Abstract
In this research, we investigate the existence and multiplicity of solutions for fractional differential equations on infinite intervals. By using monotone iteration, we identify two solutions, and the multiplicity of solutions is demonstrated by the Leggett–Williams fixed point theorem.
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1 Introduction
Fractional differential equations’ theory has been widely employed in astronomy, biology, economics, and other domains, such as described in [1, 2, 4, 8, 11–13, 15, 17, 19, 20, 26]. In recent years, many authors have combined fractional derivative operators with problems of p-Laplacian type, Kirchhoff-type, etc. [21–24]. Their work made important contributions to enriching the study of fractional derivative problems. Research has investigated many difficulties of solutions of fractional differential equations on infinite intervals in addition to those on finite intervals.
In [29], the authors studied the BVP
The authors discovered the presence of solutions by employing the Leray–Schauder nonlinear theorem. In [25], Guotao Wang studied the BVP
where \(\varsigma \in K=[0,\infty )\), \(k\in C(K\times \mathbb{R},\mathbb{R})\), \(\rho \in \mathbb{R}\), \(\varrho \in K\). The author obtained the existence and uniqueness of solutions by the monotone iterative technique. In [18], the Leggett–Williams fixed point theorem and the Guo–Krasnoselskii fixed point theorem were used by Phollakrit Thiramanus et al. to investigate
where \(\upsilon \in (1,\infty )\), \(\gamma _{p}\), \(p=1,\dots ,n\), and \(\zeta _{p}\ge 0\), \(p=1,\dots ,n\) are given constants. In [9], the authors studied the existence of solutions to the problem
where \(2<\alpha < 3\), \(0<\delta _{1} <\delta _{2} <\cdots <\delta _{n-2} <\infty \), \(\varsigma _{i} \ge 0\), \(i=1,\dots , n-2\) satisfy \(0<\sum_{i=1}^{n-2} \varsigma _{i} \delta _{i}^{ \alpha -1} <\Gamma (\alpha )\). From the Leggett–Williams fixed point theorem, the existence of at least three positive solutions was demonstrated. In [27], the authors investigated the family of BVPs
where \(1<\upsilon <\zeta _{1} <\zeta _{2} <\cdots <\zeta _{p}<\infty \), b, \(\alpha _{m}\), \(\zeta _{n}\ge 0\), \(m=1,\dots ,p\), \(n=1,\dots ,q\) are given constants. Various fixed point theorems were used to prove the results. The generalized Avery–Henderson fixed point theorem was used to demonstrate the presence of many positive solutions to problem (5) in [28].
In this paper, motivated by the previous results, we investigate the BVP
where \(m\ge 2\), \(D^{\delta }\) is the Riemann–Liouville fractional derivative of order δ, \(D^{\delta -1 } x(\infty )=\lim_{t \to \infty} D^{\delta -1 }x(t)\), \(I^{\beta _{j} }\) is the Riemann–Liouville fractional integral of order \(\beta _{i}>0\), \(i=1,2,\dots ,q\), \(0<\varrho <\infty \), \(v_{i}\), λ, \(\kappa _{j}\), \(\varsigma _{j}\ge 0\), \(i=1,\dots ,q\), \(j=1,\dots ,p\) are given constants; \(r:I\times \mathbb{R^{+} } \to \mathbb{R^{+} }\), with \(I=[0,\infty )\), is continuous and \(b:I\to \mathbb{R^{+} } \) is integrable, and
Using the monotone iteration method, we prove that two solutions of problem (6) exist. Using the Leggett–Williams fixed point theorem, we determine that problem (6) has at least three solutions.
Compared with [27, 28], we use different methods to study multiple solutions. Compared with [9, 18, 25, 27–29], we study fractional differential equations of arbitrary order \(m\ge 2\). It is obvious that our problem is more general. Our boundary conditions have more general forms and the boundary conditions of [9, 18, 25, 29] are our special cases. When \(m=2\), \(v_{i} =0\), \(\lambda =1\), \(j=1\), we know that problem (1) is a special case of problem (6). When \(m=3\), \(i=1\), \(\lambda =0\), we have that problem (2) is a special case of problem (6). When \(\lambda =0\), we obtain that the boundary conditions of problem (3) are a special case of the boundary conditions of problem (6). When \(m=3\), \(v_{i} =0\), \(\lambda =1\), \(j=1,2,\dots ,p-2\), we get that the boundary conditions of problem (4) are a special case of the boundary conditions of problem (6).
