Research

24 pages, 358 KiB

Open AccessFeature PaperArticle

Existence of Solutions to a System of Fractional q-Difference Boundary Value Problems

by Alexandru Tudorache and Rodica Luca

Mathematics 2024, 12(9), 1335; https://doi.org/10.3390/math12091335 (registering DOI) - 27 Apr 2024

Viewed by 150

We are investigating the existence of solutions to a system of two fractional

q

-difference equations containing fractional

q

-integral terms, subject to multi-point boundary conditions that encompass

q

-derivatives and fractional

q

-derivatives of different orders. In our main results, we rely [...] Read more.

We are investigating the existence of solutions to a system of two fractional

q

-difference equations containing fractional

q

-integral terms, subject to multi-point boundary conditions that encompass

q

-derivatives and fractional

q

-derivatives of different orders. In our main results, we rely on various fixed point theorems, such as the Leray–Schauder nonlinear alternative, the Schaefer fixed point theorem, the Krasnosel’skii fixed point theorem for the sum of two operators, and the Banach contraction mapping principle. Finally, several examples are provided to illustrate our findings. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

17 pages, 946 KiB

Open AccessFeature PaperArticle

Bifurcation Analysis for an OSN Model with Two Delays

by Liancheng Wang and Min Wang

Mathematics 2024, 12(9), 1321; https://doi.org/10.3390/math12091321 - 26 Apr 2024

Viewed by 231

Abstract

In this research, we introduce and analyze a mathematical model for online social networks, incorporating two distinct delays. These delays represent the time it takes for active users within the network to begin disengaging, either with or without contacting non-users of online social [...] Read more.

In this research, we introduce and analyze a mathematical model for online social networks, incorporating two distinct delays. These delays represent the time it takes for active users within the network to begin disengaging, either with or without contacting non-users of online social platforms. We focus particularly on the user prevailing equilibrium (UPE), denoted as

P^{*}

, and explore the role of delays as parameters in triggering Hopf bifurcations. In doing so, we find the conditions under which Hopf bifurcations occur, then establish stable regions based on the two delays. Furthermore, we delineate the boundaries of stability regions wherein bifurcations transpire as the delays cross these thresholds. We present numerical simulations to illustrate and validate our theoretical findings. Through this interdisciplinary approach, we aim to deepen our understanding of the dynamics inherent in online social networks. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

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9 pages, 235 KiB

Open AccessArticle

The Blow-Up of the Local Energy Solution to the Wave Equation with a Nontrivial Boundary Condition

by Yulong Liu

Mathematics 2024, 12(9), 1317; https://doi.org/10.3390/math12091317 - 25 Apr 2024

Viewed by 224

Abstract

In this study, we examine the wave equation with a nontrivial boundary condition. The main target of this study is to prove the local-in-time existence and the blow-up in finite time of the energy solution. Through the construction of an auxiliary function and [...] Read more.

In this study, we examine the wave equation with a nontrivial boundary condition. The main target of this study is to prove the local-in-time existence and the blow-up in finite time of the energy solution. Through the construction of an auxiliary function and the imposition of appropriate conditions on the initial data, we establish the both lower and upper bounds for the blow-up time of the solution. Meanwhile, based on these estimates, we obtain the result of the local-in-time existence and the blow-up of the energy solution. This approach enhances our understanding of the dynamics leading to blow-up in the considered condition. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

20 pages, 316 KiB

Open AccessFeature PaperArticle

A Signed Maximum Principle for Boundary Value Problems for Riemann–Liouville Fractional Differential Equations with Analogues of Neumann or Periodic Boundary Conditions

by Paul W. Eloe, Yulong Li and Jeffrey T. Neugebauer

Mathematics 2024, 12(7), 1000; https://doi.org/10.3390/math12071000 - 27 Mar 2024

Viewed by 447

Abstract

Sufficient conditions are obtained for a signed maximum principle for boundary value problems for Riemann–Liouville fractional differential equations with analogues of Neumann or periodic boundary conditions in neighborhoods of simple eigenvalues. The primary objective is to exhibit four specific boundary value problems for [...] Read more.

Sufficient conditions are obtained for a signed maximum principle for boundary value problems for Riemann–Liouville fractional differential equations with analogues of Neumann or periodic boundary conditions in neighborhoods of simple eigenvalues. The primary objective is to exhibit four specific boundary value problems for which the sufficient conditions can be verified. To show an application of the signed maximum principle, a method of upper and lower solutions coupled with monotone methods is developed to obtain sufficient conditions for the existence of a maximal solution and a minimal solution of a nonlinear boundary value problem. A specific example is provided to show that sufficient conditions for the nonlinear problem can be realized. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

14 pages, 304 KiB

Open AccessFeature PaperArticle

Periodic Solutions to Nonlinear Second-Order Difference Equations with Two-Dimensional Kernel

by Daniel Maroncelli

Mathematics 2024, 12(6), 849; https://doi.org/10.3390/math12060849 - 14 Mar 2024

Viewed by 514

Abstract

In this work, we provide conditions for the existence of periodic solutions to nonlinear, second-order difference equations of the form

y (t + 2) + b y (t + 1) + c y (t) = g (y (t))

, where b and c are real parameters,

c \neq 0

, and

g : R \to R

is continuous. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

28 pages, 398 KiB

Open AccessArticle

Chains with Connections of Diffusion and Advective Types

by Sergey Kashchenko

Mathematics 2024, 12(6), 790; https://doi.org/10.3390/math12060790 - 07 Mar 2024

Viewed by 448

Abstract

The local dynamics of a system of oscillators with a large number of elements and with diffusive- and advective-type couplings containing a large delay are studied. Critical cases in the problem of the stability of the zero equilibrium state are singled out, and [...] Read more.

