Abstract
The existence of a solution for a second-order p-Laplacian boundary value problem at resonance with two dimensional kernel will be considered in this paper. A semi-projector, the Ge and Ren extension of Mawhin’s coincidence degree theory, and algebraic processes will be used to establish existence results, while an example will be given to validate our result.
Highlights
1 Introduction The following second-order p-Laplacian boundary value problem will be considered in this work: (φp(u (t))) + g(t, u(t), u (t)) = 0, t ∈ (0, +∞), φp(u (0)) =
Boundary value problems are said to be at resonance if the solution of the corresponding homogeneous boundary value problem is non-trivial
Let Ω ⊃ Ω1 ∪ Ω2 be a nonempty, open and bounded set, u ∈ dom M ∩ ∂Ω, H(u, λ) = –λu + (1 – λ)JQNu, and J be as defined in Lemma 3.2 H(u, λ) = 0
Summary
Definition 2.4 ([20]) Let U be the space of all continuous and bounded vector-valued functions on [0, +∞) and X ⊂ U. The operator Nλ is said to be M-compact in Ω if there exist a vector subspace Z1 ∈ Z such that dim Z1 = dim U1 and a compact and continuous operator R : Ω ×[0, 1] → U2 such that, for λ ∈ [0, 1], the following holds: (i) (I – Q)Nλ(Ω) ⊂ Im M ⊂ (I – B)Z, (ii) QNλu = 0 ⇔ QNu = 0, λ ∈ (0, 1), (iii) R(·, u) is the zero operator and R(·, λ)|Σλ = (I – P)|Σλ , (iv) M[P + R(·, λ)] = (I – Q)Nλ.
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