Existence of Positive Solutions for Hadamard-Type Fractional Boundary Value Problems at Resonance on an Infinite Interval

Abstract

This paper investigates a class of resonance boundary value problems for Hadamard-type fractional differential equations on an infinite interval. Utilizing the Leggett-Williams norm-type theorem proposed by O’Regan and Zima, the existence of positive solutions is established. The main conclusions are illustrated with an example.

1 Introduction

The study of boundary value problems (BVPs) for differential equations on infinite intervals originated from the discussion of radially symmetric solutions for nonlinear elliptic equations.

For example, when analyzing the radially symmetric solutions of the following elliptic equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta \varpi = G(||\Im ||,\varpi ),\;\;\;||\Im || \ge \mathfrak {c} \ge 0,\;\;\;\varpi \in {\mathbb {R}^N}, \\ \ell (\varpi , - \frac{{\partial \varpi }}{{\partial n}}) = 0,\;\;\;||\Im || = \mathfrak {c}, \\ \end{array}\right. } \end{aligned}$$

(1)

(1) can be equivalently transformed into the following BVP for second-order differential equations defined on an infinite interval:

$$\begin{aligned} {\left\{ \begin{array}{ll} \varpi '' + \frac{{N - 1}}{\Im }\varpi ' = G(\Im ,\varpi ),\;\;\;\Im \ge \mathfrak {c}, \\ \ell (\varpi (\mathfrak {c}),\varpi '(\mathfrak {c})) = 0,\;\;\;\varpi (\Im )\;\text {is bounded on}\;[0, + \infty ), \\ \end{array}\right. } \end{aligned}$$

(2)

where $\Im$ represents the radial coordinate [1].

BVPs for differential equations on infinite intervals have a widespread use in real world problems, such as solid phase transition problems, plasma physics research, colloid theory, nonlinear mechanics, laminar flow theory, and non-Newtonian fluid problems [1]. Due to the practical application background of BVPs for differential equations on infinite intervals, researchers have conducted in-depth studies on the existence of solutions of fractional differential equations (FDEs) on infinite intervals and obtained a series of interesting results [2,3,4,5,6,7,8]. Alongside Riemann-Liouville and Caputo derivatives in scholarly literature, another variant of fractional derivatives is the Hadamard fractional derivative, first introduced in 1892 [9]. This derivative is characterized by the inclusion of a logarithmic function with an arbitrary exponent within the kernel of the integral in its definition.

Ioakimidis [10] highlighted the potential application of singular integrals in solving crack problems in elastic materials, which has sparked interest among scholars in Hadamard fractional calculus [11,12,13]. There are at least two distinctions between Hadamard-type fractional calculus and Riemann-Liouville fractional calculus. Firstly, regarding their definitions: the kernel of the integral in the Hadamard derivative is the power of $(\ln ({t / s}))$, whereas in the Riemann-Liouville derivative, the kernel is the power of $t-s$. Additionally, the Hadamard derivative is viewed as a generalization of the operator ${\left( {t\frac{d}{{dt}}} \right) ^n}$, while the Riemann-Liouville derivative is considered an extension of the classical operator ${\left( {\frac{d}{{dt}}} \right) ^n}$. Secondly, in terms of application: due to the logarithmic kernel of Hadamard calculus, the decay rate of solutions to Hadamard fractional differential equations is slower than that of Riemann-Liouville fractional differential equations, making them more suitable for describing ultra slow kinetics processes. For instance, Garra et al. [14] utilized Hadamard fractional calculus to study the Lomnitz logarithmic creep law of igneous rocks. The authors pointed out that although the mathematical model described by Hadamard fractional calculus appears complex, it is indeed effective in characterizing creep behavior.

Given the particularity of Hadamard fractional calculus, extensive research has been conducted in recent years on the existence of solutions to Hadamard fractional BVPs. Notably, there has been a growing interest among researchers in exploring the existence of solutions to Hadamard fractional BVPs on infinite intervals [15,16,17,18,19,20,21,22,23,24].

In [22], Wang et al employed the monotone iterative method to investigate the existence of positive solutions for the nonlinear Hadamard FDE subject to nonlocal Hadamard integral and discrete boundary conditions on an infinite interval:

$$\begin{aligned} {\left\{ \begin{array}{ll} {}^H{D^\nu }\Im (\varrho ) + \sigma (\varrho )\hbar (\varrho ,\Im (\varrho )) = 0,\;\;\;\varrho \in (1, + \infty ), \\ \Im (1) = \Im '(1) = 0,\;\;{}^H{D^{\nu - 1}}\Im (\infty ) = c{}^H{I^\delta }\Im (\xi ) + d\sum \limits _{i = 1}^{j - 2} {{\gamma _i}\Im ({\eta _i}),} \\ \end{array}\right. }\ \end{aligned}$$

(3)

where ${}^H{D^\nu }$ is Hadamard fractional derivative, $2 < \nu \le 3$, $1< \xi< {\eta _1}< {\eta _2}< \cdots< {\eta _{j - 2}} < + \infty .$

In [23], Deren and Cerdik used the monotone iterative method to explore the existence of positive solutions for the nonlinear Hadamard fractional differential systems supplemented with multipoint boundary conditions on an infinite interval:

$$\begin{aligned} {\left\{ \begin{array}{ll} {}^HD_{1{+ }}^\kappa \Im (\varrho ) + {\xi _1}(\varrho )\hbar \big (\varrho ,\varpi (\varrho ),{}^HD_{1{+ }}^{\epsilon - 1}\varpi (\varrho )\big ) = 0,\;\;\;\varsigma - 1< \kappa \le \varsigma ,\;\;\varrho \in (1, + \infty ), \\ {}^HD_{1{+ }}^\epsilon \varpi (\varrho ) + {\xi _2}(\varrho )\ell \big (\varrho ,\Im (\varrho ),{}^HD_{1{+ }}^{\kappa - 1}\Im (\varrho )\big ) = 0,\;\;\;\zeta - 1 < \epsilon \le \zeta ,\;\;\varrho \in (1, + \infty ), \\ \Im (1) = \Im '(1) = \cdots = {\Im ^{(\varsigma - 2)}}(1) = 0,\;\;{}^HD_{1{+ }}^{\kappa - 1}\Im (\infty ) = \sum \nolimits _{i = 1}^{{k_1}} {{c_i}{}^HD_{1{+ }}^{{\tau _1}}\Im ({\varsigma _i}),} \\ \varpi (1) = \varpi '(1) = \cdots = {\varpi ^{(\zeta - 2)}}(1) = 0,\;\;{}^HD_{1{+ }}^{\epsilon - 1}\varpi (\infty ) = \sum \nolimits _{j = 1}^{{k_2}} {{d_j}{}^HD_{1{+ }}^{{\tau _2}}\varpi ({\zeta _j}),} \\ \end{array}\right. }\ \end{aligned}$$

(4)

where $\varsigma ,\zeta \in \mathbb {N},~\varsigma ,\zeta \ge 3,$ ${}^HD_{1+}^\vartheta$ are Hadamard fractional derivatives of order $\vartheta \in \{ \kappa ,\epsilon ,{\tau _1},{\tau _2}\}$, ${\tau _1} \in [0,\kappa - 1],~{\tau _2} \in [0,\epsilon - 1],~{c_i} \ge 0\;(i = 1,2, \cdots , {k_1}),~{d_j} \ge 0\;(j = 1,2, \cdots , {k_2}),~1< {\varsigma _1}< {\varsigma _2}< \cdots< {\varsigma _{{k_1}}} < + \infty ,$ and $1< {\zeta _1}< {\zeta _2}< \cdots< {\zeta _{{k_2}}} < + \infty .$

Note that the existing literature on BVPs of Hadamard FDEs on infinite intervals mainly focuses on the non-resonant case [15,16,17,18,19,20,21,22,23]. In [24], Zhang and Liu proposed to study the resonant BVP of Hadamard FDEs, and used Mawhin’s continuation theorem to explore the existence of solutions for the following Hadamard FDE with integral boundary conditions at resonance on an infinite interval:

$$\begin{aligned} {\left\{ \begin{array}{ll} {}^HD_{1 + }^\varpi \Im (\varrho ) + a(\varrho )\hbar \big (\varrho ,\Im (\varrho ),{}^HD_{1 + }^{\varpi - 2}\Im (\varrho ),{}^HD_{1 + }^{\varpi - 1}\Im (\varrho )\big ) = 0,\;\;\;\varrho \in (1, + \infty ), \\ \Im (1) = \Im '(1) = 0,\;\;\;{}^HD_{1 + }^{\varpi - 1}\Im ( + \infty ) = \int _1^{ + \infty } {\ell (\varrho ){}^HD_{1 + }^{\varrho - 1}\Im (\varrho )\frac{{d\varrho }}{\varrho },} \\ \end{array}\right. } \end{aligned}$$

(5)

where $2<\varpi \le 3$, ${}^HD_{1 + }^\varpi$ is Hadamard fractional derivative, $\ell (\varrho ) \ge 0$ and $\big (1/a(\varrho )\big ) > 0$ on $[1, + \infty )$, $\hbar :[1, + \infty ) \times {\mathbb {R}^3} \rightarrow \mathbb {R}$ satisfies $a-$Carathéodory condition.

Inspired by the mentioned articles, this paper employs the Leggett-Williams norm-type theorem to discuss the existence of positive solutions for Hadamard fractional BVP at resonance on an infinite interval as follows:

$$\begin{aligned}&\begin{aligned} \left\{ \begin{array}{l} {}^HD_{1 + }^\varpi \Im (\varrho ){\text { = }}\hbar \big (\varrho ,\Im (\varrho )\big ),\;\;\;\;\;\;\;\varrho \in (1, + \infty ), \\ \Im (1) = \Im '(1) = \Im ''(1) = 0,\;\;\;{}^HD_{1 + }^{\varpi - 1}\Im (1) =\mathop {\lim }\limits _{\varrho \rightarrow + \infty } {}^HD_{1 + }^{\varpi - 1}\Im (\varrho ), \\ \end{array} \right. \end{aligned}\ \end{aligned}$$

(6)

where ${}^HD_{1 + }^\varpi$ is Hadamard fractional derivative, $3< \varpi < 4,$ and the function $\hbar :[1, + \infty ) \times \mathbb {R} \rightarrow \mathbb {R}$ satisfies the following condition:

• (H) The function $\hbar :[1, + \infty ) \times \mathbb {R} \rightarrow \mathbb {R}$ is continuous, and for each constant $\jmath > 1,$ there exists a nonnegative function ${\phi _\jmath } \in C[1, + \infty )$ satisfy the conditions $\mathop {\sup }\limits _{\varrho \ge 1} |{\phi _\jmath }(\varrho )| < + \infty$ and $\displaystyle \int _1^{+ \infty } {{\phi _\jmath }} (\varrho )\frac{{d\varrho }}{\varrho } < + \infty$, such that

  $$\begin{aligned} |\Im | < \jmath \Rightarrow \big |\hbar \big (\varrho ,(1 + {(\ln \varrho )^{\varpi - 1}})\Im \big )\big | \le {\phi _\jmath }(\varrho ),\;\;\;\;a.e.\;\; \varrho \ge 1. \end{aligned}$$

The main challenges and innovations of this paper can be summarized as follows:

• This paper studies the fractional BVP on an infinite interval. Owing to the non-compactness of the infinite domain, the classical Arzelá-Ascoli theorem cannot be directly applied to determine the compactness of the corresponding operator, which brings direct difficulties to the study of problem (6).

