arXiv subjects
Publications and source records attributed to Alberto Cabada.
Using the theory of fixed point index, we establish new results for the existence of nonzero solutions of Hammerstein integral equations with reflections. We apply our results to a first order periodic boundary value problem with reflections.
This work is devoted to the study of the first order operator $x'(t)+m\,x(-t)$ coupled with periodic boundary value conditions. We describe the eigenvalues of the operator and obtain the expression of its related Green's function in the non resonant case. We also obtain the range of the values of the real parameter $m$ for which the integral kernel, which provides the unique solution, has constant sign. In this way, we automatically establish maximum and anti-maximum principles for the equation. Some applications to the existence of nonlinear periodic boundary value problems are showed.
In this paper we analyse some possibilities of finding positive solutions for second order boundary value problems with Dirichlet and periodic boundary conditions, for which the correspondent Green's functions change sign. The obtained results can also be adapted to the Neumann and Mixed boundary conditions.
In this work we will consider integral equations defined on the whole real line and look for solutions which satisfy some certain kind of asymptotic behavior. To do that, we will define a suitable Banach space which, to the best of our knowledge, has never been used before. In order to obtain fixed points of the integral operator, we will consider the fixed point index theory and apply it to this new Banach space.
This paper is devoted to the study of the parameter's set where the Green's function related to a general linear $n^{\rm th}$-order operator, depending on a real parameter, $T_n[M]$, coupled with many different two point boundary value conditions, is of constant sign. This constant sign is equivalent to the strongly inverse positive (negative) character of the related operator on suitable spaces related to the boundary conditions. This characterization is based on spectral theory, in fact the extremes of the obtained interval are given by suitable eigenvalues of the differential operator with different boundary conditions. Moreover, we also obtain a characterization of the strongly inverse positive (negative) character on some sets, where non homogeneous boundary conditions are considered. In order to see the applicability of the obtained results, some examples are given along the paper. This method avoids the explicit calculation of the related Green's function.
The aim of this paper is to study the following fourth-order operator: T[p,c]\,u(t)\equiv u^{(4)}(t)-p\,u"(t)+c(t)\,u(t)\,,\quad t\in I\equiv [a,b]\,, coupled with the non-homogeneous simply supported beam boundary conditions: u(a)=u(b)=0\,,\quad u"(a)=d_1\leq0\,,\ u"(b)=d_2\leq 0\,. First, we prove a result which makes an equivalence between the strongly inverse positive (negative) character of this operator with the previously introduced boundary conditions and with the homogeneous boundary conditions, given by: T[p,c]\,u(t)=h(t)(\geq0)\,, u(a)=u(b)=u"(a)=u"(b)=0\,, Once that we have done that, we prove several results where the strongly inverse positive (negative) character of $T[p,c]$ it is ensured. Finally, there are shown a couple of result which say that under the hypothesis that $h>0$, we can affirm that the problem for the homogeneous boundary conditions has a unique constant sign solution.
This paper is devoted to prove the existence of one or multiple solutions of a wide range of nonlinear differential boundary value problems. To this end, we obtain some new fixed point theorems for a class of integral operators. We follow the well-known Krasnoselski\uı's fixed point Theorem together with two fixed point results of Leggett-Williams type. After obtaining a general existence result for a one parameter family of nonlinear differential equations, are proved, as particular cases, existence results for second and fourth order nonlinear boundary value problems.
In this work we study differential problems in which the reflection operator and the Hilbert transform are involved. We reduce these problems to ODEs in order to solve them. Also, we describe a general method for obtaining the Green's function of reducible functional differential equations and illustrate it with the case of homogeneous boundary value problems with reflection and several specific examples.
