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This paper considers the difference operator Aht,x approximation of the following differential operator At,xu(t,x) = -a11(t,x)utt(t,x)-a22(t,x)uxx(t,x)+σu(t,x), defined over the region R+×R with the boundary condition u(0,x) = 0,x∈R. Here, the coefficients aii(t,x),i = 1,2 are continuously differentiable and satisfy the uniform ellipticity a112(t,x)+a222(t,x)≥δ>0, and σ > 0. It investigates the structure of the fractional spaces generated by this difference operator. Moreover, the positivity of Aht,x operator in difference analogy Ho¨lder spaces is proved. [ABSTRACT FROM AUTHOR]

We study a structure of fractional spaces Eα(Lp[0,1],Ax) generated by the positive differential operator Ax defined by the formula Axu = -a(x)d2udx2+δu with domain D(Ax) = {u∈C(2)[0,1]:u(0) = u(μ),u′(0) = u′(1),1/2≤μ≤1}. Here, a(x) is a smooth function defined on the segment [0,1] and a(x)≥a>0,δ>0. It is established that for any 0 < α < 1/2, 1 ≤ p < ∞, the norms in the spaces Eα(Lp[0,1],Ax) and Wp2α[0,1] are equivalent. The positivity of the differential operator Ax in Wp2α[0,1],(0≤α≤1/2,1≤p<∞) is established. In applications, well-posedness of nonlocal boundary problems for elliptic equations is established. [ABSTRACT FROM AUTHOR]

In the present paper, the differential operator A defined by ... with domain ... is considered. The Green function for A is constructed. The estimates for the Green function are obtained and the positivity of A in the Banach space C ([0,1] x R) is established. The structure of fractional spaces generated by A is investigated, the equivalence of the norm of these fractional spaces and Holder spaces is established. The positivity of A in the Holder space ... is obtained. [ABSTRACT FROM AUTHOR]

This paper considers the operator Au(t,x) = -a11(t,x)utt(t,x)-a22(t,x)uxx(t,x)+σu(t,x), defined over the region R+×R with the boundary condition u(0,x) = 0,x∈R. Here, the coefficients aii(t,x), i = 1,2 are continuously differentiable and satisfy the uniform ellipticity condition a112(t,x)+a222(t,x)≥δ>0, and σ > 0. It investigates the structure of the fractional spaces generated by this operator. Moreover, the positivity of the operator in Ho¨lder spaces is proved. [ABSTRACT FROM AUTHOR]