• For the resonance BVPs of ordinary differential equations, a common research method is to use the continuation theorem. However, the integration on the infinite interval requires the convergence of the improper integral, which brings additional difficulties to the construction of the projection operator in the process of using the continuation theorem to deal with the resonance BVPs on the infinite intervals. In particular, the Hadamard fractional integral kernel function is a logarithmic function, which further increases the complexity of the problem.

• There are few literature on the resonance BVPs of Hadamard FDEs on the infinite interval, especially on the existence of positive solutions for this problem. We have not found any related research work. In this paper, we use the Leggett-Williams norm-type theorem to give a adequate condition for the problem (6) exists a positive solutions. Therefore, our results are new and contribute significantly to the existing literature on the topic.

The structure of this paper is organized as follows. In Sect. 2, we introduce the fundamental definitions and properties of Hadamard fractional calculus, along with the Leggett-Williams norm-type theorem and criteria for operator compactness on infinite intervals. In Sect. 3, we apply the Leggett-Williams norm-type theorem to demonstrate the existence of positive solutions for problem (6). In Sect. 4, we validate the applicability of our theoretical results by presenting a specific example.

2 Preliminaries

In this part, we present a number of definitions, lemmas of Hadamard fractional calculus, the Leggett-Williams norm-type theorem and the lemma of compactness determination on infinite interval, all of which will be used in our forthcoming discussions.

Definition 2.1

[13, 25] The Hadamard fractional integral of order $\varpi (\varpi > 0)$ of a function $\Im :[1, + \infty ) \rightarrow \mathbb {R}$ is given by

$$\begin{aligned} {}^HI_{1 + }^\varpi \Im (\varrho ) = \frac{1}{{\Gamma (\varpi )}}\int _1^\varrho {\Big (\ln \frac{\varrho }{\varsigma }} {\Big )^{\varpi - 1}}\Im (\varsigma )\frac{{d\varsigma }}{\varsigma }, \end{aligned}$$

provided the integral exists.

Definition 2.2

[13, 25] The Hadamard fractional derivative of order $\varpi (\varpi > 0)$ of a function $\Im :[1, + \infty ) \rightarrow \mathbb {R}$ is given by

$$\begin{aligned} {}^HD_{1 + }^\varpi \Im (\varrho ) = \frac{1}{{\Gamma (j - \varpi )}}{\Big (\varrho \frac{d}{{d\varrho }}\Big )^\mathfrak {n}}\int _1^\varrho {\Big (\ln \frac{\varrho }{\varsigma } \Big )^{\mathfrak {n} - \varpi - 1}}\Im (\varsigma )\frac{{d\varsigma }}{\varsigma }, \end{aligned}$$

where $\mathfrak {n}-1<\varpi <\mathfrak {n}$, $\mathfrak {n} = [\varpi ] + 1$, $[\varpi ]$ denotes the integer part of the real number $\varpi$.

Lemma 2.1

[13, 25] Let $\varpi > 0$ and $\Im \in C[1, + \infty ) \cap {L^1}[1, + \infty )$. Then the solution of Hadamard fractional differential equation ${}^HD_{1 + }^\varpi \Im (\varrho ) =0$ is given by

$$\begin{aligned} \Im (\varrho )= \sum \limits _{i = 1}^\mathfrak {n}{{c_i}} {(\ln \varrho )^{\varpi - i}}, \end{aligned}$$

and the following formula holds:

$$\begin{aligned} {}^{H}I_{1+}^{\varpi \,H}D_{1+ }^\varpi \Im (\varrho ) = \Im (\varrho ) + \sum \limits _{i = 1}^{\mathfrak {n}}{{c_i}} {(\ln \varrho )^{\varpi - i}}, \end{aligned}$$

where ${c_i} \in \mathbb {R},~i = 1,2, \cdots ,\mathfrak {n}$, and $\mathfrak {n} - 1< \varpi < \mathfrak {n}.$

Lemma 2.2

[13, 25] Let $\varpi> 0,~\epsilon > 0,$ then

$$\begin{aligned} {}^HI_{1 + }^\varpi {(\ln \varrho )^{\epsilon - 1}} = \frac{{\Gamma (\epsilon )}}{{\Gamma (\varpi +\epsilon )}}{(\ln \varrho )^{\varpi + \epsilon - 1}},\quad {}^HD_{1 + }^\varpi {(\ln \varrho )^{\epsilon - 1}} = \frac{{\Gamma (\epsilon )}}{{\Gamma (\epsilon - \varpi )}}{(\ln \varrho )^{\epsilon - \varpi - 1}}, \end{aligned}$$

in particular, ${}^HD_{1 + }^\varpi {(\ln \varrho )^{\varpi - i}} = 0,~i = 1,2, \cdots , [\varpi ] + 1.$

Let $(X, {\left\| \cdot \right\| _X})$ and $(Y, {\left\| \cdot \right\| _Y})$ are two Banach spaces. Let $N:X\rightarrow Y$ be a nonlinear operator, $L:\text{ dom }L \subset X \rightarrow Y$ be a Fredholm operator with index zero, i.e., ${\text {Im}} L$ is closed and $\dim {\text {Ker}}L = {\text {co}}\dim {\text {Im}} L < \infty$, which is also implies that there exist continuous projections $P:X \rightarrow X$ and $Q:Y \rightarrow Y$ such that

$$\begin{aligned} {\text {Im}} P = \text{ Ker }L,\quad {\text {Im}} L = \text{ Ker }Q,\quad X = \text{ Ker }L \oplus \text{ Ker }P,\quad Y = {\text {Im}} L \oplus {\text {Im}} Q, \end{aligned}$$

and $L\left| {_{{\text {dom}}L \cap {\text {Ker}}P}} \right. :\text{ dom }L \rightarrow {\text {Im}} L$ is invertible. We denote by ${K_P}{\text { = }}{(L\left| {_{{\text {dom}}L \cap {\text {Ker}}P}} \right. )^{ - 1}}$. Let $\Omega$ be an open bounded subset of X and $\text{ dom }L \cap \bar{\Omega }\ne \emptyset$. The operator $N:X \rightarrow Y$ is called L-compact on $\bar{\Omega },$ if $QN\left( {\bar{\Omega }} \right)$ is bounded and $K_P (I - Q)N:\bar{\Omega }\rightarrow X$ is compact. On account of $\dim {\mathop \textrm{Im}\nolimits } Q = \text {co}\dim {\mathop \textrm{Im}\nolimits } L$, there exists an isomorphism $J:{\text {Im}} Q \rightarrow {\text {Ker}}L.$ Then the equation $L\Im = \lambda N\Im$ is equivalent to

$$\begin{aligned} \Im = (P + JQN)\Im + \lambda {K_P}(I - Q)N\Im , \end{aligned}$$

for all $\lambda \in (0,1]$ ([26]).

Definition 2.3

[26] A nonempty convex closed set $\mathcal {C}\subset X$ is named as a cone if

${\text {(i)} }$:

$\lambda \Im \in \mathcal {C}$ for all $\Im \in \mathcal {C}$ and $\lambda \ge 0$;

${\text {(ii)} }$:

$\Im ,-\Im \in \mathcal {C}$ implies $\Im =\theta$.

Remark 2.1

[26] The cone $\mathcal {C}$ induces a partial order in X by

$$\begin{aligned} \Im \preceq \zeta ~\text {if and only if}~ \zeta -\Im \in \mathcal {C}. \end{aligned}$$

Lemma 2.3

[26] Let $\mathcal {C}$ be a cone in X. Then for every $\zeta \in \mathcal {C}\backslash \{ \theta \}$ there exists a positive number $\tau (\zeta )$ such that

$$\begin{aligned} ||\Im + \zeta || \ge \tau (\zeta )||\Im ||, \end{aligned}$$

for all $\Im \in \mathcal {C}.$

Let $\gamma :X \rightarrow \mathcal {C}$ be a retraction, i.e., a continuous mapping such that $\gamma (\Im ) = \Im$ for all $\Im \in \mathcal {C}$. Define

$$\begin{aligned} \Xi : = P + JQN + {K_P}(I - Q)N, \end{aligned}$$

and

$$\begin{aligned} {\Xi _\gamma }: = \Xi \circ \gamma . \end{aligned}$$

Theorem 2.1

[26] Let $\mathcal {C}$ be a cone in X and let ${\Omega _1}$, ${\Omega _2}$ be open bounded subsets of X with ${\bar{\Omega }_1} \subset {\Omega _2}$ and $\mathcal {C} \cap ({\bar{\Omega }_2}\backslash {\Omega _1}) \ne \emptyset$. Suppose that

${{1^ \circ } }$:

L is a Fredholm operator of index zero;

${ {2^ \circ }}$:

$QN:X \rightarrow Y$ is continuous and bounded and ${K_P}(I - Q)N:X \rightarrow X$ is compact on every bounded subset of X;

${ {3^ \circ }}$:

$||N\Im || \le ||L\Im ||$ for $\Im \in \mathcal {C} \cap \partial {\Omega _2} \cap {\text {dom}}L$ (or $L\Im \ne \lambda N\Im ,$ for any $\Im \in \mathcal {C} \cap \partial {\Omega _2} \cap {\text {dom}}L$ and $\lambda \in (0,1)$);

${ {4^ \circ }}$:

$\gamma$ maps subsets of ${\bar{\Omega }_2}$ into bounded subsets of $\mathcal {C}$;

${ {5^ \circ }}$:

${d_B}\big (\big [I - (P + JQN)\gamma \big ]{|_{{\text {Ker}}L}},{\text {Ker}}L \cap {\Omega _2},0\big ) \ne 0,$ where ${d_B}$ represent the Brouwer degree;

${ {6^ \circ }}$:

there exists ${\Im _0} \in \mathcal {C}\backslash \{ 0\}$ such that $||\Im || \le \tau ({\Im _0})||\Xi \Im ||$ for $\Im \in \mathcal {C}({\Im _0}) \cap \partial {\Omega _1}$, where $\mathcal {C}({\Im _0}) = \{ \Im \in \mathcal {C}:\lambda {\Im _0}\underset{{\smash {\scriptscriptstyle -}}}{ \prec } \Im ~\text {for some}~ \lambda > 0\}$ and $\tau ({\Im _0})$ is such that $||\Im + {\Im _0}|| \ge \tau ({\Im _0})||\Im ||$ for every $\Im \in \mathcal {C}$;

${ {7^ \circ }}$:

$(P + JQN)\gamma (\partial {\Omega _2}) \subset \mathcal {C}$;

${ {8^ \circ }}$:

${\Xi _\gamma }({\bar{\Omega }_2}\backslash {\Omega _1}) \subset \mathcal {C}$.

