1 Introduction

Fractional differential equations are becoming more and more popular recently in several journals and books due to their applications in a number of fields such as physics, biophysics, mechanical systems, electrical-analytical, and thermal systems [1–6]. For some recent development of this topic, see for example [7–13] and the references therein.

In 1892 [14], Hadamard presented a concept of fractional derivatives, which is different from Caputo and Riemann-Liouville type fractional derivatives and involves a logarithmic function of an arbitrary exponent in the integral kernel. It is significant that the study of Hadamard type fractional differential equations is still in its infancy and deserves further study. A detailed presentation of Hadamard fractional derivative is available in [3] and [15–23].

As was pointed out in [22], Hadamard’s construction is more appropriate for problems on half axes. In this situation, we consider the following Hadamard fractional integro-differential equations with Hadamard fractional integral boundary conditions on an infinite interval:

$$ \left \{ \textstyle\begin{array}{ll} {}^{H}D^{\gamma }u(t) +f(t,u(t),{{}^{H}I}^{q}u(t))=0, \quad 2< \gamma < 3, t\in (1,+\infty ), \\ u(1)=u^{\prime}(1)=0, \qquad {{}^{H}D^{\gamma -1}u(\infty )}=\sum_{i=1}^{m}\lambda_{i}{{}^{H}I ^{\beta_{i}}u(\eta ),} \end{array}\displaystyle \right . $$
(1.1)

where \({{}^{H}D}^{\gamma }\) denotes Hadamard fractional derivative of order γ, \(\eta \in (1,\infty )\), and \({{}^{H}I}^{(\cdot )}\) is the Hadamard fractional integral, \(q,\beta_{i}>0\) \((i=1,2,\ldots,m), \lambda _{i}\geq 0\) \((i=1,2,\ldots,m)\) are given constants and γ, η, \(\beta _{i}\), \(\lambda_{i}\) satisfy \(\Gamma (\gamma )>\sum_{i=1}^{m}\frac{ \lambda_{i}\Gamma (\gamma )}{\Gamma (\gamma +\beta_{i})} (\log \eta )^{ \gamma +\beta_{i}-1}\).

We recall that the monotone iterative technique represents a powerful tool for seeking the solution of a nonlinear problem. For more details as regards the application of this method in fractional differential equations, see [24–38] and the references therein.

We organize the rest of our manuscript as follows: In Section 2, we show some useful preliminaries and the key lemmas that are used in subsequent part of the manuscript. Then, in Section 3, the main results and proofs are provided. Section 4, exhibits two examples to illustrate our main results.

2 Preliminaries

Below, we will present some useful definitions and related lemmas.

Define

$$ E=\biggl\{ u\in C\bigl([1,\infty ),\mathbb{R}\bigr):\sup_{t\in [1,\infty )} \frac{\vert u(t) \vert }{1+( \log t)^{\gamma -1}}< \infty \biggr\} , $$
(2.1)

then E denotes a Banach space equipped with norm \(\Vert u \Vert _{E}=\sup_{t\in [1,\infty )}\frac{\vert u(t) \vert }{1+(\log t)^{\gamma -1}}\).

Denote

$$ \Omega =\Gamma (\gamma )-\sum_{i=1}^{m} \frac{\lambda_{i}\Gamma (\gamma )}{\Gamma (\gamma +\beta_{i})} (\log \eta )^{\gamma +\beta _{i}-1} ; $$
(2.2)

obviously \(\Omega >0\).

Definition 2.1

[3] For a function g, the Hadamard fractional integral of order γ has the following form:

$$ {}^{H}I^{\gamma } g(t)=\frac{1}{\Gamma (\gamma )} \int_{1}^{t}\biggl(\log \frac{t}{s} \biggr)^{\gamma -1}\frac{g(s)}{s}\,ds,\quad \gamma >0, $$

provided the integral exists.

Definition 2.2

[3] The Hadamard fractional derivative of fractional order γ for a function \(g:[1,\infty ) \rightarrow \mathbb{R} \) has the following form:

$$ {}^{H}D^{\gamma } g(t)=\frac{1}{\Gamma (n-\gamma )}\biggl(t\frac{d}{dt} \biggr)^{n} \int_{1}^{t}\biggl(\log \frac{t}{s} \biggr)^{n-\gamma -1}\frac{g(s)}{s}\,ds,\quad n-1< \gamma < n,n=[\gamma ]+1, $$

where \([\gamma ]\) means the integer part of the real number γ and \(\log (\cdot )=\log_{e}(\cdot )\).

