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Acta Technologica Agriculturae 1/2016Dušan Páleš et al.The most effective way for determination of curves for practical use is to use a set of control points. These control points can be accompanied by other restriction for the curve, for example boundary conditions or conditions for curve continuity (Sederberg, 2012). When a smooth curve runs only through some control points, we refer to curve approximation. The B-spline curve is one of such approximation curves and is addressed in this contribution. A special case of the B-spline curve is the Bézier curve Rédl et al., 2014). The B-spline curve is applied to a set of control points in a space, which were obtained by measurement of real vehicle movement on a slope (Rédl, 2007(Rédl, , 2008. Data were processed into the resulting trajectory (Rédl, 2012;Rédl and Kučera, 2008). Except for this, the movement of the vehicle was simulated using motion equations (Rédl, 2003;Rédl and Kročko, 2007). B-spline basis functionsBézier basis functions known as Bernstein polynomials are used in a formula as a weighting function for parametric representation of the curve (Shene, 2014). B-spline basis functions are applied similarly, although they are more complicated. They have two different properties in comparison with Bézier basis functions and these are: 1) solitary curve is divided by knots, 2) basis functions are not nonzero on the whole area. Every B-spline basis function is nonzero only on several neighbouring subintervals and thereby it is changed only locally, so the change of one control point influences only the near region around it and not the whole curve.These numbers are called knots, the set U is called the knot vector, and the half-opened interval 〈u i , u i + 1 ) is the i-th knot span. Seeing that knots u i may be equal, some knot spans may not exist, thus they are zero. If the knot u i appears p times, hence u i = u i + 1 = ... = u i + p -1 , where p >1, u i is a multiple knot of multiplicity p, written as u i (p). If u i is only a solitary knot, it is also called a simple knot. If the knots are equally spaced, i.e. (u i + 1 -u i ) = constant, for every 0 ≤ i ≤ (m -1), the knot vector or knot sequence is said uniform, otherwise it is non-uniform.Knots can be considered as division points that subdivide the interval 〈u 0 , u m 〉 into knot spans. All B-spline basis functions are supposed to have their domain on 〈u 0 , u m 〉. We will use u 0 = 0 and u m = 1.To define B-spline basis functions, we need one more parameter k, which gives the degree of these basis functions. Recursive formula is defined as follows:This definition is usually referred to as the Cox-de Boor recursion formula. If the degree is zero, i.e. k = 0, these basis functions are all step functions that follows from Eq. (1). N i, 0 (u) = 1 is only in the i-th knot span 〈u i , u i + 1 ). For example, if we have four knots u 0 = 0, u 1 = 1, u 2 = 2 and u 3 = 3, knot spans 0, 1 and 2 are 〈0, 1), 〈1, 2) and 〈2, 3), and the basis functions of degree 0 are N 0, 0 (u) = 1 on interval 〈0, 1) Acta In this co...

Abstract. We consider the nonlinear eigenvalue problemin Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R N with smooth boundary and p, q are continuous functions on Ω such that 1 < inf Ω q < inf Ω p < sup Ω q, sup Ω p < N, and q(x) < Np(x)/ (N − p(x)) for all x ∈ Ω. The main result of this paper establishes that any λ > 0 sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle. Introduction and preliminary resultsA basic result in the elementary theory of linear partial differential equations asserts that the spectrum of the Laplace operator in H In this paper we are concerned with the nonhomogeneous eigenvalue problemwhere Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary, λ > 0 is a real number, and p, q are continuous on Ω.The case p(x) = q(x) was considered by Fan, Zhang and Zhao in [14] who, using the Ljusternik-Schnirelmann critical point theory, established the existence of a sequence of eigenvalues. Denoting by Λ the set of all nonnegative eigenvalues, Fan, Zhang and Zhao showed that sup Λ = +∞, and they pointed out that only under additional assumptions we have inf Λ > 0. We remark that for the p-Laplace operator (corresponding to p(x) ≡ p) we always have inf Λ > 0.In this paper we study problem (1.1) under the basic assumptionOur main result establishes the existence of a continuous family of eigenvalues for problem (1.1) in a neighborhood of the origin. More precisely, we show that there exists λ > 0 such that any λ ∈ (0, λ ) is an eigenvalue for problem (1.1). We start with some preliminary basic results on the theory of Lebesgue-Sobolev spaces with variable exponent. For more details we refer to the book by Musielak [21] and the papers by Edmunds et al. [8,9,10], Kovacik and Rákosník [17], and Samko and Vakulov [24].Assume that p ∈ C(Ω) andFor any h ∈ C + (Ω) we defineFor any p(x) ∈ C + (Ω), we define the variable exponent Lebesgue spaceu is a measurable real-valued function andWe define a norm, the so-called Luxemburg norm, on this space by the formulaLicense or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON A NONHOMOGENEOUS QUASILINEAR EIGENVALUE PROBLEM 2931We remember that the variable exponent Lebesgue spaces are separable and reflexive Banach spaces. If 0 < |Ω| < ∞ and p 1 , p 2 are variable exponent, so that p 1 (x) ≤ p 2 (x) almost everywhere in Ω, then there exists the continuous embeddingWe denote by(Ω) the Hölder type inequalityholds true. An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the L p(x) (Ω) space, which is the mapping ρ p(x) :, then the following relations hold true: (Ω), · ) is a separable and reflexive Banach space. We note that if s(x) ∈ C + (Ω) and s(x) < p (x) for all x ∈ Ω, then the embedding W(Ω) is compact and continuous, whereWe refer to Kováčik and Rákosník [17] for more properties of Lebesgue and Sobolev spaces with variable exponen...

The content of this paper is at the interplay between function spaces L p(x) and W k,p(x) with variable exponents and fractional Sobolev spaces W s,p. We are concerned with some qualitative properties of the fractional Sobolev space W s,q(x),p(x,y) , where q and p are variable exponents and s ∈ (0, 1). We also study a related nonlocal operator, which is a fractional version of the nonhomogeneous p(x)-Laplace operator. The abstract results established in this paper are applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.

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This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.