Then the equation $L\Im = N\Im$ has a solution in the set $\mathcal {C}\cap ({\bar{\Omega }_2}\backslash {\Omega _1})$.

Lemma 2.4

[27] Let $\aleph \subset X$ be a bounded set. Then $\aleph$ is relatively compact in X if the following conditions are satisfied:

${\text {(i)} }$:

For any $\Im (\varrho )\in \aleph ,\;\frac{{\Im (\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}$ is equicontinuous on any compact interval of $[1,+\infty )$;

${\text {(ii)} }$:

For any $\varepsilon >0,$ there exists a constant $\mathcal {T}=\mathcal {T}(\varepsilon )>0$ such that, for any ${\varrho _1},{\varrho _2} \ge \mathcal {T}$ and $\Im \in \aleph$, it holds

$$\begin{aligned} \left| {\frac{{\Im ({\varrho _2})}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} - \frac{{\Im ({\varrho _1})}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}} \right| <\varepsilon . \end{aligned}$$

3 Main Results

Firstly, we define two Banach spaces, and within the framework of these spaces, we use the Leggett-Williams norm-type theorem to successfully establish the results of the existence of positive solutions for problem (6).

Define two spaces

$$\begin{aligned} X = \Bigg \{ \Im :[1, + \infty ) \rightarrow \mathbb {R}\bigg |\Im \in C[1, + \infty ),\mathop {\sup }\limits _{\varrho \in [1, + \infty )} \frac{{|\Im (\varrho )|}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}< + \infty \Bigg \}, \end{aligned}$$

equipped with the norm

$$\begin{aligned} ||\Im |{|_X} = \mathop {\sup }\limits _{\varrho \in [1, + \infty )} \frac{{|\Im (\varrho )|}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}, \end{aligned}$$

and

$$\begin{aligned} Y = \Bigg \{ \zeta :[1, + \infty ) \rightarrow \mathbb {R}\bigg |\int _1^{ + \infty } {|\zeta (\varrho )|\frac{{d\varrho }}{\varrho }} < + \infty \Bigg \}, \end{aligned}$$

equipped with the norm

$$\begin{aligned} ||\zeta |{|_Y} = \int _1^{ + \infty } {|\zeta (\varrho )|} \frac{{d\varrho }}{\varrho }. \end{aligned}$$

It is easily to verify that $(X,{\left\| \cdot \right\| _X})$ and $(Y,{\left\| \cdot \right\| _Y})$ are two Banach spaces.

Define the linear operator $L:{\text {dom}}L \rightarrow Y$ by

$$\begin{aligned}&\begin{aligned} L\Im = {}^HD_{1 + }^\varpi \Im ,\quad \Im \in {\text {dom}}L, \end{aligned}\ \end{aligned}$$

(7)

where

$$\begin{aligned} {\text {dom}}L&= \Big \{ \Im \in X|{}^HD_{1 + }^\varpi \Im (\varrho ) \in Y,\Im (1) =\Im '(1) \\&\quad = \Im ''(1) = 0,{}^HD_{1 + }^{\varpi - 1}\Im (1){\text { = }}\mathop {\lim }\limits _{\varrho \rightarrow + \infty } {}^HD_{1 + }^{\varpi - 1}\Im (\varrho )\Big \} . \end{aligned}$$

Define the nonlinear operator $N:X \rightarrow Y$ by

$$\begin{aligned} N\Im (\varrho ) = \hbar \big (\varrho ,\Im (\varrho )\big ),\quad \Im \in X, \end{aligned}$$

then the BVP (6) is equivalent to the operator equation

$$\begin{aligned} L\Im = N\Im ,\quad \Im \in {\text {dom}}L. \end{aligned}$$

Lemma 3.1

Let $L:{\text {dom}}L \subset X \rightarrow Y$ is defined by (7), then

$$\begin{aligned}{} & {} {\text {Ker}}L = \big \{ \Im \in \text {dom}L|\Im (\varrho ) = c{(\ln \varrho )^{\varpi - 1}},~\varrho \ge 1,~c \in \mathbb {R}\big \}, \end{aligned}$$

(8)

$$\begin{aligned}{} & {} {\text {Im}} L = \Big \{\zeta \in Y\Big |\int _1^{ + \infty } {\zeta (\varsigma )\frac{{d\varsigma }}{\varsigma }} = 0\Big \}. \end{aligned}$$

(9)

Proof

By Lemma 2.1, we know that ${}^HD_{1 + }^\varpi \Im (\varrho ) = 0$ has following solution

$$\begin{aligned} \Im (\varrho ) = {c_1}{(\ln \varrho )^{\varpi - 1}} + {c_2}{(\ln \varrho )^{\varpi - 2}} + {c_3}{(\ln \varrho )^{\varpi - 3}} + {c_4}{(\ln \varrho )^{\varpi - 4}},\;\;\;{c_1},{c_2},{c_3},{c_4} \in \mathbb {R}. \end{aligned}$$

From boundary conditions $\Im (1) = \Im '(1) = \Im ''(1) = 0,$ we have ${c_2} = {c_3} = {c_4} = 0$, that is, $\Im (\varrho ) = {c_1}{(\ln \varrho )^{\varpi - 1}}$. Conversely, for $\Im \in \text {dom}L$ and $\Im (\varrho )=c(\ln \varrho )^{\varpi -1}$, it follows from Lemma 2.2 that ${}^HD_{1 + }^\varpi \Im (\varrho ) = 0$. Hence, (8) holds. If $\zeta \in {\text {Im}} L,$ there exists a function $\Im \in {\text {dom}}L$ such that $\zeta (\varrho ) = {}^HD_{1 + }^\varpi \Im (\varrho ).$ By Lemma 2.1, Lemma 2.2 and boundary condition ${}^HD_{1 + }^{\varpi - 1}\Im (1)= \mathop {\lim }\limits_{\varrho\to + \infty }{}^HD_{1 + }^{\varpi - 1}\Im (\varrho )$, we know that

$$\begin{aligned}&\int _1^{ + \infty } {\zeta (\varsigma )\frac{{d\varsigma }}{\varsigma }} = 0. \end{aligned}$$

(10)

On the other hand, if $\zeta \in Y$ and satisfies (10). Let

$$\begin{aligned} \Im (\varrho ) = {}^HI_{1 + }^\varpi \zeta (\varrho ), \end{aligned}$$

then $\Im (\varrho ) \in {\text {dom}}L$ and $L\Im (\varrho ) = \zeta (\varrho )$. Therefore, (9) holds. The proof is completed. $\square$

Lemma 3.2

Assume that L is given by (7), then L is a Fredholm operator of index zero. Define the linear operators $P:X \rightarrow X$ and $Q:Y \rightarrow Y$ as follows

$$\begin{aligned}&P\Im (\varrho ) = \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}\Im (\varsigma )\frac{{d\varsigma }}{\varsigma }},\quad \varrho \in [1,+\infty ), \\&Q\zeta (\varrho ) = {e^{ - \ln \varrho }}\int _1^{ + \infty } {\zeta (\varsigma )\frac{{d\varsigma }}{\varsigma }},\quad \varrho \in [1,+\infty ). \end{aligned}$$

Proof

By the definition of the operator P, we can get ${\text {Im}} P = {\text {Ker}}L$ and $P\Im (\varrho ) = {P^2}\Im (\varrho )$. For any $\Im \in X$, then $\Im = (\Im - P\Im ) + P\Im$, it follows

$$\begin{aligned} X = {\text {Ker}}P + {\text {Ker}}L. \end{aligned}$$

Furthermore, it is easily to verify that ${\text {Ker}}P \cap {\text {Ker}}L = \left\{ \theta \right\}$. Then, we obtain

$$\begin{aligned} X = {\text {Ker}}P \oplus {\text {Ker}}L. \end{aligned}$$

Therefore, the operator $P:X \rightarrow X$ is a projection operator. By the definition of operator Q, we have

$$\begin{aligned} {Q^2}\zeta = Q(Q\zeta ) = {e^{ - \ln \varrho }}\int _1^{ + \infty } {Q\zeta (\varsigma )\frac{{d\varsigma }}{\varsigma }} = Q\zeta \int _1^{ + \infty } {{e^{ - \ln \varsigma }}\frac{{d\varsigma }}{\varsigma }} = Q\zeta . \end{aligned}$$

For any $\zeta \in Y$, then $\zeta = (\zeta - Q\zeta ) + Q\zeta$. This implies that

$$\begin{aligned} Y = {\text {Im}} Q + {\text {Im}} L. \end{aligned}$$

It follows from ${\text {Ker}}Q = {\text {Im}} L$ and ${Q^2}\zeta = Q\zeta$ that ${\text {Im}} Q \cap {\text {Im}} L = \left\{ \theta \right\} .$ We also get

$$\begin{aligned} Y = {\text {Im}} Q \oplus {\text {Im}} L. \end{aligned}$$

Hence, the operator $Q:Y \rightarrow Y$ is a projection operator. Moreover,

$$\begin{aligned} \dim {\text {Ker}}L = \dim {\text {Im}} Q = {\text {codim}}{\text {Im}} L = 1, \end{aligned}$$

that is, L is a Fredholm operator of index zero. This ends the proof. $\square$

Lemma 3.3

Define the operator ${K_P}:{\text {Im}} L \rightarrow {\text {dom}}L \cap {\text {Ker}}P$ by

$$\begin{aligned} {K_P}\zeta (\varrho ) = - \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}\zeta (\varsigma )\frac{{d\varsigma }}{\varsigma }} + \frac{1}{{\Gamma (\varpi )}}\int _1^\varrho {\Big (\ln \frac{\varrho }{\varsigma }} {\Big )^{\varpi - 1}}\zeta (\varsigma )\frac{{d\varsigma }}{\varsigma }. \end{aligned}$$

