Disconjugacy characterization by means of spectral of (k,n−k)(k,n-k) problems

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}Comment: 6 page

Positive solutions of a nonlocal Caputo fractional BVP

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.Comment: 8 page

Existence of solutions for nthn^\mathrm{th}-order nonlinear differential boundary value problems by means of new fixed point theorems

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\u{\i}'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.Comment: 31 pages, 12 figure

Constant sign solution for simply supported beam equation with non-homogeneous boundary conditions

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.Comment: 18 page

The eigenvalue Characterization for the constant Sign Green's Functions of (k,n−k)(k,n-k) problems

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

Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear fractional differential equation with mixed boundary value conditions. Riemann-Liouville fractional derivative is considered. An exhaustive study of the sign of the related Green's function is done. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed theory of compact operators defined in suitable cones, it is proved the existence of at least one solution of the considered problem. Moreover it is developed the method of lower and upper solutions and it is deduced the existence of solutions by a combination of both techniques. In some particular situations, the Banach contraction principle is used to ensure the uniqueness of solutions

An Alternative Explicit Expression of the Kernel of the One Dimensional Heat Equation with Dirichlet Conditions

This paper is devoted to the study of the one dimensional non homogeneous heat equation coupled to Dirichlet Boundary Conditions. We obtain the explicit expression of the solution of the linear equation by means of a direct integral in an unbounded domain. The main novelty of this expression relies in the fact that the solution is not given as a series of infinity terms. On our expression the solution is given as a sum of two integrals with a finite number of terms on the kernel. The main novelty is that, on the contrary to the classical method, where the solutions are derived by a direct application of the separation of variables method, on the basis of the spectral theory and the Fourier Series expansion, the solution is obtained by means of the application of the Laplace Transform with respect to the time variable. As a consequence, for any $t \ge 0$ fixed, we must solve an Ordinary Differential Equation on the spatial variable, coupled to Dirichlet Boundary conditions. The solution of such a problem is given by the construction of the related Green's function

Solutions and Green's function of the first order linear equation with reflection and initial conditions

This work is devoted to the study of the existence and sign of Green's functions for first order linear problems with constant coefficients and initial (one point) conditions. We first prove a result on the existence of solutions of $n$-th order linear equations with involutions via some auxiliary functions to later prove a uniqueness result in the first order case. We study then different situations for which a Green's function can be obtained explicitly and derive several results in order to obtain information about the sign of the Green's function. Once the sign is known, optimal maximum and anti-maximum principles follow

Comparison results for first order linear operators with reflection and periodic boundary value conditions

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

Periodic solutions for some phi-Laplacian and reflection equations

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.Comment: Preprin