High order boundary value problems on unbounded domains: Types of solutions, functional problems and applications

Abstract

The relative scarcity of results that guarantee the existence of solutions for boundary value problems on unbounded domains, contrasts with the high applicability on real problems with differential equations defined on the half-line or on the whole real line. This gap is the main reason that led to this work. The book contains four parts with different problems composed by differential equations, from second to higher orders, and integral Hammerstein equations, several types of boundary conditions, for example, Sturm-Liouville, Lidstone and functional conditions, and solutions with diverse qualitative properties, such as, impulsive, homoclinic, and heteroclinic solutions. The non-compactness of the time interval and the possibility of study unbounded functions will require the definition of adequate Banach spaces. In fact the space considered, the functional framework assumed and the set of admissible solutions for each problem are defined under a main goal: the functions must remain bounded for the space and the norm in consideration. This is achieved by defining some weight functions (polynomial or exponential) in the space or assuming some asymptotic behavior. We underline some new features of the content:

∙ relation between some properties of the Green´s functions defined on the real line, the existence of homoclinic solutions and the solvability of Lidstone-type problems; ∙ existence of heteroclinic solutions for semi-linear problems without growth or asymptotic assumptions on the nonlinearity; ∙ solvability of Hammerstein integral equations on the whole line, with discontinuous and sign-changing kernels and with nonlinearly dependence on several derivatives.

In addition to the existence, solutions will be localized in a strip. The lower and upper solutions method will play an important role, and combined with other tools like the one-sided Nagumo growth conditions, Green's functions or Schauder's fixed point theorem, provide the existence and location results for differential equations with various boundary conditions. Different applications to real phenomena will be presented, most of them translated into classical equations as Duffing, Bernoulli-Euler v. Karman, Fisher-Kolmogorov, Swift-Hohenberg, Emden-Fowler or Falkner-Skan-type equations. All these applications have a common denominator: they are defined in unbounded intervals and the existing results in the literature are scarce or proven only numerically in discrete problems. Hugo Carrasco Feliz Minhós

Introduction

[addcontentsline]<LaTeX>\addcontentsline{toc}{chapter}{Introduction}</LaTeX>[thispagestyle]<LaTeX>\thispagestyle{empty}</LaTeX> [markboth]<LaTeX>\markboth{}{}</LaTeX>The leitmotiv of this book is related with higher order boundary value problems (BVPs) defined on unbounded domains, more precisely on the half-line or on the whole real line. Roughly speaking, we can say that BVPs are rather different of initial (or final) value problems as they have not a continuous dependence of the boundary data. In fact, small perturbations on boundary values may cause vital changes on the qualitative properties of the corresponding solutions, and even on existence, non-existence or multiplicity of solutions. The following example will illustrate this fact: Consider the second order homogeneous differential equation

y′′+y=0. #(1) <label>EqHom</label>

The initial value problem, known as Cauchy problem, composed by (<ref>EqHom</ref>) and the initial values

y(0)=k₁, y′(0)=k₂

has a unique solution given by y(x)=k₁cos x + k₂ sin x, for every real k₁,k₂. However the BVP with (<ref>EqHom</ref>) and the Dirichlet boundary conditions

y(0)=0, y(π)=ε(≠0)

has no solution, but the Dirichlet BVP with (<ref>EqHom</ref>) and

y(0)=0, y(β)=ε, with 0<β<π,

has a unique solution, y(x)=((εsin x)/(sinβ)), and the BVP composed by (<ref>EqHom</ref>) together with the boundary conditions

y(0)=0, y(π)=0,

has infinite solutions, of the type y(x)=c sin x, with arbitrary c∈R. Last decades the study of BVPs defined on compact intervals has been considered by many authors with application of a huge variety of methods and techniques. However BVPs defined on unbounded intervals are scarce, as they require other type of techniques to overcome the lack of compactness. Historically, these problems began at the end of 19^{th} century with A. Kneser. This pioneer work described monotone solutions of second order ordinary differential equations. Others followed his results and different techniques have been studied, namely the lower and upper solutions method (see <cite>andres</cite> and the references therein). Several real problems were modeled by BVPs defined on infinite intervals. As examples, we refer the study of unsteady flow of a gas through a semi-infinite porous medium, the discussion of electrostatic probe measurements in solid-propellant rocket exhausts, the analysis of the mass transfer on a rotating disk in a non-Newtonian fluid, the heat transfer in the radial flow between parallel circular disks, the investigation of the temperature distribution in the problem of phase change of solids with temperature dependent thermal conductivity, as well as numerous problems arising in the study of draining flows, circular membranes, plasma physics, radially symmetric solutions of semilinear elliptic equations, nonlinear mechanics, and non - Newtonian fluid flows, the bending of infinite beams and its applications in the railways and highways. More details and examples can be seen in <cite>agarwal6</cite> and the references therein. This book is divided in four parts, each one related to some type of BVPs on unbounded intervals. The first part, Boundary value problems on the half-line, is dedicated to higher order BVPs , defined on the half-line, and it is composed by three chapters:

