Nontrivial solutions of systems of Hammerstein integral equations with first derivative dependence

By means of classical fixed point index, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of Hammerstein integral equations where the nonlinearities are allowed to depend on the first derivative. As a byproduct of our theory we discuss the existence of positive solutions of a system of third order ODEs subject to nonlocal boundary conditions. Some examples are provided in order to illustrate the applicability of the theoretical results.Comment: 18 page

Fourth order functional boundary value problems: Existence results and extremal solutions

In this work we present two types of results for some fourth order functional boundary value problems. The first one presents an existence and location result for a problem where every boundary conditions have functional dependence. The second one states sufficient conditions for the existence of extremal solutions for functional problems with more restrict boundary functions. The arguments make use of lower and upper solutions technique, a Nagumo-type condition,an adequate version of Bolzano’s theorem and existence of extremal fixed points for a suitable mapping

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

Location results: an under used tool in higher order boundary value problems

The method of lower and upper solutions provides, as well as residts of existence, other important properties such as location of solution, extremal solutions,..., which have been under used and, moreover, its potential has not been optimized, either in theory either in applications. This work will present some cases to emphasize both items: two fourth order problems with functional boundary conditions (including an application to a continuous model for the deformation of the human spine under the action of some forces) and a third order periodic problem where unbounded nonlinearities are allowed, provided that an one-sided Nagumo-type condition is verified

Periodic solutions for some fully nonlinear fourth order differential equations

In this paper we present sufficient conditions for the existence of periodic solutions to some nonlinear fourth order boundary value problems u(4)(x) = f(x; u(x); u′(x); u′′(x); u′′′(x)) u(i)(a) = u(i)(b); i = 0; 1; 2; 3; 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

EXISTENCE OF EXTREMAL SOLUTIONS FOR SOME FOURTH

In this work we present sufficient conditions for the existence of extremal solutions for some fourth order functional problem with the nonlinearity and boundary functions not necessarily continuous, but satisfying some monotonicity assumptions. The arguments make use of lower and upper solutions technique, a version of Bolzano’s theorem and existence of extremal fixed points for a suitable mapping

On the solvability of third-order three point systems of differential equations with dependence on the first derivative

This paper presents sufficient conditions for the solvability of the third order three point boundary value problem \begin{equation*} \left\{ \begin{array}{c} -u^{\prime \prime \prime }(t)=f(t,\,v(t),\,v^{\prime }(t)) \\ -v^{\prime \prime \prime }(t)=h(t,\,u(t),\,u^{\prime }(t)) \\ u(0)=u^{\prime }(0)=0,u^{\prime }(1)=\alpha u^{\prime }(\eta ) \\ v(0)=v^{\prime }(0)=0,v^{\prime }(1)=\alpha v^{\prime }(\eta ). \end{array} \right. \end{equation*} The arguments apply Green's function associated to the linear problem and the Guo--Krasnosel'ski\u{\i} theorem of compression-expansion cones. The dependence on the first derivatives is overcome by the construction of an adequate cone and suitable conditions of superlinearity/sublinearity near $0$ and $+\infty .$ Last section contains an example to illustrate the applicability of the theorem.Comment: 21 page

Existence, nonexistence and multiplicity results for some beam equations

This paper studies some fourth order nonlinear fully equations with a parameter s ∈ R, with two point boundary conditions. These problems model several phenomena, such as, a cantilevered beam with a linear relation between the curvature and the shear force at both endpoints. For some values of the real constants, it will be presented an Ambrosetti–Prodi type discussion on s. The arguments used apply lower and upper solutions technique, a priori estimations and topological degree theory

On some third order nonlinear boundary value problems: existence, location and multiplicity results

We prove an Ambrosetti–Prodi type result for some third order fully nonlinear equations, with a parameter s,under several two-point separated boundary conditions. From a Nagumo-type growth condition, an a priori estimate on u'' is obtained. An existence and location result will be proved, by degree theory, for s ∈ R such that there exist lower and upper solutions. The location part can be used to prove the existence of positive solutions if a non-negative lower solution is considered. The existence, nonexistence and multiplicity of solutions will be discussed as s varies