Solvability for a Fully Elastic Beam Equation with Left-End Fixed and Right-End Simply Supported

The aim of the present paper is to consider a fully elastic beam equation with left-end ﬁxed and right-end simply supported, i.e., ( 1 ) � 0 􏼚 , where f : [ 0 , 1 ] × R 4 ⟶ R is a continuous function. By applying Leray–Schauder ﬁxed point theorem of the completely continuous operator, the existence and uniqueness of solutions are obtained under the conditions that the nonlinear function satisﬁes the linear growth and superlinear growth. For the case of superlinear growth, a Nagumo-type condition is introduced to limit that f ( t,x 0 , x 1 ,x 2 , x 3 ) is quadratical


Introduction
is paper focuses on the existence and uniqueness of solutions for the boundary value problems (BVP) of the fourth-order ordinary differential equation: where f: [0, 1] × R 4 ⟶ R is continuous and involves all derivatives below the fourth order of unknown functions. e problem simulates the deformation of an elastic beam with the left-end fixed and the right simply supported.
e boundary value problems for the nonlinear fourthorder differential equations are the mathematical models used to characterize the deflection of elastic beams under external forces. Elastic beams are one of the most basic structures in architecture and engineering, and some different boundary conditions are derived due to the diversity of stress states on its two ends.
In the past few years, owing to its actual mathematical model and wide application background, the research on nonlinear fourth-order two-point BVP has been very active. Its solvability has attracted the close attention of many scholars, and some profound results have been obtained through various nonlinear analysis methods and techniques (See  and its references).
ere are many results on the solvability for the special cases of BVP (1) whose nonlinear term f is independent of the third-order derivative term of u (see [2,8,10,15]). However, only few articles have studied the existence of solutions for BVP (1). It is worth noting that, in [12], Yao proved the existence of the solutions for BVP (1) by calculating the maximum value of Green function and its partial derivatives as well as constructing height functions of f on bounded sets. In addition, the research on its solvability under some excellent growth conditions, especially the superlinear growth conditions, is even more rare.
In the mechanical analysis of beams, the physical meaning of the derivatives u′(t), u ″ (t), u ‴ (t), and u (4) of u(t) are slope, bending moment, shear force, and load density, respectively (see [1,2,[17][18][19]). e existence of slope, bending moment, shear force, and load density is undoubtedly very beneficial to the complete stress analysis of beams. Nevertheless, the dependence of f on the third derivative u ‴ increases the difficulty for our study, but this is also a fundamental difference from the previous problems. In recent years, the research on the solvability of the elastic beam equation that f involves all lower-order derivatives of deformation function u has become a hot topic (see [5,6,[11][12][13][14][15][16][17][18][19][20][21][22]). For example, the elastic beam equation whose both ends are simply supported (see [14,16,20]): and one end is simply supported, and the other end is sliding clamped (see [5,11,22]): and one end is fixed, and the other end is free (see [13,[17][18][19]21]): e solvability of equations (2)-(4) has been studied by various nonlinear functional methods, including fixed point index theory in cone, fixed point theorem, lower and upper solutions' method, topological degree theory, and variational method. In particular, in [16], Li and Liang researched the existence and uniqueness of solutions for BVP (2) under the condition that f(t, x 0 , x 1 , x 2 , x 3 ) is linear growth; in [19], by supplementing the Nagumo-type condition to limit the growth of f on x 3 , Li and Chen obtained the existence and uniqueness of solutions for BVP (4) under the condition that f satisfies one-side superlinear growth.
Inspired by the literature listed above, in the present article, we discuss the existence and uniqueness of solutions for BVP (1) under the linear growth and one-side superlinear growth condition. e results of existence and uniqueness under the linear growth are presented in Section 3, and the existence results under the one-side superlinear growth are presented in Section 4. It should be noted that, in this paper, the estimation of the maximum value of Green function and its partial derivatives is no longer needed. In addition, with the help of efficient norm estimation and Leray-Schauder fixed point theorem of completely continuous operator, our discussion is carried out in the whole workspace without the restrictions of boundedness of u and its derivatives. erefore, our conclusions greatly improve and generalize the case of the bounded domain in the existing literature, which is new and significant. In order to prove the conclusion, we introduce some necessary properties of the solutions for the corresponding linear equation in the following section.

