Higher order m-point boundary value problems on time scales

In this paper, we investigate the existence of positive solutions for nonlinear even-order m-point boundary value problems on time scales by means of fixed point theorems. © 2011 Elsevier Ltd. All rights reserved

Positive solutions for even-order multi-point boundary value problems on time scales

In this paper, we consider the nonlinear even-order m-point boundary value problems on time scales. We establish the criteria for the existence of at least one and three positive solutions for higher order nonlinear m-point boundary value problems on time scales by using Krasnosel’skii’s fixed point theorem and Leggett-Williams’ fixed point theorem, respectively. © 2017, Springer International Publishing. All rights reserved

The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory

Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither {\it ad hoc\/} assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially-extended systems near bifurcation points, deriving both amplitude equations and the center manifold.Comment: 44 pages, 2 Postscript figures, macro \uiucmac.tex available at macro archives or at ftp://gijoe.mrl.uiuc.edu/pu

Variational approach to second-order impulsive dynamic equations on time scales

The aim of this paper is to employ variational techniques and critical point theory to prove some conditions for the existence of solutions to nonlinear impulsive dynamic equation with homogeneous Dirichlet boundary conditions. Also we will be interested in the solutions of the impulsive nonlinear problem with linear derivative dependence satisfying an impulsive condition.Comment: 17 page

The Variational Calculus on Time Scales

The discrete, the quantum, and the continuous calculus of variations, have been recently unified and extended by using the theory of time scales. Such unification and extension is, however, not unique, and two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we review a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases.Comment: 15 pages; Published in: Int. J. Simul. Multidisci. Des. Optim. 4 (2010), 11--2

Interacting String Multi-verses and Holographic Instabilities of Massive Gravity

Products of large-N conformal field theories coupled by multi-trace interactions in diverse dimensions are used to define quantum multi-gravity (multi-string theory) on a union of (asymptotically) AdS spaces. One-loop effects generate a small O(1/N) mass for some of the gravitons. The boundary gauge theory and the AdS/CFT correspondence are used as guiding principles to study and draw conclusions on some of the well known problems of massive gravity - classical instabilities and strong coupling effects. We find examples of stable multi-graviton theories where the usual strong coupling effects of the scalar mode of the graviton are suppressed. Our examples require a fine tuning of the boundary conditions in AdS. Without it, the spacetime background backreacts in order to erase the effects of the graviton mass.Comment: 51 pages, 3 figures; v2 typos corrected, version published in NPB; v3 added appendix E on general class of fixed points in multi-trace deformation

Constraints on Theories With Large Extra Dimensions

Recently, a number of authors have challenged the conventional assumption that the string scale, Planck mass, and unification scale are roughly comparable. It has been suggested that the string scale could be as low as a TeV. In this note, we explore constraints on these scenarios. We argue that the most plausible cases have a fundamental scale of at least 10 TeV and five dimensions of inverse size 10 MeV. We show that a radial dilaton mass in the range of proposed millimeter scale gravitational arises naturally in these scenarios. Most other scenarios require huge values of flux and may not be realizable in M Theory. Existing precision experiments put a conservative lower bound of 6-10 TeV on the fundamental energy scale. We note that large dimensions with bulk supersymmetry might be a natural framework for quintessence, and make some other tentative remarks about cosmology.Comment: 31 pp. latex. Minor changes, ref. problems fixe