Involvement of the fixed point technique for solving a fractional differential system

1.   Background materials

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable, see ^[1,2,3,4]. The fractional calculus has allowed to formulate the operations of integration and differentiation to for fractional order. The order may take on any real or imaginary value. This fact enables us to consider the order of the fractional integrals and derivatives to be a function of time or of some other variable.

Recently, many authors have addressed differential equations with fractional derivatives for a different category of problems. FDEs arise in various engineering and scientific areas, where they are found in the mathematical modeling of systems and processes in the specialists of aerodynamics and electrodynamics of complex medium, physics, biophysics, chemistry, economics, blood flow phenomena, quantum theory, signal and image processing, polymer functional science, see, for example ^[5,6,7,8,9].

It should be noted that FDEs are better prepared to depict the genetic characteristics of different materials and processes than proper differential equations. Based on this advantage, fractional differential models become more realistic, practical and precise in obtaining the objective of classical models in differential. For better understanding of some real world problems, some researchers suggested recently discovered fractional operators. Among these operators, we mention the ones considered in ^{[10,11,12,13,14,15,16,17,18,19,20,21]}.

Fixed point theory (FPT) is an important pillar of non-linear analysis due to its many applications in various mathematical disciplines. The fixed-point style shined after Banach launched his famous principle, known as the Banach contraction principle. It is mainly involved in fractional differential equations by which the existence and uniqueness of solutions of many differential and integral equations with initial and boundary stipulations can be studied. For more details, see ^{[22,23,24,25,26,27]}.

One of the most important theorem of nonlinear functional analysis is the Leray-Schauder alternative, proved in 1934 by the topological degree ^[28]. A lot of authors proved several kinds of Leray-Schauder type alternatives by different methods, not based on topological degree and applied this methodology in many applications to ordinary differential equations, for more details, see ^[29,30].

Similar to earlier, by the standard fixed-point principle and Leray-Schauder alternative, the existence and unique solutions for a tripled system of FDEs via R-L integral boundary stipulations of different order are studied. The system takes the form:

where $^{c}D _{0^{+}}^{\omega },$ $^{c}D _{0^{+}}^{\varkappa }$ and $^{c}D _{0^{+}}^{\varrho }$ represent the Caputo Fractional Differentials (CFDs), $0 < \omega, \varkappa, \varrho \leq 1,$ $X, Y, Z \in C\left([0, 1]\times \mathbb{R} ^{3}, \mathbb{R} \right),$ and $\rho, \sigma, \varsigma, e, f, g\in \mathbb{R}.$ Ultimately, an example to support the results is given. Further, classical derivatives are in local nature, i.e., using classical derivatives we can describe changes in a neighborhood of a point but using fractional derivatives (our system) we can describe changes in an interval. Namely, the fractional derivative is in non-local nature. This property makes these derivatives suitable to simulate more physical phenomena such as earthquake vibrations, polymers, etc. Moreover, it is explain time delay and some fractal properties. Therefore, the search for solutions to these systems has received great attention from researchers.

2.   Fundamental facts

Assume that $\Xi = \left\{ a(s):a(s)\in C^{1}([0, 1])\right\}$ is equipped with $\left\Vert a\right\Vert = \max_{s \in \lbrack 0, 1]}\left\{ \left\vert a(s)\right\vert \right\}.$ Clearly $(\Xi, \left\Vert.\right\Vert)$ is a Banach space (BS).

Again, let $\Lambda = \left\{ b(s):b(s)\in C^{1}([0, 1])\right\}$ be endowed with $\left\Vert b\right\Vert = \max_{s \in \lbrack 0, 1]}\{\left\vert b(s)\right\vert \}.$ It is clear that the product $(\Xi \times \Lambda, \left\Vert (a, b)\right\Vert)$ is also a BS with $\left\Vert (a, b)\right\Vert = \left\Vert a\right\Vert +\left\Vert b\right\Vert.$

Also, consider $\wp = \left\{ c(s):c(s)\in C^{1}([0, 1])\right\}$ under the norm $\left\Vert c\right\Vert = \max_{s \in \lbrack 0, 1]}\{\left\vert c(s)\right\vert \}.$ Then $(\Xi \times \Lambda \times \wp, \left\Vert (a, b, c)\right\Vert)$ is a BS too with $\left\Vert (a, b, c)\right\Vert = \left\Vert a\right\Vert +\left\Vert b\right\Vert +\left\Vert c\right\Vert.$

The following definitions and lemmas are follows immediately from ^[5,8].

