Field Of Nonlinear Analysis Research Articles

Existence of Solutions for a Perturbed N-Laplacian Boundary Value Problem with Critical Growth

In this paper, we investigate a perturbed elliptic boundary value problem that exhibits critical growth characterized by a Trudinger–Moser-type inequality. Our primary focus is to establish the existence of two nontrivial solutions. This is achieved by employing a combination of the Trudinger–Moser-type inequality and a linking theorem based on the Z2-cohomological index. The main feature and novelty of this paper lies in extending the equation to N-Laplacian boundary value problems utilizing the aforementioned methods. This extension not only broadens the applicability of these techniques but also enriches the research outcomes in the field of nonlinear analysis.

Certain New Development to the Orthogonal Binary Relations

In this study, inspired by the concept of B-metric-like space (BMLS), we introduce the concept of orthogonal B-metric-like space (OBMLS) via a hybrid pair of operators. Additionally, we establish the concept of orthogonal dynamic system (ODS) as a generalization of the dynamic system (DS), which improves the existing results for analysies such as those presented here. By applying this, some new refinements of the F⊥-Suzuki-type (F⊥-ST) fixed-point results are presented. These include some tangible instances, and applications in the field of nonlinear analysis are given to highlight the usability and validity of the theoretical results.

A General Picard-Mann Iterative Method for Approximating Fixed Points of Nonexpansive Mappings with Applications

Fixed point theory provides an important structure for the study of symmetry in mathematics. In this article, a new iterative method (general Picard–Mann) to approximate fixed points of nonexpansive mappings is introduced and studied. We study the stability of this newly established method which we find to be summably almost stable for contractive mappings. A number of weak and strong convergence theorems of such iterative methods are established in the setting of Banach spaces under certain geometrical assumptions. Finally, we present a number of applications to address various important problems (zero of an accretive operator, mixed equilibrium problem, convex optimization problem, split feasibility problem, periodic solution of a nonlinear evolution equation) appearing in the field of nonlinear analysis.

Tarig Projected Differential Transform Method to solve fractional nonlinear partial differential equations

Recent advancement in the field of nonlinear analysis and fractional calculus help to address the rising challenges in the solution of nonlinear fractional partial differential equations. This paper presents a hybrid technique, a combination of Tarig transform and Projected Differential Transform Method (TPDTM) to solve nonlinear fractional partial differential equations. The effectiveness of the method is examined by solving three numerical examples that arise in the field of heat transfer analysis. In this proposed scheme, the solution is obtained as a convergent series and the result is used to analyze the hyper diffusive process with pre local information regarding the heat transfer for different values of fractional order. In order to validate the results, a comparative study has been carried out with the solution obtained from the two methods, the Laplace Adomian Decomposition Method (LADM) and Homotophy Pertubation Method (HPM) and the result thus observed coincided with each other. Inspite of the uniformity between the solutions, the proposed hybrid technique had to overcome the complexity of manupulation of Adomian polynomials and evaluation of integrals in LADM and HPM respectively. The methodology and the results presented in this paper clearly reveals the computational efficiency of the present method. Due to its computational efficiency, the TPDTM has the potential to be used as a novel tool not only for solving nonlinear fractional differential equations but also for analysing the prelocal information of the system.

Dynamic structural parameters verification on the example of theoretical analysis and in situ tests of suspension footbridge

The 21st century is a period of rapid development of computer technology, which allowed designers to create complex, three-dimensional models of engineering structures. Thanks to these solutions, it is possible to perform complex analyses, for example modal or dynamic ones of cable-stayed or suspension structures. For such objects, verification of the correct work of structural elements takes place in the field of non-linear analysis. The presented paper is an example of a comparative analysis concerning modal and static analysis - Natural Frequency with Nonlinear Material Models and Static Stress with Nonlinear Material Models, carried out in the Autodesk Simulation Multiphysics program with dynamic in situ tests of a suspension footbridge. The main purpose of the research was to evaluate the value of pre-tension forces in the cables of the load-bearing structure.

A glance into the beauty of Fixed Point Theory: Professor Emeritus Ioan A. Rus on his 80th anniversary

Professor Emeritus Ioan A. Rus, a prominent and unfatiguable mathematician with an impressive oeuvre in the field of Nonlinear Analysis, particularly in Fixed Point Theory, will be 80 on 28th of August 2016. On this occasion, we would like to congratulate him and wish him good health, happiness, the same impressive physical and mental vigour, as well as new bright results in mathematics. We also take this opportunity to present some information on the visibility and impact of Professor Rus’ main scientific contributions, which would thus complement the information given in the following anniversary articles [Berinde, V. and Petrus¸el, A., Professor Ioan A. Rus on his 70th birthday: a complete scientist, an accomplished mathematician, Fixed Point Theory 7 (2006), No. 2, 167–174; Berinde, V. and Pacurar, M., The joy of doing mathematics. Interview with Professor Ioan A. Rus, Eur. Math. Soc. Newsl., 76 (2010), 47–50; Petrus¸el, A., Professor Ioan A. Rus on his 75th birthday, Fixed point theory and its applications, 33–38, Casa Cartii de Stiinta, Cluj-Napoca, 2013].

