Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions

This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.

Ji and Li [14] studied the following impulsive differential evolution equations with nonlocal conditions: where A is the generator of a strongly continuous semigroup T β ðtÞ; sufficient conditions for the existence of mild solutions have been established by the Hausdorff measure of noncompactness and fixed point theorems.
Zhu et al. [15] investigated the fractional semilinear integrodifferential equations of mixed type with nonlocal conditions: where 0 < β ≤ 1, AðtÞ is a closed linear operator with domain DðAÞ defined on a Banach space E; the existence and uniqueness of mild solutions have been established by k-set contraction and β-resolvent family. Gou and Li [16] studied the fractional impulsive integrodifferential equations in Banach space E; local and global existences of mild solutions have been proved by measure of noncompactness and Sadovskii's fixed point theorem: where 0 < β < 1, A : DðAÞ ⊂ E ⟶ E is a closed linear operator and −A generates a uniformly bounded C 0 -semigroup TðtÞ. Inspired by these contributions, we consider the following impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions: where c D β t is the Caputo's fractional derivative of order β, β ∈ ð0, 1, AðtÞ is a closed linear operator with domain DðAÞ defined on a Banach space E, and two integral operators G and H are defined by and ω are to be specified later, I k : E ⟶ Eðk = 1, 2, ⋯, mÞ are continuous impulsive functions, the prefixed numbers t k ðk = 1, 2, ⋯, mÞ satisfy 0 = t 0 < t 1 < t 2 < ⋯< t m < t m+1 = T, xðt k Þ = xðt − k Þ, and xðt − k Þ = lim h⟶0 − xðt k + hÞ represent the left limit of xðtÞ at t = t k .
In this paper, we demonstrate the existence of PC-mild solutions for problem (4) via the theory of semigroup and fixed point theorem under the condition of compact resolvent operator. Meanwhile, the uniqueness of PC-mild solutions is proved in the case of noncompact resolvent operator. The kernels g and h of the integral operators G and H are nonlinear functions; the function ω of the nonlocal conditions is noncompact. In addition, the closed linear operator AðtÞ is dependent on t. The rest of this paper is organized as follows. In Section 2, some basic definitions and lemmas are collected that will be needed throughout the remaining sections. The existence and uniqueness of PC-mild solutions are shown in Section 3 via the theories of resolvent operators and various fixed point theorems. Finally, the summary of our results comes in Section 4.
Definition 4 (see [20,21]). Let AðtÞ be a closed and linear operator with domain DðAÞ defined on a Banach space E and β > 0. Let ρ½AðtÞ be the resolvent set of AðtÞ; AðtÞ is called the generator of a β-resolvent family if there exist ω ≥ 0 and a strongly continuous function U β : ℝ 2 + ⟶ BðEÞ such that fλ β : Re λ > ωg ⊂ ρðAÞ and In this case, U β ðt, sÞ is called the β-resolvent family generated by AðtÞ.
Lemma 5 (see [21,22]). U β ðt, sÞ satisfies the following properties: 2 Journal of Function Spaces sÞ is compact for t, s > 0, then the U β ðt, sÞ is continuous in the uniform operator topology Definition 6. A function x ∈ PCðJ, EÞ is said to be a PC-mild solution of problem (4) if xðtÞ satisfies the integral equation:

Existence and Uniqueness of Mild Solution
Theorem 7. Assume that the conditions ðH 1 Þ-ðH 3 Þ hold true and the resolvent operator U β ðt, sÞðt, s > 0Þ is compact.
Proof. Let us consider the operator Q : PCðJ, EÞ ⟶ PCðJ, EÞ as follows: It is easy to see that the operator Q is well defined in PC ðJ, EÞ.

Journal of Function Spaces
For all x ∈ T R , s ∈ J, we get ∥xðsÞ∥≤R, by the condition meanwhile, According to the condition ðH 1 Þ and the above inequalities, for all s ∈ J, we get where Obviously, a 1 ðsÞ and b 1 ðsÞ are nonnegative Lebesgue integrable functions, then Þ ds, Þ ds, In view of Lemma 5, the compactness of the resolvent operator U β ðt, sÞðt, s > 0Þ implies the continuity in the uniform operator topology. As a result, from the above inequalities, we deduce that In the end, we demonstrate that QðT R Þ ⊂ PCðJ, EÞ is precompact.
For any tð0 < t ≤ TÞ, 0 < ε < t, and x ∈ T R , the operator Q ε x is defined by Since U β ðt, sÞ is compact resolvent operator, the set Y ε ðtÞ = fðQ ε xÞðtÞ: x ∈ T R g is relatively compact in E for every ε (0 < ε < t).
Moreover, for any x ∈ T R , t ∈ J, one can show that Thus, YðtÞ = fðQxÞðtÞ: x ∈ T R g is totally bounded. Hence, YðtÞ is relatively compact in E, and so, with the help of the Arzelà-Ascoli theorem, Q : PCðJ, EÞ ⟶ PCðJ, EÞ is completely continuous.
For 0 < λ < 1, let x = λðQxÞ, we get Journal of Function Spaces Then, using the conditions ðH 1 Þ-ðH 3 Þ, it follows that That is, kxðtÞk ≤ ρ for t ∈ J, then there exists a constant ρ 1 > ρ such that kxk PC ≠ ρ 1 . Let V = fx ∈ PCðJ, EÞ: kxk PC < ρ 1 g, obviously, there is no x ∈ ∂V such that x = λðQxÞ for 0 < λ < 1. Therefore, thanks to Lemma 1, one gets that Q has at least one fixed point x in V, which is a PC-mild solution of problem (4). This completes the proof. ☐ Remark 8. Theorem 7 is proved under the condition that U β ðt, sÞ is compact for t, s > 0 and the functions f , g, h meet corresponding conditions; in the case that the resolvent operator U β ðt, sÞ is noncompact, we would obtain Theorem 9 and Theorem 10.
Proof. It follows from the conditions ðH 4 Þ-ðH 6 Þ, for any u, v ∈ PCðJ, EÞ, t ∈ J, one can derive Based on the assumption, we have kQu − Qvk PC < ku − vk PC , which means that the operator Q is a contraction mapping. Hence, the operator Q has a unique fixed point x ⋆ ∈ PCðJ, EÞ, which implies that problem (4) has a unique PC-mild solution. This completes the proof.