Abstract
By using the coincidence degree theorem, we obtain a new result on the existence of solutions for a class of fractional differential equations with periodic boundary value conditions, where a certain nonlinear growth condition of the nonlinearity needs to be satisfied. Furthermore, we study another class of differential equations of fractional order with periodic boundary conditions at resonance. A new result on the existence of positive solutions is presented by use of a Leggett–Williams norm-type theorem for coincidences. Two examples are given to illustrate the main result at the end of this paper.
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1 Introduction
Fractional calculus is the emerging mathematical field which is devoted to studying convolution-type pseudo-differential operators, specifically integrals and derivatives of any arbitrary real or complex order. In recent years, the fractional calculus has been considered as the best tool for the generalization of fractional differential equations. It has become more and more important in many fields of science and engineering, such as chemistry, biology, electricity, control theory, and image processing (see [1,2,3,4]). In addition, a considerable amount of progress has recently been made in the study of fractional calculus, and a number of results on this subject have been now achieved. For readers new to this subject, we cite a few proper ones of the books, and a comprehensive treatment of this subject and its applications can be found in [5,6,7,8].
In the past few decades, boundary value problems of fractional order involving a variety of boundary conditions have been studied by several researchers. We refer the readers to [9,10,11,12,13,14,15] and the references cited therein. Moreover,the existence of solutions to the fractional differential equations with anti-periodic boundary value conditions has been studied by many authors (see [16,17,18,19,20,21]). But the periodic boundary value problems for nonlinear fractional differential equations are seldom considered. Recently, the existence of solutions to nonlinear integer order periodic boundary value problems has been discussed in many articles (see [22,23,24,25]). Here, we point out that a few authors have recently considered fractional problems. In these formulations, the first order derivatives are replaced by fractional derivatives, which causes many difficulties in solving the resulting problems. In [26], Chen and Liu investigated the existence of solutions for the following periodic boundary value problem:
where \(0 < \alpha < 2\) is a real number, \(D_{0+}^{\alpha }\) is a Caputo fractional derivative, and \(f : [0, 1] \times \mathbb{R}^{2}\rightarrow \mathbb{R}\) is continuous.
In [27], Hu and Zhang gained the existence of positive solutions of fractional differential equation with periodic boundary value conditions of the form:
where \(2 < \alpha < 3\) is a real number, \(D_{0+}^{\alpha }\) is a Caputo fractional derivative, and \(f : [0, 1] \times \mathbb{R}\rightarrow \mathbb{R}\) is continuous.
Motivated by the work mentioned previously, this paper investigates the existence of solutions for two kinds of periodic boundary value problems (PBVP for short) of nonlinear fractional differential equations. The first one is described in the following form:
where \(0<\alpha ,\beta \leq 1\), \(D^{\alpha }_{0^{+}}\) is the Caputo fractional derivative, \(f : [0,T]\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) is continuous, \(p(t)\in C^{1}[0,T]\), \(p(0)=p(T)\), and there exists a positive constant M such that \(p(t)\geq M\) for all \(t\in [0,T]\).
However, the PBVP
is not solvable for each \(h\in C([0,T], \mathbb{R})\), and, when solvable, has no unique solution because \(x(t)+c\), \(\forall c\in \mathbb{R}\) is a solution together with \(x(t)\). In this case, a trivial necessary condition for the solvability of PBVP (2) is that
Furthermore, we change the range of α and take \(p(t)=1\), i.e., consider the following PBVP:
where \(0<\beta \leq 1\), \(1<\alpha \leq 2\), \(\alpha +\beta \geq 2\), \(g : [0,T]\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) is continuous. Then the existence of solutions for this PBVP is obtained under some assumptions of function g.
This paper is organized as follows. In Sect. 2, we establish an existence theorem of solutions for PBVP (1) under nonlinear growth restriction of f. The key is an analytic technique from the theory of coincidence degree. In Sect. 3, we obtain the existence of positive solutions of (3) by Theorem 3. Two illustrative examples of nonlinear fractional problems with periodic boundary conditions are shown in Sect. 4.
2 Existence for PBVP (1)
2.1 Preliminaries
In this subsection, as preliminaries, we firstly present some basic definitions and formulations on fractional calculus. For further background knowledge of fractional calculus, we refer the readers to [6].
Definition 1
The Riemann–Liouville fractional integral operator of order \(\alpha >0\) of a function \(u:(0,+\infty )\rightarrow \mathbb{R}\) is given by
provided that the right-hand side integral is pointwise defined on \((0,+\infty )\).
Definition 2
The Caputo fractional derivative of order \(\alpha >0\) of a function \(u:(0,+\infty )\rightarrow \mathbb{R}\) is given by
where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined on \((0,+\infty )\).
Lemma 1
([7])
The fractional differential equation \(D^{\alpha } _{0+}y(t)=0\) has solution \(y(t) = c_{0} +c_{1}t+\cdots +c _{n-1}t^{n-1}\), \(c_{i} \in \mathbb{R}\), \(i = 0, 1, \ldots , n-1\), \(n = [\alpha ]+1\). Furthermore, for \(y \in AC^{n}[0, T]\),
and
Lemma 2
([7])
The relation
is valid in the following case: \(\beta > 0\), \(\alpha +\beta > 0\), \(f(x) \in L_{1}(a, b)\).