The aim of this paper consists on the study of the following fourth-order operator: \begin{equation}\label{Ec::T4} T[M]\,u(t)\equiv u^{(4)}(t)+p_1(t)\,u"'(t)+p_2(t)\,u"(t)+M\,u(t)\,,\ t\in I \equiv [a,b]\,, \end{equation} coupled with the two point boundary conditions: \begin{equation}\label{Ec::cf} u(a)=u(b)=u"(a)=u"(b)=0\,. \end{equation} So, we define the following space: \begin{equation}\label{Ec::esp} X=\left\lbrace u\in C^4(I)\quad\mid\quad u(a)=u(b)=u"(a)=u"(b)=0 \right\rbrace \,. \end{equation} Here $p_1\in C^3(I)$ and $p_2\in C^2(I)$. By assuming that the second order linear differential equation \begin{equation}\label{Ec::2or} L_2\, u(t)\equiv u"(t)+p_1(t)\,u'(t)+p_2(t)\,u(t)=0\,,\quad t\in I, \end{equation} is disconjugate on $I$, we characterize the parameter's set where the Green's function related to operator $T[M]$ in $X$ is of constant sign on $I \times I$. Such characterization is equivalent to the strongly inverse positive (negative) character of operator $T[M]$ on $X$ and comes from the first eigenvalues of operator $T[0]$ on suitable spaces.
By topological arguments, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions of a class of perturbed nonlinear integral equations. These type of integral equations arise, for example, when dealing with boundary value problems where nonlocal terms occur in the differential equation and/or in the boundary conditions. Some examples are given to illustrate the theoretical results.
We study the existence and multiplicity of positive solutions for a nonlinear fourth-order two-point boundary value problem. The approach is based on critical point theorems in conical shells, Krasnoselskii's compression-expansion theorem, and unilateral Harnack type inequalities.
The aim of this paper is to show certain properties of the Green's functions related to the Hill's equation coupled with different two point boundary value conditions. We will obtain the expression of the Green's function of Neumann, Dirichlet, Mixed and anti-periodic problems as a combination of the Green's function related to periodic ones. As a consequence we will prove suitable results in spectral theory and deduce some comparison results for the solutions of the Hill's equation with different boundary value conditions.
This paper is devoted to the study of the sign of the Green's function related to a general linear $n^{\rm th}$-order operator, depending on a real parameter, $T_n[M]$, coupled with the $(k,n-k)$ boundary value conditions. If operator $T_n[\bar M]$ is disconjugate for a given $\bar M$, we describe the interval of values on the real parameter $M$ for which the Green's function has constant sign. One of the extremes of the interval is given by the first eigenvalue of operator $T_n[\bar M]$ satisfying $(k,n-k)$ conditions. The other extreme is related to the minimum (maximum) of the first eigenvalues of $(k-1,n-k+1)$ and $(k+1,n-k-1)$ problems. Moreover if $n-k$ is even (odd) the Green's function cannot be non-positive (non-negative). To illustrate the applicability of the obtained results, we calculate the parameter intervals of constant sign Green's functions for particular operators. Our method avoids the necessity of calculating the expression of the Green's function. We finalize the paper by presenting a particular equation in which it is shown that the disconjugation hypothesis on operator $T_n[\bar M]$ for a given $\bar M$ cannot be eliminated.
This paper is devoted to the description of the interval of parameters for which the general linear $n^{\rm th}$-order equation \begin{equation} \label{e-Ln} T_n[M]\,u(t) \equiv u^{(n)}(t)+a_1(t)\, u^{(n-1)}(t)+\cdots +a_{n-1}(t)\, u'(t)+(a_{n}(t)+M)\,u(t)=0 \,,\quad t\in I\equiv[a,b], \end{equation} with $a_i\in C^{n-i}(I)$, is disconjugate on $ I $. Such interval is characterized by the closed to zero eigenvalues of this problem coupled with $(k,n-k)$ boundary conditions, given by \begin{equation} \label{e-k-n-k} u(a)=\cdots=u^{(k-1)}(a)=u(b)=\cdots=u^{(n-k-1)}(b)=0\,,\quad 1\leq k\leq n-1\,. \end{equation}
This work is devoted to the study of the existence and periodicity of solutions of initial differential problems, paying special attention to the explicit computation of the period. These problems are also connected with some particular initial and boundary value problems with reflection, which allows us to prove existence of solutions of the latter using the existence of the first.
We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of positive solutions for systems of nonlinear Hammerstein integral equations. An example is also presented to show the applicability of our results.
We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.
We discuss the existence of multiple positive solutions for a nonlocal fractional problem recently considered by Nieto and Pimental. Our approach relies on classical fixed point index.