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4 pages, 207 KiB

Open AccessEditorial

Editorial: Overview and Some New Directions

by Shengda Zeng, Stanislaw Migórski and Yongjian Liu

Axioms 2023, 12(6), 553; https://doi.org/10.3390/axioms12060553 - 4 Jun 2023

Viewed by 1109

Abstract

The Special Issue contains eleven accepted and published submissions to a Special Issue of the MDPI journal Axioms on the subject of “Nonlinear Dynamical Systems with Applications” [...] Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

Research

Jump to: Editorial

15 pages, 328 KiB

Open AccessArticle

A Class of Janowski-Type (p,q)-Convex Harmonic Functions Involving a Generalized q-Mittag–Leffler Function

by Sarem H. Hadi, Maslina Darus and Alina Alb Lupaş

Axioms 2023, 12(2), 190; https://doi.org/10.3390/axioms12020190 - 11 Feb 2023

Cited by 6 | Viewed by 1595

Abstract

This research aims to present a linear operator

L_{p, q}^{ρ, σ, μ} f

utilizing the q-Mittag–Leffler function. Then, we introduce the subclass of harmonic

(p, q)

-convex functions [...] Read more.

This research aims to present a linear operator

L_{p, q}^{ρ, σ, μ} f

utilizing the q-Mittag–Leffler function. Then, we introduce the subclass of harmonic

(p, q)

-convex functions

{HT}_{p, q} (ϑ, W, V)

related to the Janowski function. For the harmonic p-valent functions f class, we investigate the harmonic geometric properties, such as coefficient estimates, convex linear combination, extreme points, and Hadamard product. Finally, the closure property is derived using the subclass

{HT}_{p, q} (ϑ, W, V)

under the q-Bernardi integral operator. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

13 pages, 293 KiB

Open AccessArticle

Levinson’s Functional in Time Scale Settings

by Josipa Barić

Axioms 2023, 12(2), 141; https://doi.org/10.3390/axioms12020141 - 30 Jan 2023

Cited by 1 | Viewed by 1208

Abstract

We introduce the Levinson functional on time scales using integral inequality of Levinson’s type in the terms of

Δ

-integral for convex (concave) functions on time scale sets and investigate its properties such as superadditivity and monotonicity. The obtained properties are used to [...] Read more.

We introduce the Levinson functional on time scales using integral inequality of Levinson’s type in the terms of

Δ

-integral for convex (concave) functions on time scale sets and investigate its properties such as superadditivity and monotonicity. The obtained properties are used to derive the bounds of the given Levinson’s functional and those results provide a refinement and the converse of the known Levinson’s inequality on time scales. Further, we define new types of functionals using weighted generalized and power means on time scales, and prove their properties which can be employed in future works to obtain refinements and converses of known integral inequalities on time scales. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

19 pages, 395 KiB

Open AccessArticle

A Study on Fixed-Point Techniques under the α-ϝ-Convex Contraction with an Application

by Gunasekaran Nallaselli, Arul Joseph Gnanaprakasam, Gunaseelan Mani, Ozgur Ege, Dania Santina and Nabil Mlaiki

Axioms 2023, 12(2), 139; https://doi.org/10.3390/axioms12020139 - 29 Jan 2023

Cited by 2 | Viewed by 1468

Abstract

In this paper, we consider several classes of mappings related to the class of

α

-

ϝ

-contraction mappings by introducing a convexity condition and establish some fixed-point theorems for such mappings in complete metric spaces. The present result extends and generalizes the [...] Read more.

In this paper, we consider several classes of mappings related to the class of

α

-

ϝ

-contraction mappings by introducing a convexity condition and establish some fixed-point theorems for such mappings in complete metric spaces. The present result extends and generalizes the well-known results of

α

-admissible and convex contraction mapping and many others in the existing literature. An illustrative example is also provided to exhibit the utility of our main results. Finally, we derive the existence and uniqueness of a solution to an integral equation to support our main result and give a numerical example to validate the application of our obtained results. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

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13 pages, 1354 KiB

Open AccessArticle

The Design of State-Dependent Switching Rules for Second-Order Switched Linear Systems Revisited

by Xiaolan Yuan and Yusheng Zhou

Axioms 2022, 11(10), 566; https://doi.org/10.3390/axioms11100566 - 19 Oct 2022

Cited by 2 | Viewed by 1696

Abstract

This paper focuses on the asymptotic stability of second-order switched linear systems with positive real part conjugate complex roots for each subsystem. Compared with available studies, a more appropriate state-dependent switching rule is designed to stabilize a switched system with the phase trajectories [...] Read more.

