Existence, uniqueness, variation-of-constant formula and controllability for linear dynamic equations with Perron Δ-integrals

In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron [Formula: see text]-integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for linear dynamic equations. Since we work in the framework of Perron [Formula: see text]-integrals, we can handle functions not only having many discontinuities, but also being highly oscillating. Our results require weaker conditions than those in the literature. We include some examples to illustrate our main results.

Existence and uniqueness results for Riemann–Stieltjes integral boundary value problems of nonlinear implicit Hadamard fractional differential equations

In this paper, we study the existence and uniqueness of solutions for Riemann–Stieltjes integral boundary value problems of nonlinear implicit Hadamard fractional differential equations. The investigation of the main results depends on Schauder’s fixed point theorem and Banach’s contraction principle. An illustrative example is given to show the applicability of theoretical results.

Existence Solution for Coupled System of Langevin Fractional Differential Equations of Caputo Type with Riemann–Stieltjes Integral Boundary Conditions

By employing Shauder fixed-point theorem, this work tries to obtain the existence results for the solution of a nonlinear Langevin coupled system of fractional order whose nonlinear terms depend on Caputo fractional derivatives. We study this system subject to Stieltjes integral boundary conditions. A numerical example explaining our result is attached.

On a Nonlocal Boundary Value Problem of a State-Dependent Differential Equation

In this paper, the existence of absolutely continuous solutions and some properties will be studied for a nonlocal boundary value problem of a state-dependent differential equation. The infinite-point boundary condition and the Riemann–Stieltjes integral condition will also be considered. Some examples will be provided to illustrate our results.

The quantum-mechanical Coulomb propagator in an L2 function representation

The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss–Pollaczek quadrature, the Gauss–Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss–Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L$$^2$$ 2 basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures.

Belonging of Laplace-Stieltjes integrals to convergence classes

For positive continuous functions $\alpha$ and $\beta$ increasing to $+\infty$ on $[x_0,+\infty)$ and the Laplace--Stieltjes integral $I(\sigma)=\int\limits_{0}^{\infty}f(x)e^{x\sigma}dF(x),\,\sigma\in{\Bbb R}$, a generalized convergence $\alpha\beta$-class is defined by the condition $$\int\limits_{\sigma_0}^{\infty}\dfrac{\alpha(\ln\,I(\sigma))}{\beta(\sigma)}d\sigma<+\infty.$$ Under certain conditions on the functions $\alpha$, $\beta$, $f$, and $F$, it is proved that the integral $I$ belongs to the generalized convergence $\alpha\beta$-class if and only if $\int\limits_{x_0}^{\infty}\alpha'(x)\beta_1 \left(\dfrac1{x}\ln\dfrac1{f(x)}\right)<+\infty,\,\beta_1(x)= \int\limits_{x}^{+\infty}\dfrac{d\sigma}{\beta(\sigma)}$. For a positive, convex on $(-\infty,\,+\infty)$ function $\Phi$ and the integral $I$, a convergence $\Phi$-class is defined by the condition $\int\limits_{\sigma_0}^{\infty}\dfrac{\Phi'(\sigma)\ln\,I(\sigma)}{\Phi^2(\sigma)}d\sigma<+\infty$, and it is proved that under certain conditions on $\Phi$, $f$ and $F$, the integral $I$ belongs to the convergence $\Phi$-class if and only if $\int\limits_{x_0}^{\infty}\dfrac{dx}{\Phi'\left(({1/x)\ln\,(1/f(x))}\right)}<+\infty$. Conditions are also found for the integral of the Laplace--Stieltjes type $\int\limits_{0}^{\infty} f(x)g(x\sigma)dF(x)$ to belong to the generalized convergence $\alpha\beta$-class if and only if the function $g$ belongs to this class.