Involvement of the topological degree theory for solving a tripled system of multi-point boundary value problems

1.   Introduction

Fractional calculus is a powerful tool concerned with investigating integrals and derivatives of arbitrary order and their applications in physics, engineering, fluid mechanics, optics, signals processing, biology, etc. (see ^[1,2,3,4,5]). Several remarkable mathematicians made significant contributions to the field of fractional calculus. The first mention of the fractional calculus was provoked by Lacroix ^[6] in 1819 by introducing the common $n$-th derivative of the power function $y = x^p$ using the Gamma function. In 1822, Fourier ^[7] introduced the derivative of arbitrary order via the Fourier transform of a function. Two years later, Fourier presented the fractional derivative of a function in terms of its Fourier transform. In 1832, Liouville ^[8] published two definitions of fractional derivatives of a fairly restrictive class of functions. In 1847, Riemann ^[9] developed a theory of fractional integration. Later in 1869, one of the first treatments of the Riemann-Liouville definition of the fractional integral was considered by Ya Sonin ^[10]. It is worth mentioning the fractional Riemann-Liouville version is one of the most frequently used in the literature. An alternative definition of a fractional derivative was also initiated by Caputo. For more related details, see ^{[11,12,13,14]}.

The theory of fractional differential equations is a fruitful branch of mathematics by which various real-life phenomena in several fields of engineering and science can be expressed. In this context, enormous contributions to exploring useful applications have been attained in recent times. For instance, boundary value problems of nonlinear fractional differential equations with various boundary conditions have been investigated. Fractional differential equations represent an essential point of research. Nonlinear boundary value problems (BVPs for short) arise in several fields of physics, biology, chemistry, and applied mathematics. They are related to the theory of nonlinear diffusion generated by nonlinear sources, in chemical or biological problems, in the theory of elastics stability, and in thermal ignition of gases (for details, see ^{[15,16,17,18,19]}). Many phenomena in viscoelasticity, electrochemistry, electromagnetism, control theory, etc., can be expressed as fractional differential equations. Consequently, nonlinear BVPs are of great importance. The methods of nonlinear analysis, such as the Leray-Schauder continuation theorem, the coincidence degree theory of Mawhin, the fixed point theorems of Krasnoselskii and Schauder, fixed point theorems for mixed monotone operators, and others, are frequently used to solve fractional boundary value problems. For instance, see ^{[20,21,22,23]}.

Only a few studies have used the degree theory arguments to prove the (EU) to boundary value problems (BVPs) ^{[24,25,26,27,28,29,30]}. However, to the best our knowledge, no previous research has discussed at the (EU) of solutions to tripled systems of (M-PBVPs) for (FODEs) using the topological degree technique. By this technique, Wang et al. ^[27] investigated the (EU) of solutions to a class of nonlocal Cauchy problems below:

where $D^{\ell }$ is the Caputo fractional derivative (CFD) of order $\ell \in (0, 1),$ $\varpi _{0}\in \mathbb{R},$ $\Lambda :[0, T]\times \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. Chen et al. ^[25] explored necessary criteria for existence results for the following two-point boundary value problem (BVP) and extended the result above to the case of the (BVP):

where $D_{0^{+}}^{u}$ and $D_{0^{+}}^{v}$ are (CFDs), $0 < u, v\leq 1$, and $1 < u+v\leq 2.$ The following two-point (BVP) for (FDEs) with various boundary conditions was investigated by Wang et al. ^[26]:

where $D_{0^{+}}^{u}$, and $D_{0^{+}}^{v}$ are (CFDs), $0 < u, v\leq 1$ and $1 < u+v\leq 2.$ Motivated by the previous discussion, in this paper, we use a coincidence degree theory approach for condensing maps to derive appropriate criteria for the (EU) of solutions to more broad tripled systems of nonlinear (M-PBVPs). There are also nonlinear boundary conditions. The system's structure can be described as follows:

where $\ell, \gamma, \varkappa \in (1, 2],$ $D$ refers to the standard (CFD), $\eta _{1}, \eta _{2}, \eta _{3}\in (0, 1)$ are parameters so that $\eta _{1}\xi _{1}^{\ell } < 1,$ $\eta _{2}\xi _{2}^{\gamma } < 1,$ $\eta _{3}\xi _{3}^{\varkappa } < 1,$ $\vartheta _{1}, \vartheta _{2}, \vartheta _{3}\in C(I, \mathbb{R})$, and $\Lambda _{1}, \Lambda _{2}, \Lambda _{3}:[0, 1]\times \mathbb{R} ^{3}\rightarrow \mathbb{R}$ are boundary continuous functions.

