Existence of Subharmonic Periodic Solutions to a Class of Second-Order Non-Autonomous Neutral Functional Differential Equations

and Applied Analysis 3 Let us consider the functional I x defined on H1 0 0, 2γτ by I x ∫2γτ 0 [ x′ t x′ t − τ − F t, x t , x t − τ dt. 2.4 For all x, y ∈ H1 0 0, 2γτ and ε > 0, we know that I ( x εy ) I x ε (∫2γτ 0 [ x′ t y′ t − τ x′ t − τ y′ t −Ft, x t εy t , x t − τ εy t − τ ) − F t, x t , x t − τ dt ) ε2 ∫2γτ 0 y′ t y′ t − τ dt. 2.5 It is then easy to see that 〈 I ′ x , y 〉 ∫2γτ 0 [ x′ t y′ t − τ x′ t − τ y′ t − F ′ 1 t, x t , x t − τ y t −F ′ 2 t, x t , x t − τ y t − τ ] dt, 2.6 where I ′ x denotes the Frechet differential of the function I x . By the periodicity of F t, u1, u2 , x t , and y t , we have ∫2γτ 0 x′ t y′ t − τ dt ∫2γτ 0 x′ t dy t − τ x′ t y t − τ ∣∣2γτ 0 , − ∫2γτ 0 x′′ t y t − τ dt − ∫2γτ 0 x′′ t τ y t dt, ∫2γτ 0 x′ t − τ y′ t dt − ∫2γτ 0 x′′ t − τ y t dt. 2.7 Similarly, we have ∫2γτ 0 F ′ 2 t, x t , x t − τ y t − τ dt ∫ 2γ−1 τ −τ F ′ 2 t τ, x t τ , x t y t dt ∫2γτ 0 F ′ 2 t, x t τ , x t y t dt. 2.8 4 Abstract and Applied Analysis Hence, 〈 I ′ x , y 〉 ∫2γτ 0 −x′′ t τ − x′′ t − τ −F ′ u1 t, x t x t − τ − F ′ u2 t, x t τ , x t ] y t dt. 2.9 Therefore, the Euler equation corresponding to the functional I x is x′′ t τ x′′ t − τ F ′ u1 t, x t , x t − τ F ′ u2 t, x t τ , x t ] 0. 2.10 It is not difficult to see that 2.10 is equivalent to 1.1 . Thus, system 1.1 is the Euler equation of the functional I x . It follows that it is possible to obtain 2γτ-periodic solutions of system 1.1 by seeking critical points of the functional I x . Since I x has neither a supremum nor an infimum, we do not seek critical points of the functional I x by the extremum method. But we may use operator equation theory. First via the dual variational principle, we obtain new operator equations see 4.16 related to 1.1 . Then solutions to system 1.1 are obtained by seeking critical points of operator equation 4.16 . In this paper, our main tool is the following. Lemma 2.1 Maintain Pass Theorem . Let H be a real Banach space. If I · ∈ C1 H,R satisfies the Palais-Smale condition as well as the following additional conditions: 1 there exist constants ρ > 0 and a > 0 such that I x ≥ a, for all x ∈ ∂Bρ, where Bρ {x ∈ H : ‖x‖H < ρ}, 2 I θ ≤ 0 and there exists x0 ∈Bρ such that I x0 ≤ 0, then c infh∈Γsups∈ 0,1 I h s is a positive critical value of I, where Γ {h ∈ C 0, 1 ,H | h 0 θ, h 1 x0}. 2.11 The rest of this paper is organized as follows. Subdifferentiability of lower semicontinuous convex function φ x t , x t − τ and its conjugate function are introduced in Section 3. In Section 4, we first give the definition of the weak solution to 1.1 , then we establish the new operator equation 4.16 related to 1.1 by the conjugate function of F t, x t , x t − τ and show that we can obtain solution to 1.1 from the solution to operator equation 4.16 . In Section 5, by seeking critical points of operator equation 4.16 , we obtain the result that there exist multiple subharmonic periodic solutions to system 1.1 . Finally in Section 6, an example and a remark are given to illustrate our result. 3. The Subdifferentiability and the Conjugate Function of the Lower Semicontinuous Convex Function φ x t , x t − τ Let X be a space of all given n × τ-periodic functions in t and a Banach space, where n ∈ N is a positive integer. Denote R R ∪ { ∞}. Let φ : X2 → R be a lower semicontinuous Abstract and Applied Analysis 5 convex function. Generally, φ is not always differentiable in conventional sense, but we may generalize the definition of “derivative” as follows. Definition 3.1. Let x∗ 1, x ∗ 2 ∈ X∗ × X∗. We say that x∗ 1, x∗ 2 is a subgradient of φ at point x0 t , x0 t − τ ∈ X ×X if φ x0 t , x0 t − τ 〈 x∗ 1, x t − x0 t 〉 〈 x∗ 2, x t − τ − x0 t − τ 〉 ≤ φ x t , x t − τ . 