Browsing by Subject "fixed points"

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Access status: Open Access ,
Difference equations with impulses
(Wydawnictwa AGH, 2019) Danca, Marius; Fečkan, Michal; Pospíšil, Michal
Difference equations with impulses are studied focussing on the existence of periodic or bounded orbits, asymptotic behavior and chaos. So impulses are used to control the dynamics of the autonomous difference equations. A model of supply and demand is also considered when Li-Yorke chaos is shown among others.
Access status: Open Access ,
Fixed points and stability in neutral nonlinear differential equations with variable delays
(2012) Ardjouni, Abdelouaheb; Djoudi, Ahcene
By means of Krasnoselskii’s fixed point theorem we obtain boundedness and stability results of a neutral nonlinear differential equation with variable delays. A stability theorem with a necessary and sufficient condition is given. The results obtained here extend and improve the work of C.H. Jin and J.W. Luo [Nonlinear Anal. 68 (2008), 3307–3315], and also those of T.A. Burton [Fixed Point Theory 4 (2003), 15-32; Dynam. Systems Appl. 11 (2002), 499–519] and B. Zhang [Nonlinear Anal. 63 (2005), e233–e242]. In the end we provide an example to illustrate our claim.
Access status: Open Access ,
Integral and fractional equations, positive solutions, and Schaefer’s fixed point theorem
(2016) Becker, Leigh C.; Burton, Theodore Allen; Purnaras, Ioannis K.
This is the continuation of four earlier studies of a scalar fractional differential equation of Riemann-Liouville type $D^qx(t) = -f(t,x(t)), \quad \lim_{t\to 0^+} t^{1-q} x(t) = x^0 \in\Re \quad (0 \lt q \lt 1), \tag {a}$ in which we first invert it as a Volterra integral equation $x(t)=x^0 t^{q-1} -\frac{1}{\Gamma (q)}\int\limits^t_0 (t-s)^{q-1}f(s,x(s))\,ds \tag {b}$ and then transform it into $\begin{multline}x(t)=x^0t^{q-1}-\int\limits^t_0 R(t-s)x^0s^{q-1}ds\\+\int\limits^t_0R(t-s) \bigg[x(s)-\frac{f(s,x(s))}{J} \bigg] ds, \tag {c}\end{multline}$ where $R$ is completely monotone with $\int^{\infty}_0 R(s)\,ds =1$ and $J$ is an arbitrary positive constant. Notice that when x is restricted to a bounded set, then by choosing J large enough, we can frequently change the sign of the integrand in going from $(b)$ to $(c)$. Moreover, the same kind of transformation will produce a similar effect in a wide variety of integral equations from applied mathematics. Because of that change in sign, we can obtain an a priori upper bound on solutions of $(b)$ with a parameter $\lambda \in (0,1]$ and then obtain an a priori lower bound on solutions of $(c)$. Using this property and Schaefer's fixed point theorem, we obtain positive solutions of an array of fractional differential equations of both Caputo and Riemann-Liouville type as well as problems from turbulence, heat transfer, and equations of logistic growth. Very simple results establishing global existence and uniqueness of solutions are also obtained in the same way.