Solvability of a Coupled System of Hadamard Fractional p-Laplacian Differential Equations with Infinite-Point Boundary Conditions at Resonance on an Unbounded Interval

Abstract

This paper investigates a coupled system of Hadamard fractional p-Laplacian differential equations defined on an unbounded interval, subject to infinitely many points boundary conditions and formulated under a resonance framework. Under suitable growth assumptions imposed on the nonlinear terms of the system, the existence of solutions is established by means of the Ge–Mawhin’s continuation theorem. Moreover, an example is constructed to demonstrate the applicability of the main results.

1. Introduction

The formulation of boundary value problems (BVPs) on unbounded intervals emerged as a response to the challenges of modeling nonlinear behaviors commonly observed in applied sciences. In [1], Agarwal and O’Regan presented a systematic treatment of the theoretical foundations and analytical techniques associated with such problems. Compared to BVPs defined on finite intervals, those posed on semi-infinite domains or the entire real line are better suited for capturing essential features of nonlinear diffusion, heat conduction, and stability boundary layer development, while also posing significantly greater analytical challenges. For example, in the study of phase transition phenomena in solids with temperature-dependent thermal conductivity, the function $\vartheta$, which describes the temperature distribution, is derived through solving a BVP posed over an unbounded interval:

$$\left\{\begin{array}{c}{\displaystyle \frac{d}{d\eta }\left\{(1+\mu \vartheta )\frac{d\vartheta }{d\eta }\right\}+2\eta \frac{d\vartheta }{d\eta }=0,}\hfill \\ \vartheta (0)=0,\vartheta (\infty )=1,\hfill \end{array}\right.$$

where $\mu$ is a physical constant [1].

Fractional differential equations (FDEs) have emerged as an active area of research in mathematical analysis due to their capacity to characterize the nonlocality and memory effects inherent in complex systems across natural and engineering sciences. Fractional calculus-based models, as generalizations of classical integer-order frameworks, demonstrate superior capability in depicting complex dynamical behaviors. Their extensive use in mechanics, control, and life sciences highlights their growing importance in modern theoretical investigations [2,3,4,5,6]. Due to the wide applicability of BVPs for differential equations defined on unbounded intervals in modeling real-world phenomena, substantial research has focused on the existence of solutions for fractional boundary value problems (FBVPs) on unbounded intervals, yielding numerous meaningful findings [7,8,9,10].

The Riemann–Liouville and Caputo fractional derivatives are among the most commonly employed definitions, owing to their suitability for handling classical mathematical models. In contrast, the Hadamard fractional derivative, as a distinct class of nonlocal operators with notable structural characteristics, has garnered increasing attention in recent years. Initially introduced by Hadamard in 1892, this operator employs a logarithmic kernel with an exponential weight, endowing it with unique advantages in theoretical modeling and practical applications [11]. Owing to the intricate structure of its integral kernel, significant advances have been made in the study of BVPs involving Hadamard derivatives, particularly on infinite intervals, where issues such as theoretical analysis and the existence of solutions have received considerable attention. For recent developments in this direction, the reader is referred to [12,13,14,15,16,17,18,19]. For instance, Nyamoradi and Ahmad [12] explored integral BVPs involving Hadamard FDEs defined on an infinite interval. Utilizing the Leggett–Williams fixed-point theorem combined with iterative methods, they demonstrated the existence of at least two or three positive solutions. Zhai and Liu [13] investigated BVPs characterized by integral and multi-point boundary conditions (BCs). By applying fixed-point techniques for sum-type operators within partially ordered Banach spaces, they established the local existence and uniqueness of positive solutions and developed iterative schemes for their approximation. Moreover, Luca and Tudorache [14] examined a nonlinear coupled system with Hadamard-type fractional derivatives subject to Riemann–Stieltjes integral BCs. Their analysis, which employed the Guo–Krasnoselskii and Leggett–Williams fixed-point theorems, verified the existence of positive solutions. In addition, Deren and Cerdik [15] focused on multi-point BVPs for coupled Hadamard-type systems on unbounded domains, where they utilized a monotone iterative technique to prove the existence of extremal positive solutions. In [17], Cerdik examined FDEs featuring integral and multi-point BCs on unbounded domains. By combining the Avery–Peterson fixed-point theorem with the nonlinear alternative of Leray–Schauder, the work established the existence and multiplicity results for positive solutions. Meanwhile, Xu, Cui, and O’Regan [18] studied Hadamard-type FDEs under three-point BCs. Employing fixed-point index techniques alongside growth conditions linked to the spectral radius of relevant linear operators, they formulated sufficient conditions guaranteeing the existence of positive solutions.

Traditional two-point BVPs usually impose conditions at the endpoints of the interval. However, with increasing complexity in applied models and the emergence of multi-scale boundary structures, multi-point BVPs have drawn growing attention due to their structural advantages in characterizing distributed parameter systems and piecewise control mechanisms. In particular, infinite-point BCs significantly enhance the ability to describe complex boundary information and encompass generalized boundary frameworks such as product-type and sequence-type dependencies, showing notable potential in both theoretical analysis and mathematical modeling. For example, modeling diffusion processes in heterogeneous media may involve BCs influenced by infinitely many point, thereby offering a more accurate depiction of the complexity inherent in physical boundaries. Currently, there exists a foundational body of research concerning infinite-point BVPs on bounded intervals. Nevertheless, extending such problems to the context of unbounded intervals introduces considerable analytical challenges. On one hand, it necessitates a precise characterization of the solution’s asymptotic behavior at infinity. On the other hand, methodological development relies on adaptable function spaces and operator frameworks to ensure the applicability and convergence of the theory. Given these analytical challenges and methodological demands, conducting systematic investigations into infinite-point BVPs for FDEs on unbounded intervals holds significant importance for enriching the theoretical framework of nonlinear BVPs and enhancing the expressive power of mathematical models. Although notable progress has been made in recent years regarding the existence of solutions to fractional infinite-point BVPs on bounded intervals [20,21,22,23,24,25], the corresponding existence theory for such problems in unbounded domains remains insufficiently developed [26,27,28] and continues to demand deeper exploration. Ge, Zhou, and Kou [26] applied the coincidence degree theory to examine a resonance BVP arising in a coupled fractional differential system defined on an infinite domain and subject to infinite-point BCs:

$$\left\{\begin{array}{c}{\mathfrak{D}}_{0+}^{\gamma }{\chi }_1(\mathfrak{s})={\mathfrak{h}}_1(\mathfrak{s},{\chi }_1(\mathfrak{s}),{\mathfrak{D}}_{0+}^{\sigma -1}{\chi }_2(\mathfrak{s})),\mathfrak{s}\in (0,+\infty ),\hfill \\ {\mathfrak{D}}_{0+}^{\sigma }{\chi }_2(\mathfrak{s})={\mathfrak{h}}_2(\mathfrak{s},{\chi }_2(\mathfrak{s}),{\mathfrak{D}}_{0+}^{\gamma -1}{\chi }_1(\mathfrak{s})),\mathfrak{s}\in (0,+\infty ),\hfill \\ {\chi }_1(0)=0,\underset{\mathfrak{t}\to \infty }{\lim }{\mathfrak{D}}_{0+}^{\gamma -1}{\chi }_1(\mathfrak{s})={\displaystyle \sum _{j=1}^{+\infty }}{\alpha }_j{\chi }_1({\delta }_j),\hfill \\ {\chi }_2(0)=0,\underset{\mathfrak{s}\to \infty }{\lim }{\mathfrak{D}}_{0+}^{\sigma -1}{\chi }_2(\mathfrak{s})={\displaystyle \sum _{j=1}^{+\infty }}{\beta }_j{\chi }_2({\zeta }_j),\hfill \end{array}\right.$$

where $1<\gamma ,\sigma \le 2$, ${\mathfrak{D}}_{0+}^{\gamma },{\mathfrak{D}}_{0+}^{\sigma }$ are the Riemann–Liouville fractional derivatives, $0<{\delta }_1<{\delta }_2<\cdots <{\delta }_j<\cdots$, $0<{\zeta }_1<{\zeta }_2<\cdots <{\zeta }_j<\cdots$, ${\lim }_{j\to \infty }{\delta }_j=\infty$, ${\lim }_{j\to \infty }{\zeta }_j=\infty$, ${\mathfrak{h}}_1,{\mathfrak{h}}_2:[0,+\infty )\times {\mathbb{R}}^2\to \mathbb{R}$ satisfy Carathéodory conditions.

Despite substantial advances in the study of BVPs for Hadamard FDEs on the unbounded interval, particularly regarding the existence, multiplicity, and extremal structure of positive solutions, existing results are predominantly confined to nonresonant settings. By contrast, investigations into resonant configurations for such equations remain scarce, with only a limited number of works addressing this direction as referenced in [29,30,31,32]. Notably, no results have yet been reported for Hadamard FBVPs involving the p-Laplacian operator on infinite domains at resonance. Motivated by the gap identified in the current literature, this study seeks to broaden the theoretical framework for Hadamard-type FBVPs posed on an unbounded domain at resonance. Specifically, we investigate the following coupled p-Laplacian fractional differential systems subject to infinite-point BCs at resonance on an unbounded interval:

$$\left\{\begin{array}{c}{({\varphi }_p({}^{H}D_{1+}^{\alpha }x(t)))}^{\prime }+a(t)g(t,y(t),{}^{H}D_{1+}^{\beta -1}y(t),{}^{H}D_{1+}^{\beta }y(t))=0,t\in (1,+\infty ),\hfill \\ {({\varphi }_p({}^{H}D_{1+}^{\beta }y(t)))}^{\prime }+a(t)f(t,x(t),{}^{H}D_{1+}^{\alpha -1}x(t),{}^{H}D_{1+}^{\alpha }x(t))=0,t\in (1,+\infty ),\hfill \\ x(1)={}^{H}D_{1+}^{\alpha -1}x(1)=0,{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))={\displaystyle \sum _{i=1}^{+\infty }}{\alpha }_i{\varphi }_p({}^{H}D_{1+}^{\alpha }x({\xi }_i)),\hfill \\ y(1)={}^{H}D_{1+}^{\beta -1}y(1)=0,{\varphi }_p({}^{H}D_{1+}^{\beta }y(+\infty ))={\displaystyle \sum _{i=1}^{+\infty }}{\beta }_i{\varphi }_p({}^{H}D_{1+}^{\beta }y({\eta }_i)),\hfill \end{array}\right.$$

(1)

where $1<\alpha ,\beta \le 2$, ${}^{H}D_{1+}^{\kappa }$ denotes the Hadamard fractional derivatives of order $\kappa \in \{\alpha ,\beta \}$, $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_i<\cdots$, $0<{\eta }_1<{\eta }_2<\cdots <{\eta }_i<\cdots$, ${\lim }_{i\to +\infty }{\xi }_i=+\infty$, ${\lim }_{i\to +\infty }{\eta }_i=+\infty$, ${\displaystyle \sum _{i=1}^{+\infty }}{\alpha }_i=1,{\displaystyle \sum _{i=1}^{+\infty }}{\beta }_i=1$, $(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a(t)$}\right.)>0$ on $\mathfrak{I}=:[1,+\infty )$ and ${\displaystyle {\int }_1^{+\infty }a(t)\frac{dt}{t}<+\infty }$, ${\varphi }_p(\mathfrak{s})$ is p-Laplacian operator, ${\varphi }_p(\mathfrak{s})=|\mathfrak{s}{|}^{p-2}\mathfrak{s}$, ${\varphi }_p(0)=0$, $p>1$, ${\varphi }_p$ is invertible, ${\varphi }_p^{-1}={\varphi }_q$, with $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$q$}\right.=1$, the functions f and g satisfy the following condition:

$({\mathrm{A}}_1)$

$f,g:\mathfrak{I}\times {\mathbb{R}}^3\to \mathbb{R}$ satisfy a-Carathéodory condition, where the function $\hslash (\hslash =f,g)$ is said to be a-Carathéodory if and only if the following hold:

• For every $(\mathfrak{u},\mathfrak{v},\mathfrak{w})\in {\mathbb{R}}^3$, the function $t\mapsto \hslash (t,\mathfrak{u},\mathfrak{v},\mathfrak{w})$ is measurable with respect to the Lebesgue measure;
• Almost everywhere on the interval $\mathfrak{I}$, the mapping from $(\mathfrak{u},\mathfrak{v},\mathfrak{w})$ to $\hslash (t,\mathfrak{u},\mathfrak{v},\mathfrak{w})$ remains continuous on ${\mathbb{R}}^3$;
• Given any $r>0$, one can find a function ${\phi }_r:\mathfrak{I}\to [0,+\infty )$ that satisfies the integrability condition ${\displaystyle {\int }_1^{+\infty }a(t){\phi }_r(t)dt<+\infty }$, such that

  $$\max \left\{{\displaystyle \frac{|\mathfrak{u}|}{1+{(\ln t)}^{\kappa }}},{\displaystyle \frac{|\mathfrak{v}|}{1+\ln t}},|\mathfrak{w}|\right\}\le r,$$

  implies $|\hslash (t,\mathfrak{u},\mathfrak{v},\mathfrak{w})|\le {\phi }_r(t)$ for almost every $t\in \mathfrak{I}$.

