Abstract
We establish the existence of positive solutions to a class of a singular nonlinear Hadamard-type fractional differential equations with infinite-point boundary conditions (BCs) or integral BCs. Our analysis is based on Leray–Schauder type continuation. Several examples are given to illustrate our results.
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1 Introduction
Our aim in this article is to study the problem of existence of continuous solutions of the following singular Hadamard fractional differential equation:
together with either the functional integral boundary condition given by
or the infinite-point boundary conditions given by
where \(f: [1, e] \times R^{+} \times R^{2}\) is an \(L^{p}\)-Carathéodory positive function, \(p>\frac{1}{\gamma -1}\), \({}_{\mathrm{H}}D^{\delta }\) is the Hadamard fractional derivative of order δ, and \(1<\gamma <2\), \(0<\delta <1\), \(1\leq \gamma - \delta <2\). The constants \(a_{j}\), λ, and \(v_{0}\) are nonnegative, the function \(\phi :[1, e] \rightarrow [1, e]\), \(\phi (t) \leq t \) is continuous and the singularity occurring in our problems is associated with \(v' \in C(1,e] \) at the left endpoint \(t=1\).
Due to the fact that fractional-order models are more accurate than integer-order models (that is, there are more degrees of freedom in fractional-order models), the subject of fractional differential equations has recently evolved into an interesting subject for many researchers due to its multiple applications in economics, engineering, physics, chemistry, mechanics. However, most of the results for fractional differential equations are concerned with the Riemann–Liouville fractional derivative or the Caputo fractional derivative (see for example Agarwal et al. [1], Akcan and Çetin [4], Bai and Qiu [5], Bai and Sun [6], Callegari and Nachman [9], Chalishajar and Kumar [10], El-Saka et al. [11], El-Sayed et al. [12], Kosmatov [19], Li and Zhang [21], Liu et al. [22], Qiao and Zhou [26], Qiu and Bai [27], Rida et al. [28], Song et al. [30], Staněk [31], Tian and Chen [33].
In 1892, Hadamard introduced another kind of fractional derivatives, i.e., Hadamard-type fractional differential equations, which differs from the preceding ones in the sense that the kernel of the integral and derivative contain logarithmic function of arbitrary exponent, which is presented as a quite different kind of weakly singular kernel. Details and properties of Hadamard fractional derivatives and integrals can be found in Kilbas et al. [18], Butzer et al. [8], Gambo et al. [13]. Recently, there were some results on Hadamard-type fractional differential equations; see Ahmad et al. [3], Ahmad and Ntouyas [2], Benchohra et al. [7], Lyons and Neugebauer [24], Matar [25], Shammakh [29], Thiramanus et al. [32], Yang [35], Zhang et al. [37], and the references cited therein.
The study of boundary value problems (BVPs) involving infinite-point BCs has become attractive recently, many significant and interesting cases of BVPs of fractional order were considered with infinite-point BCs by (for example) Gao and Han [14], Ge et al. [15], Guo et al. [16], Hu and Zhang [17], Li et al. [20], Liu et al. [23], Zhang and Zhong [38] and Zhang [39] (see also to the references cited therein). In the year 2016, Xu and Yang [34] proposed a generalization of the PID controller and studied two kinds of fractional-order differential equations arising in control theory together with the infinite-point boundary conditions. Their results can describe the corresponding control system accurately and also provide a platform for the understanding of our environment. However, investigations on the infinite-point problems for differential equations of fractional or integer order have gradually aroused people’s attentions and interests, but such investigations are still few.
Motivated by the above-mentioned developments and results, we consider the BVP given by (1) and (3) or by (1) and (2). In each case, we determine sufficient conditions on f guaranteeing that these problems has a continuous positive solution. We first find the existence of positive solutions of the problem (1) subject to the multi-point boundary conditions
The main new features presented in this paper are as follows. First, the boundary value problem has a more general form, in which f is not continuous, but only Carathéodory and allowed to be singular at \(t=1\). Second, the nonlocal boundary conditions of the unknown function are more general cases, which include two-point, three-point, multi-point, infinite point and integral boundary conditions, and some nonlocal problems as special cases. Third, positive continuous solutions of problems (1), (3) or (1), (2) or (1), (4) are obtained.
