On a nonlinear Volterra integrodifferential equation in Banach spaces

Abstract

Abstract. The objective of the present paper is to study the local existence, global existence, uniqueness, continuous dependence, asymptotic stability and other properties of solutions of a nonlinear Volterra integrodifferential equation in Banach spaces of more general type. The technique employed in our analysis is based on treating the equation in the domain of the infinitesimal generator of semigroups of linear operators in a Banach space with graph norm and using results from linear semigroup theory. (2000): 45N05, 45J05. Mathematics subject classification

Piecewise constant bounds for the solution of nonlinear Volterra-Fredholm integral equations

“…Diekmann and Thieme described and developed some of these models in [7,24]. The existence and uniqueness of the global solution of the equations have been investigated by Burton [4] and Dhakne et al [6]. Also some of the schemes which have considered the solution of the linear and nonlinear cases of (1.1) numerically are projection methods, continuous time and discrete time spline collocation method S. YAZDANI and M. HADIZADEH 307 [2,16], Euler Nystrom and trapezoidal Nystrom methods [13], the Adomian decomposition method [18], the time-stepping methods by a certain choice of direct quadrature (DQ) [3] and Sinc collocation method based on DE transformation in [14].…”

On nonlinear mixed fractional integrodifferential equations with nonlocal condition in Banach spaces

“…Recently, in an interesting paper [7], XiWang Dong, JinRong Wang and Yong Zhou have investigated the existence, uniqueness and boundedness of solutions of a special form of (1.1)-(1.2). Also in [5], Dhakne and Kendre have studied the existence, uniqueness and other properties of solutions of a special form of (1.1)-(1.2) when = 1 with initial condition.…”

“…The problems of existence, uniqueness and other properties of solutions of (1.1)-(1.2) or their special forms have been studied by many authors by using di erent techniques, see [1,8,14,15] and the references given therein; also, see [5,6,11]. Equations of the form (1.1)-(1.2) arise in various applications like chemical reaction kinetics, population dynamics, heat ow in material with memory, viscoelastic and reaction di usion problems, see [3,4,10] and some of the references cited therein.…”

Convergence of solutions for functional integro-differential equations with nonlinear boundary conditions

“…Integro-differential equations are widely used in many fields such as control theory, biology, and mechanics, and the qualitative theory of integro-differential equations creates an important branch of nonlinear analysis; see, for instance, the monographs [2,6,7] and the papers [5, 10-12, 16, 24, 26, 31, 33, 34]. For the results of existence of solutions and existence of extremal solutions for such equations under different boundary conditions, we refer the reader to the monographs by Guo et al [13] and Lakshmikantham and Rama Mohana Rao [23], the related literature for integro-differential equations [1,4,8,15,27,30], and for functional integro-differential equations [3,14,17,20,21,28,35,36], and the references cited therein.…”