Enhanced Qualitative Understanding of Solutions to Fractional Boundary Value Problems via Alternative Fixed-Point Methods

Abstract

In this work, we study Caputo fractional boundary value problems and contribute to the theory of fractional differential equations by improving the results of Ferreira. Specifically, we establish sharper bounds for the Green’s functions associated with the problems and apply Rus’s fixed-point theorem. Our results hold under a less restrictive assumption, thereby extending the class of problems for which the existence and uniqueness of solutions can be ensured. This is demonstrated through numerical validation presented in the final stage of our analysis. An important aspect of this approach is that it avoids the need for strong contraction conditions, suggesting potential applicability to a broader range of differential equations.

1. Introduction

Let us consider the following fractional boundary value problem (FBVP), defined for all $\alpha \in ]1,2]$ by

$$\left\{\begin{array}{c}{}^{c}D^{\alpha }u\left(t\right)=-f(t,u\left(t\right)),\forall t\in ]a,b[\\ \\ u\left(a\right)=A,u\left(b\right)=B,A,B\in \mathbb{R},\end{array}\right.$$

(1)

where ${}^{c}D^{\alpha }$ represents the Caputo fractional derivative of order $\alpha$, and f belongs to the set of continuous real-valued functions defined on $[a,b]\times \mathbb{R}$, which we refer to as $C\left(\right[a,b]\times \mathbb{R})$.

In [1], Bahsir proposed some key properties of the Green’s function associated with the FBVP (1). By applying the Banach fixed-point theorem [2], he established an explicit sufficient condition ([1], Theorem 3.1) claiming the existence and uniqueness of solutions to the problem (1). However, two years later, Ferreira discovered an error in Bashir’s condition and published a corrected version of the sufficient condition ([3], Theorem 2.1). Bashir’s original condition was expressed as an inequality involving the parameters a, b, $\alpha$, and the Lipschitz constant L of the function f. In contrast, Ferreira’s correction is not explicit; it requires the numerical maximization of a function $M(\alpha ,a,b,t)$ with respect to the variable $t\in [a,b]$. As Ferreira’s condition involves a numerical maximization procedure, he did not apply it to a range of examples. Instead, he provided an analytical proof demonstrating that, for $\alpha =2$, his condition coincides with that proposed by Kelly and Peterson ([4], Theorem 7.7).

This work aims to build upon and extend the results obtained by Ferreira [3]. To achieve this, we derive novel estimates for integrals involving the Green’s function introduced in [1]. These estimates are subsequently utilized in the FBVP (1) through the application of Rus’s contraction theorem [5], within a function space endowed with two distinct metrics. This refinement enables a more precise control of the Lipschitz constants that ensure the existence and uniqueness of solutions to the FBVP (1). As a result, the proposed approach provides a more general and robust framework than traditional methods, such as the Banach fixed-point theorem, for determining sufficient conditions for the existence and uniqueness of solutions for the FBVP (1). Moreover, the condition we present differs from that of Ferreira not only in its methodological foundation but also in its explicit formulation, as it is expressed as an inequality involving the parameters a, b, $\alpha$, and L introduced earlier. For recent developments and applications of Rus’s theorem, we refer the reader to [6,7,8].

Let us first introduce the key definitions, useful results, and metrics needed to understand both the previous and upcoming parts of this article.

The following is the definition for the Euler gamma function.

Definition 1

([9]). We denote by $\mathbb{C}$ the set of complex numbers and by $\mathcal{R}\left(z\right)$ the real part of $z.$ The Euler gamma function $\Gamma \left(z\right)$ is defined by

$$\Gamma \left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt,\forall z\in \mathbb{C}|\mathcal{R}\left(z\right)>0.$$

(2)

The following is the definition for the fractional integrals and derivatives.

Definition 2

([9]). Let $\alpha >0$. The α-th Riemann–Liouville fractional integral of a function u is defined by

$$I^{\alpha }u\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(t-s)}^{\alpha -1}u\left(s\right)ds,t>a,$$

provided the right-hand side exists. Let $\lceil \alpha \rceil$ denote the ceiling value of α. The Riemann–Liouville fractional derivative of order α is defined as

$$D^{\alpha }u\left(t\right)=D^{\lceil \alpha \rceil }{I}^{\lceil \alpha \rceil -\alpha }u\left(t\right),t>a,$$

where $D^{\lceil \alpha \rceil }$ denotes the classical $\lceil \alpha \rceil$ th order derivative operator, if the right-hand side exists. If a function u is such that

$$D^{\alpha }\left(u\left(t\right)-\sum _{i=0}^{\lceil \alpha \rceil -1}{\displaystyle \frac{u^{\left(i\right)}\left(a\right)}{i!}}{(t-a)}^i\right)$$

exists, then the Caputo fractional derivative of order α of the function u is defined by

$${}^{c}D^{\alpha }u\left(t\right):=D^{\alpha }\left(u\left(t\right)-\sum _{i=0}^{\lceil \alpha \rceil -1}{\displaystyle \frac{y^{\left(i\right)}\left(a\right)}{i!}}{(t-a)}^i\right).$$

The following is the definition of the Gauss hypergeometric function.

Definition 3

([9,10]). The $Gausshypergeometricfunction$ ${}_{2}F_1(a_1,b_1;c_1;z)$ is defined in the unit disk as the sum of the hypergeometric series as follows:

$${\displaystyle {}_{2}F_1(a_1,b_1;c_1;z)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(a_1\right)}_k{\left(b_1\right)}_k}{{\left(c_1\right)}_k}}{\displaystyle \frac{z^k}{k!}}}$$

(3)

$$(a_1,b_1\in \mathbb{C};c_1\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};z\in \overline{B}:=\{z\in \mathbb{C}|\left|z\right|\le 1\};\mathcal{R}(c_1-b_1-a_1)>0),$$

where ${\left(z\right)}_k$ is the $Pochhammersymbol$ defined for $z\in \mathbb{C}$ and $k\in {\mathbb{N}}^{*}:=\mathbb{N}\setminus \left\{0\right\}$ by

$${\left(z\right)}_0=1\mathit{and}{\left(z\right)}_k=z(z+1)\dots (z+k-1),\forall k\in {\mathbb{N}}^{*}.$$

(4)

If $\mathbb{R}(c_1-b_1-a_1)>0$, then the value of the hypergeometric function at $z=1$ is given by

$${}_{2}F_1(a_1,b_1;c_1;1^{-})={\displaystyle \frac{\Gamma \left(c_1\right)\Gamma (c_1-a_1-b_1)}{\Gamma (c_1-a_1)\Gamma (c_1-b_1)}}.$$

(5)

Moreover, it is well-known that when $a_1=-m,$ $m=0,1,2,\dots ,$ and $c_1\ne 0,-1,-2,\dots ,$ the hypergeometric function ${}_{2}F_1$ becomes a terminating hypergeometric series [9,10], which is a polynomial of degree m:

$${}_{2}F_1\left(-m,b_1;c_1;z\right)=\sum _{k=0}^m{\displaystyle \frac{{\left(-m\right)}_k{\left(b\right)}_k}{{\left(c\right)}_{k}k!}}{z}^k.$$

(6)

Finally, the Gauss hypergeometric function ${}_{2}F_1(a_1,b_1;c_1;z)$ satisfies the differentiation formula, see [9,10]:

$${\displaystyle \frac{d}{dz}}{}_{2}F_1(a_1,b_1;c_1;z)={\displaystyle \frac{a_1{b}_1}{c_1}}·{}_{2}F_1(a_1+1,b_1+1;c_1+1;z).$$

(7)

In connection to the integration of the Green’s function in Section 2, we present a recent result ([11], Theorem 4.2) by Zaidi and Almuthaybiri.

