Existence Theory for Pseudo-Symmetric Solution to p-Laplacian Differential Equations Involving Derivative

and Applied Analysis 3 We first list the definition of cone and the compression and expansion fixed point theorem 20, 22 . Definition 2.1. Let E be a real Banach space. A nonempty, closed, convex set P ⊂ E is said to be a cone provided the following conditions are satisfied: i if x ∈ P and λ ≥ 0, then λx ∈ P , ii if x ∈ P and −x ∈ P , then x 0. Lemma 2.2 see 20, 22 . Let P be a cone in a Banach spaceE. Assume thatΩ1,Ω2 are open bounded subsets of E with 0 ∈ Ω1,Ω1 ⊂ Ω2. If A : P ∩ Ω2 \ Ω1 → P is a completely continuous operator such that either i ‖Ax‖ ≤ ‖x‖, ∀x ∈ P ∩ ∂Ω1 and ‖Ax‖ ≥ ‖x‖, ∀x ∈ P ∩ ∂Ω2, or ii ‖Ax‖ ≥ ‖x‖, ∀x ∈ P ∩ ∂Ω1 and ‖Ax‖ ≤ ‖x‖, ∀x ∈ P ∩ ∂Ω2. Then, A has a fixed point in P ∩ Ω2 \Ω1 . Given a nonnegative continuous functional γ on a cone P of a real Banach space E, we define, for each d > 0, the set P γ, d {x ∈ P : γ x < d}. Let γ and θ be nonnegative continuous convex functionals on P , α a nonnegative continuous concave functional on P , and ψ a nonnegative continuous functional on P respectively. We define the following convex sets: P ( γ, α, b, d ) { x ∈ P : b ≤ α x , γ x ≤ d, P ( γ, θ, α, b, c, d ) { x ∈ P : b ≤ α x , θ x ≤ c, γ x ≤ d, 2.1 and a closed set R γ, ψ, a, d {x ∈ P : a ≤ ψ x , γ x ≤ d}. Next, we list the fixed point theorem due to Avery-Peterson 21 . Lemma 2.3 see 21 . Let P be a cone in a real Banach space E and γ, θ, α, ψ defined as above; moreover, ψ satisfies ψ λ′x ≤ λ′ψ x for 0 ≤ λ′ ≤ 1 such that for some positive numbers h and d, α x ≤ ψ x , ‖x‖ ≤ hγ x , 2.2 for all x ∈ P γ, d . Suppose that A : P γ, d → P γ, d is completely continuous and there exist positive real numbers a, b, c with a < b such that i {x ∈ P γ, θ, α, b, c, d : α x > b}/ ∅ and α A x > b for x ∈ P γ, θ, α, b, c, d , ii α A x > b for x ∈ P γ, α, b, d with θ A x > c, iii 0 / ∈ R γ, ψ, a, d and ψ A x < a for all x ∈ R γ, ψ, a, d with ψ x a. Then, A has at least three fixed points x1, x2, x3 ∈ P γ, d such that γ xi ≤ d for i 1, 2, 3, b < α x1 , a < ψ x2 , α x2 < b with ψ x3 < a. 2.3 4 Abstract and Applied Analysis Now, let E C1 0, 1 ,R . Then, E is a Banach space with norm ‖u‖ max { max t∈ 0,1 |u t |, max t∈ 0,1 ∣ ∣u′ t ∣ ∣ } . 2.4 Define a cone P ⊂ E by P { u ∈ E | u 0 0, u is concave, nonnegative on 0, 1 and u is symmetriconη, 1. 2.5 The following lemma can be founded in 11 , which is necessary to prove our result. Lemma 2.4 see 11 . If u ∈ P , then the following statements are true: i u t ≥ u ω1 /ω1 min{t, 1 η − t} for t ∈ 0, 1 , here ω1 η 1 /2, ii u t ≥ η/ω1 u ω1 for t ∈ η,ω1 , iii maxt∈ 0,1 u t u ω1 . Lemma 2.5. If u ∈ P , then the following statements are true: i u t ≤ maxt∈ 0,1 |u′ t |, ii ‖u t ‖ maxt∈ 0,1 |u′ t | max{|u′ 0 |, |u′ 1 |}, iii mint∈ 0,ω1 u t u 0 and mint∈ ω1,1 u t u 1 . Proof. i Since