A survey and some new results on the existence of solutions of IPBVPs for first order functional differential equations

Yuji Liu

Applications of Mathematics (2009)

  • Volume: 54, Issue: 6, page 527-549
  • ISSN: 0862-7940

Abstract

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This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation x ' ( t ) = f ( t , x ( t ) , x ( α 1 ( t ) ) , ⋯ , x ( α n ( t ) ) ) for a.e. t ∈ [ 0 , T ] , Δ x ( t k ) = I k ( x ( t k ) ) , k = 1 , ⋯ , m , x ( 0 ) = x ( T ) . We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin’s continuation theorem. Examples are presented to illustrate the main results.

How to cite

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Liu, Yuji. "A survey and some new results on the existence of solutions of IPBVPs for first order functional differential equations." Applications of Mathematics 54.6 (2009): 527-549. <http://eudml.org/doc/37835>.

@article{Liu2009,
abstract = {This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation \[ \{\left\lbrace \begin\{array\}\{ll\} x^\{\prime \}(t)=f(t,x(t),x(\alpha \_1(t)),\cdots ,x(\alpha \_n(t))) \text\{for a.e.\} \ t\in [0,T], \Delta x(t\_k)=I\_k(x(t\_k)), \ k=1,\cdots ,m, x(0)=x(T). \end\{array\}\right.\} \] We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin’s continuation theorem. Examples are presented to illustrate the main results.},
author = {Liu, Yuji},
journal = {Applications of Mathematics},
keywords = {periodic boundary value problem; impulsive differential equation; fixed-point theorem; growth condition; periodic boundary value problem; impulsive differential equation; fixed-point theorem; growth condition},
language = {eng},
number = {6},
pages = {527-549},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A survey and some new results on the existence of solutions of IPBVPs for first order functional differential equations},
url = {http://eudml.org/doc/37835},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Liu, Yuji
TI - A survey and some new results on the existence of solutions of IPBVPs for first order functional differential equations
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 6
SP - 527
EP - 549
AB - This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation \[ {\left\lbrace \begin{array}{ll} x^{\prime }(t)=f(t,x(t),x(\alpha _1(t)),\cdots ,x(\alpha _n(t))) \text{for a.e.} \ t\in [0,T], \Delta x(t_k)=I_k(x(t_k)), \ k=1,\cdots ,m, x(0)=x(T). \end{array}\right.} \] We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin’s continuation theorem. Examples are presented to illustrate the main results.
LA - eng
KW - periodic boundary value problem; impulsive differential equation; fixed-point theorem; growth condition; periodic boundary value problem; impulsive differential equation; fixed-point theorem; growth condition
UR - http://eudml.org/doc/37835
ER -

References

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