2.   Basic concepts

This part provides some basic definitions and findings from fractional calculus and topological degree theory. We suggest ^{[31,32,33,34,35]} for a more in-depth investigation.

Definition 2.1. For the function $\varpi \in L^{1}([a, b], \mathbb{R}),$ the fractional integral of order $\ell$ is described as

The CFD is given as

where $m = [\ell]+1$ and $[\ell]$ refers to the integer part of $\ell.$

Lemma 2.1. For (FDEs), the following result holds:

for arbitrary $a_{j}\in \mathbb{R},$ $j = 1, 2, ..., m-1.$

Let $E = C\left([0, 1], \mathbb{R} \right),$ $\widetilde{E} = C\left([0, 1], \mathbb{R} \right)$ and $\widehat{E} = C\left([0, 1], \mathbb{R} \right)$ be the space of all continuous functions on $[0, 1].$ Clearly, theses spaces are Banach spaces under the norms $\left\Vert \varpi \right\Vert = \sup\limits_{z\in \lbrack 0, 1]}\left\vert \varpi (z)\right\vert,$ $\left\Vert \rho \right\Vert = \sup\limits_{z\in \lbrack 0, 1]}\left\vert \rho (z)\right\vert$ and $\left\Vert \varrho \right\Vert = \sup\limits_{z\in \lbrack 0, 1]}\left\vert \varrho (z)\right\vert,$ respectively. Moreover, the product $E\times \widetilde{E}\times \widehat{E}$ is a Banach space equipped with the norm $\left\Vert \left(\varpi, \rho, \varrho \right) \right\Vert = \left\Vert \varpi \right\Vert +\left\Vert \rho \right\Vert +\left\Vert \varrho \right\Vert$ or $\left\vert \left(\varpi, \rho, \varrho \right) \right\vert = \max \{\left\Vert \varpi \right\Vert, \left\Vert \rho \right\Vert, \left\Vert \varrho \right\Vert \}.$

Assume that $\Delta$ is the class of bounded sets of $\mho (E\times \widetilde{E}\times \widehat{E}),$ where $E\times \widetilde{E}\times \widehat{E}$ is a Banach space. In what follows, we present basis notions and results which are very essential in the sequel (see ^[36]).

Definition 2.2. The Kuratowski measure of non-compactness $\mu :\Delta \rightarrow \mathbb{R} ^{+}$ is described as

where $\upsilon \in \Delta.$

Proposition 2.1. The Kuratowski measure $\mu$ fulfills the hypotheses below:

(1) $\upsilon$ is relatively compact if and only if $\mu \left(\upsilon \right) = 0,$

(2) $\mu \left(\beta \upsilon \right) = \left\vert \beta \right\vert \mu \left(\upsilon \right),$ $\beta \in \mathbb{R}$, and $\mu \left(\upsilon _{1}+\upsilon _{2}\right) \leq \mu \left(\upsilon _{1}\right) +\mu \left(\upsilon _{2}\right),$ that is, $\upsilon$ is a seminorm,

(3) $\upsilon _{1}\subset \upsilon _{2}$ implies $\mu \left(\upsilon _{1}\right) \leq \mu \left(\upsilon _{2}\right)$, and $\mu \left(\upsilon _{1}\cup \upsilon _{2}\right) = \max \left\{ \mu \left(\upsilon _{1}\right), \mu \left(\upsilon _{2}\right) \right\},$ $\mu \left(conv\mathit{\text{}}\upsilon \right) = \mu \left(\upsilon \right),$ $\mu \left(\overline{\upsilon }\right) = \mu \left(\upsilon \right).$

Definition 2.3. Assume that the mapping $\phi :\nabla \rightarrow E$ is continuous and bounded, where $\nabla \subset E.$ Then $\phi$ is $\mu$- Lipschitzian if there is $U\geq 0$ so that for all $\upsilon \subset \nabla$ bounded, $\mu \left(\phi (\upsilon)\right) \leq U\mu \left(\upsilon \right).$ In addition, $\phi$ is called a strict $\upsilon$-contraction if $U < 1$.

