Abstract
In this paper, by using the Leggett-Williams norm-type theorem, we consider a m-point boundary value problem for a class of fractional differential equations at resonance. A new result on the existence of solutions for above problem is obtained.
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1 Introduction
The subject of fractional calculus has gained significant interest and been a valuable tool for both science and engineering (see [1–3]). In recent years, the fractional boundary value problems (FBVPs for short) have been considered by many authors (see [4–10] and the references therein). For example, Bai studied a FBVP at non-resonance with \(1<\alpha\leq2\) (see [10]). FBVPs at resonance were studied by Kosmatov (see [11]) and Jiang (see [12]). But the positive solutions for FBVPs at resonance were studied very few. In [13], Yang and Wang considered the positive solutions of the following FBVP:
In [14], Chen and Tang studied the positive solution of FBVP as follows:
However, to the best of our knowledge, the fractional differential equations with m-point boundary conditions at resonance have not been considered. Motivated by the papers above, we consider the existence of positive solutions for a m-point FBVP of the form
where \(D_{0^{+}}^{\alpha}\) denotes the standard Caputo fractional differential operator of order α, \(1< \alpha\leq2\), \(\beta_{i} \in\mathbb{R^{+}}\), \(\sum_{i=1}^{m-2}\beta _{i}=1\), \(0<\eta_{1}<\eta_{2}<\cdots<\eta_{m-2}<1\), and \(f:[0,1]\times \mathbb{R} \rightarrow\mathbb{R} \) is continuous. Obviously, FBVP (1.1) happens to be at resonance under the condition \(\sum_{i=1}^{m-2}\beta_{i}=1\).
The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions and lemmas. In Section 3, we establish a theorem on the existence of positive solutions for FBVP (1.1) under some restrictions of f, basing on the coincidence degree theory due to [15]. Finally, in Section 4, an example is given to illustrate the main result.
2 Preliminaries
For convenience of the reader, we present some definitions, notations, and preliminary statements, which can be found in [2, 16, 17].
Let X and Y be real Banach spaces, \(L : \operatorname{dom} L\subset X\rightarrow Y\) be a Fredholm operator with index zero, where the index of a Fredholm operator L is defined by
Suppose \(P: X\rightarrow X\), \(Q: Y\rightarrow Y\) be continuous linear projectors such that
Thus, we see that
is invertible. We denote the inverse by \(K_{P}\). Moreover, by virtue of \(\operatorname{dim} \operatorname{Im}Q=\operatorname{codim} \operatorname{Im}L\), there exists an isomorphism \(J: \operatorname{Im}Q\rightarrow\operatorname{Ker}L\). Then we know that the operator equation \(Lx=Nx\) is equivalent to
where \(N: X\rightarrow Y\) be a nonlinear operator.
If Ω is an open bounded subset of X such that \(\operatorname{dom} L\cap\overline{\Omega} \neq\emptyset\), then the map \(N:X\rightarrow Y\) will be called L-compact on \(\overline{\Omega}\) if \(QN:\overline {\Omega}\rightarrow Y\) is bounded and \(K_{P}(I-Q)N:\overline{\Omega }\rightarrow X\) is compact.
Let C be a cone in X. Then C induces a partial order in X by
Lemma 2.1
(see [15])
Let C be a cone in X. Then for every \(u\in C\setminus\{0\}\) there exists a positive number \(\sigma(u)\) such that
for all \(x\in C\).
Let \(\gamma: X\rightarrow C\) be a retraction, that is, a continuous mapping such that \(\gamma(x)=x\) for all \(x\in C\). Set
and
Lemma 2.2
(see [15])
Let C be a cone in X and \(\Omega_{1}\), \(\Omega_{2}\) be open bounded subsets of X with \(\overline{\Omega}_{1} \subset\Omega_{2}\) and \(C\cap (\overline{\Omega}_{2}\setminus\Omega_{1})\neq\emptyset\). Assume that the following conditions are satisfied:
-
(1)
\(L : \operatorname{dom} L\subset X\rightarrow Y\) be a Fredholm operator of index zero and \(N:X\rightarrow Y\) be L-compact on every bounded subset of X,
-
(2)
\(Lx\neq\lambda Nx\) for every \((x,\lambda)\in[C\cap\partial \Omega_{2} \cap\operatorname{dom} L]\times(0,1)\),
-
(3)
γ maps subsets of \(\overline{\Omega}_{2}\) into bounded subsets of C,
-
(4)
\(\operatorname{deg}([I-(P+JQN)\gamma]|_{\operatorname{Ker} L }, \operatorname{Ker} L \cap \Omega_{2} , 0)\neq0\),
-
(5)
there exists \(u_{0}\in C\setminus\{0\}\) such that \(\|x\| \leq \sigma(u_{0})\|\Psi x\|\) for \(x \in C(u_{0})\cap\partial\Omega_{1}\), where \(C(u_{0})=\{x\in C: \mu u_{0} \leq x \textit{ for some } \mu>0\}\) and \(\sigma(u_{0})\) is such that \(\|x+u_{0}\|\geq\sigma(u_{0})\|x\|\) for every \(x\in C\),
-
(6)
\((P+JQN)\gamma(\partial\Omega_{2}) \subset C\),
-
(7)
\(\Psi_{\gamma}(\overline{\Omega}_{2}\setminus\Omega_{1})\subset C\).
