Existence Results for Second Order Differential Equations with Nonlocal Conditions in Banach Spaces

Abstract

Abstract. This work is concerned with implicit second order abstract di¤erential equations with nonlocal conditions. Assuming that the involved operators satisfy some compactness properties, we establish the existence of local mild solutions, the existence of global mild solutions and the existence of asymptotically almost periodic solutions.

Global existence of mild solutions of second order nonlinear Volterra integrodifferential equations in Banach spaces

“…The main tool used in our analysis is based on the contraction mapping principle and the cosine function theory. The theorem proved in this paper generalize the some results obtained by Eduardo Hernandez M. in [13]. We are motivated by the work of Eduardo Hernandez M. in [13] and infiuenced by the work of Byszewski [5].…”

Nonlocal semilinear second-order differential inclusions in abstract spaces without compactness

“…In [14,15,16], the left-hand side is of type d dt ( ẋ(t) − p(t, x, ẋ)) or the right-hand side is of type F (t, x, x(a(t)), ẋ, ẋ(b(t))) or F (t, N (t)x) and g and h may depend also on ẋ. However, the existence results there are proved assuming that S(t) is compact, for every t, or that F maps bounded sets into relatively compact ones, eventually that it satisfies a condition involving the Hausdorff measure of noncompactness.…”

Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion

“…It developed greater interest in various nonlocal issues related to differential equations and stochastic differential equations. Many of the basic results for nonlocal problems have been obtained, see [18][19][20][21][22][23][24] and references therein for more comments and citations. Recently, some authors have drawn attention to the Cauchy problems driven by differential equations.…”