Caratheodory Function Research Articles

On positive solutions for some semilinear periodic parabolic eigenvalue problems

For a bounded domain Ω in RN, N⩾2, satisfying a weak regularity condition, we study existence of positive and T-periodic weak solutions for the periodic parabolic problem Luλ=λg(x,t,uλ) in Ω×R, uλ=0 on ∂Ω×R. We characterize the set of positive eigenvalues with positive eigenfunctions associated, under the assumptions that g is a Caratheodory function such that ξ→g(x,t,ξ)/ξ is nonincreasing in (0,∞) a.e. (x,t)∈Ω×R satisfying some integrability conditions in (x,t) and ∫0Tesssupx∈Ωinfξ>0g(x,t,ξ)ξdt>0.

On a Special Caratheodory Subclass and the Set of Solvability of the Markov Trigonometric Moment Problem with Periodic Gaps

The classical Markov trigonometric moment problem is considered in the case when the moment sequence has periodically repeating gaps. The suggested approach represent a development of the N. I. Akhieser’s and M.G. Krein’s idea on the relation between the trigonometric moment problem and the Caratheodory coefficients problem. We associate with the gaps distribution law M a certain subclass C(M ) of the Caratheodory functions class C. The multiplicative representation of the class C(M ) is obtained. On this basis the complete description of the solvability set of the Markov trigonometric moment problem with gaps as well as a method of searching of its canonical solutions are given.

On the existence of periodic solutions for nonlinear evolutions in Hilbert spaces

A periodic solutions for nonlinear evolution equations of the form du/dt ∈ -Au(t) + F(t,u(t)), t ∈ R, where F : R × H → 2 is a set-valued Caratheodory function are considered when both F and A are set-valued mappings and not necessarily linear.

Elliptic problems with nonmonotone discontinuities at resonance (Erratum)

We do not assume that f is a Caratheodory function. First of all we must change Hypothesis 3.1(ii) to 3.1(ii)′, namely, we suppose the following. Hypothesis 3.1(ii)′. There exists θ > p and r0 > 0 such that for all |x| ≥ r0 and all v ∈ [ f1(z,x), f2(z,x)], we have 0 0 has a nonempty interior, but this is not true. We can construct such a set with an empty interior (e.g., the e-Cantor set). For a coercive energy functional, we can prove the existence of a solution of type II (see, e.g., [2] for a Neumann problem). But as far as we know for the noncoercive case, we do not have such a result, so it remains an open problem.

New Generalizations of the Scorza-Dragoni Theorem

We consider Caratheodory functions f : T × X → Y, where T is a topological space with regular σ-finite measure, the spaces X and Y are metrizable and separable, and X is locally compact. We show that every function of this sort possesses the Scorza-Dragoni property. A similar result is also established in the case where the space T is locally compact and X = ℝ∞.

Continued Fractions and Szegö Polynomials in Frequency Analysis and Related Topics

This paper is an expository survey of recent research on the application of Szego polynomials and PPC-continuedfractions to the frequency analysis problem described as follows: We want to determine the unknown frequencies ω1, ω2, ..., ω I from a sample of N observed values x N (m), m = 0, 1, ..., N− 1, arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with frequencies ω1,ω2, ..., ω I . The method is based on the property that certain zerosof the Szego polynomials (and poles of the PPC-fraction approximants) converge (as N → ∞) to the frequency points ei ω j , j = ±1, ± 2, ..., ± I. The remaining zeros are bounded away from the unit circle |z|=1, asN → ∞. The Levinson algorithm is used to construct the Szego polynomials and PPC-fractions from the values x N (m). A discussion is given on connections between the topics: Caratheodory functions,the trigonometric moment problem, Szego polynomials and PPC-fractions. We also describe applications to Doppler radar, medicine, speech processing, speech therapy, meteorology and ocean tides.

