Eigenvalue intervals for a two-point boundary value problem on a measure chain

AbstractWe study the existence of eigenvalue intervals for the second-order differential equation on a measure chain, xΔΔ(t)+λp(t)f(xσ(t))=0,t∈[t1,t2], satisfying the boundary conditions αx(t1)−βxΔ(t1)=0 and γx(σ(t2))+δxΔ(σ(t2))=0, where f is a positive function and p a nonnegative function that is allowed to vanish on some subintervals of [t1,σ(t2)] of the measure chain. The methods involve applications of a fixed point theorem for operators on a cone in a Banach space

Positive Solutions, Existence Of Smallest Eigenvalues, And Comparison Of Smallest Eigenvalues Of A Fourth Order Three Point Boundary Value Problem

The existence of smallest positive eigenvalues is established for the linear differential equations $u^{(4)}+\lambda_{1} q(t)u=0$ and $u^{(4)}+\lambda_{2} r(t)u=0$, $0\leq t \leq 1$, with each satisfying the boundary conditions $u(0)=u\u27(p)=u\u27\u27(1)=u\u27\u27\u27(1)=0$ where $1-\frac{\sqrt{3}}{3}\le p \u3c 1$. A comparison theorem for smallest positive eigenvalues is then obtained. Using the same theorems, we will extend the problem to the fifth order via the Green\u27s Function and again via Substitution. Applying the comparison theorems and the properties of $u_0$-positive operators to determine the existence of smallest eigenvalues. The existence of these smallest eigenvalues is then applied to characterize extremal points of the differential equation $u^{(4)} + q(t)u = 0$ satisfying boundary conditions $u(0) = u\u27(p) = u\u27\u27(b) = u\u27\u27\u27(b)= 0$ where $1-\frac

Eigenvalue comparisons for differential equations on a measure chain

[[abstract]]The theory of u0-positive operators with respect to a cone in a Banach space is applied to eigenvalue problems associated with the second order Δ-differential equation (often referred to as a differential equation on a measure chain) given by yΔΔ(t) + λp(t) y(σ(t)) = 0, t ∈ [0, 1], satisfying the boundary conditions y(0) = 0 = y(σ2(1)). The existence of a smallest positive eigenvalue is proven and then a theorem is established comparing the smallest positive eigenvalues for two problems of this type.[[notice]]補正完

Smallest Eigenvalues For A Fractional Boundary Value Problem With A Fractional Boundary Condition

We establish the existence of and then compare smallest eigenvalues for the fractional boundary value problems D_(0^+)^α u+λ_1 p(t)u=0 and $D_(0^+)^α u+λ_2 q(t)u=0,0\u3c t\u3c 1, satisfying the boundary conditions when n-1\u3cα≤ n. First, we consider the case when 0\u3c

Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings

Our monograph presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Our work unifies and extends a long list of results by many authors. We make it a point to avoid any assumption of properness/compactness, keeping in mind the motivating example of $\mathbb H^\infty$, the infinite-dimensional rank-one symmetric space of noncompact type over the reals. The monograph provides a number of examples of groups acting on $\mathbb H^\infty$ which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. Such examples often demonstrate the optimality of our theorems. We introduce a modification of the Poincar\'e exponent, an invariant of a group which gives more information than the usual Poincar\'e exponent, which we then use to vastly generalize the Bishop--Jones theorem relating the Hausdorff dimension of the radial limit set to the Poincar\'e exponent of the underlying semigroup. We give some examples based on our results which illustrate the connection between Hausdorff dimension and various notions of discreteness which show up in non-proper settings. We construct Patterson--Sullivan measures for groups of divergence type without any compactness assumption. This is carried out by first constructing such measures on the Samuel--Smirnov compactification of the bordification of the underlying hyperbolic space, and then showing that the measures are supported on the bordification. We study quasiconformal measures of geometrically finite groups in terms of (a) doubling and (b) exact dimensionality. Our analysis characterizes exact dimensionality in terms of Diophantine approximation on the boundary. We demonstrate that some Patterson--Sullivan measures are neither doubling nor exact dimensional, and some are exact dimensional but not doubling, but all doubling measures are exact dimensional.Comment: A previous version of this document included Section 12.5 (Tukia's isomorphism theorem). The results of that subsection have been split off into a new document which is available at arXiv:1508.0696

Multiple positive solutions for functional dynamic equations on time scales

AbstractIn this paper, we study the following functional dynamic equation on time scales: {[Φ(uΔ(t))]∇+a(t)f(u(t),u(μ(t)))=0,t∈(0,T)T,u(t)=φ(t),t∈[−r,0)T,u(0)−B0(uΔ(0))=0,uΔ(T)=0, where Φ:R→R is an increasing homeomorphism and a positive homomorphism and Φ(0)=0. By using the well-known Leggett–Williams fixed point theorem, existence criteria for multiple positive solutions are established. An example is also given to illustrate the main results

Positive Solutions of Two-point right focal boundary value problems on time scales

AbstractWe consider the following boundary value problem,(−1)n−1yΔn(t)=(−1)p+1F(t,y(σn−1(t))),t∈[a,b]∩T,yΔi(a)=0,0≤i≤p−1,yΔi(σ(b))=0,p≤i≤n−1,where n ≥ 2, 1 ⩽ p ⩽ n - 1 is fixed and T is a time scale. Criteria for the existence of single, double, and multiple positive solutions of the boundary value problem are developed. Upper and lower bounds for these positive solutions are established for two special cases that arise from some physical phenomena. We also include several examples to illustrate the usefulness of the results obtained