In this paper, we introduce the notion of generalized permuting tri-derivations, g-derivations, and g-tri-derivations on lattices, and we investigate and generalize some properties discussed in [1] and [2]. We also provided several properties that characterize the g-derivations, g-tri-derivations, and the generalized permuting g-tri-derivations and their trace.

In this manuscript, we investigate a boundary value problem governed by a complex system of partial differential equations given by: - div [ℒ_ϑ( z,ϑ ,∇ϑ) ]+M̂|ϑ| ^p-2ϑ =ℒ_σ (z, ħ *σ (ϑ ), ∇ (ħ *σ (ϑ )) ), under the following non-homogeneous Neumann boundary condition: ℒ_ϑ( z,ϑ ,∇ϑ) .ν =Θ (z,ϑ ) . Here, the functions ℒ_ϑ , ℒ_σ , and Θ exhibit Carathéodory characteristics. The operator σ : W^1, p(ℬ) → W^1, p( ℝ^N) operates as an extension operator correlated with the domain ℬ , while ħ denotes an integrable function defined over ℝ^N . Notably, the problem introduces a nonlocal operator, manifesting as the convolution ħ *σ (ϑ ) of ħ with σ (ϑ ) , associated with the variable ϑ . Under conditions necessitating thorough examination, we establish the existence of a weak solution to the mentioned problem by employing the topological degree method.

The given problem involves a nonlocal and nonhomogeneous anisotropic elliptic equation of the form : -K (integral(Omega) Sigma(N)(i=1) A(i)(z, partial derivative(zi) theta) + Theta(z)/pK(z)vertical bar theta vertical bar(pK(z)) dz) x (Sigma(N)(i=1) partial derivative(zi) a(i) (z, partial derivative(zi) theta) - Theta(z)/(theta)(pK(z)-2) theta) = mu f(z, theta), where Omega subset of R-N (N >== 2) is a bounded domain with Lipschitz boundary partial derivative Omega, K : R-0(+) -> R+ be a nondecreasing and continuous Kirchhoff function, and f : Omega x R -> R is a Caratheodory 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,(p) over right arrow)(z) (Omega). The viability of this approach is contingent upon the fulfillment of specific assumptions.

In this paper, by using the Mittag–Leffler operators {ℒ_α(-t^α𝕀):t≥ 0} and {ℒ_α ,α(-t^α𝕀):t≥ 0} we will prove the mild soltion of the time fractional magneto-hydrodynamics system with a fractional derivative of Caputo. Furthermore, by Itô integral, we will establish the mild solution of stochastic time fractional magneto-hydrodynamics system in ℰ𝒩_p^λ∩N_p,λ^2α .

In this article, we investigate the presence of weak solutions for obstacle problems integral(ohm)A(z,u,Du) :D(v-u)+phi(u) :D(v-u) dz >= 0, for v belonging to the following convex set K-psi,K-theta, applying the Young measure theory and a theorem by Kinderlehrer and Stampacchia, the desired outcome is achieved.

This article investigates the existence of weak solutions for a class of nonlo cal problems with Dirichlet boundary conditions. The proof of the existence result relies on Galerkin's approximation and Young's measure theory.

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.

. In this article, we apply the Ito integral to obtain the global solutions for stochastic quasi-geostrophic equations in Fourier-Besov-Morrey spaces. For comparison we also give the corresponding results of the deterministic quasi-geostrophic equations. We assume the initial data is F0 measurable and the right-hand side is a random function in a Morrey space, to obtain the well posedness of stochastic quasi-geostrophic equations.

This paper deals with the existence and uniqueness of weak solution for a class of obstacle problem of the form {[ ∫ _Ω𝒱(x,Dw):D(ϑ -w) dx+∫ _Ω⟨ w| w| ^p(x)-2,ϑ - w⟩ dx; ≥∫ _Ω𝒰(x,w)(ϑ -w) dx,; ; ϑ∈ _Λ , h, ]. where _Λ , h is a convex set defined below. By using the Young measure theory and Kinderlehrer and Stampacchia Theorem, we prove the existence and uniqueness result of the considered problem in the framework of generalized Sobolev space.