Theorem 1

([11], Theorem 4.2). For all $a<t<b$, and $\alpha >{\displaystyle \frac{1}{2}}$, we have

$$\begin{array}{cc}{I}_2^{\alpha }\left(t\right)\hfill & {\displaystyle ={\int }_a^t(t-a){(t-s)}^{\alpha -1}{(b-s)}^{\alpha -1}ds}\hfill \\ \hfill & =\left[{\displaystyle \frac{{(b-a)}^{\alpha -1}{(t-a)}^{\alpha +1}}{\alpha }}\right]{}_{2}F_1(1-\alpha ,1;\alpha +1;g\left(t\right)),\hfill \end{array}$$

(8)

where $g\left(t\right)={\displaystyle \frac{t-a}{b-a}}.$

$$\begin{array}{c}{\displaystyle {\int }_a^b{I}_2^{\alpha }\left(t\right)dt={\displaystyle \frac{{(b-a)}^{2\alpha +1}}{\alpha }}\left[{\displaystyle \frac{1}{\alpha +1}}+{\displaystyle \frac{1}{2\alpha }}-{\displaystyle \frac{2}{2\alpha +1}}\right].}\hfill \end{array}$$

(9)

Remark 1.

It is worth noting that the identity presented in Equation (8) represents a specific case of a more general result previously established in [10] ((2.46), p. 41), that is,

$${\int }_a^t{(t-s)}^{\alpha -1}{(s-a)}^{\beta -1}{(b-s)}^{\gamma -1}ds=\left[{\displaystyle \frac{\Gamma \left(\beta \right){(b-a)}^{\gamma -1}{(t-a)}^{\alpha +\beta -1}}{\Gamma (\alpha +\beta )}}\right]{}_{2}F_1(1-\gamma ,\beta ;\alpha +\beta ;g\left(t\right)),$$

where $\alpha ,\beta >0$, and γ are arbitrary. Although the proof provided in [11] was developed independently and presented in a manner suited to the specific context of that work, the above general identity appears to have been known earlier. We respectfully acknowledge this prior contribution.

We now present the following results, which will be useful to our analysis in Section 2.

Lemma 1.

Let $1<\alpha \le 2$ and $z\in (0,1)$. Define

$$h\left(z\right)={}_{2}F_1(1-\alpha ,1;\alpha +1;z).$$

(10)

Then, $h\left(z\right)$ is strictly decreasing on the interval $(0,1)$ and satisfies

$${\displaystyle \frac{\alpha }{2\alpha -1}}\le h\left(z\right)\le 1.$$

(11)

Proof of Lemma 1. 

We divide the proof into two cases based on the value of $\alpha$.

• Case 1: $1<\alpha <2$. Let $a_1=1-\alpha$, $b_1=1$, and $c=\alpha +1$; then, by (7) the standard derivative identity for hypergeometric functions, we obtain

  $$h^{\prime }\left(z\right)={\displaystyle \frac{1-\alpha }{\alpha +1}}·{}_{2}F_1(2-\alpha ,2;\alpha +2;z).$$

  (12)

  Since the hypergeometric function ${}_{2}F_1(2-\alpha ,2;\alpha +2;z)$ has strictly positive coefficients in its power series expansion and converges on $z\in (0,1)$, it follows that

  $${}_{2}F_1(2-\alpha ,2;\alpha +2;z)>0\mathrm{for}\mathrm{all}z\in (0,1).$$

  Therefore,

  $$h^{\prime }\left(z\right)<0\mathrm{for}\mathrm{all}z\in (0,1),$$

  since $1-\alpha <0$, and so $h\left(z\right)$ is strictly decreasing on $(0,1).$
• Case 2: $\alpha =2$. Using (6), the definition of a terminating hypergeometric series, we have

  $$h\left(z\right)={}_{2}F_1(-1,1;3;z)=\sum _{k=0}^1{\displaystyle \frac{{(-1)}_k{\left(1\right)}_k}{{\left(3\right)}_{k}k!}}{z}^k=1-{\displaystyle \frac{z}{3}}.$$

  (13)

  This expression is clearly strictly decreasing on $(0,1)$. From both cases, we conclude that $h\left(z\right)$ is strictly decreasing on $(0,1)$ for all $1<\alpha \le 2.$ Moreover, ${}_{2}F_1(1-\alpha ,1;\alpha +1;z)$ is well-defined and analytic on $(0,1)$, as its hypergeometric series converges for all $z\in (0,1)$. Then, by the Monotone Limit Theorem and the monotonicity of $h\left(z\right)$, see [12] (Theorem 3.3.15), we obtain:

  $$\underset{z\in (0,1)}{sup}h\left(z\right)=\underset{z\to 0^{+}}{lim}h\left(z\right)={}_{2}F_1(1-\alpha ,1;\alpha +1;0)=1,$$

  and by (5) we obtain

  $$\underset{z\in (0,1)}{inf}h\left(z\right)=\underset{z\to 1^{-}}{lim}h\left(z\right)={}_{2}F_1(1-\alpha ,1;\alpha +1;1^{-})={\displaystyle \frac{\Gamma (\alpha +1)\Gamma (2\alpha -1)}{\Gamma \left(2\alpha \right)\Gamma \left(\alpha \right)}}={\displaystyle \frac{\alpha }{2\alpha -1}}.$$

  This completes the proof.    □

Remark 2.

