Abstract
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.
Similar content being viewed by others
1 Introduction
In this paper, we study the following initial boundary value problem of nonlinear fractional partial integro-differential equations of mixed type with non-instantaneous impulses:
where \(\alpha \in (1,2)\), \(f:[0,b]\times \mathbb{R}^{3}\rightarrow \mathbb{R}\), \(l_{k}:[0,b]\times \mathbb{R}\rightarrow \mathbb{R}\), \(k=1,2,\ldots,m\), \(\varphi \in L^{2}( [0,\pi ])\), \(\mathcal{G}\) and \(\mathcal{S}\) are defined by
\(\mathcal{K}:D\times \mathbb{R}\rightarrow \mathbb{R}_{+}\) and \(\mathcal{H}:D_{0}\times \mathbb{R}\rightarrow \mathbb{R}_{+}\) are continuous and nonlinear functions, \(D=\{(t,s)\in \mathbb{R}^{2}:0\leq s\leq t\leq b\}\), \(D=\{(t,s)\in \mathbb{R}^{2}:0 \leq t, s\leq b\}\), \(\mathbb{R}_{+}=[0,+\infty )\). The pre-fixed numbers \(s_{i}\) and \(t_{i}\) satisfy \(0=s_{0}< t_{1}\leq s_{1}< t_{2}\leq \cdot \cdot \cdot <t_{m}\leq s_{m}< t_{m+1}=b\). The operator \(\mathcal{G}\) is an integral with variable upper limit, the operator \(\mathcal{H}\) is an ordinary definite integral. Therefore, problem (1.1) is called the mixed type integro-differential equations.
The theory of differential equations with instantaneous impulses often describes some processes which have a sudden change in their states at certain times, especially in biology, dynamics, physics, engineering etc. In recent years, a lot of researchers have obtained numerous good results about the fractional differential equations, for example, see [1–17] and the references therein. In [1, 5], the authors studied the stability for impulsive systems; in [2–4, 6, 13, 14], the authors studied the existence results for impulsive differential equations. Yan [7], Chen, Zhang, and Li [8, 11] studied the approximate controllability of the fractional evolution equations.
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 [18–24] and the references therein. In [18], Hernandez and O’Regan firstly studied the integer differential equations with non-instantaneous 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 [21–24], the authors studied the controllability for the fractional differential systems with non-instantaneous impulses. In [25–27], 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:
where
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 \(\mathcal{G}\) and \(\mathcal{S}\) 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 \(\mathcal{K}\) and \(\mathcal{H}\) of the operators \(\mathcal{G}\) and \(\mathcal{S}\) 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.
2 Preliminaries
Let \(E=L^{2}([0,\pi ])\) be a Banach space, \(J=[0,b]\), \(C(J,E)=\{u:J\rightarrow E\text{ is continuous}\}\), \(PC(J,E)=\{u:J\rightarrow E:u\in C((s_{k},t_{k+1}],E),\text{ and there exist } u(t_{k}^{+})\text{ and }u(t_{k}^{-})\text{ with }u(t_{k}^{-})=u(t_{k}), k=1,2,\ldots,m\}\) with the PC-norm \(\|u\|_{PC}=\sup \{\|u(t)\|:t\in J\}\). \(A: D(A)\subset E\rightarrow E\) defined by \(Au=\frac{\partial ^{2}}{\partial x^{2}}u\) with the domain \(D(A)=\{u\in E:u^{\prime \prime }\in E, u(0)=u(\pi )=0\}\), then A is a sectorial operator of type μ. Let \(u(x,t)=u(\cdot ,t)\),
then problem (1.1) can be rewritten as the following abstract form (2.1):
If there exist constant \(0<\theta <\pi /2\), \(\widetilde{M}>0\), \(\mu \in \mathbb{R}\) 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 \(0<\theta <\pi (1-\alpha /2)\).
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 \(\mathcal{T}_{\alpha }:\mathbb{R}_{+}\rightarrow \mathcal{L}(E) \) such that
and
where \(\mathcal{L}(E)\) means the space of bounded linear operators from E to E, then \(\mathcal{T}_{\alpha }(t)\) is called the α-order solution operator generated by A.