The present work contributes several key innovations in both theoretical construction and methodological generalization:

• This paper extends the study of Hadamard fractional BVPs at resonance to systems involving p-Laplacian operators, thereby generalizing the results established in prior literature [29,30,31,32] and enriching the existing theory on Hadamard FDEs.
• The boundary configuration is formulated under infinite-point BCs on an unbounded interval, which can be regarded as a natural generalization of the classical finitely multi-point conditions. For instance, by setting ${\xi }_{i+1}={\xi }_{i+2}=\cdots =0$ and ${\eta }_{i+1}={\eta }_{i+2}=\cdots =0$, one recovers the finitely multi-point structure. Therefore, this work not only extends the results of [9] within the framework of coupled systems, but also generalizes the BCs from finitely multi-point cases to the case of infinite-point.
• An a-Carathéodory control function is introduced to characterize the nonlinear terms, ensuring regulated growth behavior and facilitating effective control of the nonlinear coupling components within the system. This provides a solid foundation for constructing a priori estimates.
• To address the analytic challenges posed by the unbounded domain, a weighted Banach space is employed to build an operator framework with high adaptability. This choice supports the formulation of suitable projection operators and the verification of compactness conditions, thereby enhancing the methodological robustness.
• The existence results derived herein encompass general Hadamard fractional differential systems as well as coupled systems involving the p-Laplacian operators. Applying the Ge–Mawhin’s continuation theorem, the solvability of the system is established, and illustrative example is provided to demonstrate the main results.

2. Preliminaries

This section focuses on a brief recollection of the essential theoretical preliminaries required in Section 3, in preparation for the subsequent analysis.

Definition 1

([33]). Consider a function $u:\mathfrak{I}\to \mathbb{R}$. The Hadamard fractional integral of order α ($\alpha >0$) is formulated as

$$\begin{array}{c}\hfill {}^{H}I_{1+}^{\alpha }u(t)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_1^t{\left(\ln {\displaystyle \frac{t}{s}}\right)}^{\alpha -1}u(s){\displaystyle \frac{ds}{s}},\end{array}$$

provided the integral on the right-hand side is well defined.

Definition 2

([33]). Let $u:\mathfrak{I}\to \mathbb{R}$ be a real-valued function. The Hadamard-type fractional derivative of order α ($\alpha >0$) is expressed as

$$\begin{array}{c}\hfill {}^{H}D_{1+}^{\alpha }u(t)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\left(t{\displaystyle \frac{d}{dt}}\right)}^n{\int }_1^t{\left(\ln {\displaystyle \frac{t}{s}}\right)}^{n-\alpha -1}u(s){\displaystyle \frac{ds}{s}},\end{array}$$

where $n=\left[\alpha \right]+1$, and the integral on the right-hand side exists.

Lemma 1 

([33]). Let $\alpha ,\beta >0$. Then the Hadamard-type fractional integral and derivative satisfy the following identities

$$\begin{array}{cc}\hfill {}^{H}I_{1+}^{\alpha }{(\ln t)}^{\beta -1}=& {\displaystyle \frac{\Gamma (\beta )}{\Gamma (\alpha +\beta )}}{(\ln t)}^{\alpha +\beta -1},{}^{H}D_{1+}^{\alpha }{(\ln t)}^{\beta -1}={\displaystyle \frac{\Gamma (\beta )}{\Gamma (\beta -\alpha )}}{(\ln t)}^{\beta -\alpha -1},\hfill \end{array}$$

in particular, for $j=1,2,\cdots ,\left[\alpha \right]+1$, one has

$${}^{H}D_{1+}^{\alpha }{(\ln t)}^{\alpha -j}=0.$$

Lemma 2

([33]). Let $\alpha >0$, $n=\left[\alpha \right]+1$, and suppose that the function u belongs to the space $C[1,\infty )\cap L^1[1,\infty )$. Then the general solution to the homogeneous Hadamard-type FDE

$${}^{H}D_{1+}^{\alpha }u(t)=0,$$

is given by

$$\begin{array}{c}\hfill u(t)=\sum _{i=1}^n{c}_i{(\ln t)}^{\alpha -i},c_i\in \mathbb{R}.\end{array}$$

Furthermore, the following relation holds:

$$\begin{array}{c}\hfill {}^{H}I_{1+}^{\alpha }{}^{H}D_{1+}^{\alpha }u(t)=u(t)+\sum _{i=1}^n{c}_i{(\ln t)}^{\alpha -i}.\end{array}$$

Definition 3

([34]). Let $(X,||·|{|}_X)$ and $(H,||·|{|}_H)$ denote two Banach spaces. A continuous operator $M:X\cap \mathrm{dom}M\to H$ is referred to as quasi-linear if it fulfills the following two conditions:

(¶)

$\mathrm{Im}M:=M(X\cap \mathrm{dom}M)$ forms a closed set in H;

(¶)

$\mathrm{Ker}M:=\{u\in X\cap \mathrm{dom}M:Mu=0\}$ is linearly isomorphic to ${\mathbb{R}}^n$ for some finite integer n.

Let $X_0=\mathrm{Ker}M$, and let $X_2$ be a complementary subspace of $X_0$ in X, so that $X=X_0\oplus X_2$. Likewise, let $H_0$ be a subspace of H, and let $H_2$ denote the complement of $H_0$ in H yielding the decomposition $H=H_0\oplus H_2$. Define two projection operators: $P:X\to X_0$ and $Q:H\to H_0$. Let $\Omega$ be an open and bounded subset of X that contains the origin $\theta$.

Definition 4

([34]). Let $N_{\lambda }:\overline{\Omega }\to H$ be a continuous operator depending on the parameter $\lambda \in [0,1]$, where we denote $N=N_1$. The associated solution set is defined by ${\sum }_{\lambda }=\{u\in \overline{\Omega }:Mu=N_{\lambda }u\}$. The operator $N_{\lambda }$ is said to be M-compact on $\overline{\Omega }$ with respect to M if the following hold:

(¶)

There exists a subspace $H_0\subset H$ such that $\dim H_0=\dim X_0$;

(¶)

There exists a continuous and completely continuous mapping $R:\overline{\Omega }\times [0,1]\to X_2$;

and for every $\lambda \in [0,1]$, the following conditions are satisfied:

($a_1$)

$(I-Q)N_{\lambda }(\overline{\Omega })\subset \mathrm{Im}M\subset (I-Q)H$;

($a_2$)

$Q{N}_{\lambda }u=\theta ,\lambda \in (0,1)\iff QNu=\theta$;

($a_3$)

The operator $R(·,0)$ vanishes identically, and on the solution set ${\Sigma }_{\lambda }$, one has $R(·,\lambda ){|}_{\sum \lambda }=(I-P){|}_{\sum \lambda }$;

($a_4$)

The operator identity $M[P+R(·,\lambda )]=(I-Q)N_{\lambda }$ holds throughout $\overline{\Omega }$.

Theorem 1

([34] Ge–Mawhin’s Continuation Theorem). Let $(X,||·|{|}_X)$ and $(H,||·|{|}_H)$ be two real Banach spaces, and let $\Omega \subset X$ be a bounded, nonempty open subset. Consider a quasilinear operator $M:X\cap \mathrm{dom},M\to H$, and let $N_{\lambda }:\overline{\Omega }\to H$ be a continuous family of operators parameterized by $\lambda \in [0,1]$, which is assumed to be M-compact on $\overline{\Omega }$. Suppose the following conditions are met:

(i) 

For all $(u,\lambda )\in \partial \Omega \times (0,1)$, the equation $Mu=N_{\lambda }u$ does not hold;

(ii) 

For every $u\in \mathrm{Ker}M\cap \partial \Omega$, one has $QNu\ne (0,0)$;

(iii) 

The topological degree of the map $JQN$ over the set $\mathrm{Ker}M\cap \Omega$ with respect to the origin is nonzero, i.e., $deg(JQN,KerM\cap \Omega ,0)\ne 0,$ where $N=N_1$, $Q:H\to \mathrm{Im}Q$ is a projection, and $J:\mathrm{Im}Q\to \mathrm{Ker}M$ is a homeomorphism satisfying $J(\theta )=\theta$.

Then the operator equation $Mu=Nu$ possesses at least one solution in the $\overline{\Omega }$.

3. Main Result

In this section, we investigate the solvability of the fractional BVP (1). By utilizing Theorem 1, we demonstrate that at least one solution exists. To proceed, we begin by introducing two appropriate Banach spaces. Define

$$X=X_1\times X_2,H=Z\times Z,$$

where

$$\begin{array}{cc}& X_1=\left\{x\left|x,{}^{H}D_{1+}^{\alpha }x\right.\in C(\mathfrak{I}),\underset{t\in \mathfrak{I}}{\sup }{\displaystyle \frac{|x(t)|}{1+{(\ln t)}^{\alpha }}}<+\infty ,\right.\hfill \\ & \left.\underset{t\in \mathfrak{I}}{\sup }{\displaystyle \frac{|{}^{H}D_{1+}^{\alpha -1}x(t)|}{1+\ln t}}<+\infty ,\underset{t\in \mathfrak{I}}{\sup }|{}^{H}D_{1+}^{\alpha }x(t)|<+\infty \right\},\hfill \\ & X_2=\left\{y\left|y,{}^{H}D_{1+}^{\beta }y\right.\in C(\mathfrak{I}),\underset{t\in \mathfrak{I}}{\sup }{\displaystyle \frac{|y(t)|}{1+{(\ln t)}^{\beta }}}<+\infty ,\right.\hfill \\ & \left.\underset{t\in \mathfrak{I}}{\sup }{\displaystyle \frac{|{}^{H}D_{1+}^{\beta -1}y(t)|}{1+\ln t}}<+\infty ,\underset{t\in \mathfrak{I}}{\sup }|{}^{H}D_{1+}^{\beta }y(t)|<+\infty \right\},\hfill \\ & Z=\left\{h:\mathfrak{I}\to \mathbb{R}\left|{\int }_1^{+\infty }a(t)|h(t)|dt<+\infty \right.\right\},\hfill \end{array}$$

endowed with the norms

$$\begin{array}{cc}& {||x||}_{X_1}=\max \left\{{||x||}_0,{||{}^{H}D_{1+}^{\alpha -1}x||}_1,{||{}^{H}D_{1+}^{\alpha }x||}_{\infty }\right\},\hfill \\ & {||y||}_{X_2}=\max \left\{{||y||}_0,{||{}^{H}D_{1+}^{\beta -1}y||}_1,{||{}^{H}D_{1+}^{\beta }y||}_{\infty }\right\},\hfill \\ & {||(x,y)||}_X={||x||}_{X_1}+{||y||}_{X_2},{||(h_1,h_2)||}_H={||h_1||}_z+{||h_2||}_z,\hfill \end{array}$$

where

$$\begin{array}{cc}& {||{}^{H}D_{1+}^{\alpha }x||}_{\infty }=\underset{t\in \mathfrak{I}}{\sup }|{}^{H}D_{1+}^{\alpha }x(t)|,{||{}^{H}D_{1+}^{\beta }y||}_{\infty }=\underset{t\in \mathfrak{I}}{\sup }|{}^{H}D_{1+}^{\beta }y(t)|,\hfill \\ & {||{}^{H}D_{1+}^{\alpha -1}x||}_1=\underset{t\in \mathfrak{I}}{\sup }\left|{\displaystyle \frac{{}^{H}D_{1+}^{\alpha -1}x(t)}{1+\ln t}}\right|,{||{}^{H}D_{1+}^{\beta -1}y||}_1=\underset{t\in \mathfrak{I}}{\sup }\left|{\displaystyle \frac{{}^{H}D_{1+}^{\beta -1}y(t)}{1+\ln t}}\right|,\hfill \\ & {||x||}_0=\underset{t\in \mathfrak{I}}{\sup }\left|{\displaystyle \frac{x(t)}{1+{(\ln t)}^{\alpha }}}\right|,{||y||}_0=\underset{t\in \mathfrak{I}}{\sup }\left|{\displaystyle \frac{y(t)}{1+{(\ln t)}^{\beta }}}\right|,\hfill \\ & {||h||}_z={\int }_1^{+\infty }a(t)|h(t)|dt.\hfill \end{array}$$

It is readily verified that both $(X,||·|{|}_X)$ and $(H,||·|{|}_H)$ are Banach spaces.

