Abstract
This paper is concerned with the following boundary value problem of nonlinear fractional differential equation with integral boundary conditions:
where \(2< q\leq 3\), \(0<\sigma \leq 1\), \(\alpha , \gamma , \delta \geq 0\), and \(\beta >0\) satisfying \(0<\rho :=(\alpha +\beta )\gamma + \frac{\alpha \delta }{\varGamma (2-\sigma )}<\beta [\gamma + \frac{\delta \varGamma (q)}{\varGamma (q-\sigma )} ]\). \({}^{C}D_{0+}^{q}\) denotes the standard Caputo fractional derivative. First, Green’s function of the corresponding linear boundary value problem is constructed. Next, some useful properties of the Green’s function are obtained. Finally, existence results of at least one positive solution for the above problem are established by imposing some suitable conditions on f and \(h_{i}\) (\(i=1,2\)). The method employed is Guo–Krasnoselskii’s fixed point theorem. An example is also included to illustrate the main results of this paper.
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1 Introduction
Fractional calculus has widespread applications in many fields of science and engineering, for example, physics, viscoelasticity, continuum mechanics, bioengineering, rheology, electrical networks, control theory of dynamical systems, optics and signal processing, and so on [1, 2].
Since the discussion of many problems can be summed up in the study of boundary value problems (BVPs for short) to nonlinear fractional differential equations, recently, the existence of solutions or positive solutions of BVPs for nonlinear fractional differential equations has received considerable attention from many authors, see [3–26] and the references therein.
In particular, in 2009, by using nonlinear alternative of Leray–Schauder type and Guo–Krasnoselskii’s fixed point theorem, Bai and Qiu [5] obtained the existence of a positive solution to the singular BVP
where \(2< q\leq 3 \) is a real number, \(f:(0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous, and \(\lim _{t\rightarrow 0^{+}}f(t,\cdot )=+\infty \).
In 2012, Cabada and Wang [7] studied the existence of a positive solution for the BVP with integral boundary conditions
where \(2< q<3\), \(0<\lambda <2\), and \(f:[0,1] \times [0,+\infty )\rightarrow [0,+\infty )\) is continuous. Their analysis relied on Guo–Krasnoselskii’s fixed point theorem.
In 2014, Cabada and Hamdi [25] investigated the BVP with integral boundary conditions
where \(2< q\leq 3\), \(D_{0+}^{q}\) denotes the Riemann–Liouville fractional derivative, \(0<\lambda <q\), and \(f:[0,1] \times [0,+\infty )\rightarrow [0,+\infty )\) is a continuous function. The authors proved the existence of a positive solution to BVP (3) by employing Guo–Krasnoselskii’s fixed point theorem.
As it has been stated in [7], BVPs with integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermo-elasticity, underground water flow, population dynamics, and so forth. Motivated by the above-mentioned works, in this paper, we consider the existence of a positive solution for the following BVP of nonlinear fractional differential equation with integral boundary conditions:
Throughout this paper, we always assume that \(2< q\leq 3\), \(0<\sigma \leq 1\), \(\alpha , \gamma , \delta \geq 0\), and \(\beta >0\) satisfying \(0<\rho :=(\alpha +\beta )\gamma + \frac{\alpha \delta }{\varGamma (2-\sigma )}<\beta [\gamma + \frac{\delta \varGamma (q)}{\varGamma (q-\sigma )} ]\), \(f :[0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) and \(h_{i}\) (\(i=1,2\)): \([0,1] \rightarrow [0,+\infty )\) are continuous.
The main tool used is the following well-known Guo–Krasnoselskii’s fixed point theorem [27, 28].
Theorem 1.1
LetEbe a Banach space andKbe a cone in E. Assume that\(\varOmega _{1}\)and\(\varOmega _{2}\)are bounded open subsets ofEsuch that\(0 \in \varOmega _{1}\), \(\overline{\varOmega }_{1}\subset \varOmega _{2}\), and let\(T:K\cap (\overline{\varOmega }_{2}\backslash \varOmega _{1} ) \rightarrow K\)be a completely continuous operator such that either
- (1)
\(\Vert Tu \Vert \leq \Vert u \Vert \)for\(u\in K\cap \partial \varOmega _{1}\)and\(\Vert Tu \Vert \geq \Vert u \Vert \)for\(u\in K\cap \partial \varOmega _{2}\), or
- (2)
\(\Vert Tu \Vert \geq \Vert u \Vert \)for\(u\in K\cap \partial \varOmega _{1}\)and\(\Vert Tu \Vert \leq \Vert u \Vert \)for\(u\in K\cap \partial \varOmega _{2}\).
ThenThas a fixed point in\(K\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1} )\).
2 Preliminaries
Let \([a,b] \) (\(-\infty < a< b<+\infty \)) be a finite interval on the real axis \(\mathbb{R}\), \(\mathbb{N}=\{1, 2, 3,\ldots\}\), \(\mu >0\) and \([\mu ]\) be the integer part of μ.
First, we present definitions of some spaces.
