- Research
- Open Access
Existence of solutions of abstract non-autonomous second order integro-differential equations
- Hernán R Henríquez^{1}Email author and
- Juan C Pozo^{2}
https://doi.org/10.1186/s13661-016-0675-7
© Henríquez and Pozo 2016
- Received: 17 December 2015
- Accepted: 6 September 2016
- Published: 15 September 2016
Abstract
In this paper the existence of solutions of a non-autonomous abstract integro-differential equation of second order is considered. Assuming the existence of an evolution operator corresponding to the associate abstract non-autonomous Cauchy problem of second order, we establish the existence of a resolvent operator for the homogeneous integro-differential equation and the existence of mild solutions to the inhomogeneous integro-differential equation. Furthermore, we study the existence of classical solutions of the integro-differential equation. Finally, we apply our results to the study of the existence of solutions of a non-autonomous wave equation.
Keywords
- abstract integro-differential equations of second order
- abstract Cauchy problem of second order
- differential equations in abstract spaces
- resolvent operator
- cosine function of operators
MSC
- 45N05
- 45K05
- 34G25
- 47D09
1 Introduction
Abstract integro-differential equations have been used to model various physical phenomena. For this reason, this type of equations has attracted the attention of authors in recent years. We refer the reader to the recent papers of Diagana [1], Vijayakumar et al. [2] and the references therein.
This paper is devoted to the study of the existence of mild and classical solutions for initial value problems described as an abstract non-autonomous second order integro-differential equation in Banach spaces.
This paper has five sections. In the next section we study the existence of solutions of (1.1)-(1.2) when \(P(t,s)\) is a bounded linear map from X into X. In Section 3, we discuss the existence of a resolvent operator for the homogeneous integro-differential equation. In Section 4, we are concerned with the existence of mild and classical solutions of the inhomogeneous non-autonomous integro-differential equation (1.1)-(1.2). Finally, in Section 5 we apply our results to the study of the existence of solutions to non-autonomous wave equations.
The general terminology and notations used in this text are the following. When \((Y,\|\cdot\|_{Y})\) and \((Z,\|\cdot\|_{Z})\) are Banach spaces, we denote by \(\mathcal{L}(Y,Z)\) the Banach space of the bounded linear operators from Y into Z endowed with the norm of operators and we abbreviate this notation to \(\mathcal{L}(Y)\) whenever \(Z=Y\). For a compact set K, we denote by \(C(K,X)\) the space of continuous functions from K into X provided with the norm of uniform convergence. For a closed linear operator \(A : D(A) \subseteq X \to X\), we denote by \(\sigma (A)\) (resp. \(\rho(A)\)) its spectrum (resp. its resolvent set). Moreover, we represent by \([D(A)]\) the domain of A endowed with the graph norm \(\| x \|_{A} = \| x \| + \| Ax \|\), \(x \in D(A)\). Finally, \(\operatorname{Im}(T)\) denotes the range space of a linear operator T.
2 Existence of mild solutions
In this section we study the existence of mild solutions of problem (1.1)-(1.2). Throughout this section we assume that \(P: \Delta= \{(t,s): 0 \leq s \leq t \leq a \}\to\mathcal{L}(X)\) is a strongly continuous map.
Motivated by (1.6), we consider the following concept of solution.
Definition 2.1
The following result is an immediate consequence of this definition. We next abbreviate the notation by writing \({ \|P\| = \sup_{0 \leq s \leq t \leq a} \|P(t,s)\|}\).
Theorem 2.1
Assume the function \(f : [\sigma , a] \to X\) is integrable. Then for each \(y, z \in X\), problem (1.1)-(1.2) has a unique mild solution.
Proof
Now, we proceed to define the resolvent operator corresponding to problem (2.1)-(2.2). Let \(x(t, \sigma , y, z)\) be the mild solution of problem (2.1)-(2.2) for \(f = 0\). We define \(R(t, \sigma ) z = x(t, \sigma , 0, z)\).
