Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations

1.   Introduction

Boundary value problems (for short BVPs) for nonlinear fractional order differential equation (for short NLFDE) have been addressed by several researchers during last few decades. The necessity of fractional order differential equations (for short FDEs) lies in the fact that fractional order models are more accurate than integer order models, that is, there are more degree of freedom in the fractional order models. Furthermore, fractional order derivatives provide an excellent mechanism for the description of memory and hereditary properties of various materials and processes. In applied sense, FDEs arise in various engineering and scientific disciplines for mathematical modeling in the fields of physics, chemistry, biology, mechanics, control theory of dynamical system, electrical network, statistics and economics, see for instance ^{[1,2,3,4,5,6]} and their references.

Consequently, day by day the topics in FDEs are taking an important part in various applied research. Some recent development of FDEs can be seen in ^{[7,8,9,10,11,12,13,14,15,16]} and in their references.

Now a days, many researchers devoted themselves to determine the solvability of system of nonlinear fractional order differential equations (for short SNLFDEs) with different boundary conditions, specifically to the study of existence of positive solutions to BVPs for SNLFDEs, see for instance ^{[10,12,13,14,17,18,19,20,21,22,23,24,25,26,27,28,29]} and their references.

Inspired by the above-mentioned works on existence of positive solutions to BVPs for SNLFDEs, in this paper, we establish the existence criteria of at least one positive solution to the following boundary value problem (for short BVP) for coupled system of Riemann-Liouville type nonlinear fractional order differential equations (for short NLFDEs) applying Schauder's fixed point theorem ^[30]:

where, $D_{{0^ + }}^{{\alpha _i}}, \, \, D_{{0^ + }}^{{\beta _i}}, \, \, D_{{0^ + }}^{{\gamma _i}}\, {\rm{and}}\, D_{{0^ + }}^{{\delta _i}}$ are standard Riemann-Liouville fractional differential operators of order ${\alpha _i} \in \left( {3, 4} \right], \, \, {\beta _i} \in \left( {0, \, 1} \right), \, \, {\gamma _i} \in \left( {1, \, 2} \right), \, \, {\delta _i} \in \left( {2, \, 3} \right), \, \, \left( {i = 1, \, 2} \right),$ respectively, ${\eta _i}, \, \, {\xi _i} \in \left( {0, \, 1} \right)$ with $0 < {\eta _i}{\xi _i}^{{\alpha _i} - 1} < 1$, $\, \left( {i = 1, \, 2} \right)$ and ${f_i}, \, \, {g_i}, \, \, {a_i}\, {\rm{and }}\, {\lambda _i}, \, \left( {\, i = 1, \, 2} \right)$ satisfy the following hypothesis:

To the best of our knowledge there is no any works considering the BVP for coupled system of Riemann-Liouville type NLFDEs given by (1) applying Schauder's fixed point theorem.

The rest of this work is furnished as follows. In section 2, we will provide some basic ideas of fractional calculus, certain lemmas and state Schauder's fixed point theorem. Section 3 is used to state and prove our main results, which provide some techniques to check the existence of at least one positive solutions of coupled system of Riemann-Liouville-type NLFDEs with three-point boundary conditions given by (1). In section 3 we also give some illustrative examples. Finally, we conclude this paper.

2.   Preliminary notes

In this section, we introduce some necessary definitions and preliminary facts which will be used throughout this paper.

Definition 1 (^[3,4,5]). Let $f:(0, \infty ) \to \mathbb{R}$ be a continuous function and $\alpha > 0$. Then the Riemann-Liouville fractional integral of order $\alpha$ is defined as follows:

where $\Gamma \left( \alpha \right)$ is the Euler Gamma function of $\alpha$ and provided that the integral exists.

Definition 2 (^[3,4,5]). Let $f:(0, \infty ) \to \mathbb{R}$ be a continuous function and $\alpha > 0$. Then the Riemann-Liouville fractional derivative of order $\alpha$ is defined as follows:

where $\, \, n = \left[ \alpha \right] + 1$ and $\left[ \alpha \right]$ denotes the integer part of real number $\alpha$ and provided that the right-hand side is point-wise defined on $\left( {0, \, \infty } \right)$.