Lemma 2.1

[3] If \(a,\gamma,\beta >0\) then

$$ {}^{H}I^{\gamma }_{a} \biggl(\log \biggl( \frac{t}{a}\biggr)^{\beta -1}\biggr) (x)=\frac{\Gamma (\beta )}{\Gamma (\beta +\gamma )}\biggl(\log \frac{x}{a}\biggr)^{\beta +\gamma -1}. $$

Lemma 2.2

[3] If \(a,\gamma,\beta >0\) then

$$ {}^{H}D^{\gamma }_{a} \biggl(\log \biggl( \frac{t}{a}\biggr)^{\beta -1}\biggr) (x)=\frac{\Gamma (\beta )}{\Gamma (\beta -\gamma )}\biggl(\log \frac{x}{a}\biggr)^{\beta -\gamma -1}. $$

Lemma 2.3

[3] Given \(\gamma >0\) and \(x\in C[1,\infty )\cap L^{1}[1,\infty )\), then the solution of the Hadamard fractional differential equation \({{}^{H}D}^{\gamma }x(t)=0\) is

$$ x(t)=\sum_{i=1}^{n}c_{i}(\log t)^{\gamma -i} $$
(2.3)

and

$$ {{}^{H}I}^{\gamma } {{}^{H}D}^{\gamma }x(t)=x(t)+\sum _{i=1}^{n}c _{i}(\log t)^{\gamma -i} $$
(2.4)

where \(c_{i}\in \mathbb{R}, i=1,2,\ldots,n\), and \(n-1<\gamma <n\).

Lemma 2.4

Let \(h\in C[1,\infty )\) with \(0<\int_{1}^{\infty }h(s)\frac{ds}{s}< \infty \), then the Hadamard fractional integral boundary value problem

$$ \left \{ \textstyle\begin{array}{ll} {{}^{H}D}^{\gamma }u(t)+h(t)=0, \quad 2< \gamma < 3, t\in (1,+\infty ), \\ u(1)=u^{\prime}(1)=0, \qquad {{}^{H}D^{\gamma -1}u(\infty )}=\sum_{i=1}^{m}\lambda_{i}{{}^{H}I ^{\beta_{i}}u(\eta ),} \end{array}\displaystyle \right . $$
(2.5)

has the unique solution

$$ u(t)= \int_{1}^{\infty }G(t,s)h(s)\frac{ds}{s}, $$
(2.6)

where

$$ G(t,s)=g(t,s)+\sum_{i=1}^{m} \frac{\lambda_{i}(\log t)^{\gamma -1}}{\Omega \Gamma (\gamma +\beta_{i})}g_{i}(\eta,s), $$
(2.7)

and

$$\begin{aligned}& g(t,s) = \frac{1}{\Gamma (\gamma )} \textstyle\begin{cases} (\log t)^{\gamma -1}-(\log (\frac{t}{s}))^{\gamma -1},&1\leq s\leq t< \infty, \\(\log t)^{\gamma -1},&1\leq t\leq s< \infty, \end{cases}\displaystyle \end{aligned}$$
(2.8)
$$\begin{aligned}& g_{i}(\eta,s) = \textstyle\begin{cases} (\log \eta )^{\gamma +\beta_{i}-1}-(\log (\frac{\eta }{s}))^{\gamma + \beta_{i}-1},&1\leq s\leq \eta < \infty, \\ (\log\eta )^{\gamma +\beta_{i}-1},&1\leq\eta \leq s< \infty. \end{cases}\displaystyle \end{aligned}$$
(2.9)

Proof

We apply the Hadamard fractional integral of order γ to

$$\begin{aligned} {{}^{H}D}^{\gamma }u(t)+h(t)=0, \end{aligned}$$

and we conclude that

$$ u(t)=c_{1}(\log t)^{\gamma -1}+c_{2}(\log t)^{\gamma -2}+c_{3}(\log t)^{ \gamma -3}-\frac{1}{\Gamma (\gamma )} \int_{1}^{t}\biggl(\log \frac{t}{s} \biggr)^{ \gamma -1}h(s)\frac{ds}{s}, $$
(2.10)

where \(c_{1},c_{2},c_{3}\in \mathbb{R}\).