Define the operators $M:\mathrm{dom}M\subset X\to H$ and $N_{\lambda }:X\to H$ as follows:

$$\begin{array}{cc}& M(x(t),y(t))=(M_{1}x(t),M_{2}y(t)),(x(t),y(t))\in \mathrm{dom}M,\hfill \\ & N_{\lambda }(x(t),y(t))=\left(N_{1\lambda }y(t),N_{2\lambda }x(t)\right),(x(t),y(t))\in X,\lambda \in [0,1],\hfill \end{array}$$

where

$$\begin{array}{cc}& M_{1}x(t)={\displaystyle \frac{1}{a(t)}}{({\varphi }_p({}^{H}D_{1+}^{\alpha }x(t)))}^{\prime },N_{1\lambda }y(t)=-\lambda g(t,y(t),{}^{H}D_{1+}^{\beta -1}y(t),{}^{H}D_{1+}^{\beta }y(t)),\hfill \\ & M_{2}y(t)={\displaystyle \frac{1}{a(t)}}{({\varphi }_p({}^{H}D_{1+}^{\beta }y(t)))}^{\prime },N_{2\lambda }x(t)=-\lambda f(t,x(t),{}^{H}D_{1+}^{\alpha -1}x(t),{}^{H}D_{1+}^{\alpha }x(t)),\hfill \\ & \mathrm{dom}M=\left\{(x,y)\in X|{\varphi }_p({}^{H}D_{1+}^{\alpha }x)\in AC(\mathfrak{I}),{\varphi }_p({}^{H}D_{1+}^{\beta }y)\in AC(\mathfrak{I}),\right.\hfill \\ & x(1)={}^{H}D_{1+}^{\alpha -1}x(1)=0,{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))=\sum _{i=1}^{+\infty }{\alpha }_i{\varphi }_p({}^{H}D_{1+}^{\alpha }x({\xi }_i)),\hfill \\ & \left.y(1)=D_{1+}^{\beta -1}y(1)=0,{\varphi }_p({}^{H}D_{1+}^{\beta }y(+\infty ))=\sum _{i=1}^{+\infty }{\beta }_i{\varphi }_p({}^{H}D_{1+}^{\beta }y({\eta }_i))\right\}.\hfill \end{array}$$

Therefore, solving BVP (1) reduces to solving the corresponding operator equation

$$\begin{array}{c}\hfill M(x,y)=N(x,y),(x,y)\in \mathrm{dom}M,\end{array}$$

where $N=N_1$.

Lemma 3.

The operator $M:\mathrm{dom}M\cap X\to H$ is a quasilinear operator and satisfies

$$\begin{array}{c}\mathrm{Ker}M=\left\{(x,y)\in \mathrm{dom}M\right|x(t)=c_1{(\ln t)}^{\alpha },\hfill \\ y(t)=c_2{(\ln t)}^{\beta },\forall t\in \mathfrak{I},c_1,c_2\in \mathbb{R}\},\hfill \end{array}$$

(2)

$$\begin{array}{c}\mathrm{Im}M=\{(h_1,h_2)\in H|\sum _{i=1}^{+\infty }{\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)h_1(s)ds=0,\hfill \\ \sum _{i=1}^{+\infty }{\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)h_2(s)ds=0\}.\hfill \end{array}$$

(3)

Proof. 

For $(x,y)\in \mathrm{Ker}M,M(x,y)=(0,0)$. Then,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{a(t)}}{({\varphi }_p({}^{H}D_{1+}^{\alpha }x))}^{\prime }=0,{\displaystyle \frac{1}{a(t)}}{({\varphi }_p({}^{H}D_{1+}^{\beta }y))}^{\prime }=0.\end{array}$$

Taking into account the boundary conditions

$$\begin{array}{cc}& {\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))=\sum _{i=1}^{+\infty }{\alpha }_i{\varphi }_p({}^{H}D_{1+}^{\alpha }x({\xi }_i)),\hfill \\ & {\varphi }_p({}^{H}D_{1+}^{\beta }y(+\infty ))=\sum _{i=1}^{+\infty }{\beta }_i{\varphi }_p({}^{H}D_{1+}^{\beta }y({\eta }_i)),\hfill \end{array}$$

it follows that

$$\begin{array}{cc}& {\varphi }_p({}^{H}D_{1+}^{\alpha }x(t))={\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))=\sum _{i=1}^{+\infty }{\alpha }_i{\varphi }_p({}^{H}D_{1+}^{\alpha }x({\xi }_i)),\hfill \\ & {\varphi }_p({}^{H}D_{1+}^{\beta }y(t))={\varphi }_p({}^{H}D_{1+}^{\beta }y(+\infty ))=\sum _{i=1}^{+\infty }{\beta }_i{\varphi }_p({}^{H}D_{1+}^{\beta }y({\eta }_i)).\hfill \end{array}$$

By Lemma 2 and combined with $x(1)={}^{H}D_{1+}^{\alpha -1}x(1)=0$, $y(1)={}^{H}D_{1+}^{\beta -1}y(1)=0$, one immediately deduces that

$$\begin{array}{cc}\hfill x(t)& =c_1{(\ln t)}^{\alpha -1}+c_2{(\ln t)}^{\alpha -2}+{\displaystyle \frac{{\varphi }_q\left({\displaystyle \sum _{i=1}^{+\infty }}{\alpha }_i{\varphi }_p({}^{H}D_{1+}^{\alpha }x({\xi }_i))\right)}{\Gamma (\alpha +1)}}{(\ln t)}^{\alpha }\hfill \\ & ={\displaystyle \frac{{\varphi }_q\left({\displaystyle \sum _{i=1}^{+\infty }}{\alpha }_i{\varphi }_p({}^{H}D_{1+}^{\alpha }x({\xi }_i))\right)}{\Gamma (\alpha +1)}}{(\ln t)}^{\alpha }.\hfill \end{array}$$

Similarly, it can be concluded that

$$\begin{array}{c}\hfill y(t)={\displaystyle \frac{{\varphi }_q\left({\displaystyle \sum _{i=1}^{+\infty }}{\beta }_i{\varphi }_p({}^{H}D_{1+}^{\beta }y({\eta }_i))\right)}{\Gamma (\beta +1)}}{(\ln t)}^{\beta }.\end{array}$$

Conversely, if $x=c_1{(\ln t)}^{\alpha }$, and $y=c_2{(\ln t)}^{\beta }$, then $M_{1}x=0$ and $M_{2}y=0$, implying that Equation (2) holds. For any $(h_1,h_2)\in \mathrm{Im}M$, $\exists (x,y)\in \mathrm{dom}M$ such that

$$\begin{array}{c}\hfill h_1(t)={\displaystyle \frac{1}{a(t)}}{({\varphi }_p({}^{H}D_{1+}^{\alpha }x(t)))}^{\prime },h_2(t)={\displaystyle \frac{1}{a(t)}}{({\varphi }_p({}^{H}D_{1+}^{\beta }y(t)))}^{\prime }.\end{array}$$

Hence, we conclude that

$$\begin{array}{cc}\hfill {\varphi }_p({}^{H}D_{1+}^{\alpha }x(t))& ={\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))-{\int }_t^{+\infty }a(s)h_1(s)ds\hfill \\ & =\sum _{i=1}^{+\infty }{\alpha }_i{\varphi }_p({}^{H}D_{1+}^{\alpha }x({\xi }_i))-{\int }_t^{+\infty }a(s)h_1(s)ds\hfill \\ & =\sum _{i=1}^{+\infty }{\alpha }_i\left[{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))-{\int }_{{\xi }_i}^{+\infty }a(s)h_1(s)ds\right]\hfill \\ & -{\int }_t^{+\infty }a(s)h_1(s)ds,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\varphi }_p({}^{H}D_{1+}^{\beta }x(t))& ={\varphi }_p({}^{H}D_{1+}^{\beta }y(+\infty ))-{\int }_t^{+\infty }a(s)h_2(s)ds\hfill \\ & =\sum _{i=1}^{+\infty }{\beta }_i{\varphi }_p({}^{H}D_{1+}^{\beta }y({\eta }_i))-{\int }_t^{+\infty }a(s)h_2(s)ds\hfill \\ & =\sum _{i=1}^{+\infty }{\beta }_i\left[{\varphi }_p({}^{H}D_{1+}^{\beta }y(+\infty ))-{\int }_{{\eta }_i}^{+\infty }a(s)h_2(s)ds\right]\hfill \\ & -{\int }_t^{+\infty }a(s)h_2(s)ds.\hfill \end{array}$$

It then follows that

$$\begin{array}{c}\hfill \sum _{i=1}^{+\infty }{\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)h_1(s)ds=0,\sum _{i=1}^{+\infty }{\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)h_2(s)ds=0.\end{array}$$

(4)

On the other hand, if $(h_1,h_2)\in H$ satisfies Equation (4), let

$$\begin{array}{c}\hfill x(t)={}^{H}I_{1+}^{\alpha }{\varphi }_q\left(-{\int }_t^{+\infty }a(s)h_1(s)ds\right),y(t)={}^{H}I_{1+}^{\beta }{\varphi }_q\left(-{\int }_t^{+\infty }a(s)h_2(s)ds\right).\end{array}$$

Then $(x,y)\in \mathrm{dom}M$ and $M(x,y)=(h_1,h_2)$, thereby Equation (3) holds. Evidently, $\dim \mathrm{Ker}M=2<+\infty$, and $\mathrm{Im}M:=M(\mathrm{dom}M\cap X)$ is closed in H. Hence, M is a quasilinear operator. □

Lemma 4.

Consider the subset $V=\{(x,y)\in X:||(x,y)|{|}_X\le r,r>0\}\subset X$. This set is relatively compact in X provided the following criteria are met:

(¶)

All elements of V form an equicontinuous family on every compact subinterval of $\mathfrak{I}$;

(¶)

All functions in V exhibit uniform convergence behavior as $t\to +\infty$.

Proof. 

The proof of Lemma 4 can be established analogously to that of Lemma 4.4 in [31]. Accordingly, the detailed argument is omitted here. □

Lemma 5.

Let $\Omega \subset X$ be a nonempty bounded open set. Then the operator $N_{\lambda }$ is M-compact on $\overline{\Omega }$.

Proof. 

