scholarly journals Stability Conditions of Second Order Integrodifferential Equations with Variable Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Dingheng Pi
Keyword(s):  
Fixed Point Theory ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Point Theory ◽  
Stability Conditions ◽  
Asymptotically Stable ◽  
Variable Delay ◽  
Functional Differential ◽  
Necessary And Sufficient

We investigate integrodifferential functional differential equationsẍ+f(t,x,ẋ)ẋ+∫t-r(t)t‍a(t,s)g(x(s))ds=0with variable delay. By using the fixed point theory, we obtain conditions which ensure that the zero solution of this equation is stable under an exponentially weighted metric. Then we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable. We will give an example to apply our results.

scholarly journals Fixed Points and Stability of a Class of Integrodifferential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Dingheng Pi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Point Theory ◽  
Asymptotically Stable ◽  
Variable Delay ◽  
Functional Differential ◽  
Necessary And Sufficient

We study a class of integrodifferential functional differential equationsx¨+f(t,x,x˙)x˙+∑j=1N∫t-rj(t)taj(t,s)gj(s,x(s))ds=0with variable delay. By using the fixed point theory, we establish necessary and sufficient conditions ensuring that the zero solution of this equation is asymptotically stable.

Existence, uniqueness and stability of impulsive stochastic neutral functional differential equations driven by Rosenblatt process with varying-time delays

2019 ◽  
Vol 27 (4) ◽  
pp. 213-223
Author(s):  
El Hassan Lakhel ◽  
Abdelmonaim Tlidi
Keyword(s):  
Differential Equations ◽  
Fixed Point Theory ◽  
Functional Differential Equations ◽  
Point Theory ◽  
Uniqueness Result ◽  
Similar Process ◽  
Neutral Functional Differential Equations ◽  
Rosenblatt Process ◽  
Self Similar ◽  
Functional Differential

Abstract Hermite processes are self-similar processes with stationary increments; the Hermite process of order 1 is fractional Brownian motion (fBm) and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of impulsive neutral stochastic functional differential equations with variable delays driven by Rosenblatt process with index {H\in(\frac{1}{2},1)} , which is a special case of a self-similar process with long-range dependence. More precisely, we prove an existence and uniqueness result, and we establish some conditions, ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point theory. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

A star-shaped condition for stability of linear retarded functional differential equations†

1979 ◽  
Vol 83 (3-4) ◽  
pp. 189-198 ◽  
Author(s):  
Richard Silkowski
Keyword(s):  
Asymptotic Stability ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Retarded Functional Differential Equations ◽  
Region Of Stability ◽  
Linear Functional Differential Equation ◽  
Functional Differential ◽  
Image Position ◽  
Necessary And Sufficient

SYNOPSISSufficient conditions are developed for asymptotic stability of the autonomous linear functional differential equation of retarded type. If the asymptotic stability ofimplies the asymptotic stability ofthen these conditions are also necessary. Necessary and sufficient conditions are developed for the largest cone in the region of stability. These results are illustrated with the example

scholarly journals On periodic solutions to autonomous retarded functional differential equations

1990 ◽  
Vol 41 (3) ◽  
pp. 347-354
Author(s):  
Zhanyuan Hou
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Retarded Functional Differential Equations ◽  
Retarded Functional Differential Equation ◽  
Functional Differential ◽  
Necessary And Sufficient

Under the assumption that Ca = C([−r, 0], Sn−1(a)) is positively invariant for a > 0, two necessary and sufficient conditions are obtained for an autonomous retarded functional differential equation to have a non-trivial periodic solution in Ca. Moreover, a feasible sufficient condition is given, which is better for n = 2 than that given by Dos Reis and Baroni.

scholarly journals Fixed points and stability in nonlinear neutral integro-differential equations with variable delay

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 781-795
Author(s):  
Imene Soualhia ◽  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Initial Function ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Stability Theorem ◽  
Variable Delay ◽  
Necessary And Sufficient

The nonlinear neutral integro-differential equation x'(t) = -?t,t-?(t) a (t,s) g(x(s))ds+c(t)x'(t-?(t)), with variable delay ?(t) ? 0 is investigated. We find suitable conditions for ?, a, c and g so that for a given continuous initial function ? mapping P for the above equation can be defined on a carefully chosen complete metric space S0? in which P possesses a unique fixed point. The final result is an asymptotic stability theorem for the zero solution with a necessary and sufficient conditions. The obtained theorem improves and generalizes previous results due to Burton [6], Becker and Burton [5] and Jin and Luo [16].

scholarly journals Fixed point theory in the setting of $(\alpha,\beta,\psi,\phi)$-interpolative contractions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Erdal Karapınar ◽  
Andreea Fulga ◽  
Antonio Francisco Roldán López de Hierro
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Necessary And Sufficient Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

AbstractIn this manuscript we introduce the notion of $(\alpha,\beta,\psi,\phi)$ ( α , β , ψ , ϕ ) -interpolative contraction that unifies and generalizes significant concepts: Proinov type contractions, interpolative contractions, and ample spectrum contraction. We investigate the necessary and sufficient conditions to guarantee existence and uniqueness of the fixed point of such mappings.

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