Nontrivial solutions of boundary value problems for second order functional differential equations

In this paper we present a theory for the existence of multiple nontrivial solutions for a class of perturbed Hammerstein integral equations. Our methodology, rather than to work directly in cones, is to utilize the theory of fixed point index on affine cones. This approach is fairly general and covers a class of nonlocal boundary value problems for functional differential equations. Some examples are given in order to illustrate our theoretical results.Comment: 19 pages, revised versio

Oscillation criteria for third order delay nonlinear differential equations

The purpose of this paper is to give oscillation criteria for the third order delay nonlinear differential equation \begin{equation*} \lbrack a_{2}(t)\{(a_{1}(t)(x^{\prime }(t))^{\alpha _{1}})^{\prime}\}^{\alpha _{2}}]^{\prime }+q(t)f(x(g(t)))=0, \end{equation*} via comparison with some first differential equations whose oscillatory characters are known. Our results generalize and improve some known results for oscillation of third order nonlinear differential equations. Some examples are given to illustrate the main results

Oscillation of a class of higher order neutral differential equations

summary:In this paper, we investigate a class of higher order neutral functional differential equations, and obtain some new oscillatory criteria of solutions

Oscillation theorems for second order neutral differential equations

In this paper new oscillation criteria for the second order neutral differential equations of the form \begin{equation*} \left(r(t)\left[x(t)+p(t)x(\tau(t))\right]'\right)'+q(t)x(\sigma(t))+v(t)x(\eta(t))=0 \tag{$E$}\end{equation*} are presented. Gained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. Obtained comparison principles essentially simplify the examination of the studied equations. We cover all possible cases when arguments are delayed, advanced or mixed

Asymptotic and oscillatory behavior of higher order quasilinear delay differential equations

In the paper, we offer such generalization of a lemma due to Philos (and partially Staikos), that yields many applications in the oscillation theory. We present its disposal in the comparison theory and we establish new oscillation criteria for $n-$th order delay differential equation \begin{equation*} \left(r(t)\left[x'(t)\right]^{\gamma}\right)^{(n-1)}+q(t)x^{\gamma}(\tau(t))=0.\tag{$E$} \end{equation*} The presented technique essentially simplifies the examination of the higher order differential equations

Oscillatory theorems of a class of even-order neutral equations

AbstractA class of even-order nonlinear neutral differential equations with distributed deviating arguments is studied, and oscillatory criteria for solutions of such equations are established

Oscillation of Nonlinear First Order Neutral Differential Equations

In this paper, the author established some new integral conditions for the oscillation of all solutions of nonlinear first order neutral delay differential equations. Examples are inserted to illustrate the results

An existence result for first-order impulsive functional differential equations in banach spaces

AbstractIn this paper, the Leray-Schauder nonlinear alternative is used to investigate the existence of solutions to first-order impulsive initial value problems for functional differential equations in Banach spaces

Oscillation criteria for third-order neutral differential equations with continuously distributed delay

AbstractThe purpose of this paper is to study the oscillation of a certain class of third-order neutral differential equations with continuously distributed delay. By using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero