Advanced Search

Query:

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

Format:

Display: entries per page entries

Zbl 1166.45002
Luo, Zhiguo; Nieto, Juan J.
New results for the periodic boundary value problem for impulsive integro-differential equations. (English)
[J] Nonlinear Anal., Theory Methods Appl. 70, No. 6, A, 2248-2260 (2009). ISSN 0362-546X

Consider $J= [0,T]$, $T> 0$, the continuous function $f: J\times\Bbb R^3\to\Bbb R$, the continuous functions $I_k:\Bbb R\to\Bbb R$, $1\le k\le m$, $0= t_0< t_1<\cdots< t_m< t_{m+1}= T$, the set $D= \{(t,s)\in J\times J$; $t\ge s\}$, the functions $K\in C(D,[0,+\infty))$, $H\in C(J\times J,[0,+\infty))$ and the functions $$[{\Cal T}u](t)= \int^t_0 K(t,s)u(s)\,ds,\quad t\in J,\qquad [{\Cal S}u](t)= \int^T_0 H(t,s)u(s)\,ds,\quad t\in J,$$ where $u: J\to\Bbb R$.\par Suppose that there exist the limits $$u(t^+_k)= \lim\Sb t\to t_k\\ t< t_k\endSb u(t),\quad u(t^-_k)= \lim\Sb t\to t_k\\ t> t_k\endSb u(t),\quad 1\le k\le m,$$ and denote $\Delta u(t_k)= u(t^+_k)- u(t^-_k)$, $1\le k\le m$. The authors consider the first-order impulsive integrodifferential equation $$u'(t)= f(t,u(t), [{\Cal T}u](t), [{\Cal S}u](t)),\quad t\in J\setminus\{t_1,\dots, t_m\}\tag 1$$ with periodic boundary value conditions $$\cases \Delta u(t_k)= I_k(u(t_k)),\quad & 1\le k\le m,\\ u(0)= u(T)\endcases\tag2$$ and prove some comparison principles and establish existence results for extremal solutions $u$ of the problem $(1)\wedge (2)$ using these principles and the monotone iterative technique. For example, they consider the Banach spaces $(PC(J),\Vert.\Vert_{PC})$ and $(PC^1(J),\Vert.\Vert_{PC^1})$, where $$\multline PC(J)= \{u: J\to\Bbb R; u|_{(t_k,t_{k+1}]}\in C((t_k, t_{k+1}[,\Bbb R),\ 0\le k\le m,\ \exists u(t^+_k),\\ \exists u(t^-_k)= u(t_k),\ 1\le k\le m\},\endmultline$$ $$\multline PC^1(J)= \{u\in PC(J); u|_{(t_k, t_{k+1})}\in C^1((t_k, t_{k+1}],\Bbb R),\ 0\le k\le m,\ \exists u'(0^+),\\ \exists u'(T^-),\ \exists u'(t^+_k),\ \exists u'(t^-_k),\ 1\le k\le m\}\endmultline$$ with the norms $\Vert u\Vert_{PC}= \sup\{|u(t)|; t\in J\}$, respectively, $\Vert u\Vert_{PC^1}= \Vert u\Vert_{PC}+\Vert u'\Vert_{PC}$ and if there exist the functions $\alpha$ and $\beta$ in $PC^1(J)$, $\alpha\le \beta$, satisfying some hypotheses, then there exist monotone sequences $(\alpha_n)_n$, $(\beta_n)_n$ of functions with $$\alpha= \alpha_0\le \alpha_n\le \beta_n\le \beta_0= \beta,\quad n\in\bbfN,$$ which converge uniformly on $J$ to the extremal solutions $u$ of the problem $(1)\wedge (2)$ in $$[\alpha,\beta]= \{u\in PC(J);\, \alpha(t)\le u(t)\le \beta(t),\,t\in J\}.$$
[Dan-Mircea Borş (Iaşi)]

MSC 2000:: *45J05 Integro-ordinary differential equations
45L05 Theoretical approximation of solutions of integral equations

Keywords: periodic boundary value problem; impulsive integro-differential equations; comparison principle; monotone iterative technique; extremal solutions

Cited in: Zbl 1190.45005

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]