On consequences of certain boundary conditions on the unit circle

Let V denote the well-known class of Caratheodory functions of the form p(z) = 1+ciz-i , z e A = {z 6 C: \z < 1}, with positive real part in the unit disc and let H(M) stand for the class of holomorphic functions commonly bounded by M in A. In 1992, J. Fuka and Z. J. Jakubowski began an investigation of families of mappings p6^ fulfilling certain additional boundary conditions on the unit circle T. At first, the authors examined the class V(B, b; a) of functions defined by conditions given by the upper limits for two disjoint open arcs of T. After that, such boundary conditions given, in particular, by the nontangential limits, were assumed for different subsets of the unit circle. In parallel, G. Adamczyk started to search for properties of families, contained in H(M) and satisfying certain similar conditions on T. The present article belongs to the above series of papers. In the first section we will consider subclasses of V of functions satisfying some inequalities on several arcs of T, whereas in Sections 2 and 3—families of mappings / € H(M) with conditions given for measurable subsets of the unit circle T.

Regularity for entropy solutions of parabolic p-Laplacian type equations

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation $u_t -\operatorname{div} {\mathbf a}_p (x,\nabla u) = f$ in $]0,T[\times \Omega$ with initial datum in $L^1(\Omega)$ and assuming Dirichlet's boundary condition, where ${\mathbf a}_p(.,.)$ is a Caratheodory function satisfying the classical Leray-Lions hypotheses, $f\in L^1(]0,T[\times \Omega)$ and $\Omega$ is a domain in ${\mathbb R}^N$. We find spaces of type $L^r(0,T; {\cal M}^q(\Omega))$ containing the entropy solution and its gradient. We also include some summability results when $f = 0$ and the $p$-Laplacian equation is considered.

Lower semicontinuity for quasiconvex integrals of higher order

We consider a functional of the type \( {\cal F}(u) = \int_\Omega F(x,u,\ldots,{D^k}u)dx \), where \( \Omega \) is an open bounded set of \( {\Bbb R}^n \) and F is a Caratheodory function. By an approximation argument we prove the lower semincontinuity of \( \cal F \) with respect to the weak topology of \( W^{k,p} (\Omega; {\Bbb R}^m) \) under p-growth conditions for the integrand F.

Some remarks on critical point theory for nondifferentiable functionals

In this paper we study the existence of critical points of nondifferentiable functionals J of the kind $J(v) =\int_\Omega A(x,v) |\nabla v|^2 - F(x,v)$ with A (x,z) a Caratheodory function bounded between positive constant and with bounded derivative respect to the variable z, and F (x,z) is the primitive of a (Caratheodory) nonlinearity f (x,z) satisfying suitable hipotheses. Since J is just differentible along bounded directions, a suitable compactness condition is introduced. Its connection with coercivity is discussed. In addition, the case of concave-convex nonlinearities f (x,z), unbounded coefficients A (x,z) and related problems are also studied.

Existence and uniqueness for a degenerate parabolic equation with 𝐿¹-data

In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in L 1 ( Ω ) L^{1}(\Omega ) , u t = d i v a ( x , D u ) in ( 0 , ∞ ) × Ω , \begin{equation*}u_{t} = \mathrm {div} \mathbf {a} (x,Du) \quad \text {in } (0, \infty ) \times \Omega , \end{equation*} − ∂ u ∂ η a ∈ β ( u ) on ( 0 , ∞ ) × ∂ Ω , \begin{equation*}-{\frac {{\partial u} }{{\partial \eta _{a}}}} \in \beta (u) \quad \text {on } (0, \infty ) \times \partial \Omega ,\end{equation*} u ( x , 0 ) = u 0 ( x ) in Ω , \begin{equation*}u(x, 0) = u_{0}(x) \quad \text {in }\Omega ,\end{equation*} where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, ∂ / ∂ η a \partial / {\partial \eta _{a}} is the Neumann boundary operator associated to a \mathbf {a} , D u Du the gradient of u u and β \beta is a maximal monotone graph in R × R {\mathbb {R}}\times {\mathbb {R}} with 0 ∈ β ( 0 ) 0 \in \beta (0) .

PBVP of integrodifferential equations with Carathéodory functions

We study the periodic boundary value problems for nonlinear integro-differential equations of Volterra type with Caratheodory functions. For two situations relative to lower and upper solutions α and β∶α≤s or s≤α, the existence of solutions and the monotone iterative method for establishing extreme solutions are considered.

