General existence principles for nonlocal boundary value problems with φ -laplacian and their applications (Abstract and Applied Analysis)
The paper presents general existence principles which can be used for a large class of nonlocal boundary value problems of the form ( φ ( x ′ ) ) ′ = f 1 ( t , x , x ′ ) + f 2 ( t , x , x ′ ) F 1 x + f 3 ( t , x , x ′ ) F 2 x , α ( x ) = 0 , β ( x ) = 0 , where f j satisfy local Carathéodory conditions on some [ 0 , T ] × 𝒟 j ⊂ ℝ 2 , f j are either regular or have singularities in their phase variables ( j = 1 , 2 , 3 ) , F i : C 1 [ 0 , T ] → C 0 [ 0 , T ] ( i = 1 , 2 ) , and α , β : C 1 [ 0 , T ] → ℝ are 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.