Lemma 1 (^[10]). Suppose that $h\left( t \right) \in C\left[ {0, \, 1} \right]$ and $\left( {{H_1}} \right)$ holds, then the unique solution of the BVP

is provided by

where the Green's function ${G_1}\left( {t, \, s} \right)$ is defined by

Remark 1. Similar as Lemma 1 the unique solution of the BVP

is provided by

where the Green's function ${G_2}\left( {t, \, s} \right)$ is defined by

Remark 2. In view of Lemma 1 and Remark 1, the couple system of BVPs defined by (1) is equivalent to the following couple system of integral equations:

where the Green's functions ${G_i}\left( {t, \, s} \right), \, \, \left( {i = 1, \, 2} \right)$ are given by (3) and (5).

Lemma 2 (^[10]). The Green's functions ${G_i}\left( {t, \, \, s} \right), \, \, \left( {i = 1, \, 2} \right)$ defined as in (3) and (5) satisfy the following properties:

(ⅰ) ${G_i}\left( {t, \, \, s} \right), \, \, \left( {i = 1, 2} \right)$ are continuous on the unit square $\left[ {0, \, 1} \right] \times \, \left[ {0, \, 1} \right]$,

i.e., ${G_i}\left( {t, \, \, s} \right) \in C\left( {\left[ {0, \, 1} \right] \times \, \left[ {0, \, 1} \right]} \right)$ and ${G_i}\left( {t, \, s} \right) \geqslant 0, \, \, \, \forall \, \, t, s \in \, \left[ {0, \, 1} \right];$

(ⅱ) $\max{_{t \in \left[ {0, \, 1} \right]}}{G_i}\left( {t, \, s} \right) = {G_i}\left( {1, \, s} \right), \, \, \left( {i = 1, \, 2} \right);$

(ⅲ) $\min{_{t \in \left[ {\lambda , \, 1 - \lambda } \right]}}{G_i}\left( {t, \, s} \right) \geqslant {\theta _i}\left( s \right)\max{_{t \in \left[ {0, \, 1} \right]}}{G_i}\left( {t, \, s} \right) = {\theta _i}\left( s \right){G_i}\left( {1, \, s} \right), \, \, \lambda \in \left( {0, \, 1} \right), \, \, \left( {i = 1, \, 2} \right).$

Lemma 3. If the Green's functions ${G_i}\left( {t, \, \, s} \right), \, \, \left( {i = 1, \, 2} \right)$ are given as in (3) and (5), then there exist constants ${\kappa _i} \in \left( {0, \, 1} \right), \, \, \left( {i = 1, \, 2} \right)$ such that

Proof. Since $t \in \left[ {{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}, \, \, 1} \right]$, then from (3) we obtain that

If we take $0 \leqslant s \leqslant t \leqslant {\xi _1} \leqslant 1$, then

and

Let ${\sigma _1}$ be a positive number such that $\min{_{t \in \left[ {{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}, \, 1} \right]}}{G_1}\left( {t, \, s} \right) \geqslant {\sigma _1}{G_1}\left( {1, \, s} \right)$. Then we have

This means that ${\sigma _1} \in \left( {0, \, 1} \right)$.

If we take $0 \leqslant t \leqslant s \leqslant {\xi _1} \leqslant 1,$ then

and

Let ${\sigma _2}$ be a positive number such that $\min{_{t \in \left[ {{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}, \, 1} \right]}}{G_1}\left( {t, \, s} \right) \geqslant {\sigma _2}{G_1}\left( {1, \, s} \right)$. Then we have

This means that ${\sigma _2} \in \left( {0, \, 1} \right)$.

If we take $0 \leqslant {\xi _1} \leqslant s \leqslant t \leqslant 1,$ then

and

Let ${\sigma _3}$ be a positive number such that $\min{_{t \in \left[ {{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}, \, 1} \right]}}{G_1}\left( {t, \, s} \right) \geqslant {\sigma _3}{G_1}\left( {1, \, s} \right)$. Then we have

This means that ${\sigma _3} \in \left( {0, \, 1} \right)$.

