On a mixed nonlinear boundary value problem with the right Caputo fractional derivative and multipoint closed boundary conditions

1.   Introduction

The topic of boundary value problems is an interesting area of research in view of its applications in applied and technical sciences. In the recent years, the class of nonlocal fractional order boundary value problems involving different fractional derivatives (such as Riemann-Liouville, Caputo, etc.) received an overwhelming interest from many researchers. For the details of a variety of nonlocal single-valued and multivalued boundary value problems involving different types of fractional order derivative operators, we refer the reader to the text ^[1], articles ^{[2,3,4,5,6,7]} and the references cited therein. There has been shown a great enthusiasm in developing the existence theory for Hilfer, $\psi$-Hilfer and $(k, \psi)$ Hilfer type fractional differential equations equipped with different types of boundary conditions, for instance, see ^{[8,9,10,11,12,13,14,15,16]}.

Nonlocal boundary conditions are found to be more plausible and practical in contrast to the classical boundary conditions in view of their applicability to describe the changes happening within the given domain. Closed boundary conditions are found to be of great help in describing the situation when there is no fluid flow along the boundary or through it. The free slip condition is also a type of the closed boundary conditions which describes the situation when there is a flow along the boundary, but there is no flow perpendicular to it. Such conditions are also useful in the study of sandpile model ^[17,18], honeycomb lattice ^[19], deblurring problems ^[20], closed-aperture wavefield decomposition in solid media ^[21], vibration analysis of magneto-electro-elastic cylindrical composite panel ^[22], etc.

Now we review some works on the boundary value problems with closed boundary conditions. In ^[23], the authors studied the single-valued and multivalued fractional boundary value problems with open and closed boundary conditions. A three-dimensional Neumann boundary value problem with a generalized boundary condition in a domain with a smooth closed boundary was discussed in ^[24]. For some interesting results on impulsive fractional differential equations with closed boundary conditions, see the articles ^[25,26].

The objective of the present work is to investigate a new class of mixed nonlinear boundary value problems involving a right Caputo fractional derivative, mixed Riemann-Liouville fractional integral operators, and multipoint variant of closed boundary conditions. In precise terms, we consider the following fractional order nonlocal and nonlinear problem:

where $^CD^{\alpha}_{T-}$ denote the right Caputo fractional derivative of order $\alpha\in(1, 2]$, $I^{\rho}_{T-}$ and $I^{\sigma}_{0+}$ represent the right and left Riemann-Liouville fractional integral operators of orders $\rho, \sigma > 0$ respectively, $f, h :[0, T]\times\mathbb{R}\to\mathbb{R}$ are given continuous functions and $\lambda, p_i, q_i, r_i, v_i \in \mathbb{R}, i\in\{1, 2, 3, ..., m\}$, and $\xi_i\in(0, T)$. Notice that the integro-differential Eq (1.1) contains the usual and mixed Riemann-Liouville integrals type nonlinearities. The boundary conditions (1.2) can be interpreted as the values of the unknown function and its derivative at the right end-point $T$ of the interval $[0, T]$ are proportional to a linear combination of these values at arbitrary nonlocal positions $\xi_i \in (0, T).$ Physically, the nonlocal multipoint closed boundary conditions provide a flexible mechanism to close the boundary at arbitrary positions in the given domain instead of the left end-point of the domain.

Here we emphasize that much of the literature on fractional differential equations contains the left-sided fractional derivatives and there are a few works dealing with the right-sided fractional derivatives. For instance, the authors in ^[27,28] studied the problems involving the right-handed Riemann–Liouville fractional derivative operators, while a problem containing the right-handed Caputo fractional derivative was considered in ^[29]. The problem studied in the present paper is novel in the sense that it solves an integro-differential equation with a right Caputo fractional derivative and mixed nonlinearities complemented with a new concept of nonlocal multipoint closed boundary conditions. The results accomplished for the problems (1.1) and (1.2) will enrich the literature on boundary value problems involving the right-sided fractional derivative operators. The present work is also significant as it produces several new results as special cases as indicated in the last section.

