Affiliation:
1. Department of Mathematical Analysis and Applications of Mathematics , Faculty of Science , Palacký University , 17. listopadu 12 771 46 Olomouc Czech Republic
Abstract
Abstract
The paper deals with the boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses of the form
ϕ
(
z
′
(
t
)
)
′
=
f
(
t
,
z
(
t
)
,
z
′
(
t
)
)
for a.e.
t
∈
[
0
,
T
]
⊂
R
,
Δ
z
′
(
t
)
=
M
(
z
(
t
)
,
z
′
(
t
−
)
)
,
t
=
γ
(
z
(
t
)
)
,
z
(
0
)
=
z
(
T
)
=
0.
$$\begin{array}{}
\left(\phi(z'(t))\right)' = f(t,z(t),z'(t))\qquad \text{ for a.e. } t\in [0,T]\subset\mathbb R,\\
\Delta z'(t) = M(z(t),z'(t-)),\qquad t=\gamma (z(t)),\\ z(0) = z(T) = 0.
\end{array} $$
Here, T > 0, ϕ : ℝ → ℝ is an increasing homeomorphism, ϕ(ℝ) = ℝ, ϕ(0) = 0, f : [0, T] × ℝ2 → ℝ satisfies Carathéodory conditions, M : ℝ → ℝ is continuous and γ : ℝ → (0, T) is continuous, Δ z′(t) = z′(t+) − z′(t−). Sufficient conditions for the existence of at least one solution to this problem having no pulsation behaviour are provided.
Reference38 articles.
1. Agarwal, R. P.—O’Regan, D.: Multiple nonnegative solutions for second order impulsive differential equations, Appl. Math. Comput. 114 (2000), 51–59.
2. Azbelev, N. V.—Maksimov, V. P.—Rakhmatullina, L. F.: Introduction to the Theory of Functional Differential Equations: Methods and Applications. Contemp. Math. Appl. 3, Hindawi Publ. Corp., New York, 2007.
3. Bai, L.—Dai, B.: An application of variational method to a class of Dirichlet boundary value problems with impulsive effects, J. Franklin Inst. 348 (2011), 2607–2624.
4. Bai, L.—Dai, B.: Three solutions for a p-Laplacian boundary value problem with impulsive effects, Appl. Math. Comput. 217 (2011), 9895–9904.
5. Bainov, D. D.—Simeonov, P. S.: Impulsive Differential Equations: Periodic Solutions and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics 66, Longman Scientific and Technical, Essex, England, 1993.
Cited by
9 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献