This paper focuses on the asymptotic stability of second-order switched linear systems with positive real part conjugate complex roots for each subsystem. Compared with available studies, a more appropriate state-dependent switching rule is designed to stabilize a switched system with the phase trajectories of two subsystems rotating outward in the same direction or the opposite direction. Finally, several numerical examples are used to illustrate the effectiveness and superiority of the proposed method. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

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16 pages, 327 KiB

Open AccessArticle

Stability for a Class of Differential Set-Valued Inverse Variational Inequalities in Finite Dimensional Spaces

by Xinyue Zhu, Wei Li and Xueping Luo

Axioms 2022, 11(9), 475; https://doi.org/10.3390/axioms11090475 - 16 Sep 2022

Cited by 2 | Viewed by 1687

Abstract

In this paper, we introduce and study a new class of differential set-valued inverse variational inequalities in finite dimensional spaces. By applying a result on differential inclusions involving an upper semicontinuous set-valued mapping with closed convex values, we first prove the existence of [...] Read more.

In this paper, we introduce and study a new class of differential set-valued inverse variational inequalities in finite dimensional spaces. By applying a result on differential inclusions involving an upper semicontinuous set-valued mapping with closed convex values, we first prove the existence of Carathéodory weak solutions for differential set-valued inverse variational inequalities. Then, by the existence result, we establish the stability for the differential set-valued inverse variational inequality problem when the constraint set and the mapping are perturbed by two different parameters. The closedness and continuity of Carathéodory weak solutions with respect to the two different parameters are obtained. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

10 pages, 294 KiB

Open AccessArticle

Stability of Fractional-Order Quasi-Linear Impulsive Integro-Differential Systems with Multiple Delays

by Mathiyalagan Kalidass, Shengda Zeng and Mehmet Yavuz

Axioms 2022, 11(7), 308; https://doi.org/10.3390/axioms11070308 - 25 Jun 2022

Cited by 15 | Viewed by 2083

Abstract

In this paper, some novel conditions for the stability results for a class of fractional-order quasi-linear impulsive integro-differential systems with multiple delays is discussed. First, the existence and uniqueness of mild solutions for the considered system is discussed using contraction mapping theorem. Then, [...] Read more.

In this paper, some novel conditions for the stability results for a class of fractional-order quasi-linear impulsive integro-differential systems with multiple delays is discussed. First, the existence and uniqueness of mild solutions for the considered system is discussed using contraction mapping theorem. Then, novel conditions for Mittag–Leffler stability (MLS) of the considered system are established by using well known mathematical techniques, and further, the two corollaries are deduced, which still gives some new results. Finally, an example is given to illustrate the applications of the results. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

13 pages, 301 KiB

Open AccessArticle

Nonlinear Eigenvalue Problems for the Dirichlet (p,2)-Laplacian

by Yunru Bai, Leszek Gasiński and Nikolaos S. Papageorgiou

Axioms 2022, 11(2), 58; https://doi.org/10.3390/axioms11020058 - 30 Jan 2022

Cited by 2 | Viewed by 1957

Abstract

We consider a nonlinear eigenvalue problem driven by the Dirichlet

(p, 2)

-Laplacian. The parametric reaction is a Carathéodory function which exhibits

(p - 1)

-sublinear growth as

x \to + \infty

and as

x \to 0^{+}

[...] Read more.

We consider a nonlinear eigenvalue problem driven by the Dirichlet

(p, 2)

-Laplacian. The parametric reaction is a Carathéodory function which exhibits

(p - 1)

-sublinear growth as

x \to + \infty

and as

x \to 0^{+}

. Using variational tools and truncation and comparison techniques, we prove a bifurcation-type theorem describing the “spectrum” as

λ > 0

varies. We also prove the existence of a smallest positive eigenfunction for every eigenvalue. Finally, we indicate how the result can be extended to

(p, q)

-equations (

q \neq 2

). Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

15 pages, 1356 KiB

Open AccessArticle

Hopf Bifurcation Analysis of a Diffusive Nutrient–Phytoplankton Model with Time Delay

by Ruizhi Yang, Liye Wang and Dan Jin

Axioms 2022, 11(2), 56; https://doi.org/10.3390/axioms11020056 - 29 Jan 2022

Cited by 4 | Viewed by 2445

Abstract

In this paper, we studied a nutrient–phytoplankton model with time delay and diffusion term. We studied the Turing instability, local stability, and the existence of Hopf bifurcation. Some formulas are obtained to determine the direction of the bifurcation and the stability of periodic [...] Read more.