Definition 2.1. The standard CFD of order $f$ for continuously differentiable function $L:[0, \infty)\rightarrow \mathbb{R}$ is described by

where $[f]$ represents the integer part of the real number $f.$

Definition 2.2. The R-L fractional integral of order $f$ is described by

provided the integral exists.

The lemmas below illustrate some properties of CFDs and R-L fractional integrals ^[5].

Lemma 2.3. Suppose that $f, g\geq 0,$ $X \in L_{1}[a, b].$ Then $I ^{f}I ^{g}X(s) = I ^{f+g}X (s)$ and $^{c}D ^{f}I ^{f}X (s) = X(s),$ $\forall s\in \lbrack 0, 1].$

Lemma 2.4. Assume that $\varkappa > \omega > 0,$ $X \in L_{1}[a, b].$ Then $^{c}D^{\omega }I ^{\varkappa }X (s) = I ^{\varkappa -\omega }(s),$ $\forall s\in \lbrack 0, 1].$

Lemma 2.5. Let $\rho \neq \frac{\Gamma \left(1+e\right) }{\eta ^{e}},$ then for $X\in C([0, 1], \mathbb{R})$ be a given function, the solution of the FDE

under the boundary stipulation

is constructed by

3.   Main theorems

We will start our results with the following assumptions:

and

where $\alpha _{i}, \beta _{i}, \gamma _{i}\geq 0,$ $(i = 1, 2, 3).$

Assume that $\Omega :\Xi \times \Lambda \times \wp \rightarrow \Xi \times \Lambda \times \wp$ be an operator described by:

The results of this part are based on two rules: The first rule based on Leray-Schauder alternative.

Lemma 3.1. ^[31] Assume that $\nabla$ is a normed linear spaces and the mapping $\varphi :\nabla\rightarrow \nabla$ is a completely continuous mapping if

Then either $\mho (\varphi)$ is at the boundary, or $\varphi$ has at least one fixed point.

Theorem 3.2. Assume that $\rho \neq \frac{\Gamma \left(1+e\right) }{\eta ^{e}},$ $\sigma \neq \frac{\Gamma \left(1+f\right) }{\theta ^{f}}$ and $\varsigma \neq \frac{\Gamma \left(1+g\right) }{\vartheta ^{g}}.$ Supposethat there are real constants $\alpha _{i}, \beta _{i}, \gamma _{i}\geq 0$ $(i = 1, 2, 3)$ and $\alpha _{0} > 0,$ $\beta _{0} > 0,$ $\gamma _{0} > 0$ so that foreach $\delta _{i}\in \mathbb{R}$ $(i = 1, 2, 3),$ we get

and

Furthermore, suppose

where $A_{1}$–$A_{3}$ are described in (3.1)–(3.3). Then the boundary value problem (BVP) (1.1) has at least one solution.

Proof. In the beginning, it must be proved the completely continuous for $\Omega :\Xi \times \Lambda \times \wp \rightarrow \Xi \times \Lambda \times \wp$. Because the functions $X, Y$ and $Z$ are continuous, then $\Omega$ is continuous too. Suppose that $\psi \subset \Xi \times \Lambda \times \wp$ is a bounded set, then there exists positive coefficients $\ell _{1}, \ell _{2}$ and $\ell _{3}$ so that, for all $(a, b, c)\in \psi.$

Then for any $(a, b, c)\in \psi,$ we can get

Similarly, one can obtain that

and

It follows from (3.5)–(3.7) that $\Omega$ is uniformly bounded.

Thereafter, we prove that $\Omega$ is equi-continuous. Consider $0\leq s _{1}\leq s_{2}\leq 1,$ so, we get

analogously, we see that

and

This proves that $\Omega \left(a, b, c\right)$ is equicontinuous, and thus the operator $\Omega \left(a, b, c\right)$ is completely continuous.

Ultimately, we shall check the set $\mho = \left\{ \left(a, b, c\right) \in \Xi \times \Lambda \times \wp :\left(a, b, c\right) = \beta \Omega (a, b, c), \beta \in \lbrack 0, 1]\right\}$ is bounded. Consider $\left(a, b, c\right) \in \mho,$ then $\left(a, b, c\right) = \beta \Omega (a, b, c).$ For each $0\leq s\leq 1,$ we get

Then

and

The above three inequalities can be written as

and

which implies that

this leads to

where $A_{0}$ is given by (3.4), which illustrates that $\mho$ is bounded. Hence according to Lemma 3.1 there is at least one FP for the operator $\Omega,$ which is a solution to the BVP (1.1). This finishes the proof.