2 Preliminaries
Throughout the paper \(\|x\|_{p}=(\int _{1}^{e}|x(t)|^{p} \,dt)^{ \frac{1}{p}}\) is the norm in \(L^{p}[1, e]\), \(\|u\|_{0}=\max \{|u(t)|: t\in [1, e]\}\) be the norm in the space \(C[1, e]\) and \(AC[1, e]\) denote the space of absolute continuous functions on \([1, e]\). We denoted by \(L^{p}_{\mathrm{loc}}(1,e]\) the space of functions on \([1,e]\) defined by
We make the following assumptions:
- (\(\mathrm{H}_{1}\)):
-
\(\eta _{j}\in (1, e)\), \(j=1, 2 , \ldots , m\), \(1< \eta _{1}<\eta _{2}< \cdots <\eta _{m}< e\), \(a_{j}\), λ, and \(v_{0}\) are nonnegative, \(1<\gamma <2\), \(0<\delta <1\) and \(1\leq \gamma - \delta <2\).
- (\(\mathrm{H}_{2}\)):
-
The function \(\phi :[1, e] \rightarrow [1, e]\), \(\phi (t) \leq t\) is continuous.
- (\(\mathrm{H}_{3}\)):
-
f is \(L^{p}\)-Carathéodory function on \([1,e]\times R^{+} \times R^{2}\) i.e., for each \((v_{1}, v_{2}, v_{3}) \in R^{+} \times R^{2}\), the function \(f(\cdot,v_{1} , v_{2}, v_{3}):[1, e] \rightarrow R \) is Lebesgue measurable and for each \(t \in [1, e] \), the function \(f(t,\cdot,\cdot,\cdot): R^{+} \times R^{2} \rightarrow R\) is continuous.
- (\(\mathrm{H}_{4}\)):
-
There exist \(p(t), q(t), s(t) \in L^{p} [1,e]\) and \(r(t)\in L^{p}_{\mathrm{loc}}(1,e]\) with \((\log t)^{\gamma -2}r(t) \in L^{p}[1,e]\), with \(p>\frac{1}{\gamma -1}\) such that
$$\begin{aligned}& \bigl\vert f(t, v_{1}, v_{2}, v_{3}) \bigr\vert \leq p(t) \vert v_{1} \vert +q(t) \vert v_{2} \vert +r(t) \vert v _{3} \vert + s(t), \\& \quad \mbox{a.e. } t \in [1,e], \mbox{and all } (v_{1}, v_{2}, v_{3}) \in R^{+} \times R^{2}. \end{aligned}$$(5)
Definition 1
([18])
The Hadamard fractional integral of order γ for a function \(v\in L^{p}[1, e]\), \(1 \leq p < \infty \), is defined as
provided the integral exists, where \(\log (\cdot ) =\log _{e} (\cdot )\).
Definition 2
([18])
The Hadamard derivative of fractional order γ for a function \(v: [1, \infty )\to \mathbb{R}\) is defined as
The relationship between fractional integration (6) and derivatives (7) is stated in the next theorem [18].
Theorem 1
Let \(\gamma >0\), \(n-1 <\gamma < n\), then:
- (\(\mathrm{d}_{1}\)):
-
The Hadamard fractional differential equation \({}_{\mathrm{H}}D^{ \gamma }v(t) = 0\) is valid if and only if
$$ v(t) =\sum_{i=1}^{n} c_{i}(\log t)^{\gamma -i}, $$where \(c_{i} \in R\) (\(i = 1,\ldots , n\)) are arbitrary constants.
In particular, when \(1 < \gamma < 2\), the relation \({}_{\mathrm{H}}D^{\gamma }v(t) = 0\) holds, if and only if
$$ v(t) = c_{1}(\log t)^{\gamma -1}+ c_{2}(\log t)^{\gamma -2} \quad \textit{for any } c_{1}, c_{2} \in R. $$ - (\(\mathrm{d}_{2}\)):
-
The quality \({}_{\mathrm{H}}D^{\gamma } {}_{\mathrm{H}}J^{\gamma } v(t)= v(t) \) holds for every \(v \in L^{p}[1, e]\).
