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 .

Abstract The paper considers the numerical stability of the backward recurrence algorithm for computing approximants of branched continued fraction expansions for the Lauricella–Saran’s hypergeometric function F K ratios. For the first time, estimates of the relative errors of approximants computations are obtained, showing the dependence of the error of the approximant on the magnitude of the rounding errors of its elements and the values of the coefficients of the branched continued fractions. Also, for the above-mentioned approximants, sufficient conditions for the sets of numerical stability are established for the first time. Numerical experiments are conducted comparing the efficiency of the backward recurrence algorithm with the forward recurrence algorithm and the Lentz algorithm. It is shown that the backward recurrence algorithm provides high accuracy of computations even for a high order of approximants. Numerical experiments allowed us to evaluate the practical effectiveness of the proposed theoretical results.

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 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 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 In the present paper, utilizing a wide class of fractional integral operators (namely the Riemann-Liouville fractional integral operator) and some functions whose higher-order derivatives are absolutely continuous, we develop novel fractional integral inequalities of the Ostrowski type. Furthermore, the results obtained are found to correlate with the findings of previous studies which have identified inequalities. Finally, novel composite quadrature rules are investigated for estimating the remainder term of fractional integral of a function. In conclusion, the methodology described in this article is expected to stimulate further research in this area.

Abstract This research investigates the generalized Hyers–Ulam stability of the wave equation in an n -dimensional space, evaluated using the L 2 -norm. Typically, the results of Hyers–Ulam stability problems for differential equations are established using either the supremum norm or L ∞ -norm, with a focus on initial conditions or forcing terms to estimate error terms. In this study, we employ an integral approach utilizing the Fourier transform and Parseval’s equality to derive the L 2 -bound for the generalized Hyers–Ulam stability of the governing equation, specifically within the framework of the L 2 -norm. Furthermore, to validate the analytical estimates, we conduct numerical experiments incorporating various types of control functions based on the obtained results.

Abstract This work presents a study of the oscillatory behavior of solutions to a general class of neutral differential equations of second order. Our study considers the noncanonical case and constructs oscillation criteria using several approaches, including Riccati and comparison methods. We also improve the oscillation criteria for this class of equations by improving some of the relations used in the study. The new results prove their effectiveness and novelty in testing oscillation through application to a special case of the equation under consideration. It can be said that our results complement and extend previous related results.

Abstract In this paper we derive a nonlinear theory of Mindlin’s Form II gradient for thermoelastic materials where the heat propagation model proposed by Green and Naghdi, taking into account micro-inertia effects as well. The elastic behavior is assumed to be consistent with Mindlin’s Form II gradient elasticity theory, while the thermal behavior is based on type III entropy balance, where the second gradient of the thermal displacement is included in the set of independent constitutive variables. This leads to a fourth-order equations for the temperature. The equations of the linear theory are also obtained. Semigroup theory is then used to prove the well-posedness of the obtained problem. We show that, in general, the one-dimensional problem is not exponentially stable but that there exists polynomial stability with rates that depend on the micro-inertia parameter h and the regularity of the initial data. Using a resolvent criterion developed by Borichev and Tomilov, we prove that the polynomial decay rate is t −1/3 when h > 0, while it is t −1 when h = 0. By following a result due to Arendt-Batty, we show that the considered problem is strongly stable whenever the value of h .

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.