Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0939.45004
Liu, Xinzhi; Guo, Dajun
Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space. (English)
[J] Comput. Math. Appl. 38, No. 3-4, 213-223 (1999). ISSN 0898-1221

The initial value problem is investigated for the second-order nonlinear impulsive integro-differential equations of Volterra type $$x''=f(t,x,Tx), \quad\forall t\in J,\ t\ne t_i,$$ $$\Delta x|_{t= t_i}=- \sum^i_{j=1} \alpha_{ij} x(t_j)+ \sum^i_{j=1} \beta_{ij} x'(t_j),$$ $$\Delta x'|_{t=t_i}= -\sum^i_{j=1} \gamma_{ij} x(t_j),$$ $$x(0)=x_0,\ x'(0) =x_1,$$ where $f\in C[J\times E\times E,E]$, $J=[0,a](a>0)$, $0<t_1< \cdots< t_i < \cdots< t_m<a$, $\alpha_{ij}$, $\beta_{ij}$, $\gamma_{ij}(i\ge j,\ i=1,2, \dots, m)$ are nonnegative constants $x_0,x_1\in E$, and $$(Tx)(t)= \int^t_0k (t,s) x(s)ds,\quad forall t\in J,$$ $k\in C[D,R_+]$, $D=\{(t,s)\in J\times J: t\ge s\}$, $R_+$ is the set of all nonnegative numbers, in a real Banach space by means of upper and lower solutions. Conditions for the existence of maximal and minimal solutions are established.
[I.Foltyńska (Poznań)]

MSC 2000:: *45J05 Integro-ordinary differential equations
45N05 Integral equations in abstract spaces
45G10 Nonsingular nonlinear integral equations
45L05 Theoretical approximation of solutions of integral equations

Keywords: initial value problem; second-order nonlinear impulsive integro-differential equations of Volterra type; Banach space; upper and lower solutions; maximal and minimal solutions

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]