$Open Mathematics$

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Volume 10 Issue 6 - Topics in Nonlinear Analysis: dedicated to Kazimierz Goebel and Lech Górniewicz on the occasion of their 70th birthday

December 2012

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Open Access October 12, 2012

Editors’ preface for the topical issue “Topics in Nonlinear Analysis”

Wojciech Kryszewski, Robert Skiba

Page range: 1915-1919

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Open Access October 12, 2012

Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3

Joanna Janczewska, Jakub Maksymiuk

Page range: 1920-1927

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Abstract

We consider a conservative second order Hamiltonian system $$\ddot q + \nabla V(q) = 0$$ in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ {0} = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.

Open Access October 12, 2012

The shadowing chain lemma for singular Hamiltonian systems involving strong forces

Marek Izydorek, Joanna Janczewska

Page range: 1928-1939

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Abstract

We consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 \{ζ}→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits — the shadowing chain lemma — via minimization of action integrals and using simple geometrical arguments.

Open Access October 12, 2012

A new method of proof of Filippov’s theorem based on the viability theorem

Sławomir Plaskacz, Magdalena Wiśniewska

Page range: 1940-1943

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Abstract

Filippov’s theorem implies that, given an absolutely continuous function y: [t 0; T] → ℝd and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x 0, there exists a solution x(·) to the differential inclusion x′(t) ∈ F(t, x(t)) starting from x 0 at the time t 0 and satisfying the estimation $$\left| {x(t) - y(t)} \right| \leqslant r(t) = \left| {x_0 - y(t_0 )} \right|e^{\int_{t_0 }^t {l(s)ds} } + \int_{t_0 }^t \gamma (s)e^{\int_s^t {l(\tau )d\tau } } ds,$$ where the function γ(·) is the estimation of dist(y′(t), F(t, y(t))) ≤ γ(t). Setting P(t) = {x ∈ ℝn: |x −y(t)| ≤ r(t)}, we may formulate the conclusion in Filippov’s theorem as x(t) ∈ P(t). We calculate the contingent derivative DP(t, x)(1) and verify the tangential condition F(t, x) ∩ DP(t, x)(1) ≠ ø. It allows to obtain Filippov’s theorem from a viability result for tubes.

Open Access October 12, 2012

Filippov Lemma for certain second order differential inclusions

Grzegorz Bartuzel, Andrzej Fryszkowski

Page range: 1944-1952

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Abstract

In the paper we give an analogue of the Filippov Lemma for the second order differential inclusions with the initial conditions y(0) = 0, y′(0) = 0, where the matrix A ∈ ℝd×d and multifunction is Lipschitz continuous in y with a t-independent constant l. The main result is the following: Assume that F is measurable in t and integrably bounded. Let y 0 ∈ W 2,1 be an arbitrary function fulfilling the above initial conditions and such that where p 0 ∈ L 1[0, 1]. Then there exists a solution y ∈ W 2,1 to the above differential inclusions such that a.e. in [0, 1], .

Open Access October 12, 2012

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

Stanisław Migórski

Page range: 1953-1968

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Abstract

We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.

Open Access October 12, 2012

On variational impulsive boundary value problems

Marek Galewski

Page range: 1969-1980

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Abstract

Using the variational approach, we investigate the existence of solutions and their dependence on functional parameters for classical solutions to the second order impulsive boundary value Dirichlet problems with L1 right hand side.

Open Access October 12, 2012

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Małgorzata Klimek, Marek Błasik

Page range: 1981-1994

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Abstract

Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.

Open Access October 12, 2012

Uniqueness of solutions to an Abel type nonlinear integral equation on the half line

Wojciech Mydlarczyk

Page range: 1995-2002

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Abstract

We consider a convolution-type integral equation u = k ⋆ g(u) on the half line (−∞; a), a ∈ ℝ, with kernel k(x) = x α−1, 0 < α, and function g(u), continuous and nondecreasing, such that g(0) = 0 and 0 < g(u) for 0 < u. We concentrate on the uniqueness problem for this equation, and we prove that if α ∈ (1, 4), then for any two nontrivial solutions u 1, u 2 there exists a constant c ∈ ℝ such that u 2(x) = u 1(x +c), −∞ < x. The results are obtained by applying Hilbert projective metrics.

Open Access October 12, 2012

Measures of noncompactness in the study of solutions of nonlinear differential and integral equations

Józef Banaś

Page range: 2003-2011

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Abstract

The aim of this paper is to make an overview of some existence results for nonlinear differential and integral equations. Those results were obtained by the author and his co-workers during last years with some help of the technique of measures of noncompactness and a fixed point theorem of Darbo type.

Open Access October 12, 2012

A generalization of Krasnosel’skii fixed point theorem for sums of compact and contractible maps with application

Bogdan Przeradzki

Page range: 2012-2018

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Abstract

The existence of a fixed point for the sum of a generalized contraction and a compact map on a closed convex bounded set is proved. The result is applied to a kind of nonlinear integral equations.

