An existence theorem for ordinary differential equations in Banach spaces

Boudourides, Moses A.

doi:10.1017/s0004972700006766

Cited by 6 publications

(5 citation statements)

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“…For the important case L(t) = 0 we have, as a special case, a generalization of the existence theorems of Gomaa [13], Ibrahim-Gomaa [15], Papageorgiou [23], Cramer-Lakshmikantham-Mitchell [7], Szep [25] and Boundourides [2] in all of which the results are stated without delay. Szep in [25] studied the special case of problem (P) in a reflexive Banach space, Boundourides [2] and Cramer-Lakshmikantham-Mitchell [7] studied the special case of problem (P) in a nonreflexive Banach space, Papageorgiou [23] found weak solutions for the special case of problem (P) on a finite interval I with 0 < T < ∞, Ibrahim-Gomaa [15] found weak solutions for the special case of problem (P) on a finite interval I and in [13] we give a generalization to recent results on the Cauchy problem by using weak and strong measures of noncompactness. Moreover in [11], [12] we study the nonlinear differential equations with and without delay while in [9] we study the differential inclusions with moving constraints.…”

Section: G(t S)h N (S) Dsmentioning

confidence: 99%

On Bounded Weak and Strong Solutions of Non Linear Differential Equations with and without Delay in Banach Spaces

Gomaa¹

2013

MATH. SCAND.

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Assume that $E$ is a Banach space, $B_{r}=\{x\in E:\Vert x\Vert \le r\}$ and $C([-d,0],B_{r})$ is the Banach space of continuous functions from $[-d,0]$ into $B_{r}$. Consider $f:\mathbf{R}^+\times E\to E$; $f^{d}:[0,T]\times C([-d,0],B_{r})\to E$; for each $t\in [0,T]$ the mapping $\theta_{t}\in C([-d,0],B_{r})$ is defined by $\theta_{t}x(s)= x(t+s)$, $s\in [-d,0]$ and let $A(t)$ be a linear operator from $E$ into itself. In this paper we give existence theorems for bounded weak and strong solutions of the nonlinear differential equation 26767 \dot{x}(t)=A(t)x+f(t,x),\qquad t\in \mathbf{R}^+, 26767 and we prove that, with certain conditions, the differential equation with delay 26767 \dot{x}(t)=L(t)x(t)+f^{d}(t,\theta_{t}x),\qquad \text{if}\quad t\in [0,T] \qquad\qquad(\mathrm{Q}) 26767 has at least one weak solution where $L(t)$ is a linear operator from $E$ into $E$. Moreover, under suitable assumptions, the problem $(\mathrm{Q})$ has a solution. Furthermore under a generalization of the compactness assumptions, we show that $(\mathrm{Q})$ has a solution too.

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Section: G(t S)h N (S) Dsmentioning

confidence: 99%

On Bounded Weak and Strong Solutions of Non Linear Differential Equations with and without Delay in Banach Spaces

Gomaa¹

2013

MATH. SCAND.

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“…The first equality can be found in [3] and its proof is based on the "weak" Arzela-Ascoli theorem [5, Theorem 1.2] .…”

Section: Strong Topology Then And(e) = Sup And(e(t)) = And(e(t)) Where T Ementioning

confidence: 99%

“…Both papers based their existence result on a compactness type condition, involving the weak measure of noncompactness introduced by DeBlasi [6]. It should be noted however that the result of CramerLakshmikantham-Mitchell [51 is more general than that of Boudourides [3].…”

Section: Introductionmentioning

confidence: 99%

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Weak solutions of differential equations in Banach spaces

Papageorgiou

1986

Bull. Austral. Math. Soc.

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“…The result of Szep was extended to nonreflexive Banach spaces by Boundourides [11] and Cramer-Lakshmikantham-Mitchell [12]. Recently, in [14], [15], the authors studied the existence of weak solution x ∈ C[I, E] in a reflexive Banach space for the nonlinear Volterra-Stieltjes integral equation (1) where f is assumed to be weakly-weakly continuous.…”

Section: Introductionmentioning

confidence: 99%