The positivity of the second order difference operator with periodic conditions in Hölder spaces and its applications

2014 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Fatih Sabahattin Tetikoglu
Keyword(s):  
Difference Operator ◽  
Second Order ◽  
Hölder Spaces ◽  
Order Difference ◽  
Holder Spaces

scholarly journals Moduli of continuity, functional spaces,\break and elliptic boundary value problems. The full regularity spaces Cα0,λ(Ω̅)

2018 ◽  
Vol 7 (1) ◽  
pp. 15-34 ◽  
Author(s):  
Hugo Beirão da Veiga
Keyword(s):  
Modulus Of Continuity ◽  
Elliptic Boundary ◽  
Continuous Functions ◽  
Second Order ◽  
Hölder Spaces ◽  
Elliptic Boundary Value ◽  
New Class ◽  
Holder Spaces ◽  
Derivatives Of ◽  
Full Regularity

AbstractLet {\boldsymbol{L}} be a second order uniformly elliptic operator, and consider the equation {\boldsymbol{L}u=f} under the boundary condition {u=0}. We assume data f in generical subspaces of continuous functions {D_{\overline{\omega}}} characterized by a given modulus of continuity{\overline{\omega}(r)}, and show that the second order derivatives of the solution u belong to functional spaces {D_{\widehat{\omega}}}, characterized by a modulus of continuity{\widehat{\omega}(r)} expressed in terms of {\overline{\omega}(r)}. Results are optimal. In some cases, as for Hölder spaces, {D_{\widehat{\omega}}=D_{\overline{\omega}}}. In this case we say that full regularity occurs. In particular, full regularity occurs for the new class of functional spaces {C^{0,\lambda}_{\alpha}(\overline{\Omega})} which includes, as a particular case, the classical Hölder spaces {C^{0,\lambda}(\overline{\Omega})=C^{0,\lambda}_{0}(\overline{\Omega})}. Few words, concerning the possibility of generalizations and applications to non-linear problems, are expended at the end of the introduction and also in the last section.

scholarly journals Spectrum of Discrete Second-Order Difference Operator with Sign-Changing Weight and Its Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Ruyun Ma ◽  
Chenghua Gao
Keyword(s):  
Eigenvalue Problems ◽  
Difference Operator ◽  
Second Order ◽  
Order Difference ◽  
Sign Changing Solutions

LetT>1be an integer, and let𝕋=1,2,…,T. We discuss the spectrum of discrete linear second-order eigenvalue problemsΔ2ut-1+λmtut=0, t∈𝕋,  u0=uT+1=0, whereλ≠0is a parameter,m:𝕋→ℝchanges sign andmt≠0on𝕋. At last, as an application of this spectrum result, we show the existence of sign-changing solutions of discrete nonlinear second-order problems by using bifurcate technique.

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