Semilinear evolution equations with distributed measures
- Adrian Petruşel^{1}Email author and
- Bianca Satco^{2}
https://doi.org/10.1186/s13663-015-0392-4
© Petruşel and Satco 2015
Received: 24 February 2015
Accepted: 3 August 2015
Published: 19 August 2015
Abstract
Working with Kurzweil-Stieltjes integrals and using a measure of non-compactness allows us to relax the assumptions on the semigroup, on f and g comparing to some already known results.
Keywords
MSC
1 Introduction
The aim of the paper is to provide existence results for the semilinear evolution problem with distributed measures (1), where −A is the infinitesimal generator of a continuous semigroup \(\{ T(t),t\geq0\}\) of bounded linear operators, \(f:[0,1]\times X\to X\) and \(g:[0,1]\to X\).
This problem was firstly considered in [1] for a continuous function f and a bounded variation function g, \(\{T(t),t\geq0\}\) being a compact \(C_{0}\) semigroup of contractions. The importance of allowing the occurrence of the very general term dg on the right-hand side of the equation was clearly exposed and exemplified in [1]; the equation models situations that arise, e.g., in optimal control problems with state constraints.
We are now interested in discussing the matter of existence of mild solutions (defined in a similar way to the classical case) for the above problem under less restrictive assumptions: a regulated function g (not necessarily of bounded variation) and a possibly discontinuous function f.
At the first step, \(\{T(t),t\geq0\}\) is a uniformly continuous semigroup of bounded linear operators. The proof is based on a fixed point argument, via the Mönch fixed point theorem, using the Hausdorff measure of non-compactness. We would like to mention that our approach uses the Kurzweil-Stieltjes integral in a Banach space and this allows us to relax the assumptions on f and g comparing to the already known results (such as Theorem 8.1 in [1]).
An illustrating example (Example 25), of a parabolic problem with dynamic boundary conditions showing the applicability of the main result, is also described.
Finally, we discuss the more general case when −A is the infinitesimal generator of a strongly continuous semigroup \(\{ T(t),t\geq0\}\) of bounded linear operators. In this setting, the problem becomes complicated due to the possibility for the Stieltjes integral to be not well defined in the Banach space. Working with Kurzweil-type integration theory we are able to prove that if the semigroup has bounded \(\mathcal{B}\)-variation and the Banach space is reflexive, then an existence theory can be developed by the same method.
Our results extend and generalize some other recent theorems in the literature as well, see [2, 3] or [4–6] (for the linear case).
2 Kurzweil integration in Banach spaces
Let \([a,b]\) be an interval of the real line equipped with the usual topology and the Lebesgue measure dt. Throughout this paper X is a Banach space with norm \(\Vert \cdot \Vert \).
Let us now introduce the definition of Kurzweil integral in Banach spaces, which is one of the possible extensions of the notion of Henstock-Kurzweil integral for real-valued functions (the reader is referred to [7, 8] or [9]).
A partition of \([a,b]\) is a finite collection of pairs \(\{ ([t_{i-1},t_{i}],c_{i}),i=1,\ldots,n\}\), where \([t_{i-1},t_{i}]\) are non-overlapping subintervals of \([a,b]\), \(c_{i} \in[t_{i-1},t_{i}]\), \(i=1,\dots,n\), and \(\bigcup^{n}_{i=1}[t_{i-1},t_{i}]=[a,b]\). A gauge δ on \([a,b]\) is a positive function on \([a,b]\). For a given gauge δ, we say that a partition is δ-fine if \([t_{i-1},t_{i}] \subset (c_{i}-\delta(c_{i}),c_{i}+\delta(c_{i}))\) for any \(i\in\{1,\dots,n\}\).
Definition 1
([10])
If f is Kurzweil-integrable, then it has the same feature on any sub-interval of \([a,b]\). The function \(t\mapsto\mathrm{(K)}\int_{a}^{t} f(s)\,ds\) is called the Kurzweil-primitive of f on \([a,b]\).
Remark 2
The Kurzweil primitive is continuous (see [10]). The connection between this integral, other gauge integrals (Henstock-Lebesgue and Henstock-Kurzweil-Pettis integrals) and the classical ones (namely, Bochner and Pettis integrals) can be found in [11, 12] or [13].