Let $z=g\left(t\right)$ as defined in Theorem 1. By Lemma 1, we have $h\left(z\right)={}_{2}F_1(1-\alpha ,1;\alpha +1;z)>0$ for all $z\in (0,1)$, since

$$h\left(z\right)\ge {\displaystyle \frac{\alpha }{2\alpha -1}}>0,\mathit{for}1<\alpha \le 2.$$

Furthermore, since

$${\displaystyle \frac{{(b-a)}^{\alpha -1}{(t-a)}^{\alpha +1}}{\alpha }}\ge 0,\mathit{for}\mathit{all}t\in [a,b],$$

it follows that the integral $I_2^{\alpha }\left(t\right)$ defined in (8) satisfies

$$I_2^{\alpha }\left(t\right)\ge 0,\mathit{for}\mathit{all}t\in [a,b].$$

Lemma 2

([13], Fact 1.18.2. p. 69). Let $z_1$ and $z_2$ be complex numbers. If $q\ge 1,$ then

$$|z_1+z_2{|}^q\le 2^{q-1}\left(\right|z_1{|}^q+|z_2{|}^q).$$

(14)

Remark 3.

It is known that if $q\ge 1,$ then the following identity holds for all $x,y\in (0,+\infty )$:

$${|x-y|}^q\le {(x+y)}^q,$$

thus in the light of (14), we have

$${|x-y|}^q\le {(x+y)}^q\le 2^{q-1}(x^q+y^q).$$

(15)

The result of Rus [5], stated by the following theorem, plays a central role in our analysis, which aims to improve Ferreira’s conditions. To aid the reader’s understanding not only of how this theorem operates but also of some extensions and variations, we refer to [6].

Theorem 2

(Rus, [5]). Let $\mathbb{U}$ be a non-empty set, and let μ and ν be two metrics on $\mathbb{U}$ such that $(\mathbb{U},\mu )$ forms a complete metric space. If the following conditions are satisfied:

(i)

the mapping $M:\mathbb{U}\to \mathbb{U}$ is continuous on the metric space $(\mathbb{U},\mu )$,

(ii)

$\exists 0<k<\infty$ such that $\mu \left(M\right[u],M[v\left]\right)\le k\nu (u,v)$, for all $(u,v)\in \mathbb{U}$,

(iii)

$\exists 0<c<1$ such that $\nu \left(M\right[u],M[v\left]\right)\le c\nu (u,v)$, for all $(u,v)\in \mathbb{U}$,

then there exists a unique $w\in \mathbb{U}$ such that $M\left[w\right]=w$.

It is readily seen that Rus’s fixed-point theorem involves two metrics, where only the first is assumed to be complete. This fact is expressed in the following quote:

“Rus’s theorem relies on the existence of two metrics (which are not necessarily equivalent). The space $\left[\mathbb{U}\right]$ is assumed to be complete with respect to $\left[\mu \right]$, but $\left[\mathbb{U}\right]$ is not necessarily complete under $\left[\nu \right]$. The operator $\left[M\right]$ is assumed to be continuous with respect to the metric $\left[\mu \right]$ and contractive with respect to $\left[\nu \right]$, but not necessarily contractive with respect to $\left[\mu \right]$. As we shall see, these kinds of assumptions can be applied to operators associated with BVPs, sharpening traditional results by showing that a larger class of problems admit a unique solution.”

[S. S. Almuthaybiri and C. C. Tisdell, Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis https://doi.org/10.1515/math-2020-0056].

Therefore, we define them as follows:

• $\mathbb{U}$ denotes the set of continuous real-valued functions defined on the closed interval $[a,b]$, which we refer to as $C\left(\right[a,b\left]\right)$.
• $\mu$ denotes the ${\mathbf{L}}^{\infty }$ distance on $C\left(\right[a,b\left]\right)$, such that

  $${\displaystyle \mu (u,v)=\underset{t\in [a,b]}{max}|u\left(t\right)-v\left(t\right)|,\forall u,v\in C\left([a,b]\right).}$$

  (16)
• $\nu$ denotes the ${\mathbf{L}}^p$ distance on $C\left(\right[a,b\left]\right)$, such that

  $${\displaystyle \nu (u,v)={\left({\int }_a^b{|u\left(t\right)-v\left(t\right)|}^{p}dt\right)}^{1/p},\forall u,v\in C\left([a,b]\right).}$$

  (17)

It is well known that

• The metric space $\left(C\right([a,b],\mu )$ is complete (see [14], corollary 4.11, p. 60).
• The metric space $\left(C\right([a,b],\nu )$ is not complete (see [14], p. 89).
• For all $u,v\in C\left(\right[a,b\left]\right)$,

  $$\nu (u,v)\le {(b-a)}^{1/p}\mu (u,v).$$

  (18)

The remainder of the manuscript is organized as follows. In Section 2, we establish novel estimates for the integral of the Green’s function. In Section 3, we apply these estimates to obtain new sufficient conditions for the existence and uniqueness of solutions to the FBVP (1), using Rus’s fixed-point theorem. Section 4 presents a comparative analysis between our condition and that proposed by Ferreira [3]. We then conclude with some final remarks.

2. Estimates for Green’s Function

In this section, we present a result concerning the integration of the Green’s function. This result will be instrumental in the next section, where we derive a new sufficient condition for the existence and uniqueness of solutions to the FBVP (1).

To define a suitable operator M whose fixed points correspond to solutions of the FBVP (1), we invoke a result from Bahsir [1] (Lemma 2.1), which establishes the equivalence of the FBVP (1) to the following integral equation:

$${\displaystyle u\left(t\right)=A+{\displaystyle \frac{(t-a)(B-A)}{b-a}}+{\int }_a^{b}G(t,s)f(s,u\left(s\right))ds}$$

(19)

where G is the Green’s function defined by

$$G(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{(t-a){(b-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)(b-a)}}-{\displaystyle \frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}},\hfill & \hfill \mathrm{for}\mathrm{all}a\le s\le t\le b;\\ \\ {\displaystyle \frac{(t-a){(b-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)(b-a)}},\hfill & \hfill \mathrm{for}\mathrm{all}a\le t\le s\le b.\end{array}\right.$$

(20)

We proceed to establish and prove novel estimates for the integral of the Green’s function. These results will be used in Section 3.

Lemma 3.

For all $1<\alpha \le 2$ and $q>1$, the Green’s function $G(t,s)$ satisfies the following property:

$$\left(\mathbf{1}\right)\underset{t\in [a,b]}{max}\left({\int }_a^b{\left|G(t,s)\right|}^{q}ds\right)\le {\displaystyle \frac{2^q{(b-a)}^{q(\alpha -1)+1}}{{\Gamma }^q\left(\alpha \right)(q(\alpha -1)+1)}}.$$

(21)

Proof of Lemma 3. 