Definition 2.2
A function \(u\in PC(J,E)\) is called a PC-mild solution of Eqs. (2.1), if \(u(0)=u_{0}\), and
Lemma 2.1
Let \(0<\varrho <1\), \(\gamma >0\),
Then, for all constant \(0<\xi <1\)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 \(\Phi _{1}\), \(\Phi _{2}\)be maps of D into E such that \(\Phi _{1}x+\Phi _{2}y\in D\)for all \(x,y\in D\). If \(\Phi _{1}\)is a contraction and \(\Phi _{2}\)is completely continuous, then the operator \(\Phi _{1}+\Phi _{2}\)has a fixed point on D.
3 Main results
We assume that there exists a constant \(M>0\) such that \(\|\mathcal{T}_{\alpha }\|\leq M\) for all \(t\in J\). Define an operator \(\Phi : PC(J;E)\rightarrow PC(J;E)\) by
where
Firstly, we give the following hypotheses:
\((H_{1})\) The function \(f:J\times E^{3}\rightarrow E\) is continuous and there exist nonnegative Lebesgue integrable functions \(l_{i}\in L^{1}(J,\mathbb{R}^{+})\) (\(i=1,2,3\)) such that, for all \(t\in J\), \(u_{i},v_{i} \in E\) (\(i=1,2,3\)), we have
\((H_{1}')\) The function \(f:J\times E^{3}\rightarrow E\) is continuous, for a constant \(r>0\), there exist a positive constant ℑ, a Lebesgue integrable function \(\psi \in L^{1}(J,\mathbb{R}^{+})\), and a continuous nondecreasing function \(\Theta :\mathbb{R}^{+}\rightarrow (0,+\infty )\) such that, for any \(t\in J\) and \(u_{i}\in E\) (\(i=1,2,3\)), we have
\((H_{2})\) The functions \(l_{k}:J\times E\rightarrow E\) are continuous and there exist nonnegative constants \(l_{l_{k}}\) such that, for all \(t\in J\), \(u,v\in E \), we have
\((H_{3})\) The functions \(\mathcal{K}:J\times J \times E\rightarrow E\) and \(\mathcal{H}:J\times J \times E\rightarrow E\) are continuous, and there exist nonnegative constants \(l_{\mathcal{K}}\), \(l_{\mathcal{H}}\) such that, for all \(t,s\in J\), \(u,v\in E \), we have
Theorem 3.1
If hypotheses \((H_{1})\)–\((H_{3})\)hold and \(0\leq \tau <1\) (\(\tau =\max \{l_{l_{k}},Ml_{l_{k}}\}\)), then problem (2.1) has a unique PC-mild solution \(u^{*}\in PC(J,E)\), which means that problem (1.1) has a unique PC-mild solution.
Proof
For any \(u,v\in PC(J,E)\), by (3.2) we have
which means
where \(t\in [0,t_{1}]\cup (t_{k},s_{k}] \cup (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\). Then we obtain
where \(t\in [0,t_{1}]\cup (t_{k},s_{k}] \cup (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\). It is clear that we have
where \(t\in [0,t_{1}]\cup (t_{k},s_{k}] \cup (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\).
For any real number \(0<\varepsilon <1\), there exists a continuous function \(\phi (s)\) such that \(\int _{0}^{b}|l(s)-\phi (s)|\,ds<\varepsilon \), where \(l(s)=M(l_{1}(s)+bl_{\mathcal{K}}l_{2}(s)+bl_{\mathcal{H}}l_{3}(s))\) is a Lebesgue integrable function. For any \(t\in [0,t_{1}]\), \(u,v \in PC(J;E)\), by hypotheses \(H_{1}\), \(H_{3}\) and formula (3.3), we have
where \(\max_{t\in J}|\phi (t)|=\nu \). Assume that, for any natural number k, we have
By the formula \(C_{k+1}^{m}=C_{k}^{m}+C_{k}^{m-1}\) and (3.7), we get
By mathematical methods of induction, for any natural number n, we get
where \(\varsigma =\nu b\). By Lemma 2.1, we have
where \(0<\eta <1\), \(\lambda >1\). It is easy to see that the above Eq. (3.9) holds for \(t\in (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\). By (3.5) and (3.9), we obtain
Thus, for any fixed constant \(\lambda >1\), we can find a positive integer \(n_{0}\) such that, for any \(n>n_{0}\), we get \(0<\tau ^{n}+\frac{1}{n^{\lambda }}<1\). Therefore, for any \(u,v\in PC(J,E)\), we have
By the general Banach contraction mapping principle, we get that the operator Φ defined by (3.1) has a unique fixed point \(u^{*}\in PC(J,E)\), 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 \(\Phi _{2}\). Therefore, Theorem 3.1 improves some results that have been studied by the Banach contraction mapping principle.
