Geodesic
Neutral set differential equations
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 593-615

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The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type\begin {equation*} \begin {cases} D_{H}X(t)=F(t,X_{t},D_{H}X_{t}), \\ \kern .25em X|_{[-r,0]}=\Psi , \end {cases} \end {equation*} where $F\colon [0,b]\times \mathcal {C}_{0}\times \mathfrak {L}_{0}^{1}\rightarrow K_{c}(E)$ is a given function, $K_{c}(E)$\ is the family of all nonempty compact and convex subsets of a separable Banach space $E$, $\mathcal {C}_{0}$ denotes the space of all continuous set-valued functions $X$ from $[-r,0]$ into $K_{c}(E)$, $\mathfrak {L}_{0}^{1}$ is\ the space of all integrally bounded set-valued functions $X\colon [-r,0]\rightarrow K_{c}(E)$, $\Psi \in \mathcal {C}_{0}$\ and $D_{H}$ is the Hukuhara derivative. The continuous dependence of solutions on initial data and parameters is also studied.
The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type\begin {equation*} \begin {cases} D_{H}X(t)=F(t,X_{t},D_{H}X_{t}), \\ \kern .25em X|_{[-r,0]}=\Psi , \end {cases} \end {equation*} where $F\colon [0,b]\times \mathcal {C}_{0}\times \mathfrak {L}_{0}^{1}\rightarrow K_{c}(E)$ is a given function, $K_{c}(E)$\ is the family of all nonempty compact and convex subsets of a separable Banach space $E$, $\mathcal {C}_{0}$ denotes the space of all continuous set-valued functions $X$ from $[-r,0]$ into $K_{c}(E)$, $\mathfrak {L}_{0}^{1}$ is\ the space of all integrally bounded set-valued functions $X\colon [-r,0]\rightarrow K_{c}(E)$, $\Psi \in \mathcal {C}_{0}$\ and $D_{H}$ is the Hukuhara derivative. The continuous dependence of solutions on initial data and parameters is also studied.
DOI : 10.1007/s10587-015-0199-9
Classification : 34A12, 34K40
Keywords: neutral type; existence; uniqueness; continous dependence 
Abbas, Umber; Lupulescu, Vasile; O'Regan, Donald; Younus, Awais. Neutral set differential equations. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 593-615. doi: 10.1007/s10587-015-0199-9
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