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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.

The given problem involves a nonlocal and nonhomogeneous anisotropic elliptic equation of the form :$ \begin{aligned} &-K\left(\int_{\Omega} \sum\limits_{i = 1}^{N} \mathcal{A}_{i}\left(z, \partial_{z_{i}} \vartheta\right)+\frac{\Theta(z)}{p_{K}(z)}|\vartheta|^{p_{K}(z)} \mathrm{\; d} z\right) \\ & \quad\times \left(\sum\limits_{i = 1}^{N} \partial_{z_{i}} a_{i}\left(z, \partial_{z_{i}} \vartheta\right)-\Theta(z)|\vartheta|^{p_{K}(z)-2} \vartheta\right) = \mu f(z, \vartheta), \end{aligned} $where $ \Omega\subset \mathbb{R}^N \; (N\geq 2) $ is a bounded domain with Lipschitz boundary $ \partial \Omega $, $ K: \mathbb{R}_0^{+}\longrightarrow \mathbb{R}^{+} $ be a nondecreasing and continuous Kirchhoff function, and $ f:\Omega\times \mathbb{R} \longrightarrow \mathbb{R} $ is a Carathéodory function. The existence of weak solutions to this problem is demonstrated through the application of Berkovits and Mustonen's topological degree theory in the framework of anisotropic Sobolev spaces with variable exponent $ W_{0}^{1, \vec{p}(z)}(\Omega) $. The viability of this approach is contingent upon the fulfillment of specific assumptions. [ABSTRACT FROM AUTHOR]

In this article, we study the structure of 3-prime near-rings through the action of derivations and left multipliers. It should be noted that the combination of these types of maps allows us to obtain more precise results. Thus, and under appropriate additional assumptions, the commutativity and properties of a near-ring are discussed. Furthermore, an example is given to illustrate that the 3-primeness hypothesis cannot be omitted. [ABSTRACT FROM AUTHOR]

In this paper, we generalize some results from [M. Ashraf and S. Ali, On left multipliers and the commutativity of prime rings, Demonstr. Math. 41 2008, 4, 763–771] and present our contribution to the study of endomorphisms. In particular, we prove that a prime ring must be an integral domain if it admits endomorphisms and multipliers satisfying some algebraic identities on non-zero ideals. As a consequence of our main theorems, many known results can either be generalized or deduced. Furthermore, an example is given to show that the necessity of the primeness assumption imposed on the hypotheses of various theorems cannot be unnecessary. [ABSTRACT FROM AUTHOR]

In this paper, we will extend some results on the commutativity of Jordan ideals proved in [8] and [5]. However, instead of left generalized derivations, we will consider generalized left derivations, which are sufficient to obtain good results with respect to the structure of near-rings. We will also show that the conditions imposed in the paper cannot be removed. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the notion of multiplicative semiderivation and prove some theorems in the setting of a semigroup ideal of a 3-prime near-ring admitting a multiplicative semiderivation. Thereby, it is shown that under appropriate additional hypotheses a near-ring must be a commutative ring. Furthermore, an example is given to illustrate that the 3-primeness hypothesis is not superfluous. [ABSTRACT FROM AUTHOR]

In this work, we investigate the commutativity of 3-prime near-rings satisfying some differential identities on Jordan ideals involving multiplicative derivation. Some well-known results characterizing commutativity of 3-prime near-rings by derivations have been generalized by using multiplicative derivation. Further, we discuss an example to prove that the necessity of the 3-primeness hypothesis imposed on the various theorems cannot be marginalized. [ABSTRACT FROM AUTHOR]