ∈ B_R(u₀) and let δ(u) = min⁡{δ₁(u), δ₂(u)}. Then for any v ∈ Ω, ‖u − v‖ < δ(u) and choose

(27)

Then we have

(28)

Thus, we proved that Q : Ω → Ω is a continuous operator.

Now, we demonstrate that the operator Q : Ω → Ω is equicontinuous. For any u ∈ Ω and 0 ≤ t₁ < t₂ ≤ τ₁, we get that

(29)

where

(30)

Here we calculate

(31)

Therefore, we inspect that ‖I_i‖ tend to 0, when t₂ − t₁ → 0, i = 1,2, …, 6.

For I₁, by Lemma 5(iii), ‖I₁‖ → 0 as t₂ − t₁ → 0.

For I₂, by Lemma 5(iii), we have

(32)

For I₃, by Lemma 5(i), we have

(33)

For I₄, by Lemma 5(i), we have

(34)

For I₅, by Lemma 5(iii), we have

(35)

For I₆, by Lemma 5(iii), we have

(36)

In conclusion, ‖(Qu)(t₂)−(Qu)(t₁)‖ tends to 0 as t₂ − t₁ → 0, which implies that Q(Ω) is equicontinuous.

Let . Then it is easy to verify that Q maps B into itself and B ⊂ PC(J, E) is equicontinuous. Now, we prove that Q : B → B is a condensing operator. For any D ⊂ B, by Lemma 7, there exists a countable set D₀ = {u_n} ⊂ D, such that

(37)

By the equicontinuity of B, we know that D₀ ⊂ B is also equicontinuous.

By the fact that

(38)

we have

(39)

Thus, by (H1), (H2), (39), and Lemma 13, we have

(40)

Since Q(D₀) ⊂ B is bounded and equicontinuous, we know from Lemma 6 that

(41)

Therefore, from (37), (40), and (41), we know that

(42)

Thus, Q : B → B is a condensing operator. It follows from Lemma 10 that Q has at least one fixed point u(t₀) ∈ B, which is just a solution of problem (4) on the interval [0, τ₁]. This completes the proof.

Corollary 15. Let E be a Banach space. A : D(A) ⊂ E → E be a closed linear operator and −A generates an equicontinuous C₀-semigroup T(t) (t ≥ 0) of uniformly bounded operators in E. Suppose that the conditions (H1)–(H5) are satisfied. Then for every u₀ ∈ PC(J, E) there exists a τ₁ = τ₁(u₀), 0 < τ₁ < a such that the nonlocal problem

(43)

has a solution u ∈ PC([0, τ₁], E).

Proof. Let the function ; by the similar way one can easily verify the conditions (H1)–(H5) by properly choosing c_i. Hence, by Theorem 14, problem (43) has a solution.

Theorem 16. Let ξ : E × E → R⁺ be a given function. Assume that the following conditions hold:

(A)
there exists ψ ∈ Ψ such that
(44)
for all t ∈ J and for all a, b ∈ E with ξ(a, b) ≥ 0;
(B)
there exists u₀ ∈ PC(J, E) such that ξ(u₀(t), Qu₀(t)) ≥ 0 for all t ∈ J, where a mapping Q : PC(J, E) → PC(J, E) is defined by
(45)
(C)
for each t ∈ J, and u, v ∈ PC(J, E), ξ(u(t), v(t)) ≥ 0 implies that ξ(Qu(t), Qv(t)) ≥ 0;
(D)
for each t ∈ J, if {u_n} is a sequence in PC(J, E) such that u_n → u in PC(J, E) and ξ(u_n(t), u_n+1(t)) ≥ 0 for all , then
(46)
for all

Proof. First of all, let E = PC(J, E). It is easy to see that u ∈ E is a solution of (4) if and only if u ∈ E is a solution of the integral equation

(47)

Then problem (4) is equivalent to finding u^∗ ∈ E which is a fixed point of Q.