Then the equation \(Lx=Nx\) has at least one solution in \(C\cap(\overline {\Omega}_{2}\setminus\Omega_{1})\).
Definition 2.3
(see [17])
The Riemann-Liouville fractional integral operator of order \(\alpha>0\) of a function x is given by
provided that the right side integral is pointwise defined on \((0,+\infty)\).
Definition 2.4
(see [17])
The Caputo fractional derivative of order \(\alpha>0\) of a continuous function x is given by
where n is the smallest integer greater than or equal to α, provided that the right side integral is pointwise defined on \((0,+\infty)\).
Lemma 2.5
(see [18])
For \(\alpha>0\), the general solution of the Caputo fractional differential equation
is
where \(c_{i}\in{\mathbb{R}}\), \(i=0,1,\ldots,n-1\), here n is the smallest integer greater than or equal to α.
Lemma 2.6
(see [18])
Suppose that \(D_{0^{+}}^{\alpha}x\in C[0,1]\), \(\alpha>0\). Then
where \(c_{i}\in{\mathbb{R}}\), \(i=0,1,\ldots,n-1\), here n is the smallest integer greater than or equal to α.
In this paper, we take \(X=Y=C[0,1]\) with the norm \(\|x\|_{\infty}=\max_{t\in[0,1]} |x(t)|\).
Define the operator \(L:\operatorname{dom}L\subset X\rightarrow Y\) by
where
Let \(N:X\rightarrow Y\) be the Nemytskii operator
Then FBVP (1.1) is equivalent to the operator equation
3 Main result
In this section, a theorem on the existence of positive solutions for FBVP (1.1) will be given.
For simplicity of notation, we set
and
Obviously, \(\max_{0\leq s\leq1}\sum_{i=1}^{m-2}\beta _{i}l_{i}(s)\leq1\). We denote
Thus, one has
Theorem 3.1
Let \(f:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) be continuous. Suppose that:
- (H1):
-
there exist nonnegative functions \(a,b\in X\) with \(\frac {\Gamma(\alpha+1)}{2} b_{1}< 1\) such that
$$\begin{aligned} \bigl|f(t,u)\bigr|\leq a(t)+b(t)|u|, \quad\forall t\in[0,1], u\in\mathbb{R}, \end{aligned}$$where \(b_{1}=\|b\|_{\infty}\),
- (H2):
-
there exists a constant \(B>0\) such that
$$uf(t,u)< 0,\quad \forall t\in[0,1], |u|>B, $$ - (H3):
-
\(f(t,u)> -\kappa u\), for all \((t,u)\in[0,1]\times[0,\infty)\),
- (H4):
-
there exist \(r\in(0,+\infty)\), \(t_{0}\in[0,1]\), \(M\in(0,1)\) and continuous function \(h:(0,r]\rightarrow[0,\infty)\) such that \(f(t,u)\geq h(u)\) for all \(t\in[0,1]\), \(u\in(0,r]\), and \(\frac{h(u)}{u}\) is non-increasing on \((0,r]\) with
$$ \frac{h(r)}{r}\int_{0}^{1}G(t_{0},s) \,ds \geq\frac{1-M}{M}. $$
Then FBVP (1.1) has at least one solution in X.
Now, we begin with some lemmas that are useful in what follows.
Lemma 3.2
Let L be defined by (2.1), then
Proof
By Lemma 2.5, \(D_{0^{+}}^{\alpha}x(t)=0\) has solution
Combining with the boundary conditions of FBVP (1.1), one sees that (3.2) holds.