- (\(\mathrm{d}_{3}\)):
-
Let \(v \in C[1, \infty ) \cap L^{p}[1, \infty )\). The following formula holds:
$$ {}_{\mathrm{H}}J^{\gamma } {}_{\mathrm{H}}D^{\gamma } v(t) = v(t) - \sum_{i=1}^{n} c_{i}( \log t)^{\gamma -i}. $$ - (\(\mathrm{d}_{4}\)):
-
\({}_{\mathrm{H}}J^{\gamma } {}_{\mathrm{H}}J^{\beta } v(t)= {}_{\mathrm{H}}J^{\gamma + \beta } v(t)\), \(\beta >0 \).
- (\(\mathrm{d}_{5}\)):
-
\({}_{\mathrm{H}}J^{1-\delta } (\log t)^{\gamma -1}=\frac{ \varGamma (\gamma )}{\varGamma (\gamma -\delta +1)} (\log t)^{\gamma - \delta }\), \(\delta \in (0, 1)\).
Our results in this article are based upon the Leray–Schauder continuation principle.
Theorem 2
(Leray–Schauder continuation principle; see e.g. [36])
Let X be a Banach space and \(T : X \rightarrow X\) be a compact map. Assume that there exists an \(D >0\) so that if \(v = \lambda T v\) for \(\lambda \in [0, 1]\) then \(\|v\|_{X} \leq D\). Then \(v=Tv\) is solvable.
Lemma 1
Suppose that \(\rho \in L^{p}[1, e]\), \(p>\frac{1}{\gamma -1}\), \(1 \leq t_{1} < t_{2} \leq e\) and \(\frac{1}{p}+\frac{1}{q}=1\), then we have
where \(b=(\gamma -2)q+1\).
Proof
Using the Hölder inequality with \(\frac{1}{p}+\frac{1}{q}=1\), we get
Using again the Hölder inequality and the inequality \((1-y)^{q} \leq 1-y^{q}\), \(|y| <1\), we get
Hence the inequality (9) holds for all \(1 \leq t_{1} < t_{2} \leq e\). □
Lemma 2
Suppose that \(\rho \in L^{p}[1, e]\), \(p>\frac{1}{\gamma -1}\), then we see that
Proof
The result follows from the inequality (9), since \(0< b=( \gamma -2)q+1<1\), then as \(t_{1} \rightarrow t_{2}\) in (9), the left-hand side tends to 0. □
Lemma 3
Suppose that \(\rho \in L^{p}[1, e]\), then we have:
-
(i)
For \(t\in [1, e]\),
$$ \frac{d}{dt} \int _{1}^{t} \biggl(\log \frac{t}{\theta } \biggr)^{\gamma -1} \frac{\rho (\theta )}{\theta } \,d\theta =\frac{(\gamma -1)}{t} \int _{1}^{t} \biggl(\log \frac{t}{\theta } \biggr)^{\gamma -2} \frac{ \rho (\theta )}{\theta } \,d\theta . $$(10) -
(ii)
Let \(\{\rho _{k}\}\subset L^{p}[1, e]\) be \(L^{p}\)-convergent sequence and let \(\lim_{k\rightarrow \infty }\rho _{k}=\rho \). Then
$$ \lim_{k\rightarrow \infty } \int _{1}^{t} \biggl(\log \frac{t}{\theta } \biggr)^{\gamma -2} \frac{ \rho _{k}(\theta )}{\theta } \,d\theta = \int _{1}^{t} \biggl(\log \frac{t}{\theta } \biggr)^{\gamma -2} \frac{\rho ( \theta )}{\theta } \,d\theta . $$
Proof
(i) We have by interchanging the order of integration
For \(t\in [1, e]\), since \(\int _{1}^{s} (\log \frac{s}{\theta } )^{\gamma -2} \frac{\rho (\theta )}{\theta } \,d\theta \) is a continuous function by Lemma 2, by differentiating both sides, the equality (10) follows.