Lemma 3
([28])
Let X, Y be real Banach spaces, \(L : \operatorname{dom} L \subset X \rightarrow Y\) be a Fredholm operator with index zero, and \(P : X \rightarrow X\), \(Q : Y \rightarrow Y\) be projectors such that
It follows that \(L|_{\operatorname{dom} L \cap \operatorname{ker} P}: \operatorname{dom} L \cap \operatorname{ker} P\rightarrow \operatorname{Im} L\) is invertible.
Denote \(Y = C([0, T], \mathbb{R})\) with the norm \(\|y\|_{\infty }= \max_{t\in [0,T]}|y(t)|\), \(X=\{ x|x, D^{\alpha }_{0^{+}}x\in Y\} \) and
with the norm \(\|x\|_{X}=\max \{ \|x\|_{\infty }, \|D^{\alpha } _{0^{+}}x\|_{\infty }\} \). It is easy to see that X and \(X_{T}\) are Banach spaces.
Define an operator \(L : \operatorname{dom} L \subset X \rightarrow Y\) by
where
Let \(N_{f} : X \rightarrow Y\) be the Nemytskii operator
Then PBVP (1) is equivalent to the operator equation
2.2 Main result
In this subsection, by using the coincidence degree theorem, we establish a new existence result on PBVP (1) for the nonlinear fractional differential equation under the nonlinear growth restriction of f.
First, we show some lemmas which will play important roles in the proof of the main result.
Consider PBVP (2) with \(h\in Y\) such that \(\overline{h}=0\), and let x be a solution of PBVP (2). From Lemma 1, we have
which together with the periodic boundary condition \(x(0)=x(T)\) implies that
For any fixed \(l\in Y\), define the function \(G_{l}(a) : \mathbb{R} \rightarrow \mathbb{R} \) by
Then we have the following lemma.
Lemma 4
The function \(G_{l} (a)\) has the following properties:
-
(i)
for any fixed \(l \in Y\), the equation
$$ G_{l}(a)=0 $$(8)has a unique solution \(\tilde{a}(l)\);
-
(ii)
the function \(\tilde{a} : Y\rightarrow \mathbb{R}\), defined in (1), is continuous and sends bounded sets into bounded sets.
Proof
(i) By (7), we have
hence the solution of (8) is unique. To prove the existence, we will show that \(C_{l}(a)\cdot a>0\) for \(|a|\) sufficiently large. Since
then we have
From the property of \(p(t)\), we have
for any \(y\in \mathbb{R}\). Thus, from (9) and (10), we obtain
Since \(|a|\rightarrow \infty \) implies that \(\vert \frac{a+l(t)}{p(t)} \vert \rightarrow \infty \) uniformly for \(t\in [0,T]\), we find from (11) that there exists \(r>0\) such that
for all \(a\in \mathbb{R}\) with \(|a|=r\). By an elementary topological degree argument, it follows that the equation \(G_{l} (a)=0\) has a solution for each \(l \in Y\), which by our previous argument is unique. In this way, for any \(l\in Y\), we define a function \(\tilde{a}: Y \rightarrow \mathbb{R}\) which satisfies
To prove (ii), let Λ be a bounded subset of Y and \(l\in \varLambda \). Then, from (12), we have
and hence
Suppose that \(\{ \tilde{a}(l),l\in \varLambda \} \) is not bounded. Then, for an arbitrary \(A>0\), there is \(l\in \varLambda \) with \(\|l\|_{\infty }\) sufficiently large so that
uniformly in \(t \in [0,T]\). Hence, by using (10) and (13), we find that
Thus \(A\leq \|l\|_{\infty }\), which is a contradiction. Therefore ã sends bounded sets in Y into bounded sets in \(\mathbb{R}\).
Finally, we show the continuity of ã. Let \(\{l_{n}\}\) be a convergent sequence in Y, say \(l_{n}\rightarrow l\), as \(n\rightarrow \infty \). Since \(\{ a(l_{n})\} \) is a bounded sequence, any subsequence of it contains a convergent subsequence denoted by \(\{ a(l_{n_{j}})\} \). Let \(a ( l_{n_{j}} ) \), as \(j\rightarrow \infty \). By letting \(j\rightarrow \infty \) in
we find that
and hence \(\tilde{a}(l)=\hat{a}\), which shows the continuity of ã.
The proof is complete. □
Let \(a:Y\rightarrow \mathbb{R}\) be defined by
Then, based on Lemma 4, a is a completely continuous mapping. Furthermore, by (6) and Lemma 1, we obtain that
Lemma 5
Let L be defined by (4), then
Proof
By Lemma 1, \(\forall b, c\in \mathbb{R}\), the solution of \(D^{\beta }_{0^{+}} ( p(t)D^{\alpha }_{0^{+}}x(t) ) =0\) satisfies
Combining the property of \(p(t)\) with periodic boundary value conditions
we have \(b=0\). That is, (15) holds.
If \(y\in \operatorname{Im} L\), then there exists a function \(x \in \operatorname{dom} L\) such that \(y(t)=D^{\beta }_{0^{+}} ( p(t)D^{ \alpha }_{0^{+}}x(t) ) \). By Lemma 1, we have
From the boundary condition \(D^{\alpha }_{0^{+}}x(0)=D^{\alpha }_{0^{+}}x(T)\), it follows that
On the other hand, let \(y \in Y\) satisfy (17) and
then \(D^{\alpha }_{0^{+}}x(0)=D^{\alpha }_{0^{+}}x(T)\). From the definition of mapping a, we have
Then we have \(x \in \operatorname{dom} L\) and \(Lx(t)=D^{\beta }_{0^{+}} ( p(t)D^{\alpha }_{0^{+}}x(t) ) =y(t)\). So \(y \in \operatorname{Im} L\). The proof is complete. □
Define projectors \(P : X \rightarrow X\) and \(Q : Y \rightarrow Y\) by
By (14), we can infer that the solution \(x\in X_{T}\) of PBVP (2) satisfies the following abstract equation:
According to the proof of Lemma 5, we can also infer that the solution x of (20) is also a solution of PBVP (2).
Notice that \(a(0)=\tilde{a}(0)=0\), we get \(\mathcal{K}(0)=0\).
Lemma 6
The operator \(\mathcal{K}\) is a completely continuous operator.
Proof
In fact, by the definition of \(\mathcal{K}\), it follows that
Based on the continuity of Q, it follows that \(\mathcal{K}\) and \(D^{\alpha }_{0^{+}}\mathcal{K}\) are continuous in Y. That is, \(\mathcal{K}\) is a continuous operator.
Let \(\varOmega \subset Y\) be an arbitrary open bounded set, then \(\mathcal{K}(\overline{\varOmega })\) and \(D^{\alpha }_{0^{+}}\mathcal{K}(\overline{ \varOmega })\) are bounded. Thus, in view of the Arzelà–Ascoli theorem, it remains to verify that \(\mathcal{K}(\overline{\varOmega })\subset X _{T}\) is equicontinuous.
In view of Lemma 4, we deduce that the operator \([a((I-Q)h)+I ^{\beta }_{0^{+}}(I-Q)h]\) is bounded. That is, there exists a positive constant \(M_{1} > 0\) such that
For \(0\leq t_{1}< t_{2}\leq T\), \(h\in \overline{\varOmega }\), we have
Since \(t^{\alpha }\) is uniformly continuous in \([0,T]\), by the definition of \(\mathcal{K}\), we can see that \(\mathcal{K}(\overline{ \varOmega })\subset Y\) is equicontinuous. Likewise, it follows that \([a(I-Q)+I^{\beta }_{0^{+}}(I-Q)](\overline{\varOmega })\subset Y\) is equicontinuous. This, together with the property of \(p(s)\), implies that \(D^{\alpha }_{0^{+}}\mathcal{K}(\overline{\varOmega })\subset Y\) is also equicontinuous. Thus we prove that the operator \(\mathcal{K} : Y \rightarrow X_{T}\) is compact. The proof is complete. □
Lemma 7
Let \(f : [0,T]\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) be continuous, L, \(N_{f}\), Q be defined respectively by (4), (5), (18), and Ω be an open bounded subset of \(X_{T}\) such that \(\operatorname{dom} L \cap \overline{\varOmega }\neq \emptyset \). Assume that the following conditions are satisfied:
- (\(C_{1}\)):
-
for each \(\lambda \in (0,1)\), the equation
$$ Lx=\lambda N_{f}x $$(21)has no solution on \((\operatorname{dom} L\setminus \operatorname{ker} L)\cap \partial \varOmega \);
- (\(C_{2}\)):
-
the equation \(QN_{f}x=0\) has no solution on \(\operatorname{ker} L \cap \partial \varOmega \);
- (\(C_{3}\)):
-
the Brouwer degree \(\operatorname{deg} ( QN_{f}|_{ \operatorname{ker} L}, \varOmega \cap \operatorname{ker} L, 0 ) \neq 0\).
Then the equation \(Lx=N_{f} x\) has at least one solution in \(\operatorname{dom} L \cap \overline{\varOmega }\).
Proof
Let us consider the homotopic equation of \(Lx =N_{f} x\) as follows:
That is,
Obviously, for \(\lambda \in (0,1]\), if x is a solution of Eq. (21) or Eq. (22), then we have
It can be seen that Eq. (21) and Eq. (22) have the same solutions. Furthermore, Eq. (22) is equivalent to the following form:
where \(G_{f} : X_{T} \times [0,1]\rightarrow X_{T}\) is defined by
In view of the continuity of f and Lemma 6, it is known that \(G_{f}\) is a completely continuous operator.
For \(\lambda =1\), we assume that Eq. (23) does not have a solution on ∂Ω. Otherwise, the proof is finished. Now, by hypothesis (\(C_{1}\)), it follows that Eq. (23) has no solutions for \((x,\lambda )\in \partial \varOmega \times (0,1]\). For \(\lambda =0\), Eq. (22) is equivalent to the following PBVP:
If x is a solution of this PBVP, we have
In view of (15), the following equality holds:
Thus we have
which together with hypothesis (\(C_{2}\)) implies that \(x=c\notin \partial \varOmega \). So we prove that (23) has no solution for \((x,\lambda )\in \partial \varOmega \times [0,1]\). Then, for each \(\lambda \in [0,1]\), the Leray–Schauder degree \(\operatorname{deg}(I-G_{f}( \cdot ,\lambda ),\varOmega ,0)\) is well defined. By the homotopy property of degree, we have that
It is clear that equation \(x=G_{f}(x,1)\) is equivalent to the equation \(Lx =N_{f} x\). Let us consider the equation \(x=G_{f}(x,1)\), which will have at least one solution if \(\operatorname{deg}(I-G_{f}(\cdot ,0),\varOmega ,0)\neq 0\) holds. From now on, we will check this. By the definition of \(G_{f}\), we have that
Obviously, we show that \(x=G_{f}(x,0)=c\) holds for \(\forall c \in \mathbb{R}\), which implies that
That is,
Then, by applying the Leray–Schauder degree theory, we have
where the right-hand side degree is the Brouwer degree.