3 Kurzweil integration for bounded linear operators
Let us recall the following concepts.
Definition 3
- (i)
([17] or [18]) A function \(g:[a,b]\to X\) is regulated if it has at most discontinuities of the first kind on \([a,b]\), i.e., for every \(t\in[a,b)\) there exists \(g(t+)\in X\) such that \(\lim_{s\to t, s>t}\Vert g(s)-g(t+) \Vert =0\) and for every \(t\in(a,b]\) there exists \(g(t-)\in X\) such that \(\lim_{s\to t, s< t}\Vert g(s)-g(t-) \Vert =0\).
- (ii)
([19, 20]) Likewise, an operator-valued function \(T:[a,b]\to L(X)\) is regulated if for every \(t\in[a,b)\) there exists \(T(t+)\in L(X)\) and for every \(t\in(a,b]\) there exists \(T(t-)\in L(X)\) in the operator norm topology;
- (iii)
An operator-valued function \(T:[a,b]\to L(X)\) is said to be \(\mathcal{B}\)-regulated ([16, 21, 22]) if \(t\in[a,b] \mapsto T(t)x\) is regulated for each \(x\in X\) with \(\Vert x\Vert \leq1\).
By \(G([a,b],X)\) we denote the space of regulated X-valued functions endowed with its natural (Banach space) norm \(\Vert f\Vert _{C}={\sup_{t\in [a,b]}}\Vert f(t)\Vert \).
One of the main tools in our work is the following concept.
Definition 4
- (i)
for any \(t_{0}-\delta< t'< t_{0}\): \(\Vert x(t')-x(t_{0}-)\Vert <\varepsilon\);
- (ii)
for any \(t_{0}< t''< t_{0}+\delta\): \(\Vert x(t'')-x(t_{0}+)\Vert <\varepsilon\)
A useful version of Ascoli’s theorem for regulated functions was proved in [18] (see also [17] in finite dimensional setting).
Lemma 5
Let \(\mathcal{A}\subset G([a,b],X)\) be equi-regulated and, for every \(t\in[a,b]\), \(\mathcal{A}(t)=\{x(t), x\in\mathcal{A} \}\) be relatively compact. Then \(\mathcal{A}\) is relatively compact in \(G([a,b],X)\).
Definition 6
- (i)A function \(g:[a,b]\to X\) is of bounded variation if its total variation on \([a,b]\) is finite, i.e.,where the supremum is taken over all finite partitions of the interval \([a,b]\).$$V_{a}^{b}(g)=\sup \Biggl\{ \sum _{i=1}^{n} \bigl\Vert g(t_{i})-g(t_{i-1}) \bigr\Vert \Biggr\} < \infty, $$
- (ii)Likewise, an operator-valued function \(T:[a,b]\to L(X)\) is said to be of bounded variation if its total variationis finite, where the supremum has the same meaning as above.$$V_{a}^{b}(T)=\sup \Biggl\{ \sum _{i=1}^{n} \bigl\Vert T(t_{i})-T(t_{i-1}) \bigr\Vert _{\mathit {op}} \Biggr\} $$
- (iii)An operator-valued function \(T:[a,b]\to L(X)\) is of \(\mathcal {B}\)-bounded variation (see [19]) or of bounded semi-variation (as in [6]) ifwhere the supremum is taken over all finite partitions of the interval \([a,b]\).$$(\mathcal{B})V_{a}^{b}(T)=\sup\sup_{\Vert x_{i}\Vert \leq1} \Biggl\{ \Biggl\Vert \sum_{i=1}^{n} \bigl(T(t_{i})-T(t_{i-1}) \bigr)x_{i} \Biggr\Vert \Biggr\} < \infty, $$
It is well known that any bounded variation vector-valued or operator-valued function is regulated. As for the corresponding \(\mathcal{B}\)-notions of bounded variation and regularity, notice that if X is a uniformly convex space, then all operator-valued functions of \(\mathcal{B}\)-bounded variation are \(\mathcal{B}\)-regulated (see Theorem 1 in [21]). In fact, in [23] (Theorem 5) even more was proved: that every function \(T:[a,b]\to L(X)\) of \(\mathcal {B}\)-bounded variation is \(\mathcal{B}\)-regulated if and only if X does not contain any copy of \(c_{0}\).