Using the fact that \(u(1)=u^{\prime}(1)=0\), we conclude that \(c_{2}=c_{3}=0\). Thus,

$$ u(t)=c_{1}(\log t)^{\gamma -1}-\frac{1}{\Gamma (\gamma )} \int_{1}^{t}\biggl( \log \frac{t}{s} \biggr)^{\gamma -1}h(s)\frac{ds}{s}. $$
(2.11)

Lemma 2.2 implies that

$$ {{}^{H}D}^{\gamma -1}u(t)=c_{1}\Gamma (\gamma )- \int_{1}^{t}h(s) \frac{ds}{s}. $$
(2.12)

Thus, the condition

$$\begin{aligned} {{}^{H}D^{\gamma -1}(\infty )}=\sum_{i=1}^{m} \lambda_{i}{{}^{H}I ^{\beta_{i}}u(\eta )} \end{aligned}$$

leads to

$$ c_{1}=\frac{1}{\Omega } \Biggl( \int_{1}^{\infty} h(s)\frac{ds}{s}- \sum _{i=1}^{m}\frac{\lambda_{i}}{\Gamma (\gamma +\beta_{i})} \int_{1}^{\eta }\biggl(\log \frac{\eta }{s} \biggr)^{\gamma +\beta_{i}-1}h(s) \frac{ds}{s} \Biggr), $$
(2.13)

where Ω is defined by (2.2). Substituting \(c_{2}=c _{3}=0\) and (2.13) into (2.10), we get the unique solution of the Hadamard fractional integral boundary value problem (2.5)

$$ \begin{aligned}[b] u(t)={}& \frac{(\log t)^{\gamma -1}}{\Omega } \Biggl( \int_{1}^{\infty} h(s) \frac{ds}{s} \\ &{}-\sum _{i=1}^{m}\frac{\lambda_{i}}{\Gamma (\gamma + \beta_{i})} \int_{1}^{\eta }\biggl(\log \frac{\eta }{s} \biggr)^{\gamma +\beta_{i}-1}h(s) \frac{ds}{s} \Biggr) \\ &{}-\frac{1}{\Gamma (\gamma )} \int_{1}^{t}\biggl(\log \frac{t}{s} \biggr)^{\gamma -1}h(s) \frac{ds}{s} \\ = {}& \int_{1}^{\infty} g(t,s)h(s)\frac{ds}{s}+\sum _{i=1}^{m}\frac{ \lambda_{i}(\log t)^{\gamma -1}}{\Omega \Gamma (\gamma +\beta_{i})} \int_{1}^{\infty} g_{i}(\eta,s)h(s) \frac{ds}{s} \\ ={} & \int_{1}^{\infty} G(t,s)h(s)\frac{ds}{s}. \end{aligned} $$
(2.14)

The proof is finished. □

Lemma 2.5

The Green’s function \(G(t,s)\) defined by (2.7) has the following properties:

\((A_{1})\)::

\(G(t,s)\) is continuous and \(G(t,s)\geq 0\) for \((t,s)\in [1,\infty )\times [1,\infty )\).

\((A_{2})\)::

\(\frac{G(t,s)}{1+(\log t)^{\gamma -1}}\leq \frac{1}{ \Gamma (\gamma )}+\sum_{i=1}^{m} \frac{\lambda_{i} g_{i}( \eta,s)}{\Omega \Gamma (\gamma +\beta_{i})}\) for all \(s,t\in [1, \infty )\).

Proof

Since \((A_{1})\) it is easy to prove, we do not present it but only prove the property \((A_{2})\).

For \(\forall s,t\in [1,\infty )\),

$$\begin{aligned} \frac{G(t,s)}{1+(\log t)^{\gamma -1}} &= \frac{1}{1+(\log t)^{\gamma -1}} \Biggl[g(t,s)+\sum_{i=1}^{m} \frac{\lambda_{i} (\log t)^{ \gamma -1} g_{i}(\eta,s)}{\Omega \Gamma (\gamma +\beta_{i})} \Biggr] \\ &\leq \frac{1}{\Gamma (\gamma )}+\sum_{i=1}^{m} \frac{ \lambda_{i} (\log t)^{\gamma -1} g_{i}(\eta,s)}{\Omega \Gamma ( \gamma +\beta_{i})(1+(\log t)^{\gamma -1})} \\ &\leq \frac{1}{\Gamma (\gamma )}+\sum_{i=1}^{m} \frac{ \lambda_{i} g_{i}(\eta,s)}{\Omega \Gamma (\gamma +\beta_{i})}. \end{aligned}$$

 □

We present the following conditions for the sake of convenience:

\((C_{1})\)::

