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On a fractional differential inclusion in Banach space under weak compactness condition. (English) Zbl 1383.34004

Kusuoka, Shigeo (ed.) et al., Advances in mathematical economics. Vol. 20. Selected papers based on the presentations at the 6th conference on mathematical analysis in economic theory, Tokyo, Japan, January 26–29, 2015. Singapore: Springer (ISBN 978-981-10-0475-9/hbk; 978-981-10-0476-6/ebook). Advances in Mathematical Economics 20, 23-75 (2016).
From the introduction: In this paper, we investigate a class of boundary value prolems governed by a fractional differential inclusion in a separable Banach space \(E\) in both Bochner and Pettis settings \[ \begin{gathered} w\text{-}D^\alpha u(t)\in F(t,u(t), w\text{-}D^{\alpha-1} u(t)),\quad t\in [0,1],\\ I^\beta u(t)|_{t=0}= 0,\quad u(1)= \int^1_0 u(t)\,dt,\end{gathered} \] where \(\alpha\in ]1,2]\), \(\beta\in]0,\infty[\) are given constants, \(w\)-\(D^\gamma\) is the fractional \(w\)-R.L derivative of order \(\gamma\in \{\alpha-1, \alpha\}\), \(F:[0,1]\times E_\sigma\times E_\sigma\hookrightarrow E_\sigma\) is a convex weakly compact valued multimapping, \(E_\sigma\) is the vector space \(E\) endowed with the weak topology and \(F(t.,.,.): E_\sigma\times E_\sigma\hookrightarrow E_\sigma\) is upper semicontinuous.
For the entire collection see [Zbl 1347.91004].

MSC:

34A08 Fractional ordinary differential equations
34A60 Ordinary differential inclusions
34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
34G25 Evolution inclusions
26A39 Denjoy and Perron integrals, other special integrals
49K21 Optimality conditions for problems involving relations other than differential equations
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