We first define the continuous linear operators $P:X\to X_0$ and $Q:H\to H_0$ as follows:

$$\begin{array}{c}\hfill P(x(t),y(t))=(P_{1}x(t),P_{2}y(t)),Q(h_1(t),h_2(t))=(Q_1{h}_1(t),Q_2{h}_2(t)),\end{array}$$

$t\in \mathfrak{I}$, where $X_0=\mathrm{Ker}M,H_0=\mathrm{Im}Q$ and

$$\begin{array}{cc}& P_{1}x(t)={\displaystyle \frac{{}^{H}D_{1+}^{\alpha }x(+\infty )}{\Gamma (\alpha +1)}}{(\ln t)}^{\alpha },Q_1{h}_1(t)={\displaystyle \frac{{\displaystyle \sum _{i=1}^{+\infty }{\displaystyle {\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)h_1(s)ds}}}{{\displaystyle t\sum _{i=1}^{+\infty }{\displaystyle {\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)\frac{ds}{s}}}}},\hfill \\ & P_{2}y(t)={\displaystyle \frac{{}^{H}D_{1+}^{\beta }y(+\infty )}{\Gamma (\beta +1)}}{(\ln t)}^{\beta },Q_2{h}_2(t)={\displaystyle \frac{{\displaystyle \sum _{i=1}^{+\infty }{\displaystyle {\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)h_2(s)ds}}}{{\displaystyle t\sum _{i=1}^{+\infty }{\displaystyle {\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)\frac{ds}{s}}}}}.\hfill \end{array}$$

For convenience in the subsequent discussion, we denote $Y_0=\mathrm{Ker}P,H_1=\mathrm{Im}M$ and

$$\begin{array}{c}\hfill l\left(t,(x,y),\lambda \right)=\left(l_1(t,(x,y),\lambda ),l_2(t,(x,y),\lambda )\right),\end{array}$$

$$\begin{array}{cc}& \phi (t)={\varphi }_q(l_1(t,(x,y),\lambda ))-{}^{H}D_{1+}^{\alpha }x(+\infty ),\hfill \\ & \psi (t)={\varphi }_q(l_2(t,(x,y),\lambda ))-{}^{H}D_{1+}^{\beta }y(+\infty ),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill l_1(t,(x,y),\lambda )={\int }_t^{+\infty }a(s)(Q_1-I)N_{1\lambda }y(s)ds+{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty )),\\ \hfill l_2(t,(x,y),\lambda )={\int }_t^{+\infty }a(s)(Q_2-I)N_{2\lambda }x(s)ds+{\varphi }_p({}^{H}D_{1+}^{\beta }y(+\infty )).\end{array}$$

The operator $R:\overline{\Omega }\times [0,1]\to Y_0$ is defined as follows:

$$\begin{array}{c}\hfill R\left((x,y),\lambda \right)(t)=\left(R_1((x,y),\lambda )(t),R_2((x,y),\lambda )(t)\right),\end{array}$$

where

$$\begin{array}{c}\hfill R_1((x,y),\lambda )(t)={}^{H}I_{1+}^{\alpha }\phi (t),R_2((x,y),\lambda )(t)={}^{H}I_{1+}^{\beta }\psi (t).\end{array}$$

Based on the preceding preparations, we are now ready to prove Lemma 5. In fact, by the definition of P, it can be verified that $\mathrm{Im}P=\mathrm{Ker}M$, $P^2\left(x(t),y(t)\right)=P\left(x(t),y(t)\right)$, and $\mathrm{Ker}M\cap \mathrm{Ker}P=\{(0,0)\}$. For any $(x,y)\in X$, using the decomposition $(x,y)=\left((x,y)-P(x,y)\right)+P(x,y)$ together with $\mathrm{Im}P=\mathrm{Ker}M$, it follows that $\left((x,y)-P(x,y)\right)\in \mathrm{Ker}P$ and $P(x,y)\in \mathrm{Ker}M$. Hence, we obtain the direct sum decomposition $X=\mathrm{Ker}M\oplus \mathrm{Ker}P=X_0\oplus Y_0$. Analogously, from the definition of the operator Q, we have

$$Q^2(h_1,h_2)=Q(h_1,h_2),\mathrm{Ker}Q=\mathrm{Im}M.$$

For any $(h_1,h_2)\in H$, using the identity $(h_1,h_2)=\left((h_1,h_2)-Q(h_1,h_2)\right)+Q(h_1,h_2)$ along with $\mathrm{Ker}Q=\mathrm{Im}M$ it follows that $\left((h_1,h_2)-Q(h_1,h_2)\right)\in \mathrm{Im}M$ and $Q(h_1,h_2)\in \mathrm{Im}Q$. Moreover, it can be verified that $\mathrm{Im}Q\cap \mathrm{Im}M=\{(0,0)\}$. Therefore, we conclude that $H=\mathrm{Im}Q\oplus \mathrm{Im}M=H_0\oplus H_1$, and $\dim X_0=\dim \mathrm{Ker}M=\dim \mathrm{Im}Q=\dim H_0$.

Let $(0,0)\in \Omega \subset X$ be an open subset. For any $(x,y)\in \overline{\Omega }$, since $Q(I-Q)$ is a zero operator, we have $Q\left[(I-Q)N_{\lambda }(x,y)\right]=0$. Therefore, $(I-Q)N_{\lambda }(x,y)\in \mathrm{Ker}Q=\mathrm{Im}M$, that is,

$$Q(I-Q)N_{\lambda }(\overline{\Omega })\subset \mathrm{Im}M.$$

On the other hand, for any $(h_1,h_2)\in \mathrm{Im}M$, we observe that $(h_1,h_2)=\left((h_1,h_2)-Q(h_1,h_2)\right)+Q(h_1,h_2)$, and noting that $\mathrm{Ker}Q=\mathrm{Im}M$, it follows that $(h_1,h_2)\in (I-Q)H$, i.e., $\mathrm{Im}M\subset (I-Q)H$. Evidently, $Q{N}_{\lambda }(x,y)=(0,0)$ for $\lambda \in (0,1)$ is equivalent to $QN(x,y)=(0,0)$. Hence, conditions $(a_1)$ and $(a_2)$ in Definition 4 are satisfied. From condition $(A_1)$, it follows that the functions f and g are a-Carathéodory functions. Thanks to the Lebesgue dominated convergence theorem, it can be readily confirmed that $R\left((x,y),\lambda \right)(t)$ remains continuous on $\overline{\Omega }\times [0,1]$. We proceed to demonstrate that $R\left((x,y),\lambda \right)(t)$ is compact in two steps.

Step 1. We demonstrate that $R\left((x,y),\lambda \right)(t)$ is uniformly bounded in the space X and exhibits equicontinuity over every compact subinterval of $[1,+\infty )$. Given that $\Omega \subset X$ is a nonempty bounded open subset and invoking condition $(A_1)$, for all $(x,y)\in \overline{\Omega }$, there exist constants $r_1,r_2>0$, along with non-negative functions $G_{r_1}(t),F_{r_2}(t)\in Z$, such that

$$\begin{array}{cc}& {||y||}_{X_2}\le r_1,|g(t,y(t),{}^{H}D_{1+}^{\beta -1}y(t),{}^{H}D_{1+}^{\beta }y(t))|\le G_{r_1}(t),a.e.t\in \mathfrak{I},\hfill \\ & {||x||}_{X_1}\le r_2,|f(t,x(t),{}^{H}D_{1+}^{\alpha -1}x(t),{}^{H}D_{1+}^{\alpha }x(t))|\le F_{r_2}(t),a.e.t\in \mathfrak{I}.\hfill \end{array}$$

Since

$$\begin{array}{c}\left|{\int }_s^{+\infty }a(\tau )(Q_1-I)N_{1\lambda }y(\tau )d\tau \right|\le {\int }_s^{+\infty }|a(\tau )N_{1\lambda }y(\tau )-a(\tau )Q_1{N}_{1\lambda }y(\tau )|d\tau \hfill \\ \le {\int }_1^{+\infty }|-a(\tau )g(\tau ,y(\tau ),{}^{H}D_{1+}^{\beta -1}y(\tau ),{}^{H}D_{1+}^{\beta }y(\tau ))\hfill \\ +a(\tau ){\displaystyle \frac{{\displaystyle \sum _{i=1}^{+\infty }{\displaystyle {\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)g(s,y(s),{}^{H}D_{1+}^{\beta -1}y(s),{}^{H}D_{1+}^{\beta }y(s))ds}}}{{\displaystyle \tau \sum _{i=1}^{+\infty }{\displaystyle {\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)\frac{ds}{s}}}}}|d\tau \hfill \\ \le {\int }_1^{+\infty }a(\tau )\left|g(\tau ,y(\tau ),{}^{H}D_{1+}^{\beta -1}y(\tau ),{}^{H}D_{1+}^{\beta }y(\tau ))\right|d\tau \hfill \\ +\left|{\displaystyle \frac{{\displaystyle \sum _{i=1}^{+\infty }{\displaystyle {\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)g(s,y(s),{}^{H}D_{1+}^{\beta -1}y(s),{}^{H}D_{1+}^{\beta }y(s))ds}}}{{\displaystyle \sum _{i=1}^{+\infty }{\displaystyle {\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)\frac{ds}{s}}}}}\right|{\int }_1^{+\infty }a(\tau ){\displaystyle \frac{d\tau }{\tau }}\hfill \\ \le ||G_{r_1}|{|}_Z\left(1+{\displaystyle \frac{{\displaystyle {\int }_1^{+\infty }a(\tau )\frac{d\tau }{\tau }}}{{\displaystyle \sum _{i=1}^{+\infty }{\displaystyle {\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)\frac{ds}{s}}}}}\right):=\widetilde{r_1}.\hfill \end{array}$$

(5)

Similarly,

$$\begin{array}{c}\hfill \left|{\int }_s^{+\infty }a(\tau )(Q_2-I)N_{2\lambda }x(\tau )d\tau \right|\le ||F_{r_2}|{|}_Z\left(1+{\displaystyle \frac{{\displaystyle {\int }_1^{+\infty }a(\tau )\frac{d\tau }{\tau }}}{{\displaystyle \sum _{i=1}^{+\infty }{\displaystyle {\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)\frac{ds}{s}}}}}\right):={\tilde{r}}_2.\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill |\phi (t)|& =\left|{\varphi }_q\left[{\int }_t^{+\infty }a(\tau )(Q_1-I)N_{1\lambda }y(\tau )d\tau +{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))\right]-{}^{H}D_{1+}^{\alpha }x(+\infty )\right|\hfill \\ & \le {\varphi }_q[{\tilde{r}}_1+{\varphi }_p(r_1)]+r_1:=m_1,\hfill \\ \hfill |\psi (t)|& =\left|{\varphi }_q\left[{\int }_t^{+\infty }a(\tau )(Q_2-I)N_{2\lambda }x(\tau )d\tau +{\varphi }_p({}^{H}D_{1+}^{\beta }y(+\infty ))\right]-{}^{H}D_{1+}^{\beta }y(+\infty )\right|\hfill \\ & \le {\varphi }_q[{\tilde{r}}_2+{\varphi }_p(r_2)]+r_2:=m_2,\hfill \end{array}$$

and hence, for any $(x,y)\in \overline{\Omega }$, the following estimates hold,

$$\begin{array}{cc}\hfill ||R_1((x,y),\lambda )|{|}_0& =\underset{t\in \mathfrak{I}}{\sup }\left|{\displaystyle \frac{R_1((x,y),\lambda )(t)}{1+{(\ln t)}^{\alpha }}}\right|\hfill \\ & =\underset{t\in \mathfrak{I}}{\sup }{\displaystyle \frac{1}{\Gamma (\alpha )}}\left|{\int }_1^t{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t)}^{\alpha }}}\phi (s){\displaystyle \frac{ds}{s}}\right|\hfill \\ & \le {\displaystyle \frac{m_1}{\Gamma (\alpha )}}\underset{t\in \mathfrak{I}}{\sup }{\int }_1^t{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t)}^{\alpha }}}{\displaystyle \frac{ds}{s}}\le {\displaystyle \frac{m_1}{\Gamma (\alpha +1)}}\le m_1,\hfill \end{array}$$

$$\begin{array}{cc}& \begin{array}{cc}\hfill ||{}^{H}D_{1+}^{\alpha -1}{R}_1((x,y),\lambda )|{|}_1& =\underset{t\in \mathfrak{I}}{\sup }\left|{\displaystyle \frac{{}^{H}D_{1+}^{\alpha -1}{R}_1((x,y),\lambda )(t)}{1+\ln t}}\right|\hfill \\ & =\underset{t\in \mathfrak{I}}{\sup }\left|{\int }_1^t{\displaystyle \frac{1}{1+\ln t}}\phi (s){\displaystyle \frac{ds}{s}}\right|\hfill \\ & \le m_1\underset{t\in \mathfrak{I}}{\sup }{\int }_0^t{\displaystyle \frac{1}{1+\ln t}}{\displaystyle \frac{ds}{s}}\le m_1,\hfill \end{array}\hfill \\ & \begin{array}{cc}\hfill ||{}^{H}D_{1+}^{\alpha }{R}_1((x,y),\lambda )|{|}_{\infty }& =\underset{t\in \mathfrak{I}}{\sup }|{}^{H}D_{1+}^{\alpha }{R}_1((x,y),\lambda )(t)|\hfill \\ & =\underset{t\in \mathfrak{I}}{\sup }|\phi (t)|\le m_1,\hfill \end{array}\hfill \end{array}$$

and analogously,

$$\begin{array}{cc}& ||R_2((x,y),\lambda )|{|}_0\le m_2,||{}^{H}D_{1+}^{\beta -1}{R}_2((x,y),\lambda )|{|}_1\le m_2,\hfill \\ & ||{}^{H}D_{1+}^{\beta }{R}_2((x,y),\lambda )|{|}_{\infty }\le m_2.\hfill \end{array}$$