For \(y\in\operatorname{Im} L\), there exists \(x\in\operatorname{dom} L\) such that \(y=Lx\in Y\). By Lemma 2.6, we have
Then we get
By the boundary conditions of FBVP (1.1), we see that y satisfies
That is,
On the other hand, suppose \(y\in Y\) and satisfies (3.4). Let \(x(t)=-I_{0^{+}}^{\alpha}y(t)+x(0)\), then \(x\in\operatorname{dom}L\) and \(D_{0^{+}}^{\alpha}x(t)=-y(t)\). Thus, \(y\in\operatorname{Im}L\). Hence (3.3) holds. The proof is complete. □
Lemma 3.3
Let L be defined by (2.1), then L is a Fredholm operator of index zero, and the linear continuous projector operators \(P:X\rightarrow X\) and \(Q:Y\rightarrow Y\) can be defined as
Furthermore, the operator \(K_{P}:\operatorname{Im}L\rightarrow\operatorname{dom}L \cap\operatorname{Ker}P\) can be written by
where
Proof
Obviously, \(\operatorname{Im} P=\operatorname{Ker} L\) and \(P^{2}x=Px\). It follows from \(x=(x-Px)+Px\) that \(X=\operatorname{Ker} P+\operatorname {Ker} L\). By a simple calculation, one obtain \(\operatorname{Ker} L\cap\operatorname{Ker} P=\{0\}\). Thus, we get
For \(y\in Y\), we have
Let \(y=(y-Qy)+Qy\), where \(y-Qy\in\operatorname{Ker} Q\), \(Qy\in\operatorname{Im} Q\). It follows from \(\operatorname{Ker} Q=\operatorname{Im} L\) and \(Q^{2}y=Qy\) that \(\operatorname{Im} Q\cap\operatorname{Im} L=\{ 0 \}\). Then one has
Thus, we obtain
That is, L is a Fredholm operator of index zero.
Now, we will prove that \(K_{P}\) is the inverse of \(L|_{\operatorname{dom} L\cap \operatorname{Ker} P}\). In fact, for \(y\in\operatorname{Im}L\), we have
where
It is easy to see that \(LK_{P}y=y\). Moreover, for \(x\in\operatorname{dom} L\cap\operatorname{Ker} P\), we get \(x'(0)=0\) and
Combining (3.6) with (3.7), we know that \(K_{P}=(L|_{\operatorname {dom}L\cap\operatorname{Ker}P})^{-1}\). The proof is complete. □
Lemma 3.4
Assume \(\Omega\subset X\) is an open bounded subset such that \(\operatorname{dom}L\cap\overline{\Omega}\neq\emptyset\), then N is L-compact on \(\overline{\Omega}\).
Proof
By the continuity of f, we see that \(QN(\overline{\Omega})\) and \(K_{P}(I-Q)N(\overline{\Omega})\) are bounded. That is, there exist constants \(A,B>0\) such that \(|(I-Q)Nx|\leq A\) and \(|K_{P}(I-Q)Nx|\leq B\), \(\forall x\in\overline{\Omega}\), \(t\in[0,1]\). Thus, one need only prove that \(K_{P}(I-Q)N(\overline{\Omega})\subset X\) is equicontinuous.
Let \(K_{P,Q}=K_{P}(I-Q)N\), for \(0\leq t_{1}< t_{2}\leq1\), \(x\in\overline {\Omega}\), we get
Since \(t^{\alpha}\) is uniformly continuous on \([0,1]\), we see that \(K_{P,Q}N(\overline{\Omega})\subset X\) is equicontinuous. Thus, we see that \(K_{P,Q}N:\overline{\Omega}\rightarrow X\) is compact. The proof is completed. □
Lemma 3.5
Suppose (H1) and (H2) hold, then the set
is bounded.
Proof
Take \(x\in\Omega_{0}\), then \(Nx\in\operatorname{Im} L\). By (3.2), we have
Then, by the integral mean value theorem, there exists a constant \(\xi \in(0,1)\) such that \(f(\xi,x(\xi))=0\). So, from (H2), we get \(|x(\xi )|\leq B\). By Lemma 2.6, one has
Thus, we get
which together with (H1) implies that
That is,
In view of \(\frac{2}{\Gamma(\alpha+1)}b_{1}<1\), there exists a constant \(D_{2}>0\) such that
Hence, \(\Omega_{0}\) is bounded. The proof is complete. □
Proof of Theorem 3.1
Set \(C=\{x\in X: x(t)\geq 0, t\in[0,1]\}\), \(\Omega_{1}=\{x\in X: r>|x(t)|>M\|x\|_{\infty}, t\in [0,1]\}\), and \(\Omega_{2}=\{x\in X: \|x\|_{\infty}< R\}\), where \(R=\max\{ B,D_{2}\}\). Clearly, \(\Omega_{1}\), \(\Omega_{2}\) are open bounded subsets of X and
From Lemma 3.3, Lemma 3.4, and Lemma 3.5, we see that the conditions (1) and (2) of Lemma 2.2 are satisfied.
Let \(\gamma x(t)=|x(t)|\) for \(x\in X\) and \(J=I\). One can see that γ is a retraction and maps subsets of \(\overline{\Omega}_{2}\) into bounded subsets of C, which means that the condition (3) of Lemma 2.2 holds.