In this paper, we study the commutativity of 3 -prime near -rings satisfying some differential identities on Jordan ideals involving certain additive maps. Some well-known results characterizing the commutativity of 3 -prime nearrings by derivations and left multipliers are extended to right multipliers and left generalized derivations. Furthermore, an example is given showing the necessity of the 3 -primeness mentioned in the assumptions of our theorems is given.

In this paper, we show the existence and uniqueness of weak solutions to obstacle problem integral(Omega)sigma(x,Du) :D(vu) + (sic)u|u|(p(x)-2),v-u(sic)dx >= 0 for v belonging to the convex set K-psi,K-theta. The main tool used here is the Young measure theory and atheorem of Kinderlehrer and Stampacchia combined with the theory of Sobolev spaces with variable exponent.

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.

In this paper we study the existence of weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(z)-biharmonic operator with indefinite weight and no flux boundary condition. The proof of the existence result relies on employing the concept of a Fredholm-type results for a pair of nonlinear operators (𝔒,𝔖) , in conjunction with the theory of variable exponent Sobolev spaces.

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 purpose of this article is to prove the existence and uniqueness of weak solutions to the following obstacle problem of p-Laplace-type: ∫ _Ωσ _1(z,Du-ℱ(u)):D(v-u)+σ _2(z,Du):(v-u)+ ⟨ u| u| ^p-2, v- u⟩ dz≥ 0, with data belonging to the dual of Sobolev spaces. The main result is demonstrated by means of Kinderlehrer and Stampacchia’s Theorem and Young’s measure theory.

In this investigation, we study the existence of weak solution for a class of nonhomogeneous anisotropic elliptic problem. These problems involve the anisotropic operators with variable exponents. Based on the topological degree theory concerning a specific subset of demicontinuous operators under generalized (S_+) and the theory of anisotropic Sobolev spaces with variable exponent, we demonstrate the existence of weak solution for this problem. Our findings are applicable to problems with no-flux boundary conditions, and they extend and generalize several results previously reported in the literature.

In this paper, our main objective is to introduce the notion of a fuzzy semigroup ideal by using Yuan and Lee's definition of fuzzy group based on fuzzy binary operations. Also, some of its basic properties are studied analogously to the known results in the case of semigroup ideals defined in the framework of ordinary near-rings.

We consider N to be a 3 -prime field and P to be a prime ideal of N . In this paper, we study the commutativity of the quotient near -ring N / P with left multipliers and derivations satisfying certain identities on P , generalizing some well-known results in the literature. Furthermore, an example is given to illustrate the necessity of our hypotheses.

This paper is devoted to discussing the existence of solutions for a class of Kirchhoff-type variational inequalities: -ℳ (∫ _Ω𝒜(z,∇ u ) dz ) ∫ _Ω𝒢(z,∇ u).(∇ϑ -∇ u) dz ≥∫ _ΩΦ (z,u)(ϑ -u) dz , for υ belonging to the following convex set 𝒮_ψ , θ . By employing Young measure theory in conjunction with a theorem formulated by Kinderlehrer and Stampacchia, we attain the intended result.

This paper studies the existence and uniqueness of solution for the fractional Hall-magnetohydrodynamics system (HMH) and with two stochastic terms (SHMH). Based on the theory of Besov–Morrey spaces and the contraction principle, we will demonstrate tow main result. The first result shows the existence, uniqueness and the analyticity of solution for (HMH) in Besov–Morrey spaces N_p,λ^s . The second result prove the existence and uniqueness of solution for (SHMH) in ℒ_0^1 (Ω× (0,T),𝒫;ℳ_p^λ ) ∩N_p,λ^s .