Remark 7
In [16, 19] it is shown that for operator-valued functions the regularity (resp. bounded variation property) is stronger than the \(\mathcal{B}\)-regularity (resp. than \(\mathcal{B}\)-bounded variation property) and that they respectively coincide if X is finite dimensional.
It is not difficult to check ([4] or [16]) that \((\mathcal{B})V_{a}^{b}(\cdot)\) defines a seminorm on the linear space (with usual operations) of operator-valued functions of \(\mathcal {B}\)-bounded variation on \([a,b]\) and that \(\Vert A \Vert _{SV}=\Vert A(a)\Vert _{\mathit {op}}+(\mathcal{B})V_{a}^{b}(A) \) is a norm on this space (that becomes, in this way, a Banach space).
Similarly, \(\Vert A \Vert ^{\mathit {op}}_{BV}=\Vert A(a)\Vert _{\mathit {op}}+V_{a}^{b}(A) \) is a (Banach space) norm on the space of operator-valued functions of bounded variation.
The Kurzweil-Stieltjes integral for operator-valued functions is defined as follows.
Definition 8
- (i)We say that h is Kurzweil-Stieltjes integrable with respect to T if there exists an element \(\int_{a}^{b} d[T(t)]h(t)\in X\) such that for every \(\varepsilon>0\) there exists a gauge \(\delta_{\varepsilon}\) satisfyingfor every \(\delta_{\varepsilon}\)-fine partition \(\{ ([t_{i-1},t_{i}],c_{i}),i=\overline{1,n}\}\) of \([a,b]\);$$\Biggl\Vert \sum_{i=1}^{n} \bigl[T(t_{i})-T(t_{i-1}) \bigr]h(c_{i})-\int _{a}^{b} d \bigl[T(t) \bigr]h(t) \Biggr\Vert \leq \varepsilon $$
- (ii)The operator-valued function T is said to be Kurzweil-Stieltjes integrable with respect to h if there exists an element \(\int_{a}^{b} T(t)\,dh(t)\in X\) such that for every \(\varepsilon>0\) there exists a gauge \(\delta_{\varepsilon}\) satisfyingfor every \(\delta_{\varepsilon}\)-fine partition of \([a,b]\).$$\Biggl\Vert \sum_{i=1}^{n} T(c_{i}) \bigl(h(t_{i})-h(t_{i-1})\bigr)-\int _{a}^{b} T(t)\,dh(t) \Biggr\Vert \leq\varepsilon $$
If the gauge \(\delta_{\varepsilon}\) in the preceding Definition 8(i) can be chosen as a positive constant, then the function h is called Riemann-Stieltjes integrable with respect to T (and likewise for Definition 8(ii)). Theorem 2.1 in [1] contains a particular case of Proposition 2.1 in [5]: if T is uniformly continuous and h is of bounded variation, then the integral \(\int_{a}^{b} T(t)\,dh(t)\) exists as Riemann-Stieltjes integral.
Remark 9
As it can be seen in [19], Proposition 15, if T is \(\mathcal{B}\)-regulated and of \(\mathcal{B}\)-bounded variation and \(h:[a,b]\to X\) is regulated, the Kurzweil-Stieltjes integral \(\int_{a}^{b} d[T(t)]h(t)\in X\) exists. In particular, this happens when T has bounded variation.
Also, Proposition 2.1 in [5] states that if T is regulated and h has bounded variation, then the integral \(\int_{a}^{b} T(t)\,dh(t)\) is well defined. The same is available if T has \(\mathcal {B}\)-bounded variation and h is regulated (Theorem 3.3 in [24]).
Remark 25 in [19] asserts that if T is \((\mathcal {B})\)-regulated, then the KS-primitive \(\int_{a}^{\cdot} d[T(t)]h(t)\) is also regulated (and similar for \(\int_{a}^{b} T(t)\,dh(t)\) if h is regulated).
We shall need the following evaluation formulas.
