Advanced Search

In the present work we give an existence theorem for bounded weak solution of the differential equation $$\dot{x}\left(t\right)=A\left(t\right)x\left(t\right)+f(t,x\left(t\right)),t\ge 0$$ where $\{A\left(t\right):t\in I{ℝ}^{+}\}$ is a family of linear operators from a Banach space $E$ into itself, $B_r=\{x\in E:\parallel x\parallel \le r\}$ and $f:{ℝ}^{+}\times B_r\to E$ is weakly-weakly continuous. Furthermore, we give existence theorem for the differential equation with delay $$\dot{x}\left(t\right)=\widehat{A}\left(t\right)x\left(t\right)+f^d(t,{\theta }_{t}x)\text{if}t\in [0,T],$$ where $T,d0$, $C_{B_r}\left([-d,0]\right)$ is the Banach space of continuous functions from $[-d,0]$ into $B_r$, $f_d:[0,T]\times C_{B_r}\left([-d,0]\right)\to E$ weakly-weakly continuous function, $\widehat{A}\left(t\right):[0,T]\to L\left(E\right)$ is strongly measurable and Bochner integrable operator on $[0,T]$ and ${\theta }_{t}x\left(s\right)=x(t+s)$ for all $s\in [-d,0]$.

We consider the problem $\dot{x}\left(t\right)\in A\left(t\right)x\left(t\right)+F(t,{\theta }_{t}x))$ a.e. on $[0,b]$, $x=\kappa$ on $[-d,0]$ in a Banach space $E$, where $\kappa$ belongs to the Banach space, $C_E\left([-d,0]\right)$, of all continuous functions from $[-d,0]$ into $E$. A multifunction $F$ from $[0,b]\times C_E\left([-d,0]\right)$ into the set, $P_{f_c}\left(E\right)$, of all nonempty closed convex subsets of $E$ is weakly sequentially hemi-continuous, ${\theta }_{t}x\left(s\right)=x(t+s)$ for all $s\in [-d,0]$ and $\left\{A\right(t):0\le t\le b\}$ is a family of densely defined closed linear operators generating a continuous evolution operator $S(t,s)$. Under a generalization of the compactness assumptions, we prove an existence result and give some topological...

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation $$\dot{x}\left(t\right)=ℒ\left(t\right)x\left(t\right)+f(t,x\left(t\right)),t\in ℝ\left(\mathrm{P}\right)$$ where $\left\{ℒ\right(t):t\in ℝ\}$ is a family of linear operators from a Banach space $E$ into itself and $f:ℝ\times E\to E$. By $L\left(E\right)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a<b$ and $d>0$, we let $C\left(\right[-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^d:[a,b]\times C([-d,0],E)\to E$. Let $\widehat{ℒ}:[a,b]\to L\left(E\right)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define ${\tau }_{t}x\left(s\right)=x(t+s)$ for each $s\in [-d,0]$. We prove that, under certain conditions,...

We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri...