Volume 37, Issue 3

We study the nonlinear Schrödinger equation posed on product spaces ℝn×ℳk{{\mathbb{R}}^{n}\times{\mathcal{M}}^{k}}, for n≥1{n\geq 1} and k≥1{k\geq 1}, with ℳk{{\mathcal{M}}^{k}} any k -dimensional compact Riemannian manifold. The main results concern global well-posedness and scattering for small data solutions in non-isotropic Sobolev fractional spaces. In the particular case of k=2{k=2}, H1{H^{1}}-scattering is also obtained.

In this paper, we consider two-parameter singularly perturbed boundary value problems via fitted splines. We propose an exponentially fitting factor to the highest order derivative of the problem and discretize by parametric cubic spline functions on a uniform mesh. An error analysis of the method is presented. The efficiency of the method is demonstrated by test examples which support the theoretical results.

We present some initial results for detecting extrema of a function f:ℕa→ℝ{f:\mathbb{N}_{a}\rightarrow\mathbb{R}} when the only a priori information is some limited pointwise data about f as well as information about an appropriate fractional difference of f . In particular, such results provide some analogues of the well-known tests for detecting local extrema using the sign of Δ⁢f⁢(t){\Delta f(t)}. Certain of these results are consequences of some new monotonicity conditions, which relate conditions on the sign of Δaν⁢f⁢(t){\Delta_{a}^{\nu}f(t)}, for 0<ν<1{0<\nu<1}, to the monotonicity of f . Finally, we provide some numerical examples to illustrate the results.

The space BVσ{\mathrm{BV}_{\sigma}} was introduced and studied by Mursaleen [21]. In the present article we study the double sequence spaces (0BV(M,p)σI)2{{}_{2}(_{0}\mathrm{BV}{{}^{I}_{\sigma}}(M,p))}, BV2(M,p)σI{{}_{2}\mathrm{BV}{{}^{I}_{\sigma}}(M,p)} and (∞BV(M,p)σI)2{{}_{2}(_{\infty}\mathrm{BV}{{}^{I}_{\sigma}}(M,p))} with the help of the space BVσ{\mathrm{BV}_{\sigma}} and an Orlicz function M , where p=(pi⁢j){p=(p_{ij})} is the double sequence of strictly positive real numbers. We study some of its properties and prove some inclusion relations related to these new spaces.

In this paper, we obtain the fractional Hermite–Hadamard inequality on 𝕋={0=t0,t1,t2,t3,…}{{\mathbb{T}}=\{0=t_{0},t_{1},t_{2},t_{3},\dots\}}, where 𝕋{{\mathbb{T}}} is a set of positive real numbers such that ti<tj{t_{i}<t_{j}} for i<j{i<j}. Using this inequality, we present a new characterization of convexity of a function defined on 𝕋{{\mathbb{T}}}.