Let \(\operatorname{AC}[a,b]\) be the space of functions u which are absolutely continuous on \([a,b]\). For \(n\in \mathbb{N}\), we denote by \(\operatorname{AC}^{n}[a,b]\) the space of functions u which have continuous derivatives up to order \(n-1\) on \([a,b]\) such that \(u^{(n-1)}\in \operatorname{AC}[a,b]\). In particular, \(\operatorname{AC}^{1}[a,b]=\operatorname{AC}[a,b]\).
For \(m\in \mathbb{N}_{0}=\{0, 1, 2,\ldots\}\), we denote by \(C^{m}[a,b]\) the space of functions u which are m times continuously differentiable on \([a,b]\). In particular, for \(m=0\), \(C^{0}[a,b]=C[a,b]\) is the space of continuous functions u on \([a,b]\).
Next, we give the definitions of the Riemann–Liouville fractional integrals and fractional derivatives and the Caputo fractional derivatives on \([a,b]\), which may be found in [1].
Definition 2.1
The Riemann–Liouville fractional integrals \(I_{a+}^{\mu }u\) and \(I_{b-}^{\mu }u\) of order μ are defined by
and
respectively, where
Definition 2.2
The Riemann–Liouville fractional derivatives \(D_{a+}^{\mu }u\) and \(D_{b-}^{\mu }u\) of order μ are defined by
and
respectively, where \(n=[\mu ]+1\).
Definition 2.3
Let \(D_{a+}^{\mu }[u(s)](t)\equiv (D_{a+}^{\mu }u)(t)\) and \(D_{b-}^{\mu }[u(s)](t)\equiv (D_{b-}^{\mu }u)(t)\) be the Riemann–Liouville fractional derivatives of order μ, respectively. The Caputo fractional derivatives \({}^{C}D_{a+}^{\mu }u\) and \({}^{C}D_{b-}^{\mu }u\) of order μ on \([a,b]\) are defined by
and
respectively, where
Lemma 2.1
(see [2])
Let\(\nu >\mu \). Then the equation\(({}^{C}D_{0+}^{\mu }I_{0+}^{\nu }u)(t)=(I_{0+}^{\nu -\mu }u)(t)\), \(t \in [0,1]\)is satisfied for\(u\in C [0,1 ]\).
Lemma 2.2
(see [1])
Letnbe given by (5). Then the following relations hold:
- (1)
for\(k\in \{0,1,2,\ldots,n-1\}\), \({}^{C}D_{0+}^{\mu }t^{k}=0\);
- (2)
if\(\nu >n\), \({}^{C}D_{0+}^{\mu }t^{\nu -1}= \frac{\varGamma (\nu )}{\varGamma (\nu -\mu )}t^{\nu -\mu -1}\).
Lemma 2.3
(see [1])
Letnbe given by (5). If\(u\in \operatorname{AC}^{n}[0,1]\)or\(u\in C^{n}[0,1]\), then
For convenience, we denote
and
Lemma 2.4
Let\((1-Q_{1})(1-P_{2})\neq P_{1}Q_{2}\). Then, for any\(y\in C [0,1 ]\), the BVP
has a unique solution
here
where
and
Proof
In view of the equation in (6), Lemma 2.3, and \(u^{\prime \prime }(0)=0\), we have
By (7), Lemma 2.1, and Lemma 2.2, we obtain
It follows from (7), (8), and the boundary conditions in (6) that
and
which together with (7) shows that
From (9), we get
and
and so,
and
which together with (9) implies that
□
In what follows, we let
and
Lemma 2.5
\(G(t, s)\)satisfies the following properties:
- (1)
\(G(t,s)\leq g(s)\), \((t,s)\in [0,1]\times [0,1]\);
- (2)
\(G(t,s)\geq \eta (s)g(s)\), \((t,s)\in [0,1]\times [0,1]\).
Proof
Since (1) is obvious, we only need to prove that (2) holds.
First, it is clear that \(G(t,1)\geq \eta (1)g(1)\) for \(t\in [0,1]\).
Now, we verify that \(G(t,s)\geq \eta (s)g(s)\) for \((t,s)\in [0,1]\times [0,1)\). In fact, if \(s\leq t\), then
and if \(t\leq s\), then
□
By the definition of η and the condition \(0<\rho <\beta [\gamma + \frac{\delta \varGamma (q)}{\varGamma (q-\sigma )} ]\), we may obtain the following remark.
Remark 2.1
η is increasing on \([0,1]\) and \(0<\eta (s)<1\) for \(s\in [0,1]\).
In the remainder of this paper, we always assume that the following conditions are satisfied:
Lemma 2.6
\(H(t,s)\)has the following property:
where
and
Proof
On the one hand, in view of (1) of Lemma 2.5, we have
On the other hand, by (2) of Lemma 2.5, we get
□
Let \(E=C[0,1]\) be equipped with norm \(\Vert u \Vert =\max_{t\in [0,1]} \vert u(t) \vert \) and
where \(0<\theta =\frac{m\eta (0)}{M}<1\). Then it is easy to check that E is a Banach space and K is a cone in E.