(ii) Using the Hölder inequality, we obtain
Hence for \(t\in [1, e]\), we have
and the result follows. □
3 Existence of positive solutions of problem (1), (4)
For authors convenience, denote \(f_{v}\) by
Lemma 4
Suppose that the condition (\(\mathrm{H}_{1}\)) holds, then for \(f_{v}(t) \in L ^{p}[1, e]\) the boundary value problem
subject to the multi-point boundary conditions
has a unique solution \(v\in AC[1, e]\) if and only if v is a solution of the integral equation
where
and
Proof
As discussed in [18], the solution of the Hadamard-type fractional differential equation (11) can be written as
By using the BCs (12), we have \(c_{2}=0\), then
In view of condition \(v(e)=v_{0}+\lambda \sum_{j=1}^{m} a_{j} v( \phi (\eta _{j}))\), we have
and we get
Substituting in Eq. (15), we have the formula
Conversely, let \(v(t)\) be a solution of (13), we want to obtain (11) and (12). Now to obtain Eq. (11), operating on both sides of (13) by \({}_{\mathrm{H}}J^{2-\gamma }\) (using (\(\mathrm{d}_{4}\)) and (\(\mathrm{d}_{5}\))), we get
Again operating by \((t \frac{d}{dt} )^{2}\) on both sides of the last equation, we obtain
that is,
Now to check the conditions in (12) are satisfied, we can easy from (13) show that \(v(1)=0 \). Also to verify \(v(e)=v_{0}+\lambda \sum_{j=1}^{m} a_{j} v(\phi (\eta _{j}))\), and we have by a simple calculation using Eq. (13)
and
and then we get \(v(e)=v_{0}+\lambda \sum_{j=1}^{m} a_{j} v(\phi (\eta _{j}))\).
This complete the proof of the equivalent between the problem (11)–(12) and the integral equation (13).
Now to construct the function \(G(t,\theta )\), from the relation \(\frac{1}{1-\sigma }=1+\frac{\sigma }{1-\sigma }\), we have
Note that Lemma 3(i) guarantees that \(\int _{1}^{t} ( \log \frac{t}{\theta } )^{\gamma -1} \frac{f_{v}(\theta )}{ \theta } \,d\theta \in C^{1}[1, e]\).
Therefore \(v'\) exists for a.e. \(t \in [1, e]\) and, on differentiating (13), we obtain
and we have \(v' \in C(1, e]\) (cf. Lemma 2). Finally, we prove that \(v\in AC[1, e]\).
For \(f_{v}(t) \in L^{p}[1, e]\), we have
and we have
So v is an absolutely continuous function. Thus \(v'\) exists, for a.e. \(t \in [1,e]\). □
Lemma 5
The function \(G(t,\theta )\) defined by (14) satisfies the following properties:
-
(a)
\(G(t,\theta )\geq 0\), \(G(t, \theta ) \in C([1, e] \times [1, e])\) and \(G(1,\theta )=G(e,\theta )=0\) for \(\theta \in [1, e]\).
-
(b)
\(\max \{G(t,\theta ): (t, \theta ) \in [1, e] \times [1, e] \}=\mathbb{E}\), where \(\mathbb{E}=\frac{1}{\varGamma (\gamma )} (\frac{1}{4} )^{\gamma -1}\).
Proof
(a) It is clear that G is continuous on \([1, e] \times [1, e]\) and \(G(1,\theta )=G(e,\theta )=0\) for \(\theta \in [1, e]\).
By definition of the function G, for all \((t,\theta ) \in [1, e] \times [1, e]\), if \(\theta \leq t\), it can be written as
Furthermore, we conclude that
If \(t \leq \theta \), it is obvious that \(G(t,\theta ) \) and \(G(\phi (\eta _{j}),\theta )\geq 0\). Therefore, we can deduce that
(b) Let \(L(t,\theta ):=\frac{1}{\varGamma (\gamma )} (\log t )^{ \gamma -1} (\log \frac{e}{\theta } )^{\gamma -1}\), \(1\leq t \leq \theta \leq e\), then \(L(\cdot,\theta )\) is non-decreasing function on \([1,e]\).
