Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces

Pengyu Chen; Yongxiang Li

Applications of Mathematics (2014)

  • Volume: 59, Issue: 1, page 99-120
  • ISSN: 0862-7940

Abstract

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In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space E u ' ( t ) + A u ( t ) = f ( t , u ( t ) , G u ( t ) ) , t ∈ J , t ≠ t k , Δ u | t = t k = u ( t k + ) - u ( t k - ) = I k ( u ( t k ) ) , k = 1 , 2 , ⋯ , m , u ( 0 ) = g ( u ) + x 0 , where A : D ( A ) ⊂ E → E is a closed linear operator and - A generates a strongly continuous semigroup T ( t ) ( t ≥ 0 ) on E , f ∈ C ( J × E × E , E ) , J = [ 0 , a ] , 0 < t 1 < t 2 < ⋯ < t m < a , I k ∈ C ( E , E ) , k = 1 , 2 , ⋯ , m , and g constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that - A generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered.

How to cite

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Chen, Pengyu, and Li, Yongxiang. "Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces." Applications of Mathematics 59.1 (2014): 99-120. <http://eudml.org/doc/260820>.

@article{Chen2014,
abstract = {In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space $E$\[ \{\left\lbrace \begin\{array\}\{ll\} u^\{\prime \}(t)+Au(t)= f(t,u(t),Gu(t)),\quad t\in J, t\ne t\_k, \Delta u |\_\{t=t\_k\}=u(t\_k^+)-u(t\_k^-)=I\_k(u(t\_k)),\quad k=1,2,\dots ,m, u(0)=g(u)+x\_0, \end\{array\}\right.\} \] where $A\colon D(A)\subset E\rightarrow E$ is a closed linear operator and $-A$ generates a strongly continuous semigroup $T(t)$$(t\ge 0)$ on $E$, $f\in C(J\times E\times E, E)$, $J=[0,a]$, $0<t_1<t_2<\dots <t_m<a$, $I_k\in C(E,E)$, $k=1,2,\dots ,m$, and $g$ constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that $-A$ generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered.},
author = {Chen, Pengyu, Li, Yongxiang},
journal = {Applications of Mathematics},
keywords = {evolution equation; impulsive integro-differential equation; nonlocal condition; lower and upper solutions; monotone iterative technique; mild solution; evolution equation; impulsive integro-differential equation; nonlocal condition; lower and upper solutions; monotone iterative technique; mild solution},
language = {eng},
number = {1},
pages = {99-120},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces},
url = {http://eudml.org/doc/260820},
volume = {59},
year = {2014},
}

TY - JOUR
AU - Chen, Pengyu
AU - Li, Yongxiang
TI - Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces
JO - Applications of Mathematics
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 1
SP - 99
EP - 120
AB - In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space $E$\[ {\left\lbrace \begin{array}{ll} u^{\prime }(t)+Au(t)= f(t,u(t),Gu(t)),\quad t\in J, t\ne t_k, \Delta u |_{t=t_k}=u(t_k^+)-u(t_k^-)=I_k(u(t_k)),\quad k=1,2,\dots ,m, u(0)=g(u)+x_0, \end{array}\right.} \] where $A\colon D(A)\subset E\rightarrow E$ is a closed linear operator and $-A$ generates a strongly continuous semigroup $T(t)$$(t\ge 0)$ on $E$, $f\in C(J\times E\times E, E)$, $J=[0,a]$, $0<t_1<t_2<\dots <t_m<a$, $I_k\in C(E,E)$, $k=1,2,\dots ,m$, and $g$ constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that $-A$ generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered.
LA - eng
KW - evolution equation; impulsive integro-differential equation; nonlocal condition; lower and upper solutions; monotone iterative technique; mild solution; evolution equation; impulsive integro-differential equation; nonlocal condition; lower and upper solutions; monotone iterative technique; mild solution
UR - http://eudml.org/doc/260820
ER -

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