There exist two positive functions \(p(t)\) and \(q(t)\) such that

$$\begin{aligned}& \Lambda = \int_{1}^{\infty } \bigl[1+(\log t)^{\gamma -1} \bigr] \biggl[p(t)+\frac{q(t)( \log t)^{q}}{\Gamma (q)} \biggr]\frac{dt}{t}< \infty, \\& \bigl\vert f(t,u,v)-f(t,\overline{u},\overline{v}) \bigr\vert \leq p(t) \vert u-\overline{u} \vert +q(t)\vert v-\overline{v} \vert ,\quad t\in [1, \infty ), u,v, \overline{u},\overline{v}\in \mathbb{R}. \end{aligned}$$
\((C_{2})\)::
$$\begin{aligned} \lambda = \int_{1}^{\infty }\bigl\vert f(t,0,0) \bigr\vert \frac{dt}{t}< \infty. \end{aligned}$$

Lemma 2.6

If \((C_{1})\), \((C_{2})\) hold, then for any \(u\in E \)

$$ \int_{1}^{\infty }\bigl\vert f(t,u(t),{{}^{H}I}^{q}u(t) \bigr\vert \frac{dt}{t}\leq \Lambda \Vert u \Vert _{E}+ \lambda. $$
(2.15)

Proof

For any \(u\in E\), taking \(\overline{u}=0\), then \({{}^{H}I}^{q} \overline{u}=0\). Thus, by condition \((C_{1})\) we have

$$\begin{aligned} \bigl\vert f\bigl(t,u(t),{{}^{H}I}^{q}u(t) \bigr) \bigr\vert \leq{} & p(t)\bigl\vert u(t) \bigr\vert +q(t)\bigl\vert {{}^{H}I}^{q}u(t) \bigr\vert +\bigl\vert f(t,0,0) \bigr\vert \\ \leq {}& p(t)\bigl[1+(\log t)^{\gamma -1}\bigr]\frac{\vert u(t) \vert }{1+(\log t)^{\gamma -1}} \\ &{}+q(t)\frac{1}{\Gamma (q)}\int_{1}^{t}\biggl(\log \frac{t}{s}\biggr)^{q-1} \frac{\vert u(s) \vert }{s}\,ds+\bigl\vert f(t,0,0) \bigr\vert \\ \leq {}&p(t)\bigl[1+(\log t)^{\gamma -1}\bigr]\Vert u \Vert _{E} \\ &+q(t)\frac{1+(\log t)^{\gamma -1}}{\Gamma (q)} \\ &{}\times\int_{1}^{t}\frac{( \log \frac{t}{s})^{q-1}}{s}\frac{\vert u(s) \vert }{1+(\log s)^{\gamma -1}} \frac{1+( \log s)^{\gamma -1}}{1+(\log t)^{\gamma -1}}\,ds \\ &{}+\bigl\vert f(t,0,0) \bigr\vert \\ \leq{} & p(t)\bigl[1+(\log t)^{\gamma -1}\bigr]\Vert u \Vert _{E} \\ &{}+q(t)\frac{1+(\log t)^{\gamma -1}}{\Gamma (q)}\Vert u \Vert _{E} \int_{1}^{t}\frac{( \log t)^{q-1}}{s}\,ds+\bigl\vert f(t,0,0) \bigr\vert \\ \leq {}& p(t)\bigl[1+(\log t)^{\gamma -1}\bigr]\Vert u \Vert _{E}+q(t)\frac{1+(\log t)^{ \gamma -1}}{\Gamma (q)}(\log t)^{q}\Vert u \Vert _{E} \\ &{}+\bigl\vert f(t,0,0) \bigr\vert , \end{aligned}$$
(2.16)

from which, combined with \((C_{1})\) and \((C_{2})\), we can obtain

$$\begin{aligned} \int_{1}^{\infty }\bigl\vert f\bigl(t,u(t),{{}^{H}I}^{q}u(t) \bigr) \bigr\vert \frac{dt}{t} \leq{} & \int _{1}^{\infty }p(t)\bigl[1+(\log t)^{\gamma -1} \bigr]\Vert u \Vert _{E}\frac{dt}{t} \\ &{}+ \int_{1}^{\infty }q(t)\frac{1+(\log t)^{\gamma -1}}{\Gamma (q)}( \log t)^{q}\Vert u \Vert _{E}\frac{dt}{t} \\ &{}+ \int_{1}^{\infty }\bigl\vert f(t,0,0) \bigr\vert \frac{dt}{t} \\ = {}&\Lambda \Vert u \Vert _{E}+\lambda. \end{aligned}$$
(2.17)

The proof is done. □

3 Main results

Theorem 3.1

Suppose that the conditions \((C_{1})\) and \((C_{2})\) hold. Let

$$ w=\Lambda \Biggl(\frac{1}{\Gamma (\gamma )}+\sum _{i=1}^{m} \frac{ \lambda_{i} g_{i}(\eta,s)}{\Omega \Gamma (\gamma +\beta_{i})} \Biggr)< 1. $$
(3.1)