Based on hypothesis (\(C_{3}\)), the equation \(Lx =N_{f} x\) has at least one solution in \(\operatorname{dom} L \cap \overline{\varOmega }\). The proof is complete. □
Theorem 1
Let \(f : [0,T]\times \mathbb{R}^{2}\rightarrow \mathbb{R}\). Assume that
-
(Ha1)
there exist nonnegative functions \(a,b,c\in Y\) such that
$$ \bigl\vert f(t,u,v) \bigr\vert \leq a(t)+b(t) \vert u \vert +c(t) \vert v \vert ,\quad \forall t\in [0,T], (u,v) \in \mathbb{R}^{2}; $$ -
(Ha2)
there exists a constant \(B>0\) such that either
$$ uf(t,u,v)>0,\quad \forall t\in [0,T], \quad v\in \mathbb{R},\quad \vert u \vert >B $$(25)or
$$ uf(t,u,v)< 0,\quad \forall t\in [0,T], \quad v\in \mathbb{R},\quad \vert u \vert >B. $$(26)
Then PBVP (1) has at least one solution, provided that
Proof
Let
For \(x\in \varOmega_{1}\), we get \(N_{f} x \in \operatorname{Im} L\). It follows from (16) that
By the integral mean value theorem, there exists a constant \(\xi \in (0,T)\) such that
So, from (Ha2), we get \(|x(\xi )| \leq B\). By Lemma 1, we find that
which together with
and \(|x(\xi )| \leq B\) implies that
Combining hypothesis (Ha1) with (28), \(\forall t \in [0,T]\), we get
In fact, owing to the fact that \(Lx=\lambda N_{f} x\), in view of Lemma 1, we have
Then, by the boundary condition \(x(0)=x(T)\), it follows that
Thus, there exists a constant \(\eta \in (0,T)\) such that \(d_{1}+ \lambda I^{\beta }_{0^{+}}N_{f}x(\eta )=0\), which implies
As a consequence, we have
Based on (29), it follows that
Thus, from (27), we find that
which together with (28) yields that
Therefore, based on (30) and (31), we obtain that
It means that \(\varOmega_{1}\) is bounded. Next, we let \(\varOmega_{2}= \{ x \in \operatorname{ker} L| QN_{f}x=0 \} \). For \(x\in \varOmega_{2}\), we have \(x(t)=d\), \(\forall d\in \mathbb{R}\), which implies that
In view of (Ha2), it follows that \(|d|\leq B\). Thus, we obtain
That is, \(\varOmega_{2}\) is bounded. In addition, if (25) holds, set
For \(x\in \varOmega_{3}\), we have \(x(t)=c\), \(\forall c\in \mathbb{R}\) and
If \(\lambda =0\), then \(|c|\leq B\) since (25) holds. If \(\lambda \in (0,1]\), we can also show that \(|c|\leq B\). Otherwise, we get
which contradicts (32). So \(\varOmega_{3}\) is bounded. If (26) holds, let
By an argument similar to that above, we can prove that \(\varOmega '_{3}\) is also bounded.
Now, it remains to prove that all the conditions of Lemma 7 are satisfied. As for the details, we refer the readers to [29].
As a consequence of Lemma 7, the operator equation \(Lx=N_{f}x\) has at least one solution in \(\text{dom}L \cap \overline{ \varOmega }\). That is, PBVP (1) has at least one solution in \(X_{T}\). The proof is complete. □
3 Existence for PBVP (3)
3.1 Preliminaries
In the following, we provide the necessary background definitions on Fredholm operators and cones in a Banach space (see [28]).
Let \(X_{1}\), \(Y_{1}\) be real Banach spaces. Consider a linear mapping \(L_{1} : \operatorname{dom} L_{1} \subset X_{1} \rightarrow Y_{1}\) and a nonlinear operator \(N_{1} : X_{1} \rightarrow Y_{1} \). Assume that
-
(A1)
\(L_{1}\) is a Fredholm operator of index zero; that is, \(\operatorname{Im} L_{1}\) is closed and
$$ \operatorname{deg} \operatorname{ker} L_{1} = \operatorname{codim} \operatorname{Im} L_{1} < \infty . $$
This assumption implies that there exist continuous projections \(P_{1} : X_{1} \rightarrow X_{1}\) and \(Q_{1}: Y_{1} \rightarrow Y_{1}\) such that \(\operatorname{Im} P_{1} = \operatorname{ker} L_{1}\) and \(\operatorname{ker} Q _{1} = \operatorname{Im} L_{1}\). Moreover, since \(\operatorname{deg} \operatorname{Im} Q_{1} = \operatorname{codim} \operatorname{Im}L_{1}\), there exists an isomorphism \(J : \operatorname{Im} Q_{1} \rightarrow \operatorname{ker} L_{1}\). Denote by \(L_{P}\) the restriction of \(L_{1}\) to \(\operatorname{ker} P_{1} \cap \operatorname{dom} L_{1}\). Clearly, \(L_{P}\) is an isomorphism from \(\operatorname{ker} P_{1} \cap \operatorname{dom} L_{1}\) to \(\operatorname{Im} L_{1}\), we denote its inverse by \(K_{p} : \operatorname{Im} L_{1} \rightarrow \operatorname{ker} P_{1} \cap \operatorname{dom} L_{1}\). It is known that the coincidence equation \(L_{1}x = Nx\) is equivalent to
Let \(C_{1}\) be a cone in \(X_{1}\) such that
-
(i)
\(\mu x \in C_{1}\) for all \(x \in C_{1}\) and \(\mu \geq 0\),
-
(ii)
\(x,-x \in C_{1}\) implies \(x =\theta \).
