This paper addresses the oscillation problem of a class of impulsive differential equations with delays and Riemann-Stieltjes integrals that cover many equations in the literature. In the case of oscillatory potentials, both El-Sayed type and Kamenev type oscillation criteria are established by overcoming the difficulty caused by impulses and oscillatory potentials in the estimation of the dela...

In this paper a real analysis approach to stock price modelling is considered. A stock price and its return are defined in a duality to each other provided there exist suitable limits along a sequence of nested partitions of a time interval, mimicking sum and product integrals. It extends the class of stochastic processes susceptible to theoretical analysis. Also, it is shown that extended clas...

We show that a trader, who starts with no initial wealth and is not allowed to borrow money or short sell assets, is theoretically able to attain positive wealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly inno...

Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and a linear closed operator A, studied. The issues of unique solvability Cauchy problem to such considered. Under Lipschitz continuity condition phase variables two types over all nonlinear equation, we obtain versions on theorem nonlocal existe...

Convergence in probability of random Riemann sums of a Lebesgue integrable function on [0, 1) to the integral has been proved. In this article we generalize the result to an abstract probability space under some natural conditions and we show L1convergence rather than convergence in probability..

Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A) are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A) in terms of Riemann...

In this article, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, some properties of the first eigenvalue of a fractional differential equation are obtained. Based on these properties and through the fixed point index theory, the singular nonlinear fractional differential equations with Riemann–Stieltjes integral boundary conditions involving fractional deri...

M f dg1 ∧ · · · ∧ dgn in case the functions f, g1, . . . , gn are merely Hölder continuous of a certain order by extending the construction of the Riemann-Stieltjes integral to higher dimensions. More generally, we show that for α ∈ ( n n+1 , 1] the n-dimensional locally normal currents in a locally compact metric space (X, d) represent a subspace of the n-dimensional currents in (X, d). On the...

Let A be a symmetric matrix and let f be a smooth function deened on an interval containing the spectrum of A: Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A) are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A) in terms of Riemann{...

where f ,g ∈ C([0, 1]× R+ × R+,R+), α[u] = ∫ 1 0 u(t)dA(t) and β[u] = ∫ 1 0 u(t)dB(t) are linear functionals on C[0, 1] given by Riemann-Stieltjes integrals and are not necessarily positive functionals; a, b, c, d are nonnegative constants with ρ := ac + ad + bc > 0. By using the Guo-Krasnoselskii fixed point theorem, some sufficient conditions are obtained for the existence of at least one or ...