Special Issue: Nonlinear Analysis and Its Applications in Symmetry II

Nonlinear analysis has been a rapidly growing area of research. In this Special Issue, we present ten papers authored by a select group of experts in the area of nonlinear analysis. These papers cover a wide spectrum of important problems and topics of current research interest. The descent derivative-free method for solving convex constrained equations is discussed in [1]. Unconstrained optimization methods based on symmetry are under consideration in [2]. In [3], the authors study the existence of positive solutions for perturbations of the anisotropic eigenvalue problem. In [4], the authors analyzed uniformly locally nonexpansive mappings. Nonlinear Fredholm equations in Lebesgue spaces are investigated in [5]. Coupled fixed point theory in subordinate semimetric spaces is developed in [6]. The work [7] is devoted to nonlinear biharmonic equations in an annulus. Stability analysis of singularly perturbed time-delay differential systems is presented in [8]. The variation of constants formula in Lebesgue spaces with variable exponents is discussed in [9]. Approximation using activated singular integrals is considered in [10].

In the following text, we comment on the main goals and results of these contributions.

In the first paper, Descent Derivative-Free Method Involving Symmetric Rank-One Update for Solving Convex Constrained Nonlinear Monotone Equations and Application to Image Recovery, the authors propose a new descent projection iterative algorithm for solving a nonlinear system of equations with convex constraints. Their approach is based on a modified symmetric rank-one updating formula. The search direction of the proposed algorithm mimics the behavior of a spectral conjugate gradient algorithm where the spectral parameter is determined so that the direction is sufficiently descent. Based on the assumption that the underlying function satisfies monotonicity and Lipschitz continuity, the convergence result of the proposed algorithm is discussed. Subsequently, the efficiency of the new method is revealed. As an application, the proposed algorithm is successfully implemented on the image deblurring problem. The problem considered in this paper was also studied in [11,12,13].

In the second paper, Improvement of Unconstrained Optimization Methods Based on Symmetry Involved in Neutrosophy, the authors apply neutrosophy in order to improve methods for solving unconstrained optimization. They propose and investigate an improvement of line search methods for solving unconstrained nonlinear optimization models. The improvement is based on the application of symmetry involved in neutrosophic logic in determining appropriate step size for the class of descent direction methods. Theoretical analysis is performed to show the convergence of proposed iterations under the same conditions as for the related standard iterations. Mutual comparison and analysis of generated numerical results reveal better behavior of the suggested iterations compared with analogous available iterations considering the Dolan and Moré performance profiles and statistical ranking. Analogous iterative schemes were also considered in [14,15].

In the third paper, Existence and Nonexistence of Positive Solutions for Perturbations of the Anisotropic Eigenvalue Problem, the authors consider a Dirichlet problem, which is a perturbation of the eigenvalue problem for the anisotropic p-Laplacian. They assume that the perturbation is (p(z)−1)-sublinear and prove an existence and nonexistence theorem for positive solutions as the parameter λ moves on the positive semiaxis. They also show the existence of a smallest positive solution and determine the monotonicity and continuity properties of the minimal solution map. Some related results can be found in [16,17,18].

In the fourth paper, Three Convergence Results for Inexact Iterates of Uniformly Locally Nonexpansive Mappings, the authors extend their result obtained in 2006, together with D. Butnariu, which shows that if all iterates of a nonexpansive self-mapping of a complete metric space converge, then all its inexact iterates with summable computational errors converge too. They establish analogous results for uniformly local nonexpansive mappings, which take a nonempty closed subset of a complete metric space into the space. For some related results, see [19,20,21,22,23].

In the fifth paper, Examining Nonlinear Fredholm Equations in Lebesgue Spaces with Variable Exponents, the authors investigate the existence of solutions for a certain Fredholm integral equation in the setting of the modular function spaces Lp and apply their results within the framework of variable exponent Lebesgue spaces Lp(⋅) subject to specific conditions imposed on the exponent function p(⋅). For some related results, see [18,24,25,26].

In the sixth paper, Coupled Fixed Point Theory in Subordinate Semimetric Space, the authors study the coupled fixed point of a class of mixed monotone operators in the setting of a subordinate semimetric space. Using the symmetry between the subordinate semimetric space and a JS-space, they generalize the results of Senapati and Dey on JS-spaces and obtain some coupled fixed point results that are supported by examples. For some related results, see [27,28,29].

In the seventh paper, Positive Radial Symmetric Solutions of Nonlinear Biharmonic Equations in an Annulus, the authors discuss the existence of positive radial symmetric solutions of the nonlinear biharmonic equation on an annular domain in a finite-dimensional space Navier boundary conditions. They present some inequality conditions to obtain the existence results of positive radial symmetric solutions. Their study is mainly based on the fixed-point index theory in cones. For some related results, see [30,31,32,33,34,35,36,37,38,39,40].

In the eighth paper, Stability Analysis of Some Types of Singularly Perturbed Time-Delay Differential Systems: Symmetric Matrix Riccati Equation Approach, the author considers several types of linear and nonlinear singularly perturbed time-delay differential systems. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are analyzed. For the stability analysis in the linear case, a partial exact slow–fast decomposition of the original system and an application of the symmetric matrix Riccati equation method are proposed. Such an analysis yields parameter-free conditions, providing the asymptotic stability of the considered linear singularly perturbed time-delay differential systems for any sufficiently small value of the parameter of singular perturbation. Using the asymptotic stability results for the considered linear systems and the method of asymptotic stability in the first approximation, parameter-free conditions guaranteeing the asymptotic stability of the trivial solution to the considered nonlinear systems for any sufficiently small value of the parameter of singular perturbation are derived. Illustrative examples are presented. Some related results can be found in [41,42,43,44,45].

In the ninth paper, The Variation of Constants Formula in Lebesgue Spaces with Variable Exponents, the author analyzes the variation of the constants formula within the context of modular function spaces. Additionally, his research explores practical applications of the variation of constants formula in variable exponent Lebesgue spaces Lp (⋅). Specifically, the study examines these spaces under certain conditions applied to the exponent function p(·) as well as the semigroup S(t), utilizing the symmetry properties of the algebraic semigroup. This investigation sheds light on the intricate interplay between parameters and functions within these mathematical frameworks, offering valuable insights into their behavior and properties in Lp (⋅). For some related results, see [46,47,48].

In the tenth paper, Degree of Lp Approximation Using Activated Singular Integrals, the author presents the Lp, p ≥ 1 approximation properties of activated singular integral operators over the real line. He establishes their approximation to the unit operator with rates. The kernels here come from neural network activation functions, and the author employs the related density functions. The derived inequalities use the high-order Lp modulus of smoothness. Some related results can be found in [49,50,51].