It is well known that \(C_{1}\) induces a partial order in \(X_{1}\) by \(x \preceq y\) if and only if \(y - x \in C_{1}\). The following property is valid for every cone in a Banach space \(X_{1}\).
Lemma 8
Let \(C_{1}\) be a cone in \(X_{1}\). Then, for every \(u \in C_{1} \{0 \}\), there exists a positive number \(\sigma (u)\) such that
Let \(\gamma : X_{1} \rightarrow C_{1} \) be a retraction; that is, a continuous mapping such that \(\gamma (x) = x\) for all \(x \in C_{1}\). Set
Theorem 2
([30])
Let \(C_{1}\) be a cone in \(X_{1}\), and let \(\varOmega_{1}\), \(\varOmega_{2}\) be open bounded subsets of \(X_{1}\) with \(\overline{ \varOmega }_{1}\subset \varOmega_{2}\) and \(C_{1}\cap (\overline{\varOmega } _{2}\backslash \varOmega_{1}) \neq \emptyset \). Assume (A1) and the following assumptions hold:
-
(A2)
\(Q_{1}N_{1} : X _{1}\rightarrow Y_{1}\) is continuous and bounded and \(K_{P}(I - Q_{1})N_{1} : X_{1} \rightarrow X_{1}\) is compact on every bounded subset of \(X_{1}\);
-
(A3)
\(L_{1}x\neq \lambda N_{1}x\) for all \(x \in C_{1} \cap \partial \varOmega_{2}\cap \operatorname{Im} L_{1}\) and \(\lambda \in (0,1)\);
-
(A4)
γ maps subsets of \(\overline{\varOmega }_{2}\) into bounded subsets of \(C_{1}\);
-
(A5)
\(\operatorname{deg}\{[I-(P_{1} + JQ_{1}N_{1})_{\gamma }]|_{ \operatorname{ker} L_{1}}, \operatorname{ker} L_{1} \cap \varOmega_{2}, 0\} \neq 0\);
-
(A6)
there exists \(u_{0} \in C _{1} \{0\}\) such that \(\|x\|\leq \sigma (u_{0})\|\varPsi x\|\) for \(x \in C_{1}(u_{0}) \cap \partial \varOmega_{1}\), where \(C_{1}(u_{0}) = \{x \in C_{1} : \mu u_{0} \preceq x\textit{ for some }\mu > 0\}\) and \(\sigma (u_{0})\) such that \(\|x + u_{0}\|\geq \sigma (u_{0})\|x\|\) for every \(x \in C_{1}\);
-
(A7)
\((P_{1} + JQ_{1}N_{1})\gamma (\partial \varOmega_{2})\subset C_{1}\);
-
(A8)
\(\varPsi_{\gamma }(\overline{\varOmega }_{2}\backslash \varOmega _{1})\subset C_{1}\).
Then the equation \(L_{1}x = N_{1}x\) has a solution in the set \(C_{1} \cap (\overline{\varOmega }_{2} \backslash \varOmega_{1})\).
3.2 Main result
In this subsection, we prove the existence result for PBVP (3). We use the Banach space \(Y_{1} = C([0, T], \mathbb{R})\) with the norm \(\|y\|_{\infty }=\max_{t\in [0,T]}|y(t)|\) and denote \(X_{1}= \{ x|x, D^{\alpha }_{0^{+}}x\in Y_{1} \} \) with the norm \(\|x\|=\max \{ \|x\|_{\infty }, \|D^{\alpha }_{0^{+}}x\|_{\infty } \} \).
Define the operator \(L_{1} : \operatorname{dom} L_{1} \rightarrow X_{1}\) by \(L_{1}x = D^{\beta }_{0+}D^{\alpha }_{0+}x\), where
Define the operator \(N_{1}: X_{1}\rightarrow Y_{1}\) by \(N_{1}x(t)=g(t, x(t),D^{\alpha }_{0+}x(t))\). Then problem (3) can be written by \(L_{1} x = N_{1}x\), \(x \in \operatorname{dom} L_{1}\). For convenience, we set
where
Denote a constant \(\kappa \in (0, 1)\) satisfying
Lemma 9
The mapping \(L_{1} : \operatorname{dom} L_{1} \subset X_{1}\) is a Fredholm operator of index zero. Furthermore, the operator \(K_{P} : \operatorname{Im} L_{1} \rightarrow \operatorname{dom} L_{1} \cap \operatorname{ker} P _{1}\) can be written by
where
Proof
Based on Lemma 1, the solution \(x(t)\) of \(D_{0+}^{\beta }D _{0+}^{\alpha }x(t)=0\) satisfies \(D_{0+}^{\alpha }x(t)=c\). In this case, c should be zero by observing the definition of \(D^{\alpha }_{0+}\). Therefore, we have \(D_{0+}^{\alpha }x(t)=0\), which implies \(x(t)=c _{0}+c_{1}t\), \(c_{0}, c_{1}\in \mathbb{R}\). According to the boundary value conditions of (3), we have \(\operatorname{ker} L_{1}=\{c,c \in \mathbb{R}\}\cong \mathbb{R}^{1}\).
