Existence and Multiplicity of Solutions for Some Fractional Boundary Value Problem via Critical Point Theory

and Applied Analysis 3 A3 lim inf|x|→ ∞ ∇F t, x , x − 2F t, x /|x|μ ≥ Q > 0 uniformly for some Q > 0 and a.e. t ∈ 0, T , where r > 2 and μ > r − 2. We state our first existence result as follows. Theorem 1.1. Assume that (A1)–(A3) hold and that F t, x satisfies the condition (A). Then BVP 1.1 has at least one solution on E. 1.2. The Asymptotically Quadratic Case For the asymptotically quadratic case, we assume the following. A2′ lim sup|x|→ ∞F t, x /|x| ≤ M < ∞ uniformly for some M > 0 and a.e. t ∈ 0, T . A4 There exists τ t ∈ L1 0, T ;R such that ∇F t, x , x −2F t, x ≥ τ t for all x ∈ R and a.e. t ∈ 0, T . A5 lim|x|→ ∞ ∇F t, x , x − 2F t, x ∞ for a.e. t ∈ 0, T . Our second and third main results read as follows. Theorem 1.2. Assume that F t, x satisfies (A), (A1), (A2’), (A4), and (A5). Then BVP 1.1 has at least one solution on E. Theorem 1.3. Assume that F t, x satisfies (A), (A1), (A2’), and the following conditions: A4′ there exists τ t ∈ L1 0, T ;R such that ∇F t, x , x − 2F t, x ≤ τ t for all x ∈ R and a.e. t ∈ 0, T ; A5′ lim|x|→ ∞ ∇F t, x , x − 2F t, x −∞ for a.e. t ∈ 0, T . Then BVP 1.1 has at least one solution on E. 1.3. The Subquadratic Case For the subquadratic case, we give the following multiplicity result. Theorem 1.4. Assume that F t, x satisfies the following assumption: A6 F t, x : a t |x|γ , where a t ∈ L∞ 0, T ;R and 1 < γ < 2 is a constant. Then BVP 1.1 has infinitely many solutions on E. 2. Preliminaries In this section, we recall some background materials in fractional differential equation and critical point theory. The properties of space E are also listed for the convenience of readers. 4 Abstract and Applied Analysis Definition 2.1 see 1 . Let f t be a function defined on a, b and q > 0. The left and right Riemann-Liouville fractional integrals of order q for function f t denoted by aD −q t f t and tD −q b f t , respectively, are defined by aD −q t f t 1 Γ ( q ) ∫ t