If we take $0 \leqslant {\xi _1} \leqslant t \leqslant s \leqslant 1,$ then

and

Let ${\sigma _4}$ be a positive number such that $\min{_{t \in \left[ {{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}, \, 1} \right]}}{G_1}\left( {t, \, s} \right) \geqslant {\sigma _4}{G_1}\left( {1, \, s} \right)$. Then we have

This means that ${\sigma _3} \in \left( {0, \, 1} \right)$.

Now, if we set ${\kappa _1} = \min \left\{ {{\sigma _1}, \, \, {\sigma _2}, \, \, {\sigma _3}, \, {\sigma _4}} \right\},$ then we obtain that

Similarly, for the Green's function ${G_2}\left( {t, \, s} \right),$ we can prove that

This completes the proof.

Throughout this paper let $B = \left\{ {u\left( t \right):u \in C\left( {\left[ {0, \, 1} \right]} \right), \, \, t \in \left[ {0, \, 1} \right]} \right\}$ be a Banach space with the usual supremum norm $\left\| {\, \cdot \, } \right\|.$ Now, if we set $X = B \times B,$ where $X$ is equipped with the norm $\left\| {\, \left( {{u_1}, \, {u_2}} \right)\, } \right\| = \left\| {{u_1}} \right\| + \left\| {{u_2}} \right\|\, \, {\rm{for}}\, \left( {{u_1}, \, {u_2}} \right) \in X,$ then it is clear that $X$ is also a Banach space. Furthermore, we define the integral operators ${A_1}, \, {A_2}:X \to B$ by

and

where ${G_i}\left( {t, \, \, s} \right), \, \, \left( {i = 1, \, 2} \right)$ are the Green's functions given by (3) and (5). Finally, combining the operators ${A_1}\, \, {\rm{and}}\, \, {A_2}$, we define an operator $T:X \to X$

Then it is easy to see that the BVP (1) has a solution $\left( {{u_1}, \, {u_2}\, } \right) \in X$ if and only if $\left( {{u_1}, \, {u_2}\, } \right)$ is a fixed point of the operator $T$ defined by (6) and from this context, the main objective of this study is to find the existence of fixed point of the operator $T$ defined by (6).

For the brevity, we state only the Schauder's fixed point theorem ^[30], which will be used to prove the main results.

Theorem S. ^[30] (Schauder's Fixed Point Theorem) Let $X$ be a Banach space and $E$ be a nonempty closed convex subset of $X$ . Let $T$ be a continuous mapping of $E$ into a compact set $F \subset E$ . Then T has a fixed point in $X$.

3.   Main results

This section is devoted to establishing the existence criteria of at least one positive solution to the BVP given by (1).

Let ${\kappa _1}\, \, {\rm{and}}\, \, {\kappa _2}$ be the non-negative constants given by Lemma 3 associated to the Green's functions ${G_1}\left( {t, \, s} \right)$ and ${G_2}\left( {t, \, s} \right)$ respectively. Next suppose that ${f_1}\, \, {\rm{and}}\, \, {f_2}$ are Caratheodory type functions, that is

(ⅰ) for almost all $t \in [0,1],{f_1}(t, \cdot ):{\mathbb{R}^ + } \to \mathbb{R}$ and ${f_2}(t, \cdot ):{\mathbb{R}^ + } \to \mathbb{R}$ are continuous.

(ⅱ) for every $r \in {\mathbb{R}^ + },{f_1}( \cdot ,{\text{r}}):[0,1] \to \mathbb{R}$ and ${f_1}( \cdot ,{\text{r}}):[0,1] \to \mathbb{R}$ are measurable.

Throughout this paper, we use the following notations:

if for almost all $t \in \left[ {0, \, \, 1} \right], \, \, m \geqslant 0, \, m \in {L^1}\left( {0, \, 1} \right),$ then we denote $m \succ 0$,

and

Finally, we define a set $S$ as follows

We are now in position to present and prove the main results.

