Dirichlet boundary value problem for differential equations involving dry friction
 Jan Andres^{1}Email author and
 Hana Machů^{1}
Received: 10 April 2015
Accepted: 8 June 2015
Published: 24 June 2015
Abstract
Sufficient conditions in terms of growth restrictions are given for the solvability of the Dirichlet boundary value problem to forced nonlinear differential equations involving the combination of viscous and dry frictions. Explicit estimates of solutions and their derivatives allow us to restrict ourselves to a sufficiently large neighbourhood of the origin, when formulating these effective conditions. In this way, the behaviour of nonlinearities outside of this neighbourhood can be quite arbitrary. In order to get optimal solvability criteria, the problems with oneterm and complete linear differential operators will be treated separately by means of various Green’s functions. The obtained results are compared with some of their analogies of the other authors.
Keywords
MSC
1 Introduction
In the presence of a dry friction, the notion of a Carathéodory solution, i.e. the one with an absolutely continuous derivative, is insufficient. The appropriate notion is a Filippov solution which is a Carathéodory solution, but of a differential inclusion with a Filippov regularized righthand side (see e.g. [14, 15]). For the history and phenomenology of dry friction problems in general, see e.g. [16–20].
On the other hand, the estimates of solutions and derivatives are not indicated explicitly in [22].
There are also further related results concerning problems (2) and (5) obtained mainly by means of degree arguments (see e.g. [23–34]), but none of them allows us to get such simple criteria as those in [22].
For instance, according to Theorem 6.1 in [25] as well as Theorem 2 in [33], where the combination of sign and growth restriction was employed, both problems (2) and (5) are solvable only in the lack of a viscous friction, i.e. only for \(a=0\), provided \(\int_{0}^{T} p(t)\,\mathrm{d}t <\infty\) for (2), and \(\int_{0}^{T} p(t)\,\mathrm{d}t <\infty\) together with \(b \leq0\) for (5). Moreover, the estimates of solutions are again not indicated explicitly in [25, 33].
Similarly, in Theorem 4.1 in [26], Theorem 4.1 in [29] and Theorem 3.1 in [30], the viscous friction term \(a x'\) cannot be involved in (2) and (5) but, under more restrictive assumptions than those in [25, 33], the solution estimates are available there.
Those for the solvability of (5) hold, according to Corollary 4.1 in [23], without (6), i.e. with \(b<0\) and p is continuous on \([0, T ]\) (\(\Rightarrowp(t) \leq P\), \(t \in[ 0, T]\)), only.
In the absence of a viscous friction, i.e. when \(a=0\), the forcing term p can be, according to Corollary 4.3 in [23] and Theorem 3 in [30], Lebesgue measurable and essentially bounded, for both problems (2) and (5). For \(b=0\), however, it should be \(c=0\), \(p(t) \equiv0\) and \(k_{3}=0\) or \(a=0\) in order Corollaries 4.1 and 4.3 in [23] and Theorem 3 in [30] to be applied to (2) and (5) which reduces to a trivial solvability of (2) and (5).
If further possibly discontinuous nonlinearities or multivalued maps are implemented into the righthand sides of given differential equations or inclusions whose growth has, for instance, a superlinear character sufficiently far from the origin, then Theorem 3 in [22] does not any longer apply. Having, however, to our disposal explicit estimates of solutions like (7) or (8) and their derivatives, we can formulate criteria such that the implemented new terms can behave in an arbitrary way outside of the domains characterized by these estimates. In this way, all the results under our consideration can be naturally extended.
This will be therefore our main aim of the present paper. Of course, for obtaining the explicit estimates of solutions and their derivatives, we should study, unlike in [22], exclusively the secondorder Dirichlet boundary value problems. For the sake of brevity we will prove only one main theorem for the scalar problem. Nevertheless, since all the related proofs of all the cases under our consideration are quite analogous and differ just by the technical details, we can also present the related solvability criteria and some solutions estimates.