Thus,

$$||R\left((x,y),\lambda \right)|{|}_X\le m_1+m_2,$$

which shows that $R\left((x,y),\lambda \right)(\overline{\Omega })$ is uniformly bounded in X. Next, we proceed to show that $R\left((x,y),\lambda \right)(\overline{\Omega })$ is equicontinuous on any compact subinterval of $\mathfrak{I}$. Indeed, for any $K>1,t_1,t_2\in [1,K],(x,y)\in \overline{\Omega }$, and $\lambda \in [0,1]$, we have

$$\begin{array}{cc}& \left|{\displaystyle \frac{R_1((x,y),\lambda )(t_1)}{1+{(\ln t_1)}^{\alpha }}}-{\displaystyle \frac{R_1((x,y),\lambda )(t_2)}{1+{(\ln t_2)}^{\alpha }}}\right|\hfill \\ & ={\displaystyle \frac{1}{\Gamma (\alpha )}}\left[\left|{\int }_1^{t_1}{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_1$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_1)}^{\alpha }}}\phi (s){\displaystyle \frac{ds}{s}}-{\int }_1^{t_2}{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_2)}^{\alpha }}}\phi (s){\displaystyle \frac{ds}{s}}\right|\right]\hfill \\ & \le {\displaystyle \frac{m_1}{\Gamma (\alpha )}}\left[{\int }_1^{t_1}\left|{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_1$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_1)}^{\alpha }}}-{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_2)}^{\alpha }}}\right|{\displaystyle \frac{ds}{s}}+{\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$t_1$}\right.))}^{\alpha }}{1+{(\ln t_2)}^{\alpha }}}\right]\hfill \\ & \to 0,(t_1\to t_2),\hfill \end{array}$$

and

$$\begin{array}{cc}& \left|{\displaystyle \frac{{}^{H}D_{1+}^{\alpha -1}{R}_1((x,y),\lambda )(t_1)}{1+\ln t_1}}-{\displaystyle \frac{{}^{H}D_{1+}^{\alpha -1}{R}_1((x,y),\lambda )(t_2)}{1+\ln t_2}}\right|\hfill \\ & =\left[\left|{\displaystyle \frac{1}{1+\ln t_1}}{\int }_1^{t_1}\phi (s){\displaystyle \frac{ds}{s}}-{\displaystyle \frac{1}{1+\ln t_2}}{\int }_1^{t_2}\phi (s){\displaystyle \frac{ds}{s}}\right|\right]\hfill \\ & \le m_1\left[{\displaystyle \frac{\ln t_1·\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$t_1$}\right.)}{(1+\ln t_1)(1+\ln t_2)}}+{\displaystyle \frac{\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$t_1$}\right.)}{1+\ln t_2}}\right]\to 0,(t_1\to t_2).\hfill \end{array}$$

Similarly,

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{R_2((x,y),\lambda )(t_1)}{1+{(\ln t_1)}^{\beta }}}-{\displaystyle \frac{R_2((x,y),\lambda )(t_2)}{1+{(\ln t_2)}^{\beta }}}\right|\to 0,(t_1\to t_2),\end{array}$$

and

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{{}^{H}D_{1+}^{\beta -1}{R}_2((x,y),\lambda )(t_1)}{1+\ln t_1}}-{\displaystyle \frac{{}^{H}D_{1+}^{\beta -1}{R}_2((x,y),\lambda )(t_2)}{1+\ln t_2}}\right|\to 0,(t_1\to t_2).\end{array}$$

Furthermore,

$$\begin{array}{cc}& |{}^{H}D_{1+}^{\alpha }{R}_1((x,y),\lambda )(t_1)-{}^{H}D_{1+}^{\alpha }{R}_1((x,y),\lambda )(t_2)|\hfill \\ & =|\phi (t_1)-\phi (t_2)|=|{\varphi }_q(l_1(t_1,(x,y),\lambda ))-{\varphi }_q(l_1(t_2,(x,y),\lambda ))|,\hfill \end{array}$$

and likewise,

$$\begin{array}{cc}& |{}^{H}D_{1+}^{\beta }{R}_2((x,y),\lambda )(t_1)-{}^{H}D_{1+}^{\beta }{R}_2((x,y),\lambda )(t_2)|\hfill \\ & =|\psi (t_1)-\psi (t_2)|=|{\varphi }_q(l_2(t_1,(x,y),\lambda ))-{\varphi }_q(l_2(t_2,(x,y),\lambda ))|.\hfill \end{array}$$

Noting that

$$\begin{array}{cc}\hfill |l_1(t,(x,y),\lambda )|& =\left|{\int }_t^{+\infty }a(s)(Q_1-I)N_{1\lambda }y(s)ds+{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))\right|\hfill \\ & \le {\tilde{r}}_1+{\varphi }_p(r_1),(t\in [1,K],(x,y)\in \overline{\Omega }),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |l_2(t,(x,y),\lambda )|& =\left|{\int }_t^{+\infty }a(s)(Q_2-I)N_{2\lambda }x(s)ds+{\varphi }_p({}^{H}D_{1+}^{\beta }y(+\infty ))\right|\hfill \\ & \le {\tilde{r}}_2+{\varphi }_p(r_2),(t\in [1,K],(x,y)\in \overline{\Omega }),\hfill \end{array}$$

which implies

$$\begin{array}{cc}& |l_1(t_1,(x,y),\lambda )-l_1(t_2,(x,y),\lambda )|=\left|{\int }_{t_1}^{t_2}(Q_1-I)N_{1\lambda }x(s)ds\right|\hfill \\ & \le {\int }_{t_1}^{t_2}a(s)G_{r_1}(s)ds+{\displaystyle \frac{{\displaystyle ||G_{r_1}|{|}_Z{\int }_{t_1}^{t_2}a(s)\frac{ds}{s}}}{{\displaystyle \sum _{i=1}^{+\infty }{\displaystyle {\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)\frac{ds}{s}}}}}\to 0,(t_1\to t_2),\hfill \end{array}$$

and similarly,

$$\begin{array}{cc}& |l_2(t_1,(x,y),\lambda )-l_2(t_2,(x,y),\lambda )|\hfill \\ & \le {\int }_{t_1}^{t_2}a(s)F_{r_2}(s)ds+{\displaystyle \frac{{\displaystyle ||F_{r_2}|{|}_Z{\int }_{t_1}^{t_2}a(s)\frac{ds}{s}}}{{\displaystyle \sum _{i=1}^{+\infty }{\displaystyle {\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)\frac{ds}{s}}}}}\to 0,(t_1\to t_2).\hfill \end{array}$$

Since ${\varphi }_q(x)$ and ${\varphi }_q(y)$ are uniformly continuous on the intervals $[-{\tilde{r}}_1-{\varphi }_p(r_1),{\tilde{r}}_1+{\varphi }_p(r_1)]$ and $[-{\tilde{r}}_2-{\varphi }_p(r_2),{\tilde{r}}_2+{\varphi }_p(r_2)]$, respectively, it follows that

$$\begin{array}{cc}& |{}^{H}D_{1+}^{\alpha }{R}_1((x,y),\lambda )(t_1)-{}^{H}D_{1+}^{\alpha }{R}_1((x,y),\lambda )(t_2)|\hfill \\ & =|{\varphi }_q(l_1(t_1,(x,y),\lambda ))-{\varphi }_q(l_1(t_2,(x,y),\lambda ))|\to 0,(t_1\to t_2),\hfill \end{array}$$

and

$$\begin{array}{cc}& |{}^{H}D_{1+}^{\beta }{R}_2((x,y),\lambda )(t_1)-{}^{H}D_{1+}^{\beta }{R}_2((x,y),\lambda )(t_2)|\hfill \\ & =|{\varphi }_q(l_2(t_1,(x,y),\lambda ))-{\varphi }_q(l_2(t_2,(x,y),\lambda ))|\to 0,(t_1\to t_2).\hfill \end{array}$$

Through the above arguments, we have deduced that $R\left((x,y),\lambda \right)(\overline{\Omega })$ is equicontinuous on any compact subinterval of $\mathfrak{I}$.

Step 2. We proceed to demonstrate the uniform convergence of $R((x,y),\lambda )(\overline{\Omega })$ at infinity. In fact, for any $(x,y)\in \overline{\Omega }$, it follows from Equation (5) that

$$\begin{array}{cc}& \underset{s\to +\infty }{\lim }{\int }_s^{+\infty }a(\tau )(Q_1-I)N_{1\lambda }y(\tau )d\tau =0,\hfill \\ & \underset{s\to +\infty }{\lim }{\int }_s^{+\infty }a(\tau )(Q_2-I)N_{2\lambda }x(\tau )d\tau =0.\hfill \end{array}$$

Note that ${\varphi }_q(x)$ is uniformly continuous on the interval $[-{\tilde{r}}_1-{\varphi }_p(r_1),{\tilde{r}}_1+{\varphi }_p(r_1)]$, then for any $\epsilon >0$, there exists a constant $L_1>1$, such that for all $s\ge L_1$,

$$\begin{array}{cc}\hfill |\phi (s)|& =|{\varphi }_q\left[{\int }_s^{+\infty }a(\tau )(Q_1-I)N_{1\lambda }y(\tau )d\tau +{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))\right]\hfill \\ & -{\varphi }_q\left[{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))\right]|<\epsilon ,\forall (x,y)\in \overline{\Omega }.\hfill \end{array}$$

Similarly, since ${\varphi }_q(y)$ is uniformly continuous on $[-{\tilde{r}}_2-{\varphi }_p(r_2),{\tilde{r}}_2+{\varphi }_p(r_2)]$, for the same $\epsilon >0$, there exists a constant $L_2>1$ such that for all $s\ge L_2$,

$$\begin{array}{c}\hfill |\psi (s)|<\epsilon ,\forall (x,y)\in \overline{\Omega }.\end{array}$$

Therefore, for all $s\ge L_1$, it holds that $|\phi (s)|<\epsilon$, similarly, for $s\ge L_2$, we have $|\psi (s)|<\epsilon$. On the other hand, for $s<L_1$, we deduce $|\phi (s)|<m_1$, and for $s<L_2$, we have $|\psi (s)|\le m_2$. Moreover, noting the asymptotic limits

$$\begin{array}{cc}& \underset{t\to +\infty }{\lim }{\displaystyle \frac{{(\ln t)}^{\alpha -1}}{1+{(\ln t)}^{\alpha }}}=0,\underset{t\to +\infty }{\lim }{\displaystyle \frac{{(\ln t)}^{\beta -1}}{1+{(\ln t)}^{\beta }}}=0,\hfill \\ & \underset{t\to +\infty }{\lim }{\displaystyle \frac{1}{1+{(\ln t)}^{\alpha }}}=0,\underset{t\to +\infty }{\lim }{\displaystyle \frac{1}{1+{(\ln t)}^{\beta }}}=0,\hfill \end{array}$$

then for the above $\epsilon >0$, there exists a constant $L>\max \{L_1,L_2\}>1$, such that for all $t_1,t_2\ge L$, and $1\le s\le L_1$, we have