Let $h(t,s)$ denote the function by $h(t,s):=g_1(t,s)-g_2(t,s)$ where

$$g_1(t,s)={\displaystyle \frac{(t-a){(b-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)(b-a)}},g_2(t,s)={\displaystyle \frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}.$$

Ferreira [3] proved that the function $h(t,s)$ satisfies $h(t,s)<0$ on $t\in (a,b),$ so we can write:

$$\left|h(t,s)\right|=g_2(t,s)-g_1(t,s),$$

and using the inequality (15) with $q>1,$ $x:=g_2(t,s),$ and $y:=g_1(t,s)$ leads us to

$${\left|h(t,s)\right|}^q\le 2^{q-1}\left[{\displaystyle \frac{{(t-s)}^{q(\alpha -1)}}{{\Gamma }^q\left(\alpha \right)}}+{\displaystyle \frac{{(t-a)}^q{(b-s)}^{q(\alpha -1)}}{{\Gamma }^q\left(\alpha \right){(b-a)}^q}}\right].$$

(22)

Now for all $1<\alpha \le 2$, $q>1,$ and $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill {\int }_a^b{\left|G(t,s)\right|}^{q}ds& =& {\int }_a^t{\left|G(t,s)\right|}^{q}ds+{\int }_t^b{\left|G(t,s)\right|}^{q}ds\hfill \\ & =& {\int }_a^t{\left|h(t,s)\right|}^{q}ds+{\int }_t^b{\left|g_1(t,s)\right|}^{q}ds\hfill \\ & \le & 2^{q-1}{\int }_a^t\left[{\displaystyle \frac{{(t-s)}^{q(\alpha -1)}}{{\Gamma }^q\left(\alpha \right)}}+{\displaystyle \frac{{(t-a)}^q{(b-s)}^{q(\alpha -1)}}{{\Gamma }^q\left(\alpha \right){(b-a)}^q}}\right]ds\hfill \\ & +& {\int }_t^b{\displaystyle \frac{{(t-a)}^q{(b-s)}^{q(\alpha -1)}}{{\Gamma }^q\left(\alpha \right){(b-a)}^q}}ds\hfill \\ & =& {\displaystyle \frac{2^{q-1}}{{\Gamma }^q\left(\alpha \right)}}\left[{\displaystyle \frac{{(t-a)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1)}}+{\displaystyle \frac{{(t-a)}^q}{{(b-a)}^q}}·{\displaystyle \frac{{(b-a)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1)}}-{\displaystyle \frac{{(t-a)}^q}{{(b-a)}^q}}·{\displaystyle \frac{{(b-t)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1)}}\right]\hfill \\ & +& {\displaystyle \frac{{(t-a)}^q}{{\Gamma }^q\left(\alpha \right){(b-a)}^q}}·{\displaystyle \frac{{(b-t)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1)}}\hfill \\ & =& {\displaystyle \frac{2^{q-1}}{{\Gamma }^q\left(\alpha \right)}}\left[{\displaystyle \frac{{(t-a)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1)}}+{\displaystyle \frac{{(t-a)}^q}{{(b-a)}^q}}·{\displaystyle \frac{{(b-a)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1))}}\right]\hfill \\ & +& (1-2^{q-1})\left[{\displaystyle \frac{{(t-a)}^q}{{(b-a)}^q}}·{\displaystyle \frac{{(b-t)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1)}}\right].\hfill \end{array}$$

(23)

Above, we have used (22) to obtain (23), and from the above we have $(1-2^{q-1})<0,$ since $q>1.$ Therefore, we obtain:

$$\begin{array}{ccc}\hfill {\int }_a^b{\left|G(t,s)\right|}^{q}ds& \le & {\displaystyle \frac{2^{q-1}}{{\Gamma }^q\left(\alpha \right)}}\left[{\displaystyle \frac{{(t-a)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1)}}+{\displaystyle \frac{{(t-a)}^q}{{(b-a)}^q}}·{\displaystyle \frac{{(b-a)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1))}}\right]\hfill \\ & \le & {\displaystyle \frac{2^{q-1}}{{\Gamma }^q\left(\alpha \right)}}\left[{\displaystyle \frac{{(b-a)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1)}}+{\displaystyle \frac{{(b-a)}^q}{{(b-a)}^q}}·{\displaystyle \frac{{(b-a)}^{q(\alpha -1)+1}}{\left(q\right(\alpha -1)+1))}}\right]\hfill \\ & =& {\displaystyle \frac{2^q{(b-a)}^{q(\alpha -1)+1}}{{\Gamma }^q\left(\alpha \right)(q(\alpha -1)+1)}},\hfill \end{array}$$

thus, we deduce that

$$\underset{t\in [a,b]}{max}\left({\int }_a^b{\left|G(t,s)\right|}^{q}ds\right)\le {\displaystyle \frac{2^q{(b-a)}^{q(\alpha -1)+1}}{{\Gamma }^q\left(\alpha \right)(q(\alpha -1)+1)}}.$$

(24)

This completes the proof of Lemma 3.    □

Lemma 4.

For all $1<\alpha \le 2$, the Green’s function $G(t,s)$ satisfies the following properties:

$$\begin{array}{ccc}& & \left(\mathbf{i}\right)\underset{t\in [a,b]}{max}\left({\int }_a^b{\left|G(t,s)\right|}^{2}ds\right)\le {\displaystyle \frac{2{(b-a)}^{2\alpha -1}}{{\Gamma }^2\left(\alpha \right)(2\alpha -1)}}.\hfill \end{array}$$

(25)

$$\begin{array}{ccc}& & \left(\mathbf{ii}\right){\int }_a^b\left({\int }_a^b{\left|G(t,s)\right|}^{2}ds\right)dt=B(\alpha ,a,b),\hfill \end{array}$$

(26)

where

$$B(\alpha ,a,b)={\displaystyle \frac{{(b-a)}^{2\alpha }}{{\Gamma }^2\left(\alpha \right)(6\alpha -3)}}+{\displaystyle \frac{{(b-a)}^{2\alpha }}{{\Gamma }^2\left(\alpha \right)(4{\alpha }^2-2\alpha )}}-{\displaystyle \frac{2{(b-a)}^{2\alpha }}{\alpha {\Gamma }^2\left(\alpha \right)}}\left[{\displaystyle \frac{1}{\alpha +1}}+{\displaystyle \frac{1}{2\alpha }}-{\displaystyle \frac{2}{2\alpha +1}}\right].$$

(27)

Proof of Lemma 4. 