Theorem 1. Consider the BVP for coupled system of Riemann-Liouville-type NLFDEs given by (1), along with Caratheodory functions ${f_1}\, \, {\rm{and}}\, \, {f_2}$ . Suppose that there exist $m \succ 0$ and $\mu > 0$ such that the following conditions are satisfied:

$\left( {{H_4}} \right)\, \, {\lambda _1}\int_0^1 {\frac{{{G_1}\left( {1, \, s} \right){a_1}\left( s \right)m\left( s \right)}}{{{s^{\mu \left( {{\alpha ^*} - 1} \right)}}}}ds} < + \infty , \, \, {\lambda _2}\int_0^1 {\frac{{{G_2}\left( {1, \, s} \right){a_2}\left( s \right)m\left( s \right)}}{{{s^{\mu \left( {{\alpha ^*} - 1} \right)}}}}ds} < + \infty , \, \, where\, \, {\alpha ^*} = \max \left\{ {{\alpha _1}, \, {\alpha _2}} \right\}.$ If ${M_*} > 0,$ then the BVP given by (1) has at least one positive solution.

Poof. Since, the solution of the BVP given by (1) is equivalent to the fixed point of the integral operator $T$ defined by (6), so we have to prove that the integral operator $T$ defined by (6) exist a fixed point.

Let $\Psi = \left\{ {\left( {{u_1}, \, {u_2}} \right) \in S:{t^{{\alpha ^*} - 1}}p \leqslant {u_1}\left( t \right), \, \, {u_2}\left( t \right) \leqslant {t^{{\alpha _*} - 1}}P, \, \, \forall t \in \left[ {0, \, 1} \right]} \right\},$ where $\, {\alpha _*} = \min \left\{ {{\alpha _1}, \, {\alpha _2}} \right\}$ and $P > p > 0$ are undetermined positive constants. Then it is clear that $\Psi$ is a bounded closed convex subset of $X$.

It is obvious that operator $T:\Psi \to \Psi$ is continuous. To prove $T\left( \Psi \right) \subset \Psi$, let us fix $p = {M_*}$ and from assumption, we have $p > 0$. Now for all $t \in \left[ {0, \, 1} \right]$ and $\left( {{u_1}, \, {u_2}} \right) \in \Psi ,$ we yield that

On the other hand, if we put

then using $\left( {{H_3}} \right),$ we get

Now, if we set $P = \left( {\frac{{{N^*}}}{{{p^\mu }}} + {M^*}} \right),$ then we obtain that

Similarly, for the operator ${A_1}\left( {{u_1}, \, {u_2}} \right)$, we obtain that

Hence, from (6)–(8) we have

This means that $T\left( \Psi \right) \subset \Psi .$

Now for all ${t_1}, \, \, {t_2} \in \left[ {0, \, 1} \right], \, \, {t_1} > {t_2}$ and using (8) and (9), we get

This tells us that $T\left( \Psi \right)$ is equicontinuous. Hence from the Arzela-Ascoli theorem ^[31], we conclude that $T:\Psi \to \Psi$ is completely continuous operator and it ensure that $T$ is a continuous operator from a bounded closed convex subset of $X$ to the compact subset of that bounded closed convex subset.

Thus, in view of Theorem S (Schauder's Fixed Point Theorem) the integral operator $T$ given by (6) has at least one fixed point which is positive and this means that the BVP given (1) has at least one positive solution.

This completes the proof.

Theorem 2. Consider the BVP as like Theorem 1 and assume that $\left( {{H_4}} \right)$ holds. Suppose that there exist $m \succ 0$, $\hat m \succ 0$ and $0 < \mu < 1$ such that the following condition is satisfied:

If ${M_*} = 0,$ then the BVP given (1) has at least one positive solution.

Proof. To prove this theorem, we follow the proof the Theorem 1 and just search the positive constants $P > p > 0$ such that $T\left( \Psi \right) \subset \Psi$. So, from Theorem 1 we can obtain that

where ${N^*}$ is given by (7).

