A fixed point approach to the solution of singular fractional differential equations with integral boundary conditions

<jats:p>In this article, we first demonstrate a fixed point result under certain contraction in the setting of controlled<jats:italic>b</jats:italic>-Branciari metric type spaces.<jats:list list-type="bullet"><jats:list-item><jats:p>Thereafter, we specifically consider a following boundary value problem (BVP) for a singular fractional differential equation of order <jats:italic>α</jats:italic>:<jats:disp-formula><jats:alternatives><jats:tex-math>$$ \begin{aligned} &{}^{c}D^{\alpha }v(t) + h \bigl(t,v(t) \bigr) = 0,\quad 0< t< 1, \\ &v''(0) = v'''(0) = 0, \\ &v'(0) = v(1) = \beta \int _{0}^{1} v(s) \,ds, \end{aligned} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtable><mml:mtr><mml:mtd /><mml:mtd><mml:mmultiscripts><mml:mi>D</mml:mi><mml:none /><mml:mi>α</mml:mi><mml:mprescripts /><mml:none /><mml:mi>c</mml:mi></mml:mmultiscripts><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace /><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>t</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:msup><mml:mi>v</mml:mi><mml:mo>″</mml:mo></mml:msup><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>v</mml:mi><mml:mo>‴</mml:mo></mml:msup><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:msup><mml:mi>v</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mspace /><mml:mi>d</mml:mi><mml:mi>s</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></jats:alternatives></jats:disp-formula>where<jats:inline-formula><jats:alternatives><jats:tex-math>$3<\alpha <4$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn><mml:mo><</mml:mo><mml:mi>α</mml:mi><mml:mo><</mml:mo><mml:mn>4</mml:mn></mml:math></jats:alternatives></jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:tex-math>$0<\beta <2$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>β</mml:mi><mml:mo><</mml:mo><mml:mn>2</mml:mn></mml:math></jats:alternatives></jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:tex-math>${}^{c}D^{\alpha }$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>D</mml:mi><mml:none /><mml:mi>α</mml:mi><mml:mprescripts /><mml:none /><mml:mi>c</mml:mi></mml:mmultiscripts></mml:math></jats:alternatives></jats:inline-formula>is the Caputo fractional derivative and<jats:italic>h</jats:italic>may be singular at<jats:inline-formula><jats:alternatives><jats:tex-math>$v = 0$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></jats:alternatives></jats:inline-formula>.</jats:p></jats:list-item><jats:list-item><jats:p>Eventually, we investigate the existence and uniqueness of solutions of the aforementioned boundary value problem of order<jats:italic>α</jats:italic>via a fixed point problem of an integral operator.</jats:p></jats:list-item></jats:list></jats:p>


Introduction
As regards science and engineering disciplines, fractional differential equations occur in the fields of aerodynamics, chemistry, physics, or polymer rheology electrodynamics, such as the mathematical simulation of structures and processes. The derivatives of fractional order [28,35,38] are concerned in this sort of equations. Exceedingly, fractional-order differential equations often become tools for various perspectives on control systems, fluid dynamics, and so forth.
The significance of studying fractional-order differential equations arises from the factuality that fractional-order models are more precise than integer-order models; it seems to be so that the fractional-order models have more degrees of freedom. Recent findings can be found in [3,4,8,14,19,34,37,45] on fractional differential equations.
Integral boundary conditions have several applications in the areas such as problems with blood flow, thermo-elasticity, underground water supply, and population dynamics. We refer the reader to certain recent publications [11, 17, 22, 36, 42-44, 46, 48] and the references therein for a comprehensive explanation of the integral boundary conditions.
Many researchers have pointed out the importance of the existence and uniqueness of fractional differential equations of different orders [29,30,32,35,40]. The existence of positive solutions was considered especially in the context of cone metric spaces [9,12,47].
On the other hand, fixed point theory can be used as a definitive modeling method in numerous fields and/or engineering to achieve solutions and/or research findings. In general, it has become one of the most effective features of modern mathematics and in particular of functional analysis. Fixed point theorems are concerned with the existence, uniqueness and characteristics of a specified operator's fixed points. The contraction mapping theorem due to Banach [10] is a very important and valuable finding on fixed point theory. Recent advancement in exploring new generalized metric spaces (and/or related results) has provoked great attention in metric fixed point theory (see [1, 5-7, 13, 16, 18, 20, 23-27, 31, 39]).

