Fractional Coupled Hybrid Sturm–Liouville Differential Equation with Multi-Point Boundary Coupled Hybrid Condition

: The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other ﬁelds of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results. investigate a coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results.


Introduction and Preliminaries
Various papers have been published on fractional differential equations (FDEs) (see, e.g., in [1][2][3][4][5][6]). Over the years, hybrid fractional differential equations have attracted much attention. There have been many works on the hybrid differential equations, and we refer the readers to the papers in [7][8][9][10][11][12][13][14][15][16][17] and the references therein. During the history of mathematics, an important framework of problems called Sturm-Liouville differential equations has been in the spotlight of the mathematicians of applied mathematics and engineering; scientists of physics, quantum mechanics, and classical mechanics; and certain phenomena; for some examples see in [18,19] and the list of references of these papers. In such a manner, it is important that mathematicians design complicated and more general abstract mathematical models of procedures in the format of applicable fractional Sturm-Liouville differential equations, see in [20][21][22].
Define a supremum norm . in E = C(I, R) by u = sup t∈I |u(t)|, and a multiplication in E by (xy)(t) = x(t)y(t) for all x, y ∈ E. Evidently, E is a Banach algebra with respect to above supremum norm and the multiplication in it; also notice that It is well known that the Riemann-Liouville fractional integral of order α of a function f is defined by I α f (t) = 1 Γ(α) t 0 (t − s) α−1 f (s)ds(α > 0) and the Caputo derivative of order α for a function f is defined by where n = [α] + 1 (for more details on Riemann-Liouville fractional integral and Caputo derivative see in [2,4,5]).
The following hybrid fixed point result for three operators, due to Dhage [24], plays a key role in our first main theorem. Lemma 1. Let S be a closed convex, bounded, and nonempty subset of a Banach algebra E and let A, C : E → E and B : S → E be three operators such that (a) A and C is Lipschitzian with a Lipschitz constant δ and ρ, respectively; (b) B are compact and continuous; Then, the operator equation u = AuBu + Cu has a solution in S.
(D 1 ) The function f (u(t)) : R → R is defined on the interval I, ∂ f ∂u is bounded on I with | ∂ f ∂u | ≤ K and f (u(t)) is differentiable in (0, 1), right-differentiable at 0 and left-differentiable at 1. (D 2 ) The function p ∈ C(I, R) with p(t) = 0 for all t ∈ I, inf t∈I |p(t)| = p. Furthermore, q(t) and h(t) are absolutely continuous functions on I. (D 3 ) The function g : I × R → R {0} is continuous in its two variables, and there exists a function µ(t) ≥ 0 (∀t ∈ I) such that ) Two functions f , k : I × R → R are continuous in their two variables, and there are two functionsμ(t), µ * (t) ≥ 0 (∀t ∈ I) such that

Lemma 2.
Assume that the hypotheses (D 1 )-(D 2 ) are satisfied. Then, the problem (3) and (4) is equivalent to the integral equation where Au(t) = I β 1 p(t) I α (q(t)u(t)) , Bu(t) = I β 1 Proof. Equation (3) can be written as Operating by I α on both sides, we get Consequently, The above equation can be written as Operating by I β on both sides, we obtain Therefore, we can obtain and On subtracting (8) from (9) and applying . Therefore, by substituting the value of in (7), we get Conversely, to complete the equivalence between integral Equation (5) and the problem (3) and (4), we have from (6) and so Operating by I 1−α on both sides, we obtain Now, by using the definition of Caputo derivative and (iii), we get and then by applying (ii) and (iv), we have and so we get (3). Clearly, from (6), we can get Moreover, by using a simple computation and (5), we can obtain Now, assume that (B * ) holds. From (10), we know that Then, and so (6) and , Then, we can obtain . This completes the proof.

