Open Journal of Mathematical Analysis (OMA)

The Simpson’s inequality cannot be applied to a function that is twice differentiable but not four times differentiable or have a bounded fourth derivative in the interval under consideration. Loads of articles are bound for twice differentiable convex functions but nothing, to the best of our knowledge, is known yet for twice differentiable exponentially convex and quasi-convex functions. In this paper, we aim to do justice to this query. For this, we prove several Simpson’s type inequalities for exponentially convex and exponentially quasi-convex functions. Our findings refine, generalize and complement existing results in the literature. We regain previously known results by taking \(\alpha=0\). In addition, we also show the importance of our results by applying them to some special means of positive real numbers and to the Simpson’s quadrature rule. The obtained results can be extended for different kinds of convex functions.

In this article, we consider the limit cycles of a class of planar polynomial differential systems of the form
$$\dot{x}=-y+\varepsilon (1+\sin ^{n}\theta )xP(x,y)$$
$$ \dot{y}=x+\varepsilon (1+\cos ^{m}\theta )yQ(x,y),
$$
where \(P(x,y)\) and \(Q(x,y)\) are polynomials of degree \(n_{1}\) and \(n_{2}\) respectively and \(\varepsilon\) is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a linear center \(\dot{x}=-y, \dot{y}=x,\) by using the averaging theory of first order.

This work deals with a boundary value problem for a nonlinear semipositone multi-point fractional differential equation. By using the Schauder fixed point theorem, we show the existence of one solution for this problem. Our result extend some recent works in the literature.

A two-strain model of the transmission dynamics of herpes simplex virus (HSV) with treatment is formulated as a deterministic system of nonlinear ordinary differential equations. The model is then analyzed qualitatively, with numerical simulations provided to support the theoretical results. The basic reproduction number \(R_0\) is computed with \(R_0=\text{max}\lbrace R_1, R_2 \rbrace\) where \(R_1\) and \(R_2\) represent respectively the reproduction number for HSV1 and HSV2. We also compute the invasion reproductive numbers \(\tilde{R}_1\) for strain 1 when strain 2 is at endemic equilibrium and \(\tilde{R}_2\) for strain 2 when strain 1 is at endemic equilibrium. To determine the relative importance of model parameters to disease transmission, sensitivity analysis is carried out. The reproduction number is most sensitive respectively to the contact rates \(\beta_1\), \(\beta_2\) and the recruitment rate \(\pi\). Numerical simulations indicate the co-existence of the two strains, with HSV1 dominating but not driving out HSV2 whenever \(R_1 > R_2 > 1\) and vice versa.

In this paper, the \(q\)-derivative operator and the principle of subordination were employed to define a subclass \(\mathcal{B}_q(\tau,\lambda,\phi)\) of analytic and bi-univalent functions in the open unit disk \(\mathcal{U}\). For functions \(f(z)\in\mathcal{B}_q(\tau,\lambda,\phi)\), we obtained early coefficient bounds and some Fekete-Szegö estimates for real and complex parameters.

In this paper, we give characterizations of separation criteria for bitopological spaces via \(ij\)-continuity. We show that if a bitopological space is a separation axiom space, then that separation axiom space exhibits both topological and heredity properties. For instance, let \((X, \tau_{1}, \tau_{2})\) be a \(T_{0}\) space then, the property of \(T_{0}\) is topological and hereditary. Similarly, when \((X, \tau_{1}, \tau_{2})\) is a \(T_{1}\) space then the property of \(T_{1}\) is topological and hereditary. Next, we show that separation axiom \(T_{0}\) implies separation axiom \(T_{1}\) which also implies separation axiom \(T_{2}\) and the converse is true.

In this paper, we present the notion of generalized \(F\)-expansive mapping in complete rectangular metric spaces and study various fixed point theorems for such mappings. The findings of this paper, generalize and improve many existing results in the literature.

In this paper, we precise the hyper order of solutions for a class of higher order linear differential equations and investigate the exponents of convergence of the fixed points of solutions and their first derivatives for the second order case. These results generalize those of Nan Li and Lianzhong Yang and of Chen and Shon.

The Cauchy problem for a higher order modification of the nonlinear Schrödinger equation (MNLS) on the line is shown to be well-posed in Sobolev spaces with exponent \(s > \frac{1}{4}\). This result is achieved by demonstrating that the associated integral operator is a contraction on a Bourgain space that has been adapted to the particular linear symbol present in the equation. The contraction is proved by using microlocal analysis and a trilinear estimate that is shown via the \([k; Z]\)-multiplier norm method developed by Terence Tao.

In this paper, we give characterizations of orthogonality conditions in certain classes of normed spaces. We first consider Range-Kernel orthogonality in norm-attainable classes then we characterize orthogonality conditions for Jordan elementary operators.