In this paper, the following four conditions will be used:
- \((H_{1} )\):
-
\(r\in C(I\times \mathbb{R^{+} },\mathbb{R^{+} })\) and \(r(t,\cdot )\not \equiv 0\) on any subinterval of \(\mathbb{R^{+} }\), and when x is bounded, \(r(t,(1+t^{\delta -1} )x)\) is bounded on \(\mathbb{R^{+} }\).
- \((H_{2} )\):
-
\(b:I\to \mathbb{R^{+} }\) does not identically vanish on any subinterval of \(\mathbb{R^{+} }\) and
$$\begin{aligned} 0< \int _{0}^{\infty } b(s)\,ds< \infty . \end{aligned}$$ - \((H_{3} )\):
-
r is nondecreasing with respect to the second variable.
- \((H_{4} )\):
-
There exists a positive constant Λ such that
$$\begin{aligned} r\bigl(t,\bigl(1+t^{\delta -1} \bigr)x\bigr)\le \frac{\Lambda }{\frac{1}{\Delta } \int _{0}^{\infty}b(s)\,ds } \quad \text{for any } (t,x)\in [0,\infty )\times [0,\Lambda ]. \end{aligned}$$
Assumptions \((H_{1})\) and \((H_{2})\) will be applied in Lemma 3.2 and Theorem 3.5, while \((H_{1})\)–\((H_{4})\) will be used in Theorem 3.4.
The remainder of this article is organized as follows. Section 2 contains the definitions and lemmas required to prove our results. Section 3 presents the existence and multiplicity results for the boundary value problem (6). Section 4 provides examples relevant to the key findings of this paper.
2 Preliminaries
We present several definitions and lemmas here for the reader’s convenience, as they will be utilized to prove our primary results.
Definition 2.1
The Riemann–Liouville fractional derivative of order \(\delta > 0\) of a function \(h : (0, \infty ) \to \mathbb{R}\) is given by
where \(m-1< \nu \le m\).
Definition 2.2
The Riemann–Liouville fractional integral of order \(\eta > 0\) of a function \(h : (0, \infty ) \to \mathbb{R}\) is given by
Lemma 2.3
(see [5])
If \(\varrho ,\theta > 0\), then
Lemma 2.4
If \(e \in C([0,\infty ) ,\mathbb{R}) \), then the problem
has a unique solution
where
Proof
Since \(D^{\delta } x(t)+e(t) = 0\), we obtain
Due to \(x^{(n)}(0)=0\), \(n=0,\dots ,m-2\), we have \(l_{2} =\cdots =l_{m} =0\), that is,
Since \(D^{\delta -1 } x(\infty ) = \sum_{i = 1}^{q} v_{i} I^{ \beta _{i} } x(\varrho )+\lambda \sum_{j=1}^{p} \kappa _{j} x(\varsigma _{j})\), we have
Then
The proof is complete. □
Lemma 2.5
If \(\Delta >0\), the following properties apply to the Green function \(\Pi (t, s)\) defined by (8) for all \((t,s)\in I\times I\):
-
(i)
\(\Pi (t,s)\) is nonnegative and continuous.
-
(ii)
\(\Pi (t,s)\) is increasing with respect to t.
-
(iii)
\(\frac {\Pi (t,s)}{1+t^{\delta -1} } \le \frac {1}{\Delta } \).
-
(iv)
\(\Pi (t,s)\le \frac {1}{\Delta } t^{\delta -1}\).
Proof
(i) According to the definition of \(\Pi (t,s)\), this is clearly true.
