In the present paper, we first prove a new integral identity. Using that identity, we establish some fractional weighted midpoint-type inequalities for functions whose first derivatives are extended s-convex. Some special cases are discussed. Finally, to prove the effectiveness of our main results, we provide some applications to numerical integration as well as special means.

The present paper deals with well-posedness and asymptotic stability for weakly coupled wave equations with a more general internal control of diffusive type. Owing to the semigroup theory of linear operator, the well-posedness of system is proved. Furthermore, we show a general decay rate result. The method is based on the frequency domain approach combined with multiplier technique.

This paper considers a class of fractional impulsive wave equations and improves a previous results. In fact, this paper adopts a new topological approach to prove the existence of classical solutions with a complex arguments caused by impulsive perturbations. To the best of our knowledge, there is a severe lack of results related to such impulsive equations.

We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. To prove our main results we give a suitable integral representation of the solutions of the considered problem. Then, we construct two operators so that any fixed point of their sum is a solution.

The white-noise-driven KdV-type Boussinesq system is a class of stochastic partial differential equations (SPDEs) that describe nonlinear wave propagation under the influence of random noise—specifically white noise—and generalize features from both the Korteweg–de Vries (KdV) and Boussinesq equations. We consider a Cauchy problem for two stochastic systems based on the KdV-type Boussinesq equations. For these systems, we determine sufficient conditions to ensure that this problem is locally and globally well posed for initial data in Sobolev spaces by the linear and bilinear estimates and their modification together with the Banach fixed point.

We present here more general concepts of Hausdorff derivatives (structural Nabla derivatives) on a timescale. We examine structural Nabla integration on temporal scales. Using the fixed-point theorem, we establish adequate criteria for the question of existence and uniqueness of the solution to an initial value problem characterized by structural Nabla derivatives on timescales. Furthermore, some features of the new operator are proven and illustrated by using concrete examples.