Positive Solutions for a Higher-Order Nonlinear Neutral Delay Differential Equation

and Applied Analysis 3 where τ, p, g : R → R are continuous functions, B, δ, C are constants, τ and p are T -periodic, C/ 0, |B|/ 1, and T > 0. Zhou and Zhang 8 extended the results in 1 to the higher-order neutral functional differential equation with positive and negative coefficients: d dtn [ x t px t − τ ] −1 n 1 P t x t − σ −Q t x t − δ 0, t ≥ t0, 1.9 where p ∈ R \ {±1}, τ, σ, δ ∈ R and P,Q ∈ C t0, ∞ ,R . Zhou et al. 7 used the Krasnoselskii fixed point theorem and the Schauder fixed point theorem to prove the existence results of a nonoscillatory solution for the forced higher-order nonlinear neutral functional differential equation: d dtn [ x t p t x t − τ ] m ∑ i 1 qi t f x t − σi g t , t ≥ t0, 1.10 where τ, σi ∈ R , p, qi, g ∈ C t0, ∞ ,R for i ∈ {1, 2, . . . , m} and f ∈ C R,R . Zhang et al. 5 obtained some sufficient conditions for the oscillation of all solutions of the even order nonlinear neutral differential equations with variable coefficients: d dtn [ x t p t x τ t ] q t f x σ t 0, t ≥ t0, 1.11 where n is an even number, p, q, σ, τ ∈ C t0, ∞ ,R with 0 ≤ p t < 1, for all t ≥ t0, limt→ ∞τ t limt→ ∞, σi t ∞ and f ∈ C t0, ∞ ,R . The purpose of this paper is to investigate the solvability of 1.1 . By constructing appropriate mappings and using the Laray-Schauder nonlinear alternative theorem, we establish a few sufficient conditions which ensure the existence of uncountably many bounded positive solutions for 1.1 . Our results improve and generalize some corresponding results in 1, 2, 4, 6–8 . Three examples are given to illustrate the advantages of the results presented in this paper. Throughout this paper, we assume thatR,R , andN denote the sets of all real numbers, nonnegative numbers, and positive integers, respectively, and ν inf { τi t , αj t , βj t : t ∈ t0, ∞ , i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , k} } . 1.12 Let CB ν, ∞ ,R stand for the Banach space of all continuous and bounded functions in ν, ∞ with norm ‖x‖ supt≥ν |x t | for all x ∈ CB ν, ∞ ,R and E N {x ∈ CB ν, ∞ ,R : x t ≥ N for t ≥ ν}, U M {x ∈ E N : ‖x‖ < M}, 1.13 where M,N ∈ R with M > N > 0. Clearly, E N is a nonempty closed convex subset of CB ν, ∞ ,R and U M is an open subset of E N . By a solution of 1.1 , we mean a function x ∈ C ν, ∞ ,R with some T ≥ t0 |ν| such that x t ∑mi 1 pi t x τi t is n times continuously differentiable in T, ∞ and 4 Abstract and Applied Analysis f t, x α1 t , . . . , x αk t is n−1 times continuously differentiable in T, ∞ and 1.1 holds for t ≥ T . Lemma 1.1 the Leray-Schauder nonlinear alterative theorem 9 . Let E be a closed convex subset of a Banach space X and let U be an open subset of E with p ∈ U. Also, G : U → E is a continuous, condensing mapping with G U bounded, whereU denotes the closure ofU Then, A1 G has a fixed point inU, or A2 there are x ∈ ∂U and λ ∈ 0, 1 with x 1 − λ p λGx. 2. Main Results Now, we apply the Leray-Schauder nonlinear alterative theorem to investigate the existence of uncountably many bounded positive solutions of 1.1 under certain conditions. Theorem 2.1. Assume that there exist constantsM,N, p0, t1 and functions F,H ∈ C t0, ∞ ,R satisfying ∣ ∣f t, u1, . . . , uk ∣ ∣ ≤ F t , ∀ t, u1, . . . , uk ∈ t0, ∞ × N,M , 2.1 |h t, v1, . . . , vk | ≤ H t , ∀ t, v1, . . . , vk ∈ t0, ∞ × N,M , 2.2 max {∫ ∞ t0 F s ds, ∫ ∞ t0 sn−1max {∣ ∣g s ∣ ∣,H s } ds } < ∞, 2.3 0 < N < ( 1 − 2p0 ) M, m ∑ i 1 ∣ ∣pi t ∣ ∣ ≤ p0 < 1 2 , ∀t ≥ t1 ≥ t0. 2.4 Then, 1.1 has uncountably many bounded positive solutions inU M . Proof. Let L ∈ p0M N, 1−p0 M . It follows from 2.3 and 2.4 that there exists a constant T > 1 |t0| |t1| |ν| satisfying ∫ ∞