Let \(y(t) \in \operatorname{Im} L_{1}\) and assume that there exists a function \(x(t) \in \operatorname{dom} L_{1}\) satisfying \(L_{1}x(t) = y(t)\). In view of Lemmas 1 and 2, we have
From \(D_{0+}^{\alpha }x(0)=D_{0+}^{\alpha }x(T)\), it implies that \(\int_{0}^{T}(T-s)^{\beta -1}y(s)\,ds=0\). On the other hand, suppose \(y \in Y_{1}\) satisfying \(\int_{0}^{T}(T-s)^{\beta -1}y(s)\,ds=0\). Let
By a simple calculation, we can prove \(x(0) = x(T)\), \(x'(0) = x'(T)\), \(D_{0+}^{\alpha }x(0)=D_{0+}^{\alpha }x(T)\), which means \(x(t) \in \operatorname{dom} L_{1}\). To conclude, we get
Consider the linear operator \(P_{1} : X_{1} \rightarrow X_{1}\) defined by
and the operator \(Q_{1} : Y_{1} \rightarrow Y_{1}\) defined by
For \(x(t) \in X_{1}\), we get
Hence, we have \(P_{1}^{2} = P_{1}\). Similarly, we can get \(Q_{1}^{2} = Q_{1}\). Note that \(\operatorname{Im} P_{1} = \operatorname{ker} L_{1}\) and \(\operatorname{ker} Q_{1} = \operatorname{Im} L_{1}\). It follows from
that \(L_{1}\) is a Fredholm mapping of index zero.
It remains to prove that the operator \(K_{P}\) is the inverse of \(L_{1}|_{\operatorname{dom} L_{1} \cap \operatorname{ker} P_{1}}\).
In fact, for \(x(t) \in \operatorname{dom} L_{1} \cap \operatorname{ker} P_{1}\), we have \(D_{0+}^{\beta }D_{0+}^{\alpha }x(t) = y(t)\). By Lemma 1, we have \(x(t)=I_{0+}^{\alpha }(I_{0+}^{\beta }y(t)+c_{0})+c _{1}+c_{2}t\). According to \(x(0) = x(T)\), \(x'(0) = x'(T)\), \(D_{0+} ^{\alpha }x(0)=D_{0+}^{\alpha }x(T)\), we get
Since \(x(t) \in \operatorname{ker} P_{1}\), i.e., \(\frac{\beta }{T^{\beta }} \int_{0}^{T}(T-s)^{\beta -1}x(s)\,ds=0\), we obtain
Define an operator
Substituting \(c_{0}\), \(c_{1}\), \(c_{2}\) in the above equality, we obtain
It can be shown that \(L_{1}K_{P}y(t) = y(t)\), which implies \(K_{P} = (L _{1}|_{\operatorname{dom} L_{1}\cap \operatorname{ker} P_{1}})^{-1} \). This completes the proof of Lemma 9. □
Lemma 10
Assume that \(\varOmega \subset X_{1}\) is an open bounded set such that \(\operatorname{dom}(L_{1})\cap \overline{\varOmega }\neq \emptyset \), then \(N_{1}\) is L-compact on Ω̅.
Proof
Based on the continuity of g, we obtain that \(Q_{1}N_{1}(\overline{ \varOmega })\) and \(K_{P} (I-Q_{1})N_{1}(\overline{\varOmega })\) are bounded. Hence, for \(x(t) \in \overline{\varOmega }\), \(t \in [0, T]\), there exists a positive constant M such that \(\vert {(I - {Q_{1}}){N_{1}}x(t)} \vert \le M\), \(\vert {\frac{1}{{\alpha {T^{\alpha - 1}}}}I_{0 + } ^{\alpha + \beta - 1}(I - {Q_{1}}){N_{1}}x(T)} \vert \le M\) and \(\vert \frac{1}{T}I_{0 + }^{\alpha + \beta }(I - {Q_{1}}){N_{1}}x(T) - \frac{1}{\alpha }I_{0 + }^{\alpha + \beta - 1}(I - {Q_{1}}){N_{1}}x(T) \vert \le M\). In view of the Arzela–Ascoli theorem, we need only to prove that \(K_{P} (I-Q_{1})N_{1}(\overline{ \varOmega })\) is equicontinuous.