Now, we define an operator T on K by
Obviously, if u is a fixed point of T, then u is a nonnegative solution of BVP (4).
Lemma 2.7
\(T:K\rightarrow K\)is completely continuous.
Proof
Let \(u\in K\). Then, in view of Lemma 2.6, we have
which together with Lemma 2.6 and Remark 2.1 implies that
This indicates that \(Tu\in K\). Furthermore, it is easy to prove that T is completely continuous by an application of Arzela–Ascoli theorem [29]. □
3 Main results
Define
Theorem 3.1
Suppose that one of the following conditions is satisfied:
- (i)
\(f_{0}=+\infty \)and\(f^{\infty }=0\), or
- (ii)
\(f^{0}=0\)and\(f_{\infty }=+\infty \).
Then BVP (4) has at least one positive solution.
Proof
First, we consider case (i): \(f_{0}=+\infty \) and \(f^{ \infty }=0\).
In view of \(f_{0}=+\infty \), there exists \(r_{1}>0\) such that
where \(G_{1}\geq \frac{1}{m\theta \int _{0}^{1}\eta (s)g(s)\,ds}\).
Let \(\varOmega _{1}=\{u\in E: \Vert u \Vert < r_{1}\}\). Then, for any \(u\in K\cap \partial \varOmega _{1}\), by Lemma 2.6 and (10), we get
which shows that
On the other hand, since \(f^{\infty }=0\), there exists \(U_{1}>0\) such that
where \(\varepsilon _{1}>0\) satisfies \(\varepsilon _{1}\leq \frac{1}{2M\int _{0}^{1}g(s)\,ds}\).
Let \(M^{*}=\max_{(t,u)\in [0,1]\times [0,U_{1}]}f(t,u)\). Then we have
If we choose \(r_{2}=\max {\{2r_{1},2MM^{*}\int _{0}^{1}g(s)\,ds\}}\) and let \(\varOmega _{2}=\{u\in E: \Vert u \Vert < r_{2}\}\), then for any \(u\in K\cap \partial \varOmega _{2}\), from Lemma 2.6 and (12), we obtain
which indicates that
Therefore, it follows from Theorem 1.1, Lemma 2.7, (11), and (13) that T has a fixed point \(u\in K\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1} )\), which is a desired positive solution of BVP (4).
Next, we consider case (ii): \(f^{0}=0\) and \(f_{ \infty }=+\infty \).
In view of \(f^{0}=0\), there exists \(r_{3}>0\) such that
where \(\varepsilon _{2}>0\) satisfies \(\varepsilon _{2}\leq \frac{1}{M\int _{0}^{1}g(s)\,ds}\).
Let \(\varOmega _{3}=\{u\in E: \Vert u \Vert < r_{3}\}\). Then, for any \(u\in K\cap \partial \varOmega _{3}\), by Lemma 2.6 and (14), we get
which shows that
On the other hand, since \(f_{\infty }=+\infty \), there exists \(U_{2}>0\) such that
where \(G_{2}\geq \frac{1}{m\theta \int _{0}^{1}\eta (s)g(s)\,ds}\).
If we choose \(r_{4}=\max \{\frac{U_{2}}{\theta },2r_{3}\}\) and let \(\varOmega _{4}=\{u\in E: \Vert u \Vert < r_{4}\}\), then for any \(u\in K\cap \partial \varOmega _{4}\), we know
which together with Lemma 2.6 and (16) implies that
This indicates that
Therefore, it follows from Theorem 1.1, Lemma 2.7, (15), and (17) that T has a fixed point \(u\in K\cap (\overline{\varOmega }_{4}\setminus \varOmega _{3} )\), which is a desired positive solution of BVP (4). □
Example 3.1
Consider the following BVP:
In view of \(q=\frac{5}{2}\), \(\sigma =\frac{1}{2}\), \(\alpha =\gamma =\delta =1\), \(\beta =4\), \(h_{1}(s)=s\), and \(h_{2}(s)=1-s\), \(s\in [0,1]\), a simple calculation shows that
and
Obviously, \(Q_{1}<1\), \(P_{2}<1\) and
Moreover, since \(f(t,u)= [\sin (\frac{\pi t}{2})+1 ]u^{2}\), \((t,u)\in [0,1] \times [0,+\infty )\), it is easy to know that \(f:[0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous and
Therefore, it follows from Theorem 3.1 that BVP (18) has at least one positive solution.
4 Conclusion
In this paper, by applying Guo–Krasnoselskii’s fixed point theorem, we obtain the existence of at least one positive solution for a class of nonlinear boundary value problems involving fractional differential equation and integral boundary conditions. An illustrative example is also given to show the effectiveness of theoretical results.
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The authors wish to express their sincere thanks to anonymous referees for their detailed comments and valuable suggestions which have improved the paper greatly.
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This work is supported by the National Natural Science Foundation of China (Grant No. 11661049).
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Li, M., Sun, JP. & Zhao, YH. Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions. Adv Differ Equ 2020, 177 (2020). https://doi.org/10.1186/s13662-020-02618-9
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DOI: https://doi.org/10.1186/s13662-020-02618-9