Let \(K(t,\theta ):=\frac{1}{\varGamma (\gamma )} [ (\log t )^{\gamma -1} (\log \frac{e}{\theta } )^{\gamma -1} - (\log \frac{t}{\theta } )^{\gamma -1} ]\), \(1\leq \theta \leq t \leq e\). Then
which implies that \(K(\cdot,\theta )\) is non-increasing, for all \(\theta \in [1, e]\), hence, we obtain
and we have
□
Lemma 6
Suppose that (\(\mathrm{H}_{1}\)), (\(\mathrm{H}_{3}\)) hold, then we have
Proof
By applying Definition 2, Eq. (13), and Theorem 1(\(\mathrm{d}_{4}\)), (\(\mathrm{d}_{5}\)), we obtain
Hence \({}_{\mathrm{H}}D^{\delta }v(t) \in C[1,e]\) (notice that \(\int _{1}^{t} (\log \frac{t}{\theta } )^{\gamma -\delta -1} \frac{f_{v}( \theta )}{\theta } \,d\theta \in C[1,e]\)). □
Consider the Banach space
with the weighted norm \(\|v\|=\|v\|_{0}+\|{}_{\mathrm{H}}D^{\delta }v\|_{0}+\|v'\|_{1}\), where \(\|v'\|_{1}=\sup_{t \in [1,e]}|(\log t)^{2-\gamma } v'(t)|\).
For \(v \in V\), we define a nonlinear operator N by
From (\(\mathrm{H}_{4}\)), we conclude that \(N : V \rightarrow L^{p}\) is well defined. In fact
Let \(f_{v} \in L^{p}[1,e]\), \(p>\frac{1}{\gamma -1}\) for a.e. \(t \in [1,e]\), then we have the following lemma.
Lemma 7
Suppose that the assumption (\(\mathrm{H}_{1}\))–(\(\mathrm{H}_{3}\)) hold. Then the functions (13), (16) and (17) satisfy
and
where
and
Proof
Again by Hölder’s inequality and under the assumption (\(\mathrm{H}_{1}\)), for all \(t \in [1,e]\), we have
Hence
Similarly (cf. (16)), we have as before
Thus, we have
Similarly, for \(t \in [1,e]\), we obtain
Hence
□
Now, in order to prove problem (1), (4) has a positive solution, we define an integral operator T on V by the formula
We have
and
The properties of the operator T are given in the next lemma.
Lemma 8
Let (\(\mathrm{H}_{1}\))–(\(\mathrm{H}_{3}\)) hold. Then \(T: V\rightarrow V\) and T is a completely continuous operator.
Proof
Let \(v \in V\) and let \(f_{v}(t)=f (t, v(t), {}_{\mathrm{H}}D^{\delta }v(t), v'(t) )\) for a.e. \(t\in [1,e]\). Then \(f_{v}\in L^{p}[1,e]\) because f satisfies (\(\mathrm{H}_{3}\)) and \(f_{v}\) is positive. Since \(\int _{1}^{t} (\log \frac{t}{\theta } )^{\gamma -1} \frac{f_{v}(\theta )}{ \theta } \,d\theta \in C[1,e]\) and from \(G\geq 0\) by Lemma 5, it follows from the equality (24) that \(Tv \in C[1,e]\) and \(Tv \geq 0\) for \(t\in [1,e]\). Next using the equalities (25) and (26), we have \({}_{\mathrm{H}}D^{\delta }Tv\in C[1, e]\) and \(\lim_{t \rightarrow 1^{+}} (\log t )^{2-\gamma }(Tv)'(t)\) exists and is continuous.
Consequently, \(T:V \rightarrow V\).
As in the proof of Lemma 7, for all \(v \in V\) and a.e. \(t \in [1,e]\), we get
and
Thus, we see that the set \(\{Tv\}\) is uniformly bounded in \(C[1, e] \cap C^{1}(1, e]\).
In order to prove that T is a continuous operator, let \(\{v_{n}\} \subset V\) be a convergent sequence and let \(\lim_{n \rightarrow \infty }\|v_{n}-v\|=0\). Then \(v \in V\) and \(\|v_{n}\|_{0}\leq S\) for \(n \in N\), where S is a positive constant.
Since f is an \(L^{p}\)-Carathéodory function we have
By (5) and the dominated convergent theorem in \(L^{p}\)-space,
Put
then we have
Now we deduce from (24) that
and from (25), we have
Similarly, from (26) we have
Thus \(\lim_{n\rightarrow \infty } \|Tv_{n}-Tv\|=0\), which proves that T is a continuous operator.
Now, we need to prove that \(\{Tv\}\) be equicontinuous. For \(1 \leq t _{1} < t_{2} \leq e\), we have the relation (cf. (24)) in a similar way to Lemma 1
where \(d=(\gamma -1)q+1\). Similarly, it follows from (25) that
Also (cf. (26)), by putting \(h=(\gamma -\delta -1)q+1\), we get
and
As \(t_{2} \rightarrow t_{1}\), the right-hand side of the above four inequalities tends to zero. Therefore \(\{Tv\}\) is equicontinuous.
