Abstract As is well known, the embedding of all maximal subgroups of P of a group G can determine the structure of G , where p ∈ π ( G ) and P ∈ Syl p ( G ). Based on the topic, we defined two sets δ ( P , G ) = { P 1 ⋖ P ∣ P ∩ [ P , G ] ≰ P 1 } and F ( P , G ) = { P 1 ⋖ P ∣ P G ≰ P 1 } ${\mathfrak{F}}_{\left(P,G\right)}=\left\{{P}_{1}{< dot}P\mid {P}_{G}\nleqq {P}_{1}\right\}$ , which provide a new way to select “some” maximal subgroups of P instead of “all”. Further, we analyzed p -nilpotency of a group under the condition that every element in δ ( P , G ) satisfies the ICΦ-property (or P is an ICΦ s -subgroup of G and every element in F ( P , G ) ${\mathfrak{F}}_{\left(P,G\right)}$ satisfies the ICΦ-property). To some extent, the two sets will have wide application in investigating the p -nilpotency of finite groups and improving some known theorems.

Abstract This paper investigates a novel abstract system that includes a fractional differential equation of the Atangana-Baleanu type and a history-dependent evolutionary hemivariational inequality (AB-FDEHI). We demonstrate the unique solvability of this problem by applying a semi-discrete approximation (the so-called Rothe method) and utilizing the surjectivity of a multivalued pseudo-monotone operator theorem. Additionally, we derive a fully discrete approximation of the system (AB-FDEHI), present the error estimates, and demonstrate the convergence result. The results obtained are used to examine a novel frictional contact problem (FCP) involving a viscoelastic material and an obstacle, where the effects of memory terms and wear are taken into account.

Abstract This paper aims to explore the stability of a mixed-type additive-quartic functional equation in 2-Banach spaces via the direct method. We categorize mappings satisfying a certain functional inequality into odd, even, and general mappings, and establish generalized Hyers–Ulam stability for each category. For odd mappings, we demonstrate that the exact solution, represented by an additive mapping, is close to the approximate solution satisfying the functional inequality. For even mappings, the exact solution, represented by a quartic mapping, is close to the approximate solution. Furthermore, we show that for general mappings, the exact solution, represented by the sum of an additive and a quartic mapping, is close to the approximate solution.

Abstract This study is devoted to proving the existence of weak solutions for a nonlinear elliptic problem with Neumann-type boundary data. The problem is driven by a discontinuous power nonlinearity and a nonsmooth prescribed data. Additionally, we aim to derive an estimate that proves the well-posedness of the problem. This estimate serves as an evidence for the uniqueness of the existing solution when the boundary term is “smooth”.

Abstract This study addresses the problem of indirect boundary observability for a spatial semi-discretization of weakly coupled two-dimensional wave equations. The discretization is performed using a finite difference method on a uniform mesh. More specifically, the analysis focuses on the uniformity of the observability inequality with respect to the discretization parameter as it tends to zero. It is well known that high-frequency numerical solutions emerge when wave equations are discretized via finite differences on uniform grids. Consequently, the uniform observability property fails to hold, as the observability constant typically becomes unbounded when the mesh size decreases. This lack of uniform observability poses a significant challenge in numerical control theory. To mitigate this issue, we employ a Fourier filtering technique to eliminate spurious high-frequency components. By adapting calculus techniques analogous to those used in the continuous setting, we establish a uniform observability inequality for the filtered semi-discrete system.

Abstract Let ( E , F ) be a nonempty pair subsets of a metric space ( M , d ) and let T : E ∪ F → E ∪ F $\mathcal{T} : E\cup F\to E\cup F$ be a noncyclic mapping, means that, T ( E ) ⊆ E , T ( F ) ⊆ F $\mathcal{T}\left(E\right)\subseteq E,\mathcal{T}\left(F\right)\subseteq F$ . In this context, a point ( x ⋆ , y ⋆ ) ∈ E × F is called a best proximity pair for the mapping T $\mathcal{T}$ if d ( x ⋆ , y ⋆ ) = d i s t ( E , F ) , T x ⋆ = x ⋆ , T y ⋆ = y ⋆ . $$d\left({x}^{\star },{y}^{\star }\right)=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left(E,F\right),\quad {\mathcal{T}\mathcal{x}}^{\star }={x}^{\star },\quad {\mathcal{T}\mathcal{y}}^{\star }={y}^{\star }.$$ In this article, in the setting of strictly convex hyperbolic metric spaces, we deal on the existence and convergence of best proximity pairs for the noncyclic version of the Reich’s contraction mapping T $\mathcal{T}$ which satisfies the following condition: d ( T x , T y ) ≤ a 1 d ( P x , T x ) + a 2 d ( P y , T y ) + a 3 d ( P x , P y ) + ( 1 − η ) d i s t ( E , F ) , ∀ ( x , y ) ∈ E × F , $$d\left(\mathcal{T}x,\mathcal{T}y\right)\le \left[{a}_{1}d\left(\mathcal{P}x,\mathcal{T}x\right)+{a}_{2}d\left(\mathcal{P}y,\mathcal{T}y\right)+{a}_{3}d\left(\mathcal{P}x,\mathcal{P}y\right)\right]+\left(1-\eta \right)\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left(E,F\right), \forall \left(x,y\right)\in E{\times}F,$$ where P $\mathcal{P}$ is a proximal projection operator defined on the union of the proximal sets of E , F and η = a 1 + a 2 + a 3 ∈ (0, 1). We then generalize the above family of noncyclic mappings by considering η = 1 and survey some other existence theorems of best proximity points in strictly convex hyperbolic metric spaces by using a geometric property of proximal normal structure. In particular, we obtain new fixed point results for noncontinuous self-maps.

