Volume 42, Issue 4

We present an application-oriented approach to Urysohn and Hammerstein integral operators acting between spaces of Hölder continuous functions over compact metric spaces. These nonlinear mappings are formulated by means of an abstract measure theoretical integral involving a finite measure. This flexible setting creates a common framework to tackle both such operators based on the Lebesgue integral like frequently met in applications, as well as, e.g., their spatial discretization using stable quadrature/cubature rules (Nyström methods). Under suitable Carathéodory conditions on the kernel functions, properties like well-definedness, boundedness, (complete) continuity and continuous differentiability are established. Furthermore, the special case of Hammerstein operators is understood as composition of Fredholm and Nemytskii operators. While our differentiability results for Urysohn operators appear to be new, the section on Nemytskii operators has a survey character. Finally, an appendix provides a rather comprehensive account summarizing the required preliminaries for Hölder continuous functions defined on metric spaces.

In this paper, we aim to study a special type of metric called almost * {*} -η-Ricci soliton on the special class of contact pseudo-Riemannian manifold. First, we prove that a Kenmotsu pseudo-Riemannian metric as an * {*} -η-Ricci soliton is Einstein if either it is η-Einstein or the potential vector field V is an infinitesimal contact transformation. Further, we prove that if a Kenmotsu pseudo-Riemannian manifold admits an almost * {*} -η-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present an example of * {*} -η-Ricci solitons which illustrate our results.

In this paper, we investigate subclasses of univalent functions over | z | < 1 \lvert z\rvert<1 , using the characterization of starlikeness and Noor integral operator defined by convolution of analytic functions. We also give coefficient bounds for the class of 𝛼-spiral function of order 𝜌 and 𝑘-uniformly 𝛼-spirallike functions. Moreover, we provide some examples of univalent functions to illustrate our results.

Numerous computational and learning theory models have been studied using probabilistic functional equations. Especially in two-choice scenarios, the vast bulk of animal behavior research divides such situations into two different events. They split these actions into two possibilities according to the animals’ progress toward a particular decision. However, reward plays a crucial role in such experiments because, based on the selected side and the food placement, such scenarios may be classified into four distinct categories. This article aims to explore the animals’ reactions to such circumstances by presenting a generic stochastic functional equation. By using the well-known fixed point theory results, we examine the existence, uniqueness, and stability of solutions to the suggested functional equation. Moreover, an example is included to emphasize the significance of our findings.

In this paper, we study the existence of a weak solution for a class of Neumann boundary value problems for equations involving the p ⁢ ( x ) p(x) -Laplacian-like operator. Using a topological degree theory for a class of demicontinuous operators of generalized ( S + ) (S_{+}) -type and the theory of the variable exponent Sobolev spaces, we establish the existence of a weak solution of this problem. Our results extend and generalize several corresponding results from the existing literature.