Existence of Positive Solutions for a Nonlinear Higher-Order Multipoint Boundary Value Problem

We study the existence of positive solutions for a nonlinear higher-order multipoint boundary value problem. By applying a monotone iterative method, some existence results of positive solutions are obtained. The main result is illustrated with an example

Positive Solutions for the Eigenvalue Problem of Semipositone Fractional Order Differential Equation with Multipoint Boundary Conditions

We study the existence of positive solution for the eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions by using known Krasnosel'skii's fixed point theorem. Some sufficient conditions that guarantee the existence of at least one positive solution for eigenvalues  λ>0 sufficiently small and λ>0 sufficiently large are established

Higher order functional discontinuous boundary value problems on the half-line

In this paper, we consider a discontinuous, fully nonlinear, higher-order equation on the half-line, together with functional boundary conditions, given by general continuous functions with dependence on the several derivatives and asymptotic information on the (n−1)th derivative of the unknown function. These functional conditions generalize the usual boundary data and allow other types of global assumptions on the unknown function and its derivatives, such as nonlocal, integro-differential, infinite multipoint, with maximum or minimum arguments, among others. Considering the half-line as the domain carries on a lack of compactness, which is overcome with the definition of a space of weighted functions and norms, and the equiconvergence at ∞. In the last section, an example illustrates the applicability of our main result

Higher order functional boundary value problems: existence and location results

This paper considers a nth-order phi-laplacian differential equation with functional boundary conditions satisfying certain monotonicity assumptions. We present sufficient conditions on the nonlinearity and the boundary conditions to ensure the existence of solutions. Moreover, from the lower and upper solutions method, some information is given about the location of the solution and its qualitative properties. Due to the functional dependence in the boundary conditions, this work generalizes several results for higher order problems with many types of boundary conditions. The main results are illustrated with examples

Existence, localization and multiplicity results for nonlinear and functional

In this thesis several problems are addressed. The problems considered vary from second order problems up to high order problems where generaliza- tions to nth order are studied. Such problems range from problems without functional dependence up to problems where the functional dependence is featured both in the equation and on the boundary conditions. Functional boundary conditions include most of the classical conditions as multipoint cases, conditions with delay and/or advances, nonlocal or in- tegral, with maximum or minimum arguments,... Existence, nonexistence, multiplicity and localization results are then discussed in accordance with these conditions. The method used is the lower and upper solutions combined with di¤erent techniques (degree theory, Nagumo condition, iterative technique, Green s function) to obtain such results. Several applications are studied such as the periodic oscillations of the axis of a satellite and conjugate boundary value problems, to emphasize the applicability of the method used; RESUMO:Nesta tese, intitulada em português, Resultados de existência, localiza- ção e multiplicidade para problemas não lineares e funcionais de ordem su- perior com valores na fronteira , diferentes problemas são abordados. Estes problemas variam desde problemas de segunda ordem até problemas de or- dem superior, onde generalizações de ordem n são feitas e onde os problemas apresentados variam desde o caso em que não existe dependência funcional até aos em que esta dependência funcional está presente tanto na equação como nas condições de fronteira. Sobre estas condições, que incluem a maioria das condições clássicas, re- sultados de existência, não existência, multiplicidade e localização de solução são discutidos de acordo com estas condições. O método utilizado é o método da sub e sobre-solução combinado com diferentes técnicas. Várias aplicações são estudadas, nomeadamente as oscilações periódicas do eixo de um satélite e problemas conjugados, de forma a dar ênfase à aplicabilidade do método utilizado

On higher order fully periodic boundary value problems

In this paper we present sufficient conditions for the existence of periodic solutions of some higher order fully differential equation where the nonlinear part verifies a Nagumotype growth condition. A new type of lower and upper solutions, eventually non-ordered, allows us to obtain, not only the existence, but also some qualitative properties on the solution. The last section contains two examples to stress the application to both cases of n odd and n even

Positive solutions for multi point impulsive boundary value problems on time scales

In this paper, we consider nonlinear second-order multi-point impulsive boundary value problems on time scales. We establish the criteria for the existence of at least one, two and three positive solutions by using the Leray-Schauder fixed point theorem, the Avery-Henderson fixed point theorem and the five functional fixed point theorem, respectively. An example that supports the theoretical results is also provided