Definition 2.4. A function $\phi$ is called $\upsilon$-condensing if $\mu \left(\phi (\upsilon)\right) < \mu \left(\upsilon \right)$ for all $\upsilon \subset \nabla$ bounded with $\mu \left(\upsilon \right) > 0.$ Or, equivalently, $\mu \left(\phi (\upsilon)\right) \geq \mu \left(\upsilon \right)$ implies $\mu \left(\upsilon \right) = 0.$

For the bounded continuous mapping $\phi :\nabla \rightarrow E,$ $D_{\upsilon }(\nabla)$ represents the class of all strict $\upsilon$- contractions, and $\widetilde{D}_{\upsilon }(\nabla)$ refers to the class of all $\upsilon$-condensing maps.

Remark 2.1. Note that $D_{\upsilon }(\nabla)\subset \widetilde{D}_{\upsilon }(\nabla)$, and every $\phi \in \widetilde{D}_{\upsilon }(\nabla)$ is $\upsilon$- Lipschitz with constant $U = 1$. In addition, we recall that $\phi :\nabla \rightarrow E$ is Lipschitz if there is $U > 0$ such that

Moreover, $\phi$ is called a strict contraction if $U < 1.$

Seeking clarification for the reader, we present the following results, quoted from ^[34], which we rely on through this study.

Proposition 2.2. $(i)$ If $\phi, \Game :\nabla \rightarrow E$ are $\upsilon$-Lipschitz with $U$ and $U^{\ast },$ then $\phi +\Game :\nabla \rightarrow E$ is $\upsilon$-Lipschitz with $U+U^{\ast }.$

$(ii)$ If $\phi :\nabla \rightarrow E$ is $\upsilon$-Lipschitz with $U,$ then $\phi$ is $\upsilon$-Lipschitz with the same constant $U$.

$(iii)$ If $\phi :\nabla \rightarrow E$ is compact, then $\phi$ is $\upsilon$-Lipschitz with constant $U = 0$.

The following result deduced by Isaia ^[34] is crucial to our main finding.

Theorem 2.1. Assume that $\Xi :\Lambda \rightarrow \Lambda$ is $\mu$- condensing, and

If the set $\varphi$ is a bounded in $\Lambda,$ there is $s > 0$ so that $\varphi \subset U_{s}(0),$ and the degree

As a result, $\Xi$ owns at least one (FP), and the set of (FPs) is contained in $U_{s}(0).$

Now, we will state the hypotheses that will help us to achieve our objectives in this paper:

$(H_{1})$ There are constants $A_{\vartheta _{1}}, A_{\vartheta _{2}}, A_{\vartheta _{3}}$ so that, for $\varpi _{1}, \varpi _{2}, \rho _{1}, \rho _{2}, \varrho _{1}, \varrho _{2}\in \mathbb{R},$

$(H_{2})$ There are constants $D_{\vartheta _{1}}, D_{\vartheta _{2}}, D_{\vartheta _{3}}, O_{\vartheta _{1}}, O_{\vartheta _{2}}, O_{\vartheta _{3}}$ so that, for $\varpi, \rho, \varrho \in \mathbb{R},$

$(H_{3})$ There are constants $p_{i}, q_{i}, t_{i}$ $(i = 1, 2, 3)$ and $O_{\Lambda _{1}}, O_{\Lambda _{2}}, O_{\Lambda _{3}}$ so that, for $\varpi, \rho, \varrho \in \mathbb{R},$

$(H_{4})$ There are constants $\hbar _{\Lambda _{1}}, \hbar _{\Lambda _{2}}, \hbar _{\Lambda _{3}}$ so that, for $\varpi _{1}, \varpi _{2}, \rho _{1}, \rho _{2}, \varrho _{1}, \varrho _{2}\in \mathbb{R},$

3.   Main results

In this section, the (EU) of solutions to the BVP (1.1) are discussed. We start stating and proving the lemma below.