Theorem 2.2
Proof
In most of cases, even when \(f = 0\), the mild solution constructed in Theorem 2.1 does not satisfy equation (1.1). In the next sections, through the introduction of the concept of resolvent operator for the homogeneous equation, we analyze the existence of solution of problem (1.1)-(1.2). We will establish our results for a general situation when the operator \(P(t,s)\) is unbounded.
3 Existence of resolvent
To establish our results we introduce the following terminology. Let \([D]\) be the space D endowed with the graph norm corresponding to the operator \(A(0)\). Since \(A(t)\) is a closed linear operator on D, by the closed graph theorem we see that \([D]\) is a Banach space, and \(A(t) \in\mathcal{L}([D],X)\) for all \(t \in I\). Let now \([D]_{t}\) be the Banach space D endowed with the graph norm corresponding to the operator \(A(t)\). It is an immediate consequence of the previous assertion that \([D]\) and \([D]_{t}\) are equivalent Banach spaces. Moreover, since \(A : I \to\mathcal{L}([D],X)\) is strongly continuous, it follows from the uniform boundedness principle that there is a constant Ñ such that \(\|A(t)\|_{\mathcal{L}([D],X)} \leq\widetilde{N}\) for all \(t \in I\).
- (H1)For every \(0 \leq s \leq t \leq a\), \(P(t, s) : [D] \to Z\) is a bounded linear operator, and for each \(x \in D\), the function \(P(\cdot, \cdot) x \) is continuous with values in Z, andfor some constant \(b > 0\) independent of \(s, t \in\Delta\).$$\bigl\Vert P(t,s) x \bigr\Vert _{Z} \leq b \Vert x\Vert _{[D]} $$
- (H2)There exists a constant \(N_{2} \geq0\), such thatfor all \(\xi\leq t \in[0, a]\) and all \(x \in D\).$$\biggl\Vert \int_{\xi}^{t} S(t,s) P(s, \xi) x \,ds\biggr\Vert \leq N_{2} \Vert x\Vert $$
Motivated by the results in [21], we consider the following concept of solution (or classical solution).
Definition 3.1
A function \(x : [\sigma , a] \to[D]\) is said to be a solution of problem (3.1)-(3.2) if \(x(\cdot) \in C([\sigma , a], [D]) \cap C^{2}([\sigma , a], X)\) and (3.1) and (3.2) are satisfied.
We are in a position to establish our first result of the existence of solutions.
Theorem 3.1
Proof
- (H3)For every \(0 \leq s \leq t \leq a\), \(P(t, s) : [D] \to X\) is a bounded linear operator, and for each \(x \in D\), \(P(\cdot, \cdot) x \) is continuous andfor some \(b > 0\) independent of \(s, t \in\Delta\).$$\bigl\Vert P(t,s) x \bigr\Vert _{X} \leq b \Vert x\Vert _{[D]} $$
- (H4)There exists a constant \(L_{P} > 0\) such thatfor all \(x \in D\), \(0 \leq s \leq t_{1} \leq t_{2}\).$$\bigl\Vert P(t_{2}, s) x - P(t_{1}, s) x \bigr\Vert \leq L_{P} |t_{2} - t_{1} | \Vert x\Vert _{[D]} $$
Using Kozak [11], Theorem 2.1, and Henríquez [21], Corollary 3.4, we can establish the following result. We abbreviate by RNP the Radon-Nikodym property, and we refer the reader to [22] for the properties of spaces with the RNP.
Theorem 3.2
- (i)
The space X verifies the RNP.
- (ii)
For each \(x \in D\) the function \(A(\cdot) x\) is continuously differentiable.
- (iii)
There exists \(\lambda \in\mathbb{C}\) such that \(\lambda \in\rho (A(t))\) for all \(t \in[0, a]\).