On the other hand, we will discuss in detail the advantages and disadvantages of the usage of oneterm vs. complete linear differential operator for the associated Green’s functions.
2 Preliminaries
A multivalued mapping \(\varphi\colon X \multimap Y\) is called upper semicontinuous (u.s.c.) if, for each open \(U \subset Y\), the set \(\{x \in X \mid\varphi(x) \subset U \}\) is open in X.
Lemma 1
(cf., e.g., Proposition I.3.15 in [14])
If \(\varphi\colon X \multimap Y\) is u.s.c., then the graph \(\Gamma _{\varphi}\) is a closed subset of \(X \times Y\).
Lemma 2
(cf., e.g., Proposition I.3.16 in [14])
Assume that \(\varphi\colon X \multimap Y\) is a multivalued mapping such that \(\varphi(X) \subset K\), where \(K \subset Y\) is a compact set, and the graph \(\Gamma_{\varphi}\) of φ is closed. Then φ is u.s.c.
Let Y be a separable metric space and \((\Omega, \mathcal{U}, \mu)\) be a measurable space, i.e. a set Ω equipped with a σalgebra \(\mathcal{U}\) of subsets and a countably additive measure μ on \(\mathcal{U}\). For our needs here, Ω will be a bounded domain in \(\mathbb{R}^{k}\), equipped with the Lebesgue measure. A multivalued mapping \(\varphi\colon\Omega\multimap Y\) is called measurable if \(\{\omega\in\Omega\mid\varphi(\omega ) \subset V \} \in\mathcal{U}\), for each open set \(V \subset Y\).
 (i)
\(t \multimap F(t,x)\) is measurable, for every \(x \in \mathbb{R}^{n}\),
 (ii)
\(x \multimap F(t,x)\) is u.s.c., for almost all (a.a.) \(t\in J\),
 (iii)
\(y \leq r(t) (1+x)\), for every \((t,x) \in J \times \mathbb{R}^{n}\), and every \(y\in F(t,x)\), where \(r\colon J \rightarrow [0, \infty)\) is a Lebesgue integrable function.
Let \(F \colon X \multimap Y\) be a multivalued mapping and \(f \colon X \rightarrow Y\) be a singlevalued mapping. We say that f is a selection of F (written \(f \subset F\)) if \(f(x) \in F(x)\), for every \(x \in X\).
We will employ the following definitions and statements.
Proposition 1
(Castaing representation, cf. Theorem III.7 in [35])
Lemma 3
(cf., e.g., Lemma 7.1 in [36])
Let \(F\colon J\times\mathbb{R}^{n} \multimap\mathbb{R}^{n}\) be a uCarathéodory multivalued mapping. Then the composition \(F(t, q(t))\) admits, for every \(q \in C(J, \mathbb{R}^{n})\), a singlevalued measurable selection.
If \(X \subset Y\) and \(\varphi\colon X \multimap Y\), then a point \(x \in X\) is called a fixed point of φ if \(x \in\varphi (x)\). We set \(\operatorname{Fix} (\varphi) := \{ x \in X \mid x \in\varphi(x) \}\).
Definition 1
Let X and Y be subsets of normed linear spaces and \(\varphi\colon X \multimap Y\) be a multivalued mapping. If Y is convex, then φ is called a Kakutani map, provided φ is u.s.c. with (nonempty) compact, convex values.
Let X and Y be metric spaces. A multivalued mapping \(\varphi\colon X \multimap Y\) is called compact if its image \(\varphi(X)= \bigcup\{\varphi(x) \mid x \in X \}\) is contained in a compact subset of Y.
The following statement, which we state in the form of proposition, is usually called the KakutaniKy Fantype fixed point theorem.
Proposition 2
(cf., e.g., Theorem II.8.4 in [21])
Let C be a convex (not necessarily closed) subset of a normed linear space, and let \(\varphi\colon C \multimap C\) be a compact Kakutani map. Then φ has a fixed point.
Definition 2
Observe that Φ is a bounded u.s.c. mapping with (nonempty) compact and convex values.