Theorem 3.2
Assume that the solution operator \(\mathcal{T}_{\alpha }(t)\) (\(t\in J\)) generated by A is compact and functions \(l_{k}\) (\(k=1,2,\ldots,m\)) are bounded. If hypotheses \((H_{1}')\), \((H_{2})\), and \((H_{3})\)hold, then problem (2.1) has at least one PC-mild solution \(u^{*}\in PC(J,E)\), which means that problem (1.1) has at least one PC-mild solution provided that
where
Proof
Step 1. We prove that there exists a positive constant R such that the operator \(\Phi (B_{R})\subset B_{R}\). If the judgment is not right, then for any positive constant r, there would exist \(u_{r}\in B_{r}\) and \(t_{r}\in J\) such that \(\|(\Phi u_{r})\|>r\).
If \(t_{r}\in [0,t_{1}]\), then by (3.1) and \((H_{1}')\), we have
If \(t_{r}\in (t_{k},s_{k}]\), \(k=1,2,\ldots,m\), then by (3.1) and \((H_{2})\), we have
where \(M^{*}=\max_{k=1,2,\ldots,m}\sup_{t\in J}\|l_{k}(t_{r}, \theta )\|\).
If \(t_{r}\in (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\), then by (3.1), \((H_{1}')\), and \((H_{2})\), we have
We know that the inequality \(\|(\Phi u_{r})(t_{r})\|>r\) holds, by (3.10)–(3.13), we get
then
Let \(r\rightarrow +\infty \), we have \(\varpi (\Im \triangle +L)\geq 1\), which contradicts (3.10). Thus, we have that the operator \(\Phi (B_{R})\subset B_{R}\).
Step 2. We prove that \(\Phi _{1}:B_{R}\rightarrow B_{R}\) is a contraction map. For \(t\in [0,t_{1}]\) and \(u,v\in B_{R}\), by (3.2) we have
For \(t\in (t_{k},s_{k}]\), \(k=1,2,\ldots,m\), and \(u,v\in B_{R}\), by (3.2) and \((H_{2})\), we have
For \(t\in (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\), and \(u,v\in B_{R}\), by (3.2) and \((H_{2})\), we have
By (3.16)–(3.18), for any \(u,v\in B_{R}\), we have
From (3.10) we know that the operator \(\Phi _{1}:B_{R}\rightarrow B_{R}\) is a contraction map.
Step 3. We prove that \(\Phi _{2}\) is a continuous operator in \(B_{R}\). Let \(\{u_{n}\}_{0}^{\infty }\subset B_{R}\), and \(u_{n}\rightarrow u\in B_{R}\). By \((H_{1}')\) and \((H_{3})\), we have
For any \(s\in J\), by \((H_{1})\), we get
where the function \(2\psi _{R}(s)\) is Lebesgue integrable for \(s\in [0,t_{1}]\) and \(t\in (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\). Using the Lebesgue dominated convergence theorem and (3.3), (3.20), (3.21), for \(t\in [0,t_{1}]\), we have
For \(t\in (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\), we have
From (3.22) and (3.23), for \(s\in [0,t_{1}]\) and \(t\in (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\), we have
Thus, \(\Phi _{2}\) is a continuous operator in \(B_{R}\).
Step 4. We prove that the operator \(\Phi _{2}:B_{R}\rightarrow B_{R}\) is compact. Firstly, we prove that \(\{(\Phi _{2} u)(t):u\in B_{R}\}\) is relatively compact in E for any \(t\in [0,t_{1}]\) and \(t\in (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\). For any fixed t (\(0< t\leq t_{1}\)) and \(0<\epsilon <t\), let \(u\in B_{R}\) and define
Due to the compactness of \(\mathcal{T}_{\alpha }(t)\), the set \(\{(Q_{2,\epsilon } u)(t): u\in B_{R}\}\) is relatively compact in E for all ϵ (\(0<\epsilon <t\)). For any \(u\in B_{R}\), we get
which means that the set \(\{(\Phi _{2}u)(t): u\in B_{R}\}\) is totally bounded. Therefore, the set \(\{(\Phi _{2}u)(t): u\in B_{R}\}\) is relatively compact in E. Similar to the proof for \(t\in [0,t_{1}]\), we can prove that \(\Phi _{2}(B_{R})(t)\subset E\), \(t\in (s_{k},t_{k+1}]\), \(k=1,2,\ldots,m\), is precompact.
