Perturbation technique for a class of nonlinear implicit semilinear impulsive integrodifferential equations of mixed type with noncompactness measure
 Hengyou Lan^{1, 2}Email author and
 Yishun Cui^{3}
https://doi.org/10.1186/s136620140329y
© Lan and Cui; licensee Springer 2015
Received: 27 August 2014
Accepted: 19 December 2014
Published: 30 January 2015
Abstract
By using the ArzelaAscoli theorem, the Bellman inequality, and a monotone perturbation iterative technique in the presence of lower and upper solutions, we discuss the existence of mild solutions for a class of nonlinear firstorder implicit semilinear impulsive integrodifferential equations in Banach spaces. Under wide monotone conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique mild solution between lower and upper solutions.
Keywords
1 Introduction
The theory of impulsive differential equations has become an important area of investigation in recent years stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. Various evolutionary processes undergo abrupt changes of states at certain moments of time; between intervals of continuous evolution such changes can be well approximated as being instantaneous changes at state, or in the form of impulses. These process are modeled by impulse differential equations and have been the most important research directions and connections for impulsive differential equations; see, for example, [1–7] and the references therein. Subsequently, many authors have investigated the existence of solutions to impulsive differential equations or (implicit) impulsive integrodifferential equations with their strong applications in Banach spaces; see [1–27] and the references therein.
Recently, Lan and Cui [15] studied a class of initial value problems of nonlinear firstorder implicit impulsive integrodifferential equations in Banach space. By using the Mönch fixed point theorem, they obtained some new existence theorems of solutions for this class of nonlinear firstorder implicit impulsive integrodifferential equations in Banach spaces under some weaker conditions. Furthermore, some (implicit) impulsive differential equations under various initial and boundary conditions has also been studied by several authors; see, for example, [11, 13, 22, 24, 28] and the references therein. By using a monotone iterative technique in the presence of lower and upper solutions, Lan [23] discussed the existence of solutions for a new class of nonlinear firstorder implicit impulsive integrodifferential equations in Banach spaces. Under wide monotone conditions and the noncompactness measure conditions, he also obtained the existence of extremal solutions and a unique solution between lower and upper solutions. In [25], Chen and Li introduced and studied a class of semilinear impulsive evolution equations in Banach spaces by using a mixed monotone iterative technique. The presented results improved and extended some relevant results in ordinary differential equations and partial differential equations. For related works, see [9, 26, 27, 29, 30] and the references therein.
On the other hand, the monotone iterative technique, which is one of the approximation methods for finding solutions of a comparatively large class of impulsive differential equations, can be applied in practice easily; see, for example, [14, 16, 17, 23, 25–27, 29]. Further, some nice examples of the monotone iterative technique can be found in [20, 21]. As a matter of fact, Li and Liu [16] pointed out that ‘the monotone iterative technique in the presence of lower and upper solutions is an important method for seeking solutions of differential equations in abstract spaces’. Moreover, Li and Liu [16] used a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of solutions for the initial value problem of the impulsive integrodifferential equation of Volterra type in a Banach space. Under monotone conditions and the noncompactness measure condition of the nonlinearity function f, the authors also obtained the existence of extremal solutions and a unique solution between lower and upper solutions. In [14], by using a monotone iterative technique in the presence of lower and upper solutions, we discussed the existence of solutions for a new system of nonlinear mixed type implicit impulsive integrodifferential equations in Banach spaces. Under some monotonicity conditions and the noncompactness measure conditions, they also obtained the existence of extremal solutions and a unique solution between lower and upper solutions.
2 Preliminaries
Let \(C^{1}(J, {\mathbb{B}})\) denote the Banach space of all continuous differentiable \({\mathbb{B}}\)value functions on interval J with norm \(\x\_{C^{1}}=\max_{t\in J}\x^{\prime}(t)\\). Let \(\alpha(\cdot)\) denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [32]. For any \(E\subset C^{1}(J, {\mathbb{B}})\) and \(t\in J\), set \(E(t)= \{ x(t) \mid x\in E\}\subset{\mathbb{B}}\). If E is bounded in \(C^{1}(J, {\mathbb{B}})\), then \(E(t)\) is bounded in \({\mathbb{B}}\), and \(\alpha(E(t))\le\alpha(E)\).