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_1$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_1)}^{\alpha }}}-{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_2)}^{\alpha }}}\right|<\epsilon ,\left|{\displaystyle \frac{1}{1+\ln t_1}}-{\displaystyle \frac{1}{1+\ln t_2}}\right|<\epsilon .\end{array}$$

Likewise, for $1\le s\le L_2$, we also obtain

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_1$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\beta -1}}{1+{(\ln t_1)}^{\beta }}}-{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\beta -1}}{1+{(\ln t_2)}^{\beta }}}\right|<\epsilon .\end{array}$$

Hence, for any $t_1,t_2\ge L$, the following estimates hold

$$\begin{array}{cc}& \left|{\displaystyle \frac{R_1((x,y),\lambda )(t_1)}{1+{(\ln t_1)}^{\alpha }}}-{\displaystyle \frac{R_1((x,y),\lambda )(t_2)}{1+{(\ln t_2)}^{\alpha }}}\right|\hfill \\ & ={\displaystyle \frac{1}{\Gamma (\alpha )}}\left[\left|{\int }_1^{t_1}{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_1$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_1)}^{\alpha }}}\phi (s){\displaystyle \frac{ds}{s}}-{\int }_1^{t_2}{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_2)}^{\alpha }}}\phi (s){\displaystyle \frac{ds}{s}}\right|\right]\hfill \\ & \le {\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_1^{L_1}\left|{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_1$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_1)}^{\alpha }}}-{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_2)}^{\alpha }}}\right||\phi (s)|{\displaystyle \frac{ds}{s}}\hfill \\ & +{\displaystyle \frac{1}{\Gamma (\alpha )}}\left[{\int }_{L_1}^{t_1}{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_1$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_1)}^{\alpha }}}|\phi (s)|{\displaystyle \frac{ds}{s}}+{\int }_{L_1}^{t_2}{\displaystyle \frac{{(\ln (\raisebox{1ex}{$t_2$}\!\left/ \!\raisebox{-1ex}{$s$}\right.))}^{\alpha -1}}{1+{(\ln t_2)}^{\alpha }}}|\phi (s)|{\displaystyle \frac{ds}{s}}\right]\hfill \\ & \le {\displaystyle \frac{m_1\epsilon }{\Gamma (\alpha )}}\ln L_1+{\displaystyle \frac{2m_1\epsilon }{\Gamma (\alpha +1)}},\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{R_2((x,y),\lambda )(t_1)}{1+{(\ln t_1)}^{\beta }}}-{\displaystyle \frac{R_1((x,y),\lambda )(t_2)}{1+{(\ln t_2)}^{\beta }}}\right|\le {\displaystyle \frac{m_2\epsilon }{\Gamma (\beta )}}\ln L_2+{\displaystyle \frac{2m_2\epsilon }{\Gamma (\beta +1)}}.\end{array}$$

Analogously,

$$\begin{array}{cc}& \left|{\displaystyle \frac{{}^{H}D_{1+}^{\alpha -1}{R}_1((x,y),\lambda )(t_1)}{1+\ln t_1}}-{\displaystyle \frac{{}^{H}D_{1+}^{\alpha -1}{R}_1((x,y),\lambda )(t_2)}{1+\ln t_2}}\right|\le (\ln L_1)m_1\epsilon +2\epsilon ,\hfill \\ & \left|{\displaystyle \frac{{}^{H}D_{1+}^{\beta -1}{R}_2((x,y),\lambda )(t_1)}{1+\ln t_1}}-{\displaystyle \frac{{}^{H}D_{1+}^{\beta -1}{R}_2((x,y),\lambda )(t_2)}{1+\ln t_2}}\right|\le (\ln L_2)m_2\epsilon +2\epsilon .\hfill \end{array}$$

In addition,

$$\begin{array}{cc}& |{}^{H}D_{1+}^{\alpha }{R}_1((x,y),\lambda )(t_1)-{}^{H}D_{1+}^{\alpha }{R}_1((x,y),\lambda )(t_2)|\hfill \\ & =|\phi (t_1)-\phi (t_2)|\le |\phi (t_1)|+|\phi (t_2)|<2\epsilon ,\hfill \\ & |{}^{H}D_{1+}^{\beta }{R}_2((x,y),\lambda )(t_1)-{}^{H}D_{1+}^{\beta }{R}_2((x,y),\lambda )(t_2)|<2\epsilon .\hfill \end{array}$$

Therefore, the mapping $R((x,y),\lambda )(\overline{\Omega })$ is uniformly convergent at infinity. By Lemma 4, it follows that $R:\overline{\Omega }\times [0,1]\to Y_0$ is compact. We now proceed to verify that conditions $(a_3)$ and $(a_4)$ in Definition 4 are satisfied. In fact, for $(x,y)\in {\sum }_{\lambda }=\{(x,y)\in \overline{\Omega }|M(x,y)=N_{\lambda }(x,y)\}$, then

$$\begin{array}{cc}& \left({\displaystyle \frac{1}{a(t)}}{({\varphi }_p({}^{H}D_{1+}^{\alpha }x(t)))}^{\prime },{\displaystyle \frac{1}{a(t)}}{({\varphi }_p({}^{H}D_{1+}^{\beta }y(t)))}^{\prime }\right)\hfill \\ & =N_{\lambda }\left(x(t),y(t)\right)\in \mathrm{Im}M=\mathrm{Ker}Q,\hfill \end{array}$$

and

$$\begin{array}{cc}& R_1((x,y),\lambda )(t)={}^{H}I_{1+}^{\alpha }\phi (t)\hfill \\ & ={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_1^t{\left(\ln {\displaystyle \frac{t}{s}}\right)}^{\alpha -1}\left\{{\varphi }_q\right[{\int }_s^{+\infty }a(\tau )(Q_1-I)N_{1\lambda }y(\tau )d\tau \hfill \\ & +{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))\left]-{}^{H}D_{1+}^{\alpha }x(+\infty )\right\}{\displaystyle \frac{ds}{s}}\hfill \\ & ={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_1^t{\left(\ln {\displaystyle \frac{t}{s}}\right)}^{\alpha -1}\{{\varphi }_q[{\int }_s^{+\infty }-{({\varphi }_p({}^{H}D_{1+}^{\alpha }x(\tau )))}^{\prime }d\tau \hfill \\ & +{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))]-{}^{H}D_{1+}^{\alpha }x(+\infty )\}{\displaystyle \frac{ds}{s}}\hfill \\ & ={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_1^t{\left(\ln {\displaystyle \frac{t}{s}}\right)}^{\alpha -1}{}^{H}D_{1+}^{\alpha }x(s){\displaystyle \frac{ds}{s}}-(P_{1}x)(t).\hfill \end{array}$$

In light of the boundary conditions, we obtain

$$\begin{array}{c}\hfill R_1((x,y),\lambda )(t)=x(t)-(P_{1}x)(t)=\left[(I-P_1)x\right](t),\end{array}$$

and similarly,

$$\begin{array}{c}\hfill R_2((x,y),\lambda )(t)=y(t)-(P_{2}y)(t)=\left[(I-P_2)y\right](t).\end{array}$$

Clearly, $R_1((x,y),0)(t)$ and $R_2((x,y),0)(t)$ are zero operators. Moreover, for any $(x,y)\in \overline{\Omega }$, one has

$$\begin{array}{cc}& M_1[P_{1}x+R_1((x,y),\lambda )](t)\hfill \\ & =M_1\{{\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_1^t{\left(\ln {\displaystyle \frac{t}{s}}\right)}^{\alpha -1}{\varphi }_q[{\int }_s^{+\infty }a(\tau )(Q_1-I)N_{1\lambda }y(\tau )d\tau \hfill \\ & +{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))]{\displaystyle \frac{ds}{s}}\}\hfill \\ & ={\displaystyle \frac{1}{a(t)}}{\left[{\int }_t^{+\infty }a(\tau )(Q_1-I)N_{1\lambda }y(\tau )d\tau +{\varphi }_p({}^{H}D_{1+}^{\alpha }x(+\infty ))\right]}^{\prime }\hfill \\ & =(I-Q_1)N_{1\lambda }y(t),\hfill \end{array}$$

and analogously,

$$\begin{array}{c}\hfill M_2[P_{2}y+R_2((x,y),\lambda )](t)=(I-Q_2)N_{2\lambda }y(t).\end{array}$$

Therefore, the operator $N_{\lambda }$ is M-compact on $\overline{\Omega }$. □

• To utilize Theorem 1 and establish the primary findings of this study, we introduce the following assumptions:

$({\mathrm{A}}_2)$

There exist non-negative functions $a_i(t),b_i(t),c_i(t),d_i(t)\in Z$, for $i=1,2$, satisfying

$$\left[||b_1|{|}_Z+||c_1|{|}_Z+||d_1|{|}_Z\right]\left[||b_2|{|}_Z+||c_2|{|}_Z+||d_2|{|}_Z\right]<1,$$

such that for all $t\in \mathfrak{I}$ and $(\mathfrak{u},\mathfrak{v},\mathfrak{w})\in {\mathbb{R}}^3$, the following inequalities hold:

$$\begin{array}{cc}& |g(t,\mathfrak{u},\mathfrak{v},\mathfrak{w})|\hfill \\ & \le a_1(t)+b_1(t){\displaystyle \frac{|\mathfrak{u}{|}^{p-1}}{{(1+{(\ln t)}^{\beta })}^{p-1}}}+c_1(t){\displaystyle \frac{|\mathfrak{v}{|}^{p-1}}{{(1+\ln t)}^{p-1}}}+d_1(t)|\mathfrak{w}{|}^{p-1},\hfill \\ & |f(t,\mathfrak{u},\mathfrak{v},\mathfrak{w})|\hfill \\ & \le a_2(t)+b_2(t){\displaystyle \frac{|\mathfrak{u}{|}^{p-1}}{{(1+{(\ln t)}^{\alpha })}^{p-1}}}+c_2(t){\displaystyle \frac{|\mathfrak{v}{|}^{p-1}}{{(1+\ln t)}^{p-1}}}+d_2(t)|\mathfrak{w}{|}^{p-1}.\hfill \end{array}$$

$({\mathrm{A}}_3)$

There exist constants $B_1,B_2>0$ such that one of the following conditions is satisfied:

$$\left\{\begin{array}{c}{\mathfrak{w}}_{1}g(t,\mathfrak{u},\mathfrak{v},{\mathfrak{w}}_1)>0,|{\mathfrak{w}}_1|>B_2,\\ {\mathfrak{w}}_{2}f(t,\mathfrak{u},\mathfrak{v},{\mathfrak{w}}_2)>0,|{\mathfrak{w}}_2|>B_1,\end{array}\right.\mathfrak{u},\mathfrak{v}\in \mathbb{R},$$

(6)

or

$$\left\{\begin{array}{c}{\mathfrak{w}}_{1}g(t,\mathfrak{u},\mathfrak{v},{\mathfrak{w}}_1)<0,|{\mathfrak{w}}_1|>B_2,\\ {\mathfrak{w}}_{2}f(t,\mathfrak{u},\mathfrak{v},{\mathfrak{w}}_2)<0,|{\mathfrak{w}}_2|>B_1,\end{array}\right.\mathfrak{u},\mathfrak{v}\in \mathbb{R}.$$

(7)

Lemma 6.

Assume that $({\mathrm{A}}_2)$ and $({\mathrm{A}}_3)$ hold. Then the set

$${\Omega }_1=\{(x,y)\in \mathrm{dom}M\setminus \mathrm{Ker}M|M(x,y)=N_{\lambda }(x,y),\lambda \in (0,1)\}$$

is bounded.

Proof. 