(ii) We just have to prove that \(\pi (t,s,\delta )\) increases as t increases. Let
We have
Then
(iii) We have
(iv) We have
The proof is complete. □
Let X be a Banach space endowed with norm \(\Vert \cdot \Vert _{X}\). Let \(0 < v < w\) be given, and let ϑ be a nonnegative continuous concave functional on K. Define the convex sets \(K_{\mu } \) and \(K(\vartheta , v, w) \) by \(K_{\mu } = \{ x\in K: \Vert x \Vert _{X}< \mu \} \) and \(K(\vartheta ,v,w)= \{ x\in K:\vartheta (x)\ge v, \Vert x \Vert _{X} \le w \} \).
Lemma 2.6
(see [6])
Let \(H:\overline{K_{\sigma } } \to \overline{K_{\sigma } } \) be a completely continuous operator, and let ϑ be a nonnegative continuous concave functional on K such that \(\vartheta (x)\le \Vert x \Vert \) for all \(x\in \overline{K_{\sigma}} \). Assume there exist \(0 < \tau _{1} < \tau _{2} < \tau _{3} \le \sigma \) such that
- \((B_{1})\):
-
\(\{ x\in K(\vartheta ,\tau _{2} ,\tau _{3} )|\vartheta (x)> \tau _{2} \} \ne \emptyset \) and \(\vartheta (Hx)>\tau _{2} \) for \(x\in K(\vartheta ,\tau _{2} ,\tau _{3} )\);
- \((B_{2})\):
-
\(\Vert Hx \Vert \le \tau _{1} \) for \(\Vert x \Vert \le \tau _{1} \);
- \((B_{3})\):
-
\(\vartheta (Hx)>\tau _{2} \) for \(x\in K(\vartheta ,\tau _{2} ,\sigma )\) with \(\Vert Hx \Vert >\tau _{3} \).
Then H has at least three fixed points \(x_{1}\), \(x_{2}\), and \(x_{3}\) such that
3 Main results
Let \(X= \{ x\in C(I,\mathbb{R^{+} } ):\sup_{t\in I } \frac{ \vert x(t) \vert }{1+t^{\delta -1} } < \infty \} \) be the Banach space with norm \(\Vert x \Vert _{X} = \sup_{t\in I } \frac{ \vert x(t) \vert }{1+t^{\delta -1} } \). We define a cone \(\Upsilon \subset X\) by
and an operator \(\Phi :\Upsilon \to X\) by
It is simple to demonstrate that \(\Phi :\Upsilon \to \Upsilon \).
Lemma 3.1
Let \(\Omega \subset X\) be a bounded set. Then Ω is relatively compact in X if the following conditions hold:
-
(i)
For any \(x\in \Omega \), \(\frac {x(t)}{1+t^{\alpha -1} } \) is equicontinuous on any compact interval of \([0,\infty )\);
-
(ii)
For any \(\varepsilon >0\), there exists a constant \(N>0\) such that
$$\begin{aligned} \biggl\vert \frac{x(t_{1} )}{1+t_{1 }^{\alpha -1 } }- \frac{x(t_{2} )}{1+t_{2}^{\alpha -1 } } \biggr\vert < \varepsilon \end{aligned}$$for any \(t_{1}, t_{2} >N\) and \(x\in \Omega \).
Lemma 3.2
(see [7])
If \((H_{1} )\) and \((H_{2} )\) hold, then \(\Phi :\Upsilon \to \Upsilon \) is completely continuous.
Remark 3.3
To prove that Φx is equiconvergent at infinity, we will give another method. For any \(\varepsilon >0\), there exists a constant \(N_{1}>0\) such that
where
Note
Then for the above \(\varepsilon >0\), there exist constants \(N_{2}>0\), \(N_{3}>N_{1}\) such that for any \(t_{1}\), \(t_{2}>N_{2}\), we have
and for any \(t_{1}\), \(t_{2}>N_{3}\), \(0\le s\le N_{1}\), we have
Choose \(N>\max \{ N_{2} ,N_{3} \} \). Then for any \(t_{1}\), \(t_{2}>N\), we have
Thus, Φx is equiconvergent at infinity.
Theorem 3.4
If \((H_{1})\)–\((H_{4})\) hold, then two explicit monotone iterative sequences can yield two positive solutions \(x^{*}\), \(y^{*} \) of problem (6), namely
in the interval \((0,\Lambda t^{\delta -1} ]\).