Lemma 2.1
Proof
It follows from Theorem 2.9 of [31, Chapter 4] and Lemma 2.2 in [26] that this conclusion follows directly. □
Lemma 2.2
[33]
Lemma 2.3
[33]
Lemma 2.4
[34, Corollary 3.1(b)]
3 Existence and uniqueness theorems
In this section, we will prove our main results concerning the mild solutions of the nonlinear firstorder implicit impulsive integrodifferential equation (1.1) in Banach spaces.
Definition 3.1
Definition 3.2
A \(C_{0}\)semigroup \(G(t)\) (\(t\ge t_{0}\)) in \(\mathbb{B}\) is said to be positive, if the order inequality \(G(t)x\ge\theta\) holds for every \(x\ge\theta \), \(x\in{\mathbb{B}}\), and \(t\ge t_{0}\).
It is easy to see that for any \(M\ge0\), \(AMI\) also generates a \(C_{0}\)semigroup \(\Gamma(t)=e^{M(t)}G(t)\) (\(t\ge t_{0}\)) in \(\mathbb{B}\). \(\Gamma(t)\) is a positive \(C_{0}\)semigroup if \(G(t)\) is a positive \(C_{0}\)semigroup for all \(t\ge t_{0}\).
 (H_{1}):

Problem (1.1) has a lower solution \(y_{0}\in PC^{1}(J, {\mathbb{B}})\) and an upper solution \(x_{0}\in PC^{1}(J, {\mathbb{B}})\) with \(y_{0}\le x_{0}\), and there exist constants \(M\in(0,1)\) such thatfor all \(t\in J\) and \(y_{0}(t)\le u_{1}\le u_{2}\le x_{0}(t)\), \(Ty_{0}(t)\le v_{1}\le v_{2}\le Tx_{0}(t)\), and \(y_{0}^{\prime}(t)\le w_{1}\le w_{2}\le x_{0}^{\prime}(t)\).$$f(t, u_{2}, v_{2}, w_{2})f(t, u_{1}, v_{1}, w_{1})\geM(u_{2}u_{1}) $$
 (H_{2}):

\(I_{k}(x)\) is increasing on the order interval \([y_{0}(t),x_{0}(t)]\) for \(t\in J\), \(k=1, 2,\ldots, m\).
 (H_{3}):

There exist \(0<2L<1M\) such thatfor all \(t\in J\), and increasing or decreasing monotonic sequences \(\{u_{n}\}\subset[y_{0}(t),x_{0}(t)]\), \(\{v_{n}\}\subset[Ty_{0}(t), Tx_{0}(t)]\) and \(\{w_{n}\}\subset[y_{0}^{\prime}(t), x_{0}^{\prime}(t)]\).$$\alpha\bigl(\bigl\{ f\bigl(t,u_{n}(t),v_{n}(t),w_{n}(t) \bigr)\bigr\} \bigr)\le L\bigl[\alpha\bigl(\bigl\{ u_{n}(t)\bigr\} \bigr)+ \alpha\bigl(\bigl\{ v_{n}(t)\bigr\} \bigr)+\alpha\bigl(\bigl\{ w_{n}(t)\bigr\} \bigr)\bigr] $$
In the sequel, we prove the following main results of this paper.
Theorem 3.1
Let \({\mathbb{B}}\) be an ordered Banach space, whose positive cone P is normal, \(A: \operatorname{dom}(A)\subset{\mathbb{B}}\to{\mathbb{B}}\) be a closed linear operator, the positive \(C_{0}\)semigroup \(G(t)\) (\(t\ge t_{0}\)) generated by A be compact in \({\mathbb{B}}\), \(f\in C(J\times{\mathbb{B}}\times{\mathbb{B}}\times{\mathbb{B}}, {\mathbb{B}})\), and \(I_{k}\in C({\mathbb{B}}, {\mathbb{B}})\) for \(k=1, 2, \ldots, m\). Suppose that the conditions (H_{1}) ∼ (H_{3}) hold. Then problem (1.1) has minimal and maximal mild solutions between \([y_{0},x_{0}]\), which can be obtained by a monotone iterative procedure starting from \(y_{0}\) and \(x_{0}\), respectively.
