In this paper, we study the existence and controllability of Hilfer fractional differential inclusions with noninstantaneous impulses in Banach spaces. The analysis is conducted using various mathematical tools and the set-valued version of Mo & uml;nch's fixed point theorem, which relies on several properties of the Kuratowski measure of noncompactness. To demonstrate the applicability of our results, we conclude the study with a detailed example.
For N >= 4, we let Omega be a bounded domain of R-N and Gamma be a closed curve contained in Omega. We study the existence of positive solutions u is an element of H-0(1) (Omega) to the equation -Delta u + hu = lambda rho(-s1)(Gamma)u(2s1*-1) + rho(-s2)(Gamma)u(2s2*-1) in Omega,(1) where h : Omega -> R is a continuous function, lambda is a positive real parameter, 0 <= s(2) < s(1) < 2, and rho(Gamma) is the distance function to Gamma. In this paper, we prove the existence of mountain pass solutions for the Euler-Lagrange equation (1) depending on the local geometry of the curve and the potential h. We also study the existence, symmetry and decay estimates of the positive entire solutions of (1) with Omega = R-N and Gamma being the real line.
The purpose of this paper is to explore the weak demicompactness of the operator Tn, which is the restriction of a linear operator T to its range R(Tn), Tn are considered as linear operators from R(Tn) to itself, n is an element of N. As well, we present several results on upper generalized semi-Fredholm operators, focusing on the concept of weak demicompact operators. We specify conditions on certain ranges that ensure the persistence of the weak demicompactness property under restrictions. Moreover, our study provides perturbation results concerning the generalized Gustafson essential spectrum for 2 & times; 2 operator matrices.
In this paper, we prove a local rigidity of convex hypersurfaces in the spaces of constant curvature of dimension n >= 4. Namely, we show that two isometric convex hypersurfaces are congruent locally around their corresponding under the isometry points of strict convexity. This result extends the result of E.P. Senkin, who showed such rigidity under the additional assumption of C-1-smoothness of the hypersurfaces.
In the paper, the problems of approximate controllability are studied for the control system wt = Delta w, w(0, x2, t) = u(x2, t), x1 E IIB+ = (0, +oo), x2 E (0, T)) f1 L2(IIB & times; (0, T)). It is proved that each initial state belonging to L2(IIB+ & times; IIB) is approximately controllable to an arbitrary end state belonging to L2(IIB+ & times; IIB) by applying these controls. A numerical algorithm of solving the approximate controllability problem for this system is given. The results are illustrated by an example.
In this paper, we consider the nonlinear elliptic equation Delta fv(tau) +lambda v = 0 on a complete smooth metric measure space with the m-Bakry-Emery Ricci curvature bounded from below, where tau > 0 and lambda are constants. We obtain some new local and global universal log-gradient estimates for positive solutions to the equation using the Nash-Moser iteration technique. As applications of these estimates, we obtain a Liouville type theorem, a Harnack inequality and the global gradient estimates for such solutions. Our results generalize and improve the estimates established by Wang (J. Differential Equations 260 (2016), 567-585) and Zhao (Arch. Math. (Basel) 114 (2020), 457-469).
We introduce and study dynamical systems and measures on stationary generalized Bratteli diagrams B that are represented as the union of countably many classical Pascal-Bratteli diagrams. We describe all ergodic tail invariant measures on B. For every probability tail invariant measure nu p on the classical Pascal-Bratteli diagram, we approximate the support of nu p by the path space of a sub diagram. By considering various orders on the edges of B, we define dynamical systems with various properties. We show that there exist orders such that the sets of infinite maximal and infinite minimal paths are empty. This implies that the corresponding Vershik map is a homeomorphism. We also describe orders on both B and the classical Pascal-Bratteli diagram that generate either uncountably many minimal infinite and uncountably many maximal infinite paths, or uncountably many minimal infinite paths alongside countably infinitely many maximal infinite paths.
This article provides a complete characterization of all simple closed geodesics on regular spherical octahedra and cubes. Additionally, estimates for the number of such geodesics on regular spherical tetrahedra are presented.
This paper considers the problem of exact observability of a general class of linear distributed parameter systems in Hilbert spaces connected to Riesz basis properties of some families of exponential functions and the divided differences of those functions. Under some assumptions on asymptotic spectral analysis of the differential operator of the system, the conditions of exact observability are stated in the form of exact observable spaces being the direct sum of some specific Sobolev spaces. The main result consists of proving the optimality of these subspaces of observable states. The result was based on advanced non-harmonic analysis approach connected to the unusual fact that time-space Riesz basis does not consist only of exponential functions but also contains divided differences of these functions.
We consider a special class of linearly growing C0-groups from [20, 24], whose generators are essentially nonselfadjoint unbounded operators. More precisely, these generators have pure point imaginary spectrum, clustering at i infinity, and corresponding dense and minimal, but not uniformly minimal family of eigenvectors, hence this family do not form a Schauder basis. We obtain sharp two-sided estimates for the norms of C0-groups from this class on dense subsets of a phase space, namely, on D(Ak) for any k is an element of N, where A is the unbounded generator of the corresponding C0-group. Thereby we prove that these C0-groups have sub-linear growth on D(Ak). This yields the sub-linear growth of classical and all more regular solutions of the Cauchy problems for the corresponding abstract linear evolution equations.
In this paper, we consider a linear system with the control u E Q, where Q is a certain domain which does not contain the origin as an interior point. In particular, the origin may not belong to the set Q. The synthesis problem is solved, i.e. the control u(x) E Q which transfers a point x that belongs to a neighbourhood V (0) to 0 in a finite time is constructed by using the controllability function method. Moreover, this function can be found as the time of motion from a point x E V (0) to the origin. The case of the linear control system with a non-autonomous term is also considered.
Petrenko's theory of growth of meromorphic functions lies within the wider spectrum of Nevanlinna theory and was initiated by V.P. Petrenko in 1960s. This paper is focused on the achievements of an outstanding student of V.P. Petrenko, I.I. Marchenko, and his contributions to the theory. An overview of some of I.I. Marchenko's main results (and their further generalizations and applications) concerning deviations, separated maximum modulus points and strong asymptotic values constitutes the body of the paper. The final part of the paper is devoted to a generalization of an early result of I.I. Marchenko and A.I. Shcherba on the sum of deviations of functions holomorphic in the unit disc.
In the present paper, we study the criterion for univalence and quasi-conformal extensions for locally univalent analytic mappings and analytic mappings. For locally univalent analytic functions, we introduce integral operators in Loewner chain and obtain sufficient conditions for univalent and quasiconformal extensions to generalize the results of Becker, Ahlfors and Wang et al. For analytic functions, we use different proof methods to obtain a sufficient condition for univalence, which generalizes the result of Masih et al.
The paper provides a structural result about centralizers of Belavin-Drinfeld r-matrices. This result appears to be useful in computing Belavin-Drinfeld cohomology, which was introduced earlier for classificalion of certain Lie bialgebras and quantum groups.
In the paper, we prove the existence of the renormalized solution for the nonlinear degenerate parabolic equation partial derivative b(u) /partial derivative t-div(A(t, x, u)Du) = f, where the matrix A (t, x, s) = (aij(t, x, s))1 <= i <= N 1 <= j <= N is not controlled with respect to u, f is an element of L-1(Q), and b is a strictly increasing C-1-function