Open Access October 12, 2012

Elliptic problems in generalized Orlicz-Musielak spaces

Piotr Gwiazda, Piotr Minakowski, Aneta Wróblewska-Kamińska

Page range: 2019-2032

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Abstract

We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a Δ2 nor ∇2-condition for an inhomogeneous and anisotropic N-function but assume it to be log-Hölder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ∞-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.

Open Access October 12, 2012

New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba

Page range: 2033-2050

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Abstract

We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

Open Access October 12, 2012

Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains

Mikhail Borsuk, Damian Wiśniewski

Page range: 2051-2072

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Abstract

We study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.

Open Access October 12, 2012

A class of continua that are not attractors of any IFS

Marcin Kulczycki, Magdalena Nowak

Page range: 2073-2076

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Abstract

This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.

Open Access October 12, 2012

Invariant sets and Knaster-Tarski principle

Krzysztof Leśniak

Page range: 2077-2087

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Abstract

Our aim is to point out the applicability of the Knaster-Tarski fixed point principle to the problem of existence of invariant sets in discrete-time (multivalued) semi-dynamical systems, especially iterated function systems.

Open Access October 12, 2012

Global bifurcation of homoclinic trajectories of discrete dynamical systems

Jacobo Pejsachowicz, Robert Skiba

Page range: 2088-2109

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Abstract

We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving topological properties of the asymptotic stable bundles.

Open Access October 12, 2012

The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century

Krzysztof Ciesielski

Page range: 2110-2128

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Abstract

The Poincaré-Bendixson Theorem and the development of the theory are presented — from the papers of Poincaré and Bendixson to modern results.

Open Access October 12, 2012

Seiberg-Witten invariants, the topological degree and wall crossing formula

Maciej Starostka

Page range: 2129-2137

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Abstract

Following S. Bauer and M. Furuta we investigate finite dimensional approximations of a monopole map in the case b 1 = 0. We define a certain topological degree which is exactly equal to the Seiberg-Witten invariant. Using homotopy invariance of the topological degree a simple proof of the wall crossing formula is derived.

Open Access October 12, 2012

Equivariant Morse equation

Marcin Styborski

Page range: 2138-2159

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Abstract

The paper is concerned with the Morse equation for flows in a representation of a compact Lie group. As a consequence of this equation we give a relationship between the equivariant Conley index of an isolated invariant set of the flow given by .x = −∇f(x) and the gradient equivariant degree of ∇f. Some multiplicity results are also presented.

Open Access October 12, 2012

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Grzegorz Graff, Agnieszka Kaczkowska

Page range: 2160-2172

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Abstract

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.

Open Access October 12, 2012

Equivariant degree of convex-valued maps applied to set-valued BVP

Zdzisław Dzedzej

Page range: 2173-2186

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Abstract

An equivariant degree is defined for equivariant completely continuous multivalued vector fields with compact convex values. Then it is applied to obtain a result on existence of solutions to a second order BVP for differential inclusions carrying some symmetries.

Open Access October 12, 2012

The homotopies of admissible multivalued mappings

Mirosław Ślosarski

Page range: 2187-2199

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Abstract

Certain properties of homotopies of admissible multivalued mappings shall be presented, along with their applications as the tool for examining the acyclicity of a space.

Open Access October 12, 2012

Notes on tiled incompressible tori

Leonid Plachta

Page range: 2200-2210

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Abstract

Let Θ denote the class of essential tori in a closed braid complement which admit a standard tiling in the sense of Birman and Menasco [Birman J.S., Menasco W.W., Special positions for essential tori in link complements, Topology, 1994, 33(3), 525–556]. Moreover, let R denote the class of thin tiled tori in the sense of Ng [Ng K.Y., Essential tori in link complements, J. Knot Theory Ramifications, 1998, 7(2), 205–216]. We define the subclass B ⊂ Θ of typical tiled tori and show that R ⊂ B. We also describe a method allowing to construct new examples of tiled essential tori T which are outside the class B in the strong sense. In [Kazantsev A., Essential tori in link complements: detecting the satellite structure by monotonic simplification, preprint available at http://arxiv.org/abs/1005.5263], Kazantsev showed that the inclusion R ⊂ Θ is proper by giving the corresponding example of a nonthin tiled torus T. It turns out this torus T is inside the class B. We show that the inclusion B ⊂ Θ is proper. It follows that the tori from the class B do not provide the complete geometric description of the class Θ. The main results of the paper are Theorems 2.1 and 2.2 which give a constructive procedure for obtaining examples of nontypical tiled essential tori.