For \(0 \leq t_{1} < t_{2} \leq T\), \(x\in \overline{\varOmega }\), by virtue of the definition of \(K_{P}\), we have
Notice that t and \(t^{\alpha }\) are uniformly continuous on \([0, T]\). Therefore, we have \(K_{P} (I-Q_{1})N_{1}(\overline{\varOmega })\) is equicontinuous on \([0, T]\). The proof is completed. □
Theorem 3
Assume that
-
(Hb1)
for \(t \in [0, T]\) and \((u, v) \in [0,B]\times [0,B]\), one has
$$ -\kappa (u+v) \leq g(t, u, v) \leq -c_{1}u - c_{2}v+c_{3} $$and
$$ g(t, u, v) \leq -b_{1} \bigl\vert g(t, u, v) \bigr\vert + b_{2}u + b_{3}v+b_{4}, $$where \(b_{1}\), \(b_{2}\), \(b_{3}\), \(b_{4}\), \(c_{1}\), \(c_{2}\), \(c_{3}\), B are positive constants with
$$\begin{aligned}& {b_{1}} {c_{1}} {c_{2}}\beta + {b_{1}}c_{1}^{2}\beta + 8{T^{\alpha + \beta - 1}} {b_{2}}c_{2}^{2} - 8{T^{\alpha + \beta - 1}} {b_{3}} {c_{1}} {c_{2}} > 0, \end{aligned}$$(34)$$\begin{aligned}& \varGamma (3 - \alpha )\varGamma (\alpha + \beta ) - 2\kappa (\alpha - 1) {T^{\alpha + 2\beta - 2}}>0, \end{aligned}$$(35)$$\begin{aligned}& B > \max \biggl\{ {{A_{1}},{A_{2}}, \frac{c_{3}}{c_{1}}} \biggr\} , \end{aligned}$$(36)where
$$ {A_{1}} = \frac{{{b_{1}}{c_{2}}{c_{3}}\beta + 8{b_{2}}{c_{2}}{c_{3}} {T^{\alpha + \beta - 1}} + 8{b_{4}}{c_{1}}{c_{2}}{T^{\alpha + \beta - 1}}}}{{{b_{1}}{c_{1}}{c_{2}}\beta + {b_{1}}c_{1}^{2}\beta + 8{T^{ \alpha + \beta - 1}}{b_{2}}c_{2}^{2} - 8{T^{\alpha + \beta - 1}}{b _{3}}{c_{1}}{c_{2}}}} $$and
$$ {A_{2}} = \frac{{{c_{3}}(2\alpha + \beta - 2){T^{\beta }}}}{{\varGamma (3 - \alpha )\varGamma (\alpha + \beta ) - 2\kappa (\alpha - 1){T^{\alpha + 2\beta - 2}}}}. $$ -
(Hb2)
there exist \(r \in (0,B)\), \(t_{0} \in [0, T]\), \(m \in (0, 1)\), and \(h_{i}(x) : (0, r] \rightarrow [0,+\infty )\), \(i=1,2\), such that \(g(t, u, v) \geq h_{1}(u)+h_{2}(v)\) for \(t \in [0, T]\), \((u,v)\in (0, r]\times (0, r]\). Moreover, \(\frac{h_{1}(u)}{u}\) and \(\frac{h_{2}(v)}{v}\) are nonincreasing on \((0, r]\) and
$$ \frac{\beta }{{{T^{\beta }}}}\frac{{h_{i}(r)}}{r} \int_{0}^{T} {G( {t_{0}},s){{(T - s)}^{\beta - 1}}\,ds} \ge \frac{{1 - m}}{2m},\quad i=1, 2. $$
Then problem (3) has at least one positive solution on [0, T].
Proof
Firstly, conditions (A1) and (A2) of Theorem 3 are satisfied based on Lemmas 9 and 10.
Then, consider the cone \(C_{1}=\{u \in X_{1} : u(t) \geq 0, D_{0+} ^{\alpha }u(t) \geq 0, t \in [0, T] \}\). Let \(\varOmega_{1}=\{u\in X_{1}: m\|u\|<|u(t)|<r, m\|u\|<|D_{0+}^{\alpha }u(t)|<r, t\in [0,T]\}\), \(\varOmega_{2}=\{u\in X_{1}: \|u\|< B, t\in [0,T]\}\). Obviously, \(\varOmega_{1}\) and \(\varOmega_{2}\) are bounded and
Furthermore, \(C_{1} \cap (\overline{\varOmega }_{2} \backslash \varOmega _{1})\neq \emptyset \). Let \(J = I\) and \((\gamma u)(t) = |u(t)|\) for \(u\in X_{1}\), then γ is a retraction and maps subsets of \(\overline{\varOmega }_{2}\) into bounded subsets of \(C_{1}\), which means that (A4) holds.
Next, we will prove that (A3) holds. Suppose that there exist \(x_{0}\in \partial \varOmega_{2}\cap C_{1} \cap \operatorname{dom} L_{1}\) and \(\lambda_{0} \in (0, 1)\) such that \(L_{1}x_{0} = \lambda_{0} N_{1}x _{0}\), that is, \(D_{0+}^{\beta }D_{0+}^{\alpha }x_{0}(t)=\lambda_{0} g(t, x_{0}(t), D_{0+}^{\alpha }x_{0}(t))\), \(t \in [0, T]\). Then assumption (Hb1) gives
and
Since \(D_{0+}^{\beta }D_{0+}^{\alpha }x_{0}(t)=\lambda_{0} g(t, x_{0}(t), D_{0+}^{\alpha }x_{0}(t))\in \operatorname{Im} L_{1}\), based on the definition of \(\operatorname{Im} L_{1}\) and (38), we can obtain
which gives
Furthermore, (37) and (40) imply
which gives
Based on the function expression of \(k(t, s)\), we get
By virtue of (40), (41), (42), and the equation \(x_{0} = (I-P_{1})x_{0} +P_{1}x_{0} = K_{P}L_{1}(I-P_{1})x_{0} + P _{1}x_{0} = P_{1}x_{0} + K_{P} L_{1}x_{0}\), we have
In view of (Hb1), we have
and
In addition, based on the definition of function \(k(t,s)\), we obtain
Hence, on the basis of (43)–(45), we have
Therefore, by a simple calculation, we get
Based on the definition of norm \(\|\cdot \|\), we have \(B \le \max \{ {{A_{1}},{A_{2}}} \} \) with
and
which contradicts (Hb1). Hence (A3) holds.