For \(x\in\operatorname{Ker}L \cap\Omega_{2}\), we have \(x(t)\equiv c\). Let
From \(H(c,\lambda)=0\), one has \(c\geq0\). Moreover, if \(H(R,\lambda )=0\), we get
which contradicts (H2). Thus \(H(c,\lambda)\neq0\) for \(x\in\partial \Omega_{2}\), \(\lambda\in[0,1]\). Hence
So, the condition (4) of Lemma 2.2 holds.
Let \(x\in\overline{\Omega}_{2}\setminus\Omega_{1}\), \(t\in[0,1]\), we have
which together with (H3) and (3.1) yields
Thus, the condition (7) of Lemma 2.2 holds. In addition, we can prove the condition (6) of Lemma 2.2 holds too by a similar process.
Finally, we will show that the condition (5) of Lemma 2.2 is satisfied. Let \(u_{0}(t)\equiv1\), \(t\in[0,1]\), then \(u_{0} \in C\setminus\{ 0\}\), \(C(u_{0})=\{ x\in C: x(t)>0, t\in[0,1]\}\) and we can take \(\sigma (u_{0})=1\). For \(x\in C(u_{0})\cap\partial\Omega_{1}\), we have \(x(t)>0\), \(t\in[0,1]\), \(0<\|x\|_{\infty}\leq r\), and \(x(t)\geq M\|x\|_{\infty}\), \(t\in[0,1]\). So, from (H4), we obtain
Then the condition (5) of Lemma 2.2 holds.
Consequently, by Lemma 2.2, the equation \(Lx=Nx\) has at least one solution \(x^{*}\in C\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\). Namely, FBVP (1.1) has at least one positive solution in X. The proof is complete. □
4 Example
We consider the following FBVP:
Thus, we have
Moreover, \(f(t,u)\geq8-\frac{1}{10}u\geq-\frac{1}{4}u\) for all \(u\geq 0\), and \(l(s)\leq1\), \(G(t,s)\leq4\), \(\kappa=-\frac{1}{4}\). So, we can find that (H1), (H2), (H3) hold. Next, we take \(t_{0}=0\), \(h(x)=x\), and \(M=\frac{2}{3}\), thus \(G(0,s)= \frac{1}{\Gamma(\frac{5}{2})}(1-s)^{\frac{3}{2}}+\frac{3(\Gamma(\frac {7}{2})-1)}{2\Gamma(\frac{7}{2})(1-(\frac{1}{2})^{\frac{3}{2}})}l(s)\), \(0\leq s \leq1\), and \(\int_{0}^{1}G(0,s)\,ds=1\). Then (H4) is satisfied. According to the above points, by Theorem 3.1, we can conclude that FBVP (4.1) has at least one positive solution.
References
Glockle, WG, Nonnenmacher, TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46-53 (1995)
Oldham, KB, Spanier, J: The Fractional Calculus. Academic Press, New York (1974)
Diethelm, K, Freed, AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In: Keil, F, Mackens, W, Voss, H, Werther, J (eds.) Scientific Computing in Chemical Engineering II. Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217-224. Springer, Heidelberg (1999)
Agarwal, RP, O’Regan, D, Stanek, S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57-68 (2010)
Bai, Z, Lü, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005)
Kaufmann, ER, Mboumi, E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 2008, 3 (2008)
Zhang, S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006, 36 (2006)
Jafari, H, Gejji, VD: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 180, 700-706 (2006)
Liang, S, Zhang, J: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 71, 5545-5550 (2009)
Bai, Z, Zhang, Y: Solvability of fractional three-point boundary value problems with nonlinear growth. Appl. Math. Comput. 218, 1719-1725 (2011)
Kosmatov, N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 2010, 135 (2010)
Jiang, W: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal. 74, 1987-1994 (2011)
Yang, A, Wang, H: Positive solutions of two-point boundary value problems of nonlinear fractional differential equation at resonance. Electron. J. Qual. Theory Differ. Equ. 2011, 71 (2011)
Chen, Y, Tang, X: Positive solutions of fractional differential equations at resonance on the half-line. Bound. Value Probl. 2012, 64 (2012)
O’Regan, D, Zima, M: Leggett-Williams norm-type theorems for coincidences. Arch. Math. 87, 222-244 (2006)
Mainardi, F: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291-348. Springer, Wien (1997)
Metzler, F, Schick, W, Kilian, HG, Nonnenmacher, TF: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103, 7180-7186 (1995)
Lakshmikantham, V, Leela, S, Vasundhara Devi, J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009)
Acknowledgements
The authors would like to thank the referees very much for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (11271364).
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Wu, Y., Liu, W. Positive solutions for a class of fractional differential equations at resonance. Adv Differ Equ 2015, 241 (2015). https://doi.org/10.1186/s13662-015-0557-9
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DOI: https://doi.org/10.1186/s13662-015-0557-9