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...

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.

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...

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...

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

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.

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.

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

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

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.

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

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...

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

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.

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...

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...

The nonlocal boundary value problem, v′(t)

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)

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.

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

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...

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.

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...

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

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...