Solvability of higher-order BVPs in the half-line with unbounded nonlinearities
Feliz Minhós Hugo Carrasco
This work presents sufficient conditions for the existence of unbounded solutions of a Sturm-Liouville type boundary value problem on the half-line. One-sided Nagumo condition plays a special role because it allows an asymmetric unbounded behavior on the nonlinearity. The arguments are based on fixed point theory and lower and upper solutions method. An example is given to show the applicability of our results.
keywords: Infinite interval problem unbounded upper lower solutions high order fixed point theory one-sided Nagumo condition.
Periodic solutions for some fully nonlinear fourth order differential equations
Feliz Minhós
In this paper we present sufficient conditions for the existence of solutions to the periodic fourth order boundary value problem

$u^((4))(x) = f(x,u(x),u'(x),u''(x),u'''(x))$
$u^((i))(a) = u^((i))(b), i=0,1,2,3,$
for $x \in [a,b],$ and $f : [a,b] \times \mathbb{R}^4\to\mathbb{R}$ a continuous function. To the best of our knowledge it is the first time where this type of general nonlinearities is considered in fourth order equations with periodic boundary conditions.
 The difficulties in the odd derivatives are overcome due to the following arguments: the control on the third derivative is done by a Nagumo-type condition and the bounds on the first derivative are obtained by lower and upper solutions, not necessarily ordered.
 By this technique, not only it is proved the existence of a periodic solution, but also, some qualitative properties of the solution can be obtained.
keywords: fully fourth order equations Periodic solutions Nagumo condition lower and upper solutions
On higher order nonlinear impulsive boundary value problems
Feliz Minhós Rui Carapinha
This work studies some two point impulsive boundary value problems composed by a fully differential equation, which higher order contains an increasing homeomorphism, by two point boundary conditions and impulsive effects. We point out that the impulsive conditions are given via multivariate generalized functions, including impulses on the referred homeomorphism. The method used apply lower and upper solutions technique together with fixed point theory. Therefore we have not only the existence of solutions but also the localization and qualitative data on their behavior. Moreover a Nagumo condition will play a key role in the arguments.
keywords: nagumo condition ø-Laplacian differential equations generalized impulsive conditions upper and lower solutions fixed point theory.
High order periodic impulsive problems
João Fialho Feliz Minhós
The theory of impulsive problem is experiencing a rapid development in the last few years. Mainly because they have been used to describe some phenomena, arising from different disciplines like physics or biology, subject to instantaneous change at some time instants called moments. Second order periodic impulsive problems were studied to some extent, however very few papers were dedicated to the study of third and higher order impulsive problems.
    The high order impulsive problem considered is composed by the fully nonlinear equation \begin{equation*} u^{\left( n\right) }\left( x\right) =f\left( x,u\left( x\right) ,u^{\prime }\left( x\right) ,...,u^{\left( n-1\right) }\left( x\right) \right) \end{equation*} for a. e. $x\in I:=\left[ 0,1\right] ~\backslash ~\left\{ x_{1},...,x_{m}\right\} $ where $f:\left[ 0,1\right] \times \mathbb{R} ^{n}\rightarrow \mathbb{R}$ is $L^{1}$-Carathéodory function, along with the periodic boundary conditions \begin{equation*} u^{\left( i\right) }\left( 0\right) =u^{\left( i\right) }\left( 1\right) ,         i=0,...,n-1, \end{equation*} and the impulsive conditions \begin{equation*} \begin{array}{c} u^{\left( i\right) }\left( x_{j}^{+}\right) =g_{j}^{i}\left( u\left( x_{j}\right) \right) ,        i=0,...,n-1, \end{array} \end{equation*} where $g_{j}^{i},$ for $j=1,...,m,$are given real valued functions satisfying some adequate conditions, and $x_{j}\in \left( 0,1\right) ,$ such that $0 = x_0 < x_1 <...< x_m < x_{m+1}=1.$
     The arguments applied make use of the lower and upper solution method combined with an iterative technique, which is not necessarily monotone, together with classical results such as Lebesgue Dominated Convergence Theorem, Ascoli-Arzela Theorem and fixed point theory.
keywords: periodic boundary value problems diff erential equations with impulses Nagumo condition Higher order problems lower and upper solutions.
The role of lower and upper solutions in the generalization of Lidstone problems
João Fialho Feliz Minhós
In this the authors consider the nonlinear fully equation
          \begin{equation*} u^{(iv)} (x) + f( x,u(x) ,u^{\prime}(x) ,u^{\prime \prime}(x) ,u^{\prime \prime \prime}(x) ) = 0 \end{equation*} for $x\in [ 0,1] ,$ where $f:[ 0,1] \times \mathbb{R} ^{4} \to \mathbb{R}$ is a continuous functions, coupled with the Lidstone boundary conditions, \begin{equation*} u(0) = u(1) = u^{\prime \prime}(0) = u^{\prime \prime }(1) = 0. \end{equation*}
    They discuss how different definitions of lower and upper solutions can generalize existence and location results for boundary value problems with Lidstone boundary data. In addition, they replace the usual bilateral Nagumo condition by a one-sided condition, allowing the nonlinearity to be unbounded$.$ An example will show that this unilateral condition generalizes the usual one and stress the potentialities of the new definitions.
keywords: non ordered lower and upper solutions. Lidstone problems one-sided Nagumo condition
Higher order two-point boundary value problems with asymmetric growth
Feliz Minhós A. I. Santos
In this work it is studied the higher order nonlinear equation