(i) Let $H\left(t\right)$ and K denote the function and constant defined below, respectively,

$$H\left(t\right)={\displaystyle \frac{{(t-a)}^2}{{\Gamma }^2\left(\alpha \right){(b-a)}^2}},K={\displaystyle \frac{2}{{\Gamma }^2\left(\alpha \right)(b-a)}}.$$

For all $1<\alpha \le 2$, and $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill {\int }_a^b{\left|G(t,s)\right|}^{2}ds& =& {\int }_a^t{\left|G(t,s)\right|}^{2}ds+{\int }_t^b{\left|G(t,s)\right|}^{2}ds\hfill \\ & =& {\int }_a^t\left(H\left(t\right){(b-s)}^{2\alpha -2}-K(t-a){(b-s)}^{\alpha -1}{(t-s)}^{\alpha -1}+{\displaystyle \frac{{(t-s)}^{2\alpha -2}}{{\Gamma }^2\left(\alpha \right)}}\right)ds\hfill \\ & +& {\int }_t^{b}H\left(t\right){(b-s)}^{2\alpha -2}ds\hfill \\ & =& H\left(t\right){\int }_a^b{(b-s)}^{2\alpha -2}ds+{\int }_a^t{\displaystyle \frac{{(t-s)}^{2\alpha -2}}{{\Gamma }^2\left(\alpha \right)}}ds-K{I}_2^{\alpha }\left(t\right)\hfill \\ & =& {\displaystyle \frac{H\left(t\right){(b-a)}^{2\alpha -1}}{(2\alpha -1)}}+{\displaystyle \frac{{(t-a)}^{2\alpha -1}}{{\Gamma }^2\left(\alpha \right)(2\alpha -1)}}-K{I}_2^{\alpha }\left(t\right).\hfill \end{array}$$

(28)

Above, $I_2^{\alpha }\left(t\right)$ is defined in Equation (8), and in light of Remark 2, we have $K{I}_2^{\alpha }\left(t\right)\ge 0.$ Therefore, we obtain:

$$\begin{array}{ccc}\hfill {\int }_a^b{\left|G(t,s)\right|}^{2}ds& \le & {\displaystyle \frac{H\left(t\right){(b-a)}^{2\alpha -1}}{(2\alpha -1)}}+{\displaystyle \frac{{(t-a)}^{2\alpha -1}}{{\Gamma }^2\left(\alpha \right)(2\alpha -1)}}\hfill \\ & \le & {\displaystyle \frac{H\left(b\right){(b-a)}^{2\alpha -1}}{(2\alpha -1)}}+{\displaystyle \frac{{(b-a)}^{2\alpha -1}}{{\Gamma }^2\left(\alpha \right)(2\alpha -1)}}\hfill \\ & =& {\displaystyle \frac{2{(b-a)}^{2\alpha -1}}{{\Gamma }^2\left(\alpha \right)(2\alpha -1)}},\hfill \end{array}$$

(29)

so, since relation (29) is true for all $t\in [a,b]$, we deduce that

$$\underset{t\in [a,b]}{max}\left({\int }_a^b{\left|G(t,s)\right|}^{2}ds\right)\le \left({\displaystyle \frac{2{(b-a)}^{2\alpha -1}}{{\Gamma }^2\left(\alpha \right)(2\alpha -1)}}\right).$$

(30)

This concludes the proof of the first result of Lemma 4.

(ii) From relation (28), we have

$$\begin{array}{ccc}\hfill {\int }_a^b\left({\int }_a^b{\left|G(t,s)\right|}^{2}ds\right)dt& =& {\int }_a^b\left({\displaystyle \frac{{(b-a)}^{2\alpha -1}H\left(t\right)}{(2\alpha -1)}}+{\displaystyle \frac{{(t-a)}^{2\alpha -1}}{{\Gamma }^2\left(\alpha \right)(2\alpha -1)}}\right)dt-K{\int }_a^b{I}_2^{\alpha }\left(t\right)dt\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& =& {\displaystyle \frac{{(b-a)}^{2\alpha }}{{\Gamma }^2\left(\alpha \right)(6\alpha -3)}}+{\displaystyle \frac{{(b-a)}^{2\alpha }}{{\Gamma }^2\left(\alpha \right)(4{\alpha }^2-2\alpha )}}-{\displaystyle \frac{2{(b-a)}^{2\alpha }}{\alpha {\Gamma }^2\left(\alpha \right)}}\left[{\displaystyle \frac{1}{\alpha +1}}+{\displaystyle \frac{1}{2\alpha }}-{\displaystyle \frac{2}{2\alpha +1}}\right].\hfill \end{array}$$

(32)

Note that, via the direct evaluation of (31) and using Theorem 1 (9), we derive expression (32). This concludes the proof of the second result of Lemma 4.    □

3. A Fixed-Point Approach to Existence and Uniqueness Using Two Metrics

In this section, we establish new results concerning the existence and uniqueness of solutions to the FBVP (1), developed in two steps. First, we prove our general main result, Theorem 3, through an application of Rus’s fixed-point theorem (Theorem 2). In this case, we consider the two metrics defined in (16) and (17) for general values of p and q. A similar approach is then employed for the special case $p=q=2.$ We apply the estimates derived in Section 2, in conjunction with Rus’s theorem, to establish our results.

Theorem 3.

Let L be a positive constant and $f(t,u)$ be a continuous L-Lipschitz function with respect to its second variable u, that is, for all $(t,u),(t,v)\in [a,b]\times \mathbb{R}$ it satisfies

$$\left|f\right(t,u)-f(t,v\left)\right|\le L|u-v|.$$

(33)

If there exist two numbers $p>1$ and $q>1$ satisfying $\frac{1}{p}+\frac{1}{q}=1$, and

$$0<L{\left({\int }_a^b{\left({\int }_a^b{\left|G(t,s)\right|}^{q}ds\right)}^{\frac{p}{q}}dt\right)}^{\frac{1}{p}}<1,$$

(34)

then the FBVP (1) has a unique solution in $C\left(\right[a,b\left]\right)$.

Proof of Theorem 3. 

Based on the formulation (19) of the FBVP (1), we introduce the mapping M acting on $C\left(\right[a,b\left]\right)$, where each $u\in C\left(\right[a,b\left]\right)$ is mapped to $M\left[u\right]\in C\left(\right[a,b\left]\right)$ such that

$${\displaystyle M\left[u\right]\left(t\right)=A+{\displaystyle \frac{(t-a)(B-A)}{b-a}}+{\int }_a^{b}G(t,s)f(s,u\left(s\right))ds,\forall t\in [a,b].}$$

(35)

We will use Theorem 2 to prove that the mapping M has a unique fixed point $w\in C\left(\right[a,b\left]\right)$.