Open Access October 12, 2012

The minimal displacement problem in the space l ∞

Krzysztof Bolibok

Page range: 2211-2214

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Abstract

We give a lower bound for the minimal displacement characteristic in the space l ∞.

Open Access October 12, 2012

Fixed points and iterations of mean-type mappings

Janusz Matkowski

Page range: 2215-2228

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Abstract

Let (X, d) be a metric space and T: X → X a continuous map. If the sequence (T n)n∈ℕ of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit $$\mu _T (x) = \mathop {\lim }\limits_{n \to \infty } T^n (x)$$ is a fixed point of T. The problem of determining the form of µT leads to the invariance equation µT ○ T = µT, which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I p, where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping M =(M 1,...,M p) of I p. In this paper we give the explicit forms of µM for some classes of mean-type mappings. In particular, the classical Pythagorean harmony proportion can be interpreted as an important invariance equality. Some applications are presented. We show that, in general, the mean-type mappings are not non-expansive.

Open Access October 12, 2012

The Demyanov metric and some other metrics in the family of convex sets

Tadeusz Rzeżuchowski

Page range: 2229-2239

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Abstract

We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.

Open Access October 12, 2012

Perron-Frobenius and Krein-Rutman theorems for tangentially positive operators

Adam Kanigowski, Wojciech Kryszewski

Page range: 2240-2263

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Abstract

We study several aspects of a generalized Perron-Frobenius and Krein-Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency or weak range assumptions implying the resolvent invariance of the cone. The further assumptions rely on relations between the spectral and essential spectral bounds of the operator. In general we do not assume that the cone induces the Banach lattice structure into the underlying space.

Open Access October 12, 2012

On local convexity of nonlinear mappings between Banach spaces

Iryna Banakh, Taras Banakh, Anatolij Plichko, Anatoliy Prykarpatsky

Page range: 2264-2271

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Abstract

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.

Open Access October 12, 2012

Concave iteration semigroups of linear continuous set-valued functions

Andrzej Smajdor, Wilhelmina Smajdor

Page range: 2272-2282

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Abstract

Let {F t: t ≥ 0} be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) − F t (x) exist for x ∈ K and t > 0, then D t F t (x) = (−1)F t ((−1)G(x)) for x ∈ K and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and $$G(x) = \mathop {\lim }\limits_{s \to 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord{\left/ {\vphantom {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \kern-\nulldelimiterspace} {\left( { - s} \right)}}$$ for x ∈ K.

Open Access October 12, 2012

Limiting distribution for a simple model of order book dynamics

Łukasz Kruk

Page range: 2283-2295

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Abstract

A continuous-time model for the limit order book dynamics is considered. The set of outstanding limit orders is modeled as a pair of random counting measures and the limiting distribution of this pair of measure-valued processes is obtained under suitable conditions on the model parameters. The limiting behavior of the bid-ask spread and the midpoint of the bid-ask interval are also characterized.

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20 YEARS - 20 VOLUMES

Look back at the journal’s 20-year journey and browse its high-impact content. Visit the website

On this occasion seven papers of excellent and profound mathematics have been published to encourage reflection about further directions in the field.
See the content of the SPECIAL ISSUE ON "FUTURE DIRECTIONS OF FURTHER DEVELOPMENTS IN MATHEMATICS"　　　

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SPECIAL ISSUE PROPOSALS

If you have an idea for a prospective topical issue, please complete Special Issue proposal questionaire (in DOC), addressing your proposal to the Managing Editor (Dr. Justyna Zuk, Justyna.Zuk@degruyter.com).

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Open Mathematics is a fully peer-reviewed open access journal that publishes significant, original and relevant works on topics across the full span of mathematics. Coverage includes:

Algebra, Algebraic Geometry, Category Theory, Complex Analysis, Control Theory and Optimization, Differential Equations, Differential Geometry, Discrete Mathematics, Dynamical Systems and Ergodic Theory, Functional Analysis, Geometry, Mathematical Logic and Foundations, Mathematical Physics, Number Theory, Numerical Analysis, Operator Theory, Probability Theory and Statistics, Real Analysis, Topology.

In order to sustain the publishing process, each article accepted for publication in Open Mathematics is subject to an Article Processing Charge of €1000. This fee is used to cover the costs of the peer-review process, professional typesetting and copyediting, as well as online hosting, long-term preservation, and extensive promotion to potential readers. There is no submission fee. Information regarding payment of the charge will be provided following acceptance for publication. Inquiries concerning Article Processing Charges should be addressed to the Managing Editor (Dr. Justyna Zuk, Justyna.Zuk@degruyter.com) before or just after submission of a manuscript.

Detailed information on　Article Processing Charges, Editorial Policy, Instructions for Authors etc. can be found as documents (in PDF) in the Supplementary Materials below. Publishing Ethics, Open Access Services at DG, FAQ for Authors and Editors etc. can be directed from External Links.