Then the Hadamard fractional integral boundary value problem (1.1) admits an unique solution \(\widetilde{u}(t)\) in E. In addition, there exists a monotone iterative sequence \({u_{n}(t)}\) such that \(u_{n}(t)\rightarrow \widetilde{u}(t)\) \((n\rightarrow \infty )\) uniformly on any finite sub-interval of \([1,\infty )\), where

$$ u_{n}(t)= \int_{1}^{\infty }G(t,s)f\bigl(s,u_{n-1}(s),{{}^{H}I}^{q}u_{n-1}(s) \bigr) \frac{ds}{s}. $$
(3.2)

Furthermore, there exists an error estimate for the approximating sequence

$$ \Vert u_{n}-\widetilde{u} \Vert _{E}\leq \frac{w^{n}}{1-w}\Vert u_{1}-u_{0} \Vert _{E} \quad (n=1,2,\ldots). $$
(3.3)

Proof

Define the operator T by

$$ (Tu) (t)= \int_{1}^{\infty }G(t,s)f\bigl(s,u(s),{{}^{H}I}^{q}u(s) \bigr)\frac{ds}{s}. $$
(3.4)

By Lemma 2.4, the Hadamard fractional integral boundary value problem (1.1) possesses a solution u iff u is a solution of \(u=Tu\).

First, for any \(t\in [1,\infty )\), by Lemma 2.5 and Lemma 2.6, we have

$$\begin{aligned} \frac{\vert (Tu)(t) \vert }{1+(\log t)^{\gamma -1}}&\leq \int_{1}^{\infty }\frac{G(t,s)}{1+( \log t)^{\gamma -1}} \bigl\vert f \bigl(s,u(s),{{}^{H}I}^{q}u(s)\bigr) \bigr\vert \frac{ds}{s} \\ &\leq \Biggl(\frac{1}{\Gamma (\gamma )}+\sum_{i=1}^{m} \frac{ \lambda_{i} g_{i}(\eta,s)}{\Omega \Gamma (\gamma +\beta_{i})} \Biggr) \bigl[\Lambda \Vert u \Vert _{E}+ \lambda \bigr] \\ &= w\Vert u \Vert _{E}+k. \end{aligned}$$
(3.5)

This means

$$ \Vert Tu \Vert _{E}\leq w\Vert u \Vert _{E}+k,\quad \forall t \in [1,\infty ), $$
(3.6)

where w is defined in (3.1) and

$$ k=\lambda \Biggl(\frac{1}{\Gamma (\gamma )}+\sum_{i=1}^{m} \frac{ \lambda_{i} g_{i}(\eta,s)}{\Omega \Gamma (\gamma +\beta_{i})} \Biggr). $$
(3.7)

In addition, for any \(u, \overline{u}\in E\), we have

$$\begin{aligned} \frac{\vert (Tu)(t)-(T\overline{u})(t) \vert }{1+(\log t)^{\gamma -1}} \leq{} & \int _{1}^{\infty }\frac{G(t,s)}{1+(\log t)^{\gamma -1}} \bigl\vert f(s,u(s),{{}^{H}I}^{q}u(s)-f\bigl(s,\overline{u}(s),{{}^{H}I}^{q}\overline{u}(s)\bigr) \bigr\vert \frac{ds}{s} \\ \leq{} &\int_{1}^{\infty }\frac{G(t,s)}{1+(\log t)^{\gamma -1}} \bigl[p(s)\bigl\vert u(s)-\overline{u}(s)\bigl\vert \mathop{+} q(s) \bigr\vert {{}^{H}I}^{q}u(s)-{{}^{H}I}^{q}\overline{u}(s) \bigr\vert \bigr]\frac{ds}{s} \\ \leq{} & \int_{1}^{\infty }\frac{G(t,s)}{1+(\log t)^{\gamma -1}}p(s)\bigl[1+( \log s)^{\gamma -1}\bigr]\frac{\vert u(s)-\overline{u}(s) \vert }{1+(\log s)^{\gamma -1}}\,ds \\ &{}+ \int_{1}^{\infty }\frac{G(t,s)}{1+(\log t)^{\gamma -1}}q(s)\bigl\vert {{}^{H}I}^{q}u(s)-{{}^{H}I}^{q} \overline{u}(s) \bigr\vert \frac{ds}{s} \\ \leq{} & \int_{1}^{\infty }\frac{G(t,s)}{1+(\log t)^{\gamma -1}}p(s)\bigl[1+( \log s)^{\gamma -1}\bigr]\Vert u-\overline{u} \Vert _{E}\,ds \\ &{}+\int_{1}^{\infty }\frac{G(t,s)}{1+(\log t)^{\gamma -1}}q(s)\frac{[1+(\log s)^{\gamma -1}](\log s)^{\gamma }}{\Gamma (\gamma )}\Vert u-\overline{u} \Vert _{E}\frac{ds}{s} \\ \leq {}& \Biggl(\frac{1}{\Gamma (\gamma )}+\sum_{i=1}^{m} \frac{\lambda_{i}g_{i}(\eta,s)}{\Omega \Gamma (\gamma +\beta_{i})} \Biggr) \\ &{}\times\int_{1}^{\infty }\Vert u-\overline{u} \Vert _{E}\bigl[1+(\log s)^{\gamma -1}\bigr] \biggl[p(s)+ \frac{q(s)( \log s)^{\gamma }}{\Gamma (\gamma )}\biggr]\frac{ds}{s} \\ \leq{} & \Biggl(\frac{1}{\Gamma (\gamma )}+\sum_{i=1}^{m} \frac{ \lambda_{i} g_{i}(\eta,s)}{\Omega \Gamma (\gamma +\beta_{i})} \Biggr) \Lambda \Vert u-\overline{u} \Vert _{E} \\ = {}& w\Vert u-\overline{u} \Vert _{E}. \end{aligned}$$
(3.8)