Lemma 3.1. The solutions of the BVP

are equivalent to the solutions of the following Fredholm integral equation:

where $\Lambda _{1}:I\rightarrow \mathbb{R}$ is an $\ell _{1}$ times integrable function, and $\Game _{\ell _{1}}(z, r)$ is defined by

Proof. Reflecting $I^{\ell }$ on (3.1) and applying Lemma 2.1, we have

for some $a_{0}, a_{1}\in \mathbb{R}.$ It follows from the conditions $\varpi (0) = \vartheta _{1}(\varpi)$ and $\varpi (1) = \eta _{1}\varpi (\xi _{1})$ that $a_{0} = \vartheta _{1}(\varpi)$ and

Hence, we have

After a simple rearrangement, one can write

In the light of Lemma 3.1, solutions of (M-PBVPs) (1.1) are solutions of the Fredholm integral equations below:

where $\Game _{\gamma }(z, r)$ and $\Game _{\varkappa }(z, r)$ are defined by

and

It is clear that

Define the operators $\phi _{1}:E\rightarrow E,$ $\phi _{2}:\widetilde{E} \rightarrow \widetilde{E},$ $\phi _{3}:\widehat{E}\rightarrow \widehat{E}$ by

and the operators $\Game _{1}, \Game _{2}, \Game _{3}:E\times \widetilde{E} \times \widehat{E}\rightarrow E\times \widetilde{E}\times \widehat{E}$ by

Now, consider $\phi = (\phi _{1}, \phi _{2}, \phi _{3})$, $\Game = (\Game _{1}, \Game _{2}, \Game _{3})$ and $\psi = \phi +\Game.$ Then, the suggested problem (3.3) can be written as an operator equation as follows:

Hence, the solutions of the proposed problem (3.3) are FPs of $\psi.$

Lemma 3.2. Under the assumptions $(H_{1})$ and $(H_{2}),$ the operator $\phi$ satisfies the Lipschitz condition and the following condition is true:

where $D = \max \left\{ D_{\vartheta _{1}}, D_{\vartheta _{2}}, D_{\vartheta _{3}}\right\},$ $O = \max \left\{ O_{\vartheta _{1}}, O_{\vartheta _{2}}, O_{\vartheta _{3}}\right\}$.

Proof. From Hypothesis $(H_{1}),$ one can write

Applying Proposition 2.2 (ii), we have $\phi$ is $\upsilon$-Lipschitz with constant $A.$ Now, from $(H_{2}),$ we have

where $D = \max \left\{ D_{\vartheta _{1}}, D_{\vartheta _{2}}, D_{\vartheta _{3}}\right\},$ $O = \max \left\{ O_{\vartheta _{1}}, O_{\vartheta _{2}}, O_{\vartheta _{3}}\right\}.$ This completes the proof.

Lemma 3.3. The operator $\Game$ is continuous, and it satisfies the following growth condition under the postulate $(H_{3})$:

where $\Theta = \theta \left(p+q+t\right),$ $\theta = \max \left\{ \frac{1}{ \left(1-\xi _{1}\eta _{1}\right) \Gamma (\ell)}, \frac{1}{\left(1-\xi _{2}\eta _{2}\right) \Gamma (\gamma)}, \frac{1}{\left(1-\xi _{3}\eta _{3}\right) \Gamma (\varkappa)}\right\},$ $p = \max \left\{ p_{1}, p_{2}, p_{3}\right\},$ $q = \max \{q_{1}, q_{2}, q_{3}\},$ $t = \max \{t_{1}, t_{2}, t_{3}\},$ and $\Upsilon = \theta \left(O_{\Lambda _{1}}+O_{\Lambda _{2}}+O_{\Lambda _{3}}\right).$

Proof. Assume that $\{\left(\varpi _{k}, \rho _{k}, \varrho _{k}\right) \}$ is a sequence of the bounded set $V_{s} = \left\{ \left\Vert \left(\varpi, \rho, \varrho \right) \right\Vert \leq s:\left(\varpi, \rho, \varrho \right) \in E\times \widetilde{E}\times \widehat{E}\right\}$ so that $\{\left(\varpi _{k}, \rho _{k}, \varrho _{k}\right) \}\rightarrow \left(\varpi, \rho, \varrho \right)$ in $V_{s}.$ We want to show that $\left\Vert \left(\varpi _{k}, \rho _{k}, \varrho _{k}\right) -\left(\varpi, \rho, \varrho \right) \right\Vert \rightarrow 0.$ Consider

