The present paper is a study of some direct results in \(L_p\)−approximation by a linear combination of summation-integral type operators. We obtain an error estimate in terms of the higher order modulus of smoothness using some properties of the Steklov mean.

This article presents the results on existence, uniqueness and stability of mild solution for impulsive stochastic semilinear functional differential equations with non-Lipschitz condition and Lipschitz condition. The results are obtained by using the method of successive approximation and Bihari’s inequality.

In this paper we consider three point boundary value problems of second order. We introduce new and sufficient conditions that allow us to obtain the existence of a nontrivial solution by using Leray Schauder nonlinear alternative. As an application, we give some examples to illustrate our results.

In this paper, we study the existence of three solutions to the following nonlinear m-point boundary value problem \[ \begin{cases} u''(t) + \beta^2u(t) = h(t)f(t, u(t)),\,\,\,\,\, 0 < t < 1,\\ u'(0) = 0, u(1) =\Sigma^{m-2}_{i=1}\alpha_i u(\eta_i), \end{cases} \] where \(0<\beta<\frac{\pi}{2}, f\in C([0,1]\times \mathbb{R}^+, \mathbb{R}^+). h(t)\) is allowed to be singular at \(t = 0\) and \(t = 1\). The arguments are based only upon the Leggett-Williams fixed point theorem. We also prove nonexist results.

Rough sets were originally proposed in the presence of an equivalence relation. An equivalence relation is sometimes difficult to be obtained in rearward problems due to the vagueness and incompleteness of human knowledge. The purpose of this paper is to introduce and discuss the concept of T-rough semiprime ideal, T-rough fuzzy semiprime ideal and T-rough quotient ideal in a commutative ring which are a generalization of rough set and approximation theory. We compare relation between a rough ideal and a T-rough ideal and prove some theorems.

In this paper, we establish a set of sufficient conditions for the controllability of nonlocal impulsive functional integrodifferential evolution systems with finite delay. The controllability results are obtained with out assuming the compactness condition on the evolution operator by using the semigroup theory and applying the fixed point approach. An example is provided to illustrate the theory.

In this paper, we prove that an implicit random iteration process with errors which is generated by a finite family of asymptotically quasi- nonexpansive random operators converges strongly to a common random fixed point of the random operators in uniformly convex Banach spaces.

In this paper, a family of nonlinear functions is given for the exact controllability of semilinear third order dispersion equation. The obtained result has been illustrated by applying it on nonlinear Korteweg-de Vries (KdV) equation.

Stability analysis is performed and stabilization strategies are proposed for a general class of stochastic delay differential equations subjected to switching and impulses. Hybrid switching and impulses are combined to exponentially stabilize an otherwise unstable stochastic delay system. Three differential stabilization strategies are proposed, i.e. the average dwellime approach, the impulsive stabilization, and a combined strategy. Both moment stability and almost sure stability of the resulting impulsive and switched hybrid stochastic delay systems are investigated using the well-known Lyapunov- Razumikhin method in the hybrid and stochastic setting. Several examples are presented to illustrate the main results and numerical simulations are presented to demonstrate the analytical results.