**4. Examples**

Here, we illustrate the applicability of our results by constructing numerical examples.

**Example 1.** *Consider the following boundary value problem:*

$$\begin{cases} \,^cD^{13/4}\mathbf{x}(t) + \sum\_{i=1}^2 l^{\overline{p}\_i} g\_i(t, \mathbf{x}(t)) = f(t, \mathbf{x}(t)), & t \in [0, 1], \\\ \mathbf{x}(0) = \mathbf{0}, \ \mathbf{x}'(0) = \mathbf{0}, \ \mathbf{x}''(0) = \mathbf{0}, \\\ \quad a\mathbf{x}(1) + \beta\mathbf{x}'(1) = \gamma\_1 \int\_0^{1/4} \mathbf{x}(s)ds + \sum\_{j=1}^3 a\_j \mathbf{x}(\eta\_j) + \gamma\_2 \int\_{1/2}^1 \mathbf{x}(s)ds. \end{cases} \tag{18}$$

*Here, m* = 4, *q* = 13/14, *p*1 = 10/14, *p*2 = 11/14, *p*3 = 12/14, *α* = *β* = *γ*1 = *γ*2 = 1, *ζ* = 1/4, *α*1 = 1/2, *α*2 = 3/4, *α*3 = 1, *η*1 = 1/7, *η*2 = 2/7, *η*3 = 3/7, *ξ* = 1/2*.*

$$\begin{aligned} \text{(i)} \quad &\text{Let } f(t, \mathbf{x}) = \frac{e^{2t}}{70} (\arctan \mathbf{x} + \sin 5t), \ g\_1(t, \mathbf{x}) = \frac{1}{4} \left( \frac{e^{-t} \cos \mathbf{x} + t^2 + 1}{\sqrt{t^2 + 49}} \right), \text{and } g\_2(t, \mathbf{x}) = \frac{1}{34} (\sin \mathbf{x} + \sin \mathbf{x}), \\\ e^{-t} \sqrt{57}). \text{ It is easy to see that } (A\_1) \text{ is satisfied with } L = e^2/70, \ L\_1 = 1/28, \text{and } L\_2 = 1/34. \end{aligned}$$

Using the given data, we have Ω ≈ 1.932128, Ω1 ≈ 0.677039, Ω2 ≈ 1.237301, and

$$\Lambda\_1 = \left(\alpha + \beta(m - 1) - \gamma\_1 \left(\frac{\zeta^m}{m}\right) - \sum\_{j=1}^3 \alpha\_j \eta\_j^{m - 1} - \gamma\_2 \left(\frac{1 - \zeta^m}{m}\right)\right) \approx 3.666981\dots$$

Then, *L*Ω + 2 ∑ *i*=1 *Li*Ω*i* ≈ 0.203951 + 0.060571 < 1. Thus, by Theorem 1, the boundary value problem (18) has a unique solution on [0, 1].

(ii) We choose the following functions in problem (18) for illustrating Theorem 2:

$$f(t, \mathbf{x}) = \frac{2}{17} (\sin \mathbf{x} + e^{-t} \cos 7t), \ g\_1(t, \mathbf{x}) = \frac{3}{32} \left( \frac{|\mathbf{x}|}{1 + |\mathbf{x}|} \right) + 2t, \ g\_2(t, \mathbf{x}) = \frac{1}{34} (\sin \mathbf{x} + e^{-t} \sqrt{32}).\tag{19}$$

Here *L* = 2/17, *L*1 = 3/32 and *L*2 = 1/34 as | *f*(*<sup>t</sup>*, *x*) − *f*(*<sup>t</sup>*, *y*)| ≤ 217 |*x* − *y*|, |*g*1(*<sup>t</sup>*, *x*) − *g*1(*<sup>t</sup>*, *y*)| ≤ 3 32 |*x* − *y*| and |*g*2(*<sup>t</sup>*, *x*) − *g*2(*<sup>t</sup>*, *y*)| ≤ 134 |*x* − *y*|. Further,

$$\begin{aligned} \|f(t, \mathbf{x})\| &\leq \quad \frac{2}{17} |\sin \mathbf{x}| + \varepsilon^{-t} |\cos 7t| \leq \frac{2}{17} + \varepsilon^{-t} \cos 7t = \mu(t), \\\|g\_1(t, \mathbf{x})\| &\leq \quad \frac{3}{32} + 2t = \mu\_1(t), \end{aligned}$$

and

$$\|\|g\_2(t,x)\|\| \le \frac{1}{34}|\sin x| + \varepsilon^{-t}\sqrt{32} \le \frac{1}{34} + \varepsilon^{-t}\sqrt{32} = \mu\_2(t).$$

Obviously, *μ* = 19/17, *μ*1 = 67/32 and *μ*2 = 5.686266. Moreover, we have

$$\left( L \left[ \Omega - \left( \frac{1}{\Gamma(q+1)} \right) \right] + \sum\_{i=1}^{2} L\_i \left[ \Omega\_i - \left( \frac{1}{\Gamma(q+p\_i+1)} \right) \right] \right) \approx 0.1824334 < 1.5$$

As the hypothesis of Theorem 2 holds true, so there exists least one solution for problem (18) with the functions given by (19).