1. Let us prove that the mapping M is continuous from $\left(C\right([a,b]),\mu )$ to $\left(C\right([a,b]),\mu )$ and that condition (ii) of Rus’s Theorem 2 is fulfilled. Consider metrics $\mu ={\mathbf{L}}^{\infty }$ distance and $\nu ={\mathbf{L}}^p$ distance on the set $\mathbb{U}=C\left(\right[a,b\left]\right)$. Then, for all $u,v\in C\left(\right[a,b]$, $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill \left|M\right[u\left]\right(t)-M[v\left]\right(t\left)\right|& \le & {\int }_a^b\left|G(t,s)\right||f(s,u\left(s\right))-f(s,v\left(s\right)|ds\hfill \\ & \le & {\int }_a^b\left|G(t,s)\right|L|u\left(s\right)-v\left(s\right)|ds\hfill \end{array}$$

$$\begin{array}{ccc}& \le & {\left({\int }_a^b{\left|G(t,s)\right|}^{q}ds\right)}^{\frac{1}{q}}L{\left({\int }_a^b{|u\left(s\right)-v\left(s\right)|}^{p}ds\right)}^{\frac{1}{p}}\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& \le & L\underset{t\in [a,b]}{max}{\left({\int }_a^b{\left|G(t,s)\right|}^{q}ds\right)}^{\frac{1}{q}}\nu (u,v)\hfill \end{array}$$

(37)

$$\begin{array}{ccc}& \le & L{\left({\displaystyle \frac{2^q{(b-a)}^{q(\alpha -1)+1}}{{\Gamma }^q\left(\alpha \right)(q(\alpha -1)+1)}}\right)}^{\frac{1}{q}}\nu (u,v).\hfill \end{array}$$

(38)

Note that we used Hölder’s inequality [15,16] and relation (33) to derive inequality (36). Moreover, we have applied (21) to (37) to obtain (38). Thus, we deduce that condition (ii) of Rus’s theorem is fulfilled, that is,

$$\mu \left(M\right[u],M[v\left]\right)\le k\nu (u,v),$$

(39)

where

$$0<k:=L{\left({\displaystyle \frac{2^q{(b-a)}^{q(\alpha -1)+1}}{{\Gamma }^q\left(\alpha \right)(q(\alpha -1)+1)}}\right)}^{\frac{1}{q}}.$$

(40)

2. We now prove that the mapping M is continuous from $\left(C\right([a,b]),\mu )$ to $\left(C\right([a,b]),\mu )$. By combining inequalities (18) and (49), we deduce that for all $u,v\in C\left(\right[a,b\left]\right)$

$$\mu (M\left[u\right],M\left[v\right])\le k{(b-a)}^{1/p}\mu (u,v).$$

(41)

Thus,

$$\forall \epsilon >0,\exists \delta ={\displaystyle \frac{\epsilon }{k{(b-a)}^{1/p}}}\mathrm{such}\mathrm{that}\mu (u,v)<\delta ⟹\mu (M\left[u\right],M\left[v\right])<\epsilon ,$$

(42)

that is, the mapping M is Lipschitz continuous and hence continuous from $\left(C\right([a,b]),\mu )$ to $\left(C\right([a,b]),\mu )$. This proves that condition (i) of Rus’s theorem is satisfied.

3. Let us prove that M is a contraction mapping on $\left(C\right([a,b]),\nu )$. Based on inequality (36), we deduce that for each $u,v\in C\left(\right[a,b\left]\right)$

$${\left({\int }_a^b{|M\left[u\right]\left(t\right)-M\left[v\right]\left(t\right)|}^{p}dt\right)}^{\frac{1}{p}}\le L{\left({\int }_a^b\left({\int }_a^b{\left|G(t,s)\right|}^{q}ds\right)dt\right)}^{\frac{1}{q}}\nu (u,v),$$

(43)

that is,

$$\nu (M\left[u\right],M\left[v\right])\le L{\left({\int }_a^b\left({\int }_a^b{\left|G(t,s)\right|}^{q}ds\right)dt\right)}^{\frac{1}{q}}\nu (u,v).$$

(44)

Then, inequality (44) is rewritten as follows:

$$\nu \left(M\right[u],M[v\left]\right)\le c\nu (u,v),$$

(45)

where

$$0<c:=L{\left({\int }_a^b\left({\int }_a^b{\left|G(t,s)\right|}^{q}ds\right)dt\right)}^{\frac{1}{q}}<1.$$

(46)

Based on assumption (34) of Theorem 3, the constant c belongs to the interval $]0,1[$. Therefore, M is a contraction mapping on $\left(C\right([a,b]),\nu )$. Thus, condition (iii) of Rus’s theorem is fulfilled.

Given that all conditions of Theorem 2 are fulfilled, the existence and uniqueness of a fixed point for the operator M follow. In other words, for any $\alpha \in ]1,2]$, the FBVP (1) admits a unique solution $u\in C\left(\right[a,b\left]\right)$. This completes the proof of Theorem 3.    □

At this stage, the left-hand side of condition (34) does not yet have a closed-form expression for general values of p and q. Moreover, obtaining a numerical evaluation in this general setting remains a challenging and unresolved issue. Therefore, we consider the special case of Theorem 3, where $p=q=2$. In this case, from (37) for all $u,v\in C\left(\right[a,b]$, $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill \left|M\right[u\left]\right(t)-M[v\left]\right(t\left)\right|& \le & L\underset{t\in [a,b]}{max}{\left({\int }_a^b{\left|G(t,s)\right|}^{2}ds\right)}^{\frac{1}{2}}\nu (u,v)\hfill \end{array}$$

(47)

$$\begin{array}{ccc}& \le & L{\left({\displaystyle \frac{2{(b-a)}^{2\alpha -1}}{{\Gamma }^2\left(\alpha \right)(2\alpha -1)}}\right)}^{\frac{1}{2}}\nu (u,v).\hfill \end{array}$$

(48)

Above we have applied (25) to (47) to obtain (48). Thus, for this special case we deduce that condition (ii) of Rus’s theorem is fulfilled, that is,

$$\mu \left(M\right[u],M[v\left]\right)\le k\nu (u,v),$$

(49)

where k becomes

$$0<k:=L{\left({\displaystyle \frac{2{(b-a)}^{2\alpha -1}}{{\Gamma }^2\left(\alpha \right)(2\alpha -1)}}\right)}^{\frac{1}{2}}.$$

(50)

Furthermore, for this special case we have from (44),

$$\begin{array}{ccc}\hfill \nu \left(M\right[u],M[v\left]\right)& \le & L{\left({\int }_a^b\left({\int }_a^b{\left|G(t,s)\right|}^{2}ds\right)dt\right)}^{\frac{1}{2}}\nu (u,v)\hfill \end{array}$$

(51)

$$\begin{array}{ccc}& =& L\sqrt{B(\alpha ,a,b)}\nu (u,v).\hfill \end{array}$$

(52)

Above we have applied (26) to (51) to obtain (52). Thus, for this special case we deduce that condition (iii) of Rus’s theorem is fulfilled, where c becomes $0<c:=L\sqrt{B(\alpha ,a,b)}<1,$ and $B(\alpha ,a,b)$ is defined explicitly in the equation by (27) as the ${\mathbf{L}}^p$ norm squared of the Green’s function. The constant $\sqrt{B(\alpha ,a,b)}$ appears frequently in the remainder of the manuscript. For the sake of notational simplicity, we denote it by $N(\alpha ,a,b)$. Accordingly, the constant $N(\alpha ,a,b)$, as introduced above, will serve a role analogous to that of the constant $M(\alpha ,a,b)$ defined by Ferreira in [3].