Preliminaries
Let I � [0, 1], n ∈ N. We introduce the following common spaces on I: Firstly, we consider the following linear boundary value problem (LBVP) corresponding to BVP (1): Proof. For each h ∈ L 2 (I), one can easily test that is the Green function corresponding to the homogeneous linear problem, and it is obvious that G(t, s) ≥ 0, t, s ∈ I. rough specific calculations, we can get erefore, by applying the contraction mapping principle to the solution operator S and combining with (6), it is easy to prove that S has only one fixed point on L 2 (I), that is, u shown in (6) is the unique solution of LBVP (5). When h ∈ C(I), u(t) ≔ Sh ∈ C 4 (I) is a unique classical solution of LBVP (5). It is clear that G(t, s) is continuous; then, according to (6), S: L 2 (I) ⟶ H 4 (I) is a bounded linear operator.
Furthermore, S: L 2 (I) ⟶ H 3 (I) is obviously completely continuous since the Sobolev embedding H 4 (I)↪C 3 (I) is compact and the embedding C 3 e unique solution u of LBVP (5) has the following properties: Proof (a) With a simple calculation, we get en, we can verify that, for given h ∈ C(I), Since for every s ∈ I, and thus, u (10), we can check that, for given h ∈ C(I), Since for every s ∈ I, and hence, combining with (11) and (12), Considering the boundary condition of BVP (5) and using the H€ older inequality, we can obtain that, for every t ∈ I, and then, In the same way, we have erefore,

Mathematical Problems in Engineering 3
According to the conclusion of (b) and the continuity of u ‴ (t), there exists t 0 ∈ [0, 1] such that u ‴ (t 0 ) � 0. us, We can find out in just the same way that ose show that (c) is valid. At this point, the proof is finished. Finally, we introduce the famous Leray-Schsuder fixed point theorem, which will be used to establish our main theorems.
□ Lemma 3 (see [7]). Let X be a Banach space, and T: X ⟶ X is a completely continuous operator. If the solution set of the homotopy family equation is bounded in X, then there exists a fixed point of T in X.
Proof. Define operator F: H 3 (I) ⟶ L 2 (I) by and then, F is continuous and maps bounded sets of H 3 (I) into bounded sets of L 2 (I). According to the definition of S defined in Lemma 1, the solution of BVP (1) is equivalent to the fixed point of the composition operator: By Lemma 1, T: H 3 (I) ⟶ H 3 (I) is completely continuous. Now, we apply Lemma 3 to verify that T has a fixed point.
en, in accordance with Lemma 3, there exists u 0 ∈ H 3 (I) is the fixed point of T, which is the solution of BVP (1). e proof of eorem 1 is completed.

Solvability under Superlinear Growth
In this section, in order to facilitate the establishment of the theorem, we need a Nagumo-type condition to limit the growth of f on x 3 and an important Lemma for the Nagumo-type condition, such that, for every (t, Proof. Let M > 0, then by (38), there is a constant M 1 > 0 satisfying Set u be a solution of BVP (1) that satisfies ‖u‖ C 2 ≤ M; now, we check ‖u ‴ ‖ C ≤ M 1 . From Lemma 2 (b), it follows that there exists t 0 ∈ I, such that u ‴ (t 0 ) � 0. Suppose ‖u ‴ ‖ C ≠ 0, that is, ‖u ‴ ‖ C > 0, then there exists t 1 ∈ I such that erefore, t 0 ≠ t 1 . us, there are four cases: (1) u ‴ (t 1 ) > 0 and t 0 < t 1 (2) u ‴ (t 1 ) > 0 and t 0 > t 1 (3) u ‴ (t 1 ) < 0 and t 0 < t 1 Mathematical Problems in Engineering 5 (4) u ‴ (t 1 ) < 0 and t 0 > t 1 For case 1, let (see Figure 1) and according to the definition of supremum and the continuity of u ‴ (t), we have t 0 ≤ t 2 < t 1 ≤ 1 and Since hence, by BVP (1) and (39), en, Taking integral operation on (t 2 , t 1 ] for both sides of the inequality, and making variable substitution ρ � u ‴ (t) on the left. en, ereby, en, based on (40), Similarly, we can discuss cases 2°− 4°, and the conclusions are the same. □ Theorem 3. Suppose that f ∈ C(I × R 4 , R) satisfies the following: en, BVP (1) has at least one solution.