Now, if we set

where ${\kappa _1}\, \, {\rm{and}}\, \, {\kappa _2}$ be the non-negative constants given by Lemma 3, then using $\left( {{H_5}} \right),$ we get

Hence, if we consider $p\, \, {\rm{and}}\, \, P$ satisfying $\left( {\frac{{{N^*}}}{{{p^\mu }}} + {M^*}} \right) \leqslant P\, \, {\rm{and}}\, \,$ $\frac{{{{\hat N}^*}}}{{{P^\mu }}} \geqslant p$ then using the same process as like the Theorem 1, we conclude that $T$ is a continuous operator from a bounded closed convex subset of $X$ to the compact subset.

Thus, in view of Theorem S the integral operator $T$ given by (6) has at least one fixed point which is positive and this means that the BVP given (1) has at least one positive solution.

This completes the proof.

Theorem 3. Consider the BVP as like Theorem 1. and assume that there exist $m \succ 0,\hat m \succ 0$, and $0<\mu<1$ such that $\left(H_{4}\right)$ and $\left(H_{5}\right)$ hold. If $M^{*}<0$ with the following condition $\left( {{H_6}} \right){M_*} \geqslant {\left[ {\frac{{{{\hat N}^*}}}{{{{\left( {{N^*}} \right)}^\mu }}}{\mu ^2}} \right]^{\frac{1}{{1 - {\mu ^2}}}}}\left( {1 - \frac{1}{{{\mu ^2}}}} \right),$

then the BVP given (1) has at least one positive solution.

Proof. To prove this theorem, we follow the proof the Theorem 2 and just search the positive constants $P > p > 0$ such that

Now, if we fix $P = \frac{{{N^*}}}{{{p^\mu }}},$ then $\frac{{{{\hat N}^*}}}{{{{\left( {{N^*}} \right)}^\mu }}}{p^{{\mu ^2}}} + {M_*} \geqslant p$ implies that either $\frac{{{{\hat N}^*}}}{{{P^\mu }}} + {M_*} \geqslant p$ or, ${M_*} \geqslant p - \frac{{{{\hat N}^*}}}{{{{\left( {{N^*}} \right)}^\mu }}}{p^{{\mu ^2}}} = \varphi (p)$. It is clear that the minimum value of $\varphi \left( p \right)$ occur at $p = {p_0} = {\left[ {\frac{{{{\hat N}^*}{\mu ^2}}}{{{{\left( {{N^*}} \right)}^\mu }}}} \right]^{\frac{1}{{1 - {\mu ^2}}}}}$. Hence, if we put $p = {p_0},$ then we obtain that

Therefore, for ${M_*} \geqslant \varphi \left( {{p_0}} \right)$ (11) is satisfied. Consequently, $\left( {{H_6}} \right)\,$ is satisfied. Thus, in view of Theorem 2 and Theorem S the integral operator $T$ given by (6) has at least one fixed point which is positive and this means that the BVP given (1) has at least one positive solution.

This completes the proof.

Now, we give some illustrative examples.

Example 1. Consider the BVP for coupled system of Riemann-Liouville-type NLFDEs provided by