Then we get

$$ \Vert Tu-T\overline{u} \Vert _{E}\leq w\Vert u- \overline{u} \Vert _{E},\quad\forall u,\overline{u}\in E. $$
(3.9)

Through the Banach fixed point theorem, we can ensure that T has a unique fixed point ũ in E. That is, (1.1) admits a unique solution ũ in E. In addition, for any \(u_{0}\in E\), \(\Vert u_{n}-\widetilde{u} \Vert _{E}\rightarrow 0\) as \(n\rightarrow \infty \), where \(u_{n}=Tu_{n-1}\) \((n=1,2,\ldots)\).

From (3.9), we have

$$ \Vert u_{n}-u_{n-1} \Vert _{E}\leq w^{n-1}\Vert u_{1}-u_{0} \Vert _{E} $$
(3.10)

and

$$\begin{aligned} \Vert u_{n}-u_{j} \Vert _{E} &\leq \Vert u_{n}-u_{n-1} \Vert _{E}+\Vert u_{n-1}-u_{n-2} \Vert _{E}+\cdots+\Vert u_{j+1}-u_{j} \Vert _{E} \\ &\leq \frac{w^{n}(1-w^{n-j})}{1-w}\Vert u_{1}-u_{0} \Vert _{E}. \end{aligned}$$
(3.11)

Letting \(n\rightarrow \infty \) on both sides of (3.11), we conclude that

$$ \Vert u_{n}-\widetilde{u} \Vert _{E}\leq \frac{w^{n}}{1-w}\Vert u_{1}-u_{0} \Vert _{E}. $$
(3.12)

 □

4 Example

Example 4.1

In the following we discuss the Hadamard fractional integral boundary value problem

$$ \textstyle\begin{cases} {}^{H}D^{\frac{5}{2}} u(t) +\frac{e^{-3t}t}{1+(\log t)^{\frac{3}{2}}}\cos (3t^{2} \\ \quad +u(t))+\frac{3\sqrt{\pi }e^{-3t}t}{8[1+(\log t)^{\frac{3}{2}}](\log t)^{\frac{5}{2}}}\arctan ({}^{H}I^{\frac{5}{2}}u(t))=0,\\ u(1)=u^{\prime}(1)=0, \qquad {}^{H}D^{\frac{3}{2}}u(+\infty )=\lambda_{1}{{}^{H}I^{\beta_{1}}u(\eta ),} \end{cases} $$
(4.1)

where \(\gamma =\frac{5}{2}\), \(m=1\), \(q=\frac{5}{2}\), and \(\lambda_{1}\), \(\beta_{1}\), η satisfy (\(\lambda_{1}\geq 0, \beta_{1}>0, \eta >1\))

$$ 0< \frac{\Gamma (\frac{1}{2}+\beta_{1})}{3\Gamma (\frac{1}{2}+\beta _{1})-4\lambda_{1}(\log \eta )^{\frac{3}{2}+\beta_{1}}}< \frac{e^{3}\sqrt{ \pi }}{2} $$
(4.2)

(see Figure 1).

Figure 1
figure 1

Set of \(\pmb{\lambda_{1}\geq 0,\beta_{1}>0,\eta >1}\) such that \(\pmb{0< \frac{\Gamma (\frac{1}{2}+\beta_{1})}{3\Gamma (\frac{1}{2}+\beta _{1})-4\lambda_{1}(\log \eta )^{\frac{3}{2}+\beta_{1}}}<\frac{e^{3}\sqrt{ \pi }}{2}}\) .

