The present article deals with the existence of weak solutions to a class of p(z)p\left(z)-Kirchhoff-type problems. To address these problems, we employ a variational approach in conjunction with the theory of variable exponent Sobolev spaces, while imposing suitable assumptions on the source term. Furthermore, we utilize the theory of Young measures.
We consider conformal iterated function systems (CIFS) that are induced by an infinite rooted tree TT, which we call TT-conformal iterated function systems (TT-CIFS). We focus on the parameterized families of TT-CIFS that satisfy the trasversality condition and give a dimension formula for the exiting measure for a.e. parameter. We also give a sufficient condition that the exiting measure is absolutely continuous for a.e. parameter.
In this article, we study weighted Stepanov-like pseudo-almost automorphic functions with infinite delay using measure theory. We present a new concept of weighted ergodic functions, which is more general than the classical one. Then, we establish many interesting results on the space of such functions. We also study the existence and uniqueness of (μ,ν)\left(\mu ,\nu )-Stepanov-like pseudo-almost automorphic solutions of infinite class in the α\alpha -norm for some partial functional differential equations in a Banach space with unbounded delay, using the spectral decomposition of the phase space developed by Adimy and his co-authors. An example is given to illustrate this work.
This article aims to exhibit new instability results on the effects of arbitrary delay on a class of second-order evolution equations involving in a Hilbert space. More precisely, we prove that arbitrary finite, large or small delay might remarkably destroy the stability of a well-behaved (stable) system, that is by driving the resulting system with delay to generate non-trivial periodic solutions with constant energy, solutions with exponential growth rate and solutions with blow-up energy. Also, an interesting new effect of large delay is constructively established. We further supply this study with applications and numerical simulations.
This article discusses the concept of quasi-semigroups of bounded linear operators within the framework of time scales, encompassing both continuous and discrete cases. It extends the classical theory of quasi-semigroups of operators, which serves as a generalization of strongly continuous semigroups of operators in time scales. In addition, the text mentions providing applications in abstract evolution equations, indicating practical implications of this theoretical framework.