Since the set of functions \(\{Tv\}\), \(\{t^{2-\gamma }(Tv)'\}\) and \(\{{}_{\mathrm{H}}D^{\delta }Tv\}\) are bounded in \(C[1, e]\) and equicontinuous on \([1, e]\), T is relatively compact in V by the Arzelà–Ascoli theorem. Combining this fact with the continuity of T we see that T is a completely continuous operator. □
Our main result of this section is as follows.
Theorem 3
Assume that (\(\mathrm{H}_{1}\))–(\(\mathrm{H}_{4}\)) hold and let \(\sigma <1\). Suppose that the functions p, q and r satisfy
where the constants \(\mathcal{A}\), \(\mathcal{B}\) and \(\mathcal{C}\) are given by (21)–(23), respectively.
Then the multi-point boundary value problem (1), (4) has at least one positive solution.
Proof
From Lemma 4, we know that \(v \in V\) is a solution of (1), (4) if and only if
By Lemma 8, we can apply the Leray–Schauder continuation theorem to obtain the existence of a solution for (28) in V.
To do this it is suffices to verify that the set of all possible solutions of the family of problems
is, a priori, bounded in V by a constant independent of \(\lambda \in [0,1]\). Then for \(t\in [1,e]\) we have from Lemma 4
which implies that, in similar way to Lemma 7,
and
Using (\(\mathrm{H}_{4}\)), and Eqs. (18), (29) and (30), it follows that
for \(t \in [1,e]\). Thus
It follows from the assumption (27) that there is a constant \(\mathcal{D}\), independent of \(\lambda \in [0, 1]\), such that
This together with (29) and (30) implies that
Therefore,
This completes the proof of the theorem. □
4 Positive solutions for boundary value problem (1), (2)
Let \(v \in AC[1,e]\) be the solution of the multi-point problem given by (1) and (4). Then we have the following theorem.
Theorem 4
Suppose that the assumptions (\(\mathrm{H}_{3}\)) and (\(\mathrm{H}_{4}\)) are satisfied. If
and \(\phi (t):[1, e] \rightarrow [1, e]\) is a deviated and continuous differentiable function in \([1, e]\) with \(\phi '(t)>0\) or \(\phi (t)\) is a deviated and monotonically increasing function.
Then there exists a positive solution \(v\in AC[1,e]\) of the problem (1) with integral boundary condition (2) represented by
where
and
Proof
Let \(v\in AC[1,e]\) be a solution of the multi-point boundary value problem (1) and (4) given by (13).
Let \(a_{j}=\frac{(t_{j}-t_{j-1}) \phi '(\eta _{j})}{\phi (\eta _{j})}\), \(\eta _{j} \in (t_{j-1}, t_{j}) \subset (1, e)\) and \(1=t_{0}< t_{1}< t _{2}<\cdots <t_{m}=e\). Then the multi-point boundary conditions in (4) will be
From the continuity of the solution v of (1), (4), we can obtain
that is, the nonlocal condition (4) is transformed to the integral condition
The constant σ will be as \(m\rightarrow \infty \)
Also, the constants \(\mathcal{A}\), \(\mathcal{B}\), \(\mathcal{C}\) will be
Similarly
and
Now from the continuity of the solution v (cf. (13)), we have
Hence, the continuous positive solution of integral boundary problem (1), (2) is given by (32). □
5 Positive solutions for infinite-point boundary problem (1), (3)
Our second main result of this paper is presented as an existence result for problem (1), (3).
We have the following theorem.
Theorem 5
Let the assumptions (\(\mathrm{H}_{2}\))–(\(\mathrm{H}_{4}\)) and the following conditions hold:
- (\(\mathrm{H}_{5}\)):
-
\(1< \eta _{1}<\eta _{2}<\cdots<\eta _{j}< \cdots <e\), \(j=1, 2, \ldots \) and \(\sigma _{2}= \lambda \sum_{j=1}^{\infty } a_{j}( \log \phi (\eta _{j}))^{\gamma -1}<1\).