Now, let u, v ∈ E such that ξ(u(t), v(t)) ≥ 0 for all t ∈ J. By condition (A), we have

(48)

This implies that for each u, v ∈ E with ξ(u(t), v(t)) ≥ 0 for all t ∈ J, we obtain that

(49)

for all u, v ∈ E. Now, we define the function γ : E × E → [0, ∞) by

(50)

and also we define the w-distance p on E by p(u, v) = ‖u − v‖. From (49), we have

(51)

for all u, v ∈ E. This implies that Q is an (γ, ψ, p)-contractive mapping. From condition (B), there exists u₀ ∈ E such that γ(u₀, Qu₀) ≥ 1. Next, by using condition (C), the following assertions hold for all u, v ∈ E:

(52)

and hence Q is an γ-admissible mapping. Finally, from condition (D), we get that condition (iii) of Lemma 11 holds. Therefore, by Lemma 11, we find x^∗ ∈ E such that x^∗ = Qx^∗ and so x^∗ is a solution of problem (4), which completes the proof.

4. Applications

In this section, we present two examples, which illustrate the applicability of our main results.

Throughout this section, we let E = L²([0, π]) and consider the operator A : D(A)⊆E → E defined by

(53)

It is well known that −A generates a uniformly bounded C₀ semigroup (T(t)) t > 0 in E.

Example 1. We consider the following impulsive fractional differential equation:

(54)

where t ∈ [0,1], x ∈ (0, π), t ≠ 1/2, ∂^α/∂t^α is the Caputo fractional-order partial derivative of order α, 0 < α ≤ 1, u₀ ∈ L²[0, π]. Take J : = [0,1] and so a = 1.

Let

(55)

then the impulsive fractional differential equation (54) can be transformed into the abstract form of problem (4).

Next, let u, v ∈ C(J, E), we calculate

(56)

hence the condition (H1), (H2) holds with K₁ = 1/25, K₂ = K₃ = 2/25, M_R(t) = 1/5, Ω(r) = r, b₁ = b₂ = 2, and c₁ = c₂ = 2. Then we have

(57)

for any bounded set X, Y, Z ∈ C(J, E), I ⊂ J, we have

(58)

Hence the condition (H4), (H5) holds with μ = ρ = 1/7, L₁ = 1/25, L₂ = L₃ = 2/25, and L₄ = L₅ = 2/5. Then, by Theorem 14, problem (54) has a solution on [0,1].

Example 2. Consider the following impulsive fractional differential equation:

(59)

where t ∈ [0,1], x ∈ [0, π], t ≠ 1/2, and ∂^α/∂t^α is the Caputo fractional-order partial derivative of order α, 0 < α ≤ 1, u₀ ∈ L²[0, π], 0 < η₁ < η₂ < ⋯<1, and η_i is given positive constants with

. Take J≔[0,1] and so a = 1.

Let

(60)

then the impulsive fractional differential equation (59) can be transformed into the abstract form of problem (43).

Next, letting u, v ∈ C(J, E), we calculate

(61)

hence the condition (H1), (H2) holds with K₁ = 1/25, K₂ = K₃ = 2/25, M_R(t) = 1/5, and Ω(r) = r. Then we have

(62)

for any bounded set X, Y, Z ∈ C(J, E), I ⊂ J, we have

(63)

Hence the condition (H3), (H4) holds with μ = ρ = 1/7, L₁ = 1/25, L₂ = L₃ = 2/25, L₄ = L₅ = 2/5, and N₁ = N₂ = 2/15. Therefore, by Corollary 15, problem (59) has a solution on [0,1].

5. Conclusions

In this paper, we studied the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11061031).

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Existence of Solutions for Fractional Impulsive Integrodifferential Equations in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

5. Conclusions

Competing Interests

Acknowledgments

References

References

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Existence of Solutions for Fractional Impulsive Integrodifferential Equations in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

5. Conclusions

Competing Interests

Acknowledgments

References

References

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