The continuity of $\Lambda _{1}$ leads to $\Lambda _{1}\left(r, \varpi _{k}(r), \rho _{k}\left(r\right), \varrho _{k}\left(r\right) \right) \rightarrow \Lambda _{1}\left(r, \varpi (r), \rho \left(r\right), \varrho \left(r\right) \right)$ as $k\rightarrow \infty.$ For all $z\in \lbrack 0, 1],$ from $(H_{3}),$ we get

which leads to the integrability for $z, r\in \lbrack 0, 1].$ Applying the Lebesgue dominated convergence theorem, we have

Analogously, the rest terms tend to $0$ as $k\rightarrow \infty.$ This implies that

Similarly, one can obtain that

and

Now, for growth condition on $\Game,$ using $(H_{3})$ and (3.6), we have

and

It follows that

This finishes the desired result.

Lemma 3.4. The mapping $\Game :E\times \widetilde{E}\times \widehat{E} \rightarrow E\times \widetilde{E}\times \widehat{E}$ is compact. As a result, $\Game$ is $\upsilon$-Lipschitz with constant zero.

Proof. Consider a bounded set $\mho \subset V_{s}\subseteq E\times \widetilde{E} \times \widehat{E}$ and a sequence $\{\left(\varpi _{k}, \rho _{k}, \varrho _{k}\right) \}$ in $\mho.$ Then from (3.8), we obtain that

which implies that $\Game (\mho)$ is bounded. Now, for equi-continuity and for given $\varepsilon > 0$, put

If $z, \lambda \in \lbrack 0, 1]$, and $\lambda -z\in (0, \delta _{1}),$ then for each $\left(\varpi _{k}, \rho _{k}, \varrho _{k}\right) \in \mho,$ we claim that

Consider

Now, we realize the following cases:

($\bullet$) If $\delta _{1}\leq z < \lambda < 1,$ we have

($\bullet \bullet)$ If $0\leq z < \delta _{1},$ $\lambda < 2\delta _{1},$ we get

and

This proves that $\Game (\mho)$ is equi-continuous. In light of the Arzelà-Ascoli theorem, $\Game (\mho)$ is compact. Based on Proposition 2.2 (iii), $\Game$ is a $\upsilon$-Lipschitz with a constant zero.

Theorem 3.1. The tripled system of nonlinear (M-PBVPs) (1.1) has at least one solution $\left(\varpi, \rho, \varrho \right) \in E\times \widetilde{E}\times \widehat{E}$ provided that the assumptions $(H_{1})$–$(H_{3})$ hold and $D+\Theta < 1.$ In addition, the set of solutions of problem (1.1) is bounded in $E\times \widetilde{E}\times \widehat{E }.$

Proof. Based on Lemmas 3.2 and 3.4, $\phi$ is $\upsilon$-Lipschitz with constant $A\in \lbrack 0, 1)$, and $\Game$ is $\upsilon$-Lipschitz with constant zero, respectively. From Proposition 2.2 (i), we have, $\psi$ is a strict $\upsilon$-contraction with constant $A.$ Define

In order to prove that $G$ is bounded, assume that $\left(\varpi, \rho, \varrho \right) \in G.$ Then, in light of growth stipulations as in Lemmas 3.2 and 3.3, we get

which implies that $G$ is bounded in $E\times \widetilde{E}\times \widehat{E}.$ Hence, according to Theorem 2.1, we conclude that $\psi$ has at least one (FP), and the set of (FPs) is bounded in $E\times \widetilde{E}\times \widehat{E}.$

Theorem 3.2. The tripled system of nonlinear (M-PBVPs) (1.1) has a unique solution $\left(\varpi, \rho, \varrho \right) \in E\times \widetilde{ E}\times \widehat{E}$ provided that the assumptions $(H_{1})$–$(H_{4})$ are true and $A+\theta \left(\hbar _{\Lambda _{1}}+\hbar _{\Lambda _{2}}+\hbar _{\Lambda _{3}}\right) < 1.$

Proof. For $\left(\varpi, \rho, \varrho \right), \left(\varpi ^{\ast }, \rho ^{\ast }, \varrho ^{\ast }\right) \in \mathbb{R} ^{3},$ it follows from the Banach FP theorem and (3.7) that

From $(H_{4})$ and (3.6), we have

which yields that

Analogously, we can obtain

and

Combining (3.10)–(3.12), we get

Using (3.9) and (3.13), we have

This proves that $\psi$ is a contraction mapping. From the Banach FP theorem, the suggested problem has a unique solution.

