ON LOCAL INTEGRATED $C$-COSINE FUNCTION AND WEAK SOLUTION OF SECOND ORDER ABSTRACT CAUCHY PROBLEM

Approximate Solution of the Second Order Initial Value Problem by Using Epsilon Modified Block-Pulse Function

The present work approaches the problem of achieving the approximate solution of the second order initial value problems (IVPs) via its conversion into a Volterra integral equation of the second kind (VIE2). Therefore, we initially solve the IVPs using Runge–Kutta of the forth–order method (RK), and then convert it into VIE2, and apply the εmodified block–pulse functions (εMBPFs) and their oper...

A Second-order Impulsive Cauchy Problem

We study the existence of mild and classical solutions for an abstract second-order impulsive Cauchy problem modeled in the form ü(t) = Au(t)+f(t,u(t),u̇(t)), t ∈ (−T0,T1), t ≠ ti; u(0) = x0, u̇(0) = y0; u(ti) = I1 i (u(ti)), u̇(ti) = I2 i (u̇(t+ i )), where A is the infinitesimal generator of a strongly continuous cosine family of linear operators on a Banach space X and f , I1 i , I 2 i are appro...

Inhomogeneous Two-Parameter Abstract Cauchy Problem

Existence and blow-up of solution of Cauchy problem for the sixth order damped Boussinesq equation

‎In this paper‎, ‎we consider the existence and uniqueness of the global solution for the sixth-order damped Boussinesq equation‎. ‎Moreover‎, ‎the finite-time blow-up of the solution for the equation is investigated by the concavity method‎.

Integrated Semigroups and Their Applications to the Abstract Cauchy Problem

This paper is concerned with characterizations of those linear, closed, but not necessarily densely defined operators A on a Banach space E with nonempty resolvent set for which the abstract Cauchy problem u'(t) = Au(t), u(0) = x has unique, exponentially bounded solutions for every initial value x e D(A). Investigating these operators we are led to the class of "integrated semigroups". Among o...