Beschreibung: We consider the problem to control a vibrating string to rest in a given finite time. The string is fixed at one end and controlled by Neumann boundary control at the other end. We give an explicit representation of the L 2 -norm minimal control in terms of the given initial state. We show that if the initial state is sufficiently regular, the same control is also L p -norm minimal for p 〉 2.

Beschreibung: We define coring objects in the category of algebras over a perfect field of characteristic p (with connected underlying Hopf algebra) and the corresponding notion for Dieudonné modules, and prove the equivalence of the two resulting categories, extending thus the methods of Dieudonné theory for Hopf rings from Ravenel (Reunión Sobre Teoría de Homotopía, volume 1 of Serie notas de matemática y simposia, 177–194, 1975 ), Schoeller (Manusc Math 3:133–155, 1970 ), Goerss (Homotopy invariant algebraic structures: a conference in honor of J. Michael Boardman, 115–174, 1999 ) and Saramago (Dieudonné theory for ungraded and periodically graded Hopf rings, Ph.D. thesis, The Johns Hopkins University, 2000 ).

Beschreibung: The aim of this work is the resolution of a non-autonomous abstract differential equation of elliptic type set on unbounded domain. The study is performed in the framework of Hölder spaces. An example for a concrete elliptic problem in nonsmooth cylindrical domains will illustrate the theory.

Beschreibung: We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic boundary conditions on domains or metric graphs. In particular, our results on the heat equation with general Wentzell-type boundary conditions complement those previously obtained by, among others, Bandle–von Below–Reichel and Vázquez–Vitillaro.

Beschreibung: In this paper, we investigate some properties of semi-Fredholm operators on Banach spaces. These results are applied to the determination of the stability of various essential spectra of closed densely defined linear operators. Also, we generalize some results in the literature and we extend and unify those obtained in Jeribi (J Math Anal Appl 271:343–358, 2002 ), Jeribi (J Math Anal Appl 275:222–237, 2002 ), Jeribi (Ser Math Inf 17:35–55, 2002 ), Jeribi (Arch Inequal Appl 2:123–140, 2004 ), Latrach and Dehici (J Math Anal Appl 259:277–301, 2001 ), Latrach and Paoli (J Aust Math Soc 77:73–89, 2004 ).

Beschreibung: We give a simple criterion so that a countable infinite direct sum of trace (evaluation) maps is a trace map. An application to the theory of self-adjoint extensions of direct sums of symmetric operators is provided; this gives an alternative approach to results recently obtained by Malamud–Neidhardt and Kostenko–Malamud using regularized direct sums of boundary triplets.

Beschreibung: The p -adic semistable laws are characterized as weak limits of scaled summands of p -adic-valued, rotation-symmetric, independent, and identically distributed random variables whose tail probability satisfies some condition. In this article it is verified such scaled sums do not converge in probability, and some more precise estimates, corresponding to the law of iterated logarithm in the real-valued setting, are given to the asymptotic growth rate of the sum. The critical scaling order is explicitly given, over which the scaled sum almost surely converges to 0. On the other hand, under the critical order, the limit superior of the p -adic norm of the scaled sum diverges almost surely. Furthermore it is shown that, at the critical order, a crucial change of the asymptotic behavior of the scaled sum occurs according to the decay of the tail probability of the random variables. In this situation, the critical value for the order of the tail probability is also found.

Beschreibung: In this paper, we continue to investigate some important results in generalized topological groups and we prove extension closed property for connectedness, compactness, and separability of generalized topological groups. Last, we define generalized topological group actions on generalized topological spaces and we establish a homeomorphism between action group and action space.

Beschreibung: Let f be an analytic function on the unit disc \({\mathbb{D}}\) and F ( a , b ; c ; z ) be the Gaussian hypergeometric function. We consider the operator T a , b , c on H ( p , q , α ) defined as T a , b , c f ( z ) =  f ( z ) * F ( a , b ; c ; z ), where * denotes the usual Hadamard/convolution product. We prove that the Taylor coefficients of F ( a , b ; c ; z ) are a multiplier from H ( p , q , α ) to H ( p , q , α + a + b − c − 1) under certain conditions on a , b and c . As a consequence, we generalize some well-known results on fractional derivatives and integrals. Furthermore, we supply some conditions on a , b and c under which F ( a , b ; c ; z ) lies in H ( p , q , α ).

Beschreibung: We address in this paper the Fredholm and compactness issues for the variational problem ( J , C β ), Bahri (Pitman Research Notes in Mathematics Series No. 173. Scientific and Technical, London, 1988 ), Bahri (C. R. Acad. Sci. Paris 299, Serie I, 15, 757–760, 1984 ). We prove that the intersection operator restricted to periodic orbits of the Reeb vector-field ∂ per does not mix with the intersection operator ∂ ∞ of the critical points at infinity. The Fredholm issues are extensively discussed in the Introduction and solved in Bahri (Arab J Maths, 2014 ). We also address in this paper the issue of existence of periodic orbits for three-dimensional Reeb vector-fields, the Weinstein conjecture (Weinstein, J Differ Equ 33:353–358, 1979 ) on S 3 , solved in dimension 3 throughout the works of Rabinowitz (Commun Pure Appl Math 31:157–184, 1978 ) and Hofer (Invent Math 114:515–563, 1993 ); see also Hutchings (Proc. 2010 ICM 46:1022–1041, 2010 ) and Taubes (Geom Topol 11:2117–2202, 2007 ) for the full Weinstein conjecture in dimension 3. Following our previous work (Bahri, Adv Nonlinear Stud 8:1–17, 2008 ), we devise a new method to find these periodic orbits when they are of odd index. We conjecture that this method, when combined with the other results described above about the intersection operator, gives rise to a homology that is specific of the contact structure and that is invariant by deformation. The existence result, as derived here, is weaker than the one announced by Taubes (Geom Topol 11:2117–2202, 2007 ). After appropriate generalization, it provides a new proof, via variational theory, of the Weinstein conjecture on three-dimensional closed contact manifolds with finite fundamental group.