Let $(x,y)\in {\Omega }_1$. Then $M(x,y)=N_{\lambda }(x,y)$, which implies

$$\begin{array}{c}\hfill Q_1{N}_{1\lambda }y=0,Q_2{N}_{2\lambda }x=0,\end{array}$$

that is,

$$\begin{array}{cc}& \sum _{i=1}^{+\infty }{\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)g(s,y(s),{}^{H}D_{1+}^{\beta -1}y(s),{}^{H}D_{1+}^{\beta }y(s))ds=0,\hfill \\ & \sum _{i=1}^{+\infty }{\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)f(s,x(s),{}^{H}D_{1+}^{\alpha -1}x(s),{}^{H}D_{1+}^{\alpha }x(s))ds=0.\hfill \end{array}$$

From assumption $({\mathrm{A}}_3)$, there exist constants $s_{01},s_{02}\in [1,+\infty )$ such that $|{}^{H}D_{1+}^{\alpha }x(s_{01})|\le B_1$ and $|{}^{H}D_{1+}^{\beta }y(s_{02})|\le B_2$. On the other hand, by Lemmas 1 and 2, combined with the boundary conditions, it follows that

$$\begin{array}{cc}& x(t)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_1^t{\left(\ln {\displaystyle \frac{t}{s}}\right)}^{\alpha -1}{}^{H}D_{1+}^{\alpha }x(s){\displaystyle \frac{ds}{s}},\hfill \\ & y(t)={\displaystyle \frac{1}{\Gamma (\beta )}}{\int }_1^t{\left(\ln {\displaystyle \frac{t}{s}}\right)}^{\beta -1}{}^{H}D_{1+}^{\beta }y(s){\displaystyle \frac{ds}{s}}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}& ||x|{|}_0\le ||{}^{H}D_{1+}^{\alpha }x|{|}_{\infty },||{}^{H}D_{1+}^{\alpha -1}x|{|}_1\le ||{}^{H}D_{1+}^{\alpha }x|{|}_{\infty },\hfill \\ & ||y|{|}_0\le ||{}^{H}D_{1+}^{\beta }y|{|}_{\infty },||{}^{H}D_{1+}^{\beta -1}y|{|}_1\le ||{}^{H}D_{1+}^{\beta }y|{|}_{\infty }.\hfill \end{array}$$

Noting that $M(x,y)=N_{\lambda }(x,y)$, and utilizing assumption $({\mathrm{A}}_2)$, we deduce

$$\begin{array}{cc}& |{\varphi }_p({}^{H}D_{1+}^{\alpha }x(t))|=|{\varphi }_p({}^{H}D_{1+}^{\alpha }x(s_{01}))\hfill \\ & -\lambda {\int }_{s_{01}}^{t}a(s)g(s,y(s),{}^{H}D_{1+}^{\beta -1}y(s),{}^{H}D_{1+}^{\beta }y(s))ds|\hfill \\ & \le |{\varphi }_p({}^{H}D_{1+}^{\alpha }x(s_{01}))|+\left|{\int }_{s_{01}}^{t}a(s)g(s,y(s),{}^{H}D_{1+}^{\beta -1}y(s),{}^{H}D_{1+}^{\beta }y(s))ds\right|\hfill \\ & \le {\varphi }_p(B_1)+{\int }_1^{+\infty }a(s)[a_1(s)+b_1(s){\displaystyle \frac{|y{|}^{p-1}}{{(1+{(\ln s)}^{\beta })}^{p-1}}}\hfill \\ & +c_1(s){\displaystyle \frac{|{}^{H}D_{1+}^{\beta -1}y{|}^{p-1}}{{(1+\ln s)}^{p-1}}}+d_1(s)|{}^{H}D_{1+}^{\beta }y{|}^{p-1}]ds\hfill \\ & \le {\varphi }_p(B_1)+||a_1|{|}_Z+||b_1|{|}_Z{\varphi }_p(||y|{|}_0)\hfill \\ & +||c_1|{|}_Z{\varphi }_p(||{}^{H}D_{1+}^{\beta -1}y|{|}_1)+||d_1|{|}_Z{\varphi }_p(||{}^{H}D_{1+}^{\beta }y|{|}_{\infty })\hfill \\ & \le {\varphi }_p(B_1)+||a_1|{|}_Z+\left[||b_1|{|}_Z+||c_1|{|}_Z+||d_1|{|}_Z\right]{\varphi }_p(||{}^{H}D_{1+}^{\beta }y|{|}_{\infty }).\hfill \end{array}$$

Hence,

$$\begin{array}{cc}& ||{\varphi }_p({}^{H}D_{1+}^{\alpha }x)|{|}_{\infty }={\varphi }_p(||{}^{H}D_{1+}^{\alpha }x|{|}_{\infty })\hfill \\ & \le {\varphi }_p(B_1)+||a_1|{|}_Z+\left[||b_1|{|}_Z+||c_1|{|}_Z+||d_1|{|}_Z\right]{\varphi }_p(||{}^{H}D_{1+}^{\beta }y|{|}_{\infty }).\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}& ||{\varphi }_p({}^{H}D_{1+}^{\beta }y)|{|}_{\infty }={\varphi }_p(||{}^{H}D_{1+}^{\beta }y|{|}_{\infty })\hfill \\ & \le {\varphi }_p(B_2)+||a_2|{|}_Z+\left[||b_2|{|}_Z+||c_2|{|}_Z+||d_2|{|}_Z\right]{\varphi }_p(||{}^{H}D_{1+}^{\alpha }x|{|}_{\infty }).\hfill \end{array}$$

It then follows that

$$\begin{array}{cc}& {\varphi }_p(||{}^{H}D_{1+}^{\alpha }x|{|}_{\infty })\hfill \\ & \le {\displaystyle \frac{{\varphi }_p(B_1)+||a_1|{|}_Z+\left[||b_1|{|}_Z+||c_1|{|}_Z+||d_1|{|}_Z\right]\left[{\varphi }_p(B_2)+||a_2|{|}_Z\right]}{1-\left[||b_1|{|}_Z+||c_1|{|}_Z+||d_1|{|}_Z\right]\left[||b_2|{|}_Z+||c_2|{|}_Z+||d_2|{|}_Z\right]}}:={\tilde{A}}_1,\hfill \end{array}$$

and

$$\begin{array}{cc}& {\varphi }_p(||{}^{H}D_{1+}^{\beta }y|{|}_{\infty })\hfill \\ & \le {\displaystyle \frac{{\varphi }_p(B_2)+||a_2|{|}_Z+\left[||b_2|{|}_Z+||c_2|{|}_Z+||d_2|{|}_Z\right]\left[{\varphi }_p(B_1)+||a_1|{|}_Z\right]}{1-\left[||b_1|{|}_Z+||c_1|{|}_Z+||d_1|{|}_Z\right]\left[||b_2|{|}_Z+||c_2|{|}_Z+||d_2|{|}_Z\right]}}:={\tilde{A}}_2,\hfill \end{array}$$

and thus,

$$\begin{array}{c}\hfill ||{}^{H}D_{1+}^{\alpha }x|{|}_{\infty }\le {\varphi }_q({\tilde{A}}_1),||{}^{H}D_{1+}^{\beta }y|{|}_{\infty }\le {\varphi }_q({\tilde{A}}_2).\end{array}$$

Therefore,

$$\begin{array}{c}\hfill ||(x,y)|{|}_X=||x|{|}_{X_1}+||y|{|}_{X_2}\le {\varphi }_q({\tilde{A}}_1)+{\varphi }_q({\tilde{A}}_2),\end{array}$$

which implies that ${\Omega }_1$ is bounded. □

Lemma 7.

Assume that $({\mathrm{A}}_3)$ holds. Then the set

$$\begin{array}{c}\hfill {\Omega }_2=\{(x,y)\in \mathrm{Ker}M|N(x,y)\in \mathrm{Im}M\}\end{array}$$

is bounded.

Proof. 

For $(x,y)\in {\Omega }_2$. Then the functions admit the form $x(t)=c_1{(\ln t)}^{\alpha },y(t)=c_2{(\ln t)}^{\beta },c_1,c_2\in \mathbb{R}$, and satisfy $QN(x,y)=(0,0)$. Consequently,

$$\begin{array}{cc}& \sum _{i=1}^{+\infty }{\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)g(s,c_2{(\ln s)}^{\beta },c_2\Gamma (\beta +1)\ln s,c_2\Gamma (\beta +1))ds=0,\hfill \\ & \sum _{i=1}^{+\infty }{\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)f(s,c_1{(\ln s)}^{\alpha },c_1\Gamma (\alpha +1)\ln s,c_1\Gamma (\alpha +1))ds=0.\hfill \end{array}$$

From assumption $({\mathrm{A}}_3)$, it follows that

$$\begin{array}{c}\hfill |c_2\Gamma (\beta +1)|\le B_2,|c_1\Gamma (\alpha +1)|\le B_1.\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill ||(x,y)|{|}_X& =\max \left\{\left|{\displaystyle \frac{c_1{(\ln t)}^{\alpha }}{1+{(\ln t)}^{\alpha }}}\right|,\left|{\displaystyle \frac{c_1\Gamma (\alpha +1)\ln t}{1+\ln t}}\right|,|c_1\Gamma (\alpha +1)|\right\}\hfill \\ & +\max \left\{\left|{\displaystyle \frac{c_2{(\ln t)}^{\beta }}{1+{(\ln t)}^{\beta }}}\right|,\left|{\displaystyle \frac{c_2\Gamma (\beta +1)\ln t}{1+\ln t}}\right|,|c_2\Gamma (\beta +1)|\right\}\hfill \\ & \le B_1+B_2:=C.\hfill \end{array}$$

Thus, ${\Omega }_2$ is bounded. □

Lemma 8.

Assume that the second set of inequalities in assumption $({\mathrm{A}}_3)$, namely (7), is satisfied. Then the set

$$\begin{array}{c}\hfill {\Omega }_3=\{(x,y)\in \mathrm{Ker}M|\lambda J^{-1}(x,y)+(1-\lambda )QN(x,y)=(0,0),\lambda \in [0,1]\}\end{array}$$

is bounded, where the mapping $J^{-1}:\mathrm{Ker}M\to \mathrm{Im}Q$ is defined by

$$\begin{array}{c}\hfill J^{-1}(c_1{(\ln t)}^{\alpha },c_2{(\ln t)}^{\beta })=(c_2{t}^{-1},c_1{t}^{-1}),c_1,c_2\in \mathbb{R},t\in \mathfrak{I},\end{array}$$

is a homeomorphism.

Proof. 

For $(x,y)\in {\Omega }_3$, then $x(t)=c_1{(\ln t)}^{\alpha }$, $y(t)=c_2{(\ln t)}^{\beta },c_1,c_2\in \mathbb{R}$, and in addition,

$$\begin{array}{cc}& \lambda c_2-(1-\lambda ){\displaystyle \frac{{\displaystyle \sum _{i=1}^{+\infty }{\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)g(s,c_2{(\ln s)}^{\beta },c_2\Gamma (\beta +1)\ln s,c_2\Gamma (\beta +1))ds}}{{\displaystyle \sum _{i=1}^{+\infty }{\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)\frac{ds}{s}}}}=0,\hfill \end{array}$$

$$\begin{array}{cc}& \lambda c_1-(1-\lambda ){\displaystyle \frac{{\displaystyle \sum _{i=1}^{+\infty }{\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)f(s,c_1{(\ln s)}^{\alpha },c_1\Gamma (\alpha +1)\ln s,c_1\Gamma (\alpha +1))ds}}{{\displaystyle \sum _{i=1}^{+\infty }{\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)\frac{ds}{s}}}}=0.\hfill \end{array}$$

If $\lambda =1$, it immediately follows that $c_1=c_2=0$. In the case $\lambda \in [0,1)$, we infer $|c_1|\le B_1,|c_2|\le B_2$. Otherwise, if these bounds are violated, then from (7) it follows that

$$\begin{array}{cc}& {\displaystyle \frac{\lambda \Gamma (\beta +1)}{1-\lambda }}{c}_2^2\hfill \\ & ={\displaystyle \frac{{\displaystyle \sum _{i=1}^{+\infty }{\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)c_2\Gamma (\beta +1)g(s,c_2{(\ln s)}^{\beta },c_2\Gamma (\beta +1)\ln s,c_2\Gamma (\beta +1))ds}}{{\displaystyle \sum _{i=1}^{+\infty }{\alpha }_i{\int }_{{\xi }_i}^{+\infty }a(s)\frac{ds}{s}}}}<0,\hfill \\ & {\displaystyle \frac{\lambda \Gamma (\alpha +1)}{1-\lambda }}{c}_1^2\hfill \\ & ={\displaystyle \frac{{\displaystyle \sum _{i=1}^{+\infty }{\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)c_1\Gamma (\alpha +1)f(s,c_1{(\ln s)}^{\alpha },c_1\Gamma (\alpha +1)\ln s,c_1\Gamma (\alpha +1))ds}}{{\displaystyle \sum _{i=1}^{+\infty }{\beta }_i{\int }_{{\eta }_i}^{+\infty }a(s)\frac{ds}{s}}}}<0,\hfill \end{array}$$

this leads to a contradiction, implying that ${\Omega }_3$ is bounded. □

Remark 1.

