We develop Trapezoid, Midpoint, and Simpson’s rules for the Riemann-Stieltjes integral, the latter two being new. These rules are completely natural when the notion of relative convexity is used. Mathematics subject classification (2010): 65D30.

Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral R b a f eit du (t) of continuous complex valued integrands f : C (0; 1)! C de…ned on the complex unit circle C (0; 1) and various subclasses of integrators u : [a; b] [0; 2 ] ! C of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well. 1. Introduction The ...

It has been proposed that the arbitrage possibility in the fractional Black–Scholes model depends on the definition of the stochastic integral. More precisely, if one uses the Wick– Itô–Skorohod integral one obtains an arbitrage-free model. However, this integral does not allow economical interpretation. On the other hand it is easy to give arbitrage examples in continuous time trading with sel...

We prove an inequality of the Loéve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces, and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any [Formula: see text], we introduce the space of regulated signals [Formula: see text] ([Formula: see text] are rea...

In the present paper we investigate the problem of approximating the Riemann-Stieltjes integral R b a f ( ) du ( ) in the case when the integrand f is (n+ 1)-time di¤erentiable (n 0) and the derivative f (n+1) is continuous on [a; b], while the integrator u is Riemann integrable on [a; b] : A priory error bounds for di¤erent classes of functions are provided.

We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants γk(a). These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials Ln to investigate a certain sum Sγ(n) involv...

Keywords: Volterra–Stieltjes integral equation Fractional integral–differential equations Riemann–Liouville fractional operators Existence and stability of solutions Fixed point a b s t r a c t Our aim in this paper is to study the existence and the stability of solutions for Riemann–Liouville Volterra–Stieltjes quadratic integral equations of fractional order. Our results are obtained by using...

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants {λk}k=1, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate a related set of constants cn, n = 1, 2, . . ., showing in detail that the leading behaviour (1/2) ln n of λn/n is absent in cn. Additional results are presented, including a novel explicit...

We study the existence and multiplicity of positive solutions for a system Riemann–Liouville fractional differential equations with sequential derivatives, parameters sign-changing singular nonlinearities, subject to nonlocal coupled boundary conditions which contain Riemann–Stieltjes integrals various derivatives. In proof our main results we use nonlinear alternative Leray–Schauder type Guo–K...

We study the existence of positive solutions for a Riemann–Liouville fractional differential equation with sequential derivatives, parameter and sign-changing singular nonlinearity, subject to nonlocal boundary conditions containing varied derivatives general Riemann–Stieltjes integrals. also present associated Green functions some their properties. In proof main results, we apply Guo–Krasnosel...