The second rule based on Banach's FP theorem ^[32]. By using it, we prove the existence and uniqueness of solutions to the BVP (1.1).

Theorem 3.3. Let the functions $X, Y, Z :[0, 1]\times \mathbb{R} ^{3}\rightarrow \mathbb{R}$ be continuous and there are coefficients $p_{i}, q_{i}, r_{i},$ $i = 1, 2, 3$ so that for each $s\in \lbrack 0, 1]$ and $a_{i}, b_{i}\in \mathbb{R},$ $i = 1, 2, 3,$

and

In addition, suppose that

where $A_{1}$–$A_{3}$ are described in (3.1)–(3.3). Then there exists a unique solution for the BVP (1.1).

Proof. Consider

and

so that

Now, we shall show that $\Omega \digamma _{\xi }\subset \digamma _{\xi },$ where $\digamma _{\xi } = \left\{ \left(a, b, c\right) \in \Xi \times \Lambda \times \wp :\left\Vert \left(a, b, c\right) \right\Vert \leq \xi \right\}.$

For $\left(a, b, c\right) \in \digamma _{\xi },$ we get

Hence

By the same manner, we can get

and

Thus, $\left\Vert \Omega \left(a, b, c\right) \left(s\right) \right\Vert \leq \xi.$

Finally, we show that the operator $\Omega$ is a contraction. Indeed for $\left(a_{2}, b_{2}, c_{2}\right), \left(a_{1}, b_{1}, c_{1}\right) \in \Xi \times \Lambda \times \wp$ and for any $s\in \lbrack 0, 1],$ we can write

consequently, we get

Analogously, we obtain

and

Inequalities (3.8)–(3.10) implies that

Because

then $\Omega$ is a contraction. So, according to Banach's contraction principle, there is a unique FP of the operator $\Omega,$ which is a unique solution of Problem (1.1). This complete the required.

The example below support the theoretical results.

Example 3.4. Assume that the system of fractional BVP below:

Here, $\omega = \varkappa = \varrho = \frac{1}{3},$ $\rho = \sqrt{5},$ $\sigma = \sqrt{3},$ $\varsigma \sqrt{2},$ $\eta = \frac{1}{3},$ $Z = \frac{1}{2},$ $\vartheta = \frac{3}{4},$ $e = \frac{5}{2},$ $f = \frac{3}{2},$ $g = \frac{1}{2},$

and

It should be noted that

and

Furthermore,

and

Moreover,

Thus, all requirements of Theorem 3.3 are fulfilled, hence Problem (3.11) has a unique solution.

4.   Conclusions

Fractional derivatives do not take into account only local characteristics of the dynamics but considers the global evolution of the system; for that reason, when dealing with certain phenomena, they provide more accurate models of real-world behavior than standard derivatives. Nonlinear systems describing different phenomena can be modeled with fractional derivatives. Chaotic behavior has also been reported in some fractional models. There exist theoretical results related to existence and uniqueness of solutions to initial and boundary value problems with fractional differential equations; for the nonlinear case, there are still few of them. So, in this manuscript, we were able to study existence of a unique solution to a system of FDEs with nonlocal integral boundary conditions using Banach contraction principle. Ultimately, theoretical results were supported by an illustrative example. As a future work, our method can be applied to obtain existence of solutions for two fractional $q-$differential inclusions under some integral boundary value conditions as the work of ^[33,34]. Moreover, the kernel can be taken as a singular one to solve partial integro-differential equations and to study Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel motivated by the work of ^[35,36,37]. In addition, we can replace Caputo fractional derivatives with conformable derivative functions to obtain a solution to fractional-order differential equations. These new investigations and applications would enhance the impact of the new setup.

Abbreviations

● FDEs        Fractional derivative equations

● CFDs        Caputo fractional derivatives

● FPT        Fixed point technique

● BS        Banach space

● R-L        Riemann-Liouville

● BVP        Boundary value problem

Availability of data and material

The data used to support the findings of this study are available from the corresponding author upon request.

Acknowledgments

The authors thank the Spanish Government and the European Fund of Regional Development FEDER for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and the Basque Government for Grant IT1207-19.

Conflict of interest

The authors declare that they have no competing interests concerning the publication of this article.