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We study the existence of global weak solutions for a hyperbolic differential inclusion with a source term, and then investigate the asymptotic stability of the solutions by using Nakao lemma.

This paper studies the reducibility of almost-periodic Hamiltonian systems with small perturbation near the equilibrium which is described by the following Hamiltonian system: $$\frac{dx}{dt} = J \bigl[{A} +\varepsilon{Q}(t,\varepsilon) \bigr]x+ \varepsilon g(t,\varepsilon)+h(x,t,\varepsilon). $$ It is proved that, under some non-resonant conditions, non-degeneracy conditions, the suitable hypothesis of analyticity and for the sufficiently small ε, the system can be reduced to a constant coefficients system with an equilibrium by means of an almost-periodic symplectic transformation.

In this article, we give some identities on the twisted (h, q)-Genocchi numbers and polynomials and q-Bernstein polynomials with weighted α.

This study introduces extended Branciari quasi-b-distance spaces, a novel implicit contractive condition in the underlying space, and basic fixed-point results, a weak well-posed property, a weak limit shadowing property and generalized Ulam–Hyers stability. The given notions and results are exemplified by suitable models. We apply these results to obtain a sufficient condition ensuring the existence of a unique positive-definite solution of a nonlinear matrix equation (NME) $\mathcal{X}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G(X)}\mathcal{A}_{i}$ , where $\mathcal{Q}$ is an $n\times n$ Hermitian positive-definite matrix, $\mathcal{A}_{1}$ , $\mathcal{A}_{2}$ , …, $\mathcal{A}_{m}$ are $n \times n$ matrices, and $\mathcal{G}$ is a nonlinear self-mapping of the set of all Hermitian matrices that are continuous in the trace norm. We demonstrate this sufficient condition for the NME $\mathcal{X}= \mathcal{Q} +\mathcal{A}_{1}^{*}\mathcal{X}^{1/3} \mathcal{A}_{1}+\mathcal{A}_{2}^{*}\mathcal{X}^{1/3} \mathcal{A}_{2}+ \mathcal{A}_{3}^{*}\mathcal{X}^{1/3}\mathcal{A}_{3}$ , and visualize this through convergence analysis and a solution graph.

Let be the coordination quadruple of the projective Klingenberg plane (PK-plane) coordinated with dual quaternion ring , where Q is any quaternion ring over a field. In this paper, we define addition and multiplication of points on the line geometrically, also we give the algebraic correspondences of them. Finally, we carry over some well-known properties of ordinary addition and multiplication to our definition. MSC:51C05, 51N35, 14A22, 16L30.

The two-component μ-Hunter–Saxton system is considered in the spatially periodic setting. Firstly, two wave-breaking criteria are derived by employing the transport equation theory and the localization analysis method. Secondly, a sufficient condition of the blow-up solutions is established by using the classic method. The results obtained in this paper are new and different from those in previous works.

We investigate locally compact topological groups for which a generalized analog of the Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\mathbb{R}^{n} \times K$ (where K is a separable unimodular locally compact group of type I), Euclidean motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, 2-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups.

Two classes of functions are hereby considered; namely, η-quasiconvex, and strongly η-quasiconvex functions. For the former, we establish some novel integral inequalities of the trapezoid kind for functions with second derivatives, while, for the latter, we obtain some new estimates of the integral $\int _{\mathfrak {\alpha }}^{\beta }(\mathfrak {r}-\mathfrak {\alpha })^{p}(\beta -\mathfrak {r})^{q}\mathcal {K}(\mathfrak {r}) \,d\mathfrak {r}$ when $|\mathcal {K}(\mathfrak {r})|$ , to some powers, is strongly η-quasiconvex. Results obtained herein contribute to the development of these new classes of functions by providing broader generalizations to some well-known results in the literature. Furthermore, we employ our results to deduce some estimates for the perturbed version of the trapezoidal formula. Finally, applications to some special means are also presented.