Proof
We define \(W= \{ x\in X: \Vert x \Vert _{X} \le \Lambda \}\), while Φ is defined by (10). Then we show that \(\Phi (W)\subset W\). For any \(x\in W\), by \((H_{4})\) and (9), we have
Due to the definition of the operator Φ and assumption \((H_{3})\), we have that Φ is a nondecreasing operator. Define \(x_{0} (t)=0\), \(x_{1} =\Phi x_{0}\), \(x_{2} =\Phi x_{1}=\Phi ^{2}x_{0}\) for any \(t\in I\). In view of \(x_{0} (t)=0\in W\) and \(\Phi (W)\subset W\), we have \(x_{1} \in W\), \(x_{2} \in W\) and
Considering the nondecreasing nature of the operator Φ, we get
Define a sequence \(\Phi x_{n} =x_{n+1}\), \(n\in \mathbb{N} \). Clearly, the sequence \(\{ x_{n} \} _{n=1}^{\infty } \subset W\) and it satisfies
Because of the complete continuity of the operator Φ, there exists a subsequence \(\{ x_{n_{k} } \} _{k=1}^{\infty } \subset W\), \(x^{*} \subset W\) such that \(x_{n_{k} }\to x^{*}\), \(k\to \infty \). This, together with the monotone nature of \(\{ x_{n} \} _{n=1}^{\infty } \), implies that \(\lim_{n \to \infty} x_{n} =x^{*} \). Since Φ is continuous and \(\Phi x_{n} =x_{n+1}\), we have \(\Phi x^{*} =x^{*} \), i.e., \(x^{*}\) is a fixed point of the operator Φ.
Define \(y_{0} (t)=\Lambda t^{\alpha -1}\), \(y_{1} =\Phi y_{0}\), \(y_{2} =\Phi y_{1} =\Phi ^{2} y_{0}\) for any \(t\in I\). In view of \(y_{0} (t)=\Lambda t^{\alpha -1} \in W\) and \(\Phi (W)\subset W\), we have \(y_{1}\in W\), \(y_{2} \in W\). By Lemma 2.5 and \((H_{4})\), we obtain
Due to the nondecreasing nature of the operator Φ, we have
Define a sequence \(\Phi y_{n} =y_{n+1}\), \(n\in \mathbb{N}\). Clearly, the sequence \(\{ y_{n} \} _{n=1}^{\infty } \subset W\) and it satisfies
As before, we can conclude that there exists \(y^{*} \in W\) such that \(\lim_{n \to \infty} y_{n} =y^{*}\). Since Φ is continuous and \(\Phi y_{n} =y_{n+1}\), we have \(\Phi y^{*} =y^{*} \), i.e., \(y^{*}\) is a fixed point of the operator Φ.
Since for any \(t\in I\), \(r(t,\cdot )\not \equiv 0\), 0 is not a solution of problem (6). According to the above process, we know that \(x^{*}\) and \(y^{*} \) are two positive solutions of problem (6) in \((0,\Lambda t^{\delta -1} ]\), which can be established using two explicit monotonic iterative sequences (11), respectively. □
Theorem 3.5
If \((H_{1})\), \((H_{2})\) hold, then there exist numbers \(\delta _{1}\), \(\delta _{2}\), \(\delta _{3}>0\), and \(0< \theta < 1\) such that \(0< \delta _{1}< \delta _{2} <\frac{\delta _{2}}{\theta } \le \delta _{3}\). In addition, assume
- \((A_{1})\):
-
\(r(t,(1+t^{\delta -1})x)\le \delta _{3}M_{1} \), \((t,x)\in I\times [ 0,\delta _{3} ] \), where \(M_{1} = ( \frac{1}{\Delta } \int _{0}^{\infty }b(s)\,ds ) ^{-1}\);
- \((A_{2})\):
-
\(r(t,(1+t^{\delta -1})x)\le \delta _{1}M_{1} \), \((t,x)\in I\times [ 0,\delta _{1} ] \);
- \((A_{3})\):
-
\(r(t,(1+t^{\delta -1})x)\ge \delta _{2}M_{2} \), \((t,x)\in [ \frac{1}{k} ,k ] \times [ \delta _{2},\delta _{3} ] \), where \(M_{2} = ( \frac{k^{1-\delta } }{1+k^{\delta -1} } \int _{ \frac{1}{k} }^{k}b(s)\,ds ) ^{-1}\).
