General Existence Principle for Singular BVPs and Its Application

2004 ◽  
Vol 11 (3) ◽  
pp. 549-565
Author(s):  
Irena Rachůnková ◽  
Svatoslav Staněk
Keyword(s):  
Boundary Value Problems ◽  
Large Class ◽  
Closed Subset ◽  
Boundary Value ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Carathéodory Conditions ◽  
Existence Principle ◽  
General Existence

Abstract We present a general existence principle which can be used for a large class of singular boundary value problems of the form where 𝑓 satisfies the local Carathéodory conditions on [0, 𝑇] × 𝐷, a set 𝐷 ⊂ is not closed, 𝑓 has singularities in its phase variables on the boundary ∂𝐷 of 𝐷, and 𝑆 is a closed subset in 𝐶𝑛–1([0, 𝑇]). The proof is based on the regularization and sequential techniques. An application of the general existence principle to singular conjugate (𝑝, 𝑛–𝑝) BVPs is also given.

scholarly journals General existence principles for nonlocal boundary value problems withφ-Laplacian and their applications

2006 ◽  
Vol 2006 ◽  
pp. 1-30 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O'Regan ◽  
Svatoslav Stanek
Keyword(s):  
Boundary Value Problems ◽  
Large Class ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Degree Theory ◽  
Nonlocal Boundary Value Problems ◽  
Carathéodory Conditions ◽  
General Existence

The paper presents general existence principles which can be used for a large class of nonlocal boundary value problems of the form(φ(x′))′=f1(t,x,x′)+f2(t,x,x′)F1x+f3(t,x,x′)F2x,α(x)=0,β(x)=0, wherefjsatisfy local Carathéodory conditions on some[0,T]×𝒟j⊂ℝ2,fjare either regular or have singularities in their phase variables(j=1,2,3),Fi:C1[0,T]→C0[0,T](i=1,2), andα,β:C1[0,T]→ℝare continuous. The proofs are based on the Leray-Schauder degree theory and use regularization and sequential techniques. Applications of general existence principles to singular BVPs are given.

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