Existence of positive solutions for singular higher-order fractional differential equations with infinite-point boundary conditions

In this paper, we consider the following class of nonlinear singular fractional differential equations: where \(\alpha,\mu_{i}\in\mathbb{R}^{1}_{+}=[0, +\infty)\), \(n\in \mathbb{N}\) (the set of natural numbers), and \(n-1<\alpha\leq n\), \(n\geq4\), \(i-1<\mu_{i}\leq i\) (\(i=1,2,\ldots,n-2\)), \(f(t,x_{1},x_{2},\ldots ,x_{n-1})\) may have singularity at \(t=0,1\), \(x_{i}=0\) (\(i=1,2,\ldots ,n-1\)), \(0<\xi_{1}<\xi_{2}<\cdots<\xi_{j}<\cdots<1\), \(\eta_{j}>0\) (\(j=1,2,\ldots\)), \(\sum_{j=1}^{\infty}\eta_{j}\xi_{j}^{\alpha-1}<\Theta\), \(\Theta =(\alpha-1)(\alpha-2)\cdots(\alpha-n+2)\), and \(D_{0^{+}}^{\alpha}u\), \(D^{\mu _{i}}_{0^{+}}u\) (\(i=1,2,\ldots,n-2\)) are the Riemann-Liouville derivatives.

Boundary value problems for nonlinear fractional differential equations arise from the studies of complex problems in many disciplinary areas such as fluid flows, electrical networks, rheology, biology chemical physics, and so on. Fractional-order models have been shown to be more accurate than integer-order models, and in applications of these models, it is important to theoretically establish conditions for the existence of positive solutions. In recent years, many authors investigated the existence of positive solutions for fractional equation boundary value problems (see [1–24] and the references therein), and a great deal of results have been developed for differential and integral boundary value problems. In [14], the author considered the following fractional differential equation: where \(\alpha\in\mathbb{R}^{1}_{+}\), \(n\in\mathbb{N}\), \(n-1<\alpha \leq n\), \(n\geq3\), \(\alpha_{j}\geq0\), \(0<\xi_{1}<\xi_{2}<\cdots<\xi _{j-1}<\xi _{j}<\cdots<1\) (\(j=1,2,\ldots\)), and \(D_{0^{ +}}^{\alpha}\) is the standard Riemann-Liouville derivative, and \(f\in C((0, 1)\times(0,+\infty), \mathbb{R}^{1}_{+})\) allows singularities with respect to both time (\(t=0,1\)) and space variables (\(u=0\)). The author established the existence and multiplicity of positive solutions. In [15], the authors investigated the fractional differential equation where \(\alpha,v,\mu\in\mathbb{R}^{1}_{+}\), \(3<\alpha\leq4\), \(0< v\leq1\), \(0<\mu\leq1\) are real numbers, \(D^{\alpha}_{0+}\), \(D^{v}_{0^{+}}\), and \(D^{\mu}_{0^{+}}\) are the Riemann-Liouville fractional derivatives, \(f(t,x,y,z)\) is a Carathéodory function singular at \(x,y,z=0\). The authors obtained the existence and multiplicity of positive solutions by means of Krasnoselskii’s fixed point theorem. As there do not exists \(0 < L < M\) such that (3.13) of [15] holds, the results of the multiple solutions are not correct in [15]. In [16], the authors investigated the fractional differential equation where \(\alpha,\mu\in\mathbb{R}^{1}_{+}\), \(1<\alpha\leq2\), \(\mu >0\) are real numbers, \(\alpha-\mu\geq1\), \(D^{\alpha}_{0+}\) and \(D^{\mu}_{0^{+}}\) are the Riemann-Liouville fractional derivatives, f is a Carathéodory function, and \(f(t,x,y)\) is singular at \(x=0\). The authors obtained the existence of positive solutions by means of the Krasnoselskii’s fixed point theorem. In [17], the author investigated the fractional differential equation where \(\alpha,\mu\in\mathbb{R}^{1}_{+}\), \(2<\alpha\leq3\), \(0<\mu \leq1\) are real numbers, \(D^{\alpha}_{0+}\) and \(D^{\mu}_{0^{+}}\) are the Riemann-Liouville fractional derivatives, f is a Carathéodory function, and \(f(t,x,y,z)\) is singular at \(x,y,z=0\). The authors obtained the existence of positive solutions by means of the Krasnoselskii’s fixed point theorem. In [18], the author investigated the singular problem where \(\alpha\in\mathbb{R}^{1}_{+}\), \(n-1<\alpha\leq n\), \(n\geq2\), the nonlinear function \(f(t,x_{1},x_{2},\ldots,x_{n-1})\) may be singular at \(x_{i}=0\) (\(i=1,2,\ldots,n-1\)), and \(q(t)\) may be singular at \(t=0\). The existence results of positive solutions are obtained by a fixed point theorem for a mixed monotone operator.