4.   Supportive example

Consider the tripled system of nonlinear (M-PBVPs) below

From the problem (4.1) we take $\ell = \gamma = \varkappa = \frac{4}{3} \in (1, 2],$ $\eta _{1} = \xi _{1} = \frac{1}{3},$ $\eta _{2} = \xi _{2} = \frac{1}{4},$ $\eta _{3} = \xi _{3} = \frac{1}{5}$ with $\eta _{1}\xi _{1}^{\ell } = \frac{1}{ 3}\left(\frac{1}{3}\right) ^{\frac{3}{4}} = 0.1462 < 1,$ $\eta _{2}\xi _{2}^{\gamma } < 1,$ $\eta _{3}\xi _{3}^{\varkappa } < 1$ and $s = 3 > 0$. The solution of the BVP (4.1) can be written as

where $\Game _{\ell },$ $\Game _{\gamma }$\ and $\Game _{\varkappa }$ are the Green's functions, and they may be simply obtained as shown in (3.2), (3.4) and (3.5), respectively. Let us consider $\varsigma = \frac{1}{3},$ and then according to Theorem 3.2, we have $\hbar _{\Lambda _{1}} = \hbar _{\Lambda _{2}} = \hbar _{\Lambda _{3}} = \frac{1}{75} = p_{i} = q_{i} = t_{i}$ $(i = 1, 2, 3),$ taking $A_{\vartheta _{1}} = A_{\vartheta _{2}} = A_{\vartheta _{3}} = \frac{1}{3}.$ Then, the hypotheses $(H_{1})$–$(H_{4})$ are fulfilled. We get

The continuity and boundedness of $\phi _{1}, \phi _{2}, \phi _{3}, \Game _{1}, \Game _{2}$ and $\Game _{3}$ imply that $\phi = \left(\phi _{1}, \phi _{2}, \phi _{3}\right)$ and $\Game = \left(\Game _{1}, \Game _{2}, \Game _{3}\right)$ are also. Hence, $\psi = \phi +\Game$ is continuous and bounded. Moreover,

This illustrates that, if $\Game$ is $\upsilon$-Lipschitz with constant $\frac{1}{3}$ and $\phi$ is $\upsilon$-Lipschitz with constant $0,$ then $\psi$ is a strict $\upsilon$-contraction with constant $\frac{1}{3}.$ Furthermore, it is easy to see that $\theta = 1.259845459.$ Since

the solution

implies that $G$ is bounded. Using Theorem 3.1, the tripled system of nonlinear (M-PBVPs) (4.1) has a solution $\left(\varpi, \rho, \varrho \right)$ in $C\left([0, 1]\times \mathbb{R} ^{3}, \mathbb{R} \right)$. In addition, $A+\theta \left(\hbar _{\Lambda _{1}}+\hbar _{\Lambda _{2}}+\hbar _{\Lambda _{3}}\right) = 0.38373 < 1.$ Therefore, by Theorem 3.2, the suggested problem (4.1) has a unique solution.

5.   Conclusions

The technique of a coincidence degree theory for condensing maps has been incorporated to obtain suitable conditions for the (EU) of positive solutions to tripled systems of nonlinear (M-PBVPs) under nonlinear boundary conditions. We provided an example to illustrate the obtained results. Our findings can be applied to further arbitrary fractional order differential equations, linear and nonlinear fractional integro-differential systems, Hadamard fractional derivatives, and other topics as future work.

Acknowledgments

Mohra Zayed extends appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for supporting this work through the research groups program under grant R.G.P.2/207/43.

Conflict of interest

All authors declare that they have no conflicts of interest.