The rest of the paper is arranged as follows. In Section 2, we present an auxiliary lemma which is used to transform the given nonlinear problem into a fixed-point problem. Section 3 contains the main results and illustrative examples. Some interesting observations are presented in the last Section 4.

2.   A preliminary result

Let us begin this section with some definitions ^[30].

Definition 2.1. The left and right Riemann-Liouville fractional integrals of order $\beta > 0$ for $g \in L_1[a, b],$ existing almost everywhere on $[a, b],$ are respectively defined by

Definition 2.2. For $g \in AC^{n}[a, b],$ the right Caputo fractional derivative of order $\beta \in (n-1, n], n \in \mathbb{N},$ existing almost everywhere on $[a, b],$ is defined by

In the following lemma, we solve a linear variant of the fractional integro-differential equation (1.1) supplemented with multipoint closed boundary conditions (1.2).

Lemma 2.1. Let $H, F \in C[0, T]$ and $\Delta\ne 0.$ Then the linear problem

is equivalent to the integral equation

where

Proof. Applying the right fractional integral operator $I^{\alpha}_{T-}$ to the integro-differential equation in (2.1), we get

where $c_0$ and ${c_1}$ are unknown arbitrary constants. Using (2.4) in the nonlocal closed boundary conditions of (2.1), we obtain

where $S_i, \; i = 1, \dots, 10,$ are given in (2.3).

Solving the system (2.5) for $c_0$ and ${c_1}$, we find that

where $\Delta$ is given in (2.3). Substituting the above values of $c_0$ and $c_1$ in (2.4) together with the notation (2.3), we obtain the solution (2.2). The converse of this lemma can be obtained by direct computation. This completes the proof.

3.   Existence and uniqueness results

This section is devoted to our main results concerning the existence and uniqueness of solutions for the problems (1.1) and (1.2).

In order to convert the problems (1.1) and (1.2) into a fixed point problem, we define an operator $\mathcal{V}: \mathcal{X} \rightarrow \mathcal{X}$ by using Lemma 2.1 as follows:

where $\mathcal{X} = C([0, T], \mathbb{R})$ denotes the Banach space of all continuous functions from $[0, T]\rightarrow \mathbb{R}$ equipped with the norm $\|y\| = \sup{\{|y(t)|: t \in[0, T]\}}.$ Notice that the fixed point problem $\mathcal{V}y(t) = y(t)$ is equivalent to the boundary value problems (1.1) and (1.2) and the fixed points of the operator $\mathcal{V}$ are its solutions.

In the forthcoming analysis, we use the following estimates:

where we have used $u^{\sigma}\leq T^\sigma, \; \rho, \sigma > 0.$

In the sequel, we set

where

3.1.   Existence results

In the following, Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$'s fixed point theorem ^[31] is applied to prove our first existence result for the problems (1.1) and (1.2).

Theorem 3.1. Assume that:

$(H_1)$ There exists $\mathcal{L} > 0$ such that $|f(t, x)-f(t, y)|\le \mathcal{L} |x-y|, \, \forall t\in [0, T], \, x, y \in\mathbb{R};$

$(H_2)$ There exists $\mathcal{K} > 0$ such that $|h(t, x)-h(t, y)|\le \mathcal{K} |x-y|, \, \forall t\in [0, T], \, x, y \in\mathbb{R};$

$(H_3)$ $|f(t, y)|\leq \delta(t)$ and $|h(t, y)|\leq \theta(t)$, where $\delta, \theta\in C([0, T], \mathbb{R}^+).$

Then, the problems (1.1) and (1.2) has at least one solution on $[0, T]$ if $\mathcal{L}\gamma_1+\mathcal{K}\gamma_2 < 1,$ where

Proof. Introduce the ball $B_{\eta} = \{ y \in \mathcal{X} : \|y\|\leq \eta \},$ with

Now we verify the hypotheses of Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$'s fixed point theorem in three steps by splitting the operator $\mathcal{V}: \mathcal{X} \rightarrow \mathcal{X}$ defined by (3.1) on $B_{\eta}$ as $\mathcal{V} = \mathcal{V}_1+\mathcal{V}_2$, where

(i) For $y, x \in B_{\eta}$, we have

where we used (3.4). Thus $\mathcal{V}_1y+\mathcal{V}_2x\in B_{\eta}$.