Proof
It is easy to see that \(F: PC^{1}(J, {\mathbb{B}})\to PC^{1}(J, {\mathbb{B}})\) is continuous. By Lemma 2.1, we know that the mild solution of problem (1.1) is equivalent to the fixed point of F. Since \(G(t)\) (\(t\ge t_{0}\)) is a positive \(C_{0}\)semigroup, \(G(0)=I\) ([31]) and it follows from assumptions (H_{1}) and (H_{2}) that F is increasing in \([y_{0},x_{0}]\) and maps any bounded set in \([y_{0},x_{0}]\) into a bounded set.
Similarly, we can prove the compactness of Ψ in (3.3).
For any \(t\in J\), since \(\{Fu(t) \mid u\in[y_{0},x_{0}]\}=\{\Gamma (tt_{0})+\Phi u(t)+\Psi u(t) \mid u\in[y_{0},x_{0}]\}\), and \(Fu(t_{0})=u_{0}\) is precompact in \(\mathbb{B}\), we know that \(F([y_{0},x_{0}])\) is precompact in \(C^{1}(J_{k}, {\mathbb{B}})\) by using the ArzelaAscoli theorem. Thus, \(F: [y_{0},x_{0}]\to[y_{0},x_{0}]\) is completely continuous.
Finally, we show that problem (1.1) has minimal and maximal mild solutions between \([y_{0},x_{0}]\), which can be obtained by a monotone iterative procedure starting from \(y_{0}\) and \(x_{0}\), respectively.
It follows from the completely continuity of F that F has minimal and maximal fixed points \(\underline{u}\) and \(\overline{u}\) in \([y_{0},x_{0}]\), and so they are the minimal and maximal mild solutions of problem (1.1) in \([y_{0},x_{0}]\), respectively.
Again by the Bellman inequality, we know that \(\phi(t)\equiv0\) in \(J_{2}^{\prime}\), from which we obtain \(\alpha(E_{0}(t_{2}))=0\) and \(\alpha(I_{2}(E_{0}(t_{2})))=0\).
Continuing such a process interval by interval up to \(J_{m+1}^{\prime}\), we can prove that \(\phi(t)\equiv0\) in every \(J_{k}^{\prime}\), \(k=1, 2, \ldots, m+1\).
For any \(J_{k}\), if we modify the value of \(y_{n}\) at \(t=t_{k1}\) via \(y_{n}(t_{k1})=y_{n}(t^{+}_{k1})\), \(n\in{\mathbb{N}}\), then \(\{y_{n}\}\subset C^{1}(J_{k}, {\mathbb{B}})\) and it is equicontinuous. Since \(\alpha(\{y_{n}(t)\})\equiv0\), \(\{y_{n}(t)\}\) is precompact in \({\mathbb{B}}\) for every \(t\in J_{k}\). By the ArzelaAscoli theorem, we know that \(\{y_{n}\}\) is precompact in \(C^{1}(J_{k}, {\mathbb{B}})\). Hence, \(\{y_{n}\}\) has a convergent subsequence in \(C^{1}(J_{k}, {\mathbb{B}})\). Combining this with the monotonicity (3.8), we easily prove that \(\{y_{n}\}\) itself is convergent in \(C^{1}(J_{k}, {\mathbb{B}})\). In particular, \(\{y_{n}(t)\}\) is uniformly convergent in \(J_{k}^{\prime}\). Consequently, \(\{y_{n}(t)\}\) is uniformly convergent over the whole of J.
By the monotonicity of F, it is easy to see that \(\underline{u}\) and \(\overline{u}\) are the minimal and maximal fixed points of F in \([y_{0},x_{0}]\), and therefore they are the minimal and maximal mild solutions of problem (1.1) in \([y_{0},x_{0}]\), respectively. This completes the proof. □
Remark 3.1
From Theorem 3.1, we obtain the following result.