In order to prove (A5), we consider \(x(t) \in \operatorname{ker} L_{1} \cap \overline{\varOmega }_{2}\), then \(x(t) \equiv c\). For \(c \in [-B,B]\) and \(\lambda \in [0, 1]\), we obtain
By use of the proof by contradiction, it can be shown that \(H(c, \lambda ) = 0\) implies \(c \geq 0\). Suppose \(H(B, \lambda ) = 0\) for some \(\lambda \in (0, 1]\), then we have
In view of (Hb1), we have
which is a contradiction. In addition, if \(\lambda = 0\), then \(B = 0\), which is impossible. As a result, for \(x \in \operatorname{ker} L _{1}\cap \partial \varOmega_{2}\) and \(\lambda \in [0, 1]\), we have \(H(x, \lambda ) \neq 0 \). Thus,
So (A5) holds. It remains to prove (A6). Let \(x_{0}(t) \equiv 1\), \(t \in [0, T]\), then \(x_{0} \in C_{1} \backslash \{0\}\), \(C_{1}(x_{0}) = \{x \in C_{1} : x(t) > 0, t \in [0, T]\}\). We take \(\sigma (x_{0}) = 1\) and let \(x\in C_{1}(x_{0})\cap \partial \varOmega_{1}\), then \(0 < \|x\|\leq r\) and \(x(t) \geq m\|x\| \) on \([0, T]\).
For \(u \in C_{1}(x_{0}) \cap \varOmega_{1}\), (Hb2) implies
To conclude, for all \(x\in C_{1}(x_{0}) \cap \partial \varOmega_{1}\), we have \(\|x\|\leq \sigma (x_{0})\|\varPsi x\|_{\infty }\leq \sigma (x_{0}) \|\varPsi x\|\), i.e., (A6) holds. For \(x\in \partial \varOmega_{2}\), (Hb2) implies
Thus, for \(x\in \partial \varOmega_{2}\), one has \([(P_{1}+JQ_{1}N_{1}) \circ \gamma ]x(t)\subset C_{1}\). Then (A7) holds.
Finally, we prove (A8). For \(x(t) \in \overline{\varOmega }_{2} \backslash \varOmega_{1}\), based on (H2) and (33), we have
Hence, \(\varPsi_{\gamma }(\overline{\varOmega }_{2}\backslash \varOmega_{1}) \subset C_{1}\), that is, (A8) holds. Hence, applying Theorem 2, PBVP (3) has a positive solution \(x^{\ast }(t)\) on \([0, T]\) with \(r \leq \|x^{\ast }(t)\|\leq B\). This completes the proof. □
4 Examples
In this section, two examples will be given to illustrate our main result.
Example 1
Consider the following PBVP for the nonlinear fractional differential equation:
According to PBVP (1), we get that \(p(t)=t^{2}-t+\frac{5}{4}\), \(M=1\), \(\alpha =\frac{3}{4}\), \(\beta =\frac{1}{2}\), \(T=1\), and
Let \(a(t)=4\), \(b(t)=\frac{1}{24}\), \(c(t)=0\), \(B=8\). A simple calculation shows that \(\|b\|_{\infty }=\frac{1}{24}\), \(\|c\|_{\infty }=0\), and
All assumptions of Theorem 1 are satisfied. Hence PBVP (46) admits at least one solution.
Example 2
Consider the fractional periodic boundary value problem
where \(g(t,x,D^{1.5}_{0^{+}}x)=\frac{2}{5}(1+t^{2})(-\frac{1}{2}x- \frac{1}{2}D^{1.5}_{0^{+}}x+\frac{5}{4})\).
Corresponding to PBVP (47), we have that \(\beta =0.5\), \(\alpha =1.5\), \(T=1\), and
By a simple calculation, we obtain \(G(t,s)<2.5\). Hence, we take \(\kappa =\frac{2}{5}\) based on (33). In addition, we find that if \(t\in [0,1]\), \(x\in [0,60]\), and \(D_{0+}^{1.5}x\in [0,60]\), the following inequality holds:
So we can choose \(B=60\), \(c_{1}=c_{2}=\frac{1}{5}\), \(c_{3}=1\), \(b_{1}=1\), \(b_{2}=b_{3}=\frac{6}{5}\), \(b_{4}=1\).
Furthermore, it is easy to verify that
Therefore, (Hb1) is satisfied.
We take \(r=0.5\in [0,60]\), \(h_{1}(x)=\frac{x}{10}\), \(h_{2}(D_{0+}^{1.5}x)=\frac{D _{0+}^{1.5}x}{10}\). By calculation, we obtain
and
which is nonincreasing on \((0, 0.5]\).
Let \(t_{0} = 0\), then we have
Using the given data, we have
holds for \(m=0.8\). One sees that (Hb2) is satisfied. In consequence, the conclusion of Theorem 3 implies that problem (47) has a positive solution on \([0, 1]\).
5 Conclusion
We have proved the existence of solutions for two classes of fractional differential equations with periodic boundary value conditions, where certain nonlinear growth conditions of the nonlinearity need to be satisfied. The problem is issued by applying the Leggett–Williams norm-type theorem for coincidences. We also provide examples to make our results clear.
Abbreviations
- PBVP:
-
Periodic boundary value problems.
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This work is partially supported by the National Science Foundation of China (51476047, U1637208) and the Natural Scientific Foundation of Heilongjiang Province in China (A2016003).
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Yao, W., Guo, Z. & Sun, J. Periodic boundary value problems for two classes of nonlinear fractional differential equations. Bound Value Probl 2018, 172 (2018). https://doi.org/10.1186/s13661-018-1092-x
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DOI: https://doi.org/10.1186/s13661-018-1092-x