Abstract In this paper, we study real solutions of the nonlinear Helmholtz equation − Δ u − k 2 u = γ M ( x ) | u | q − 2 u + f ( x , u ) for x ∈ R N , $$-{\Delta}u-{k}^{2}u=\gamma M\left(x\right)\vert u{\vert }^{q-2}u+f\left(x,u\right)\quad \,\text{for}\,x\in {\mathbb{R}}^{N},$$ subject to the asymptotic conditions u ( x ) = O | x | 1 − N 2 and ∂ 2 u ∂ r 2 ( x ) + k 2 u ( x ) = o | x | 1 − N 2 , as r = | x | → ∞ , $$u\left(x\right)=O\left(\vert x{\vert }^{\frac{1-N}{2}}\right)\quad \text{and}\quad \frac{{\partial }^{2}u}{\partial {r}^{2}}\left(x\right)+{k}^{2}u\left(x\right)=o\left(\vert x{\vert }^{\frac{1-N}{2}}\right),\quad \text{as\,}r=\vert x\vert \to \infty ,$$ where N ≥ 1, k > 0, q ∈ (1, 2). Assuming that f ( x , t ) behaves as a convex function of the form | t | p −1 with p ∈ (2, 2*), then the existence and multiplicity of real solutions are obtained via the variational methods.

Abstract In this paper, we develop a new common fixed-point theorem tailored specifically to digital metric spaces, where the discrete nature of pixel grids demands specialized distance and contraction concepts. To demonstrate the breadth and utility of our result, we first provide a concrete example illustrating the theorem’s hypotheses and conclusions, and then showcase a practical application: a fractal-based image compression algorithm built around the digital Sierpinski triangle. This not only validates our theoretical contribution but also highlights its potential for efficient representation and processing of self-similar structures in digital imagery.

Abstract Let F n be the n × n Fourier matrix on the cyclic group Z n ${\mathbb{Z}}_{n}$ , a renowned theorem of Chebotarëv asserts that all minors in F n for prime n are non-zero. In this short note it is shown that (i) all principal minors in the Kronecker product F p ⊗ F q are non-vanishing (principal non-singularity) for distinct odd primes p , q if q is large enough and generates the multiplicative group Z p * ${\mathbb{Z}}_{p}^{{\ast}}$ ; (ii) the Fourier matrix on Z 2 k × Z q ${\mathbb{Z}}_{2}^{k}{\times}{\mathbb{Z}}_{q}$ is principally non-singular upon permutation (in particular, for k = 1 the identity permutation suffices) for odd prime q and k = 1, 2, 3. The proof is just an exposition of existing techniques reorganized in a unified way. The result will have implications in combining Riesz bases of exponentials.

Abstract This paper is committed to the existence of multiple solutions for the critical Schrödinger–Poisson system involving ( p , q )-Laplacian on the Heisenberg group: − Δ H , p u − Δ H , q u + ϕ | u | p − 2 u = β | u | r − 2 u + μ ∫ Ω | u ( η ) | p λ * | η − 1 ξ | λ d η | u | p λ * − 2 u in Ω , − Δ H ϕ = | u | p in Ω , u = ϕ = 0 on ∂ Ω , $$\begin{cases}-{{\Delta}}_{H,p}u-{{\Delta}}_{H,q}u+\phi \vert u{\vert }^{p-2}u=\beta \vert u{\vert }^{r-2}u+\mu \left({\int }_{{\Omega}}\frac{\vert u\left(\eta \right){\vert }^{{p}_{\lambda }^{{\ast}}}}{\vert {\eta }^{-1}\xi {\vert }^{\lambda }}\mathrm{d}\eta \right)\vert u{\vert }^{{p}_{\lambda }^{{\ast}}-2}u\hfill & \text{in} {\Omega},\hfill \\ -{{\Delta}}_{H}\phi =\vert u{\vert }^{p}\hfill & \text{in} {\Omega},\hfill \\ u=\phi =0\hfill & \text{on} \partial {\Omega},\hfill \end{cases}$$ where Ω ⊂ H N ${\Omega}\subset {\mathbb{H}}^{N}$ is a smooth bounded domain, Q = 2 N + 2 is the homogeneous dimension of the Heisenberg group H N ${\mathbb{H}}^{N}$ , Δ H , ℘ φ = div H | ∇ H φ | H ℘ − 2 ∇ H φ ${{\Delta}}_{H,\wp }\varphi ={\text{div}}_{H}\left(\vert {\nabla }_{H}\varphi {\vert }_{H}^{\wp -2}{\nabla }_{H}\varphi \right)$ is the ℘ -sub-Laplacian, for ℘ ∈ { p , q }, 1 ≤ p < q < 2 p < r < p λ * $1\le p{< }q{< }2p{< }r{< }{p}_{\lambda }^{{\ast}}$ , λ ∈ (0, Q ) and p λ * = p ( 2 Q − λ ) 2 ( Q − p ) ${p}_{\lambda }^{{\ast}}=\frac{p\left(2Q-\lambda \right)}{2\left(Q-p\right)}$ is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, and μ , β > 0 are some real parameters. Under different parameter control conditions, we show the compactness condition via the concentration compactness principle on the Heisenberg group, and the existence and multiplicity of solutions for above system is obtained by the Krasnoselskii genus theory and variational methods. The main features and novelty of this system lies in the simultaneous appearance of double phase operators and nonlocal critical terms. As far as we know, our results even new for the case p = q .