Proposition 3
(cf., e.g., Theorem 3.2.1 in [38])
It will also be convenient to recall the following lemmas.
Lemma 4
(cf., e.g., Theorem 0.3.4 in [39])

the set \(\{x_{k}(t) \mid k \in\mathbb{N}\}\) is bounded \(\forall t \in J\),

there is a Lebesgue integrable function \(\alpha\colon J \rightarrow\mathbb{R}\) such that$$\bigl\vert x'_{k}(t)\bigr\vert \leq\alpha(t), \quad \textit{for almost all }t \in J, \forall k \in\mathbb{N}. $$
 (i)
\(\{x_{k}\}\) uniformly converges to x,
 (ii)
\(\{x'_{k}\}\) weakly converges in \(L^{1}(J,\mathbb{R}^{n})\) to \(x'\).
Lemma 5
(cf. [40], p.88)
3 Main theorem
Let us consider at first the scalar Dirichlet problem for differential equations involving a dry friction (4), where a, b, c, \(x_{0}\), \(x_{T}\) and \(T >0\) are real constants and \(p\colon J \rightarrow \mathbb{R}\), \(J=[0,T]\), is a Lebesgue integrable function.
Since the function \(\operatorname{sgn}(\cdot)\) is discontinuous in the spatial variable, problem (4) need not have a Carathéodory solution, i.e. a function \(x \colon J \rightarrow\mathbb {R}\) with an absolutely continuous derivative, satisfying (4), for almost all \(t \in J\). Therefore, we need another notion of an appropriate solution, namely the one in the sense of Filippov. For this goal, we use the concept of the Filippovlike regularization (see [15]) of spatially discontinuous maps. More precisely, applying Definition 2 to the righthand side involving spatial discontinuities, we can speak about a solution in the sense of Filippov of the original problem, provided it is a Carathéodory solution of a multivalued problem with a Filippovlike regularized righthand side.
In our situation, the discontinuous function to be regularized is the function signum. On the basis of the Filippovlike regularization of \(\operatorname{sgn}(\cdot)\), we obtain the multivalued mapping Signum defined in (3), i.e. \(\operatorname{Sgn}(\cdot)\).
One can readily check that the Signum mapping is u.s.c. with compact and convex values. Hence, after the described Filippovlike regularization, problem (4) with a discontinuous function \(\operatorname{sgn}(\cdot)\) becomes multivalued, i.e. (5).
By a Filippov solution of (4), we understand a function \(x(\cdot)\colon J \rightarrow\mathbb{R}\) with absolutely continuous derivative, satisfying problem (5), almost everywhere on J.
For our needs, it will be convenient to consider still the problem involving more multivalued terms, namely (9), where \(F_{1} \colon\mathbb{R}\multimap\mathbb{R}\), \(F_{2} \colon\mathbb {R}\multimap\mathbb{R}\) are u.s.c. maps with compact, convex values and \(P \colon J \multimap\mathbb{R}\) is an Aumann integrable mapping, \(J=[0,T]\).
One can also easily check that \(v(t) \leq k_{2} := \max\{x_{0}, x_{T}\} \), for all \(t\in J\), and \(v'(t)= k_{3} :=\frac{x_{T}x_{0}}{T}\), for all \(t \in J\).
Obviously, \(F_{1} \vert_{\overline{B}_{D}} \colon\overline{B}_{D} \multimap\mathbb{R}\) and \(F_{2} \vert_{\overline{B}_{D}} \colon \overline{B}_{D} \multimap\mathbb{R}\) are u.s.c. maps with compact and convex values, where \(\overline{B}_{D} := \{ z \in\mathbb{R}\midz \leq D\}\) is the closed ball, constant D is such that \(D = D_{0}+k_{2}+k_{3}\), where \(D_{0}>0\) is a suitable constant which will be specified later, and \(k_{2}\), \(k_{3}\) are defined above. Thus, the same is true for \(F_{1}^{*}, F_{2}^{*} \colon\mathbb{R}\multimap\mathbb{R}\) defined as above.
