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This article is devoted to enhancing a class of generalized Suzuki-type nonlinear contractions following Pant to a class of Suzuki–Ćirić-type nonlinear contractions via comparison functions via a locally ζ-transitive relation and implemented the same to ascertain certain fixed-point results. The outcomes presented herewith unify and generalize a few existing findings. An illustrative examples is offered to explain our findings. Our outcomes assist us in figuring out the unique solution to a boundary value problem.

We study a boundary value problem for a system of the third order semi-linear partial differential equations with nonlocal boundary conditions. We establish sufficient conditions of existence, uniqueness, regularity and sign-preserving property of solutions of the studied problem and construct an iterative method for its approximation. [ABSTRACT FROM AUTHOR]

Bidirectional reflectance distribution functions (BRDFs) describe how light interacts with a point on a surface. To propose a new BRDF formulation or to compare different reflectance representations, it is necessary to confront the results obtained with those functions against previous work or measured data. Despite the importance of using reliable techniques to evaluate a BRDF, there is a lack of works in the literature that gathers and compares those. This paper proposes a compilation of techniques used to evaluate BRDF representations along with their formal definitions. Those techniques were classified into three different groups—comparison functions, rendered images, and plots—and, to illustrate their use, three classical and widely adopted models and one state-of-the-art BRDF representation were evaluated regarding their capacity to preserve the appearance of measured materials. Based on our research regarding comparison functions, a stable and robust BRDF evaluation technique is proposed. It has been observed during both literature review and experiments that each group of techniques provides complementary information about the evaluated BRDFs, which suggests that at least one model from each category should be adopted during the choice of criteria to evaluate a BRDF. [ABSTRACT FROM AUTHOR]

In this paper, we extend the relation-theoretic contraction principle due to Alam and Imdad (J. Fixed Point Theory Appl. 17 (2015) 693-702) for Matkowski contractions employing a locally T-transitive binary relation. Our results improve and enrich several fixed point theorems of the existing literature. [ABSTRACT FROM AUTHOR]

In this article, we establish a second-order estimate for the solutions to the infinity Laplace equation $$ -\Delta_{\infty} u=b(x)g(u), \quad u>0, \quad x \in \Omega,\; u|_{\partial \Omega}=0, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $g\in C^1((0,\infty),(0,\infty))$, $g$ is decreasing on $(0,\infty)$ with $\lim_{s \to 0^+}g(s)=\infty$ and g is normalized regularly varying at zero with index $-\gamma$ ($\gamma>1$), $b \in C({\bar{\Omega}})$ is positive in $\Omega$, may be vanishing on the boundary. Our analysis is based on Karamata regular variation theory.

This technical note focuses on positive definite functions defined on compact sets which contain the origin in their interior. It provides a class- K function bounding inequality for the given positive definite function which is satisfied for its entire domain of definition. An application of the result is also given to analyze the stability of perturbed nonlinear systems whose nominal parts contain an asymptotically stable equilibrium point at the origin. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of ultra distance function. Based on the notion of ultra distance function, we introduce the definitions of (k, ψ, L)-quasi contractions of type (I) and type (II) in the frame of quasi metric spaces. We employ our new definitions to construct and prove many fixed and common fixed point results in the frame of quasi metric spaces. Our results extend and improve many exciting results in the literatures. Also, we introduce some examples and some applications in order to support the usability of our work.

The task of probabilistic record linkage is to find and link records that refer to the same entity across several disparate data sources. The accurate linking of records (entity resolution) is an important task for the healthcare industry, government, law enforcement, and the private sector, for obvious reasons. However, finding exact matches of an entity can be challenging due to records with typographical, phonetical or other types of errors (noise) found across real-world data sources. Over the years, many comparison functions have been developed to relate pairs of records and produce a similarity score. With a pair of predefined thresholds, one may decide if records pairs match, do not match, or if they require further clerical review. Nevertheless, finding appropriate comparison functions, identity descriptors (fields), threshold values, and efficient classifiers remains a challenging task. In this study, we propose a supervised gradient-based learning model that can adjust its structure and parameters based on matching scores coming from many comparison functions (and applied to many fields), to efficiently classify the records. The design of this structure is transparent, and can potentially allow us to locate which comparison functions and fields are more significant to correctly link the records. To train this structure, we propose a novel performance index that can help learn how to separate matched from non-matched records. Results completed with the use of synthetic datasets affected by different levels of noise and real-world datasets show the effectiveness of the algorithm, which can significantly reduce the number of false positives, false negatives, and the number of records selected for review. [ABSTRACT FROM AUTHOR]

In this paper we obtain a generalization of Matkowski's fixed point theorem and Istrăţescu's fixed point theorem concerning convex contractions. More precisely, given a complete b-metric space ( X, d) we prove that every continuous function $$ f:X\rightarrow X$$ is a Picard operator, provided that there exist $$ m\in \mathbb {N}^{*}$$ and a comparison function $$\varphi $$ such that $$d(f^{[m]}(x),f^{[m]}(y))\le \varphi (\max \{d(x,y),d(f(x),f(y)),\ldots ,d(f^{[m-1]}(x),f^{[m-1]}(y))\})$$ for all $$x,y\in X$$ . In addition, we point out that if $$m=1$$ , taking into account that a metric space is a b-metric space, we obtain a generalization of Matkowski's fixed point theorem. Moreover, we prove that Istrăţescu's fixed point theorem concerning convex contractions is a particular case of our result for $$m=2$$ . By providing appropriate examples we show that the above-mentioned two generalizations are effective. [ABSTRACT FROM AUTHOR]