Preface

As we all know, many biological and physical systems, such as neuronal systems and disease systems, are featured by certain nonlinear and complex patterns in their elements and networks. These phenomena carry significant biological and physical information and regulate down-stream mechanism in many instances. This issue of Discrete and Continuous Dynamical Systems, Series S, comprises a collection of recent works in the general area of nonlinear differential equations and dynamical systems, and related applications in mathematical biology and engineering. The common themes of this issue include theoretical analysis, mathematical models, computational and statistical methods on dynamical systems and differential equations, as well as applications in fields of neurodynamics, biology, and engineering etc. <br>&nbsp; &nbsp Research articles contributed to this issue explore a large variety of topics and present many of the advances in the field of differential equations, dynamical systems and mathematical modeling, with emphasis on newly developed theory and techniques on analysis of nonlinear systems, as well as applications in natural science and engineering. These contributions not only present valuable new results, ideas and techniques in nonlinear systems, but also formulate a few open questions which may stimulate further study in this area. We would like to thank the authors for their excellent contributions, the referees for their tireless efforts in reviewing the manuscripts and making suggestions, and the chief editors of DCDS-S for making this issue possible. We hope that these works will help the readers and researchers to understand and make future progress in the field of nonlinear analysis and mathematical modeling.

Two notions which affected nonlinear analysis (Bernard Bolzano lecture)

General mathematical theories usually originate from the investigation of particular problems and notions which could not be handled by available tools and methods. The Fucik spectrum and the $p$-Laplacian are typical examples in the field of nonlinear analysis. The systematic study of these notions during the last four decades led to several interesting and surprising results and revealed deep relationship between the linear and the nonlinear structures. This paper does not provide a complete survey. We focus on some pioneering works and present some contributions of the author. From this point of view the list of references is by no means exhaustive.

A Note on Legendre–Fenchel Conjugate of the Product of Two Positive-Definite Quadratic Forms

The Legendre-Fenchel conjugate of the product of two positive-definite quadratic forms was posted as an open question in the field of nonlinear analysis and optimization by Hiriart-Urruty ('Question 11' in SIAM Review 49, 255-273, 2007). Under a convexity assumption on the function, it was answered by Zhao (SIAM J. Matrix Analysis & Applications 31(4), 1792-1811, 2010). In this note, we answer the open question without making the convexity assumption.

The Legendre–Fenchel Conjugate of the Product of Two Positive Definite Quadratic Forms

It is well known that the Legendre-Fenchel conjugate of a positive definite quadratic form can be explicitly expressed as another positive definite quadratic form and that the conjugate of the sum of several positive definite quadratic forms can be expressed via inf-convolution. However, the Legendre-Fenchel conjugate of the product of two positive definite quadratic forms is not clear at present. Hiriart-Urruty posted it as an open question in the field of nonlinear analysis and optimization [Question 11 in SIAM Rev., 49 (2007), pp. 255-273]. From a convex analysis point of view, it is interesting and important to address such a question. The purpose of this paper is to answer this question and to provide a formula for the conjugate of the product of two positive definite quadratic forms. We prove that the computation of the conjugate can be implemented via finding a root of a certain univariate polynomial equation, and we also identify the situations in which the conjugate can be explicitly expressed as a single function without involving any parameter. Some other issues, including the convexity condition for the product function, are also investigated as well. Our analysis shows that the relationship between the matrices of quadratic forms plays a vital role in determining whether the conjugate can be explicitly expressed or not.

A note on the existence of solutions to some nonlinear functional integral equations

Substituting the usual growth condition by an assumption that a specific initial value problem has a maximal solution, we obtain existence results for functional nonlinear integral equations with variable delay. Appli- cation of the technique to initial value problems for differential equations as well as to integrodifferential equations are given. Nonlinear integral equations and nonlinear functional integral equations have been some topics of great interest in the field of nonlinear analysis for a long time. Since the pioneering work of Volterra up to our days, integral equations have attracted the interest of scientists not only because of their mathematical context but also because of their miscellaneous applications in various fields of science and technology. In particular, existence theory for nonlinear integral equations, strongly related with the evolution on fixed point theory, has been boosted ahead after the remarkable work of Krasnoselskii (6) which signaled a new era in the research of the subject. The present note is motivated by a recent paper by Dhage and Ntouyas (3) presenting some results on the existence of solutions to the nonlinear functional integral equation (E)

Coherent structures in strongly interacting many-body systems. I. Derivation of dynamics

Numerous examples in condensed matter physics possess an effective Hamiltonian structure involving two-body anharmonic interactions between elementary excitations. The effective second-quantised Hamiltonian used to model these systems is essentially of the same form except for the particulars of the interaction parameters and the statistics of the particles which are interacting. The author investigates such systems in the limit of strongly anharmonic interactions which may possibly lead to symmetry breaking. He introduces quantum field operators and calculates their equations of motion. The interaction coefficients are expanded in a Taylor series about the critical mode to second order to illustrate the method, but he gives the expansion to infinite order for completeness. It is demonstrated that both in first and second order the equations of motion are reduced to a system of highly nonlinear coupled partial differential equations (PDE) for the field operator. He shows that the form of the second-order equation is exact due to renormalisation theory, in the case where the Hamiltonian coefficients represent operators in real space which are isotropic, and incorporates symmetries and topologies which would appear in an expansion to infinite order. This special situation arises in a very wide range of physical examples. These equations can be solved exactly using a particular ansatz in conjunction with recent mathematical developments in the field of non-linear analysis. In particular, the method of symmetry reduction for PDE provides a complete set of symmetry variables for the type of equations used in the study.

The hierarchical concept in finite element analysis

The hierarchical concept for finite element shape functions was introduced many years ago as a convenient device for mixed order interpolation. Its full advantages have not been realized until a much later time—and these include in addition 1. (a) improved conditioning; 2. (b) ease of introducing error indicators if successive refinement is sought. Further, it is possible to use the ideas to construct a range of error estimators which compare well with alternatives and are ideally suited for adaptive refinements of analysis. As the hierarchical elements are equally simple to implement as “standard” fixed order elements it is felt that more programs will, in the future, turn to utilize their advantages. This is especially true in the field of nonlinear analysis where even today computational economies are necessary.