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On the fractional partial integro-differential equations of mixed type with non-instantaneous impulses

In this paper, we consider the initial boundary value problem for a class of nonlinear fractional partial integro-differential equations of mixed type with non-instantaneous impulses in Banach spaces. Sufficient conditions of existence and uniqueness of PC-mild solutions for the equations are obtained via general Banach contraction mapping principle, Krasnoselskii’s fixed point theorem, and α-order solution operator.

Meanwhile, fractional differential equations with non-instantaneous impulsive effects have been applied widely as mathematical models to consider many phenomena in biology, dynamics, physics, control model, etc., see [18][19][20][21][22][23][24] and the references therein. In [18], Hernandez and O'Regan firstly studied the integer differential equations with noninstantaneous impulses. In [19,20], Chen, Zhang, and Li studied the non-autonomous evolution equations with non-instantaneous impulses and obtained the main results of the existence. In [21][22][23][24], the authors studied the controllability for the fractional differential systems with non-instantaneous impulses. In [25][26][27], the authors studied the initial boundary value problem for time fractional partial differential equations with delay and discussed the existence and uniqueness of the mild solutions. In [28,29], the authors also studied the differential equations of mixed type. Guo [28] studied the existence and uniqueness of the following integer nonlinear integro-differential equations of mixed type in a Banach space E: the kernels K and H are linear functions. Chen, Zhang, and Li [19] studied the existence of the following fractional non-autonomous integro-differential evolution equations of mixed type: where the operators G and S are the same as in (1.4), and the kernels K and H are also linear functions.
To the best of our knowledge, we have not found the relevant results that study the initial boundary value problem for the fractional partial integro-differential equations of mixed type with non-instantaneous impulses. Therefore, motivated by the above-mentioned papers, we study the existence of PC-mild solutions for problem (1.1). In this paper, the kernels K and H of the operators G and S are nonlinear functions. The nonlinear term f satisfies the Lipschitz condition, where the Lipschitz coefficients are Lebesgue integrable functions. In the proof of the main results by the general Banach contraction mapping principle, we do not need extra conditions to ensure the contraction coefficients less than one. Our main results of this paper generalize and improve some corresponding results.

Preliminaries
then problem (1.1) can be rewritten as the following abstract form (2.1): If there exist constant 0 < θ < π/2, M > 0, μ ∈ R such that its resolvent exists outside the sector then the operator A is called sectorial operator of type μ, where the linear operator A in problem (2.1) is sectorial of type μ with 0 < θ < π(1α/2).

Definition 2.1 ([30]) Let A be a closed and linear operator with domain D(A) defined on a Banach space E.
If there exist a real number μ and a strongly continuous function and where L(E) means the space of bounded linear operators from E to E, then T α (t) is called the α-order solution operator generated by A.
Then, for all constant 0 < ξ < 1 and all real number s > 1, we get

Lemma 2.2 (Krasnoselskii's fixed point theorem) Let D be a bounded closed and convex
subset of a Banach space E, and let 1 , 2 be maps of D into E such that 1 x + 2 y ∈ D for all x, y ∈ D. If 1 is a contraction and 2 is completely continuous, then the operator 1 + 2 has a fixed point on D.

Main results
We assume that there exists a constant M > 0 such that T α ≤ M for all t ∈ J. Define an operator : Firstly, we give the following hypotheses: (H 1 ) The function f : J × E 3 → E is continuous, for a constant r > 0, there exist a positive constant , a Lebesgue integrable function ψ ∈ L 1 (J, R + ), and a continuous nondecreasing function : R + → (0, +∞) such that, for any t ∈ J and u i ∈ E (i = 1, 2, 3), we have (H 2 ) The functions l k : J × E → E are continuous and there exist nonnegative constants l l k such that, for all t ∈ J, u, v ∈ E, we have (H 3 ) The functions K : J × J × E → E and H : J × J × E → E are continuous, and there exist nonnegative constants l K , l H such that, for all t, s ∈ J, u, v ∈ E, we have Proof For any u, v ∈ PC(J, E), by (3.2) we have which means where max t∈J |φ(t)| = ν. Assume that, for any natural number k, we have By the formula C m k+1 = C m k + C m-1 k and (3.7), we get By mathematical methods of induction, for any natural number n, we get n 2 u -n 2 v PC ≤ C 0 n ε n + C 1 n ε n-1 ς 1 1! + · · · + C n n ε n-n ς n n! uv PC , (3.8) where ς = νb. By Lemma 2.1, we have where 0 < η < 1, λ > 1. It is easy to see that the above Eq. (3.9) holds for t ∈ (s k , t k+1 ], k = 1, 2, . . . , m. By (3.5) and (3.9), we obtain Thus, for any fixed constant λ > 1, we can find a positive integer n 0 such that, for any n > n 0 , we get 0 < τ n + 1 n λ < 1. Therefore, for any u, v ∈ PC(J, E), we have By the general Banach contraction mapping principle, we get that the operator defined by (3.1) has a unique fixed point u * ∈ PC(J, E), which means that problem (1.1) has a unique PC-mild solution.
Remark 3.1 In Theorem 3.1, we prove the existence and uniqueness of the PC-mild solutions for problem (1.1) using the general Banach contraction mapping principle. Note that we do not need extra conditions to ensure the contraction constant 0 < k < 1 for the operator 2 . Therefore, Theorem 3.1 improves some results that have been studied by the Banach contraction mapping principle.

Proof
Step 1. We prove that there exists a positive constant R such that the operator (B R ) ⊂ B R . If the judgment is not right, then for any positive constant r, there would exist u r ∈ B r and t r ∈ J such that ( u r ) > r.
Step 3. We prove that 2    Thus, 2 is a continuous operator in B R .

Conclusion
In this paper, we turn the initial boundary value problem for the fractional partial integrodifferential equations of mixed type with non-instantaneous impulses into the abstract form. The kernels K and H of the operators G and S are nonlinear functions. The nonlinear term f satisfies the Lipschitz condition, where the Lipschitz coefficients are Lebesgue integrable functions. The main results are obtained via general Banach contraction mapping principle, Krasnoselskii's fixed point theorem, and α-order solution operator. In the proof of the main results by the general Banach contraction mapping principle, we do not need extra conditions to ensure the contraction coefficients less than one. Our main results of this paper generalize and improve some corresponding results.