On strong singular fractional version of the Sturm–Liouville equation

The Sturm–Liouville equation is among the significant differential equations having many applications, and a lot of researchers have studied it. Up to now, different versions of this equation have been reviewed, but one of its most attractive versions is its strong singular version. In this work, we investigate the existence of solutions for the strong singular version of the fractional Sturm–Liouville differential equation with multi-points integral boundary conditions. Also, the continuity depending on coefficients of the initial condition of the equation is examined. An example is proposed to demonstrate our main result.

In 2015, the fractional problem c D α x(t) = f (t, x(t), D β x(t)) with boundary value conditions x(0) + x (0) = y(x), 1 0 x(t) dt = m and x (0) = x (3) = · · · = x (n-1) (0) = 0 was investigated, where 0 < t < 1, m is a real number, n ≥ 2, α ∈ (n -1, n), 0 < β < 1, D α and D β are the Caputo fractional derivatives, y ∈ C([0, 1], R) and f : (0, 1] × R × R → R is continuous with f (t, x, y) may be singular at t = 0 [29]. In 2019, the fractional Sturm-Liouville differential equation D α (ρ(t)D β y (t)) + θ (t)y(t) = h(t)κ(y(t)) with boundary conditions y (0) = 0 and m k=1 ξ k y(a k ) = y n i=1 η j y(b j ) was considered, where α ∈ (0, 1], ρ(t) ∈ C 1 (J, R), and θ (t) and h(t) are absolute continuous functions on J = [0, T ], T < ∞ with ρ(t) = 0 for all t ∈ J ; κ(y(t)) : R → R is defined and differentiable on the interval J , 0 ≤ a 1 < · · · < a m < c, d ≤ b 1 < b 2 < · · · < b n ≤ T and ξ k , η j and v ∈ R [14]. The hybrid version of this problem has © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. been studied recently in [13]. From the background of the research, it became clear to us that there are different methods for solving weakly singular equations, but generally these methods are not able to solve the strongly singular case (see [30,31]). Thus, it is very important to study the strong singular fractional differential equations with new techniques [32]. Therefore, considering the existing gap, we intend to introduce a new method for solving strongly singular equations in this research, which has not been presented so far. Regarding the main idea of the works, we examine the existence of solutions for the strong singular pointwisely defined fractional Sturm-Liouville differential equation is n -1 times differentiable and can be zero at some points in [0, 1], D β is the Caputo derivative of fractional order β, and I p i is the Riemann-Liouville integral of fractional order p i .
By carefully checking the used techniques in related works, we find that equation (1) is singular at t 0 ∈ [0, 1] whenever p(t 0 ) = 0 or q or h is singular at the point t 0 . Problem (1) is strong singular at the point t 0 whenever at least one of the functions 1 p(t) or q(t) or h(t) is singular at the point t 0 , but is not integrable on the interval [0, 1]. In this article, we use · 1 for the norm of L 1 [0, 1] and · for the sup norm of X = C[0, 1].
The Riemann-Liouville integral of fractional order υ with the lower limit s ≥ 0 for a function g : (s, ∞) → R is defined by dζ provided that the right-hand side is pointwisely defined on (s, ∞). We denote I υ g(t) for I υ 0 + g(t) [33]. Also, the Caputo fractional derivative of order α > 0 of the function g is defined by [33]. We need the following two statements to prove our main results. Lemma 1.1 ([34]) Let m -1 < σ ≤ m and ν ∈ C(0, 1). Then I σ D σ ν(t) = ν(t) + m-1 i=0 e i t i for some real constants e 0 , . . . , e m-1 .

Lemma 1.2 ([35])
Let C be a closed and convex subset of a Banach space X, be a relatively open subset of C with 0 ∈ , and F : → C be a continuous and compact mapping. Then either i) the mapping F has a fixed point in¯ , or ii) there exist y ∈ ∂ and λ ∈ (0, 1) with y = λF y.

Main results
We first provide our key lemma.
Proof By using the same strategy in [32], one can find Lemma 1.1 is valid on L 1 [0, 1]. Let ν(t) be a solution for the fractional boundary value problem (FBVP). Via Lemma 1.1, there are some real constants e 0 , . . . , e n-1 such that Since D β ν(0) = 0, we get e 0 = 0. Also since d dt (I α (q(t)f (ν(t)))) = I α-1 (q(t)f (ν(t))), by derivation from the last equality, we have Since I α (q(t)f (ν(t)))| t=0 = 0, it results that e 1 = (p (t)D β ν(t) + p(t)D β+1 ν(t))| t=0 . Thus, e 1 = 0. By continuing this way, one can check that e 2 = · · · = e n-1 = 0 and so If and then it is evolved that Once again for the above equality, by using Lemma 1.1, it is concluded that there are some real constants d 0 , . . . , d k-1 such that Since ν (i) (0) = 0 for 1 ≤ i ≤ k -1, we get d 1 = · · · = d k-1 = 0 therefore it is concluded that Hence, Also, by integration of order p i from (2), for each 1 ≤ i ≤ n 0 , we have p(s) ds which implies that Thus it results in and so ). This indicates that One can obtain the other part by using some calculations. This completes the proof.
Note that the generalized boundary conditions of the Sturm-Liouville problem lead us to attaining a different integral equation to consider. Also, as we have a strong singularity in the problem, we need to investigate the equation by a novel method.
Designate the space X = C[0, 1] with the supremum norm. Define the map H : X → X by (1), then ν 0 is a fixed point of the map H. Vice versa, ν 0 ∈ X is a solution for the problem when ν 0 is a fixed point of the mapping. In the next result, we suppose that the maps q, h : [0, 1] → R may be singular at some points in [0, 1] and the function p (1) is n -1 times differentiable but can be zero at some points in [0, 1]. In the next theorem, using inequalities for controlling singular points by some functions that are called control functions, and by the fixed point method, we will investigate the existence of a solution for the singular fractional differential problem (SFDP).
Proof First, we show that H is continuous. Let ν, ν * ∈ X and t ∈ [0, 1]. Then we have Let > 0 be given. Since lim z→0 + (z) Obviously,p(t, ξ ) is increasing with respect to t and is decreasing with respect to ξ . Therefore, we get

Example 2.3 Consider the strong singular Sturm-Liouville equation
with boundary conditions u (0) = 0 and u( 1 Now, by using Theorem 2.2, the Sturm-Liouville problem (5) has a solution. Also for a better graphical understanding of the problem, the graph of q(t) is shown in Fig. 1.

Continuous dependence
In this part, according to the topics raised in [14], we verify continuous dependence of the solution for the fractional Sturm-Liouville differential equation (1).
Also, we have