(ii) Using ($H_1$) and ($H_2$), it is easy to show that

which, in view of the condition $\mathcal{L}\gamma_1+\mathcal{K}\gamma_2 < 1$, implies that the operator $\mathcal{V}_1$ is a contraction.

(iii) Continuity of the functions $f, h$ implies that the operator $\mathcal{V}_2$ is continuous. In addition, $\mathcal{V}_2$ is uniformly bounded on $B_{\eta}$ as

where $\Omega_i,$ and $\gamma_i,$ $i = 1, 2,$ are defined in (3.2) and (3.3), respectively. To show the compactness of $\mathcal{V}_2$, we fix $\sup_{(t, y)\in{[0, T]\times B_{\eta}}}|f(t, y)| = \overline{f}$, $\sup_{(t, y)\in{[0, T]\times B_{\eta}}}|h(t, y)| = \overline{h}$. Then, for $0 < t_1 < t_2 < T$, we have

which tends to zero, independent of $y,$ as $t_2\to t_1$. This shows that $\mathcal{V}_2$ is equicontinuous. It is clear from the foregoing arguments that the operator $\mathcal{V}_2$ is relatively compact on $B_{\eta}$. Hence, by the Arzelá-Ascoli theorem, $\mathcal{V}_2$ is compact on $B_{\eta}$.

In view of the foregoing arguments (i)–(iii), the hypotheses of the Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$'s fixed point theorem ^[31] are satisfied. Hence, the operator $\mathcal{V}_1+\mathcal{V}_2 = \mathcal{V}$ has a fixed point, which implies that the problems (1.1) and (1.2) has at least one solution on $[0, T]$. The proof is finished.

Remark 3.1. Interchanging the roles of the operators $\mathcal{V}_1$ and $\mathcal{V}_2$ in the previous result, the condition $L\gamma_1+K\gamma_2 < 1$ changes to the following one:

where $\Omega_1, \Omega_2$ and $\gamma_1, \gamma_2$ are defined in (3.2) and (3.3) respectively.

The following existence result relies on Leray-Schauder nonlinear alternative ^[32].

Theorem 3.2. Suppose that the following conditions hold:

$(H_4)$ There exist continuous nondecreasing functions $\phi_1, \phi_2 : [0, \infty) \to (0, \infty)$ such that $\forall (t, y) \in [0, 1] \times {\mathbb R}$, $|f(t, y)|\le \omega_1(t)\phi_1(\|y\|)$ and $|h(t, y)|\le \omega_2(t)\phi_2(\|y\|),$ where $\omega_1, \omega_2 \in C([0, T], \mathbb{R}^+)$;

$(H_5)$There exists a constant $M > 0$ such that

Then, the problems (1.1) and (1.2) has at least one solution on $[0, T].$

Proof. We firstly show that the operator $\mathcal{V}: \mathcal{X} \rightarrow \mathcal{X}$ defined by (3.1) is completely continuous.

(i) $\mathcal{V}$ maps bounded sets into bounded sets in $\mathcal{X}.$

Let $y \in \mathcal{B}_r = \{y\in\mathcal{X}:\|y\|\leq r\}$, where $r$ is a fixed number. Then, using the strategy employed in the proof of Theorem 3.1, we obtain

(ii) $\mathcal{V}$ maps bounded sets into equicontinuous sets.

Let $0 < t_1 < t_2 < T$ and $y\in \mathcal{B}_r.$ Then, we obtain

Notice that the right-hand side of the above inequality tends to $0$ as $t_2 \to t_1$, independent of $y\in \mathcal{B}_r$. Thus, it follows by the Arzelá–Ascoli theorem that the operator $\mathcal{ V}: \mathcal{X} \to \mathcal{X}$ is completely continuous.