Then problem (6) has at least three positive solutions \(x_{1}^{*}\), \(x_{2}^{*}\), and \(x_{3}^{*}\) such that
Proof
Let \(k>1\), \(\omega (x)=\min_{t\in [ \frac{1}{k} ,k ] } \frac{x(t)}{1+t^{\delta -1} } \), while Φ is defined by (10). The proof will be broken down into four steps.
Step 1.
For any \(x\in \overline{\Upsilon _{\delta _{3}} } \), by the condition \((A_{1} )\) and (9), we have
Then \(\Phi :\overline{\Upsilon _{\delta _{3}} } \to \overline{\Upsilon _{\delta _{3}} } \). According to Lemma 3.2, we get that \(\Phi :\overline{\Upsilon _{\delta _{3}} } \to \overline{\Upsilon _{\delta _{3}} } \) is completely continuous.
Step 2.
Let \(x_{0}(t)=0.5 (\delta _{2}+\frac{\delta _{2}}{\theta } ) (1+t^{\delta -1} ) \), then we obtain that
and
which shows that
and thus
By the condition \((A_{3} )\), Lemma 2.5, and (8), for any \(x\in \Upsilon (\omega ,\delta _{2},\frac{\delta _{2}}{\theta } )\), we have
Step 3.
By the condition \((A_{2} )\) and (9), we have
Step 4.
Similar to (14), for any \(x\in \Upsilon (\omega ,\delta _{2},\delta _{3})\) and \(\Vert \Phi x \Vert _{X} > \frac{\delta _{2}}{\theta } \), by condition \((A_{3} )\), Lemma 2.5, and (8), we have \(\omega (\Phi x)>\delta _{2}\).
Conclusion.
Thus, by Lemma 2.6, problem (6) has at least three positive solutions \(x_{1}^{*}\), \(x_{2}^{*}\), and \(x_{3}^{*}\) such that \(\Vert x_{1}^{*} \Vert _{X} \le \delta _{1}\), \(\omega ( x_{2}^{*})\ge \delta _{2} \), \(\Vert x_{3}^{*} \Vert _{X} \ge \delta _{1}\), and \(\omega (x_{3}^{*} )< \delta _{2}\). □
4 Examples
Example 4.1
We consider the problem
where
and
We can show that
When \(x\in [0,1]\), we have
Thus, we have that \((H_{1} )\)–\((H_{4} )\) hold. We can obtain that problem (15) has two positive solutions \(x^{*}\), \(y^{*} \) in \((0, t^{\alpha -1} ]\) by Theorem 3.4, which can be approximated by the iterative sequences
Example 4.2
We consider the problem
where
and
We have
Choosing \(\delta _{1}=\frac{1}{2}\), \(\delta _{2}=1.01\), \(\delta _{3}=101\), \(\theta =\frac{1}{10} \), we have
From Theorem 3.5, the BVP (16) has at least three positive solutions \(x_{1}^{*}\), \(x_{2}^{*}\), and \(x_{3}^{*}\) such that \(\Vert x_{1}^{*} \Vert _{X} \le \frac{1}{2} \), \(\omega ( x_{2}^{*})\ge 1.01 \), \(\Vert x_{3}^{*} \Vert _{X} \ge \frac{1}{2} \), and \(\omega (x_{3}^{*} )< 1.01\).
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The authors would like to thank the referee for pertinent comments and valuable suggestions.
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This research is supported by NSFC (11871302), the Fund of the Natural Science of Shandong Province (ZR2014AM034), and colleges and universities of Shandong province science and technology plan projects (J13LI01).
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Zhou, W., Hao, Z. & Bohner, M. Existence and multiplicity of solutions of fractional differential equations on infinite intervals. Bound Value Probl 2024, 26 (2024). https://doi.org/10.1186/s13661-024-01832-8
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DOI: https://doi.org/10.1186/s13661-024-01832-8