Stability, fixed points and inverses of delays

2006 ◽  
Vol 136 (2) ◽  
pp. 245-275 ◽  
Author(s):  
Leigh C. Becker ◽  
T. A. Burton
Keyword(s):  
Existence And Uniqueness ◽  
Initial Function ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Zero Solution ◽  
Asymptotically Stable ◽  
Variable Delay ◽  
Uniqueness Of Solutions ◽  
The Stability ◽  
Image Position

The scalar equation with variable delay r(t) ≥ 0 is investigated, where t−r(t) is increasing and xg(x) > 0 (x ≠ 0) in a neighbourhood of x = 0. We find conditions for r, a and g so that for a given continuous initial function ψ a mapping P for (1) can be defined on a complete metric space Cψ and in which P has a unique fixed point. The end result is not only conditions for the existence and uniqueness of solutions of (1) but also for the stability of the zero solution. We also find conditions ensuring that the zero solution is asymptotically stable by changing to an exponentially weighted metric on a closed subset of Cψ. Finally, we parlay the methods for (1) into results for

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 Torsion Discriminance for Stability of Linear Time-Invariant Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 386
Author(s):  
Yuxin Wang ◽  
Huafei Sun ◽  
Yueqi Cao ◽  
Shiqiang Zhang
Keyword(s):  
Lebesgue Measure ◽  
Linear Time ◽  
Zero Solution ◽  
Positive Lebesgue Measure ◽  
Asymptotically Stable ◽  
Linear Time Invariant ◽  
Measurable Set ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

scholarly journals A Fractional Order Dengue Fever Model in the Context of Protected Travellers

2021 ◽  
Author(s):  
E. Bonyah ◽  
M. L. Juga ◽  
C. W. Chukwu ◽  
Fatmawati
Keyword(s):  
Dengue Fever ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Climate Changes ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Qualitative Information ◽  
Uniqueness Of Solutions ◽  
Vector Borne Diseases ◽  
Vector Borne

AbstractClimate changes are affecting the control of many vector-borne diseases, particularly in Africa. In this work, a dengue fever model with protected travellers is formulated. Caputo-Fabrizio operator is utilized to obtain some qualitative information about the disease. The basic properties and the reproduction number is studied. The two steady states are determined and the local stability of the states are found to be asymptotically stable. The fixed pointed theory is made use to obtain the existence and uniqueness of solutions of the model. The numerical simulation suggests that the fractional-order affects the dynamics of dengue fever.

scholarly journals On the existence and uniqueness of solutions of parabolic equations

1975 ◽  
Vol 13 (3) ◽  
pp. 451-456 ◽  
Author(s):  
K.L. Teo
Keyword(s):  
Continuous Function ◽  
Boundary Value Problem ◽  
Boundary Problem ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Divergence Form ◽  
Parabolic Operator ◽  
Uniqueness Of Solutions ◽  
Compact Closure ◽  
Image Position

Recently, Eklund (Proc. Amer. Math. Soc. 47 (1975), 137–142) has shown that to each continuous function F on ∂pQ −⊲ {∂Ω × [0, T]} ∪ {Ω × (0)} there is a unique solution to the boundary value problemwhere L is a linear second order parabolic operator in divergence form, Ω ⊂ Rn is a bounded domain with compact closure and ∂Ω denotes its boundary. In this note, it is shown that the existence theorem of Eklund remains valid for the following boundary problem

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.

Hopf Bifurcation of an Age-Structured Epidemic Model with Quarantine and Temporary Immunity Effects

2021 ◽  
Vol 31 (12) ◽  
pp. 2150183
Author(s):  
Lili Liu ◽  
Jian Zhang ◽  
Ran Zhang ◽  
Hongquan Sun
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Reproduction Number ◽  
Exponential Polynomial ◽  
Uniqueness Of Solutions ◽  
Fourth Degree ◽  
Age Structured ◽  
The Stability ◽  
Distribution Of Roots

In this paper, we investigate an epidemic model with quarantine and recovery-age effects. Reformulating the model as an abstract nondensely defined Cauchy problem, we discuss the existence and uniqueness of solutions to the model and study the stability of the steady state based on the basic reproduction number. After analyzing the distribution of roots to a fourth degree exponential polynomial characteristic equation, we also derive the conditions of Hopf bifurcation. Numerical simulations are performed to illustrate the results.

On a class of linear integro-differential equations

1947 ◽  
Vol 43 (2) ◽  
pp. 153-163 ◽  
Author(s):  
H. R. Pitt
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
General Equation ◽  
Uniqueness Of Solutions ◽  
Solution Of Equations ◽  
Image Position

This paper is a sequel to a previous one(1) with the same title which dealt with the general solution of equations of the typeWe consider here the more general equationwhere g(x) is a given function. We are interested particularly in the existence and uniqueness of solutions of the latter equation and show how these are related to the closure and completeness properties of sets of functions {xneωnx} derived from the kernels kr(y).

scholarly journals An analysis on the stability of a state dependent delay differential equation

2016 ◽  
Vol 14 (1) ◽  
pp. 425-435 ◽  
Author(s):  
Sertaç Erman ◽  
Ali Demir
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Delay Term ◽  
State Dependent ◽  
State Dependent Delay ◽  
The Stability ◽  
Delay Differential ◽  
Equation With Delay

AbstractIn this paper, we present an analysis for the stability of a differential equation with state-dependent delay. We establish existence and uniqueness of solutions of differential equation with delay term $\tau (u(t)) = \frac{{a + bu(t)}}{{c + bu(t)}}.$ Moreover, we put the some restrictions for the positivity of delay term τ(u(t)) Based on the boundedness of delay term, we obtain stability criterion in terms of the parameters of the equation.

Existence and Uniqueness of Weak Solutions to a Thin Film Type Equation

2015 ◽  
Vol 738-739 ◽  
pp. 465-468
Author(s):  
Bo Liang ◽  
Hui Ying Shen ◽  
Xi Ting Peng ◽  
Mei Shan Wang
Keyword(s):  
Thin Film ◽  
Boundary Condition ◽  
Parabolic Equation ◽  
Existence And Uniqueness ◽  
Initial Function ◽  
Type Equation ◽  
Multidimensional Space ◽  
Discrete Problem ◽  
Uniqueness Of Solutions ◽  
Fourth Order Parabolic Equation

This paper is concerned with a fourth order parabolic equation in multidimensional spacewith boundary condition u=Δu=0 and initial function u0. The minimizer method yields the existence and uniqueness for the elliptic equation. Finally, the existence and uniqueness of solutions of the corresponding parabolic equation are obtained from the semi-discrete problem.

scholarly journals Existence and uniqueness of solutions for a nonlinear second order differential equation in Hilbert space

1990 ◽  
Vol 33 (1) ◽  
pp. 89-95 ◽  
Author(s):  
Dang Dinh Hai
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Order Differential Equation ◽  
Real Hilbert Space ◽  
Boundary Value ◽  
Uniqueness Of Solutions ◽  
Second Order Differential Equation ◽  
Image Position

This paper is concerned with the existence and uniqueness of solutions for the Picard boundary value problemin a real Hilbert space. Our theorems improve corresponding results of Mawhin for |k| large.

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