The Study of Higher-order Resonant and Non-resonant Boundary Value Problems

The existence of at least one solution to a nonlinear n th order differential equation x (n) = f (t, x, x ,. .. , x (n−1)), 0 < t < 1, under both non-resonant and resonant boundary conditions, is proved. The methods involve the characterization of the R δ-set and an application of a new generalization for a multi-valued version of the Miranda Theorem.


Introduction
Higher order differential equations have been extensively studied in recent years.A variety of results ranging from the theoretical aspects of existence and uniqueness of solutions to analytic and numerical methods for finding solutions have appeared in the literature [1,6,[11][12][13]18].Such differential equations can be written in the form Lx = Nx, where L is a linear differential operator defined in appropriate Banach spaces and N is a nonlinear operator.When L is a linear Fredholm operator of index 0 under certain boundary conditions, then the kernel of the linear part of the above equation is trivial, and in this case, the corresponding BVP is called non-resonant.This means that there exists an integral operator; then, topological methods can be applied to prove existence theorems.If the kernel of L is nontrivial, then the problem is said to be at resonance, and then the problem can be managed by using coincidence degree theory.Such boundary value problems for higher order differential equations have been studied by standard methods in many papers; see, for instance, the papers [2,4,[7][8][9][10][15][16][17].
Motivated in this paper by the above research, by applying a generalized Miranda Theorem [14] and a technique completely different from the methods mentioned above, we obtain results devoted to the study of the following higher-order nonlinear differential equation Corresponding author.Email: yangaij2004@163.comunder the non-resonant boundary conditions and under the resonant conditions respectively, where is a continuous vector function and satisfies appropriate growth conditions; in particular, we assume: The rest of this paper is organized as follows.In Section 2, we give some preliminary definitions and theorems on the topological structure of certain sets in metric spaces, which will be employed to obtain the main results.In Section 3, we study the non-resonant BVP (1.1)-(1.2).By introducing an auxiliary initial value problem and characterizing the upper semicontinuous set-valued R δ -map, we show that this problem has at least one solution.Finally, in Section 4, we deal with the resonant BVP (1.1)- (1.3).By a differential transformation, we get the desired results by adopting the techniques used in Section 3.

Preliminaries
First, we present some notations and terminologies.Definition 2.1.A metric space X is an absolute neighborhood retract (written ANR) if, given a space Y and a homeomorphic embedding i : X → Y of X onto a closed subset i(X) ⊂ Y, i(X) is a neighborhood retract of Y, i.e. there is an open neighborhood U of i(X) in Y and a retraction r : U → i(X).A map r : U → i(X) is a retraction provided that r(y) = y for y ∈ i(X).Definition 2.2.A nonempty space X is contractible provided there exist x 0 ∈ X and a homotopy h : X × [0, 1] → X such that h(x, 0) = x and h(x, 1) = x 0 for every x ∈ X.
Definition 2.5.By a decomposable map we mean a pair (D, F) consisting of a set-valued map Remark 2.6.In our case, Z will be a Banach space, which is ANR.Moreover, notice that a decomposable map (D, F) is an admissible map in the sense of Górniewicz (see [5]).
Remark 2.7.A superposition of a set-valued map with compact values and continuous function is an USC map, so any decomposable map is USC.
Definition 2.8.We say the two decomposable maps (D 0 , F 0 ), (D 1 , F 1 ) where Next, we present a result from [3] about the topological structure of the set of solutions for some nonlinear functional equations.Theorem 2.9.Let X be a space, (B, • ) a Banach space and h : X → B a proper map, i.e. h is continuous and for every compact E ⊂ B the set h −1 (E) is compact.Assume further that for each ε > 0 a proper map h ε : X → B is given and the following two conditions are satisfied: (b) for any ε > 0 and u ∈ B such that u ≤ ε, the equation h ε (x) = u has exactly one solution.
Next, we present a generalization of the Miranda Theorem proven in [14], which will be of crucial importance.Theorem 2.10.Let M i > 0, i = 1, . . ., k, and F be an admissible map from ∏ k i=1 [−M i , M i ] to R k , i.e. there exist a Banach space E, dim E ≥ k, a linear, bounded and surjective map ϕ : )

Solutions for BVP (1.1)-(1.2)
In this section, we discuss the BVP (1.1)-(1.2).First, in setting the stage for the application of Theorem 2.10, let the Banach space (B, • B ) be defined by Next, we consider the equation (1.1) under the following initial conditions: where c ∈ R k is fixed.Notice that the IVP (1.1)-(3.1) is equivalent to Moreover, we have Note that Then, by applying (3.3) and (H1), for t ∈ [0, 1], we get where Now, in view of Gronwall's Lemma, we have where By (3.4), we get the following estimate From above, the Leray-Schauder Alternative implies that the IVP (1.1)-(3.1)has a bounded global solution for every t ∈ [0, 1] and fixed c ∈ R k .Now, given c ∈ R k , consider the nonlinear operator T : R k × B → B, (c, x) → T c (x), defined as Taking any sequence {c n }, c n → c 0 and {x n } ⊆ Φ(c n ), we have Thus, x 0 ∈ Φ(c 0 ) and the proof is complete.
Proof.By Lemma 3.2, the map Φ is USC.It remains to be shown that for any a compact map from Lemma 3.1, h is a compact vector field associated with T c (•).Then we shall show that there exists a sequence h n : E → B of continuous proper mappings satisfying conditions (a) and (b) of Theorem 2.9 with respect to h.
For the proof it is sufficient to define a sequence x) is a one-to-one map.To do this, we define auxiliary mappings r n : R + → R + by Now, we can define the sequence {T n c (•)} as follows: We see that T n c (•) are continuous and compact.Since |r n (t) − t| ≤ 1 n , we deduce from the compactness of T n c (•) and (3.8) that T n c (x) → T c (x) uniformly in E. Next, we shall prove that h n is a one-to-one map.Assume that for some u, v ∈ E, we have Thus, we obtain u(t) = v(t) for t ∈ [0, r n (r n (t)) = 0. Hence, by the property of operator T c (•) mentioned above, we have . By repeating this procedure n times we infer that u(t) = v(t) for t ∈ [0, 1].Therefore, h n is a one-to-one map.Hence the assumptions of Theorem 2.9 hold and h Consider a multifunction F : R k R k given by Now, let ϕ : B → R k be such that ϕ(x) = x(1).
By Theorem 2.10, there exists c ∈ R k such that 0 ∈ F(c).This completes the proof.where c ∈ R k .Now, we consider the equation (4.1) under the following initial conditions:

Lemma 3 . 1 .Lemma 3 . 2 .
by applying (H1),(3.4)and(3.5)one can easily show that the image of{(c, x) ∈ R k × B : (c, x) R k ×B ≤ L}under T is relatively compact.we obtain the following results.Let assumption (H1) hold.Then the operator T is completely continuous.Notice that the solutions of the IVP (1.1)-(3.1)are fixed points of the operator T defined by(3.6).Let Fix T c (•) denote the set of fixed points of operator T c , where c ∈ R k is given.Let assumption (H1) hold and Φ : R k