Fourth-Order Differential Equation with Deviating Argument

and Applied Analysis 3 Motivated by 14, 15 , here we study the existence of AL-solutions for 1.1 . The approach is completely different from the one used in 15 , in which an iteration process, jointly with a comparison with the linear equation y 4 q t y 2 0, is employed. Our tools are based on a topological method, certain integral inequalities, and some auxiliary functions. In particular, for proving the continuity in the Fréchet space C t0,∞ of the fixed point operators here considered, we use a similar argument to that in the Vitali convergence theorem. Our results extend to the case with deviating argument analogues ones stated in 15 for 1.7 when n 4. We obtain sharper conditions for the existence of unbounded ALsolutions of 1.1 , and, in addition, we show that under additional assumptions on q, r, these conditions become also necessary for the existence of AL-solutions, in both the bounded and unbounded cases. In the final part, we consider the particular case f u |u| sgn u λ > 0 1.8 and we study the possible coexistence of bounded and unbounded AL-solutions. The role of deviating argument and the one of the growth of the nonlinearity are also discussed and illustrated by some examples. 2. Unbounded Solutions Here we study the existence of unbounded AL-solutions of 1.1 . Our first main result is the following. Theorem 2.1. For any c, 0 < c < ∞, there exists an unbounded solution x of 1.1 such that lim t→∞ x′ t c, lim t→∞ x i t 0, i 2, 3, 2.1 provided ∫∞ 0 |r t |F ( φ t ) dt < ∞, 2.2 where for u > 0 F u max { f v : |v − u| ≤ 1 2 u } . 2.3 Proof. Without loss of generality, we prove the existence of solutions of 1.1 satisfying 2.1 for c 1. Let u and v be two linearly independent solutions of 1.3 with Wronskian d 1. Denote w s, t u s v t − u t v s , z s, t ∂ ∂t w s, t . 2.4 4 Abstract and Applied Analysis As claimed by the assumptions on q, all solutions of 1.3 and their derivatives are bounded. Thus, put M sup{|w s, t | |z s, t | : s ≥ 0, t ≥ 0}, L 2 2M 1 q0 . 2.5 Let t ≥ t0 be such that φ t ≥ t0 for t ≥ t. Define φ t ⎧ ⎨ ⎩ φ t if t ≥ t, φ ( t ) if t0 ≤ t ≤ t, 2.6 and choose t0 ≥ 0 large so that ∫∞ t0 |r s |F ( φ s ) ds ≤ 1 2L , 1 q0 ∫∞ t0 ∣q′ t ∣dt ≤ 1 2 . 2.7 Denote by C t0,∞ the Fréchet space of all continuous functions on t0,∞ , endowed with the topology of uniform convergence on compact subintervals of t0,∞ , and consider the set Ω ⊂ C t0,∞ given by Ω { x ∈ C t0,∞ : t 2 ≤ x t ≤ 3t 2 } . 2.8 Let T > t0 and define on t0, T the function g t γ ′′ t q t γ t , 2.9 where γ t − ∫T