Periodic Solutions to Second-Order Nonlinear Differential Equations in Banach Spaces

In this article, we deal with the existence of solutions for the following second-order differential equation: u′′(t)=f(t,u(t))+h(t)u(a)-u(b)=u′(a)-u′(b)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&u''(t)=f(t,u(t))+h(t)\\&u(a)-u(b)= u'(a)-u'(b)=0, \end{aligned}\right. \end{aligned}$$\end{document}where B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {B}}$$\end{document} is a reflexive real Banach space, f:[a,b]×B→B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:[a,b]\times {\mathbb {B}}\rightarrow {\mathbb {B}}$$\end{document} is a sequentially weak–strong continuous mapping, and h:[a,b]→B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h:[a,b]\rightarrow {\mathbb {B}}$$\end{document} is a continuous function on B.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {B}}.$$\end{document} The proofs are obtained using a recent generalization of the well-known Bolzano–Poincaré–Miranda theorem to infinite-dimensional Banach spaces. In the last section, we present three examples of application of the general result.


Introduction
The main purpose of this article is to study the existence of solutions of the second-order dynamical equation  [6,10,13], and degree theory and fixed point theory [8,12]. In this sense, in [20], the author gave a generalization of Miranda-Poincaré theorem and using this generalization proved theorems about the 51 Page 2 of 17 D. Ariza-Ruiz and J. Garcia-Falset MJOM existence for systems of k equations x = f (t, x, x ), where f : [0, 1] × R k × R k → R k is a vector function, subject to various boundary conditions. Recently, the existence of periodic solutions of second-order differential equations on a general Banach space has been treated in several papers; for instance, see [14,17,19]. In [14], operator-valued Fourier multiplier theorems are used to study this problem. In [17], the authors developed an infinitedimensional version of Poincaré-Miranda theorem and they showed their relation with viability theory for differential inclusions and how to apply them, in particular, in the context of constrained PDEs. On the other hand, in [19], variational methods are used to study the existence of two solutions of superlinear second-order Hamiltonian systems that have damped and impulsive effects.
In this paper, we are able to give sufficient conditions to obtain existence of periodic solutions for Problem (1.1). Namely, we obtain that when B is a reflexive Banach space and f : [a, b] × B → B is a sequentially weak-strong continuous mapping satisfying conditions (3.2) and (3.4) (see them below), then, using a generalization of Bolzano-Poincaré-Miranda theorem [1] and Leray-Schauder Alternative theorem (for instance, see [4]), Problem (1.1) admits a classical solution.

Notations and Preliminaries
In this section, we shall recall some definitions and results that are needed later on.
Throughout this article, B will denote a real reflexive Banach space with norm · and we shall denote its dual by B * . The closed ball of center x 0 and radius r is denoted by B r [x 0 ] and the sphere of center x 0 and radius r by ∂B r [x 0 ]. The symbol " is used to denote the convergence of a sequence in B with respect to the weak topology σ(B, B * ), as well as the symbol " → is used to denote the convergence of a sequence in B with respect to the norm topology.
It is well known (for example, see [18] The hypothesis that is used to produce a more convenient form of Schauder's fixed point theorem is called Leray-Schauder boundary condition: Given a mapping F : B → B, the mapping F satisfies this condition if there exists r > 0, such that x = r implies F (x) = λx for all λ > 1. The result that replaces the requirement in Schauder's fixed point theorem is the following one (see, e.g., [4]): We state below Ascoli-Arzelà's theorem (see [7, p.34]) exactly in the form we need later as follows: We finish this section recalling several basic definitions and relevant results related to Sobolev spaces.
Let Ω be a measurable subset of R N which for simplicity will be assumed to be bounded. The Sobolev space W m,p (Ω) (see [3,18]) is the Banach space of all functions in L p (Ω) whose weak derivatives up to order m also belong to L p (Ω). The norm in W m,p (Ω) is given by or by the equivalent norm The next inequality is known as "Poincaré's inequality" and it is very useful in the study of Dirichlet elliptic equations (for instance, see [18, Section 13.2]) as follows: As a consequence of the above inequality, we get to ∇u 2 is an equivalent norm on W 1,2 0 (Ω), namely where we are considering that w 1,2 = w 2 2 + ∇w 2 2 , and ∇w 2 =