Then ${K_P} = {(L{|_{{\text {dom}}L \cap {\text {Ker}}P}})^{ - 1}}$ and

$$\begin{aligned} ||{K_P}\zeta |{|_X} \le ||\zeta |{|_Y},\quad \forall \zeta \in {\text {Im}} L. \end{aligned}$$

Proof

Firstly, we will show that the definition of the ${K_P}$ is resonbale. For $\zeta \in {\text {Im}} L,$ we have

$$\begin{aligned} \begin{aligned}&{K_P}\zeta (\varrho ){|_{\varrho = 1}} = 0,\;\;\;\big ({K_P}\zeta (\varrho )\big )'{|_{\varrho = 1}} = 0,\;\;\;\big ({K_P}\zeta (\varrho )\big )''{|_{\varrho = 1}} = 0,\\&{}^HD_{1 + }^{\varpi - 1}{K_P}\zeta (\varrho ){|_{\varrho = 1}} = - \int _1^{ + \infty } {{e^{ - \ln \varsigma }}} \zeta (\varsigma )\frac{{d\varsigma }}{\varsigma } \\&= - \int _1^{ + \infty } {{e^{ - \ln \varsigma }}} \zeta (\varsigma )\frac{{d\varsigma }}{\varsigma } + \int _1^{ + \infty } \zeta (\varsigma )\frac{{d\varsigma }}{\varsigma } =\mathop {\lim }\limits _{\varrho \rightarrow + \infty } {}^HD_{1 + }^{\varpi - 1}{K_P}\zeta (\varrho ), \end{aligned}\ \end{aligned}$$

(11)

and

$$\begin{aligned}&\begin{aligned}&P{K_P}\zeta (\varrho ) \\&= \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} \Bigg [ - \frac{{{{(\ln \varsigma )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \imath }}\zeta (\imath )\frac{{d\imath }}{\imath }} \\&\quad + \frac{1}{{\Gamma (\varpi )}}\int _1^\varsigma {\Big (\ln \frac{\varsigma }{\imath }} \Big )^{\varpi - 1}\zeta (\imath )\frac{{d\imath }}{\imath }\Bigg ]\frac{{d\varsigma }}{\varsigma } \\&= \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\Bigg [ - \frac{{\int _1^{ + \infty } {{e^{ - \ln \imath }}\zeta (\imath )\frac{{d\imath }}{\imath }} }}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} {(\ln \varsigma )^{\varpi - 1}}\frac{{d\varsigma }}{\varsigma } \\&\quad + \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} \int _1^\varsigma {\Big (\ln \frac{\varsigma }{\imath }} {\Big )^{\varpi - 1}}\zeta (\imath )\frac{{d\imath }}{\imath }\frac{{d\varsigma }}{\varsigma }\Bigg ] \\&= \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\Bigg [ - \int _1^{ + \infty } {{e^{ - \ln \imath }}\zeta (\imath )\frac{{d\imath }}{\imath }} + \int _1^{ + \infty } {{e^{ - \ln \imath }}\zeta (\imath )\frac{{d\imath }}{\imath }} \Bigg ] = 0. \end{aligned} \end{aligned}$$

(12)

From (11) and (12), we obtain ${K_P}$ is well defined. Next, we will prove that ${K_P} = {(L{|_{{\text {dom}}L \cap {\text {Ker}}P}})^{ - 1}}$. Actually, if $\zeta \in {\text {Im}} L$, by Lemma 2.2, we have

$$\begin{aligned}&\begin{aligned} L{K_P}\zeta (\varrho ) = {}^HD_{1 + }^{\varpi \,H}I_{1 + }^\varpi \zeta (\varrho ) + {}^HD_{1 + }^\varpi \Bigg [- \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} \zeta (\varsigma )\frac{{d\varsigma }}{\varsigma }\Bigg ] = \zeta (\varrho ). \end{aligned}\ \end{aligned}$$

(13)

On the other hand, for any $\Im (\varrho )\in {{\text {dom}}L \cap {\text {Ker}}P}$. Since ${K_P}L\Im (\varrho ) \in {\text {Ker}}P$ and $\Im (\varrho ) \in {\text {Ker}}P,$ it follows that

$$\begin{aligned} 0 = P{K_P}L\Im (\varrho )&= P\Bigg [\Im (\varrho )+{c_1}{(\ln \varrho )^{\varpi - 1}} - \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}\big ({}^HD_{1 + }^\varpi \Im (\varsigma )\big )\frac{{d\varsigma }}{\varsigma }} \Bigg ]\\&=\Bigg [{c_1} - \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}\big ({}^HD_{1 + }^\varpi \Im (\varsigma )\big )\frac{{d\varsigma }}{\varsigma }} \bigg ]{(\ln \varrho )^{\varpi - 1}}. \end{aligned}$$

This combined with Lemma 2.1 and $\Im (\varrho )\in {\text {dom}}L$, we deduce that

$$\begin{aligned}&\begin{aligned} {K_P}L\Im (\varrho )&= {}^HI_{1 + }^{\varpi \,H}D_{1 + }^\varpi \Im (\varrho ) - \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}({}^HD_{1 + }^\varpi \Im (\varsigma ))\frac{{d\varsigma }}{\varsigma }} \\&= \Im (\varrho ) + {c_1}{(\ln \varrho )^{\varpi - 1}} - \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}({}^HD_{1 + }^\varpi \Im (\varsigma ))\frac{{d\varsigma }}{\varsigma }}\\&=\Im (\varrho ). \end{aligned}\ \end{aligned}$$

(14)

Form (13) and (14), we obtain ${K_P} = {(L{|_{{\text {dom}}L \cap {\text {Ker}}P}})^{ - 1}}$. Moreover,

$$\begin{aligned}&||{K_P}\zeta |{|_X} = \mathop {\sup }\limits _{\varrho \in [1, + \infty )} \frac{{|{K_P}\zeta |}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} \\&= \mathop {\sup }\limits _{\varrho \in [1, + \infty )} \frac{1}{{\Gamma (\varpi )}}\bigg [\frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\bigg |\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} \zeta (\varsigma )\frac{{d\varsigma }}{\varsigma }\bigg | + \bigg |\int _1^\varrho {\frac{{{{(\ln \frac{\varrho }{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}} \zeta (\varsigma )\frac{{d\varsigma }}{\varsigma }\bigg |\bigg ] \\&\le \frac{{2||\zeta |{|_Y}}}{{\Gamma (\varpi )}}\le ||\zeta |{|_Y}. \end{aligned}$$

The proof is complete.

Lemma 3.4

Suppose that the condition (H) holds, let $\Omega \subset X$ be an open bounded subset and satisfy $\text {dom}L \cap \bar{\Omega }\ne \emptyset$, then N is L-compact on ${\bar{\Omega }}$.

Proof

It follows from $\Omega \subset X$ is an open bounded subset that there exists a constant $\jmath > 0$, such that $||\Im |{|_X}\le \jmath$, for any $\Im \in \Omega$. By (H), we obtain

$$\begin{aligned} |QN\Im | \le {e^{ - \ln \varrho }}\int _1^{ + \infty } {|N\Im (\varsigma )|\frac{{d\varsigma }}{\varsigma }} \le {e^{ - \ln \varrho }}||{\phi _\jmath }|{|_Y},\quad \varrho \in [1,+\infty ). \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned} ||QN\Im |{|_Y} = \int _1^{ + \infty } {|QN\Im } (\varrho )|\frac{{d\varrho }}{\varrho } \le ||{\phi _\jmath }|{|_Y}\int _1^{ + \infty } {{e^{ - \text {ln}\varrho }}}\frac{{d\varrho }}{\varrho } = ||{\phi _\jmath }|{|_Y} < + \infty . \end{aligned} \end{aligned}$$

(15)

By Lemma 3.3, we also have

$$\begin{aligned} \begin{aligned} ||{K_P}(I - Q)N\Im |{|_X}&\le ||(I - Q)N\Im |{|_Y} \le \int _1^{ + \infty } {|N\Im (\varrho )|\frac{{d\varrho }}{\varrho } + } \int _1^{ + \infty } {|QN\Im (\varrho )|\frac{{d\varrho }}{\varrho }} \\&\le ||{\phi _\jmath }|{|_Y} + ||QN\Im |{|_Y} \le 2||{\phi _\jmath }|{|_Y} < + \infty . \end{aligned}\ \end{aligned}$$

(16)

Form (15) and (16), we get $QN(\bar{\Omega })$ and ${K_P}(I - Q)N(\bar{\Omega })$ are uniformly bounded. For $\Im (\varrho ) \in \bar{\Omega }$, let

$$\begin{aligned} \Re (\varrho ) = (I - Q)N\Im (\varrho ),\quad \varrho \in [1, + \infty ), \end{aligned}$$

then

$$\begin{aligned} ||\Re |{|_Y} = ||(I - Q)N\Im |{|_Y} \le 2||{\phi _\jmath }|{|_Y} < + \infty . \end{aligned}$$

