The Existence Of Symmetric Positive Solutions To A Three-Point Boundary Value Problem

In this paper, we are concerned with the existence and multiplicity of symmetric positive solutions for the following three-point boundary value problem with one-dimensional p-Laplace equationwhereφ_p(u) = |u|^(P-2)u, p > 1, q(t) : [0,1 ]→[0, +∞) is a nonnegative continuous and symmetric function in [0,1], f(t,u,v) : t∈[0,1], u∈[0,+∞), v∈(-∞,+∞) is also a nonnegative continuous and symmetric function in [0,1].The ordinary differential equations arise in a variety of different areas of applied math-ematics and physics. Firstly, the paper presents some results of the ordinary differential equations in the recent years. In the second chapter , we introduce some necessary definitions, theorems and lemmas that will be used to prove our main results. In the third chapter, we construct an integral operator so that to find the solution of our problem is equivalent to find the fixed point of the operator, and hence to obtain the existence of multiple solutions by repeatedly applying the Avery-Peterson fixed point theorem.In the second chapter, we first consider the following equationwhereφ_p(u) = |u|^P-2u, p > 1, e(t) : [0,1 ]→[0, +∞) is a nonnegative continuous and symmetric function in [0,1]. The solution of the equation (2) can be expressedIt is easy to verify that the function is symmetric and positive We define the following convex sets:and a closed setHereγandθare nonnegative continuous convex functionals, a is a nonnegative continuous concave functionals andψis a nonnegative continuous functionals, a, b, c, d are positive real numbers.In this chapter, to prove our results we need the following theorems:Theorem 1 (Schauder Fixed-point Theorem) Let X be a real linear space, (?) is a closed nonempty set. F :(?) is a completely continuous operator; then F has a fixed-point in (?), i.e., F(x) = x, x∈Ω.Theorem 2 Let K be a real Banach space E.In K, letγandθbe nonnegative continuous convex functional,αis a nonnegative continuous concave functional andψis a nonnegative continuous functional which satisfies (?) for 0≤λ≤1, such that forsome positive numbers M,d,α(u)≤ψ(u), ||u||≤Mγ(u) for all u∈(?). Suppose thatT : (?) is completely continuous and there exist positive numbers a, b, c with a < b such thatThen T has at least three fixed-point (?), such that In the third chapter, we gave the sufficient conditions that guarantee the existence of symmetric positive solutions to the boundary value problem (1). We assumeA₁) q(t) : [0,1]→[0, +∞) is a nonnegative continuous and symmetric function in [0,1], i.e., q(t) = q(1 - t) for t∈[0, 1]. It also satisfies (?);A₂) f(t, u, v) : t∈[0,1], u∈[0, +∞), v∈(-∞, +∞) is a nonnegative continuous and symmetric function in [0,1], i.e., f(t, u. v) = f(1 - t. u, -v) when t∈[0,1].According to the relations between the two equations, we define the operator T : K→K bywhere K is a cone in Banach space E = C¹ [0,1] and K = {u∈E | u(t)≥0, u'(0) -u'(1) = u(?), u is a concave function and symmetric in[0,1]}.Assume A₁), A₂) hold , we can prove that the operator T is completely continuous and has (?). By means of the Schauder fixed-point theorem , we obtain that the boundary value problem (1) has a symmetric positive solution.For convenience, we introduce the following notations . Let (?), N = (?). In order to get the existence of at least three positive solutions to the boundary value problem (1), we gave the following theorem.Theorem 3 Assume A₁), A₂) hold and 0 < a < b≤(?), and suppose that f satisfies the following conditions:S₁) f(t, u, v)≤(?), where (t,u, v)∈[0.1]×[0.3d] x [-d,d];S₂) f(t, u, v)≥(?), where (t. u, v)∈(?)×[b, kb]×[-d, d]; S₃) f(t. u, v) < (?), where (t. u. v)∈[0. 1]×[0, a]×[-d, d]. Then the boundary value problem (1) has at least three positive solutions u₁, u₂, u₃ such thatCorollary Assume A₁),A₂) hold and (?), and suppose that f satisfies the following conditions : P₁) (?), where (t, u,v)∈[0,1]×[0,3d_i]×[-d_i,d_i];P₂) (?), where (t, u, v)∈(?);P₃) (?), where (t, u, v)∈[0,1]×[0, a_i]×[-d_i, d_i].Where i = 1,2,3,…, then the boundary value problem (1) has at least 2n + 1 positive solutions.