Full size image

For example, we can take \(\lambda_{1}=\frac{1}{10},\beta_{1}= \frac{3}{2},\eta =\frac{5}{2}\),

$$\begin{aligned}& \bigl\vert f\bigl(t,u,{{}^{H}I}^{\frac{5}{2}}u(t) \bigr)-f\bigl(t,\overline{u},{{}^{H}I}^{\frac{5}{2}}\overline{u}(t)\bigr) \bigr\vert \\& \quad \leq \frac{e^{-3t}t}{1+(\log t)^{ \frac{3}{2}}}\bigl\vert \cos \bigl(3t^{2}+u(t) \bigr)-\cos \bigl(3t^{2}+\overline{u}(t)\bigr) \bigr\vert \\& \qquad {}+\frac{3\sqrt{\pi }e^{-3t}t}{8[1+(\log t)^{\frac{3}{2}}](\log t)^{ \frac{5}{2}}}\bigl\vert \arctan \bigl({}^{H}I^{\frac{5}{2}}u(t) \bigr)-\arctan \bigl({{}^{H}I}^{\frac{5}{2}}\overline{u}(t)\bigr) \bigr\vert \\& \quad \leq \frac{e^{-3t}t}{1+(\log t)^{\frac{3}{2}}}\bigl\vert u(t)-\overline{u}(t) \bigr\vert \\& \qquad {}+\frac{3\sqrt{\pi }e^{-3t}t}{8[1+(\log t)^{\frac{3}{2}}](\log t)^{ \frac{5}{2}}}\bigl\vert {}^{H}I^{\frac{5}{2}}u(t)-{{}^{H}I}^{\frac{5}{2}} \overline{u}(t) \bigr\vert . \end{aligned}$$

Since \(p(t)=\frac{e^{-3t}t}{1+(\log t)^{\frac{3}{2}}}\) and \(q(t)=\frac{3\sqrt{\pi }e^{-3t}t}{8[1+(\log t)^{\frac{3}{2}}]( \log t)^{\frac{5}{2}}}\), we can show that

$$\begin{aligned}& \Lambda = \int_{1}^{\infty }\bigl[1+(\log t)^{\frac{3}{2}}\bigr] \biggl[\frac{e^{-3t}t}{1+( \log t)^{\frac{3}{2}}}+\frac{3\sqrt{\pi }e^{-3t}t(\log t)^{ \frac{5}{2}}}{8[1+(\log t)^{\frac{3}{2}}](\log t)^{\frac{5}{2}}\Gamma (\frac{5}{2})}\biggr]\frac{dt}{t} \\& \hphantom{\Lambda}= \frac{1}{2e^{3}}< \infty, \\& \lambda = \int_{1}^{\infty }\bigl\vert f(t,0,0) \bigr\vert \frac{dt}{t}= \int_{1}^{\infty }e^{-3t}\,dt=\frac{1}{3e^{3}}< \infty. \end{aligned}$$

Then \((C_{1})\) and \((C_{2})\) hold. At last, by a simple computation, we have

$$\begin{aligned}& \Omega =\Gamma (\gamma )-\frac{\lambda_{1}\Gamma (\gamma )}{\Gamma (\gamma +\beta_{1})}(\log \eta )^{\gamma +\beta_{1}-1}= \frac{3\sqrt{ \pi }}{4}-\frac{\sqrt{\pi }\lambda_{1}(\log \eta )^{\frac{3}{2}+ \beta_{1}}}{\Gamma (\frac{1}{2}+\beta_{1})}>\frac{1}{2e^{3}}>0, \\& w=\Lambda \biggl(\frac{1}{\Gamma (\gamma )}+\frac{\lambda_{1} g_{1}( \eta,s)}{\Omega \Gamma (\gamma +\beta_{i})} \biggr)= \frac{2}{e^{3}\sqrt{ \pi }} \biggl(\frac{\Gamma (\frac{1}{2}+\beta_{1})}{3\Gamma (\frac{1}{2}+ \beta_{1})-4\lambda_{1}(\log \eta )^{\frac{3}{2}+\beta_{1}}} \biggr)< 1. \end{aligned}$$

As a result, the conditions of Theorem 3.1 hold. Thus, the conclusion of Theorem 3.1 implies that (4.1) possesses a unique solution.

