OMA-Vol. 2 (2018), Issue 2, pp. 129–141 | Open Access Full-Text PDF

Abstract: In this paper, we find a solution of a new type of Langevin equation involving Hilfer fractional derivatives with impulsive effect. We formulate sufficient conditions for the existence and uniqueness of solutions. Moreover, we present Hyers-Ulam stability results.

OMA-Vol. 2 (2018), Issue 2, pp. 114–128 | Open Access Full-Text PDF

Abstract:Let \(\mathcal{A}\) be the class of analytic and univalent functions in the open unit disc \(\Delta\) normalized such that \(f(0)=0=f^{\prime }(0)-1.\) In this paper, for \(\psi \in \mathcal{A}\) of the form \(\frac{z}{f(z)}, f(z)=1+\sum\limits_{n=1}^{\infty }a_{_{n}}z^{n}\) and \(0\leq \alpha \leq 1,\) we introduce and investigate interesting subclasses \(\mathcal{H}_{\sigma }(\phi ), \;S_{\sigma }(\alpha ,\phi ), \; M_{\sigma }(\alpha ,\phi ),\) \( \Im _{\alpha} (\alpha ,\phi )\) and \(\beta _{\alpha}(\lambda ,\phi ) \left( \lambda \geq 0 \right)\) of analytic and bi-univalent Ma-Minda starlike and convex functions. Furthermore, we find estimates on the coefficients \(\left\vert a_{1}\right\vert\) and \(\left\vert a_{2}\right\vert\) for functions in these classess. Several related classes of functions are also considered.

OMA-Vol. 2 (2018), Issue 2, pp. 93–113 | Open Access Full-Text PDF

Abstract:In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish both the existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.