In this paper, we studied a nutrient–phytoplankton model with time delay and diffusion term. We studied the Turing instability, local stability, and the existence of Hopf bifurcation. Some formulas are obtained to determine the direction of the bifurcation and the stability of periodic solutions by the central manifold theory and normal form method. Finally, we verify the above conclusion through numerical simulation. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

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14 pages, 416 KiB

Open AccessArticle

The Darboux Transformation and N-Soliton Solutions of Gerdjikov–Ivanov Equation on a Time–Space Scale

by Huanhe Dong, Xiaoqian Huang, Yong Zhang, Mingshuo Liu and Yong Fang

Axioms 2021, 10(4), 294; https://doi.org/10.3390/axioms10040294 - 5 Nov 2021

Cited by 4 | Viewed by 2003

Abstract

The Gerdjikov–Ivanov (GI) equation is one type of derivative nonlinear Schrödinger equation used widely in quantum field theory, nonlinear optics, weakly nonlinear dispersion water waves and other fields. In this paper, the coupled GI equation on a time–space scale is deduced from Lax [...] Read more.

The Gerdjikov–Ivanov (GI) equation is one type of derivative nonlinear Schrödinger equation used widely in quantum field theory, nonlinear optics, weakly nonlinear dispersion water waves and other fields. In this paper, the coupled GI equation on a time–space scale is deduced from Lax pairs and the zero curvature equation on a time–space scale, which can be reduced to the classical and the semi-discrete GI equation by considering different time–space scales. Furthermore, the Darboux transformation (DT) of the GI equation on a time–space scale is constructed via a gauge transformation. Finally, N-soliton solutions of the GI equation are given through applying its DT, which are expressed by the Cayley exponential function. At the same time, one-solition solutions are obtained on three different time–space scales (

X

=

R

,

X

=

C

and

X

=

K p

). Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

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14 pages, 328 KiB

Open AccessArticle

Degenerated and Competing Dirichlet Problems with Weights and Convection

by Dumitru Motreanu

Axioms 2021, 10(4), 271; https://doi.org/10.3390/axioms10040271 - 22 Oct 2021

Cited by 9 | Viewed by 1640

Abstract

This paper focuses on two Dirichlet boundary value problems whose differential operators in the principal part exhibit a lack of ellipticity and contain a convection term (depending on the solution and its gradient). They are driven by a degenerated [...] Read more.

This paper focuses on two Dirichlet boundary value problems whose differential operators in the principal part exhibit a lack of ellipticity and contain a convection term (depending on the solution and its gradient). They are driven by a degenerated

(p, q)

-Laplacian with weights and a competing

(p, q)

-Laplacian with weights, respectively. The notion of competing

(p, q)

-Laplacians with weights is considered for the first time. We present existence and approximation results that hold under the same set of hypotheses on the convection term for both problems. The proofs are based on weighted Sobolev spaces, Nemytskij operators, a fixed point argument and finite dimensional approximation. A detailed example illustrates the effective applicability of our results. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

17 pages, 343 KiB

Open AccessArticle

Global Directed Dynamic Behaviors of a Lotka-Volterra Competition-Diffusion-Advection System

by Lili Chen, Shilei Lin and Yanfeng Zhao

Axioms 2021, 10(3), 195; https://doi.org/10.3390/axioms10030195 - 20 Aug 2021

Cited by 2 | Viewed by 1721

Abstract

This paper investigates the problem of the global directed dynamic behaviors of a Lotka-Volterra competition-diffusion-advection system between two organisms in heterogeneous environments. The two organisms not only compete for different basic resources, but also the advection and diffusion strategies follow the dispersal towards [...] Read more.

This paper investigates the problem of the global directed dynamic behaviors of a Lotka-Volterra competition-diffusion-advection system between two organisms in heterogeneous environments. The two organisms not only compete for different basic resources, but also the advection and diffusion strategies follow the dispersal towards a positive distribution. By virtue of the principal eigenvalue theory, the linear stability of the co-existing steady state is established. Furthermore, the classification of dynamical behaviors is shown by utilizing the monotone dynamical system theory. This work can be seen as a further development of a competition-diffusion system. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

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