Editorial for the Special Issue “Advances in Differential and Difference Equations and Their Applications”

This Special Issue of Mathematics titled “Advances in Differential and Difference Equations and Their Applications” presents a collection of articles that highlight the significant progress in the study of differential and difference equations. These contributions cover a wide range of topics, from boundary value problems and the asymptotic behavior of solutions to complex mathematical models and the application of fractional and difference equations in various scientific and engineering fields.

Differential equations are a core mathematical tool for modeling dynamic systems in a diverse range of areas, such as physics, biology, economics, and engineering. Likewise, difference equations provide discrete counterparts to continuous models, which are crucial in understanding phenomena occurring in digital systems and various computational models. The papers presented in this issue reflect the latest theoretical advancements and applications in these important research areas.

• Radially Symmetric Positive Solutions of the Dirichlet Problem for the p-Laplace Equation, by Bo Yang, presents new lower estimates for positive solutions of the p-Laplace boundary value problem, offering significant insights into the existence and nonexistence conditions for such solutions (https://doi.org/10.3390/math12152351).
• Minimum Principles for Sturm–Liouville Inequalities and Applications, by Phuc Ngo and Kunquan Lan, introduces a minimum principle for Sturm–Liouville inequalities, providing a framework to understand solution behavior in boundary-value contexts (https://doi.org/10.3390/math12132088).
• Global Existence of Small Data Solutions to Weakly Coupled Systems of Semi-Linear Fractional $\sigma$-Evolution Equations, by Seyyid Ali Saiah et al., explores the existence of a long-term solution in fractional evolution equations, analyzing memory terms and initial conditions (https://doi.org/10.3390/math12131942).
• Stability and Bifurcation Analysis in a Discrete Predator–Prey System, by Luyao Lv and Xianyi Li, investigates a Leslie-type predator–prey model, focusing on the influence of functional responses on stability, bifurcation, and dynamics (https://doi.org/10.3390/math12121803).
• Differentiation of Solutions of Caputo Boundary Value Problems, by Jeffrey W. Lyons, generalizes differentiation techniques for fractional boundary value problems, expanding upon classical results (https://doi.org/10.3390/math12121790).
• Kamenev-Type Criteria for Testing Asymptotic Behavior of Solutions, by Hail S. Alrashdi et al., examines third-order quasi-linear neutral differential equations, offering new tools for studying the asymptotic behavior of solutions (https://doi.org/10.3390/math12111734).
• Existence of Solutions to a System of Fractional q-Difference Boundary Value Problems, by Alexandru Tudorache and Rodica Luca, analyzes multi-point boundary conditions in q-difference systems, demonstrating their existence (https://doi.org/10.3390/math12091335).
• Bifurcation Analysis for an OSN Model with Two Delays, by Liancheng Wang and Min Wang, delves into the dynamics of online social networks, focusing on delays representing user activity transitions (https://doi.org/10.3390/math12091321).
• The Blow-Up of the Local Energy Solution to the Wave Equation, by Yulong Liu, studies finite-time blow-up and local existence for wave equations with nontrivial boundary conditions (https://doi.org/10.3390/math12091317).
• A Signed Maximum Principle for Riemann–Liouville Fractional Differential Equations, by Paul W. Eloe et al., derives conditions for maximum principles in fractional differential equations with periodic boundary conditions (https://doi.org/10.3390/math12071000).
• Periodic Solutions to Nonlinear Second-Order Difference Equations, by Daniel Maroncelli, offers conditions for periodic solutions in nonlinear difference equations, highlighting their computational and theoretical implications (https://doi.org/10.3390/math12060849).
• Chains with Connections of Diffusion and Advective Types, by Sergey Kashchenko, investigates oscillator chains, emphasizing their stability under diffusive and advective couplings (https://doi.org/10.3390/math12060790).
• Existence and Limit Behavior of Constraint Minimizers for a Non-Local Kirchhoff-Type Energy Functional, by Xincai Zhu and Hanxiao Wu, studies minimization problems in energy functionals, connecting them to Kirchhoff-type equations (https://doi.org/10.3390/math12050661).
• Multivalued Contraction Fixed-Point Theorem in b-Metric Spaces, by Bachir Slimani et al., extends fixed-point theorems in b-metric spaces, contributing to fixed-point theory (https://doi.org/10.3390/math12040567).
• Multiplicity Results of Solutions to Schrödinger–Kirchhoff-Type Double Phase Problems, by Yun-Ho Kim and Taek-Jun Jeong, establishes solutions for double phase problems with concave–convex nonlinearities using advanced theorems (https://doi.org/10.3390/math12010060).
• Global Existence, Blowup, and Asymptotic Behavior for a Kirchhoff-Type Parabolic Problem, by Zihao Guan and Ning Pan, investigates pseudo-parabolic equations with fractional Laplacians and logarithmic nonlinearities, analyzing the dynamics of the solution (https://doi.org/10.3390/math12010005).

We extend our gratitude to all of the authors who contributed their original work to this Special Issue, and to the reviewers whose critical evaluations ensured the high quality of this collection. Finally, we thank the Editorial team of Mathematics for their professional support and for providing a platform to publish this important body of work.

We hope that this Special Issue serves as a valuable resource for researchers, sparks new ideas, and strengthens the connections within the global mathematical community.