On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces

• Autorzy: Adel Mahmoud Gomaa
• Języki publikacji: EN

Abstrakty

EN

In the present work we give an existence theorem for bounded weak solution of the differential equation \[ \dot{x}(t) = A(t)x(t) + f (t, x(t)),\quad t \geq 0 \] where \(\{A(t) : t \in I\mathbb{R}^+ \}\) is a family of linear operators from a Banach space \(E\) into itself, \(B_r = \{x \in E : \|x\| \leq r\}\) and \(f \colon \mathbb{R}^+ \times B_r \to E\) is weakly-weakly continuous. Furthermore, we give existence theorem for the differential equation with delay \[ \dot{x}(t) = \hat{A}(t) x(t) + f^d (t, θ_t x)\quad \text{if}\ t \in [0, T], \] where \(T, d \gt 0\), \(C_{B_r} ([-d, 0])\) is the Banach space of continuous functions from \([-d, 0]\) into \(B_r\), \(f_d\colon [0, T] \times C_{B_r} ([-d, 0]) \to E\) weakly-weakly continuous function, \(\hat{A}(t)\colon [0,T] \to L(E)\) is strongly measurable and Bochner integrable operator on \([0,T]\) and \(θ_t x(s) = x(t + s)\) for all \(s \in [-d, 0]\).

Słowa kluczowe

EN

Nonlinear differential equations; weak solutions; measures of noncompactness; delay

• Wydawca: Polish Mathematical Society
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2007
• Tom: 47
• Numer: 2

Daty

wydano

2007

online

2017-12-19

Twórcy

autor

Adel Mahmoud Gomaa

Identyfikatory

DOI

10.14708/cm.v47i2.5248