Proposition 10
- (i)(Proposition 10 in [19]) If the function \(h:[a,b]\to X\) is Kurzweil-Stieltjes integrable with respect to the operator-valued function \(T:[a,b]\to L(X)\) of \(\mathcal{B}\)-bounded variation, then$$\biggl\Vert \int_{a}^{b} d \bigl[T(t) \bigr]h(t) \biggr\Vert \leq(\mathcal{B})V_{a}^{b}(T) \cdot \Vert h \Vert _{C}. $$
- (ii)(Theorem 5.1 in [16]) If \(T:[a,b]\to L(X)\) has \(\mathcal {B}\)-bounded variation and h is regulated, then$$\biggl\Vert \int_{a}^{b} T(t)\,dh(t) \biggr\Vert \leq \bigl(\bigl\Vert T(a)\bigr\Vert +\bigl\Vert T(b)\bigr\Vert +( \mathcal {B})V_{a}^{b}(T) \bigr)\cdot \Vert h\Vert _{C}. $$
We now present an integration by parts theorem that comes from some related results given in [6] and [25] (it can also be proved by combining Theorem 1.15 in [26] and Corollary 3.6 in [24]).
Theorem 11
- (i)
T is of \(\mathcal{B}\)-bounded variation;
- (ii)
g is Kurzweil-integrable.
Proof
Notice first that the primitive in Kurzweil sense \([a,b]\ni t\mapsto \mathrm{(K)}\int_{a}^{t} g(s)\,ds \in X\) is continuous (see [10]).
Lemma 12
Proof
Lemma 13
If \(\{T(t), t\geq0\}\) is a uniformly continuous semigroup and \(h:[a,b]\to X\) is regulated, then \(\int_{a}^{\cdot} T(\cdot-s)\,dh(s)\) is regulated.
Proof
We end this section with a mean value result that comes in an obvious manner (taking into account that the integral is a limit of integral sums).
Lemma 14
4 Solutions for semilinear evolution equations with distributed measures using Kurzweil-Stieltjes integration
We begin by clarifying the concept of solution that we are searching for. Notice that, in the sequel, the integrals are taken in Kurzweil sense.
Definition 15
Remark that the functions on the right-hand side of the equality in this definition are regulated as a consequence of Corollary 18 and Lemma 13.
Remark 16
When the above integrals exist in Riemann, respectively Riemann-Stieltjes sense (e.g., when f is continuous and g has bounded variation), we get the notion of solution used in [1] (p.3200).
If, moreover, \(dg=G dt\) is defined by a density \(G\in L^{1}([0,1],X)\) (e.g., when X has the Radon-Nikodym property and g is absolutely continuous), the previous definition concerns classical mild solutions.
In order to simplify the proof of the main theorem, we first present some auxiliary results.
Lemma 17
Let \(\mathcal{G}\) be a family of X-valued Kurzweil-integrable functions on \([a,b]\) such that the set of their primitives is equi-continuous, and let \(\{T(t), t\geq0\}\) be a uniformly continuous semigroup. Then the family \(\{ \mathrm{(K)}\int_{a}^{\cdot} T(\cdot-s)g(s)\,ds, g\in\mathcal{G}\}\) is equi-continuous on \([a,b]\).
Proof
By hypothesis, the collection of primitives of functions in \(\mathcal {G}\) is \(\Vert \cdot \Vert _{C}\)-bounded (by M).
For a singleton \(\mathcal{G}\), we get the following.
Corollary 18
If \(g:[a,b]\to X\) is Kurzweil-integrable and \(T:[a,b]\to L(X)\) as in the preceding lemma, then \(\mathrm{(K)}\int_{a}^{\cdot} T(\cdot-s)g(s)\,ds\) is continuous.
Since we allow the occurrence of discontinuous functions, we shall use instead a measure of non-compactness. Recall that the Hausdorff measure of non-compactness \(\beta_{H}\) is defined, for any \(A {\subset} X\), by the infimum of all \(r>0\) such that there exists a finite number of balls covering A, of radius smaller than r (we refer the reader to [27] or [28]).
Lemma 19
Proof
The Mönch fixed point theorem that we recall below will be the main tool in obtaining the existence result.