Secondly, we prove that \(\Phi _{2}(B_{R})\) is equicontinuous. Case 1. For \([0,t_{1}]\), let \(0\leq \tau _{1}<\tau _{2}\leq t_{1}\), \(u\in B_{R}\),
The operator \(\mathcal{T}_{\alpha }(t)\) is compact, which means that the operator \(\mathcal{T}_{\alpha }(t)\) is continuous in the sense of uniform operator topology. Thus, \(\|(\Phi _{2}u)(\tau _{2})-(\Phi _{2}u)(\tau _{1})\|\rightarrow 0\) as \(\tau _{2}\rightarrow \tau _{1}\).
Case 2. For \((s_{k},t_{k+1}]\) (\(k=1,2,\ldots,m\)), let \(s_{k}\leq \tau _{1}<\tau _{2}\leq t_{k+1}\), \(u\in B_{R}\). Similar to the proof for Case 1, we have \(\|(\Phi _{2}u)(\tau _{2})-(\Phi _{2}u)(\tau _{1})\|\rightarrow 0\) as \(\tau _{2}\rightarrow \tau _{1}\). Thus, \(\Phi _{2}(B_{R})\) is equicontinuous. By the Arzelá–Ascoli theorem, we get that \(\Phi _{2}:B_{R}\rightarrow B_{R}\) is completely continuous. Therefore, by Lemma 2.2 we get that the operator Φ has a fixed point \(u^{*}\) in \(B_{R}\), which is a PC- mild solution of problem (2.1). It implies that problem (1.1) has a PC-mild solution on the interval \([0,b]\). □
4 Conclusion
In this paper, we turn the initial boundary value problem for the fractional partial integro-differential equations of mixed type with non-instantaneous impulses into the abstract form. The kernels \(\mathcal{K}\) and \(\mathcal{H}\) of the operators \(\mathcal{G}\) and \(\mathcal{S}\) 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.
References
Liu, X.Z., Ramirez, C.: Stability analysis by contraction principle for impulsive systems with infinite delays. Commun. Nonlinear Sci. Numer. Simul. 82, 105021 (2020)
Chen, P.Y., Li, Y.X., Yang, H.: Perturbation method for nonlocal impulsive evolution equations. Nonlinear Anal. Hybrid Syst. 8, 22–30 (2013)
Chen, P.Y., Li, Y.X., Zhang, X.P.: Double perturbations for impulsive differential equations in Banach spaces. Taiwan. J. Math. 20, 1065–1077 (2016)
Hao, X.A., Liu, L.S.: Mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces. Math. Methods Appl. Sci. 40, 4832–4841 (2017)
Suriguga, M., Kao, Y.G., Hyder, A.A.: Uniform stability of delayed impulsive reaction-diffusion systems. Appl. Math. Comput. 372, 124954 (2020)
Zhang, T.W., Xiong, L.L.: Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative. Appl. Math. Lett. 101, 106072 (2020)
Yan, Z.M., Lu, F.X.: Approximate controllability of a multi-valued fractional impulsive stochastic partial integro-differential equation with infinite delay. Appl. Math. 292, 425–447 (2017)
Chen, P.Y., Zhang, X.P., Li, Y.X.: Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators. Fract. Calc. Appl. Anal. 23(1), 268–291 (2020)
Chen, P.Y., Zhang, X.P., Li, Y.X.: Fractional non-autonomous evolution equation with nonlocal conditions. J. Pseudo-Differ. Oper. Appl. 10(4), 955–973 (2019)
Chen, P.Y., Li, Y.X., Zhang, X.P.: Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete Contin. Dyn. Syst., Ser. B. https://doi.org/10.3934/dcdsb.2020171
Chen, P.Y., Zhang, X.P., Li, Y.X.: Approximate controllability of non-autonomous evolution system with nonlocal conditions. J. Dyn. Control Syst. 26(1), 1–16 (2020)
Chen, P.Y., Zhang, X.P., Li, Y.X.: A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Commun. Pure Appl. Anal. 17(5), 1975–1992 (2018)
Hao, X.A., Liu, L.S.: Mild solution of second-order impulsive integro-differential evolution equations of Volterra type in Banach spaces. Qual. Theory Dyn. Syst. 19(1), 1–18 (2020)
Hao, X.A., Zuo, M.Y., Liu, L.S.: Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities. Appl. Math. Lett. 82, 24–31 (2018)
Hao, X.A., Zhang, L.Y.: Positive solutions of a fractional thermostat model with a parameter. Symmetry 11(1), 122 (2019)
Hao, X.A., Sun, H., Liu, L.S., Wang, D.B.: Positive solutions for semipositone fractional integral boundary value problem on the half-line. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(4), 3055–3067 (2019)