The conclusion of the Leray-Schauder nonlinear alternative ^[32] will be applicable once it is shown that there exists an open set $U\subset C([0, T], \mathbb{R})$ with $y\neq \nu\mathcal{V}y$ for $\nu\in (0, 1)$ and $y \in \partial U$. Let $y\in C([0, T], \mathbb{R})$ be such that $y = \nu\mathcal{V}y$ for $\nu\in (0, 1)$. As argued in proving that the operator $\mathcal{V}$ is bounded, one can obtain that

which can be written as

On the other hand, we can find a positive number $M$ such that $\|y\|\neq M$ by assumption ($H_5$). Let us set

Clearly, $\partial W$ contains a solution only when $\|y\| = M$. In other words, we cannot find a solution $y\in\partial W$ satisfying $y = \nu\mathcal{V}y$ for some $\nu\in (0, 1)$. In consequence, the operator $\mathcal{V}$ has a fixed point $y\in \overline{W}$, which is a solution of the problems (1.1) and (1.2). The proof is finished.

3.2.   Uniqueness result

Here we apply Banach contraction mapping principle to establish the uniqueness of solutions for the problems (1.1) and (1.2).

Theorem 3.3. If the conditions ($H_1$) and ($H_2$) hold, then the problems (1.1) and (1.2) has a unique solution on $[0, T]$ if

where $\Omega_1$ and $\Omega_2$ are defined in (3.2).

Proof. In the first step, we show that $\mathcal{V}\mathcal{B}_\kappa \subset \mathcal{B}_\kappa,$ where $\mathcal{B}_\kappa = \{y\in\mathcal{X}:\|y\|\leq \kappa\}$ with

For $y\in \mathcal{B}_\kappa$ and using the condition ($H_1$), we have

Similarly, using ($H_2$), we get

In view of (3.6) and (3.7), we obtain

which implies that $\mathcal{V}y \in \mathcal{B}_\kappa$, for any $y \in \mathcal{B}_\kappa$. Therefore, $\mathcal{V}\mathcal{B}_\kappa\subset\mathcal{B}_\kappa$.

Next, we prove that $\mathcal{V}$ is a contraction. For that, let $x, y \in \mathcal{X}$ and $t\in[0, T]$. Then, by the conditions $(H_1)$ and $(H_2)$, we obtain

which shows that $\mathcal{V}$ is a contraction in view of the condition (3.5). Therefore, we deduce by Banach contraction mapping principle that there exists a unique fixed point for the operator $\mathcal{V}$, which corresponds to a unique solution for the problems (1.1) and (1.2) on $[0, T]$. The proof is completed.

3.3.   Examples

In this subsection, we construct examples for illustrating the abstract results derived in the last two subsections. Let us consider the following problem:

Here $\alpha = 9/8, \rho = 7/3, \sigma = 3/4, \lambda = 3, \xi_1 = 3/7, \xi_2 = 2/3, \xi_3 = 4/5, p_1 = 1/2, p_2 = 1/3, p_3 = 1/4, q_1 = -2, q_2 = -3, q_3 = -4, r_1 = 1, r_2 = -1, r_3 = 3, v_1 = -2/7, v_2 = -3/7, v_3 = -4/7.$ Using the given data, it is found that

In consequence, we get $\Omega_1\approx 2.517580993, \Omega_2\approx 0.3543113654$ ($\Omega_1$, $\Omega_2$ are defined in (3.2)).