We now show that for any $\Im \in \bar{\Omega }$, ${K_P}(I - Q)N\Im$ is equicontinuous on any compact interval of $[1,+\infty )$. In fact, for any $T>1$, $\Im \in \bar{\Omega }$ and $1 \le {\varrho _1}< {\varrho _2} < T.$ Since the uniform continuity of $\frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}$ on $[{\varrho _1},{\varrho _2}]$, and $\mu (\varrho ,\varsigma )=\frac{{{{(\ln \frac{\varrho }{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}$ is also uniform continuity on $[{\varrho _1},{\varrho _2}] \times [1,{\varrho _1}]$, then

$$\begin{aligned}&\Bigg |\frac{{{K_P}(I - Q)N\Im ({\varrho _2})}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} - \frac{{{K_P}(I - Q)N\Im ({\varrho _1})}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg | \\&= \frac{1}{{\Gamma (\varpi )}}\Bigg |\int _1^{{\varrho _2}} {\frac{{{{(\ln \frac{{{\varrho _2}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}}} \Re (\varsigma )\frac{{d\varsigma }}{\varsigma } - \frac{{{{(\ln {\varrho _2})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} \Re (\varsigma )\frac{{d\varsigma }}{\varsigma } \\&\quad - \int _1^{{\varrho _1}} {\frac{{{{(\ln \frac{{{\varrho _1}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}} \Re (\varsigma )\frac{{d\varsigma }}{\varsigma } \\&\quad + \frac{{{{(\ln {\varrho _1})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} \Re (\varsigma )\frac{{d\varsigma }}{\varsigma }\Bigg |\\&= \frac{1}{{\Gamma (\varpi )}}\Bigg |\int _1^{{\varrho _1}} \Bigg [\frac{{{{(\ln \frac{{{\varrho _2}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} - \frac{{{{(\ln \frac{{{\varrho _1}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg ]\Re (\varsigma )\frac{{d\varsigma }}{\varsigma } \\&\quad + \int _{{\varrho _1}}^{{\varrho _2}} {\frac{{{{(\ln \frac{{{\varrho _2}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}}} \Re (\varsigma )\frac{{d\varsigma }}{\varsigma } \\&\quad - \Bigg [\frac{{{{(\ln {\varrho _2})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} - \frac{{{{(\ln {\varrho _1})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg ]\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} \Re (\varsigma )\frac{{d\varsigma }}{\varsigma }\Bigg |\\&\le \frac{1}{{\Gamma (\varpi )}}\Bigg [\int _1^{{\varrho _1}} {\Bigg |\frac{{{{(\ln \frac{{{\varrho _2}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} - } \frac{{{{(\ln \frac{{{\varrho _1}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg ||\Re (\varsigma )|\frac{{d\varsigma }}{\varsigma }\\&\quad +\int _{{\varrho _1}}^{{\varrho _2}} {\frac{{{{(\ln \frac{{{\varrho _2}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}}} |\Re (\varsigma )|\frac{{d\varsigma }}{\varsigma } \\&\quad + \Bigg |\frac{{{{(\ln {\varrho _2})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} - \frac{{{{(\ln {\varrho _1})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg |\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} |\Re (\varsigma )|\frac{{d\varsigma }}{\varsigma }\Bigg ] \rightarrow 0,\;\;\text {as}\;\;{\varrho _1} \rightarrow {\varrho _2}. \end{aligned}$$

We finally prove that $\Im \in \bar{\Omega }$, ${K_P}(I - Q)N\Im$ is equiconvergent at infinity. Indeed, for any $\varepsilon >0,$ there exists a positive constant $\mathcal {T}>1$ such that

$$\begin{aligned} \int _\mathcal {T}^{ + \infty } {|\Re (\varsigma )|\frac{{d\varsigma }}{\varsigma }} < \varepsilon . \end{aligned}$$

Note that

$$\begin{aligned} \mathop {\lim }\limits _{\varrho \rightarrow +\infty } \frac{{{{(\ln \frac{\varrho }{\mathcal {T}})}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} = 1,\quad \mathop {\lim }\limits _{\varrho \rightarrow + \infty } \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} = 1, \end{aligned}$$

then for any $\varepsilon >0,$ there exists a constant $\mathcal {T}(\varepsilon )>\mathcal {T}$, such that $\varrho >\mathcal {T}(\varepsilon )$,

$$\begin{aligned} 1 - \frac{{{{(\ln \frac{\varrho }{\mathcal {T}})}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}< \varepsilon ,\quad 1 - \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} < \varepsilon . \end{aligned}$$

Hence, for any $\Im \in \bar{\Omega }$ and ${\varrho _2},{\varrho _1} > \mathcal {T}(\varepsilon )$, without loss of generality, we assume that ${\varrho _2} > {\varrho _1}$, it follows

$$\begin{aligned}&\Bigg |\frac{{{K_P}(I - Q)N\Im ({\varrho _2})}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} \\&\quad - \frac{{{K_P}(I - Q)N\Im ({\varrho _1})}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg | \\&= \frac{1}{{\Gamma (\varpi )}}\Bigg |\int _1^\mathcal {T} \Bigg [\frac{{{{(\ln \frac{{{\varrho _2}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} {-} \frac{{{{(\ln \frac{{{\varrho _1}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}} \Bigg ]\Re (\varsigma )\frac{{d\varsigma }}{\varsigma } \\&\quad+ \int _\mathcal {T}^{{\varrho _2}} {\frac{{{{(\ln \frac{{{\varrho _2}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}}\Re (\varsigma )\frac{{d\varsigma }}{\varsigma }} \\&\quad - \int _\mathcal {T}^{{\varrho _1}} {\frac{{{{(\ln \frac{{{\varrho _1}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}}\Re (\varsigma )\frac{{d\varsigma }}{\varsigma } \\&\quad - \Bigg [\frac{{{{(\ln {\varrho _2})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}}- \frac{{{{(\ln {\varrho _1})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg ]\int _1^{+ \infty } {{e^{ - \ln \varsigma }}} \Re (\varsigma )\frac{{d\varsigma }}{\varsigma }\Bigg | \\&\le \frac{1}{{\Gamma (\varpi )}}\int _1^\mathcal {T} {\Bigg |\frac{{{{(\ln \frac{{{\varrho _2}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} {-} \frac{{{{(\ln \frac{{{\varrho _1}}}{\varsigma })}^{\varpi -1}}}}{{1+{{(\ln {\varrho _1})}^{\varpi -1}}}}} \Bigg ||\Re (\varsigma )|\frac{{d\varsigma }}{\varsigma }\\&\quad + \frac{2}{{\Gamma (\varpi )}}\int _\mathcal {T}^{+\infty } {|\Re (\varsigma )|\frac{{d\varsigma }}{\varsigma }} \\&\quad + \frac{1}{{\Gamma (\varpi )}}\Bigg |\frac{{{{(\ln {\varrho _2})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} \\&\quad - \frac{{{{(\ln {\varrho _1})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg |{\int _1^{ + \infty } {{e^{ - \ln \varsigma }}} |\Re (\varsigma )|\frac{{d\varsigma }}{\varsigma }} \\&\le \frac{1}{{\Gamma (\varpi )}}\int _1^\mathcal {T} {\Bigg |1 - \frac{{{{(\ln \frac{{{\varrho _2}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}}\Bigg ||\Re (\varsigma )|\frac{{d\varsigma }}{\varsigma } + } \frac{1}{{\Gamma (\varpi )}}\int _1^\mathcal {T} {\Bigg |1 - \frac{{{{(\ln \frac{{{\varrho _1}}}{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg ||\Re (\varsigma )|\frac{{d\varsigma }}{\varsigma }} \\&\quad + \frac{{2\varepsilon }}{{\Gamma (\varpi )}} \\&\quad +\frac{{||\Re |{|_Y}}}{{\Gamma (\varpi )}}\Bigg [\Bigg |1 - \frac{{{{(\ln {\varrho _2})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}}\Bigg | \\&\quad +\Bigg |1 - \frac{{{{(\ln {\varrho _1})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}\Bigg |\Bigg ] \le (2||\Re |{|_Y} + 1)\frac{{2\varepsilon }}{{\Gamma (\varpi )}}. \end{aligned}$$

Therefore, ${K_P}(I - Q)N:\bar{\Omega }\rightarrow X$ is compact. The proof is complete. $\square$

Define a homeomorphism operator $J:{\text {Im}} Q \rightarrow {\text {Ker}}L$ by

$$\begin{aligned} J(c{e^{ - \ln \varrho }}) = c{(\ln \varrho )^{\varpi - 1}},~\varrho \ge 1,~c \in \mathbb {R}. \end{aligned}$$

Then, $JQN + {K_P}(I - Q)N:X \rightarrow X$ can be written as

$$\begin{aligned}{}[JQN + {K_P}(I - Q)N]\Im (\varrho ) {=} \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {G(\varrho ,\varsigma )\hbar \big (\varsigma ,\Im (\varsigma )\big )\frac{{d\varsigma }}{\varsigma }},~\varrho \ge 1, \end{aligned}$$

(17)

where

$$\begin{aligned} G(\varrho ,\varsigma ) {=} {\left\{ \begin{array}{ll} 0,~\varrho {=} 1, \\ \Gamma (\varpi ) {+} \frac{1}{2} - {e^{ - \ln \varsigma }} {+} \frac{{{{\left(\ln \frac{\varrho }{\varsigma }\right)}^{\varpi - 1}}}}{{{{(\ln \varrho )}^{\varpi - 1}}}} - \int _1^\varrho {\frac{{{{\left(\ln \frac{\varrho }{\imath }\right)}^{\varpi - 1}}}}{{{{(\ln \varrho )}^{\varpi - 1}}}}{e^{ - {\ln \imath } }}\frac{{d\imath }}{\imath }},\;\varrho {\ne } 1,~1 {\le } \varsigma {\le } \varrho {<}+\infty , \\ \Gamma (\varpi ) {+} \frac{1}{2} - {e^{ - \ln \varsigma }} - \int _1^\varrho {\frac{{{{\left(\ln \frac{\varrho }{\imath }\right)}^{\varpi - 1}}}}{{{{(\ln \varrho )}^{\varpi - 1}}}}{e^{ - {\ln \imath } }}\frac{{d\imath }}{\imath },~1 {<} \varrho {\le } \varsigma {<} + \infty .}\\ \end{array}\right. } \end{aligned}$$

For any $\varrho ,\varsigma \in (1,+\infty )$, we have

$$\begin{aligned}&\Gamma (\varpi ) + \frac{1}{2} - {e^{ - \ln \varsigma }} + \frac{{{{\left(\ln \frac{\varrho }{\varsigma }\right)}^{\varpi - 1}}}}{{{{(\ln \varrho )}^{\varpi - 1}}}} - \int _1^\varrho {\frac{{{{\left(\ln \frac{\varrho }{\imath }\right)}^{\varpi - 1}}}}{{{{(\ln \varrho )}^{\varpi - 1}}}}{e^{ - {\ln \imath } }}\frac{{d\imath }}{\imath }}\\&\le \Gamma (\varpi ) + \frac{1}{2} + \frac{{{{\left(\ln \frac{\varrho }{\varsigma }\right)}^{\varpi - 1}}}}{{{{(\ln \varrho )}^{\varpi - 1}}}}\le \Gamma (\varpi ) + \frac{3}{2}, \end{aligned}$$

and

$$\begin{aligned}&\Gamma (\varpi ) + \frac{1}{2} - {e^{ - \ln \varsigma }} - \int _1^\varrho {\frac{{{{\left(\ln \frac{\varrho }{\imath }\right)}^{\varpi - 1}}}}{{{{(\ln \varrho )}^{\varpi - 1}}}}{e^{ - {\ln \imath } }}\frac{{d\imath }}{\imath }}\\&\ge \Gamma (\varpi ) + \frac{1}{2} - {e^{ - \ln \varsigma }} - \int _1^\varsigma {{e^{ - \ln \imath }}\frac{{d\imath }}{\imath }} = \Gamma (\varpi ) - \frac{1}{2}>0. \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned} 0 < \Gamma (\varpi ) - \frac{1}{2} \le G(\varrho ,\varsigma ) \le \Gamma (\varpi ) + \frac{3}{2}. \end{aligned}\ \end{aligned}$$

