Electron. J. Diff. Equ., Vol. 2012 (2012), No. 56, pp. 1-16.

Existence and topological structure of solution sets for phi-Laplacian impulsive differential equations

Johnny Henderson, Abdelghani Ouahab, Samia Youcefi

Abstract:
In this article, we present results on the existence and the topological structure of the solution set for initial-value problems for the first-order impulsive differential equation
$$\displaylines{ (\phi(y'))' = f(t,y(t)), \quad\hbox{a.e. } t\in [0,b],\cr y(t^+_{k})-y(t^-_k)=I_{k}(y(t_{k}^{-})), \quad k=1,\dots,m,\cr y'(t^+_{k})-y'(t^-_k)=\bar I_{k}(y'(t_{k}^{-})), \quad k=1,\dots,m,\cr y(0)=A,\quad y'(0)=B, }$$
where $0=t_0<t_1<\dots<t_m<t_{m+1}=b$, $m\in\mathbb{N}$. The functions $I_k, \bar I_k$ characterize the jump in the solutions at impulse points $t_k$, $k=1,\dots,m$. For the final result of the paper, the hypotheses are modified so that the nonlinearity $f$ depends on $y'$, but the impulsive conditions and initial conditions remain the same.

Submitted January 20, 2012. Published April 6, 2012.
Math Subject Classifications: 34A37, 34K45.
Key Words: phi-Laplacian; fixed point theorems; impulsive solution; compactness.

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Johnny Henderson
Department of Mathematics, Baylor University
Waco, TX 76798-7328, USA
email: Johnny_Henderson@baylor.edu
Abdelghani Ouahab
Laboratory of Mathematics, Sidi-Bel-Abbès University
PoBox 89, 22000 Sidi-Bel-Abbès, Algeria
email: agh_ouahab@yahoo.fr
  Samia Youcefi
Laboratory of Mathematics, Sidi-Bel-Abbès University
PoBox 89, 22000 Sidi-Bel-Abbès, Algeria
email: youcefi.samia@yahoo.com

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