Observe that, in this way, \(F_{1}^{*}(z+v(t)) = F_{1}(z+v(t))\), for \(z \leq D_{0}\), and \(F_{2}^{*}(z+v'(t)) = F_{2}(z+v'(t))\), for \(z \leq D_{0}\).
Hence, let us find sufficient conditions for the solvability of (13). We distinguish two cases in order to separate formally a linear differential operator and a multivalued perturbation.
Theorem 1
Proof
In order to check all the assumptions of Proposition 2, we will proceed in four steps.
(i) Since problem (16) is uniquely solvable, the set \(\varphi(Q)\) is nonempty.
(ii) Let us prove that the set \(\varphi(Q)\), i.e. the set of solutions of (15), is relatively compact. According to the well known ArzeláAscoli lemma, the set of solutions is relatively compact in \(C^{1}(J, \mathbb{R})\) if and only if it is uniformly bounded and equicontinuous, both in the \(C^{1}\)norm.
Since this estimate holds in the same way for all \(q \in Q\), this already means that the solutions \(u(\cdot)\) of (15), i.e. the set \(\varphi(Q)\), are uniformly bounded in the \(C^{1}\)norm.
Moreover, according to (20), (22), if there exists a positive constant \(D_{0}\) such that \(D_{0} \geq\Delta_{1}(D_{0})\), then the set \(\varphi(Q)\) satisfies \(\varphi(Q) \subset Q\).
Therefore, the solutions \(u(\cdot)\) of (15) are equicontinuous in the \(C^{1}\)norm. Summing up (a) and (b), the elements of the set \(\varphi(Q)\) are relatively compact in the \(C^{1}\)norm, as claimed.
(iii) We will show that the operator φ is u.s.c. In view of Lemma 2, and since φ was shown to be compact, it is sufficient to show that the graph \(\Gamma_{\varphi}\) is closed. Let \(\{(q_{k}, u_{k})\} \subset\Gamma_{\varphi}\) be a sequence such that \(\{(q_{k}, q'_{k}, u_{k})\} \rightarrow(q, q', u)\), where \(q \in Q\). For all \(k \in\mathbb{N}\) and a.a. \(t \in J\), the sequence \(\{u_{k}'\}\) is bounded and \(u_{k}''(t)\leq p(t)+M_{1}(D_{0})+M_{2}(D_{0})+k_{1} D_{0}+c+a k_{3}+b k_{2}\), for a.a. \(t \in J\). The sequence \(\{w_{k}:=u'_{k}\}\) satisfies all the assumptions of Lemma 4.
Thus, applying Lemma 4 to the sequence \(\{w_{k}:=u'_{k}\}\), we find that there exists a subsequence of \(\{u_{k}'\}\), for the sake of simplicity denoted in the same way as the sequence, which converges uniformly to \(u'\) on J and such that \(\{u''_{k}\}\) converges weakly to \(u''\) in \(L^{1}(J,\mathbb{R})\).
The set \(\varphi(Q)\) is relatively compact and graph \(\Gamma_{\varphi }\) is closed. Therefore, the mapping φ is u.s.c., compact and, in particular, with compact values.
(iv) Finally, we will show that the mapping φ has convex values.