- (\(\mathrm{H}_{6}\)):
-
\({\mathcal{A}}_{2} \|p\|_{p}+ {\mathcal{B}}_{2} \|q\|_{p}+{\mathcal{C}}_{2} \|(\log t)^{\gamma -2}r\|_{p}<1\).
- (\(\mathrm{H}_{7}\)):
-
The series \(\sum_{j=1}^{\infty }a_{j} < \infty \) is convergent.
Then there exists a positive solution \(v\in AC[0,1]\) of the infinite-point boundary problem (1) with infinite-point boundary condition (3) given by the following integral equation:
where
and
Proof
Let \(v\in AC[1,e]\) be a solution of the multi-point boundary value problem (1) and (4) given by (13). We have
and
By the comparison test, we see that the three series in (3), \(\lambda \sum_{j=1}^{\infty } a_{j} (\log \phi (\eta _{j}))^{\gamma -1}\) and \(\sum_{j=1}^{\infty }a_{j} \int _{1}^{\phi (\eta _{j})} ( \log \frac{\phi (\eta _{j})}{\theta } )^{\gamma -1} \frac{f_{v}( \theta )}{\theta } \,d\theta \) are convergent. Thus, by taking the limit as \(m \rightarrow \infty \) in (13) and by applying the properties of the Riemann sum for continuous functions, we obtain
which, satisfies the differential equation (1). Furthermore, from (33) and the relation \(\frac{1}{\sigma _{2}}=1+\frac{\sigma _{2}}{1- \sigma _{2}}\), we have \(v(1)=0\) and
This proves that the integral equation (33) satisfies the problem given by (1) under infinite-point BCs (3). □
6 Application
Example 1
Let us consider the singular Hadamard-type fractional differential problem:
Here, \(\gamma =3/2\), \(\delta =1/4\), \(p=3\), \(q=3/2\), \(\phi (\xi )= \xi \), \(\lambda =\frac{1}{100}\) and
Clearly
where \(p(t)=\frac{1}{5(t-1)^{2/7}}\), \(q(t)=\frac{1}{10(t-1)^{1/4}}\), \(r(t)=\frac{(\log t)^{1/2} }{60}\), \(s(t)=\frac{1}{(t-1)^{1/7}}\).
Indeed we have \(\|p\|_{3}\approx 0.39268\), \(\|q\|_{3}\approx 0.16606\), \(\|(\log t)^{-1/2} r\|_{3}\approx 0.019962\), \(\sigma _{1} \approx 0.00667<1\), \({\mathcal{A}}_{1} \approx 1.57228\), \({\mathcal{B}} _{1} \approx 1.816696\), \({\mathcal{C}}_{1} \approx 1.6647\);
All the assumptions of Theorem 4 hold, therefore the singular Hadamard-type fractional differential problem (35), (36) has a continuous positive solution.
Example 2
Let γ, δ, p, q, λ and \(f(t,v_{1}, v_{2},v _{3})\), be as in the previous example and consider the deviated function \(\phi (\eta _{j})=\eta _{j}=e^{\frac{1}{j}} \in [1,e]\), and let \(a_{j}=\frac{1}{j^{5/2}}\).
Then the infinite-point boundary condition 3 becomes
It follows that \(\sum_{j=1}^{\infty }a_{j} \approx 1.35556\), \(\sum_{j=1}^{\infty } a_{j} (\log \phi (\eta _{j}))^{\gamma -1}=\sum_{j=1}^{\infty }\frac{1}{j^{3}}\approx 1.20205 \), \(\sigma _{2}= 0.012021<1\), \({\mathcal{A}}_{2} \approx 1.58359\), \({\mathcal{B}}_{2} \approx 1.426999\), \({\mathcal{C}}_{2} \approx 1.671622\). Then
Therefore, all the assumptions of Theorem 5 hold and the singular Hadamard-type fractional differential problem (35), (37) has a continuous positive solution.
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El-Sayed, A.M.A., Gaafar, F.M. Positive solutions of singular Hadamard-type fractional differential equations with infinite-point boundary conditions or integral boundary conditions. Adv Differ Equ 2019, 382 (2019). https://doi.org/10.1186/s13662-019-2315-x
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DOI: https://doi.org/10.1186/s13662-019-2315-x