On weak minima of certain integral functionals

We prove a regularity result for weak minima of integral functionals of the form T Ω F (x,Du) dx where F (x, ξ) is a Caratheodory function which grows as |ξ| with some p > 1.

Generalized upper and lower solution method for the forced Duffing equation

This paper gives the generalized upper and lower solution method for the forced Duffing equation x + kx' + f(t, x) = 0, and obtains existence theorems for T-periodic solutions, where f is a Caratheodory function. Our results generalize or extend some famous results obtained by Mawhin(1985), Habets(1990), Nkashama(1989) and Nieto(1990).

Une méthode de point fixe pour le p-laplacien

Due to the compactness of the operator (-Δ p) 1 N ƒ, 1 < p < ∞, where Δ p is the p-Laplacian and N ƒ is the Nemytskii operator corresponding to a Caratheodory function ƒ: Ω × R → R, which satisfies a particular growth condition, the homotopy invariance of Leray-Schauder degree can be used in order to prove the existence of a W 0 1,p (Ω)-solution for the equation -Δ p u = N ƒ u.

On the solution set of an equation of the type f(t, ?(u(t)) = 0

We investigate the solution set Γ of an equation of the type f(t, Φ(u(t)) = 0, where Φ is a linear homeomorphism from a topological vector space X onto L1(T) and f: T×R → R is a Caratheodory function. More precisely, we characterize the property of Γ of intersecting each closed hyperplane of X.

Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order

where Ω is a bounded domain in R ,L is a linear second order elliptic operator for which the maximum principle holds, B is a linear first order boundary operator and f is a nonlinear Caratheodory function. We are concerned with the solvability of (1.1) in presence of lower and upper solutions. A classical basic result in this context says that if α is a lower solution and β is an upper solution satisfying

Remarks on Sublinear Elliptic Problems on Unbounded Cylinders

m l Ž where V s O = R , O ; R is a bounded smooth domain m, l G 0, m . w . N ' m q l G 1, V s O if m s 0, and V s R if l s 0 , f : V = 0, ` a R is a Caratheodory function, and r : V a R is continuous, r G 0. There is by now an extensive literature on this kind of problem, for w x w x w x w x instance, Amick and Toland 1 , Stuart 2 , Esteban 3 , Burton 4 , Bona, w x w x Bose, and Turner 5 , and Berestycki and Lions 20 , where different Ž . Ž . conditions on r x and f x, s , as well as several techniques, were considered. This list is, of course, far from complete and we would also refer to w x w x Schindler 6 , Costa and Miyagaki 7 and their references. Physical motiva-

Orthogonal Polynomials on Arcs of the Unit Circle, I

LetEl=∪lj=1[ϕ2j−1, ϕ2j]⊆[0, 2π], R(ϕ)=∏2lj=1sin((ϕ−ϕj)/2) and[formula]forϕ∈(ϕ2j−1, ϕ2j). Furthermore let V, W be arbitrary real trigonometric polynomials such that R=VW and let A(ϕ) be a real trigonometric polynomial which has no zero inEl. First we derive an explicit representation of the Caratheodory function associated withf(ϕ; W)=W(ϕ)/A(ϕ)r(ϕ) onEl. With the help of this result the polynomialsPn(z), which are orthogonal on the set of arcsΓEl:={eiϕ:emsp14;ϕ∈El} with respect tof(ϕ; W), are completely characterized by a quadratic equation. (In fact a more general case including Dirac-mass points is considered.) This characterization is the basis of all of our further investigations on polynomials orthogonal on several arcs as the description of that measures which generate orthogonal polynomials with periodic or asymptotically periodic reflection coefficients, the explicit representation of the orthogonality measure of the associated polynomials, the asymptotic representation of polynomials orthogonal onΓEl, etc.

Existence and regularity results for nonlinear elliptic equations with measure data

In this paper we prove some existence and regularity results for equations of the type $- \text{div}\, (a(u,u,\Delta u)+ K(x,u)) +H(x,\Delta u) =f $, in a bounded open set $\Omega$, $u=0 $ on $ \partial \Omega$, where $a$, $K$ and $H$ are Caratheodory functions and $f$ is a Radon measure.