where for all ${u_1}, \, {u_2} > 0, \, \, {f_1}\left( {t, \, {u_1}, \, {u_2}} \right) = {\left[ {{u_1}\left( t \right) + {u_2}\left( t \right)} \right]^4} > 0, \, \, \, {f_2}\left( {t, \, {u_1}, \, {u_2}} \right) = {\left[ {{u_1}\left( t \right) + {u_2}\left( t \right)} \right]^6} > 0,$ ${\alpha _1} = \frac{{10}}{3}, \, {\alpha _2} = \frac{{13}}{4} \in \left( {3, \, 4} \right], \, {\beta _1} = \frac{1}{2}, \, \, {\beta _2} = \frac{2}{3} \in \left( {0, \, 1} \right), \, \, {\gamma _1} = \frac{4}{3}, \, \, {\gamma _2} = \frac{3}{2} \in \left( {1, \, 2} \right), \, {\delta _1} = \frac{9}{4}, \, \, {\delta _2} = \frac{5}{2} \in \left( {2, \, 3} \right),$ ${\eta _1} = \frac{1}{2}, \, \, {\eta _2} = \frac{1}{3}, \, \, \, {\xi _1} = \frac{1}{2} \in \left( {0, \, 1} \right), \, \, {\xi _2} = \frac{1}{3} \in \left( {0, \, 1} \right), \, \, 0 < {\eta _1}{\xi _1}^{{\alpha _1} - 1} < 1, \, \, \, 0 < {\eta _2}{\xi _2}^{{\alpha _2} - 1} < 1$, ${\lambda _1} = 1 > 0, \, \, {\lambda _2} = 2 > 0, \,$ ${\rm{for}}\, \, {\rm{all}}\, \, t \in \left[ {0, \, 1} \right]\, \, \, {a_1}\left( t \right) = t > 0, \, \, {a_2}\left( t \right) = {t^2} > 0, \,$ and ${g_1}\left( t \right) = {t^2}, \, \, {g_2}\left( t \right) = {t^3}$. For the above values it is clear that $\left( {{H_1}} \right)\, \, {\rm{and}}\, \left( {{H_2}} \right)\,$ are satisfied.

Now if we consider $m\left( t \right) = {u_1}\left( t \right){\left[ {{u_1}\left( t \right) + {u_2}\left( t \right)} \right]^7}\, \, {\rm{and}}\, \, \mu = 1$, then by direct calculation we obtain that $\, 0 \leqslant {f_1}\left( {t, \, {u_1}, \, {u_2}} \right), \, \, {f_2}\left( {t, \, {u_1}, \, {u_2}} \right) \leqslant \frac{{m\left( t \right)}}{{{u_1}^\mu }}, \, \, \forall \, \, t \in \left[ {0, \, 1} \right]\, , \, {\lambda _1}\int_0^1 {\frac{{{G_1}\left( {1, \, s} \right){a_1}\left( s \right)m\left( s \right)}}{{{s^{\mu \left( {{\alpha ^*} - 1} \right)}}}}ds} < + \infty , \,$ and $\, {\lambda _2}\int_0^1 {\frac{{{G_2}\left( {1, \, s} \right){a_2}\left( s \right)m\left( s \right)}}{{{s^{\mu \left( {{\alpha ^*} - 1} \right)}}}}ds} < + \infty , \, \, {\rm{for}}\, \, {\alpha ^*} = \max \left\{ {{\alpha _1}, \, {\alpha _2}} \right\},$ that is the conditions $\left( {{H_3}} \right)$ and $\left( {{H_4}} \right)\,$ are satisfied. Furthermore, since ${G_1}\left( {t, \, s} \right), \, {G_2}\left( {t, \, s} \right) \geqslant 0$, then we get

Therefore, all the conditions of Theorem 1 are satisfied by BVP (12). Hence by an application of Theorem 1, we can say that the BVP (12) has at least one positive solution.

Example 2. Consider the BVP for coupled system of Riemann-Liouville-type NLFDEs provided by

where for all $\, {u_1}, \, {u_2} > 0, \, {f_1}\left( {t, \, {u_1}, \, {u_2}} \right) = {\left[ {{u_1}\left( t \right) + {u_2}\left( t \right)} \right]^{\frac{1}{4}}} > 0, \, {f_2}\left( {t, \, {u_1}, \, {u_2}} \right) = {\left[ {{u_1}\left( t \right) + {u_2}\left( t \right)} \right]^{\frac{1}{3}}} > 0,$ ${\alpha _1} = \frac{7}{2}, \, {\alpha _2} = \frac{{10}}{3} \in \left( {3, \, 4} \right], \, {\beta _1} = \, \, {\beta _2} = \frac{1}{3} \in \left( {0, \, 1} \right), \, \, {\gamma _1} = \, {\gamma _2} = \frac{3}{2} \in \left( {1, \, 2} \right), \, {\delta _1} = \, \, {\delta _2} = \frac{9}{4} \in \left( {2, \, 3} \right),$ ${\eta _1} = \, {\eta _2} = 1, \, \, {\xi _1} = \, \, {\xi _2} = \frac{1}{2} \in \left( {0, \, 1} \right), \, \, 0 < {\eta _1}{\xi _1}^{{\alpha _1} - 1} < 1, \, \, \, 0 < {\eta _2}{\xi _2}^{{\alpha _2} - 1} < 1$, ${\lambda _1} = {\lambda _2} = 1 > 0,$ for all $t \in \left[ {0, \, 1} \right]$ ${a_1}\left( t \right) = t > 0, \, \, {a_2}\left( t \right) = {t^2} > 0, \,$ and ${g_1}\left( t \right) = \left( {{t^3} - \frac{1}{4}} \right), \, \, {g_2}\left( t \right) = \left( {{t^2} - \frac{1}{3}} \right)$. For the above values it is clear that $\left( {{H_1}} \right)\, \, {\rm{and}}\, \left( {{H_2}} \right)\,$ are satisfied.