Corollary 3.1
Let \({\mathbb{B}}\) be an ordered and weakly sequentially complete Banach space, whose positive cone P is normal, \(f\in C(J\times{\mathbb{B}}\times{\mathbb{B}}\times{\mathbb{B}}, {\mathbb{B}})\), and \(I_{k}\in C({\mathbb{B}}, {\mathbb{B}})\), \(k=1, 2, \ldots, m\). If the conditions (H_{1}) and (H_{2}) are satisfied, then problem (1.1) has minimal and maximal mild solutions between \(y_{0}\) and \(x_{0}\), which can be obtained by a monotone iterative procedure starting from \(y_{0}\) and \(x_{0}\), respectively.
Next we discuss the uniqueness of the mild solution to problem (1.1) in \([y_{0},x_{0}]\). Assume we replace the assumption (H_{3}) by the following assumption.
Theorem 3.2
Let \({\mathbb{B}}\) be an ordered Banach space, whose positive cone P is normal, \(f\in C(J\times{\mathbb{B}}\times{\mathbb{B}}\times{\mathbb{B}}, {\mathbb{B}})\) and \(I_{k}\in C({\mathbb{B}}, {\mathbb{B}})\), \(k=1, 2, \ldots, m\). If the conditions (H_{1}), (H_{2}), and (H_{4}) hold, then problem (1.1) has a unique mild solution between \(y_{0}\) and \(x_{0}\), which can be obtained by a monotone iterative procedure starting from \(y_{0}\) or \(x_{0}\).
Proof
Therefore, by Theorem 3.1, problem (1.1) has a minimal solution \(\underline{u}\) and a maximal solution \(\overline{u}\) in \([y_{0},x_{0}]\). By the proof of Theorem 3.1, (3.7), (3.8), (3.10), and (3.11) are valid. Going from \(J_{1}^{\prime}\) to \(J_{m+1}^{\prime}\) interval by interval, we show that \(\underline{u}(t)\equiv\overline {u}(t)\) in every \(J^{\prime}_{k}\), \(k=1,2,\ldots,m+1\).
Continuing such a process interval by interval up to \(J_{m+1}^{\prime}\), we see that \(\underline{u}(t)\equiv\overline{u}(t)\) over the whole of J. Hence, \(u^{*}:=\underline{u}=\overline{u}\) is the unique mild solution of problem (1.1) in \([y_{0},x_{0}]\), which can be obtained by the monotone iterative procedure (3.7) starting from \(y_{0}\) or \(x_{0}\). □
Remark 3.2
(1) Using the above argument method interval by interval from \(J_{1}^{\prime}\) to \(J_{m+1}^{\prime}\), we can also improve the main results in [19] and [21], and delete some restrictive conditions there.
(2) In this study, the equicontinuity of the semigroup \(G(t)\) (\(t\ge t_{0}\)) generated by A is not required.
4 Concluding remarks
In this paper, we introduce and study the following nonlinear firstorder implicit impulsive differential equation problem in Banach space \({\mathbb{B}}\):
By using a monotone iterative technique in the presence of lower and upper solutions, the existence of extremal solutions and a unique mild solution between the lower and upper solutions are obtained under wide monotone conditions and the noncompactness measure conditions. The results presented in this paper improved and generalized some known results concerned with the integrodifferential equations and classical (abstract) differential equations.
Moreover, we remark that if the lower solution and the upper solution for problem (1.1) do not exist, then we have the following results.
Theorem 4.1
 (1)
If the \(C_{0}\)semigroup \(G(t)\) (\(t\ge t_{0}\)) is compact in \(\mathbb{B}\), and the conditions (H_{1}) and (H_{2}) in Section 3 are satisfied, then problem (1.1) has minimal and maximal mild solutions.
 (2)
Problem (1.1) has minimal and maximal mild solutions when the conditions (H_{1}) ∼ (H_{3}) in Section 3 are satisfied.
 (3)
If the positive cone P is regular, and the conditions (H_{1}) and (H_{2}) in Section 3 are satisfied, then problem (1.1) has minimal and maximal mild solutions.
 (4)
Problem (1.1) has a unique mild solution when the conditions (H_{1}), (H_{2}), and (H_{4}) in Section 3 are satisfied.
Proof
Declarations
Acknowledgements
We thank the referees’ valuable comments and suggestions to improve our paper.
This work was partially supported by Sichuan Province Cultivation Fund Project of Academic and Technical Leaders, and the Open Research Fund of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (2013WZJ01).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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