In this paper, by constructing suitable comparison functions, we mainly give the boundary behavior of solutions to boundary blow-up elliptic problems ▵ ∞ u = b ( x ) f ( u ) , x ∈ Ω , u | ∂ Ω = + ∞ , where Ω is a bounded domain with smooth boundary in R N , the operator △ ∞ is the ∞-Laplacian, b ∈ C α ( Ω ¯ ) which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary and the nonlinear term f is a Γ -varying function at infinity, whose variation at infinity is not regular. [ABSTRACT FROM AUTHOR]

In this paper, by constructing new comparison functions, we mainly study the boundary behavior of solutions to boundary blow-up elliptic problems for more general nonlinearities f (which may be rapidly varying at infinity) $\triangle_{\infty} u =b(x)f(u)$, $x\in \Omega $, $u|_{\partial \Omega }=+\infty$, where Ω is a bounded domain with smooth boundary in $\mathbb{R}^{N}$, and $b \in C(\bar{\Omega })$ which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary. [ABSTRACT FROM AUTHOR]

In this paper, by constructing suitable comparison functions, we mainly analyze the exact boundary behavior for the unique solution near the boundary to the singular Dirichlet problem − Δ ∞ u = b ( x ) g ( u ) , u > 0 , x ∈ Ω , u | ∂ Ω = 0 , where Ω is a bounded domain with smooth boundary in R N , g ∈ C 1 ( ( 0 , ∞ ) , ( 0 , ∞ ) ) , g is decreasing on ( 0 , ∞ ) with lim s → 0 + ⁡ g ( s ) = ∞ , g is normalized regularly varying at zero with index − γ ( γ > 1 ) and b ∈ C ( Ω ¯ ) which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary. [ABSTRACT FROM AUTHOR]

The use of comparison functions has become standard in systems and control theory, particularly for the purposes of studying stability properties. The use of these functions typically allows elegant and succinct statements of stability properties such as asymptotic stability and input-to-state stability and its several variants. Furthermore, over the last 20 years several inequalities involving these comparison functions have been developed that simplify their manipulation in the service of proving more significant results. Many of these inequalities have appeared in the body of proofs or in appendices of various papers. Our goal herein is to collect these inequalities in one place. [ABSTRACT FROM AUTHOR]

Abstract: Integral input-to-state stability (iISS) is a robust stability property of interest in the analysis and control of nonlinear dynamical systems affected by external inputs. The computation of tight comparison functions associated with this stability property is useful for assessing robustness in the iISS sense for specific systems. This paper presents a variational characterization of these tight comparison functions, along with an approach to computation via solution of an associated Hamilton–Jacobi–Bellman partial differential equation. A limiting case of relevance to the related input-to-state stability (ISS) property is also considered. An illustrative example highlights the application of this approach. [Copyright &y& Elsevier]

Nonlinear ℒ2-gain is a finite gain concept that generalises the notion of conventional (linear) finite ℒ2-gain to admit the application of ℒ2-gain analysis tools of a broader class of nonlinear systems. The computation of tight comparison function bounds for this nonlinear ℒ2-gain property is important in applications such as small gain design. This article presents an approximation framework for these comparison function bounds through the formulation and solution of an optimal control problem. Key to the solution of this problem is the lifting of an ℒ2-norm input constraint, which is facilitated via the introduction of an energy saturation operator. This admits the solution of the optimal control problem of interest via dynamic programming and associated numerical methods, leading to the computation of the proposed bounds. Two examples are presented to demonstrate this approach. [ABSTRACT FROM AUTHOR]

Conditions for the exponential stability of a dynamic system are established using a scalar Lyapunov function and comparison functions of the same order of increase. The result is illustrated by way of example of a nonlinear scalar dynamic equation [ABSTRACT FROM AUTHOR]

Given monotone operators on the positive orthant in n-dimensional Euclidean space, we explore the relation between inequalities involving those operators, and induced monotone dynamical systems. Attractivity of the origin implies stability for these systems, as well as a certain inequality, the no-joint-increase condition. Under the right perspective the converse is also true. In addition we construct an unbounded path in the set where trajectories of the dynamical system decay monotonically, i.e., we solve a positive continuous selection problem. [ABSTRACT FROM AUTHOR]

The article presents the modification of the Wentzel-Kramers Brillouim (WKB) method and examine its relation with the Ermakov equation. It says that the equation can be solved with the use of a certain iterative scheme and the use of such scheme is equal to the application of the WKB method. It states that when the function ɷ2 (x) have no zeros turning points, it is convenient to select the comparison function Ω2 (σ) = 1.