(18)

Theorem 3.1

Assume that condition (H) holds and there exist non-negative functions ${\vartheta _i}(\varrho )(i = 1,2,3),~{\omega _j}(\varrho )(j = 1,2)$ and $\eta (\varrho )$ such that

$$\begin{aligned}&\begin{aligned} \hbar (\varrho ,\Im ) \le - {\vartheta _1}(\varrho )|\hbar (\varrho ,\Im )| + {\vartheta _2}(\varrho )\frac{\Im }{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} + {\vartheta _3}(\varrho ),~\varrho \ge 1, \end{aligned}\ \end{aligned}$$

(19)

and

$$\begin{aligned}&\begin{aligned} - \eta (\varrho )\frac{\Im }{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} \le \hbar (\varrho ,\Im ) \le - {\omega _1}(\varrho )\frac{\Im }{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} + {\omega _2}(\varrho ),~\varrho \ge 1, \end{aligned}\ \end{aligned}$$

(20)

where $0 \le \displaystyle \frac{\Im }{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} \le M,~M > {M_0}$, ${\vartheta _1}(\varrho )$ is bounded on $[1, + \infty )$, ${\omega _1}(\varrho ) > 0$, $\varrho \ge 1$, ${\vartheta _2}(\varrho )$, ${\vartheta _3}(\varrho )$, ${\omega _1}(\varrho )$, ${\omega _2}(\varrho ) \in Y$,

$$\begin{aligned}{} & {} {\vartheta _0}: = \mathop {\inf }\limits _{\varrho \ge 1} {\vartheta _1}(\varrho )> 0,\;\;\;\int _1^{ + \infty } {{\vartheta _3}(\varrho )} \frac{{d\varrho }}{\varrho }> 0,\;\;\;\int _1^{ + \infty } {{\omega _2}(\varrho )} \frac{{d\varrho }}{\varrho } > 0, \end{aligned}$$

(21)

$$\begin{aligned}{} & {} \quad {\Lambda _0}: = \frac{2}{{{\vartheta _0}}}\mathop {\sup }\limits _{\varrho \ge 1} \Bigg [\frac{{{\vartheta _2}(\varrho )}}{{{\omega _1}(\varrho )}}+ \frac{{{\vartheta _0}{e^{ - \ln \varrho }}(1 + {{(\ln \varrho )}^{\varpi - 1}})}}{2{\omega _1}(\varrho )}\Bigg ] < + \infty ,\nonumber \\{} & {} \quad \;{\omega _0}:= \int _1^{ + \infty } {\frac{{{{(\ln \varsigma )}^{\varpi - 1}}{\omega _1}(\varsigma )}}{{1 + {{(\ln \varsigma )}^{\varpi - 1}}}}}\frac{{d\varsigma }}{\varsigma }>0, \end{aligned}$$

(22)

and

$$\begin{aligned}&\begin{aligned} \int _1^{ + \infty } {\eta (\varrho )} \frac{{d\varrho }}{\varrho }< \frac{{2\Gamma (\varpi ) + 2}}{{\big (2\Gamma (\varpi ) + 1\big )\big (2\Gamma (\varpi ) + 3\big )}},~~{e^{\ln \varrho }}\eta (\varrho ) < \frac{{2[1 + {{(\ln \varrho )}^{\varpi - 1}}]}}{{2\Gamma (\varpi ) + 3}}. \end{aligned} \end{aligned}$$

(23)

Then BVP (6) has at least one positive solution.

Proof

Define

$$\begin{aligned} {\xi _0}: = \frac{{2\Gamma (\varpi )}}{{2\Gamma (\varpi ) + 1}} + \frac{{2\Gamma (\varpi ) + 3}}{{2\Gamma (\varpi ) + 2}}\int _1^{ + \infty } {\eta (\varsigma )} \frac{{d\varsigma }}{\varsigma }, \end{aligned}$$

then from (23) that ${\xi _0}< 1$. Consider the cone

$$\begin{aligned} \mathcal {C} = \big \{ \Im |~\Im \in X,~\Im (\varrho ) \ge 0,~\varrho \ge 1\big \}. \end{aligned}$$

Let

$$\begin{aligned}{} & {} {\Omega _1} = \Big \{ \Im \in X\big |~{\xi _1}||\Im |{|_X}< \frac{{\Im (\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}< {M_1},~\varrho \ge 1\Big \}, \\{} & {} \quad {\Omega _2} = \big \{\Im \in X\big |~||\Im |{|_X} < {M_2},~\varrho \ge 1\big \}, \end{aligned}$$

where ${M_2} \in ({M_0},M),~{M_1} \in (0,{M_2}),~{\xi _1} \in ({\xi _0},1),$ and

$$\begin{aligned} {M_0}: = \max \bigg \{ \frac{{{\Lambda _0}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{\omega _2}(\varsigma )} \frac{{d\varsigma }}{\varsigma } + \frac{2}{{{\vartheta _0}\Gamma (\varpi )}}\int _1^{ + \infty } {{\vartheta _3}(\varsigma )} \frac{{d\varsigma }}{\varsigma },~\frac{1}{{{\omega _0}}}\int _1^{ + \infty } {{\omega _2}(\varsigma )} \frac{{d\varsigma }}{\varsigma }\bigg \}. \end{aligned}$$

(24)

Obviously, ${\Omega _1}$ and ${\Omega _2}$ are open bounded set of X.

Step 1. By Lemma 3.2, we obtain L is a Fredholm operator of index zero, then the proof of ${1^ \circ }$ in Theorem 2.1 is complete.

Step 2. In view of Lemma 3.4, we have $QN:X \rightarrow Y$ is continuous and bounded and ${K_P}(I - Q)N:X \rightarrow X$ is compact on each bounded subset of X, that is, the condition ${2^ \circ }$ of Theorem 2.1 holds.

Step 3. We show that ${3^ \circ }$ in Theorem 2.1 holds. Using proof by contradiction, suppose that there exists ${\Im ^ * } \in \mathcal {C} \cap \partial {\Omega _2} \cap {\text {dom}}L$ and ${\lambda ^ * } \in (0,1)$ such that $L{\Im ^ * } = {\lambda ^ * }N{\Im ^ * }.$ Note that

$$\begin{aligned} {\Im ^ * } = (I - P){\Im ^ * } + P{\Im ^ * } = {K_P}L(I - P){\Im ^ * } + P{\Im ^ * } = {K_P}L{\Im ^ * } + P{\Im ^ * }, \end{aligned}$$

then

$$\begin{aligned}&\begin{aligned}&\frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}= - \frac{1}{{\Gamma (\varpi )}}\frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}\big ({}^HD_{1 + }^\varpi {\Im ^ * }(\varsigma )\big )\frac{{d\varsigma }}{\varsigma }} \\&\quad + \frac{1}{{\Gamma (\varpi )}}\int _1^\varrho {\frac{{{{\left(\ln \frac{\varrho }{\varsigma }\right)}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}} \big ({}^HD_{1 + }^\varpi {\Im ^ * }(\varsigma )\big )\frac{{d\varsigma }}{\varsigma } \\&\quad + \frac{1}{{\Gamma (\varpi )}}\frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}{\Im ^ * }(\varsigma )\frac{{d\varsigma }}{\varsigma }} \\&< \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}\big |{}^HD_{1 + }^\varpi {\Im ^ * }(\varsigma )\big |\frac{{d\varsigma }}{\varsigma }} + \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } \big | {}^HD_{1 + }^\varpi {\Im ^ * }(\varsigma )\big |\frac{{d\varsigma }}{\varsigma }\\&\;\;\;+ \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}{\Im ^ * }(\varsigma )\frac{{d\varsigma }}{\varsigma }}\\&< \frac{2}{{\Gamma (\varpi )}}\int _1^{ + \infty } \big | {}^HD_{1 + }^\varpi {\Im ^ * }(\varsigma )\big |\frac{{d\varsigma }}{\varsigma } + \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}{\Im ^ * }(\varsigma )\frac{{d\varsigma }}{\varsigma }}. \\ \end{aligned}\ \end{aligned}$$

(25)

It follows from (19) and (20) that

$$\begin{aligned} \begin{aligned}&{}^HD_{1 + }^\varpi {\Im ^ * }(\varrho ) = {\lambda ^ * }\hbar (\varrho ,{\Im ^ * }(\varrho )) \\&\le - {\lambda ^ * }{\vartheta _1}(\varrho )\big |\hbar \big (\varrho ,{\Im ^ * }(\varrho )\big )\big | + {\lambda ^ * }{\vartheta _2}(\varrho )\frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} + {\lambda ^ * }{\vartheta _3}(\varrho ) \\&\le - {\vartheta _1}(\varrho )\big |{}^HD_{1 + }^\varpi {\Im ^ * }(\varrho )\big | + {\vartheta _2}(\varrho )\frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} + {\vartheta _3}(\varrho ), \\ \end{aligned}\ \end{aligned}$$

(26)

and

$$\begin{aligned}&\begin{aligned} {}^HD_{1 + }^\varpi {\Im ^ * }(\varrho ) = {\lambda ^ * }\hbar \big (\varrho ,{\Im ^ * }(\varrho )\big ) \le - {\lambda ^ * }{\omega _1}(\varrho )\frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} + {\lambda ^ * }{\omega _2}(\varrho ). \end{aligned} \end{aligned}$$

(27)

In view of the fact that

$$\begin{aligned}&\int _1^{ + \infty } {{}^HD_{1 + }^\varpi {\Im ^ * }(\varrho )} \frac{{d\varrho }}{\varrho } = \int _1^{ + \infty } \big ({}^HD_{1 + }^{\varpi - 1}{\Im ^ * }(\varrho )\big )'d\varrho \nonumber \\&= \mathop {\lim }\limits _{\varrho \rightarrow + \infty } {}^HD_{1 + }^{\varpi - 1}{\Im ^ * }(\varrho ) - {}^HD_{1 + }^{\varpi - 1}{\Im ^ * }(1) = 0, \end{aligned}$$

