Volume 43, Issue 1

In this paper, we prove the existence and uniqueness of Caputo time fractional pseudo-hyperbolic equations of higher order with purely nonlocal conditions of integral type. We use an a priori estimate method; the so-called energy inequalities method, based on some functional analysis tools, is developed for a Caputo time fractional of 2 ⁢ m {2m} -th and ( 2 ⁢ m + 1 ) {(2m+1)} -th order and the density of the range of the operator generated by the considered problem. Using the Laplace transform and homotopy perturbation, we find a semi-analytical solution. Finally, we give some examples for illustration.

In linear systems, it is well known that any combination of stabilizing state feedback controller and a state observer results in the output feedback controller which asymptotically stabilizes the closed-loop system. This is known as the separation principle, which does not generally hold for nonlinear systems. For nonlinear systems, the output feedback problem is usually pursued by extending the results of observer synthesis, which itself is a challenging problem. In this paper, we investigate the stabilization via an observer configuration problem for some classes of nonlinear time-varying systems by using new integral inequalities of Gronwall type. Given that the sufficient conditions of the controller and observer problem are satisfied, we show that the proposed controller with estimated state feedback from the proposed practical observer will achieve the global practical exponential stabilization. Furthermore, the effectiveness of the proposed scheme is shown through simulation for two numerical examples.

This work studies the coupled nonlinear fourth-order wave system u ¨ i + Δ 2 ⁢ u i + u i = ± ( ∑ 1 ≤ j ≤ m a i ⁢ j ⁢ | u j | p ) ⁢ | u i | p - 2 ⁢ u i . \ddot{u}_{i}+\Delta^{2}u_{i}+u_{i}=\pm\bigg{(}\sum_{1\leq j\leq m}a_{ij}\lvert u% _{j}\rvert^{p}\biggr{)}\lvert u_{i}\rvert^{p-2}u_{i}. The main goal is to develop a local theory in the energy space and to investigate some issues of the global theory. Indeed, using a standard contraction argument coupled with Strichartz estimates, one obtains a local solution in the inhomogeneous Sobolev space ( H 1 ) m {(H^{1})^{m}} for the energy sub-critical regime. Then the local solution extends to a global one in the attractive regime; also in the energy critical case there is a global solution with small data. For a repulsive source term, by using the potential well theory with a concavity argument, the local solution may concentrate in finite time or extend to a global one. Finally, in the inter-critical regime, one proves the existence of infinitely many non-global solutions with data near to the stationary solution. Here, one needs to deal with the coupled source term which gives some technical restrictions such as p ≥ 2 {p\geq 2} in order to avoid a singularity. This assumption in the inter-critical regime gives a restriction on the space dimension.

In this paper, we investigate and evaluate the analytical expressions for some definite integrals of Srinivasa Ramanujan in terms of Meijer’s G -function by using the Laplace transforms of sin ⁡ ( β ⁢ x 2 ) {\sin(\beta x^{2})} , cos ⁡ ( β ⁢ x 2 ) {\cos(\beta x^{2})} , x ⁢ sin ⁡ ( β ⁢ x 2 ) {x\sin(\beta x^{2})} and x ⁢ cos ⁡ ( β ⁢ x 2 ) {x\cos(\beta x^{2})} . In addition, we investigate a number of infinite summation formulas involving Meijer’s G -function and closed-form evaluation of some related infinite series.

In this paper, we first prove a new parameterized identity that generates a quadrature rule family similar to Simpson’s second formula, and then we establish some new Simpson-like type inequalities for functions whose first derivatives are s -convex in the second sense, from which we can deduce the famous 3 8 {\frac{3}{8}} -Simpson inequality. We end the article with some applications.