Dirichlet boundary value problem for differential equations involving dry friction

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 right-hand 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 \([0, T ]\) (\(\Rightarrow|p(t)| \leq P\), \(t \in[ 0, T]\)), 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 right-hand 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 second-order 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 one-term vs. complete linear differential operator for the associated Green’s functions.

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.

Let Y be a separable metric space and \((\Omega, \mathcal{U}, \mu)\) be a measurable space, i.e. a set Ω equipped with a σ-algebra \(\mathcal{U}\) 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 \(V \subset Y\).

Let \(F \colon X \multimap Y\) be a multivalued mapping and \(f \colon X \rightarrow Y\) be a single-valued 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.

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 \(\operatorname{Fix} (\varphi) := \{ x \in X \mid x \in\varphi(x) \}\).

The following statement, which we state in the form of proposition, is usually called the Kakutani-Ky Fan-type fixed point theorem.

Observe that Φ is a bounded u.s.c. mapping with (nonempty) compact and convex values.

It will also be convenient to recall the following lemmas.

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 Filippov-like regularization (see [15]) of spatially discontinuous maps. More precisely, applying Definition 2 to the right-hand 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 Filippov-like regularized right-hand side.

In our situation, the discontinuous function to be regularized is the function signum. On the basis of the Filippov-like 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 Filippov-like 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 \(\overline{B}_{D} := \{ z \in\mathbb{R}\mid|z| \leq D\}\) 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.

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 \(F_{1}(x) := d\sin x\).

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 (component-wise)-integrable mapping. In this way, dry friction terms can be involved, after their Filippov’s regularization, in some components of \(F_{2}\).