The local dynamics of a system of oscillators with a large number of elements and with diffusive- and advective-type couplings containing a large delay are studied. Critical cases in the problem of the stability of the zero equilibrium state are singled out, and it is shown that all of them have infinite dimensions. Applying special methods of infinite normalization, we construct quasinormal forms, namely, nonlinear boundary value problems of the parabolic type, whose nonlocal dynamics determine the behavior of the solutions of the initial system in a small neighborhood of the equilibrium state. These quasinormal forms contain either two or three spatial variables, which emphasizes the complexity of the dynamical properties of the original problem. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

14 pages, 300 KiB

Open AccessArticle

Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional

by Xincai Zhu and Hanxiao Wu

Mathematics 2024, 12(5), 661; https://doi.org/10.3390/math12050661 - 23 Feb 2024

Viewed by 404

Abstract

In this paper, we study the constrained minimization problem for an energy functional which is related to a Kirchhoff-type equation. For

s = 1

, there many articles have analyzed the limit behavior of minimizers when

η > 0

as [...] Read more.

In this paper, we study the constrained minimization problem for an energy functional which is related to a Kirchhoff-type equation. For

s = 1

, there many articles have analyzed the limit behavior of minimizers when

η > 0

as

b \to 0^{+}

or

b > 0

as

η \to 0^{+}

. When the equation involves a varying non-local term

{(\int_{R^{3}} {| \nabla u |}^{2} d x)}^{s}

, we give a detailed limit behavior analysis of constrained minimizers for any positive sequence

{η_{k}}

with

η_{k} \to 0^{+}

. The present paper obtains an interesting result on this topic and enriches the conclusions of previous works. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

11 pages, 235 KiB

Open AccessArticle

Multivalued Contraction Fixed-Point Theorem in b-Metric Spaces

by Bachir Slimani, John R. Graef and Abdelghani Ouahab

Mathematics 2024, 12(4), 567; https://doi.org/10.3390/math12040567 - 13 Feb 2024

Viewed by 574

Abstract

The authors explore fixed-point theory in b-metric spaces and strong b-metric spaces. They wish to prove some new extensions of the Covitz and Nadler fixed-point theorem in b-metric spaces. In so doing, they wish to answer a question proposed by [...] Read more.

The authors explore fixed-point theory in b-metric spaces and strong b-metric spaces. They wish to prove some new extensions of the Covitz and Nadler fixed-point theorem in b-metric spaces. In so doing, they wish to answer a question proposed by Kirk and Shahzad about Nadler’s theorem holding in strong b-metric spaces. In addition, they offer an improvement to the fixed-point theorem proven by Dontchev and Hager. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

35 pages, 444 KiB

Open AccessArticle

Multiplicity Results of Solutions to the Double Phase Problems of Schrödinger–Kirchhoff Type with Concave–Convex Nonlinearities

by Yun-Ho Kim and Taek-Jun Jeong

Mathematics 2024, 12(1), 60; https://doi.org/10.3390/math12010060 - 24 Dec 2023

Cited by 1 | Viewed by 662

Abstract

The present paper is devoted to establishing several existence results for infinitely many solutions to Schrödinger–Kirchhoff-type double phase problems with concave–convex nonlinearities. The first aim is to demonstrate the existence of a sequence of infinitely many large-energy solutions by applying the fountain theorem [...] Read more.

The present paper is devoted to establishing several existence results for infinitely many solutions to Schrödinger–Kirchhoff-type double phase problems with concave–convex nonlinearities. The first aim is to demonstrate the existence of a sequence of infinitely many large-energy solutions by applying the fountain theorem as the main tool. The second aim is to obtain that our problem admits a sequence of infinitely many small-energy solutions. To obtain these results, we utilize the dual fountain theorem. In addition, we prove the existence of a sequence of infinitely many weak solutions converging to 0 in

L^{\infty}

-space. To derive this result, we exploit the dual fountain theorem and the modified functional method. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

28 pages, 374 KiB

Open AccessArticle

Global Existence, Blowup, and Asymptotic Behavior for a Kirchhoff-Type Parabolic Problem Involving the Fractional Laplacian with Logarithmic Term

by Zihao Guan and Ning Pan

Mathematics 2024, 12(1), 5; https://doi.org/10.3390/math12010005 - 19 Dec 2023

Viewed by 574

Abstract

In this paper, we studied a class of semilinear pseudo-parabolic equations of the Kirchhoff type involving the fractional Laplacian with logarithmic nonlinearity: [...] Read more.

In this paper, we studied a class of semilinear pseudo-parabolic equations of the Kirchhoff type involving the fractional Laplacian with logarithmic nonlinearity:

\{\begin{matrix} u_{t} + M ({[u]}_{s}^{2}) {(- Δ)}^{s} u + {(- Δ)}^{s} u_{t} = {| u |}^{p - 2} u \ln | u |, & in Ω \times (0, T), \\ u (x, 0) = u_{0} (x), & in Ω, \\ u (x, t) = 0, & on \partial Ω \times (0, T), \end{matrix}

, where

{[u]}_{s}

is the Gagliardo semi-norm of u,

{(- Δ)}^{s}

is the fractional Laplacian,

s \in (0, 1)

,

2 λ < p < 2_{s}^{*} = 2 N / (N - 2 s)

,

Ω \in R^{N}

is a bounded domain with

N > 2 s

, and

u_{0}

is the initial function. To start with, we combined the potential well theory and Galerkin method to prove the existence of global solutions. Finally, we introduced the concavity method and some special inequalities to discuss the blowup and asymptotic properties of the above problem and obtained the upper and lower bounds on the blowup at the sublevel and initial level. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

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