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Boundary Value Problems 2015 2015 :106

https://doi.org/10.1186/s13661-015-0371-z

© Andres and Machů2015

Received: 10April2015

Accepted: 8June2015

Published: 24June2015

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 one-term 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.

Dirichlet problem dry friction Filippov solution Green’s functions growth restrictions Kakutani-Ky Fan fixed point theorem solution estimates
34A60 34B15 34B16 34B27 47H04
x''(t)+b \sin x(t) = p(t), \qquad x(0)=x_{0}, \qquad x(T)=x_{T},
x''(t)+ax'(t)+b\sin x(t) = p(t), \quad a \in\mathbb{R}
x''(t)+ax'(t)+bx(t) = p(t), \qquad x(0)= x_{0}, \qquad x(T)=x_{T},
x''(t) + ax'(t)+bx(t) = 0, \qquad x(0)=0, \qquad x(T)=0,

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 , 17 , 18 , New Balance Mens ML501 Classics Core Running Shoe Blue/Black MmX3k5
,
20 ].

x''(t)+ax'(t)+b\sin x(t)+c \operatorname{sgn}x'(t) = p(t),\qquad x(0)=x_{0}, \qquad x(T)=x_{T},
(1)
x''(t)+ax'(t)+b\sin x(t) \in p(t)-c \operatorname{Sgn} x'(t),\qquad x(0)=x_{0},\qquad x(T)=x_{T},
(2)
\operatorname{Sgn} z = \left \{ \textstyle\begin{array}{l@{\quad}l} -1, \text{for }z \in(-\infty, 0), \\ {[ -1, 1 ]}, \text{for }z=0, \\ 1, \text{for }z \in(0, \infty), \end{array}\displaystyle \right .
(3)
x''(t)+ax'(t)+bx(t)+c \operatorname{sgn}x'(t) = p(t),\qquad x(0)=x_{0}, \qquad x(T)=x_{T},
(4)
x''(t)+ax'(t)+bx(t) \in p(t)-c \operatorname{Sgn} x'(t),\qquad x(0)=x_{0},\qquad x(T)=x_{T}.
(5)
x''(t)+ax'(t)+bx(t)=0, \qquad x(0)=x_{0}, \qquad x(T)=x_{T}.

On the other hand, the estimates of solutions and derivatives are not indicated explicitly in [ 22 ].

Similarly, the function det ( A ) 1 / m is concave and the corresponding maximization problem can be phrased in a similar way using a geometric mean bound from Sec. 4.2.4 (Geometric mean) .

We next consider a non-square matrix A R m × p . Assume p m and denote the singular values of A by

σ 1 ( A ) σ 2 ( A ) σ p ( A ) 0.

The singular values are connected to the eigenvalues of A T A via

(6.13) AmoonyFashion Womens Solid Pu Kitten Heels Peep Toe Pull On Sandals Red hGPzof1
σ i ( A ) = λ i ( A T A ) ,

and if A is square and symmetric then σ i = | λ i | . We show next how to optimize several functions of the singular values.

Largest singular value

The epigraph σ 1 ( A ) t can be characterized using (6.13) as

A T A t 2 I

which from Schur’s lemma is equivalent to

(6.14) [ t I A A T t I ] 0.

The largest singular value σ 1 ( A ) is also called the spectral norm or the 2 -norm of A , A 2 := σ 1 ( A ) .

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