Hao, X.A., Zhang, L.Y., Liu, L.S.: Positive solutions of higher order fractional integral boundary value problem with a parameter. Nonlinear Anal., Model. Control 24(2), 210–223 (2019)
Hernandez, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141, 1641–1649 (2013)
Chen, P.Y., Zhang, X.P., Li, Y.X.: Non-autonomous evolution equations of parabolic type with non-instantaneous impulses. Mediterr. J. Math. 16, 118 (2019)
Chen, P.Y., Zhang, X.P., Li, Y.X.: Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families. J. Fixed Point Theory Appl. 21, 84 (2019)
Wang, J.R., Ibrahim, A.G., O’Regan, D.: Controllability of Hilfer fractional noninstantaneous impulsive semilinear differential inclusions with nonlocal conditions. Nonlinear Anal., Model. Control 24(6), 958–984 (2019)
Liu, S.D., Debbouche, A., Wang, J.R.: ILC method for solving approximate controllability of fractional differential equations with noninstantaneous impulses. J. Comput. Appl. Math. 339, 343–355 (2018)
Wang, J.R.: Stability of noninstantaneous impulsive evolution equations. Appl. Math. Lett. 73, 157–162 (2017)
Sousa, J.V.D., Oliveira, D.D., Oliveira, E.C.: On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation. Math. Methods Appl. Sci. 42(4), 1249–1261 (2019)
Ouyang, Z.G.: Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay. Comput. Math. Appl. 61, 860–870 (2011)
Zhu, B., Liu, L.S., Wu, Y.H.: Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay. Appl. Math. Lett. 61, 73–79 (2016)
Zhu, B., Liu, L.S., Wu, Y.H.: Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay. Comput. Math. Appl. 78, 1811–1818 (2019)
Guo, D.J.: Solutions of nonlinear integro-differential equations of mixed type in Banach spaces. J. Appl. Math. Simul. 2, 1–11 (1989)
Chen, P.Y., Zhang, X.P., Li, Y.X.: Cauchy problem for fractional non-autonomous evolution equations. Banach J. Math. Anal. 14(2), 559–584 (2020)
Agarwal, R.P., Andrade, B.D., Siracusa, G.: On fractional integro-differential equations with state-dependent delay. Comput. Math. Appl. 62, 1143–1149 (2011)
Liu, L.S., Wu, C.X., Guo, F.: Existence theorems of global solutions of initial value problems for nonlinear integro-differential equations of mixed type in Banach spaces and applications. Comput. Math. Appl. 47, 13–22 (2004)
Liu, L.S., Guo, F., Wu, C.X., Wu, Y.H.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. J. Math. Anal. Appl. 309, 638–649 (2005)
Availability of data and materials
No data were used to support this study.
Funding
This work was supported by the Project of Shandong Province Higher Educational Science and Technology Program (No.J16LI14), the National Natural Science Foundation of China (No.11871302, 11301303), the National Social Science Foundation of China (No.15BJY007), the Taishan Scholars Program of Shandong Province (No.tsqn20161041), the Humanities and Social Sciences Project of the Ministry Education of China (No. 19YJA910002), the Natural Science Foundation of Shandong Province (No.ZR2018MG002), the Fostering Project of Dominant Discipline and Talent Team of Shandong Province Higher Education Institutions (No.1716009), the Shandong Provincial Social Science Project Planning Research Project(19CQXJ08), the Risk Management and Insurance Research Team of Shandong University of Finance and Economics, the Excellent Talents Project of Shandong University of Finance and Economics, the Collaborative Innovation Center Project of the Transformation of New, and old Kinetic Energy and Government Financial Allocation.
Author information
Authors and Affiliations
Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhu, B., Han, B., Liu, L. et al. On the fractional partial integro-differential equations of mixed type with non-instantaneous impulses. Bound Value Probl 2020, 154 (2020). https://doi.org/10.1186/s13661-020-01451-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-020-01451-z