Assume that the first set of conditions in assumption $({\mathrm{A}}_3)$, namely (6), holds. Then the set

$$\begin{array}{c}\hfill {\tilde{\Omega }}_3=\{(x,y)\in \mathrm{Ker}M|-\lambda J^{-1}(x,y)+(1-\lambda )QN(x,y)=(0,0),\lambda \in [0,1]\}\end{array}$$

is bounded.

Proof. 

This conclusion can be derived directly by an argument analogous to that of Lemma 8. The details are omitted. □

• Utilizing the previously stated lemmas, we now move on to derive the principal results.

Theorem 2.

Supposing that conditions $({\mathrm{A}}_1)$–$({\mathrm{A}}_3)$ are met, the BVP (1) possesses at least one solution within the space X.

Proof. 

Let

$$\begin{array}{c}\hfill \Omega =\left\{(x,y)\in X|||(x,y)|{|}_X<\max \left\{{\varphi }_q(A_1)+{\varphi }_q(A_2),C\right\}+1\right\}.\end{array}$$

Lemmas 3 and 5 imply that M is a quasi-linear operator, and $N_{\lambda }$ is M-compact on $\overline{\Omega }$. Furthermore, according to Lemmas 6 and 7, the following conditions hold:

(i)

$M(x,y)\ne N_{\lambda }(x,y),\mathrm{for}\mathrm{all}((x,y),\lambda )\in \partial \Omega \times (0,1);$

(ii)

$QN(x,y)\ne (0,0),\mathrm{for}\mathrm{all}(x,y)\in \mathrm{Ker}M\cap \partial \Omega .$

We now proceed to verify condition (iii) of Theorem 1. Assuming condition (7) holds without loss of generality, we construct the following homotopy

$$H\left((x,y),\lambda \right)=\lambda (x,y)+(1-\lambda )JQN(x,y),$$

for all $(x,y)\in \overline{\Omega }\cap \mathrm{Ker}M,\lambda \in [0,1]$. By Lemma 8, it holds that

$$\begin{array}{c}\hfill H\left((x,y),\lambda \right)\ne (0,0),(x,y)\in \partial \Omega \cap \mathrm{Ker}M,\lambda \in [0,1].\end{array}$$

If condition (6) holds instead, one may define the homotopy

$$\begin{array}{c}\hfill H\left((x,y),\lambda \right)=\lambda (x,y)-(1-\lambda )JQN(x,y),\end{array}$$

and similarly verify that

$$\begin{array}{c}\hfill H\left((x,y),\lambda \right)\ne (0,0),(x,y)\in \partial \Omega \cap \mathrm{Ker}M,\lambda \in [0,1].\end{array}$$

Thus, by the homotopy invariance of the topological degree, we obtain

$$\begin{array}{cc}& \mathrm{deg}\{JQN{|}_{\mathrm{Ker}M},\Omega \cap \mathrm{Ker}M,(0,0)\}=\mathrm{deg}\{H((·,·),0),\Omega \cap \mathrm{Ker}M,(0,0)\}\hfill \\ & =\mathrm{deg}\{H((·,·),1),\Omega \cap \mathrm{Ker}M,(0,0)\}=\mathrm{deg}\{I,\Omega \cap \mathrm{Ker}M,(0,0)\}\ne 0,\hfill \end{array}$$

which confirms that condition (iii) holds. Consequently, by applying Theorem 1, we conclude that the equation $M(x,y)=N(x,y)$ has at least one solution in $\mathrm{dom}M\cap \overline{\Omega }$, indicating that the BVP (1) admits at least one solution. Therefore, Theorem 2 is confirmed. □

4. Example

Example 1.

Consider the following Hadamard FBVP,

$$\left\{\begin{array}{c}{({\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(t)))}^{\prime }+\frac{t^{-5}\sin \sqrt{|y(t)|}}{4\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}+t^{-5}{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(t))+\frac{1}{4}{t}^{-5}=0,t\in (1,+\infty ),\hfill \\ {({\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(t)))}^{\prime }+\frac{t^{-4}\sin \sqrt{|x(t)|}}{3\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}+t^{-4}{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(t))+\frac{1}{3}{t}^{-4}=0,t\in (1,+\infty ),\hfill \\ x(1)={}^{H}D_{1+}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(1)=0,{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(+\infty ))={\displaystyle \sum _{i=1}^{+\infty }}{\alpha }_i{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x({\xi }_i)),\hfill \\ y(1)={}^{H}D_{1+}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(1)=0,{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(+\infty ))={\displaystyle \sum _{i=1}^{+\infty }}{\beta }_i{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y({\eta }_i)),\hfill \end{array}\right.$$

(8)

where $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_i<\cdots$, $0<{\eta }_1<{\eta }_2<\cdots <{\eta }_i<\cdots$, $\underset{i\to +\infty }{\lim }{\xi }_i=+\infty$, $\underset{i\to +\infty }{\lim }{\eta }_i=+\infty$, $a(t)={\displaystyle \frac{1}{t}}$, ${\alpha }_i={\displaystyle \frac{1}{2^i}}$, ${\beta }_i={\displaystyle \frac{1}{i(i+1)}}$, ${\displaystyle \sum _{i=1}^{+\infty }}{\alpha }_i=1$, ${\displaystyle \sum _{i=1}^{+\infty }}{\beta }_i=1$. Corresponding to problem (1.1), here $\alpha =\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, $\beta =\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$, $p=\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, and

$$\begin{array}{cc}& g(t,y(t),{}^{H}D_{1+}^{\beta -1}y(t),{}^{H}D_{1+}^{\beta }y(t))={\displaystyle \frac{t^{-4}\sin \sqrt{|y(t)|}}{4\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}}+t^{-4}{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(t))+{\displaystyle \frac{1}{4}}{t}^{-4},\hfill \\ & f(t,x(t),{}^{H}D_{1+}^{\alpha -1}x(t),{}^{H}D_{1+}^{\alpha }x(t))={\displaystyle \frac{t^{-3}\sin \sqrt{|x(t)|}}{3\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}+t^{-3}{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(t))+{\displaystyle \frac{1}{3}}{t}^{-3}.\hfill \end{array}$$

Let ${\phi }_{r_1}(t)=(2\sqrt{r_1}+1)t^{-4}$ and ${\phi }_{r_2}(t)=(2\sqrt{r_2}+1)t^{-3}$. Then it follows that the functions g and f satisfy condition $({\mathrm{A}}_1)$. Let

$$\begin{array}{c}\hfill a_1(t)=b_1(t)={\displaystyle \frac{1}{4}}{t}^{-4},a_2(t)=b_2(t)={\displaystyle \frac{1}{3}}{t}^{-3},\\ \hfill c_1(t)=c_2(t)=0,d_1(t)=t^{-4},d_2(t)=t^{-3}.\end{array}$$

Then, it follows that

$$\begin{array}{cc}& \begin{array}{cc}& |g(t,y(t),{}^{H}D_{1+}^{\beta -1}y(t),{}^{H}D_{1+}^{\beta }y(t))|=\left|{\displaystyle \frac{t^{-4}\sin \sqrt{|y(t)|}}{4\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}}+t^{-4}{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{(}^H{D}_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(t))+{\displaystyle \frac{1}{4}}{t}^{-4}\right|\hfill \\ & \le {\displaystyle \frac{1}{4}}{t}^{-4}+{\displaystyle \frac{1}{4}}{t}^{-4}{\displaystyle \frac{\sqrt{|y(t)|}}{\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}}+t^{-4}\sqrt{|{}^{H}D_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(t)|},t\in \Im ,\hfill \end{array}\hfill \\ & \begin{array}{cc}& |f(t,x(t),{}^{H}D_{1+}^{\alpha -1}x(t),{}^{H}D_{1+}^{\alpha }x(t))|=\left|{\displaystyle \frac{t^{-3}\sin \sqrt{|x(t)|}}{3\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}+t^{-3}{\varphi }_{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(t))+{\displaystyle \frac{1}{3}}{t}^{-3}\right|\hfill \\ & \le {\displaystyle \frac{1}{3}}{t}^{-3}+{\displaystyle \frac{1}{3}}{t}^{-3}{\displaystyle \frac{\sqrt{|x(t)|}}{\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}+t^{-3}\sqrt{|{}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(t)|},t\in \Im ,\hfill \end{array}\hfill \end{array}$$

and

$$\begin{array}{cc}& ||b_1|{|}_Z={\displaystyle \frac{1}{4}}{\int }_1^{+\infty }{t}^{-5}dt={\displaystyle \frac{1}{16}},||c_1|{|}_Z=0,||d_1|{|}_Z={\int }_1^{+\infty }{t}^{-5}dt={\displaystyle \frac{1}{4}},\hfill \\ & ||b_2|{|}_Z={\displaystyle \frac{1}{3}}{\int }_1^{+\infty }{t}^{-4}dt={\displaystyle \frac{1}{9}},||c_2|{|}_Z=0,||d_2|{|}_Z={\int }_1^{+\infty }{t}^{-4}dt={\displaystyle \frac{1}{3}},\hfill \\ & [||b_1|{|}_Z+||c_1|{|}_Z+||d_1|{|}_Z\left]\right[||b_2|{|}_Z+||c_2|{|}_Z+||d_2|{|}_Z]={\displaystyle \frac{5}{36}}<1,\hfill \end{array}$$

that is, condition $({\mathrm{A}}_2)$ is satisfied. Moreover, let $B_1=3$, $B_2=4$, we obtain

$$\begin{array}{cc}& {}^{H}D_{1+}^{\beta }y(t)g(t,y(t),{}^{H}D_{1+}^{\beta -1}y(t),{}^{H}D_{1+}^{\beta }y(t))\hfill \\ & ={}^{H}D_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(t){\displaystyle \frac{t^{-4}\sin \sqrt{|y(t)|}}{4\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}}+t^{-4}|{}^{H}D_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(t){|}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+{\displaystyle \frac{1}{4}}{}^{H}D_{1+}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}y(t)t^{-4}>0,\hfill \\ & {}^{H}D_{1+}^{\alpha }x(t)f(t,x(t),{}^{H}D_{1+}^{\alpha -1}x(t),{}^{H}D_{1+}^{\alpha }x(t))\hfill \\ & ={}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(t){\displaystyle \frac{t^{-3}\sin \sqrt{|x(t)|}}{3\sqrt{1+{(\ln t)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}+t^{-3}|{}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(t){|}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+{\displaystyle \frac{1}{3}}{t}^{-3}{}^{H}D_{1+}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x(t)>0.\hfill \end{array}$$

Hence, condition $({\mathrm{A}}_3)$ is satisfied. Consequently, by Theorem 2, we conclude that the BVP (8) admits at least one solution.

5. Concluding Remarks

In this study, we conduct a thorough investigation of a coupled system of Hadamard fractional p-Laplacian differential equations defined on an unbounded interval, within the context of infinite-points BCs at resonance. The analytical process yields several notable advances: we generalize existence results previously established for linear resonant operators to the nonlinear framework incorporating quasilinear p-Laplacian structures within Hadamard fractional differential systems. The infinite-points BCs introduced herein represent a natural extension of finitely many-point BCs, thereby offering a more flexible and realistic modeling of complex BCs. To address the compactness deficiency inherent to the unbounded domain, we construct suitable weighted Banach spaces and utilize appropriate projection operators. Combined with the a-Carathéodory growth conditions and the Ge–Mawhin continuation theorem, this framework successfully establishes the existence of solutions. Future research directions may include extending the current analysis to coupled Hadamard fractional p-Laplacian systems with high-dimensional kernel structures, particularly under infinite-point BCs on unbounded intervals at resonance, and investigating the solvability of Hadamard fractional p-Laplacian equations with both integral-type and infinite-point BCs at resonance, formulated on unbounded intervals.