Now if we consider $m(t) = {u_1}(t){\left[ {{u_1}(t) + {u_2}(t)} \right]^{\frac{1}{2}}}, \hat m(t) = {u_1}(t){\left[ {{u_1}(t) + {u_2}(t)} \right]^{\frac{1}{5}}}$ and $u = \frac{1}{2}$, then by direct calculation we obtain that $0 \leqslant {f_1}\left( {t, {u_1}, {u_2}} \right), {f_2}\left( {t, {u_1}, {u_2}} \right) \leqslant \frac{{m(t)}}{{u_1^\mu }}, \forall t \in [0, 1]$, $\frac{{\hat m(t)}}{{u_1^\mu }} \leqslant {f_1}\left( {t, {u_1}, {u_2}} \right), {f_2}\left( {t, {u_1}, {u_2}} \right) \leqslant \frac{{m(t)}}{{u_1^\mu }}, \forall t \in [0, 1], {\lambda _1}\int_0^1 {\frac{{{G_1}(1, s){a_1}(s)m(s)}}{{{s^{\mu \left( {{\alpha ^*} - 1} \right)}}}}} ds < + \infty$ and $\, {\lambda _2}\int_0^1 {\frac{{{G_2}\left( {1, \, s} \right){a_2}\left( s \right)m\left( s \right)}}{{{s^{\mu \left( {{\alpha ^*} - 1} \right)}}}}ds} < + \infty , \, \, {\rm{for}}\, \, {\alpha ^*} = \max \left\{ {{\alpha _1}, \, {\alpha _2}} \right\},$ that is the conditions $\left( {{H_3}} \right), \, \left( {{H_4}} \right)$ and $\left( {{H_5}} \right)\,$ are satisfied. Furthermore, since $\max{_{t \in \left[ {0, \, 1} \right]}}{G_i}\left( {t, \, s} \right) = {G_i}\left( {1, \, s} \right), \, \, \left( {i = 1, \, 2} \right)$, then we have

Therefore, all the conditions of Theorem 2 are satisfied by BVP (13). Hence by an application of Theorem 2, we can say that the BVP (13) has at least one positive solution.

Example 3. Consider the BVP for coupled system of Riemann-Liouville-type NLFDEs provided by