Preliminaries
In the year 1993, Czerwik [15] initiated the idea of b-metric spaces as a generalization of metric spaces by multiplying a constant b on the right side of the equation of triangle inequality. Definition 2.1 Let X = ∅ and b ≥ 1 be a given real number. A function d b : X × X → [0, +∞) is a b-metric if and only if for each r, s, t ∈ X the following conditions are satisfied: (1) d b (r, s) ≥ 0 for all r, s ∈ X and d b (r, s) = 0 if and only if r = s; Nabil Mlaiki et al. [33] defined a new type of generalized b-metric spaces, namely controlled metric type spaces, as follows: s) for all r, s, t ∈ X. The pair (X, d ω ) is called a controlled metric type space.
By concatenating the concepts of controlled metric type spaces [33] and extended-Branciari b-distance spaces [2], the authors in [41] proposed a new sort of metric spaces, namely controlled b-Branciari metric type spaces, that are defined now. (1) d C (r, s) = 0 if and only if r = s for all r, s ∈ X; (2) d C (r, s) = d C (s, r) for all r, s ∈ X; for all r, s ∈ X and for all distinct points t, w ∈ X, each distinct from r and s, respectively. The pair (X, d C ) is named a controlled b-Branciari metric type space.
The major aspect of controlled b-Branciari metric type space is the extension of the rectangular inequality. Hence (X, d C ) is a controlled b-Branciari metric type space. However, we can see that For the reader's convenience, we present some necessary definitions and lemmas from the theory of fractional calculus.
provided that such an integral exists.
Ying He [21] takes into consideration the problem (2.4) for a continuous function h and β ∈ (0, α). The outcomes in the paper correspond to the positive solutions to this problem. Ying He [21] first developed the accurate estimation of the below BVP of the Green's function, and unveiled some of its properties;

Fixed point theorem
Throughout this section, in the sense of controlled b-Branciari metric type spaces, we give a fixed point result under specific contraction condition.
be a complete controlled b-Branciari metric type space with coefficient function C(r, s) > 1 for any r, s ∈ X and F : X → X be a mapping satisfying and lim sup n,m→+∞ and we assume lim sup n→+∞ C(r, r n ) and lim sup n→+∞ C(r n , r) exist for any r ∈ X. Then F has a fixed point in X. Moreover, suppose that, for any r, s ∈ X, we have where F n r = F n-1 (Fr). Then the fixed point of F is unique.
Proof Let r 0 ∈ X and define an iterative sequence {r n } by r 0 , Fr 0 = r 1 , Now to show that {r n } is Cauchy, we consider d C (r n , r n+p ) in two cases.
Passing n → +∞ in the aforementioned inequality, we find By employing Eq. (3.9) and by the hypothesis of the theorem, we get d C (Fr, r) ≤ 0 as n → +∞. Therefore d C (r, Fr) = 0 i.e., Fr = r. As a result we see that r is a fixed point of F. Unicity:Let r, s be the two fixed points of F where r = s, then Fr = r and Fs = s. Consider Letting n → +∞ in the equation above and utilizing (3.4), we obtain d C (r, s) < d C (r, s), which is a contradiction. Thereby, r is a unique fixed point of F.

Existence-uniqueness of the solution of the BVP (2.4)
In this section, we confirm the existence and uniqueness of the solution of the nonlinear BVP (2.4) within controlled b-Branciari metric type spaces. We commence this section by proposing the Green's function developed in Ying He [21] relevant to the BVP corresponding to the linear fractional equation (2.5). Thereafter we present an inequality fulfilled by the Green's function to be utilized on the nonlinear BVP (2.4) in our existence-unicity result. It is proved in Ying He [21] that linear problem (2.5) has a unique solution in C[0, 1] given by v(t) = 1 0 G(t, s)y(s) ds, (4.1) where G(t, s) is the Green's function defined by where 3 < α < 4 and 0 < β < 2. For the properties of the Green's function, we refer to Ying He [21].

G(t, s)h s, v(s) ds = Fv(t),
which implies Fv ∈ C[0, 1]. Hence the map F : Then d C is a complete controlled b-Branciari metric type on C[0, 1] with a controlled function Therefore v(t) satisfies the differential equation ( We propose the following existence-uniqueness theorem for the solution of the problem (2.4). Proof Through utilizing the Cauchy-Schwarz inequality and the definition of the map F described in Eq. (4.6), we get (4.10) Taking the supremum over [0, 1] along with the definition of the metric (4.7) and a controlled function C, we obtain

Example
In this section, we add an example to Theorem 3.1.
Example 5.1 Consider the following differential equation of fractional order: which is singular at v = 0 with the boundary conditions The solution of the BVP provided in the example can be verified to fulfil the following  For v, z ∈ [-1, 1), we see that Similarly, we can prove the other cases. Therefore the map F specified by (4.6) has a unique fixed point and perhaps the BVP provided in the example does have a unique solution in C[0, 1].

Conclusion
Following up Sevinik Adigüzel et al. [40], in this study, we dealt with problem (2.4) in the context of controlled b-Branciari metric type spaces, which is a stronger concept than the concept of extended-Branciari b-distance spaces [2] and controlled metric type spaces [33], providing a disparate approach to the existence and uniqueness of the solution. This method can also be used for different α-derivative intervals. The theorem of existenceuniqueness in this study strengthens the current research as it provides quite requisites not only for positive solutions but also for any continuous solutions to the problem.