We present a new theorem on the differential inequality $u^{(m)} ≤ w(u)$. Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f(t,x)$.
The integral equation of Urysohn type is considered, for the deterministic and stochastic cases. We show, using the fixed point theorem of Darbo type that under some assumptions the equations have solutions belonging to the space of continuous functions. The main tool used in our paper is the technique associated with measures of noncompactness.
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 \(s \in [-d, 0]\).
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Sufficient conditions for normal structure of a Banach space are given. One of them implies reflexivity for Banach spaces with an unconditional basis, and also for Banach lattices.
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Résumé. On présente une fonction continue f: c₀ → c₀ qui satisfait à une condition lipschitzienne par rapport à la mesure de non-compacité de Hausdorff (ou Kuratowski), mais telle que f n'est pas la somme d'une fonction dissipative et d'une fonction compacte. Cet exemple attache de l'importance au théorème d'existence de Sabina Schmidt (1989) pour des équations différentielles dans les espaces de Banach.
In this paper we prove a theorem for the existence of pseudo-solutions to the Cauchy problem, x' = f(t,x), x(0) = x₀ in Banach spaces. The function f will be assumed Pettis-integrable, but this assumption is not sufficient for the existence of solutions. We impose a weak compactness type condition expressed in terms of measures of weak noncompactness. We show that under some additionally assumptions our solutions are, in fact, weak solutions or even strong solutions. Thus, our theorem is an essential generalization of previous results.
We consider the problem \(\dot{x}(t) \in A(t)x(t) + F (t, θ_t x))\) a.e. on \([0, b]\), \(x = \kappa\) on \([-d, 0]\) in a Banach space \(E\), where \(\kappa\) belongs to the Banach space, \(C_E ([-d, 0])\), of all continuous functions from \([-d, 0]\) into \(E\). A multifunction \(F\) from \([0, b] \times C_E ([-d, 0])\) into the set, \(P_{f_c} (E)\), of all nonempty closed convex subsets of \(E\) is weakly sequentially hemi-continuous, \(θ_t x(s) = x(t + s)\) for all \(s \in [-d, 0]\) and \(\{A(t) : 0 \leq t \leq b\}\) 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 properties of our solution sets that generalizes earlier theorems by Papageorgiou, Rolewicz, Deimling, Frankowska and Cichoń.
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We define a modulus for the property (β) of Rolewicz and study some useful properties in fixed point theory for nonexpansive mappings. Moreover, we calculate this modulus in $l^p$ spaces for the main measures of noncompactness.
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