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Karakostas fixed point theorem and semilinear neutral differential equations with impulses and nonlocal conditions

Leiva Hugo, Riera Lenin, Lalvay Sebastián

Journal of Mathematics and Applications

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2022

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Vol. 45

107--128

EN

This paper is concerned with the existence and uniqueness of solutions for a semilinear neutral differential equation with impulses and nonlocal conditions. First, we assume that the nonlinear terms are locally Lipschitz, and to achieve the existence of solutions, Karakostas Fixed Point Theorem is applied. After that, under some additional conditions, the uniqueness is proved as well. Next, assuming some bound on the non-linear terms the global existence is proved by applying a generalization of Gronwall inequality for impulsive differential equations. Then, we suppose stronger hypotheses on the nonlinear functions, such as globally Lipschitz conditions, that allow us to appy Banach Fixed Point Theorem to prove the existence and uniqueness of solutions. Finally, we present an example as an application of our method.

2

Lie symmetries of first order neutral differential equations

Lobo Jervin Zen, Valaulikar Y.S.

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 1

29--40

EN

In this paper we extend the method of obtaining symmetries of ordinary differential equations to first order non-homogeneous neutral differential equations with variable coefficients. The existing method for delay differential equations uses a Lie-Bäcklund operator and an Invariant Manifold Theorem to define the operators which are used to obtain the infinitesimal generators of the Lie group. In this paper, we adopt a different approach and use Taylor’s theorem to obtain a Lie type invariance condition and the determining equations for a neutral differential equation. We then split this equation in a manner similar to that of ordinary differential equations to obtain an over-determined system of partial differential equations. These equations are then solved to obtain corresponding infinitesimals, and hence desired equivalent symmetries. We then obtain the symmetry algebra admitted by this neutral differential equation.

3

On the exponential stability of a neutral differential equation of first order

Gözen M., Tunç C.

Journal of Mathematics and Applications

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2018

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Vol. 41

95--107

EN

In this work, we establish some assumptions that guaranteeing the global exponential stability (GES) of the zero solution of a neutral differential equation (NDE). We aim to extend and improves some results found in the literature.

4

Oscillation of higher-order half-linear neutral differential equations

Tang S., Li T., Thandapani E.

Demonstratio Mathematica

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2013

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Vol. 46, nr 1

101--109

EN

In this paper, we establish some new oscillation criteria for the higher-order half-linear neutral differential equation (…). An example is given to illustrate the main results.

5

Bounded oscillation for a class of even order neutral difference equations

Lin X., Shen J.

Fasciculi Mathematici

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2002

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Nr 33

37-47

EN

We investigate bounded oscillation for the even order neutral delay difference equation delta'' (x(n) -cx(n-m) = pn(x-k), where u is even. The sufficient conditions obtained in this paper improve and generalize the results in related literature.