Abstract and Applied Analysis

https://www.hindawi.com/journals/aaa/

List of Papers (Total 3,563)

On the discreteness of the spectra of the Dirichlet and Neumann p-biharmonic problems

We are interested in a nonlinear boundary value problem for (|u″|p−2u″)′​′=λ|u|p−2u in [0,1], p>1, with Dirichlet and Neumann boundary conditions. We prove that eigenvalues of the Dirichlet problem are positive, simple, and isolated, and form an increasing unbounded sequence. An eigenfunction, corresponding to the nth eigenvalue, has precisely n−1 zero points in (0,1...

An Ambrosetti-Prodi-type problem for an elliptic system of equations via monotone iteration method and Leray-Schauder degree theory

In this paper we employ the Monotone Iteration Method and the Leray-Schauder Degree Theory to study an ℝ2-parametrized system of elliptic equations. We obtain a curve dividing the plane into two regions. Depending on which region the parameter is, the system will or will not have solutions. This is an Ambrosetti-Prodi-type problem for a system of equations.

The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics

We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE's which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domain Ω, coupled to a “parabolic–like” beam equation holding on ∂Ω, and wherein the coupling is accomplished through velocity terms on the...

Embedding functions and their role in interpolation theory

The embedding functions of an intermediate space A into a Banach couple (A0,A1) are defined as its embedding constants into the couples (1αA0,1βA1), ∀α,β>0. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak-type interpolation to any Banach...

Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators

Let iAj(1≤j≤n) be generators of commuting bounded strongly continuous groups, A≡(A1,A2,…,An). We show that, when f has sufficiently many polynomially bounded derivatives, then there exist k,r>0 such that f(A) has a (1

A new spectral theory for nonlinear operators and its applications

In this paper, by applying (p,k)-epi mapping theory, we introduce a new definition of spectrum for nonlinear operators which contains all eigenvalues, as in the linear case. Properties of this spectrum are given and comparison is made with the other definitions of spectra. We also give applications of the new theory.

Existence of positive radial solutions for a weakly coupled system via blow up

The existence of positive solutions to certain systems of ordinary differential equations is studied. Particular forms of these systems are satisfied by radial solutions of associated partial differential equations.

Uniform asymptotic normal structure, the uniform semi-Opial property, and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part II

In this part of our paper we present several new theorems concerning the existence of common fixed points of asymptotically regular uniformly lipschitzian semigroups.

Singular nonlinear elliptic equations in Rn

This paper deals with existence, uniqueness and regularity of positive generalized solutions of singular nonlinear equations of the form −Δu

Semilinear Volterra integrodifferential equations with nonlocal initial conditions

We establish the global existence of mild solutions to a class of nonlocal Cauchy problems associated with semilinear Volterra integrodifferential equations in a Banach space.

Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t)=Au(t)

Multiplicity of positive solutions to semilinear elliptic boundary value problems

We study semilinear elliptic boundary value problems of one parameter dependence where the number of positive solutions is discussed. Our main purpose is to characterize the critical value given by the infimum of such parameters for which positive solutions exist. Our approach is based on super- and sub-solutions, and relies on the topological degree theory on the positive cones...

Solvability of a nonlinear second order conjugate eigenvalue problem on a time scale

We consider the nonlinear second order conjugate eigenvalue problem on a time scale: y ΔΔ(t)

Domain perturbation method and local minimizers to Ginzburg-Landau functional with magnetic effect

We prove the existence of vortex local minimizers to Ginzburg-Landau functional with a global magnetic effect. A domain perturbating method is developed, which allows us to extend a local minimizer on a nonsimply connected superconducting material to the local minimizer with vortex on a simply connected material.

Nonlinear ergodic theorems for asymptotically almost nonexpansive curves in a Hilbert space

We introduce the notion of asymptotically almost nonexpansive curves which include almost-orbits of commutative semigroups of asymptotically nonexpansive type mappings and study the asymptotic behavior and prove nonlinear ergodic theorems for such curves. As applications of our main theorems, we obtain the results on the asymptotic behavior and ergodicity for a commutative...

Local existence result of the single dopant diffusion including cluster reactions of high order

We consider the pair diffusion process which includes cluster reactions of high order. We are able to prove a local (in time) existence result in arbitrary space dimensions. The model includes a nonlinear system of reaction-drift-diffusion equations, a nonlinear system of ordinary differential equations in Banach spaces, and a nonlinear elliptic equation for the electrochemical...

Boundary value problems for second-order partial differential equations with operator coefficients

Let Ω T be some bounded simply connected region in ℝ 2 with ∂ Ω T=Γ¯1∩Γ¯2. We seek a function u(x,t)((x,t)∈Ω T) with values in a Hilbert space H which satisfies the equation ALu(x,t)=Bu(x,t)

Obstacles to bounded recovery

Let X be a Banach space, V⊂X is its subspace and U⊂X*. Given x∈X, we are looking for v∈V such that u (v)=u (x) for all u∈U and ‖v‖ ≤M‖x‖. In this article, we study the restrictions placed on the constant M as a function of X,V, and U.

On the curvature of nonregular saddle surfaces in the hyperbolic and spherical three-space

This paper proves that any nonregular nonparametric saddle surface in a three-dimensional space of nonzero constant curvature k, which is bounded by a rectifiable curve, is a space of curvature not greater than k in the sense of Aleksandrov. This generalizes a classical theorem by Shefel

On best proximity pair theorems and fixed-point theorems

The significance of fixed-point theory stems from the fact that it furnishes a unified approach and constitutes an important tool in solving equations which are not necessarily linear. On the other hand, if the fixed-point equation Tx=x does not possess a solution, it is contemplated to resolve a problem of finding an element x such that x is in proximity to Tx in some sense...

On the modulus of u-convexity of Ji Gao

We consider the modulus of u-convexity of a Banach space introduced by Ji Gao (1996) and we improve a sufficient condition for the fixed-point property (FPP) given by this author. We also give a sufficient condition for normal structure in terms of the modulus of u-convexity.

Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets

It is shown that if X is a Banach space and C is a union of finitely many nonempty, pairwise disjoint, closed, and connected subsets {Ci:1≤i≤n } of X, and each Ci has the fixed-point property (FPP) for asymptotically regular nonexpansive mappings, then any asymptotically regular nonexpansive self-mapping of C has a fixed point. We also generalize the Goebel-Schöneberg theorem to...

Local solvability of a constrainedgradient system of total variation

Suppose X is a real q-uniformly smooth Banach space and F,K:X→X with D(K)=F(X)=X are accretive maps. Under various continuity assumptions on F and K such that 0=u

Estimates for the Green function and singular solutions for polyharmonic nonlinear equation

We establish a new form of the 3G theorem for polyharmonic Green function on the unit ball of ℝn(n≥2) corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functions Km,n containing properly the classical Kato class Kn. We exploit properties of functions belonging to Km,n to prove an infinite existence result of singular positive...