Main Results
In 1971, Knobloch [16] showed that if f : [0, T ] × R n → R n is a continuous function which is locally Lipschitzian in u, the condition: there exists R > 0, such that f (t, u), u ≥ 0 for all t ∈ [0, T ] and u ∈ R n satisfying u = R is a sufficient condition for the existence of a solution for Problem (1.1). In our case, we have to work with conditions (3.2) and (3.4) (see below). For our subsequent study, we introduce, for each element of the Banach space C([a, b], B), an auxiliary mapping and we will prove the existence and uniqueness of zeros of this mapping.
Then, the mapping F v : B → B given by (3.1) has at least a zero in B.
. We consider the sequence of functions given by By hypothesis, f is sequentially weak-strong continuous; then, which implies, by the dominated convergence theorem, the strong convergence On the other hand, if we consider the functional [·, ·] c : B × B → R given by  Proof. Note that, by (3.4), F v is strictly accretive, that is, F v (x)−F v (y), j > 0 for all x, y ∈ B, with x = y and for all j ∈ J(x−y). Indeed, for each x, y ∈ B and j ∈ J(x − y), we have that where δ s is the constant given in (3.2), such that Proof. For the sake of clarity, the proof will be divided into five steps.
Step 1. Notice that if we consider v(t) = u (t), then Problem (1.1) is equivalent to solve the following problem: where D is any element of B depending of v.
Step 2. By Lemmas 3.1 and 3.2 , for each v ∈ C ([a, b], B), there exists a unique zero Integrating Eq. (3.6), we deduce that the previous problem is equivalent to the following integral equation: and D v is the unique zero of F v , because in this case, it is clear that b a v(t) dt = 0. Thus, we consider the operator T : C([a, b], B) → C ([a, b], B), given by and, therefore, to find a fixed point for the operator T is equivalent to find a solution of Eq. (3.6).
Step 3. Let us see that T is continuous. Let {v n } be a sequence in C([a, b], B) that converges strongly to v ∈ C ([a, b], B). Note that {D vn } is a bounded sequence, and since B is a reflexive Banach space, we may assume that D vn D. We can argue as in proof of Lemma 3.1 where now we can define g n (t) := f (t, On the other hand, for each t ∈ [a, b] dτ , for all n ∈ N. Since f is sequentially weak-strong continuous, g n (τ ) − g(τ ) → 0 as n → ∞ for all τ ∈ [a, b] and, furthermore, g n (t) ≤ K for all t ∈ [a, b] where K ≥ 0 is the constant of boundedness of f ([a, b] × B 2(b−a)+ρ ) and ρ > 0 satisfying v n ∞ ≤ ρ for all n ∈ N, because {v n } is a convergent sequence. Then, by the dominated convergence theorem, we obtain T (v n ) → T (v) as n → ∞, which means that T is continuous.
Step 4. We next prove that T is compact. Let C be a bounded subset of C ([a, b], B) and let us see that T (C) is compact. To prove this, we shall make use of the Ascoli-Arzelà Theorem.
In the first place, let 0 < = (C) := sup{ v ∞ : v ∈ C} < ∞, because C is bounded. Then, for any v ∈ C, we get to is the constant given in (3.2). Since f is sequentially weak-strong continuous, there exists K > 0, such that Then, for any t ∈ [a, b] and v ∈ C, we have that that is, (i) holds. On the other hand, for any ε > 0, if we take δ(ε) = ε/(K + h ∞ ) > 0, then, for all v ∈ C and t, s ∈ [a, b] with t > s and |t − s| ≤ δ(ε), we obtain that Therefore, from Ascoli-Arzelà Theorem, we deduce that T is a compact operator.
Step 5. Let us prove that T has a fixed point in C([a, b], B). It is easy to see that, for any v ∈ C ([a, b], B)  ([a, b], B) From the hypothesis of Γ, we deduce that there exists S > 0 such that, for all v ∞ ≥ S Then, the Leray-Schauder boundary condition on the ball B S [0] holds. Therefore, by Leray-Schauder alternative Theorem, there exists at least a fixed point of T .
Finally, we point out that the hypothesis b a h(t) dt = 0 cannot be removed in our main result (see Example 4.2 below).

Examples
In this section, we present three examples of application of Theorem 3.1. Two of them for a system of n (n ∈ N) second-order differential equations and the last one for a partial differential equation.
Remark 4.1. By carefully checking the above example, we find that other kinds of systems of equations can be solved. Just consider another norm · in R n . For example, we can ensure the existence of at least a periodic solution of a system of nonlinear second-order differential equations of the following type: The following example shows that in Theorem 3.1, neither the hypothesis b a h(t) dt = 0 nor condition (3.2) can be omitted. Indeed, suppose that u 0 is a solution of (4.2), then if we call v(t) = u 0 (t), we have This means that which is a contradiction.
On the other hand, it is not difficult to see that the function f : 1+|x| is under the conditions of Remark 4.1; therefore, hypothesis b a h(t)dt = 0 cannot be omitted.
If we call f (t, x) = x 1+|x| + 2 and h ≡ 0, we may say that in the main result, condition (3.2) cannot be removed. We next give an example in the case f is unbounded and, moreover, the inequality required for Γ in assumption (A 2 ) holds for any closed interval Here, · 2 denotes the Euclidean norm of R n . It is immediate that f weak-strong continuous, because R n is finite dimensional and f is continuous. We claim that f satisfies (3. for all x, y ∈ R n , with y 2 ≤ M and x 2 ≥ δ M = M .
We now prove that f is strictly accretive. Let x, y ∈ R n . Note that In [11, Proof of Lemma 1], the authors prove that (in general, for α ≥ 0) On the other hand, if y 2 > x 2 , from Cauchy-Bunyakovsky-Schwarz inequality Therefore, f (x) − f (y), x − y > 0 for all x, y ∈ R n with y 2 > x 2 . This one, together with the symmetry of scalar product, implies that the above inequality holds for all x, y ∈ R n , x = y, that is, f is strictly accretive.
Finally, let us see that (A 2 ) holds. Indeed, just take the function Γ : since δ s = s is the constant given in (3.2) in this case, and 0 < α < 1.