Solvability of Three-Point Boundary Value Problems at Resonance with a p-Laplacian on Finite and Infinite Intervals

and Applied Analysis 3 to BVP 1.1 when T ∞, which we call the infinite case. Some explicit examples are also given in the last section to illustrate our main results. 2. Preliminaries For the convenience of the readers, we provide here some definitions and lemmas which are important in the proof of our main results. Ge-Mawhin’s continuation theorem and the modified one are also stated in this section. Lemma 2.1. Let Φp s |s|p−2s, p > 1. Then Φp satisfies the properties. 1 Φp is continuous, monotonically increasing, and invertible. MoreoverΦ−1 p Φq with q > 1 a real number satisfying 1/p 1/q 1; 2 for any u, v 0, Φp u v Φp u Φp v , if p < 2, Φp u v 2p−2 ( Φp u Φp v ) , if p 2. 2.1 Definition 2.2. Let R2 be an 2-dimensional Euclidean space with an appropriate norm | · |. A function f : 0, T × R2 → R is called Φq-Carathéodory if and only if 1 for each x ∈ R2, t → f t, x is measurable on 0, T ; 2 for a.e. t ∈ 0, T , x → f t, x is continuous on R2; 3 for each r > 0, there exists a nonnegative function φr ∈ L1 0, T with φr,q t : Φq ∫T t φr τ dτ ∈ L1 0, T such that |x| r implies ∣f t, x ∣φr t , a.e. t ∈ 0, T . 2.2 Next we state Ge-Mawhin’s continuation theorem 3, 4 . Definition 2.3. Let X, Z be two Banach spaces. A continuous opeartor M : X ∩ domM → Z is called quasi-linear if and only if ImM is a closed subset of Z and KerM is linearly homeomorphic to R, where n is an integer. Let X2 be the complement space of KerM in X, that is, X KerM ⊕ X2. Ω ⊂ X an open and bounded set with the origin 0 ∈ Ω. Definition 2.4. A continuous operator Nλ : Ω → Z, λ ∈ 0, 1 is said to be M-compact in Ω if there is a vector subspace Z1 ⊂ Z with dimZ1 dimKerM and an operator R : Ω × 0, 1 → X2 continuous and compact such that for λ ∈ 0, 1 , I −Q Nλ ( Ω ) ⊂ ImM ⊂ I −Q Z, 2.3 QNλx 0, λ ∈ 0, 1 ⇐⇒ QNx 0, ∀x ∈ Ω, 2.4 R ·, 0 is the zero operator, R ·, λ |Σλ I − P |Σλ , 2.5 M P R ·, λ I −Q Nλ, 2.6 4 Abstract and Applied Analysis where P , Q are projectors such that ImP KerM and ImQ Z1, N N1, Σλ {x ∈ Ω, Mx Nλx}. Theorem 2.5 Ge-Mawhin’s continuation theorem . Let X, ‖ · ‖X and Z, ‖ · ‖Z be two Banach spaces, Ω ⊂ X an open and bounded set. Suppose M : X ∩ domM → Z is a quasi-linear operator and Nλ : Ω → Z, λ ∈ 0, 1 isM-compact. In addition, if i Mx/ Nλx, for x ∈ domM ∩ ∂Ω, λ ∈ 0, 1 , ii QNx/ 0, for x ∈ KerM ∩ ∂Ω, iii deg JQN,Ω ∩ KerM, 0 / 0, whereN N1. Then the abstract equation Mx Nx has at least one solution in dom M ∩Ω. According to the usual direct-sum spaces such as those in 3, 5, 7, 11–13 , it is difficult maybe impossible to define the projectorQ under the at most linearly increasing conditions. We have to weaken the conditions of Ge-Mawhin continuation theorem to resolve such problem. Definition 2.6. Let Y1 be finite dimensional subspace of Y .Q : Y → Y1 is called a semiprojector if and only ifQ is semilinear and idempotent, whereQ is called semilinear providedQ λx λQ x for all λ ∈ R and x ∈ Y . Remark 2.7. Using similar arguments to those in 3 , we can prove that when Q is a semiprojector, Ge-Mawhin’s continuation theorem still holds. 3. Existence Results for the Finite Case Consider the Banach spaces X C1 0, T endowed with the norm ‖x‖X max{‖x‖∞, ‖x′‖∞}, where ‖x‖∞ max0 t T |x t | and Z L1 0, T with the usual Lebesgue norm denoted by ‖ · ‖z. Define the operator M by M : domM ∩X −→ Z, Mx t Φp x′ t )′ , t ∈ 0, T , 3.1 where domM {x ∈ C1 0, T ,Φp x′ ∈ C1 0, T , x 0 x η , x′ T 0}. Then by direct calculations, one has KerM {x ∈ domM ∩X : x t c ∈ R, t ∈ 0, T }, ImM { y ∈ Z : ∫η 0 Φq (∫T s y τ dτ ) ds 0 } . 3.2 Obviously, KerM R and ImM is close. So the following result holds. Lemma 3.1. LetM be defined as 3.1 , thenM is a quasi-linear operator. Abstract and Applied Analysis 5 Set the projector P and semiprojector Q by P : X −→ X, Px t x 0 , t ∈ 0, T , 3.3 Q : Z −→ Z, Qy t 1 ρ Φp (∫ηand Applied Analysis 5 Set the projector P and semiprojector Q by P : X −→ X, Px t x 0 , t ∈ 0, T , 3.3 Q : Z −→ Z, Qy t 1 ρ Φp (∫η