Lemma 3.
Assume that the hypotheses (D 1 )-(D 5 ) are satisfied. Let |u(t)| ≤ r for all t ∈ I, Proof. (i) Assume that |u(t)| ≤ r for all t ∈ I. Then, we can write Similarly, we can prove that Similarly, we have Now, we are ready to state and prove our main theorem. Proof. Let E = C(I, R). From (D 5 ), we know that there exists a number r > 0 such that ζ * 1 = sup t∈I ζ 1 (t, 0), ζ * 2 = sup t∈I ζ 2 (t, 0), k 0 = sup t∈I k(t, 0) and M = f (0). Define a subset S r of E defined by Clearly, S r is a closed, convex, and bounded subset of E. From Lemma 2, we know that the problems in (3) and (4) are equivalent to the equation Define three operators A, C : E → E and B : S r → E by (11) can be written as

Now, the integral Equation
In the following steps, we will show that the operators A, B, and C satisfy all the conditions of Lemma 1.
Step 1: In this step, we show that A and C are Lipschitzian on E. Let u, v ∈ E, then by (D 3 ), we have for all t ∈ I. Taking the supremum over t, we get Similarly, by applying (D 3 ), we can obtain That is, A and C are Lipschitzian with Lipschitz constants µ and μ , respectively.
Step 2: We show that B is compact and continuous operator on S r into E. At first, we show that B is continuous on S r . Let {u n } be a sequence in S r converging to a point u ∈ S r . Then, by the Lebesgue dominated convergence theorem, for all t ∈ I. That is, B is a continuous operator on S r .
Next, we will show that the set B(S r ) is a uniformly bounded in S r . For any u ∈ S r , by using Lemma 3 (i), we have Now, as Taking supremum over t, Bu ≤ Θ for all u ∈ S r . This shows that B is uniformly bounded on S r . Now, we show that B(S r ) is an equi-continuous set in E. Let t 1 , t 2 ∈ I with t < t 2 . Then, for any u ∈ S r , by applying Lemma 3 (ii), we have Then, for ε > 0, there exist δ > 0 such that for all t 1 , t 2 ∈ I and for all u ∈ S r . This shows that B(S r ) is an equi-continuous set in E. Therefore, we proved that the set B(S r ) is uniformly bounded and equi-continuous set in E. Then, B(S r ) is compact by Arzela-Ascoli Theorem. As a consequence, B(S r ) is a completely continuous operator on S r .
Step 3: Let u ∈ E and v ∈ S r be two given elements such that u = AuBv + Cu. Then, we get and so Taking the supremum over t, we get u ≤ r.
If in Theorem 1, we take ζ 1 (t, w) = k(t, w) = ζ 2 (t, w) − 1 = 0 for all t ∈ I and w ∈ R, we have the following Corollary. Corollary 1. Let the hypotheses (D 1 )-(D 2 ) be satisfied. Assume that Then, the fractional Sturm-Liouville differential problem has a solution u ∈ C(I, R) if and only if u solves the integral equation Therefore, D β c (u(t)) ∈ C(I, R).

Continuous Dependence
The following result will be useful in this section (in fact it is a special case of Theorem 1 with ζ 2 (t, x) = 1 for all t ∈ I and x ∈ R). Corollary 2. Let the hypotheses (D 1 ), (D 2 ), and (D 4 ) be satisfied. Assume that there exists a number r > 0 such that Then, the fractional couple hybrid Sturm-Liouville differential equation with multi-point boundary couple hybrid condition has a solution u ∈ C(I, R) if and only if u solves the integral equation Furthermore, D β c (u(t)) ∈ C(I, R).
In this section, we will investigate continuous dependence (on the coefficients ξ i and η j of the multi-point boundary couple hybrid condition) of the solution of the fractional couple hybrid Sturm-Liouville differential Equation (17) with multi-point boundary couple hybrid condition (18). The main Theorem of this section generalizes Theorem 3.2 in [23] and Theorem 5 in [8].
First, we give the following Definition.

Theorem 2.
Assume that the assertions of Corollary (21) are satisfied. Then, the solution of the fractional couple hybrid Sturm-Liouville differential problem (17) and (18) is continuously dependent on the coefficients ξ i and η j of the multi-point boundary couple hybrid condition.

Conclusions
Scientists utilize various Sturm-Lioville equations for modeling various phenomena and processes. This variety factor in investigating complicates the fractional Sturm-Lioville equations and boosts scientists' ability for exact modelings of more phenomena. This methods will lead scientists to make advanced software which help them to allow more cost-free testing and less material consumption. In this paper, we investigate a coupled hybrid version of the Sturm-Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm-Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm-Liouville differential equation with integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results.