Thus, we obtain the following result as a special case of Theorem 3.

Theorem 4.

Let L be a positive constant and $f(t,u)$ be a continuous L-Lipschitz function with respect to its second variable u, that is, for all $(t,u),(t,v)\in [a,b]\times \mathbb{R}$ it satisfies

$$\left|f\right(t,u)-f(t,v\left)\right|\le L|u-v|.$$

(53)

If

$$\begin{array}{c}\hfill 0<LN(\alpha ,a,b)<1,\end{array}$$

(54)

then the FBVP (1) has a unique solution in $C\left(\right[a,b\left]\right)$.

4. Numerical Validation

The main objective of this section is to demonstrate numerically that the condition (54) stated in our Theorem 4 is less restrictive than that proposed by Ferreira. More precisely, we demonstrate that there exists an infinite class of FBVP (1) for which the existence and uniqueness of solutions are guaranteed by our Theorem 4 but not by Ferreira’s theorem [3] (Theorem 2.1).

Let

$$M(\alpha ,a,b,t)={\displaystyle \frac{1}{\Gamma (\alpha +1)}}\left(-2{(t-h\left(t\right))}^{\alpha }+2{\displaystyle \frac{(t-a){(b-h\left(t\right))}^{\alpha }}{b-a}}+{(t-a)}^{\alpha }-(t-a){(b-a)}^{\alpha -1}\right),$$

(55)

where

$$h\left(t\right)={\displaystyle \frac{r\left(t\right)}{s\left(t\right)}};r\left(t\right)={\left({\displaystyle \frac{t-a}{b-a}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}b-t;s\left(s\right)={\left({\displaystyle \frac{t-a}{b-a}}\right)}^{{\displaystyle \frac{1}{\alpha -1}}}-1,$$

(56)

is the function introduced by Ferreira in [3]. Note that the function h is continuous on the half-open interval $[a,b[$ and satisfies ${lim}_{t\to b}h\left(t\right)=b$. Hence, h is bounded on the closed interval $[a,b]$. According to Ferreira [3], the function $M(\alpha ,a,b,t)$ attains a global maximum on the interval $[a,b]$, which we denote by $M^{*}(\alpha ,a,b)$. Moreover, outside the classical case where $\alpha =2$ (for which $M^{*}(\alpha ,a,b)=\frac{{(b-a)}^2}{8}$), it is difficult—if not impossible—to derive an explicit expression for the maximum $M^{*}(\alpha ,a,b)$ as a function of $\alpha$, a, and b. Consequently, in order to compare our condition with that proposed by Ferreira, we compute $M^{*}(\alpha ,a,b)$ numerically. Since $M(\alpha ,a,b,t)$ possesses a unique global maximum on $[a,b]$, the computed value of the maximum is independent of the numerical optimization method employed. In this study, the value of $M^{*}(\alpha ,a,b)$ is computed using the function Golden Section Search [17] (defined by Algorithm 1 below), with the tolerance parameter $\epsilon ={10}^{-12}$. The Golden Section Search offers a systematic approach for selecting points within a given interval in such a way that the length of the interval containing the maximizer $x^{*}$ is reduced at each iteration—typically by a factor of approximately $\frac{2}{1+\sqrt{5}}$. It should be emphasized that the precise location of the maximizer $x^{*}$ cannot, in general, be determined exactly and can only be approximated within a specified tolerance $\epsilon$. Therefore, an approximation ${\widehat{x}}^{*}$ is deemed admissible if it satisfies the inequality $|x^{*}-{\widehat{x}}^{*}|<\epsilon$.

Algorithm 1: Golden Section Search

Table 1. Values of $M^{*}(\alpha ,0,1)$ and $N(\alpha ,0,1)$ for $\alpha \in \{\frac{5}{4},\frac{3}{2},\frac{7}{4},2\}$.

$\mathit{\alpha }$	$\frac{5}{4}$	$\frac{3}{2}$	$\frac{7}{4}$	2
$M^{*}(\alpha ,0,1)$	$0.3086$	$0.1980$	$0.1443$	$0.1250$
$N(\alpha ,0,1)$	$0.2517$	$0.1682$	$0.1261$	$0.1054$

presents the values of $M^{*}(\alpha ,0,1)$ and $N(\alpha ,0,1)$ for some special values of $\alpha \in ]1,2]$. These values may be utilized by the reader to formulate counterexamples.

Figure 1 displays $M^{*}(\alpha ,0,1)$ and $N(\alpha ,0,1)$ as functions of $\alpha$ over the interval $]1.05,2]$. As illustrated in the figure, $N(\alpha ,0,1)$ is strictly less than $M^{*}(\alpha ,0,1)$ throughout the interval $]1,2]$.

Consequently, Table 1 and Figure 1 offer visual evidence that our condition constitutes a refinement of Ferreira’s criterion.

4.1. Construction of Examples

Let $\alpha \in ]1,2]$ be a real number, and consider a continuous function $g(t,u)$ that is a K-Lipschitz with respect to the variable u. It can be observed from Figure 1 that $N(\alpha ,0,1)<M^{*}(\alpha ,0,1)$ for every $\alpha \in ]1,2]$. Therefore, for all $\beta \in ]\frac{1}{K{M}^{*}(\alpha ,0,1)},\frac{1}{KN(\alpha ,0,1)}[$, we can define the function $f(t,u)=\beta g(t,u)$. It is readily verified that f is continous and L-Lipschitz with constant $L=\beta K$. Under these assumptions, the inequalities $L{M}^{*}(\alpha ,0,1)>1$ and $LN(\alpha ,0,1)<1$ are satisfied. Therefore, by Theorem 4, the FBVP (1), where $a=0$, and $b=1$, admits a unique solution; however, this conclusion cannot be established using Ferreira’s theorem.

4.2. Concrete Examples

In both Ferreira’s Theorem 2.1 [3] and our Theorem 4, we denote the continuity condition of the function $f(t,u)$ by (i), while the Lipschitz condition of the function f with respect to the variable u is denoted by (ii). This section provides two illustrative examples in which conditions (i) and (ii), and condition (54) of Theorem 4 are satisfied, whereas Ferreira’s Theorem 2.1 fails to hold.

Example 1.