Theorem 20
([29])
Let D be a closed, convex subset of a Banach space and \(N:D\rightarrow D\) be continuous with further property that for some \(x_{0} \in D\) one has: \(C\subset D\) countable, \(\overline{C}=\overline{\operatorname{conv}} ( \{x_{0} \}\cup N(C) ) \Longrightarrow\overline{C}\) compact. Then N has a fixed point.
We proceed now to presenting the main result of the paper.
Theorem 21
- (1)for every \(x\in G([0,1],X)\), the mapping \(f(\cdot,x(\cdot))\) is Kurzweil-integrable and the functionis \(\Vert \cdot \Vert _{A}\)-continuous;$$x\in G \bigl([0,1],X \bigr)\mapsto f \bigl(\cdot,x(\cdot) \bigr)\in\mathcal{K} \bigl([0,1],X \bigr) $$
- (2)
the family \(\{ \mathrm{(K)}\int_{0}^{\cdot} f(s,x(s))\,ds, x\in G([0,1],X) \}\) is equi-continuous;
- (3)
there exists a constant \(c>0\) such that \(\beta_{H}(f([0,1]\times D))\leq c\beta_{H}(D)\) for every bounded \(D\subset X\).
Then the evolution problem (1) has at least one \(\mathcal{L}^{\infty}\)-solution on some non-zero length interval \([0,\alpha]\subset[0,1]\).
Proof
We shall prove that the preceding evolution problem has at least one \(\mathcal{L}^{\infty}\)-solution on \([0,\alpha]\).
We assert that the Mönch fixed point result can be applied in this case.
Let us now prove that for an arbitrary \(\overline{x}\in\mathcal{K}\), any countable collection \(C\subset\mathcal{K}\) satisfying the equality \(\overline{C}=\overline{\operatorname{conv}}(\{\overline{x}\}\cup N(C))\) is relatively compact. We shall apply Lemma 5.
Let C be such a subset of \(\mathcal{K}\). Then \(N(C)\) is equi-regulated by Lemma 17, therefore all we have to check is that for every \(t\in[0,\alpha]\), \(C(t)\) is relatively compact in X.
Finally, the Mönch fixed point theorem asserts that the operator N possesses fixed points, which means that our evolution problem has \(\mathcal{L}^{\infty}\)-solutions. □
Remark 22
Theorem 2.4 in [4] presents a situation where f satisfies the integrability of superpositions \(f(\cdot,x(\cdot))\) in hypothesis (1), namely when f is a Carathéodory function perturbed by a Kurzweil-integrable one. In fact, it is not difficult to see that in this case conditions (1) and (2) are both satisfied.
In particular, we deduce Theorem 8.1 in [1] as follows.
Corollary 23
If f is continuous and g has bounded variation, then the evolution problem (1) has \(\mathcal{L}^{\infty}\)-solutions (involving Riemann integral) on some non-empty interval.
Remark 24
Our study applied in the particular case \(dg=0\) gives a result more general than those in [2] or [3] (when the measure driving the equation is the Lebesgue measure), see also [30].
We complete this section by giving an example (borrowed from [1]) in order to illustrate the applicability of our result.
Example 25
Theorem 21 allows us to get the existence of \(\mathcal {L}^{\infty}\)-solutions when \(\alpha_{1}\) and \(\alpha_{2}\) are of the form of a sum of a Carathéodory function with a Kurzweil-integrable one and ϕ, ψ, η are regulated.
This can be done following the same steps as in the proof of Theorem 9.2 in [1] (that describe the way that our problem can be rewritten as a Cauchy problem of type (1)) and applying our main result. Finally, let us remark that here the conditions are weaker than those in the existence Theorem 9.2 in [1], which assumes that \(\alpha_{1}\) and \(\alpha_{2}\) are continuous and ϕ, ψ, η have bounded variation.
5 Remarks on the same problem for strongly continuous semigroup
We are facing now the matter of existence of solutions when −A is the infinitesimal generator of a strongly continuous semigroup \(\{ T(t),t\geq0\}\) of bounded linear operators.