∙ Chapter 1 - Third order boundary value problems. Third order differential equations on infinite intervals can describe the evolution of physical phenomena like draining or coating fluid flow problems. The non-compactness of the time interval and the possibility of studying unbounded functions require the redefinition of the admissible Banach space and its weighted norms. In this chapter it will be proved an existence and localization of, at least, one solution for a BVP with Sturm-Liouville type boundary conditions. The tools involved will be the one-sided Nagumo-type growth condition, Green's functions, lower and upper solution method and Schauder's fixed point theorem. An example will finish the chapter. ∙ Chapter 2 - General n^{th}-order problems. This chapter arises in the attempt to generalize the previous one to order n. In a particular case, fourth order differential equations can model the bending of an elastic beam. An example is shown to demonstrate the importance of the one-sided Nagumo-type growth condition. ∙ Chapter 3 - Impulsive problems on the half-line with infinite impulse moments. Some of the previous techniques are applied in a second order impulsive problem on the half-line, with generalized impulsive functions, depending on the unknown function and its derivative, and allowing an infinite number of impulse moments. The notion of Carathéodory sequence is a key argument in the method.

The second part, Homoclinic solutions and Lidstone problems on the whole real line, considers BVPs on the whole real line, looking for sufficient conditions on the nonlinearity to guarantee the existence of homoclinic solutions, and its relation to solutions for Lidstone-type problems. It contains three chapters:

∙ Chapter 4 - Homoclinic solutions for second order problems. In this chapter it will be used the lower and upper solutions method with unordered functions. An existence and localization result will be settled. Specific applications to Duffing-type equations and beam equations with damping will finish the chapter. ∙ Chapter 5 - Homoclinic solutions to fourth order problems. Different problems involving Bernoulli-Euler-v. Karman, Fisher - Kolmogorov or Swift-Hohenberg equations are strongly linked with fourth order differential equations. This chapter will establish existence results and examples for each particular case. ∙ Chapter 6 - Lidstone boundary value problems. The Lidstone theory, initially applied to interpolation problems, is considered, in this chapter, in the whole real line with a strong connection to the homoclinic solutions. In this final chapter of this part it will be studied a problem of an infinite beam resting on granular foundations with moving loads.

The third part, Heteroclinic solutions and Hammerstein equations, contains four chapters:

∙ Chapters 7, 8 and 9 provide sufficient conditions for the existence of heteroclinic solutions for three types of φ-Laplacian equations, sometimes named as semi-linear equations, on the real line. We point out that these heteroclinic solutions are obtained without the usual monotone or growth assumptions on the nonlinearity. ∙ Chapter 10 studies integral equations, more precisely, Hammerstein equations, defined on the whole real line, with discontinuous nonlinearities, which may depend, not only on the unknown function, but also on some derivatives, without monotone or asymptotic assumptions. Moreover, the kernels and their partial derivatives in order to the first variable, are very general functions: they may be discontinuous and may change signal. A simple criterion to see if the existent solutions are homoclinic or heteroclinic solutions is included, together with an application to a fourth order BVP.

In the last part, Functional boundary value problems, we study BVPs with functional boundary conditions, that is, with boundary data that can depend globally on the correspondent variables. In this way it contains and generalize many types of boundary conditions such as multipoint, advanced or delayed, nonlocal, integro-differential, with maximum or minimum arguments, among others. Part 4 is divided into three chapters, each one with different type of problems:

∙ Chapter11 - Second order problems. BVPs involving functional boundary conditions can model thermal conduction, semiconductor and hydrodynamic problems. An application to a problem composed by an Emden-Fowler-type equation and a infinite multipoint condition will be formulated and solved. ∙ Chapter 12 - Third order functional problems. Falkner-Skan equations are obtained from partial differential equations. They can model the behavior of a viscous flow over a plate. Until now, only numerical techniques could deal with this type of problems, however, in this chapter it will be proved an existence and localization result by topological methods. ∙ Chapter 13 - Phi-Laplacian equations with functional boundary conditions. This final chapter will deal with weighted norms, namely the Bielecki norm. This will be a fundamental tool to manage unbounded solutions. An important fact is that the homeomorphism φ does not need to be surjective.

Throughout this work, the usual Lemma of Arzèla-Ascoli can not be used due to the lack of compactness, and this issue is overcome with some methods, techniques and specific tools. We point out some of them:

∙ Weighted spaces and the corresponding weighted norms; ∙ Carathéodory functions admissible for the nonlinearities; ∙ Green's functions on unbounded domains; ∙ Equiconvergence at ∞.

The space considered and the functional framework assumed define the set of admissible solutions for each problem with a main goal: the functions must remain bounded for the space and the norm in consideration. This is achieved by defining some weight functions (polynomial or exponential) in the space or assuming some asymptotic behavior. Therefore, for each problem it is presented the specific space and norm to be used.