(28)

which combined with (26) and (28), it follows

$$\begin{aligned} 0&= \int _1^{ + \infty } {{}^HD_{1 + }^\varpi {\Im ^ * }(\varrho )} \frac{{d\varrho }}{\varrho } \\&\le - \int _1^{ + \infty } {{\vartheta _1}(\varrho )} \big |{}^HD_{1 + }^\varpi {\Im ^ * }(\varrho )\big |\frac{{d\varrho }}{\varrho } + \int _1^{ + \infty } {{\vartheta _2}(\varrho )} \frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\frac{{d\varrho }}{\varrho } + \int _1^{ + \infty } {{\vartheta _3}(\varrho )} \frac{{d\varrho }}{\varrho }, \end{aligned}$$

that is,

$$\begin{aligned} \int _1^{ + \infty } {\big |{}^HD_{1 + }^\varpi {\Im ^ * }(\varrho )\big |\frac{{d\varrho }}{\varrho }} \le \frac{1}{{{\vartheta _0}}}\int _1^{ + \infty } {{\vartheta _2}(\varrho )} \frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\frac{{d\varrho }}{\varrho } + \frac{1}{{{\vartheta _0}}}\int _1^{ + \infty } {{\vartheta _3}(\varrho )} \frac{{d\varrho }}{\varrho }. \end{aligned}$$

On the other hand, from (27) and (28), we also have

$$\begin{aligned} 0&= \int _1^{ + \infty } {{}^HD_{1 + }^\varpi {\Im ^ * }(\varrho )} \frac{{d\varrho }}{\varrho }\\&\le - \int _1^{ + \infty } {{\lambda ^ * }{\omega _1}(\varrho )\frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}}\frac{{d\varrho }}{\varrho } + \int _1^{+\infty }{{\lambda ^ * }{\omega _2}(\varrho )} \frac{{d\varrho }}{\varrho }, \end{aligned}$$

that is,

$$\begin{aligned} \int _1^{ + \infty } {{\omega _1}(\varrho )\frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}} \frac{{d\varrho }}{\varrho } \le \int _1^{ + \infty } {{\omega _2}(\varrho )} \frac{{d\varrho }}{\varrho }. \end{aligned}$$

This combined with (19), (22), (24) and (25), we obtain

$$\begin{aligned}&\frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}< \frac{2}{{\Gamma (\varpi )}}\int _1^{ + \infty } \big | {}^HD_{1 + }^\varpi {\Im ^ * }(\varrho )\big |\frac{{d\varrho }}{\varrho }\\&\quad + \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varrho }}{\Im ^ * }(\varrho )\frac{{d\varrho }}{\varrho }} \\&\le \frac{2}{{{\vartheta _0}\Gamma (\varpi )}}\int _1^{ + \infty } \frac{{{\vartheta _2}(\varrho )}{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\frac{{d\varrho }}{\varrho } + \frac{2}{{{\vartheta _0}\Gamma (\varpi )}}\int _1^{ + \infty } {{\vartheta _3}(\varrho )} \frac{{d\varrho }}{\varrho }\\&\quad + \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varrho }}{\Im ^ * }(\varrho )\frac{{d\varrho }}{\varrho }} \\&= \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } \bigg [{\frac{{2{\vartheta _2}(\varrho )}}{{{\vartheta _0}}}} \frac{{{\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} + {e^{ - \ln \varrho }}{\Im ^ * }(\varrho )\bigg ]\frac{{d\varrho }}{\varrho }\\&\quad + \frac{2}{{{\vartheta _0}\Gamma (\varpi )}}\int _1^{ + \infty } {{\vartheta _3}(\varrho )} \frac{{d\varrho }}{\varrho } \\&= \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } \frac{2}{{{\vartheta _0}}}\bigg [\frac{{{\vartheta _2}(\varrho )}}{{{\omega _1}(\varrho )}}\\&\quad +\frac{{{\vartheta _0}{e^{ - \ln \varrho }}{{\big (1 + {{(\ln \varrho )}^{\varpi - 1}}\big )}}}}{{2{\omega _1}(\varrho )}} \bigg ]\frac{{{\omega _1}(\varrho ){\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\frac{{d\varrho }}{\varrho }+ \frac{2}{{{\vartheta _0}\Gamma (\varpi )}}\int _1^{ + \infty } {{\vartheta _3}(\varrho )} \frac{{d\varrho }}{\varrho } \\&\le \frac{{{\Lambda _0}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {\frac{{{\omega _1}(\varrho ){\Im ^ * }(\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\frac{{d\varrho }}{\varrho }}\\&\quad + \frac{2}{{{\vartheta _0}\Gamma (\varpi )}}\int _1^{ + \infty } {{\vartheta _3}(\varrho )} \frac{{d\varrho }}{\varrho } \\&\le \frac{{{\Lambda _0}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{\omega _2}(\varrho )\frac{{d\varrho }}{\varrho }} \\&\quad + \frac{2}{{{\vartheta _0}\Gamma (\varpi )}}\int _1^{ + \infty } {{\vartheta _3}(\varrho )} \frac{{d\varrho }}{\varrho } \le {M_0} < {M_2}, \end{aligned}$$

which is contradict to ${\Im ^ * } \in \mathcal {C} \cap \partial {\Omega _2} \cap {\text {dom}}L$, the proof of ${3^ \circ }$ in Theorem 2.1 is complete.

Step 4. Let $(\gamma \Im )(\varrho ) = |\Im (\varrho )|$, it is easy to demonstrate that $\gamma :X \rightarrow \mathcal {C}$ is a retraction. The proof of ${4^ \circ }$ in Theorem 2.1 is now complete.

Step 5. In order to prove ${5^ \circ }$ in Theorem 2.1, let $\Im \in {\text {Ker}}L \cap {\bar{\Omega }_2},$ then $\Im (\varrho ) = c{(\ln \varrho )^{\varpi - 1}},~\varrho \ge 1,~c \in \mathbb {R}$. Define

$$\begin{aligned} H\big (c{(\ln \varrho )^{\varpi - 1}},\upsilon \big )&= \big [I - \upsilon \big (P + JQN\big )\gamma \big ]\big (c{(\ln \varrho )^{\varpi - 1}}\big ) \\&= \Big [c - \upsilon \big |c\big | - \upsilon \int _1^{ + \infty } {\hbar \big (\varsigma ,|c|{{(\ln \varsigma )}^{\varpi - 1}}\big )\frac{{d\varsigma }}{\varsigma }} \Big ]{(\ln \varrho )^{\varpi - 1}}, \end{aligned}$$

where $c \in [ - {M_2},{M_2}],~\upsilon \in [0,1].$ Define a homeomorphism operator $\mathcal {J}:{\text {Ker}}L \cap {\bar{\Omega }_2} \rightarrow \mathbb {R}$ by

$$\begin{aligned} \mathcal {J}(c{(\ln \varrho )^{\varpi - 1}}) {=} c, \end{aligned}$$

then

$$\begin{aligned}&{d_B}\big (H(c{(\ln \varrho )^{\varpi - 1}},\upsilon \big ),{\text {Ker}}L \cap {\Omega _2},0) \\&= {d_B}\big (\mathcal {J}H(\mathcal {J}^{ - 1}c,\upsilon ),\mathcal {J}({\text {Ker}}L \cap {\Omega _2}),\mathcal {J}(0)\big ) \\&= {d_B}\big ({J_1}H(\mathcal {J}^{ - 1}c,\upsilon ),\mathcal {J}({\text {Ker}}L \cap {\Omega _2}),0\big ). \end{aligned}$$

Using (20) and (23), we can show that $\mathcal {J}H(\mathcal {J}^{ - 1}c,\upsilon ) = 0$ implies $c \ge 0.$ Let ${a_0} \in \mathcal {J}({\text {Ker}}L \cap {\partial \Omega _2}),$ then $|{a_0}| = {M_2}.$ Assume that $\mathcal {J}H(\mathcal {J}^{ - 1}{a_0},\upsilon ) = 0,~\upsilon \in (0,1],$ then ${a_0} = {M_2}.$ From (20) and (22), we have

$$\begin{aligned}&{M_2} = \upsilon \Bigg ({M_2} + \int _1^{ + \infty } \hbar \big ({\varsigma }, {M_2}{(\ln \varsigma )^{\varpi - 1}}\big )\frac{{d\varsigma }}{\varsigma }\Bigg ) \\&\quad \le \upsilon \Bigg ({M_2} - {M_2}\int _1^{ + \infty } {{\omega _1}(\varsigma )} \frac{{{{(\ln \varsigma )}^{\varpi - 1}}}}{{1 + {{(\ln \varsigma )}^{\varpi - 1}}}}\frac{{d\varsigma }}{\varsigma } + \int _1^{ + \infty } {{\omega _2}(\varsigma )} \frac{{d\varsigma }}{\varsigma }\Bigg ) \\&\quad < \upsilon {M_2} \le {M_2}. \end{aligned}$$

This leads to a contradiction. Moreover, if $\upsilon = 0$, then ${M_2} = 0$, which is also a contradiction. Therefore, $\mathcal {J}H(\mathcal {J}^{ - 1}c,\upsilon ) \ne 0$, for $c \in \mathcal {J}({\text {Ker}}L \cap {\partial \Omega _2})$, $\upsilon \in [0,1].$ Consequently,

$$\begin{aligned}&{d_B}\big ([I - (P + JQN)\gamma ]{|_{{\text {Ker}}L}},~{\text {Ker}}L \cap {\Omega _2},~0\big ) \\&\quad = {d_B}\big (H( \cdot ,1),~{\text {Ker}}L \cap {\Omega _2},~0\big ) \\&\quad = {d_B}\big (\mathcal {J}H(\mathcal {J}^{ - 1}c,1),~\mathcal {J}({\text {Ker}}L \cap {\Omega _2}),~0\big ) \\&\quad = {d_B}\big (\mathcal {J}H(\mathcal {J}^{ - 1}c,0),~\mathcal {J}({\text {Ker}}L \cap {\Omega _2}),~0\big ) \\&\quad = {d_B}\big (I,~\mathcal {J}({\text {Ker}}L \cap {\Omega _2}),~0) \ne 0. \end{aligned}$$

The demonstration of ${5^ \circ }$ in Theorem 2.1 has been concluded.

