Volume 9, Issue 3

The purpose of this paper is to study Ricci semisymmetric paracontact metric manifolds satisfying ∇ξ⁡h=0{\nabla_{\xi}h=0} and such that the sectional curvature of the plane section containing ξ equals a non-zero constant c . Also, we study paracontact metric manifolds satisfying the curvature condition Q⋅R=0{Q\cdot R=0}, where Q and R are the Ricci operator and the Riemannian curvature tensor, respectively, and second order symmetric parallel tensors in paracontact metric manifolds under the same conditions. Several consequences of these results are discussed.

In this paper, we propose an iterative algorithm for approximating a common fixed point of an infinite family of quasi-Bregman nonexpansive mappings which is also a solution to finite systems of convex minimization problems and variational inequality problems in real reflexive Banach spaces. We obtain a strong convergence result and give applications of our result to finding zeroes of an infinite family of Bregman inverse strongly monotone operators and a finite system of equilibrium problems in real reflexive Banach spaces. Our result extends many recent corresponding results in literature.

The article aims to explore some new classes of differential equations and associated integral equations for some hybrid families of Laguerre polynomials. The recurrence relations and differential, integro-differential and partial differential equations for the hybrid Laguerre–Appell polynomials are derived via the factorization method. An analogous study of these results for the hybrid Laguerre–Bernoulli, Euler and Genocchi polynomials is presented. Further, the Volterra integral equations for the hybrid Laguerre–Appell polynomials and for their corresponding members are also explored.

We consider a quasilinear equation of fourth order that models the mechanical vibrations of a marine riser. We study the nonexistence of global solutions for any real value of the initial energy. To this end, we construct a new invariant set and improve previous results.

Several classes of locally compact groups have been shown to possess a qualitative uncertainty principle for the Gabor transform. These include Moore groups, the Heisenberg group ℍn{\mathbb{H}_{n}}, the group ℍn×D{\mathbb{H}_{n}\times D} (where D is a discrete group) and other low-dimensional nilpotent Lie groups.

In this paper, we consider the system {-M1⁢(∫Ω|∇⁡u|p⁢(x)+|u|p⁢(x)p⁢(x)⁢𝑑x)⁢(Δp⁢(x)⁢u-|u|p⁢(x)-2⁢u)=λ⁢a⁢(x)⁢|u|r1⁢(x)-2⁢u-μ⁢b⁢(x)⁢|u|α⁢(x)-2⁢uin ⁢Ω,-M2⁢(∫Ω|∇⁡v|q⁢(x)+|v|q⁢(x)q⁢(x)⁢𝑑x)⁢(Δq⁢(x)⁢v-|v|q⁢(x)-2⁢v)=λ⁢c⁢(x)⁢|v|r2⁢(x)-2⁢v-μ⁢d⁢(x)⁢|v|β⁢(x)-2⁢vin ⁢Ω,u=v=0on ⁢∂⁡Ω,\left\{\begin{aligned} &\displaystyle{-}M_{1}\bigg{(}\int_{\Omega}\frac{\lvert% \nabla u\rvert^{p(x)}+\lvert u\rvert^{p(x)}}{p(x)}\,dx\biggr{)}(\Delta_{p(x)}u% -\lvert u\rvert^{p(x)-2}u)=\lambda a(x)\lvert u\rvert^{r_{1}(x)-2}u-\mu b(x)% \lvert u\rvert^{\alpha(x)-2}u&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M_{2}\bigg{(}\int_{\Omega}\frac{\lvert\nabla v\rvert^{q(x)}+% \lvert v\rvert^{q(x)}}{q(x)}\,dx\biggr{)}(\Delta_{q(x)}v-\lvert v\rvert^{q(x)-% 2}v)=\lambda c(x)\lvert v\rvert^{r_{2}(x)-2}v-\mu d(x)\lvert v\rvert^{\beta(x)% -2}v&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω is a bounded domain in ℝN{\mathbb{R}^{N}} (N≥2{N\geq 2}) with a smooth boundary ∂⁡Ω{\partial\Omega}, M1⁢(t),M2⁢(t){M_{1}(t),M_{2}(t)} are continuous functions and λ,μ>0{\lambda,\mu>0}. We prove that for any μ>0{\mu>0} there exists λ*{\lambda_{*}} sufficiently small such that for any λ∈(0,λ*){\lambda\in(0,\lambda_{*})} the above system has a nontrivial weak solution. The proof relies on some variational arguments based on Ekeland’s variational principle, and some adequate variational methods.