3.1 For all x0 t ∈ X, the set of all subgradients of φ at point x0 t , x0 t − τ will be called the sub-differential of φ at point x0 t , x0 t − τ and will be denoted by ∂φ x0 t , x0 t − τ . By the definition of Subdifferentiability of function φ, we may define its conjugate function φ∗ by φ∗ ( x∗ 1, x ∗ 2 ) sup {〈 x∗ 1, x t 〉 〈 x∗ 2, x t − τ 〉 − φ x t , x t − τ , 3.2and Applied Analysis 5 convex function. Generally, φ is not always differentiable in conventional sense, but we may generalize the definition of “derivative” as follows. Definition 3.1. Let x∗ 1, x ∗ 2 ∈ X∗ × X∗. We say that x∗ 1, x∗ 2 is a subgradient of φ at point x0 t , x0 t − τ ∈ X ×X if φ x0 t , x0 t − τ 〈 x∗ 1, x t − x0 t 〉 〈 x∗ 2, x t − τ − x0 t − τ 〉 ≤ φ x t , x t − τ . 3.1 For all x0 t ∈ X, the set of all subgradients of φ at point x0 t , x0 t − τ will be called the sub-differential of φ at point x0 t , x0 t − τ and will be denoted by ∂φ x0 t , x0 t − τ . By the definition of Subdifferentiability of function φ, we may define its conjugate function φ∗ by φ∗ ( x∗ 1, x ∗ 2 ) sup {〈 x∗ 1, x t 〉 〈 x∗ 2, x t − τ 〉 − φ x t , x t − τ , 3.2 where 〈·〉 denotes the duality relation ofX∗ andX. So it is not difficult to obtain the following propositions. Proposition 3.2. φ∗ is a lower semicontinuous convex function (φ∗ may have functional value ∞, but not functional value −∞). Proposition 3.3. If φ ≤ ψ, then φ∗ ≥ ψ∗. Proposition 3.4 Yang inequality . One has φ x t , x t − τ φ∗x∗ 1, x∗ 2 ) ≥ x∗ 1, x t 〉 〈 x∗ 2, x t − τ 〉 . 3.3 Proposition 3.5. One has φ x t , x t − τ φ∗x∗ 1, x∗ 2 ) 〈 x∗ 1, x t 〉 〈 x∗ 2, x t − τ 〉 ⇐⇒ x∗ 1, x∗ 2 ) ∈ ∂φ x t , x t − τ . 3.4 Proposition 3.6. φ∗ does not always equal ∞. Proof. Let x0 t ∈ X and β ∈ R such that φ x0 t , x0 t − τ < ∞ and β0 < φ x0 t , x0 t − τ . We consider the two convex sets in X2 × R defined by A epiφ {( x t , x t − τ , β ∈ X2 × R |: φ x t , x t − τ < ∞, β > φ x t , x t − τ } , B {( x0 t , x0 t − τ , β0 )} . 3.5 6 Abstract and Applied Analysis By Hahn-Banach Theorem, we know that there exists f1, f2, k ∈ X∗ ×X∗ ×R and α ∈ R such that 〈 f1, x t 〉 〈 f2, x t − τ 〉 kβ > α, ∀x t , x t − τ , β ∈ eipφ, 〈 f1, x0 t 〉 〈 f2, x0 t − τ 〉 kβ0 < α. 3.6 So, we have 〈 f1, x0 t 〉 〈 f2, x0 t − τ 〉 kφ x0 t , x0 t − τ > α > 〈 f1, x0 t 〉 〈 f2, x0 t − τ 〉 kβ0. 3.7 Thus k > 0, and 〈 − 1 k f1, x t 〉 〈 − 1 k f2, x t − τ 〉 − φ x t , x t − τ < − k . 3.8 Since φ∗ is a lower semicontinuous convex function that does not always equal ∞, we may define its conjugate function φ∗∗ by φ∗∗ ( x∗∗ 1 , x ∗∗ 2 ) sup x∗ 1,x 2 ∈X∗×X∗ {〈 x∗∗ 1 , x ∗ 1 〉 〈 x∗∗ 2 , x ∗ 2 〉 − φx∗ 1, x∗ 2 )} , 3.9 where 〈·〉 denotes the duality relation of X∗∗ and X∗. Theorem 3.7. Let φ be a lower semicontinuous convex function that does not always equal ∞, then φ∗∗ φ. Proof. We divide our proof into two parts. First we show that φ∗∗ φ holds when φ > 0 and then φ∗∗ φ holds for all lower semicontinuous convex functions φ that do not always equal ∞. i The case when φ > 0. From the definition of φ∗∗ and Yang inequality, it is obvious that φ∗∗ ≤ φ holds. Next, to prove φ∗∗ ≥ φ holds, suppose to the contrary that there exist a point x0 t , x0 t − τ ∈ X2, such that φ∗∗ x0 t , x0 t − τ < φ x0 t , x0 t − τ holds. Consider the two convex sets A eipφ {( x t , x t − τ , β ∈ X2 × R | φ x t , x t − τ < ∞, β ≥ φ x t , x t − τ } , B0 {( x0 t , x0 t − τ , φ∗∗ x0 t , x0 t − τ )} . 3.10 By the Hahn-Banach Theorem, we know that there exist g1, g2, k∗ ∈ X∗ ×X∗ × R and α1 ∈ R such that 〈 g1, x t 〉 〈 g2, x t − τ 〉 k∗β > α1, ∀ ( x t , x t − τ , β ∈ eipφ, 3.11 〈 g1, x0 t 〉 〈 g2, x0 t − τ 〉 k∗φ∗∗ x0 t , x0 t − τ < α1. 3.12 Abstract and Applied Analysis 7 So, it follows that k∗ ≥ 0. Let ε > 0. Using φ > 0 and 3.11 , one obtains that 〈 g1, x t 〉 〈 g2, x t − τ 〉 k∗ ε φ x t , x t − τ ≥ α1, ∀ x t , x t − τ ∈ D ( φ ) . 3.13and Applied Analysis 7 So, it follows that k∗ ≥ 0. Let ε > 0. Using φ > 0 and 3.11 , one obtains that 〈 g1, x t 〉 〈 g2, x t − τ 〉 k∗ ε φ x t , x t − τ ≥ α1, ∀ x t , x t − τ ∈ D ( φ ) . 3.13