The main aim of this paper is to study the dynamics of a recurrent neural networks with different input currents in terms of asymptotic point. Under certain circumstances, we studied the existence, the uniqueness of bounded solutions and their homoclinic and heteroclinic motions of the considered system with rectangular currents input. Moreover, we studied the unpredictable behavior of the continuous high-order recurrent neural networks and the discrete high-order recurrent neural networks. Our method was primarily based on Banach’s fixed-point theorem, topology of uniform convergence on compact sets and Gronwall inequality. For the demonstration of theoretical results, we give examples and their numerical simulations.
This paper investigates existence, uniqueness, and Ulam’s stability results for a nonlinear implicit ψ-Hilfer FBVP describing Navier model with NIBCs. By Banach’s fixed point theorem, the unique property is established. Meanwhile, existence results are proved by using the fixed point theory of Leray-Schauder’s and Krasnoselskii’s types. In addition, Ulam’s stability results are analyzed. Furthermore, several instances are provided to demonstrate the efficacy of the main results.
In this paper, we prove a Sehgal–Guseman-type fixed point theorem in b-rectangular metric spaces which provides a complete solution to an open problem raised by Zoran D. Mitrović (A note on a Banach’s fixed point theorem in b-rectangular metric space and b-metric space). The result presented in the paper generalizes and unifies some results in fixed point theory.
AbstractThe main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach’s fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019].
We study the existence and uniqueness of solutions for integro-differential equations involving two fractional orders. By using the Banach’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder’s degree theory, the existence and uniqueness of solutions are obtained. Some illustrative examples are also presented.
AbstractThe aim of this article is to discuss the uniqueness and Ulam–Hyers stability of solutions for a nonlinear fractional integro-differential equation involving a generalized Caputo fractional operator. The used fractional operator is generated by iterating a local integral of the form $(I_{a}^{\rho }f)(t)=\int _{a}^{t}f(s)s^{\rho -1}\,ds$
(
I
a
ρ
f
)
(
t
)
=
∫
a
t
f
(
s
)
s
ρ
−
1
d
s
. Our reported results are obtained in the Banach space of absolutely continuous functions that rely on Babenko’s technique and Banach’s fixed point theorem. Besides, our main findings are illustrated by some examples.
In this paper, we investigate the models of the impulsive cellular neural network with generalized constant piecewise delay (IDEGPCD). To guarantee the existence, uniqueness and global exponential stability of the equilibrium state, several new adequate conditions are obtained, which extend the results of the previous literature. The method is based on utilizing Banach’s fixed point theorem and a new IDEGPCD’s Gronwall inequality. The criteria given are easy to check and when the impulsive effects do not affect, the results can be extracted from those of the non-impulsive systems. Typical numerical simulation examples are used to show the validity and effectiveness of the proposed results. We end the paper with a brief conclusion.
In this paper, we investigate the models of the impulsive cellular
neural network with piecewise alternately advanced and retarded argument
of generalized argument (in short IDEPCAG). To ensure the existence,
uniqueness and global exponential stability of the equilibrium state,
several new sufficient conditions are obtained, which extend the results
of the previous literature. The method is based on utilizing Banach’s
fixed point theorem and a new IDEPCAG’s Gronwall inequality. The
criteria given are easy to check and when the impulsive effects do not
affect, the results can be extracted from those of the non-impulsive
systems. Typical numerical simulation examples are used to show the
validity and effectiveness of proposed results. We end the article with
a brief conclusion.
<p style='text-indent:20px;'>In this work we will discuss about an approximation method for initial value problems associated to fractional order differential equations. For this method we will use Bernstein spline approximation in combination with the Banach's Fixed Point Theorem. In order to illustrate our results, some numerical examples will be presented at the end of this article.</p>
In this paper, we introduce the concept of new type of
F
-contractive type for quasipartial b-metric spaces and some definitions and lemmas. Also, we will prove a new fixed-point theorem in quasipartial
b
-metric spaces for
F
-contractive type mappings. In addition, we give an application which illustrates a situation when Banach’s fixed-point theorem for complete quasipartial
b
-metric spaces cannot be applied, while the conditions of our theorem are satisfying.
A Study of ψ-Hilfer Fractional Boundary Value Problem via Nonlinear Integral Conditions Describing Navier Model