(i) For illustrating Theorem 3.1, we consider the functions

where $m_1$ and $m_2$ are finite positive real numbers. Observe that

and $f(t, y)$ and $h(t, y)$ respectively satisfy the conditions $(H_1)$ and $(H_2)$ with $\mathcal{L} = 2m_1/25$ and $\mathcal{K} = m_2/24.$ Moreover, $\gamma_1 \approx0.9438765902$ and $\gamma_2 \approx 0.2972831604.$ By the condition $\mathcal{L}\gamma_1+\mathcal{K}\gamma_2 < 1$, we get

For the values of $m_1$ and $m_2$ satisfying the inequality (3.10), the hypothesis of Theorem 3.1 is satisfied. Hence, it follows by the conclusion of Theorem 3.1 that the problem (3.8) with $f(t, y)$ and $h(t, y)$ given in (3.9) has at least one solution on $[0, 1].$ If the values $m_1$ and $m_2$ do not satisfy the inequality (3.10), then Theorem 3.1 does not guarantee the existence of at least one solution to the problem (3.8) with $f(t, y)$ and $h(t, y)$ given in (3.9) for such values of $m_1$ and $m_2$.

(ii) In order to illustrate Theorem 3.2, we take the following functions (instead of (3.9)) in the problem (3.8):

Observe that the assumption $(H_4)$ is satisfied as $|f(t, y)| \le \omega_1(t) \phi_1(\|y\|)$ and $|h(t, y)| \le \omega_2(t) \phi_2(\|y\|),$ where $\omega_1(t) = e^{-3t}/(t^2+3)$, $\phi_1(\|y\|) = (\|y\|+1/5),$ $\omega_2(t) = 2/(7\sqrt{t^3+1})$, $\phi_2(\|y\|) = (\|y\|+\pi/4).$ It is easy to see that $\| \omega_1\| = 1/3$ and $\omega_2\| = 2/7.$ By the condition ($H_5$), we find that $M > 4.151876169$. Thus, all the conditions of Theorem 3.2 are satisfied and hence the problem (3.8) with $f(t, y)$ and $h(t, y)$ given by (3.11) has at least one solution on $[0, 1].$

(iii) The conditions $(H_1)$ and $(H_2)$ are respectively satisfied by $f(t, y)$ and $h(t, y)$ defined in (3.9) with $\mathcal{L} = 2m_1/25$ and $\mathcal{K} = m_2/24.$ By the condition (3.5), we have

Clearly, all the assumptions of Theorem 3.3 hold true with the values of $m_1$ and $m_2$ satisfying the inequality (3.12). In consequence, the problem (3.8) with $f(t, y)$ and $h(t, y)$ given in (3.11) has a unique solution on $[0, 1].$ In case, we take $m_1 = m_2 = m$ in (3.9), then the condition (3.12) implies the existence of a unique solution for the problem at hand for $m < 4.62600051.$ One can notice that Theorem 3.1 does not guarantee the existence of a unique solution to the problem (3.8) with $f(t, y)$ and $h(t, y)$ given in (3.9) for the values of $m_1$ and $m_2,$ which do not satisfy the inequality (3.12).

4.   Conclusions

In this study, we discussed the existence and uniqueness of solutions under different assumptions for a boundary value problem involving a right Caputo fractional derivative with usual and mixed Riemann-Liouville integrals type nonlinearities, equipped with nonlocal multipoint version of the closed boundary conditions. Our results are not only new in the given configuration, but also yield some new results as special cases. Here are some examples.

● If $\lambda = 0$ in (1.1), then our results correspond to the fractional differential equation ${^CD^{\alpha}_{T-}}y(t) = f(t, y(t))$ with the boundary conditions (1.2).

● In case, we take $q_i = 0, r_i = 0, \forall i = 1, \dots, m$ in the results of this paper, we obtain the ones for the Eq (1.1) supplemented with boundary conditions: $y(T) = \sum\limits_{ı = 1}^m p_iy(\xi_i), \; \; y'(T) = \sum\limits_{ı = 1}^m v_iy'(\xi_i).$

● We get the results for the Eq (1.1) complemented with boundary conditions: $y(T) = T\sum\limits_{ı = 1}^m q_iy'(\xi_i), \; \; Ty'(T) = \sum\limits_{ı = 1}^m r_i y(\xi_i)$ by taking $p_i = 0, v_i = 0, \, \forall i = 1, \dots, m$ in the obtained results.

Acknowledgments

The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia, has funded this project under grant No. (KEP-PhD: 35-130-1443).

Conflict of interest

The authors declare no conflict of interest.