Preface to the Special Issue “Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities”

Over the past century, nonlinear analysis has been widely and significantly applied in many areas of mathematics, including nonlinear ordinary and partial differential equations, functional analysis, fixed point theory, nonlinear optimization, variational analysis, convex analysis, dynamical system theory, mathematical economics, signal processing, control theory, data mining, and more. Typical problems in analytic number theory involve enumeration problems related to prime numbers, Diophantine equations, and similar number-theoretic objects. These questions are of long-standing intrinsic interest, and the answers provided by analytic number theory are often used in applied mathematics. Applications include cryptography, asymptotic and error analysis, Fourier series and transforms, contour integrals and residues, and Laplacian spectral theory, among others. Abstract mathematical inequalities are considered as important tools in mathematical and scientific research. Classical inequalities, such as Jensen’s inequality, Hermite–Hadamard’s inequality, Hölder’s inequality, Minkowski’s inequality, Grüss’s inequality, and Chebyshev’s inequality, have been widely applied across various branches of mathematics, including functional analysis, optimization theory, numerical analysis, probability and statistics, and information theory.

This Special Issue focuses on novel and real-world applications of nonlinear analysis, analytic number theory, and mathematical inequalities. We cordially invited researchers to contribute their original, high-quality research papers that could promote advancements in these areas.

To ensure the highest quality for this Special Issue, the Guest Editors organized a comprehensive review process for each submission in accordance with the journal’s policies and guidelines. This Special Issue features 12 high-quality papers, selected from 41 submissions, resulting in an acceptance rate of approximately 29%. The published contributions are as follows:

i.

Zhong, Y.; Huang, H. Cash Flow Optimization on Insurance: An Application of Fixed-Point Theory. Mathematics 2023, 11, 902. https://doi.org/10.3390/math11040902;

ii.

Wang, Y.; Li, M.; Yao, C.; Jiang, B. Two New Modified Regularized Methods for Solving the Variational Inclusion and Null Point Problems. Mathematics 2023, 11, 1469. https://doi.org/10.3390/math11061469;

iii.

Huang, H.; Pal, S.; Bera, A.; Dey, L.K. On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings. Mathematics 2023, 11, 1852. https://doi.org/10.3390/math11081852;

iv.

Hwang, Y.-A.; Liao, Y.-H. Non-Emptiness, Relative Coincidences and Axiomatic Results for the Precore. Mathematics 2023, 11, 2812. https://doi.org/10.3390/math11132812;

v.

Fadel, M.; Raza, N.; Du, W.-S. Characterizing q-Bessel Functions of the First Kind with Their New Summation and Integral Representations. Mathematics 2023, 11, 3831. https://doi.org/10.3390/math11183831;

vi.

Zhong, Z.; Zhang, G.; Yin, L.; Chen, Y. Description and Analysis of Data Security Based on Differential Privacy in Enterprise Power Systems. Mathematics 2023, 11, 4829. https://doi.org/10.3390/math11234829;

vii.

Qi, F.; Agarwal, R.P. Several Functions Originating from Fisher–Rao Geometry of Dirichlet Distributions and Involving Polygamma Functions. Mathematics 2024, 12, 44. https://doi.org/10.3390/math12010044;

viii.

Zhang, T.; Liu, J. Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight. Mathematics 2024, 12, 785. https://doi.org/10.3390/math12050785;

ix.

Filali, D.; Dilshad, M.; Akram, M. Weak $\psi$-Contractions on Directed Graphs with Applications to Integral Equations. Mathematics 2024, 12, 2675. https://doi.org/10.3390/math12172675;

x.

Eljaneid, N.H.E.; Alshaban, E.; Alatawi, A.; Ali, M.S.; Alsharari, S.S.; Khan, F.A. Generalized Weak Contractions Involving a Pair of Auxiliary Functions via Locally Transitive Binary Relations and Applications to Boundary Value Problems. Mathematics 2025, 13, 163. https://doi.org/10.3390/math13010163;

xi.

Okeke, G.A.; Udo, A.V.; Alqahtani, R.T. Novel Method for Approximating Fixed Point of Generalized $\alpha$-Nonexpansive Mappings with Applications to Dynamics of a HIV Model. Mathematics 2025, 13, 550. https://doi.org/10.3390/math13040550;

xii.

Mahmoud, M.; Alofi, A.S.; Zurayyir, M.A. New Approximation Formula of Digamma Function with Bounded Remainder. Mathematics 2025, 13, 720. https://doi.org/10.3390/math13050720.

These contributions can be categorized based on their main objectives, as follows:

• Nonlinear functional analysis (see i, ii, iii, ix, x, xi);
• Fixed point theory and applications (see iii, ix, x, xi);
• Convex analysis and optimization (see i, ii, iv, vi, v, vii, viii);
• Analytic number theory (see vii);
• Mathematical inequalities and applications (see v, vii, viii, xii);
• Mathematical means and applications (see v, vii, viii);
• Theory and applications for special functions (see v, vii, viii,xii).

We hope that interested researchers and practitioners will read these papers published in this Special Issue and find valuable inspiration for future work in these exciting areas.

In conclusion, this Special Issue has undoubtedly achieved its original goals. Notably, several papers have been recognized repeatedly as ESI Highly Cited Papers. Clearly, this Special Issue has shed new light on important topics and raised new challenges for future research. Finally, we would like to express our sincere gratitude to the editorial team and the reviewers of Mathematics, especially the Editors-in-Chief, for their invaluable support throughout the editing process.