The starting point is the discussion presented in [1], p.3184 for bounded variation function g: when working with Riemann-Stieltjes integral, in order that the integral \(\int_{a}^{b} T(t-s)\,dg(s)\) be well defined in X, some extra assumptions must be imposed on X, on g or on the semigroup (in general, the integral is an element of the sequential completion \(X^{c}\) of X in the \(\sigma (X,X^{*})\) topology). More precisely, \(\int_{a}^{b} T(t-s)\,dg(s)\in X\) if X is reflexive (Remark 2.2), or if the semigroup is uniformly continuous (Theorem 2.1).
When considering the Kurzweil-Stieltjes integral, even for regulated function g, this problem can be solved for non-reflexive spaces and non-uniformly continuous semigroups: as recalled in Remark 9, if the semigroup has bounded \(\mathcal{B}\)-variation and g is regulated, then the Kurzweil-Stieltjes integral \(\int_{a}^{t} T(t-s)\,dg(s)\in X\).
As for our method of study, it can be extended to strongly continuous semigroups if we require that the Banach space X is reflexive. Indeed, a closer look reveals that only Lemma 12, Lemma 13 and Lemma 17 need to be generalized. Thus:
Lemma 26
If \(\{T(t), t\geq0\}\) is a strongly continuous semigroup with bounded \(\mathcal{B}\)-variation on compact intervals, then \((\mathcal{B})V_{0}^{t'} (T(t''-\cdot)-T(t'-\cdot))\) tends to 0 as \(t''>t' \to t'\).
Proof
Lemma 27
If \(h:[a,b]\to X\) is regulated and \(\{T(t), t\geq0\}\) is a strongly continuous semigroup with bounded \((\mathcal {B})\)-variation on compacts such that \((\mathcal{B})V_{t}^{t'} (T)\to0\) whenever \(t'\to t\), then \(\int_{a}^{\cdot} T(\cdot-s)\,dh(s)\) is regulated.
Proof
Lemma 28
Let \(\mathcal{G}\) be a family of X-valued Kurzweil-integrable functions on \([a,b]\) such that the set of their primitives is equi-continuous. Suppose that \(\{T(t), t\geq0\}\) is a strongly continuous semigroup with bounded \(\mathcal{B}\)-variation on \([a,b]\). Then the family \(\{ \mathrm{(K)}\int_{a}^{\cdot} T(\cdot-s)g(s)\,ds, g\in \mathcal{G}\}\) is equi-continuous.
Proof
The collection of primitives of functions in \(\mathcal{G}\) is \(\Vert \cdot \Vert _{C}\)-bounded.
In the described framework (of strongly continuous semigroup of bounded linear operators on a reflexive space) the main result is given below.
Theorem 29
Let −A be the infinitesimal generator of a strongly continuous semigroup \(\{T(t),t\geq0\}\) with bounded \(\mathcal{B}\)-variation on compact intervals such that \((\mathcal{B})V_{t}^{t'} (T)\to0\) whenever \(t'\to t\).
Let \(g:[0,1] \to X\) be regulated and \(f:[0,1]\times X \to X\) satisfy the hypotheses (1), (2), (3) in Theorem 21. Then the evolution problem (1) has at least one \(\mathcal{L}^{\infty}\)-solution on some non-zero length interval \([0,\alpha]\subset[0,1]\).
Proof
We shall prove that the preceding evolution problem has at least one \(\mathcal{L}^{\infty}\)-solution on \([0,\alpha]\).
We assert that the Mönch fixed point result is suitable in this case.
6 Conclusions
We obtained the existence of mild solutions (in a generalized sense) for the semilinear evolution problem with distributed measures (1) under less restrictive assumptions comparing to similar results in literature, where the involved semigroup of bounded linear operators was a compact \(C_{0}\) semigroup, the function f in the semilinear part was continuous, and g was supposed to have bounded variation.
More precisely, in a general Banach space the existence result works for a uniformly continuous semigroup (Theorem 21), while for a reflexive Banach space the existence of solutions was proved for a \(C_{0}\) semigroup with bounded \((\mathcal{B})\)-variation on compact intervals (Theorem 29). In both situations, the function f is allowed to be discontinuous and the function g is only regulated (possibly with unbounded variation).
Declarations
Acknowledgements
The first author thanks the Visiting Professor Programming at King Saud University for funding this work. The author extends his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Prolific Research Group (PRG-1436-10). For the second author, this work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PN-II-RU-TE-2012-3-0336.
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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