Proof
Repeating this argument, we can show that \(\Gamma^{n}\) is a contraction for n large enough. If \(x(\cdot) = x(\cdot, \sigma , z)\) is the fixed point of Γ, then \(x(\cdot)\) is a solution of problem (3.1)-(3.2).
It is worth to point out that condition (3.8) implies condition (H2). We also can obtain a similar result modifying this condition.
Corollary 3.1
- (i)
The space X verifies the RNP.
- (ii)
For each \(x \in D\) the function \(A(\cdot) x\) is continuously differentiable.
- (iii)
There exists \(\lambda \in\mathbb{C}\) such that \(\lambda \in\rho (A(t))\) for all \(t \in[0, a]\).
Proof
The above theorems characterize broad classes of operators P for which there is a solution of problem (3.1)-(3.2). There are also some particular situations in which it is possible to establish the existence of solution of problem (3.1)-(3.2). Below we will mention one of them.
- (A)
The operator \(A(t) = A_{0} + B(t)\), where \(A_{0}\) is the infinitesimal generator of a cosine function \((C_{0}(t))_{t \in\mathbb{R}}\) with associated sine function \((S_{0}(t))_{t \in\mathbb{R}}\), and \(B : [0, a] \to\mathcal{L}(X)\) is a strongly differentiable map.
Lemma 3.1
Proof
Theorem 3.3
Proof
Repeating this argument, we can show that \(\Gamma^{n}\) is a contraction for n large enough. If \(x(\cdot)\) denotes the fixed point of Γ, then \(x(\cdot)\) is a solution of problem (3.1)-(3.2).
Corollary 3.2
Proof
We next apply the previous construction to the study of problem (1.1)-(1.2), where \(f :[0, a] \to X\) is an integrable function. We generalize to inhomogeneous equations the notion introduced in Definition 3.1.
Definition 3.2
A function \(x : [0, a] \to X\) is said to be a solution (or classical solution) of problem (1.1)-(1.2) if \(x \in C([0, a], [D]) \cap C^{2}([0, a], X)\) and (1.1)-(1.2) are satisfied.
Corollary 3.3
Proof
By comparing Corollary 3.3 with Definition 2.1 and Theorem 2.2, we are led to establish the following concept.
4 Existence of solutions
Lemma 4.1
Proof
It is a direct consequence of properties of evolution operator S. □
- (F)Let \(f : [0, a] \to X\) be a function such that the function \(F : [0, a] \to X\) given byis a solution of the abstract Cauchy problem (1.5) with initial conditions \(F(0) = F^{\prime}(0) = 0\).$$F(t) = \int_{0}^{t} S(t, \xi) f(\xi) \,d\xi $$
Lemma 4.2
Assume condition (F) holds. Then \(u(t) \in D\) and \(u : [0, a] \to[D]\) is continuous.
Proof
(ii) Since D is dense in X, for an integrable function \(f : [0, a] \to X\) there exists a sequence \((\varphi _{n})_{n}\) in \(C([0, a], D)\) such that \(\varphi _{n} \to f\) as \(n \to\infty\) for the norm in \(L^{1}([0, a], X)\). As a matter of fact, we can take \(\varphi _{n}\) as a trapezoidal function with values in D. Consequently, we can assume that \(\varphi _{n} \in C([0, a], [D])\). Let \(\Phi_{n}\) be the function defined by (4.2) with \(\varphi _{n}\) instead of φ. Let \(x_{n}(\cdot)\) be the solution of equation (4.3) with \(v(t) = \Phi_{n}(t)\), and let \({ u_{n}(t) = \int_{0}^{t} R(t, \xi) \varphi _{n}(\xi) \,d\xi}\).
Lemma 4.3
Proof
We are in a position to establish our first result of the existence of solutions of problem (1.1)-(1.2). This result is an immediate consequence of Kozak [11], Theorem 2.1, and Henríquez [21], Corollary 3.4.
Theorem 4.1
- (i)
The space X verifies the RNP.