$\u^{( n)} (x)=f(x,u(x),u^{'}(x),\ldots ,u^{( n-1)} (x)) $

with $n\in \mathbb{N}$ such that $n\geq 2,$ $f:[ a,b] \times \mathbb{R}^{n}\rightarrow \mathbb{R}$ a continuous function, and the two-point boundary conditions

$u^{(i)}(a) =A_{i},\text{ \ \ }A_{i}\in \mathbb{R},\text{ \ }i=0,\ldots ,n-3$,
$u^{( n-1) }(a) =u^{( n-1) }(b)=0.$

From one-sided Nagumo-type condition, allowing that $f$ can be unbounded, it is obtained an existence and location result, that is, besides the existence, given by Leray-Schauder topological degree, some bounds on the solution and its derivatives till order $(n-2)$ are given by well ordered lower and upper solutions.
    An application to a continuous model of human-spine, via beam theory, will be presented.

keywords: degree theory Higher two-point BVP positive solutions continuous model of human-spine. lower and upper solutions one-sided Nagumo-type condition
On the solvability of some fourth-order equations with functional boundary conditions
Feliz Minhós João Fialho
In this paper it is considered a fourth order problem composed of a fully nonlinear differential equation and functional boundary conditions satisfying some monotone conditions.This functional dependence on $u,u^' $ and $u^{''}$and generalizes several types of boundary conditions such as Sturm-Liouville, multipoint, maximum and/or minimum arguments, or nonlocal. The main theorem is an existence and location result as it provides not only the existence, but also some qualitative information about the solution.
keywords: lower and upper solutions human spine continuous model Fourth order functional problems beam equation Nagumo-type condition
Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities
Feliz Minhós João Fialho
In this work the authors present some existence, non-existence and location results of the problem composed of the fourth order fully nonlinear equation \begin{equation*} u^{\left( 4\right) }\left( x\right) +f( x,u\left( x\right) ,u^{\prime }\left( x\right) ,u^{\prime \prime }\left( x\right) ,u^{\prime \prime \prime }\left( x\right) ) =s\text{ }p(x) \end{equation*} for $x\in \left[ a,b\right] ,$ where $f:\left[ a,b\right] \times \mathbb{R} ^{4}\rightarrow \mathbb{R},$ $p:\left[ a,b\right] \rightarrow \mathbb{R}^{+}$ are continuous functions and $s$ a real parameter, with the boundary conditions \begin{equation*} u\left( a\right) =A,\text{ }u^{\prime }\left( a\right) =B,\text{ }u^{\prime \prime \prime }\left( a\right) =C,\text{ }u^{\prime \prime \prime }\left( b\right) =D,\text{ } \end{equation*} for $A,B,C,D\in \mathbb{R}.$ In this work they use an Ambrosetti-Prodi type approach, with some new features: the existence part is obtained in presence of nonlinearities not necessarily bounded, and in the multiplicity result it is not assumed a speed growth condition or an asymptotic condition, as it is usual in the literature for these type of higher order problems.
    The arguments used apply lower and upper solutions technique and topological degree theory.
    An application is made to a continuous model of the human spine, used in aircraft ejections, vehicle crash situations, and some forms of scoliosis.
keywords: one-sided Nagumo condition non-existence and multiplicity of solutions. Higher order Ambrosetti-Prodi problems existence
Existence and location result for a fourth order boundary value problem
Feliz Minhós T. Gyulov A. I. Santos
In the present work we prove an existence and location result for the fourth order fully nonlinear equation% \begin{equation*} u^{(iv)}=f\left( t,u,u^{\prime },u^{\prime \prime },u^{\prime \prime \prime }\right) ,\quad 0
keywords: a priori estimate degree theory. Fourth order boundary value problem a pair of lower and upper solutions Nagumo condition
Non ordered lower and upper solutions to fourth order problems with functional boundary conditions
Alberto Cabada João Fialho Feliz Minhós
In this paper, given $f : I \times (C(I))^2 \times \mathbb{R}^2 \leftarrow \mathbb{R}$ a $L^1$ Carathéodory function, it is considered the functional fourth order equation $u^(iv) (x) = f(x, u, u', u'' (x), u''' (x))$ together with the nonlinear functional boundary conditions $L_0(u, u', u'', u (a)) = 0 = L_1(u, u', u'', u' (a))$ $L_2(u, u', u'', u'' (a), u''' (a)) = 0 = L_3(u, u', u'', u'' (b}, u''' (b)):$ Here $L_i, i$ = 0; 1; 2; 3, are continuous functions satisfying some adequate monotonicity assumptions. It will be proved an existence and location result in presence of non ordered lower and upper solutions and without monotone assumptions on the right hand side of the equation.
keywords: functional boundary value problems Higher order problems Nagumo condition lower and upper solutions

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