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Fractional spaces generated by the positive differential and difference operators in a Banach space. (English) Zbl 1130.46303

Taş, K. (ed.) et al., Mathematical methods in engineering. Selected papers of the international symposium, MME06, Ankara, Turkey, April 27–29, 2006. Dordrecht: Springer (ISBN 978-1-4020-5677-2/hbk; 978-1-4020-5678-9/e-book). 13-22 (2007).
Summary: The structure of the fractional spaces \(E_{\alpha,q}(L_q[0,1],A^x)\) generated by the positive differential operator \(A^x\) defined by the formula \(A^xu = -a(x)\frac{d^2u}{dx^2} + \delta u\), with domain \(D(A^x) = \{u\in C^{(2)}[0,1] : u(0) = u(1)\), \(u'(0) = u'(1)\}\), is investigated. It is established that for any \(0 < a < 1\) the norms in the spaces \(E_{\alpha,q}(L_q[0,1],A^x)\) and \(W^{2\alpha}_q[0,1]\) are equivalent. The positivity of the differential operator \(A^x\) in \(W^{2\alpha}_q[0,1]\) \((0\leq\alpha <\frac12)\) is established. The discrete analogue of these results for the positive difference operator \(A^x\), a second order of approximation of the differential operator \(A^x\), defined by the formula
\[ A^x_hu^h = \left\{-a(x_k)\,\frac{u_{k+1}-2u_k+u_{k-1}}{h^2}+\delta u_k\right\}^{M-1}_1, \quad u_h = \{u_k\}^M_0,\;Mh = 1, \]
with \(u_0=u_M\) and \(-u_2 + 4u_1- 3u_0 = u_{M-2}- 4u_{M-1} + 3u_M\), is established. In applications, the coercive inequalities for the solutions of the nonlocal boundary-value problem for two-dimensional elliptic equations and of the second order of accuracy difference schemes for the numerical solution of this problem are obtained.
For the entire collection see [Zbl 1106.00005].

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47E05 General theory of ordinary differential operators
34B05 Linear boundary value problems for ordinary differential equations
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