- (ii)
For each \(x \in D\) the function \(A(\cdot) x\) is continuously differentiable.
- (iii)
There exists \(\lambda \in\mathbb{C}\) such that \(\lambda \in\rho (A(t))\) for all \(t \in[0, a]\).
- (iv)
The function f is Lipschitz continuous.
Proof
We finish this section with an application of Theorem 3.3.
Theorem 4.2
Let \(y, z \in D\). Assume that conditions of Theorem 3.3 are fulfilled. Assume further that \(B: [0, a] \to\mathcal{L}([D], E)\) is a strongly continuous operator valued map. Let \(f : [0, a] \to X\) be a continuously differentiable function. Then the function \(x(\cdot)\) given by (3.20) is the solution of problem (1.1)-(1.2).
Proof
5 Applications
The one dimensional wave equation modeled as an abstract Cauchy problem has been studied extensively. See for example [31]. In this section, we apply the results established previously to study the existence of solutions to a non-autonomous wave equation modeled by an integro-differential equation. To avoid technical difficulties, we will consider only a pair of simple types of wave equation.
We model this problem in the space \(X = L^{2}([0, \pi])\). For this reason, we assume that \(\varphi , z \in X\). Similarly, \(H^{2}([0, \pi])\) denotes the Sobolev space of functions \(x : \mathbb{R}\to\mathbb{C}\) such that \(x^{\prime\prime} \in L^{2}([0, \pi])\).
- (i)
For each \(\xi\in[0, \pi]\), \(\tilde{f}(\cdot, \xi )\) is continuous.
- (ii)
For \(t \geq0\), \(\tilde{f}(t, \cdot)\) is measurable, and there exists a positive function \(\eta\in L^{2}([0, \pi])\) such that \(|\tilde{f}(t, \xi)| \leq\eta(\xi)\) for \(\xi\in[0, \pi]\).
We take \(B(t) x (\xi) = b(t) x(\xi)\) defined on X. It is easy to see that \(A(t) = A_{0} + B(t)\) is a closed linear operator. Moreover, it is clear that condition (A) is fulfilled. Consequently, \(A(t)\) generates an evolution operator \(S(t,s)\). In addition, since \(B(t) : [D(A_{0})] \to[D(A_{0})]\), we see that \(B : [0, \infty) \to\mathcal{L}([D(A_{0})], E)\) is a strongly continuous map.
Using this construction, and defining \(x(t) = w(t, \cdot) \in X\), problem (5.1)-(5.3) is modeled in the abstract form (1.1)-(1.2). The following result is a consequence of Theorem 3.3 and Theorem 4.2.
Corollary 5.1
- (a)
There exists a resolvent operator \(R(t,s)\) for equation (5.1).
- (b)
- (c)If \(\varphi , z \in D(A_{0})\), and f̃ satisfies the local Lipschitz conditionfor all \(t_{2}, t_{1} \in[0, a]\) and \(\xi\in[0 \pi]\), where \(L_{a} \geq0\), then problem (5.1)-(5.3) has a classical solution given by (5.5).$$ \bigl\vert \tilde{f}(t_{2}, \xi) - \tilde{f}(t_{1}, \xi) \bigr\vert \leq L_{a} \vert t_{2} - t_{1} \vert $$(5.6)
Proof
(b) This assertion is a consequence of (i) and Definition 3.3.
Using again that every Hilbert space satisfies the RNP we get the following consequence.
Corollary 5.2
Proof
Assertion (a) is a consequence of Theorem 3.2. Conditions (H3)-(H4) are immediate consequences of the definition of P and the properties of \(\alpha (\cdot )\). It only remains to show that estimate (3.8) holds. We proceed as in the proof of Corollary 5.1.
Declarations
Acknowledgements
The research of Hernán R Henríquez was supported in part by CONICYT under Grant FONDECYT 1130144 and DICYT-USACH, and the research of Juan C Pozo was supported in part by CONICYT, under Grant FONDECYT 3140103.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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