After all, applying Proposition 2, we obtain the existence of a fixed point of the multivalued mapping φ which represents a solution of problem (13). However, because of the definitions of \(F_{1}^{*}\), \(F_{2}^{*}\), such a solution must be a solution of problem (12) as well as (9). □
Example 1
Remark 1
Remark 2
4 Discussion of further possibilities
 (1)For \(a^{2}4b>0\): \(D_{0} \geq\Delta_{2}(D_{0})\), where\(\lambda_{1}=\frac{a+\sqrt{a^{2}4b}}{2}\), \(\lambda_{2}=\frac{a\sqrt {a^{2}4b}}{2}\).$$\begin{aligned} \Delta_{2}(D_{0}) :=& \frac{e^{ (\lambda_{1}\lambda_{2} )T}}{\sqrt{a^{2}4 b}} \bigl[1+\vert \lambda_{1}\vert +\lambda_{2} \bigr] \\ &{}\times \bigl[\mathcal{P}+T \bigl(M_{1}(D_{0})+M_{2}(D_{0})+c+a k_{3}+b k_{2} \bigr) \bigr], \end{aligned}$$
 (2)For \(a^{2}4b=0\): \(D_{0} \geq\Delta_{3}(D_{0})\), where$$\begin{aligned} \begin{aligned} \Delta_{3}(D_{0}) :={} &e^{\frac{a}{2} T} \biggl[1+ \frac{T}{4} \bigl(1+2\vert a\vert \bigr) \biggr] \\ &{}\times \bigl[\mathcal{P}+T \bigl(M_{1}(D_{0})+M_{2}(D_{0})+c+a k_{3}+b k_{2} \bigr) \bigr]. \end{aligned} \end{aligned}$$
 (3)For \(a^{2}4b<0\) and \(T \neq \frac{2\pi k}{\sqrt{4ba^{2}}}\), \(k \in \mathbb{N}\): \(D_{0} \geq\Delta_{4}(D_{0})\), where$$\begin{aligned} \Delta_{4}(D) :=& \frac{e^{\frac{a}{2} T} [2+a+\sqrt {4ba^{2}} ]}{ \sqrt{4ba^{2}}\vert \sin(\frac{T}{2}\sqrt{4ba^{2}})\vert } \\ &{}\times \bigl[\mathcal{P}+T \bigl(M_{1}(D_{0})+M_{2}(D_{0})+c+a k_{3}+b k_{2} \bigr) \bigr]. \end{aligned}$$
In view of Remark 2 and the estimates above, we can immediately formulate the following corollary for a Lebesgue integrable selection \(p \subset P\) and \(F_{1}(x) := d\sin x\).
Corollary 1
Remark 3
Observe that, for \(b = 0\), the conclusion of Corollary 1 reduces to the Dirichlet problem for a forced pendulum with a dry friction while, for \(d = 0\), the result concerns the Dirichlet problem for a forced ‘linear’ oscillator with a dry friction. Moreover, the nonresonance condition \(b \neq (\frac{k \pi}{T} )^{2}\), \(k \in\mathbb{N}\), distinguished in Corollary 1 in three cases (\(b>0\), \(b<0\), \(b=0\)), is evidently optimal, when comparing it with all its analogies discussed above (cf. (20), Remark 1 and Remark 2).
Finally, we can also consider a particular case with \(c=0\) of the Dirichlet problem (9) in a vector form in \(\mathbb{R}^{n}\), i.e. let a, b (\(c=0\)) and \(T>0\) remain real constants, but \(x_{0}, x_{T} \in\mathbb{R}^{n}\), \(F_{1} \colon\mathbb{R}^{n} \multimap \mathbb{R}^{n}\), \(F_{2} \colon\mathbb{R}^{n} \multimap\mathbb{R}^{n}\) be u.s.c. with convex and compact values and \(P \colon J \multimap \mathbb{R}^{n}\) be an Aumann (componentwise)integrable mapping. In this way, dry friction terms can be involved, after their Filippov’s regularization, in some components of \(F_{2}\).
Because of the terms \(c\operatorname{sgn}x'(t)\) resp. \(c\operatorname{Sgn} x'(t)\), the related equations resp. inclusions are no longer linear but, for the sake of simplicity, we reserve for them this name.
Declarations
Acknowledgements
The first author was supported by the grant No. 1406958S ‘Singularities and impulses in boundary value problems for nonlinear ordinary differential equations’ of the Grant Agency of the Czech Republic. The second author was supported by the grant IGA Mathematical Models, IGA_PrF_2014028 and by the grant No. 1406958S ‘Singularities and impulses in boundary value problems for nonlinear ordinary differential equations’ of the Grant Agency of the Czech Republic.
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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