RIEMANN–LIOUVILLE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH FRACTIONAL NONLOCAL MULTI-POINT BOUNDARY CONDITIONS

In this paper, we investigate the existence and uniqueness of solutions for Riemann–Liouville fractional integro-differential equations equipped with fractional nonlocal multi-point and strip boundary conditions in the weighted space. The methods of our study include the well-known tools of the fixed point theory, which are commonly applied to establish the existence theory for the initial and boundary value problems after converting them into the fixed point problems. We also discuss the case when the nonlinearity depends on the Riemann–Liouville fractional integrals of the unknown function. Numerical examples illustrating the main results are presented.

Mean Convergence of Entire Interpolations in Weighted Space

We investigate the convergence of entire Lagrange interpolations and of Hermite interpolations of exponential type $$\tau $$ τ , as $$\tau \rightarrow \infty $$ τ → ∞ , in weighted $$L^p$$ L p -spaces on the real line. The weights are reciprocals of entire functions that depend on $$\tau $$ τ and may be viewed as smoothed versions of a target weight w. The convergence statements are obtained from weighted Marcinkiewicz inequalities for entire functions. We apply our main results to deal with power weights.

Magnetic WKB constructions on surfaces

This article is devoted to the description of the eigenvalues and eigenfunctions of the magnetic Laplacian in the semiclassical limit via the complex WKB method. Under the assumption that the magnetic field has a unique and non-degenerate minimum, we construct the local complex WKB approximations for eigenfunctions on a general surface. Furthermore, in the case of the Euclidean plane, with a radially symmetric magnetic field, the eigenfunctions are approximated in an exponentially weighted space.

Stability of the mixed Caputo fractional integro-differential equation by means of weighted space method

<abstract><p>In this research work, we consider a class of nonlinear fractional integro-differential equations containing Caputo fractional derivative and integral derivative. We discuss the stabilities of Ulam-Hyers, Ulam-Hyers-Rassias, semi-Ulam-Hyers-Rassias for the nonlinear fractional integro-differential equations in terms of weighted space method and Banach fixed-point theorem. After the demonstration of our results, an example is given to illustrate the results we obtained.</p></abstract>

Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces

<p style='text-indent:20px;'>A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences <inline-formula><tex-math id="M1">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. First the existence of a pullback attractor in <inline-formula><tex-math id="M2">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula> is established by utilizing the dense inclusion of <inline-formula><tex-math id="M3">\begin{document}$ \ell^2 \subset {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.</p>