Step 6. To prove ${6^ \circ }$ in Theorem 2.1, take ${\Im _0} = 1 + {(\ln \varrho )^{\varpi - 1}} \in \mathcal {C}\backslash \{ 0\}$ and $\tau ({\Im _0}) = 1$, then

$$\begin{aligned} \mathcal {C}({\Im _0}) = \Big \{ \Im \in \mathcal {C}\big |\mathop {\inf }\limits _{\varrho \ge 1} \frac{{\Im (\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} > 0\Big \}. \end{aligned}$$

Since $\mathop {\lim }\limits _{\varrho \rightarrow \infty } \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} = 1,$ there exists a ${\varrho _0} > 0,$ such that

$$\begin{aligned} \frac{{{{(\ln {\varrho _0})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _0})}^{\varpi - 1}}}} > \frac{{2\Gamma (\varpi ) + 1}}{{2\Gamma (\varpi ) + 2}}. \end{aligned}$$

For $\Im \in \mathcal {C}({\Im _0}) \cap \partial {\Omega _1}$, we have

$$\begin{aligned} ||\Im |{|_X} \le {M_1} < {M_2},\quad \frac{{\Im (\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} \ge {\xi _1}||\Im |{|_X}. \end{aligned}$$

Then, by (18), (20) and (23), we get

$$\begin{aligned}&\frac{{(\Xi \Im )({\varrho _0})}}{{1 + {{(\ln {\varrho _0})}^{\varpi - 1}}}}= \frac{1}{{1 + {{(\ln {\varrho _0})}^{\varpi - 1}}}}\Big [P\Im ({\varrho _0}) + \big (JQN + {K_P}(I - Q)N\big )\Im ({\varrho _0}) \Big ] \\&\quad = \frac{{{{(\ln {\varrho _0})}^{\varpi - 1}}}}{{1 + {{(\ln {\varrho _0})}^{\varpi - 1}}}}\Bigg [\frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}\Im (\varsigma )\frac{{d\varsigma }}{\varsigma }} + \frac{1}{{\Gamma (\varpi )}}\int _1^{ + \infty } {G({\varrho _0},\varsigma )\hbar \big (\varsigma ,\Im (\varsigma )\big )\frac{{d\varsigma }}{\varsigma }} \Bigg ] \\&\quad> \frac{{2\Gamma (\varpi ) + 1}}{{2\Gamma (\varpi ) + 2}}\Bigg [\frac{{{\xi _1}||\Im |{|_X}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}\big (1 + {{(\ln \varsigma )}^{\varpi - 1}}\big )\frac{{d\varsigma }}{\varsigma }}- \frac{||\Im |{|_X}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {G({\varrho _0},\varsigma )\eta (\varsigma )\frac{{d\varsigma }}{\varsigma }} \Bigg ]\\&\quad> ||\Im ||{|_X}\frac{{2\Gamma (\varpi ) + 1}}{{2\Gamma (\varpi )}}\Big ({\xi _1} - \frac{{2\Gamma (\varpi ) + 3}}{{2\Gamma (\varpi ) + 2}}\int _1^{ + \infty } {\eta (\varsigma )\frac{{d\varsigma }}{\varsigma }} \Big )>||\Im |{|_X}, \end{aligned}$$

that is, $||\Im |{|_X} \le \tau ({\Im _0})||\Xi \Im |{|_X},~\forall \Im \in \mathcal {C}({\Im _0}) \cap \partial {\Omega _1}$. Hence, the condition ${6^ \circ }$ of Theorem 2.1 is satisfied.

Step 7. To prove ${7^ \circ }$ in Theorem 2.1. For $\Im \in \partial {\Omega _2}$, it follows from (20) and (23) that

$$\begin{aligned}&(P + JQN)(\gamma \Im )(\varrho ) \\&= \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}|\Im (\varsigma )|\frac{{d\varsigma }}{\varsigma }} + {(\ln \varrho )^{\varpi - 1}}\int _1^{ + \infty } {\hbar \big (\varsigma ,|\Im (\varsigma )|\big )} \frac{{d\varsigma }}{\varsigma } \\&\ge \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } \big [ {e^{ - \ln \varsigma }}\big (1 + {(\ln \varsigma )^{\varpi - 1}}\big ) - \eta (\varsigma )\big ]\frac{{|\Im (\varsigma )|}}{{1 + {{(\ln \varsigma )}^{\varpi - 1}}}}\frac{{d\varsigma }}{\varsigma }\\&\ge 0. \end{aligned}$$

So, $(P + JQN)\gamma (\partial {\Omega _2}) \subset \mathcal {C}.$ Subsequently, the condition ${7^ \circ }$ of Theorem 2.1 is met.

Step 8. To prove ${8^ \circ }$ in Theorem 2.1. For $\Im \in {\bar{\Omega }_2}\backslash {\Omega _1},$ from (18), (20) and (23), we obtain

$$\begin{aligned}&{{\Xi _\gamma } }\Im (\varrho ) = \big [P + JQN + {K_P}(I - Q)N\big ]|\Im (\varrho )| \\&\quad = \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {{e^{ - \ln \varsigma }}|\Im (\varsigma )|\frac{{d\varsigma }}{\varsigma }} + \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\int _1^{ + \infty } {G(\varrho ,\varsigma )\hbar \big (\varsigma ,|\Im (\varsigma )|\big )\frac{{d\varsigma }}{\varsigma }} \\&\quad \ge \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{\Gamma (\varpi )}}\bigg [\int _1^{ + \infty } {\Big ({e^{ - \ln \varsigma }}\big (1 + {{(\ln \varsigma )}^{\varpi - 1}}\big ) - \left(\Gamma (\varpi ) + \frac{3}{2} \right)\eta (\varsigma )\Big )\frac{{|\Im (\varsigma )|}}{{1 + {{(\ln \varsigma )}^{\varpi - 1}}}}}\bigg ]\frac{{d\varsigma }}{\varsigma } \\&\quad \ge 0, \end{aligned}$$

this implies, ${\Xi _\gamma }({\bar{\Omega }_2}\backslash {\Omega _1}) \subset \mathcal {C}.$ Hence, the condition ${8^ \circ }$ of Theorem 2.1 is fulfilled. Consequently, by Theorem 2.1, we deduce that BVP (6) has at least one positive solution in $\mathcal {C} \cap ({\bar{\Omega }_2}\backslash {\Omega _1})$. This completes the proof. $\square$

4 Examples

Example 4.1

Consider the following BVP

$$\begin{aligned} {\left\{ \begin{array}{ll} {}^HD_{1 + }^{3.5}\Im (\varrho )=\hbar \big (\varrho ,\Im (\varrho )\big ),\;\;\;\;\varrho \in (1, + \infty ), \\ \Im (1) = \Im '(1) = \Im ''(1) = 0,\;\;\;\;{}^HD_{1 + }^{2.5}\Im (1)=\mathop {\lim }\limits _{\varrho \rightarrow + \infty } {}^HD_{1 + }^{2.5}\Im (\varrho ), \end{array}\right. } \end{aligned}$$

(29)

where

$$\begin{aligned} \hbar (\varrho ,\Im ) = - {\omega _1}(\varrho )\frac{\Im }{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} + {\omega _2}(\varrho ),\; {\omega _1}(\varrho ) = \frac{{1 + {{(\ln \varrho )}^{2.5}}}}{{35\varrho }},\;{\omega _2}(\varrho ) = \frac{1}{{1 + {{(\ln \varrho )}^2}}}. \end{aligned}$$

Let ${\phi _\jmath }(\varrho ) = {\omega _1}(\varrho )\jmath + {\omega _2}(\varrho ),$ it is easy to verify that $\hbar$ satisfies the condition (H). Take

$$\begin{aligned} {\vartheta _1}(\varrho ) = 2,\;{\vartheta _2}(\varrho ) = \eta (\varrho ) = \frac{{1 + {{(\ln \varrho )}^{2.5}}}}{{35\varrho }},\;{\vartheta _3}(\varrho )=\frac{3}{{1 + {{(\ln \varrho )}^2}}}, \end{aligned}$$

then

$$\begin{aligned}&- {\vartheta _1}(\varrho )|\hbar (\varrho ,\Im )| + {\vartheta _2}(\varrho )\frac{\Im }{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} + {\vartheta _3}(\varrho ) \\&\quad \ge - \frac{{2\Im }}{{35\varrho }} - \frac{2}{{1 + {{(\ln \varrho )}^2}}} + \frac{\Im }{{35\varrho }} + \frac{3}{{1 + {{(\ln \varrho )}^2}}} \\&\quad =\hbar (\varrho ,\Im ), \end{aligned}$$

and

$$\begin{aligned} - \eta (\varrho )\frac{\Im }{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} = -\frac{\Im }{{35\varrho }}. \end{aligned}$$

Therefore, $\hbar (\varrho ,\Im )$ satisfies (19) and (20). By calculation,

$$\begin{aligned} {\vartheta _0}= & {} 2,\;\;\;\int _1^{ + \infty } {{\vartheta _3}(\varrho )} \frac{{d\varrho }}{\varrho } = \frac{{3\pi }}{2}> 0,\;\;\;\int _1^{ + \infty } {{\omega _2}(\varrho )} \frac{{d\varrho }}{\varrho } = \frac{\pi }{2}> 0, \\ {\Lambda _0}:= & {} \frac{2}{{{\vartheta _0}}}\mathop {\sup }\limits _{\varrho \ge 1} \Bigg [\frac{{{\vartheta _2}(\varrho )}}{{{\omega _1}(\varrho )}}+ \frac{{{\vartheta _0}{e^{ - \ln \varrho }}(1 + {{(\ln \varrho )}^{2.5}})}}{2{\omega _1}(\varrho )}\Bigg ] = 36< + \infty , \\{} & {} \int _1^{ + \infty } {\eta (\varrho )} \frac{{d\varrho }}{\varrho } = \frac{{1 + \Gamma (3.5)}}{{35}} \approx 0.1235< \frac{{\Gamma (3.5) + 1}}{{2\big (\Gamma (3.5) + 0.5\big )\big (\Gamma (3.5) + 1.5\big )}}\approx 0.1172, \\{} & {} {\omega _0}{:=} \int _1^{ + \infty } {\frac{{{{(\ln \varsigma )}^{2.5}}{\omega _1}(\varsigma )}}{{1 + {{(\ln \varsigma )}^{2.5}}}}}\frac{{d\varsigma }}{\varsigma }\approx 0.09495{>}0,\;{e^{\ln \varrho }}\eta (\varrho ) {=} \frac{{1 + {{(\ln \varrho )}^{2.5}}}}{{35}}{<} \frac{{1 + {{(\ln \varrho )}^{2.5}}}}{{\Gamma (3.5) + 1.5}}. \end{aligned}$$

Hence, (21)–(23) hold. According to Theorem 3.1 that BVP (29) has at least one positive solution.