Consider the ideal case of Ferreira’s condition corresponding to $\alpha =2$. In this setting, the function $M(2,a,b,t)$ attains its maximum value $M^{*}(2,a,b)=\frac{{(b-a)}^2}{8}$. Note that when $\alpha =2$, the Caputo derivative ${}^{c}D^{p}u\left(t\right)$ coincides with the classical derivative $u^{″}\left(t\right)$. Thus, Problem (57) below constitutes a special case of Problem (1) corresponding to $\alpha =2$ and $f(t,u)=9sin\left(u\left(t\right)\right)+t^2+10$

$$\left\{\begin{array}{ccc}{u}^{″}\left(t\right)& =\hfill & -(9sin\left(u\left(t\right)\right)+t^2+10),\forall t\in ]0,1[\hfill \\ & \hfill & \\ u\left(0\right)& =\hfill & 0,u\left(1\right)=1.\hfill \end{array}\right.$$

(57)

In this problem, we have

$$\begin{array}{cc}\hfill \left|f(t,u)-f(t,v)\right|& =9\left|sin\left(u\right)-sin\left(v\right)\right|\hfill \\ \hfill & =18\left|cos\left(\frac{u+v}{2}\right)sin\left(\frac{u-v}{2}\right)\right|\hfill \\ \hfill & \le 18\left|sin\left(\frac{u-v}{2}\right)\right|\hfill \\ \hfill & \le 18\left|\frac{u-v}{2}\right|\hfill \\ \hfill & =9\left|u-v\right|\hfill \end{array}$$

(58)

It is worth noting that if v is the zero function, then the final inequality in relation (58) reduces to an equality. Consequently, there does not exist a constant $K<9$ such that

$$\left|f(t,u)-f(t,v)\right|<K|u-v|\mathrm{for}\mathrm{all}u,v\in C\left([0,1]\right).$$

Recall that if the function $f(t,u)$ is (i) continuous and (ii) K-Lipschitzian with respect to the second variable u on the interval $[0,1]$, then,

1. 

According to Ferreira [3], Problem (57) admits a unique solution if

$$K\frac{{(1-0)}^2}{8}<1,$$

(59)

2. 

According to our Theorem 4, Problem (57) admits a unique solution if

$$K\frac{{(1-0)}^2}{\sqrt{90}}<1.$$

(60)

Here, conditions (i) and (ii) are satisfied. Given that $K=9$, condition (60) holds, while Condition (59) fails to be satisfied. This indicates that Ferreira’s criterion does not permit the conclusion of existence and uniqueness for the solution to Problem (57). Hence, our condition provides a sharper result than that of Ferreira.

Remark 4.

From Example (57), an infinite family of related examples can be generated by replacing $f(t,u)$ with $g(t,u)=\beta f(t,u)$, where β is an arbitrary constant selected from the interval $]\frac{1}{9M^{*}(2,0,1)},\frac{1}{9N(2,0,1)}[=]0.8889,1.054[$. Let $L=9\beta$; then, g is an L-Lipschtisian function with respect to u. Since $\beta \in ]\frac{1}{9M^{*}(2,0,1)},\frac{1}{9N(2,0,1)}[$, then

$$\begin{array}{ccc}\hfill LN(2,0,1)& =& 9\beta N(2,0,1)<9{\displaystyle \frac{1}{9N(2,0,1)}}N(2,0,1)=1\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill L{M}^{*}(2,0,1)& =& 9\beta M^{*}(2,0,1)>9{\displaystyle \frac{1}{9M^{*}(2,0,1)}}{M}^{*}(2,0,1)=1.\hfill \end{array}$$

(62)

Inequalities (61) and (62) demonstrate the existence of an infinite class of fractional boundary value problems for which the existence and uniqueness of solutions are ensured by our Theorem 4 but not by the result established in Ferreira’s theorem [3].

Example 2.

We now examine Problem (1) in the specific case where $\alpha =\frac{3}{2}$ and $f(t,u)=\frac{11}{2}sin\left(u\left(t\right)\right)+t^2+{\displaystyle \frac{1}{2}}$.

$$\left\{\begin{array}{ccc}{}^{c}D^{3/2}u\left(t\right)& =\hfill & -(\frac{11}{2}sin\left(u\left(t\right)\right)+t^2+{\displaystyle \frac{1}{2}}),\forall t\in ]0,1[\hfill \\ & \hfill & \\ u\left(0\right)& =\hfill & 0,u\left(1\right)=1.\hfill \end{array}\right.$$

(63)

From Table 1, $M^{*}(\frac{3}{2},0,1)=0.1980$ and $N(\frac{3}{2},0,1)=0.1682$. Following the same approach as in Example 1, the function f is shown to be K-Lipschitzian with respect to the variable u, where $K=\frac{11}{2}$. Recall that if the function $f(t,u)$ is (i) continuous and (ii) K-Lipschitzian with respect to the second variable u on the interval $[0,1]$, then

1. 

According to Ferreira [3], Problem (63) admits a unique solution if

$$K{M}^{*}(\frac{3}{2},0,1)<1,$$

(64)

2. 

According to our Theorem 4, Problem (63) admits a unique solution if

$$KN(\frac{3}{2},0,1)<1.$$

(65)

Here, conditions (i) and (ii) are satisfied. Given that $K=\frac{11}{2}$, $M^{*}(\frac{3}{2},0,1)=0.1980$, and $N(\frac{3}{2},0,1)=0.1682$, condition (65) holds, while condition (64) fails to be satisfied. This indicates that Ferreira’s criterion does not permit the conclusion of existence and uniqueness for the solution to Problem (63). Hence, our condition provides a sharper result than that of Ferreira.

Remark 5.

Following the same reasoning as in Remark 4, one can construct an infinite class of fractional boundary value problems for which the existence and uniqueness of solutions are guaranteed by our Theorem 4 but not by the result established in Ferreira’s theorem.

5. Conclusions

In this study, we proved our first main result, Theorem 3, for general values of p and q. However, the left-hand side of condition (34) does not yet have a closed-form expression in this general setting, and obtaining a numerical evaluation remains a challenging and unresolved issue. Therefore, we focused on the special case where $p=q=2$.

Using the novel integral bounds derived in Section 2 for the Green’s functions associated with the FBVP (1) for all $1<\alpha \le 2$, and applying Rus’s theorem (Theorem 2), we established improved existence and uniqueness results: Theorem 4. These sharpen prior bounds compared to classical contraction methods and extend previous findings from [3]. Our results are further supported by numerical comparisons.

Nevertheless, it remains unclear whether alternative choices of p and q, combined with newly derived integral bounds for the Green’s functions associated with the FBVP (1), could lead to stronger results than those established in this work. Furthermore, it is worth investigating whether Rus’s fixed-point theorem offers a more effective alternative to the classical Banach fixed-point method when addressing the FBVP (1) for $\alpha >2$, under appropriate boundary conditions. These questions present an interesting direction for future research.