where for all $\, {u_1}, \, {u_2} > 0, \, {f_1}\left( {t, \, {u_1}, \, {u_2}} \right) = {\left[ {{u_1}\left( t \right) + {u_2}\left( t \right)} \right]^{\frac{1}{4}}} > 0, \, {f_2}\left( {t, \, {u_1}, \, {u_2}} \right) = {\left[ {{u_1}\left( t \right) + {u_2}\left( t \right)} \right]^{\frac{1}{3}}} > 0,$ ${\alpha _1} = \frac{7}{2}, \, {\alpha _2} = \frac{{10}}{3} \in \left( {3, \, 4} \right], \, {\beta _1} = \, \, {\beta _2} = \frac{1}{3} \in \left( {0, \, 1} \right), \, \, {\gamma _1} = \, {\gamma _2} = \frac{3}{2} \in \left( {1, \, 2} \right), \, {\delta _1} = \, \, {\delta _2} = \frac{9}{4} \in \left( {2, \, 3} \right),$ ${\eta _1} = \, {\eta _2} = 1, \, \, {\xi _1} = \, \, {\xi _2} = \frac{1}{2} \in \left( {0, \, 1} \right), \, \, 0 < {\eta _1}{\xi _1}^{{\alpha _1} - 1} < 1, \, \, \, 0 < {\eta _2}{\xi _2}^{{\alpha _2} - 1} < 1$, ${\lambda _1} = {\lambda _2} = 1 > 0,$ for all $t \in \left[ {0, \, 1} \right]$ ${a_1}\left( t \right) = t > 0, \, \, {a_2}\left( t \right) = {t^2} > 0, \,$ and ${g_1}\left( t \right) = \left( {{t^2} - 1} \right), \, \, {g_2}\left( t \right) = \left( {{t^3} - 1} \right)$. For the above values it is clear that $\left( {{H_1}} \right)\, \, {\rm{and}}\, \left( {{H_2}} \right)\,$ are satisfied.

Now if we consider $m(t) = {u_1}(t){\left[ {{u_1}(t) + {u_2}(t)} \right]^{\frac{1}{2}}}, \hat m(t) = {u_1}(t){\left[ {{u_1}(t) + {u_2}(t)} \right]^{\frac{1}{5}}}$ and $\mu = \frac{1}{2}$, then by direct calculation we obtain that $0 \leqslant {f_1}\left( {t, {u_1}, {u_2}} \right), {f_2}\left( {t, {u_1}, {u_2}} \right) \leqslant \frac{{m(t)}}{{u_1^\mu }}, \forall t \in [0, 1],$ $\frac{{\hat m(t)}}{{u_1^\mu }} \leqslant {f_1}\left( {t, {u_1}, {u_2}} \right), {f_2}\left( {t, {u_1}, {u_2}} \right) \leqslant \frac{{m(t)}}{{u_1^\mu }}, \forall t \in [0, 1], {\lambda _1}\int_0^1 {\frac{{{G_1}(1, s){a_1}(s)m(s)}}{{{s^{\mu \left( {{\alpha ^*} - 1} \right)}}}}} ds < + \infty$, $\, {\lambda _2}\int_0^1 {\frac{{{G_2}\left( {1, \, s} \right){a_2}\left( s \right)m\left( s \right)}}{{{s^{\mu \left( {{\alpha ^*} - 1} \right)}}}}ds} < + \infty , \, \, {\rm{for}}\, \, {\alpha ^*} = \max \left\{ {{\alpha _1}, \, {\alpha _2}} \right\},$ and ${M_*} \geqslant {\left[ {\frac{{{{\hat N}^*}}}{{{{\left( {{N^*}} \right)}^\mu }}}{\mu ^2}} \right]^{\frac{1}{{1 - {\mu ^2}}}}}\left( {1 - \frac{1}{{{\mu ^2}}}} \right),$ that is the conditions $\left( {{H_3}} \right), \, \left( {{H_4}} \right), \, \left( {{H_5}} \right)$ and $\left( {{H_6}} \right)\,$ are satisfied.

Furthermore, since ${G_1}\left( {t, \, s} \right), \, {G_2}\left( {t, \, s} \right) \geqslant 0$, then we have

Therefore, all the conditions of Theorem 3 are satisfied by BVP (14). Hence by an application of Theorem 3, we can say that the BVP (14) has at least one positive solution.

4.   Conclusion

In this paper, some new existence criteria of at least one positive solution to the three-point BVP for coupled system of Riemann-Liouville-type NLFDEs given by (1) have been studied by applying Schauder's fixed point theorem. Proven theorems (Theorem 1-3) of this paper have been used as the efficient method to checked the existence of at least one positive solution to the coupled system of BVP for NLFDEs given by (1). The established results provide an easy and straightforward technique to cheek the existence of positive solutions to the considered BVP given by (1). Moreover, the results of this paper extend the corresponding results of Han and Yang ^[10] and Hao and Zhai ^[27].

Conflict of interest

The authors declare that they have no conflict of interests.