Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology

Preface

Contents

In this work we prove the uniqueness of positive solution and characterize the existence of positive solutions of a wide class of elliptic BVP of Logistic type with sublinear weighted mixed boundary conditions. The results obtained in this work are an extension of the previous one found in S. Cano-Casanova⁹. Monotonicity techniques are the main technical tools, used to develop the mathematical analysis.

We review several recent results of Fernández-Cabrera, Triebel and the author on logarithmic interpolation spaces based on the real method.

This work discusses some aspects of the existence of solutions to the singular Dirichlet problem –Δu = λ(x)u – a(x)u^p, x ∈ Ω, u|∂Ω = ∞ under several assumptions on the null set {a(x) = 0} of the nonnegative coefficient a. In addition, a detailed analysis of the asymptotic profile of the solution to the finite Dirichlet problem –Δu = λ(x)u – a(x)u^p, x ∈ Ω, u|∂Ω = u₀, as u₀ → ∞ is performed.

We analyze the motion of a fluid containing a soluble substance in the interior of a closed loop under the effects of natural convection and a given external heat flux. This motion is governing by a coupled differential system. First, we show the well posedness of this system in a framework generalizing the existence results of [5]. Finally, we prove some result about the asymptotic behaviour for solutions of above system generalizing the results of [12].

In this paper we show the uniqueness and ascertain the boundary blow-up rates of the large solutions in a general class of radially symmetric semilinear elliptic boundary value problems. Our results are optimal in many circumstances and provide us with substantial generalizations of extremely sharp existing results. The main uniqueness theorem is a direct consequence of the maximum principle, rather than of the explicit knowledge of the boundary blow-up of the large solutions, as in most of the available references.

Ecology, Economy and Management require a huge interdisciplinary effort to ascertain the hidden mechanisms driving the evolution of communities and firm networks. This article shows that strategic alliances in competitive environments provoke an explosive increment of productivity and stability through a feedback mechanism promoted by cooperation, while competition causes segregation within cooperative profiles. Some further speciation and radiation mechanisms enhancing innovation, facilitated by environmental heterogeneities and specific market regulations, might explain the biodiversity of life and the high complexity of industrial and financial markets. Extinctions occur by the lack of adaptation of strongest competitors to sudden environmental stress.

Let 𝔏 be a smooth family from a neighborhood of the (real or complex) number λ₀ to the set of Fredholm operators of index zero between two Banach spaces. This paper shows that the following assertions are equivalent: (a) 𝔏 admits a local Smith form at λ₀; (b) λ₀ is an algebraic eigenvalue of 𝔏; (c) the lengths of the Jordan chains of 𝔏 at λ₀ are uniformly bounded from above; (d) the determinant of any finite-dimensional reduction of 𝔏 has a zero of finite order at λ₀. To prove this, we use two theories of algebraic multiplicity: the one given by Jordan chains and the one given by transversalization. It is also shown that two families are locally equivalent at λ₀ if and only if their multiplicity invariants at λ₀ coincide.

The problem –Δu + F(V(εx), u) = 0 is considered in ℝⁿ for small ε. A special case, –Δu + V(εx)u – u^p = 0, is equivalent to the problem of finding semi-classical stationary states of the non-linear Schrödinger equation. For small ε > 0 solutions are obtained that approach, as ε → 0, a linear combination of translates, through a distance O(1/ε), of solutions of –Δu + F(a_k, u) = 0 in W^2,2 (Rⁿ), where a_k is a finite sequence of non-degenerate critical values of V. These are the so-called multi-bump solutions. The method involves a rescaling of the variables and the use of a modified implicit function theorem. The usual implicit function theorem is inapplicable owing to lack of convergence of the derivative of the non-linear Hilbert space operator, obtained after an appropriate rescaling, in the operator-norm topology. An asymptotic formula for the solution for small ε is obtained.

This paper concerns with some elliptic equations with non-linear boundary conditions. Sub-supersolution and bifurcation methods are used in order to obtain existence, uniqueness or multiplicity of positive solutions.

The dynamical systems method (DSM), for solving operator equations, especially nonlinear and ill-posed, is developed in this paper. Consider an operator equation F(u) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation ill-posed if the operator F′(u) is not boundedly invertible, and well-posed otherwise. The DSM for solving linear and nonlinear ill-posed problems in H consists of the construction of a dynamical system, that is, a Cauchy problem, which has the following properties: (1) it has a global solution; (2) this solution tends to a limit as time tends to infinity; (3) the limit solves the original linear or non-linear problem.

The DSM is justified for:

(a) an arbitrary linear solvable equations with bounded operator,

(b) for well-posed solvable nonlinear equations with twice Fréchet differentiable operator F,

(c) for ill-posed solvable nonlinear equations with monotone operators,

(d) for ill-posed solvable nonlinear equations with non-monotone operators from a wide class of operators,

(e) for ill-posed solvable nonlinear equations with operators F such that A := F′(u) satisfies the spectral assumption of the type ||(A+sI)⁻¹|| ≤ c/s, where c > 0 is a constant, and s ∊ (0, s₀), s₀ > 0 is a fixed number, arbitrarily small, c does not depend on s and u; and

(f) for some monotone operators which are not Fréchet differentiable and for some unbounded, closed, densely defined F.

In Newton-type schemes the main difficulty is to invert the derivative of the operator. A novel scheme, based on DSM, allows one to avoid this inversion.

Global convergence theorem is obtained for the regularized continuous analog of Newton’s method for monotone operators. Global convergence means that convergence is established for an arbitrary initial approximation, not necessarily the one which is sufficiently close to the solution.

A general approach to constructing convergent iterative schemes for solving well-posed nonlinear operator equations is described and convergence theorems are obtained for such schemes.

Stopping rules for stable solution of ill-posed problems with noisy data are given.

We consider the nonlinear Sturm-Liouville equation

−(p(t)u′(t))′ + q(t)u(t) = a(t)u⁺(t) − b(t)u⁻(t) + h(t), t ∊ (0, 2π),

together with either separated or periodic boundary conditions. We assume that 1/p, q, a, b, h ∊ L¹(0, 2π). We also assume that a ≠ b (in the L¹(0, 2π) sense) — thus the nonlinearity is jumping. Solvability conditions for jumping nonlinearity problems have been expressed in terms of the ‘Fučík spectrum’ or, more recently, the set of ‘half-eigenvalues’. In the separated case the geometric structure of both these sets has been completely described, but for a general periodic problem the full structure is not completely known for either of these sets. In this paper we consider the periodic problem and describe some recent results for both these sets. In particular, we show that, for a given nodal number k ≥ 1, the set of half-eigenvalues whose associated half-eigenfunctions have this nodal number can be arbitrarily large, but must be finite when a, b, q are analytic. Also, the set of Fučík curves with associated nodal number k can be arbitrarily large. In addition, we give a non-solvability result for the problem, in terms of half-eigenvalues.

No abstract received.

Conditions of Liapunov type are given on a function V : ℝⁿ → ℝ in order that a “sub-level set” {x ∊ ℝⁿ : V(x) ≤ c} be positively invariant under the flow generated by a C¹ ODE ẋ = f(x) in ℝⁿ. The necessity and sufficiency of the weak Lyapunov condition holding only on the level set {V = c} are given in terms of the size of the set of critical points on V lying on {V = c}. The case in which V is not differentiable is also studied by means of several extensions of the concept of . Higher order derivatives of V with respect to the flow are also used.