Some Results on Fractional m-Point Boundary Value Problems

Mathematical models due to fractional differential equations can describe the natural phenomenon in physics, population dynamics, chemical technology, biotechnology, aerodynamics, electrodynamics of complex medium, polymer rheology, and control of dynamical systems (see [1–4]). Due to the nonlocal characteristics and the rapid development of the theory of fractional operators, some authors have investigated different aspects of fractional differential equations including existence of solutions, Lyapunov’s inequality, and Hyers-Ulam stability for fractional differential equations by different mathematical techniques. For example, first, many authors have discussed the existence of nontrivial solutions of fractional differential equations in nonsingular case as well as singular case. Usually, the proof is based on either the method of upper and lower solutions, fixed-point theorems, alternative principle of Leray-Schauder, topological degree theory, or critical point theory. We refer the readers to [5– 20]. Second, Lyapunov, during his study of general theory of stability of motion in 1892, introduced the stability criterion for second-order differential equations, which yielded a counter inequality be called Lyapunov inequality (see [21, 22]). Since then, we can find considerable modifications of Lyapunov-type inequality of differential equations, such as linear differential-algebraic equations, fractional differential equations, extreme Pucci equations, and dynamic equations, which are applied to study the stability and disconjugacy or oscillatory criterion for the mentioned problems, and we refer the readers to [23–32]. Finally, the stability of functional equations was originally raised by Hyers in 1941 (see [33, 34]). Thereafter, the stability properties of all kinds of equations have attracted the attention of many mathematicians. To see more details on the Ulam-Hyers stability and Ulam-Hyers-Rassias of differential equations, we refer the readers to [35–38]. Inspired by the references, this paper is mainly concerned with the existence, Lyapunov’s inequality, and Ulam-Hyers stability results for the m-point boundary value problems.


Introduction
Mathematical models due to fractional differential equations can describe the natural phenomenon in physics, population dynamics, chemical technology, biotechnology, aerodynamics, electrodynamics of complex medium, polymer rheology, and control of dynamical systems (see [1][2][3][4]). Due to the nonlocal characteristics and the rapid development of the theory of fractional operators, some authors have investigated different aspects of fractional differential equations including existence of solutions, Lyapunov's inequality, and Hyers-Ulam stability for fractional differential equations by different mathematical techniques. For example, first, many authors have discussed the existence of nontrivial solutions of fractional differential equations in nonsingular case as well as singular case. Usually, the proof is based on either the method of upper and lower solutions, fixed-point theorems, alternative principle of Leray-Schauder, topological degree theory, or critical point theory. We refer the readers to [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Second, Lyapunov, during his study of general theory of stability of motion in 1892, introduced the stability criterion for second-order differential equations, which yielded a counter inequality be called Lyapunov inequality (see [21,22]). Since then, we can find considerable modifications of Lyapunov-type inequality of differential equations, such as linear differential-algebraic equations, fractional differential equations, extreme Pucci equations, and dynamic equations, which are applied to study the stability and disconjugacy or oscillatory criterion for the mentioned problems, and we refer the readers to [23][24][25][26][27][28][29][30][31][32]. Finally, the stability of functional equations was originally raised by Hyers in 1941 (see [33,34]). Thereafter, the stability properties of all kinds of equations have attracted the attention of many mathematicians. To see more details on the Ulam-Hyers stability and Ulam-Hyers-Rassias of differential equations, we refer the readers to [35][36][37][38].
Inspired by the references, this paper is mainly concerned with the existence, Lyapunov's inequality, and Ulam-Hyers stability results for the m-point boundary value problems.
where α i , ξ i , h, and f satisfy the following assumptions: For these goals, we first convert problem (1) into an integral equation via Green function. Furthermore, we study the properties and estimates of the Green function. Then, on the basis of these properties, we apply some fixed-point theorems to establish some existence results of problem (1) under some suitable conditions. In addition, the Lyapunov inequality and Hyers-Ulam stability of the proposed problem are also considered.

Preliminaries
Before beginning the main results, we state some classic and modified definitions and lemmas from fractional calculus.
Definition 1 [4]. The fractional integral of order q > 0 of a function u : ð0,+∞Þ ⟶ R is given by provided the right-hand side is pointwise defined on ð0, +∞Þ .
Definition 2 [4]. The fractional derivative of order q > 0 of a continuous function u : ð0,+∞Þ ⟶ R is given by where n = ½q + 1, provided that the right-hand side is pointwise defined on ð0, +∞Þ.
Definition 3 [21]. Assume that q > 0, then for some C i ∈ R, i = 1, 2, ⋯, n, where n is the smallest integer greater than or equal to q.

Lemma 4.
Assume that (H1) holds. Then, for any yðtÞ ∈ L 1 ½0, 1, the boundary value problem has a unique solution uðtÞ = Proof. From Definition 3, it follows that Journal of Function Spaces Since u ″ ð0Þ = 0, it is clear that C 2 = 0. Then, On one hand, taking the derivative of u′′ðtÞ, we can get On the other hand, combining the boundary conditions uð1Þ = u ′ ð0Þ = 0, we have Furthermore, we have According to these above expressions, we have Then, from u″ð1Þ − ∑ m−2 i=1 α i u ‴ ðξ 1 Þ = 0, it follows that which yields If In the similar way, we also can get the expression of Gðt, sÞ on other intervals.

Existence Results
Theorem 7. Assume that (H1)-(H3) hold. In addition, there exists a positive constant L > 0 such that Then, problem (1) has a unique solution if L Gjhj L 1 < 1.
Proof. Let C½0, 1 = fxðtÞ: xðtÞ is continuous on ½0, 1g is a Banach space with the norm kxk = max 0≤t≤1 jxðtÞj. From Lemma 4, it is clear that solutions of (1) can be rewritten as fixed points of operator T, which is defined by Now, we show that T : B r ⟶ B r and T is a contraction map, where B r = fu ∈ E : kuk < rg with On one hand, for any u ∈ B r , we have which implies that TðB r Þ ⊂ B r . On the other hand, for any u, v ∈ E, we have which implies that T is a contraction map. ☐ Therefore, by the Banach contraction mapping principle, it follows that the operator T has a unique fixed point, which is the unique solution for problem (1).

Theorem 8.
Assume that (H1)-(H3) hold. In addition, there exists a positive constant K such that j f ðuÞj ≤ K for u ∈ R. Then, problem (1) has at least one solution.
The proof is based on the following fixed-point theorem.
Lemma 9 [39]. Let E be a Banach space, E 1 is a closed, convex subset of E, Ω an open subset of E 1 , and 0 ∈ Ω. Suppose that T : Ω ⟶ E 1 is completely continuous. Then, either (i) T has a fixed point in Ω, or (ii) there are u ∈ ∂Ω (the boundary of Ω in E 1 ) and ρ ∈ ð0, 1Þ with u = ρTu Proof of Theorem 8. First, we show that the operator T is uniformly bounded.