scholarly journals INTERACTION BETWEEN PERIODIC ELASTIC WAVES AND TWO CONTACT NONLINEARITIES

2012 ◽  
Vol 22 (04) ◽  
pp. 1150022 ◽  
Author(s):  
STÉPHANE JUNCA ◽  
BRUNO LOMBARD
Keyword(s):  
Periodic Solutions ◽  
Elastic Waves ◽  
Evolution Equations ◽  
Diophantine Condition ◽  
Nonlinear Phenomenon ◽  
Neutral Delay ◽  
Mean Values ◽  
Diffracted Waves ◽  
Delay Differential ◽  
Propagation Of Elastic Waves

Propagation of elastic waves is studied in a 1D medium containing two cracks. The latter are modeled by smooth nonlinear jump conditions accounting for the finite, non-null compressibility of real cracks. The evolution equations are written in the form of a system of two nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. Perturbation analysis indicates that, under periodic excitation, the periodic solutions oscillate around positive mean values, which increase with the forcing level. This typically nonlinear phenomenon offers non-destructive means to evaluate the cracks. Existence, uniqueness and attractivity of periodic solutions is then examined. At some particular values of the ratio between the wave travel time and the period of the source, results are obtained whatever the forcing level. With a much larger set of ratios but at small forcing levels, results are obtained under a Diophantine condition. Lastly, numerical experiments are proposed to illustrate the behavior of the periodic diffracted waves.

scholarly journals Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments

2020 ◽  
Vol 18 (1) ◽  
pp. 93-105
Author(s):  
Mukhiddin I. Muminov ◽  
Ali H. M. Murid
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Second Order ◽  
Algebraic Equations ◽  
Neutral Delay ◽  
Piecewise Constant Arguments ◽  
Piecewise Constant ◽  
Neutral Delay Differential Equations ◽  
Delay Differential

Abstract In this paper, we describe a method to solve the problem of finding periodic solutions for second-order neutral delay-differential equations with piecewise constant arguments of the form x″(t) + px″(t − 1) = qx([t]) + f(t), where [⋅] denotes the greatest integer function, p and q are nonzero real or complex constants, and f(t) is complex valued periodic function. The method reduces the problem to a system of algebraic equations. We give explicit formula for the solutions of the equation. We also give counter examples to some previous findings concerning uniqueness of solution.

scholarly journals A priori bounds for periodic solutions of neutral differential equations with delay

2007 ◽  
Vol 38 (3) ◽  
pp. 239-246
Author(s):  
Jing-Wen Li ◽  
Sui Sun Cheng
Keyword(s):  
Periodic Solutions ◽  
A Priori ◽  
Existence Results ◽  
A Priori Bounds ◽  
Neutral Delay ◽  
Neutral Differential Equations ◽  
Neutral Delay Differential Equation ◽  
Existence Of Periodic Solutions ◽  
Delay Differential ◽  
Equations With Delay

A priori bounds are established for periodic solutions of a neutral delay differential equation. Such bounds can be used to establish existence of periodic solutions. The resulting existence results improve some existing ones in the literature.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

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