Open Journal of Mathematical Analysis (OMA)

It is proved that if the problem \(\nabla^2u=1\) in \(D\), \(u|_S=0\), \(u_N=m:=|D|/|S|\) then \(D\) is a ball. There were at least two different proofs published of this result. The proof given in this paper is novel and short.

In this paper, we use Riccati transformation technique to establish some new oscillation criteria for the second order nonlinear dynamic equation with damping on time scales $$(r(t)(x^\Delta(t))^\alpha)^\Delta-p(t)(x^\Delta(t))^\alpha+q(t)f(x(t))=0.$$ Our results not only generalize some existing results, but also can be applied to the oscillation problems that are not covered in literature. Finally, we give some examples to illustrate our main results.

We define fractional transforms \(\mathscr{R}_\mu\) and \(\mathscr{H}_\mu\), \(\mu>0\) on the space \(\mathbb{R}\times\mathbb{R}^n\). First, we study these transforms on regular function spaces and we establish that these operators are topological isomorphisms and we give the inverse operators as integro differential operators. Next, we study the \(L^p\)-boundedness of these operators. Namely, we give necessary and sufficient condition on the parameter \(\mu\) for which the transforms \(\mathscr{R}_\mu\) and \(\mathscr{H}_\mu\) are bounded on the weighted spaces \(L^p([0,+\infty[\times\mathbb{R}^n,r^{2a}dr\otimes dx)\) and we give their norms.

Let \(S\) be a \(C^{1}\)-smooth closed connected surface in \(\mathbb{R}^3\), the boundary of the domain \(D\), \(N=N_s\) be the unit outer normal to \(S\) at the point \(s\), \(P\) be the normal section of \(D\). A normal section is the intersection of \(D\) and the plane containing \(N\). It is proved that if all the normal sections for a fixed \(N\) are discs, then \(S\) is a sphere. The converse statement is trivial.

Let \(D\) be an open subset of \(\mathbf R^N\) and \(f: \overline D\to \mathbf R^N\) a continuous function. The classical topological degree for \(f\) demands that \(D\) be bounded. The boundedness of domains is also assumed for the topological degrees for compact displacements of the identity and for operators of monotone type in Banach spaces. In this work, we follow the methodology introduced by Nagumo for constructing topological degrees for functions on unbounded domains in finite dimensions and define the degrees for Leray-Schauder operators and \((S_+)\)-operators on unbounded domains in infinite dimensions.

The aim of this paper is to present a viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces. The strong convergence of the viscosity rules is proved with some assumptions. This paper extend and improve results presented in [1, 2, 3, 4].

Chemical reaction network theory is an area of applied mathematics that attempts to model the behavior of real world chemical systems. Since its foundation in the 1960s, it has attracted a growing research community, mainly due to its applications in biochemistry and theoretical chemistry. It has also attracted interest from pure mathematicians due to the interesting problems that arise from the mathematical structures involved. In this report, we compute newly defined topological indices, namely, Arithmetic-Geometric index (\(AG_{1}\) index), \(SK\) index, \(SK_{1}\) index, and \(SK_{2}\) index of the Honey Comb Derived Networks. We also compute sum connectivity index and modified Randić index. Moreover we give geometric comparison of our results.

The aim of this paper is to present new sixth order iterative methods for solving non-linear equations. The derivation of these methods is purely based on variational iteration technique. Our methods are verified by means of various test examples and numerical results show that our developed methods are more effective with respect to the previously well known methods.

A line graph has many useful applications in physical chemistry. Topological indices are numerical parameters associated to a structure and, in combination, determine properties of the concerned material. In this paper, we compute the closed form of Zagreb polynomilas of all generalized class of carbon nanocones and compute important degree-based topological indices.

Unsteady free convection flow of Casson fluid over an unbounded upright plate subject to time dependent velocity \(U_{o}f(t)\) with constant wall temperature has been carried out. By introducing dimensionless variables, the general solutions are obtained by Laplace transform method. The solution corresponding to Newtonian fluid for \(\gamma \rightarrow \infty\) is obtained as a limiting case. Exact solutions corresponding to (i) \(f(t)=f H(t)\), (ii) \(f(t)=f t^{a}\), \(a > 0 \) (iii) \( f(t)=f H(t)cos(\omega t)\) are also discussed as special cases of our general solutions. Expressions for shear stress in terms of skin friction and the rate of heat transfer in the form of Nusselt number are also presented. Velocity and temperature profiles for different parameters are discussed graphically.Free convection; Time depending velocity; Exact solutions; Casson fluid; Vertical plate.