Motivated by the results mentioned, in this paper, we utilize the sequential technique and regularization to investigate the existence of positive solutions of BVP (1.1), where \(u\in C^{n-2}[0,1]\cap C^{n-1}(0,1)\) is said to be a positive solution of BVP (1.1) if and only if u satisfies (1.1) and \(u(t)>0\) for any \(t\in(0, 1]\). By using the sequential technique and regularization on a cone, some new existence results are obtained for the case where the nonlinearity is allowed to be singular with respect to both time and space variables. We emphasize here that our work presented in this paper has various new features. Firstly, we study singular nonlinear differential equation boundary value problems, that is, \(f(t,x_{1},x_{2},\ldots,x_{n-1})\) may have singularity at \(t=0,1\) and \(x_{i}=0\) (\(i=1,2,\ldots,n-1\)), which leads to many difficulties in analysis. Secondly, compared with [15–17], we complete the proof without the need of imposing the third condition of the Carathéodory conditions, that is, the condition \(|f(t,x_{1},x_{2},\ldots,x_{n-1})|\leq\varphi_{H}(t)\) is successfully removed, and, at the same time, the condition is extended to Thirdly, a special cone in a special space is established to overcome the difficulties caused by the singularity. Finally, values at infinite points are involved in the boundary conditions, and the fractional orders are involved in the nonlinearity of the boundary value problem (1.1).

For convenience, we list some conditions to be used throughout the paper.

In order to overcome the singularity, we utilize the sequential technique and regularization to testify the existence of positive solutions for problem (1.1). Next, for any \(m\in\mathbb {N}\), we define \(X_{m}:\mathbb{R}^{1}\rightarrow \mathbb{R}^{1}\) and \(f_{m}:[0,1]\times\mathbb{R}^{n-1}_{+}\rightarrow\mathbb {R}^{1}\) by the following formulas: for all \((x_{1},x_{2},\ldots,x_{n-1})\in\mathbb{R}^{n-1}\) and a.e. \(t\in[0,1]\), Then condition (H₁) gives that \(f_{m}\) satisfies the local Carathéodory condition on \([0,1]\times\mathbb{R}^{n-1}_{+}\) and \(f_{m}(t,x_{1},x_{2},\ldots,x_{n-1})\geq C\) for a.e. \(t\in[0,1]\) and all \((x_{1},x_{2},\ldots,x_{n-1})\in\mathbb{R}^{n-1}_{+}\). Condition (H₂) provides that for a.e. \(t\in[0,1]\) and all \((x_{1},x_{2},\ldots,x_{n-1})\in\mathbb {R}^{n-1}_{+}\).

Next, we discuss the regular fractional differential equation

For convenience of the reader, we first present some basic definitions and lemmas. These definitions can be found in the recent literature such as [19, 20]. In this paper, \(\|x\|_{1}=\int^{1}_{0}|x(t)|\, dt\) is the norm in \(L^{1}[0,1]\), \(\|x\|=\max\{|x(t)|:t\in[0,1]\}\) is the norm in \(C[0,1]\), and is the norm in \(E=C^{n-2}[0,1]\); \(\mathit{AC}^{k}[0,1]\) (\(k=0,1,2,\ldots\)) is the space of absolutely continuous functions having absolutely continuous kth-order derivatives on \([0,1]\).

Let \(E=C^{n-2}[0,1]\) and define the cone K in E as By Lemma 2.4 and (2.1) we have For any \(m\in\mathbb{N}\), define the operator \(Q_{m}:K\rightarrow E\) as follows:

We consider the following nonlinear singular fractional differential equation: where \(\alpha=\frac{9}{2}\), \(\mu_{1}=\frac{1}{2}\), \(\mu_{2}=\frac {3}{2}\), \(\mu _{3}=\frac{5}{2}\), \(\eta_{j}=\frac{14}{j^{2}}\), \(\xi_{j}=\frac {1}{j^{\frac {9}{7}}}\), \(a\in (0,\frac{2}{7} )\), \(b\in (0,\frac {1}{9} )\), \(c\in (0,\frac{2}{7} )\), \(d\in (0,\frac {2}{5} )\), \(b_{1},c_{1},d_{1}\in(0,1)\). Letting \(\gamma (t)=\frac {3+t^{\frac{1}{2}}}{2\text{,}100t^{\frac{1}{2}}}\) and \(\beta(t)=\frac {1+t^{\frac {1}{2}}}{t^{\frac{1}{2}}}\), we easily get Then the function satisfies the hypotheses (H₀), (H₁), and (H₂) for and so hypothesis (H₃) is also satisfied. Therefore, Theorem 1.1 guarantees that the fractional differential equation (5.1) has one positive solution u satisfying inequality (1.3) for \(t\in [0,1]\), where \(M=\frac{ 7}{(\alpha-4)\Gamma(\alpha+1)}\).

In this paper, some existence results are obtained successfully for the boundary value problem (1.1) for the case where the nonlinearity is allowed to be singular with respect to not only the time variable but also the space variable and also the boundary conditions may involve infinite number of points. Compared with previous work [15–17], we complete the proof without imposing the third Carathéodory condition, that is, the condition \(|f(t,x_{1},x_{2},\ldots,x_{n-1})|\leq\varphi_{H}(t)\) is successfully removed, and, at the same time, the condition is extended to which leads to more general results. Moreover, the results of [15] seem to be wrong when \(\lim_{x\rightarrow0}\frac{h(x,x,\ldots,x)}{x}=0\). So we have improved the result of [15–17]. The method we utilized for the analysis is the sequential technique and regularization, and the existence of positive solutions is obtained by the fixed point theorem.