An Existence Result in α-norm for Impulsive Functional Differential Equations with Variable Times
Issue:
Volume 10, Issue 1, February 2022
Pages:
1-8
Received:
12 October 2021
Accepted:
11 November 2021
Published:
23 January 2022
Abstract: The dynamics of evolving processes is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of “impulses.” In fact, there are many processes and phenomena in the real world, which are subjected during their development to the short-term external influences. Their duration is negligible compared with the total duration of the studied phenomena and processes. Impulsive differential equations take an important place in some area such that physics, chemical technology, population dynamics, biotechnology, and economics. The study of such equations is relatively less developed due to the difficulties created by the state-dependent impulses. In the case of impulses at variable times, a “beating phenomenon” may occur, that is to say, a solution of the differential equation may hit a given barrier several times (including infinitely many times). In this work, we study the existence of solutions for some partial impulsive functional differential equations with variable times in Banach spaces by using the fractional power of closed operators theory. We suppose that the undelayed part admits an analytic semigroup. The delayed part is assumed to be Lipschitz. We use Schaefer fixed-point Theorem to prove the existence of solutions for this first order equation with impulse in α-norm.
Abstract: The dynamics of evolving processes is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of “impulses.” In fact, there are many processes and phenomena in the real world, which are subjected during their development to t...
Show More
Some Characters of a Generalized Rational Difference Equation
Issue:
Volume 10, Issue 1, February 2022
Pages:
9-14
Received:
18 January 2022
Accepted:
7 February 2022
Published:
18 February 2022
Abstract: Difference equations arise in many contexts in biological, economic and social sciences., can exhibit a complicated dynamical behavior, from stable equilibria to a bifurcating hierarchy of cycles. There are a lot of fascinating problems, which are often concerned with both mathematical aspects of the fine structure of the trajectories and practical applications. In this paper, we investigate the generalized rational difference equation, a kind of fractional linear maps with two delays. Sufficient conditions for the global asymptotic stability of the zero fixed point are given. For the positive equilibrium, we find the region of parameters in which the positive equilibrium is local asymptotic stable and attracts all positive solutions. As for general solutions, two specific and easy checked conditions on the initial values are obtained to guarantee corresponding solutions to be eventually positive. The upper or lower bound are also provided according to different parameters. Of particular interest for this generalized equation would be the existence of periodic solutions and their stabilities. We get the necessary and sufficient conditions for the existence of period two solutions depending on the combination of delay terms. In addition, the sufficient conditions for the existence of 2r− and 2d−periodic solutions are obtained too.In the end of the paper, we give examples to illustrate our results.
Abstract: Difference equations arise in many contexts in biological, economic and social sciences., can exhibit a complicated dynamical behavior, from stable equilibria to a bifurcating hierarchy of cycles. There are a lot of fascinating problems, which are often concerned with both mathematical aspects of the fine structure of the trajectories and practical...
Show More