Example 4.2

Let us discuss the following Hadamard fractional integral boundary value problem:

$$ \left \{ \textstyle\begin{array}{ll} {{}^{H}D}^{\frac{9}{4}} u(t) +f(t,u(t), {{}^{H}I}^{\frac{9}{2}}u(t))=0, \\ u(1)=u^{\prime} (1)=0, \qquad {{}^{H}D}^{\frac{9}{4}}u(+\infty )=\sum_{i=1}^{3}\lambda_{i} {{{}^{H}I}^{\beta_{i}}u(\eta )}, \end{array}\displaystyle \right . $$
(4.3)

here

$$\begin{aligned} f\bigl(t,u(t),{}^{H}I^{\frac{9}{2}}u(t)\bigr)=\frac{\sin (t^{2}+u(t))}{16(1+t^{2})[1+( \log t)^{\frac{5}{4}}]}+ \frac{{}^{H}I^{\frac{9}{2}}u(t)-\sin ({}^{H}I^{ \frac{9}{2}}u(t))\cos ({}^{H}I^{\frac{9}{2}}u(t))}{128(1+t^{2})[1+( \log t)^{\frac{5}{4}}](\log t)^{\frac{9}{2}}}. \end{aligned}$$

Take \(\gamma =\frac{9}{4}\), \(m=3\), \(q=\frac{9}{2}\), \(\eta = \frac{9}{4}\), \(\lambda_{1}=\frac{1}{10}\), \(\beta_{1}=\frac{3}{4}\), \(\lambda_{2}=\frac{\sqrt{\pi }}{2}\), \(\beta_{2}=\frac{7}{4}\), \(\lambda_{3}=4 \), and \(\beta_{3}=\frac{11}{4}\).

The function f satisfies the inequality

$$\begin{aligned}& \bigl\vert f\bigl(t,u,{}^{H}I^{\frac{9}{2}}u(t) \bigr)-f\bigl(t,\overline{u},{{}^{H}I}^{\frac{9}{2}}\overline{u}(t)\bigr) \bigr\vert \\& \quad \leq \frac{1}{16(1+t^{2})[1+(\log t)^{\frac{5}{4}}]}\bigl\vert u(t)-\overline{u}(t) \bigr\vert \\& \qquad {}+ \frac{1}{64(1+t^{2})[1+(\log t)^{\frac{5}{4}}]( \log t)^{\frac{9}{2}}}\bigl\vert {}^{H}I^{\frac{9}{2}}u(t)-{{}^{H}I}^{\frac{9}{2}} \overline{u}(t) \bigr\vert . \end{aligned}$$

Since \(p(t)=\frac{1}{16(1+t^{2})[1+(\log t)^{\frac{5}{4}}]}\) and \(q(t)=\frac{1}{64(1+t^{2})[1+(\log t)^{\frac{5}{4}}](\log t)^{ \frac{9}{2}}}\), we can show that

$$\begin{aligned}& \Lambda =\int_{1}^{\infty }\bigl[1+(\log t)^{\frac{5}{4}}\bigr] \\& \qquad {}\times\biggl[\frac{1}{16(1+t^{2})[1+(\log t)^{\frac{5}{4}}]}+\frac{(\log t)^{\frac{9}{2}}}{64 \Gamma (\frac{9}{2})(1+t^{2})[1+(\log t)^{\frac{5}{4}}](\log t)^{ \frac{9}{2}}}\biggr]\frac{dt}{t} \\& \hphantom{\Lambda{}}< \frac{\pi }{32}< \infty, \\& \lambda = \int_{1}^{\infty }\bigl\vert f(t,0,0) \bigr\vert \frac{dt}{t}< \int_{1}^{\infty }\frac{1}{4(1+t^{2})}\,dt= \frac{\pi }{64}< \infty. \end{aligned}$$

Then \((C_{1})\) and \((C_{2})\) hold. At last, by a simple computation, we have

$$\begin{aligned}& \Omega =\Gamma \biggl(\frac{9}{4}\biggr)-\sum _{i=1}^{3}\frac{\lambda_{i} \Gamma (\frac{9}{4})}{\Gamma (\frac{9}{4}+\beta_{i})}(\log \eta )^{ \frac{9}{4}+\beta_{i}-1}\approx 0.8562>0, \\& w=\Lambda \Biggl(\frac{1}{\Gamma (\frac{9}{4})}+\sum_{i=1}^{3} \frac{ \lambda_{i} g_{i}(\eta,s)}{\Omega \Gamma (\frac{9}{4}+\beta_{i})}\Biggr) \approx 0.0452< 1. \end{aligned}$$

Thus, by the application of Theorem 3.1 the Hadamard fractional integral boundary value problem (4.3) admits an unique solution.