Abstract
In this paper, we study the existence of solutions for a class of nonlinear higher-order fractional differential equation with fractional nonlocal boundary condition by using the monotone iterative technique based on the method of upper and lower solutions and give a specific iterative equation about its solutions.
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1 Introduction
We consider the existence of solutions for the following nonlinear fractional differential equation with nonlocal boundary value condition:
where \(n-1<\alpha<n\) is a real number, \(n\geq2\), \(^{C}D^{ \alpha }_{0^{+}}\), \(^{C}D^{ \beta_{i}}_{0^{+}}\), \(i=1,2,\ldots,n-1\), \(i-1<\beta _{i}<i\) is the standard Caputo fractional derivative, \(I^{ \gamma }_{0^{+}}\) is the standard Riemann-Liouville integral, \(0<\gamma\), and \(0<\rho<\Gamma(n+\gamma)\). The nonlinear term \(f:[0,1]\times\mathbb {R}^{n}\rightarrow\mathbb{R}\) is continuous.
The boundary value problem of fractional equations has emerged as a new branch in the fields of differential equations for their deep backgrounds. In recent years, it is popular and important because the subject of fractional calculus frequently appears in various fields such as physics, chemistry, biology, economics, control theory, signal and image processing, and blood flow phenomenon. For more details about fractional calculus and fractional differential equations, we refer the reader to the monographs by Miller and Ross [1], Heikkila et al. [2], Podlubny [3], Hilfer [4], and Kilbas et al. [5], the survey by Agarwal et al. [6], and the papers [7–14]. Many scholars have studied the existence for nonlinear fractional differential equations with a variety of boundary conditions; see [15–23] and the references therein. However, sometimes it is better to impose integral conditions because they lead to more precise measures than those proposed by a local condition; then, it is greatly important to obtain specific solutions when a solution exists. For this reason, the aim of this paper is to study the existence of solutions for problem (1) by using the monotone iterative technique based on the method of upper and lower solutions, to obtain the existence of solutions for problem (1) by establishing a comparison theorem, and to give a specific iterative equation. For monotone iterative technique, which is based on the method of upper and lower solutions, see recent papers [24–28].
2 Preliminaries
Let \(I=[0,1]\). We denote by \(C(I)\) the Banach space of all continuous functions \(u(t)\) on I with norm \(\Vert u\Vert _{C}=\max_{t\in I}\vert u(t)\vert \). Generally, for \(n\in\mathbb{N}\), we use \(C^{n}(I)\) to denote the Banach space of all nth-order continuously differentiable functions on I with norm
Let \(C^{+}(I)\) denote the cone of all nonnegative functions in \(C(I)\). Let \(\mathrm {AC}^{n}\) be the Banach space of all absolutely continuous functions \(u(t)\) on I differentiable up to order n with norm
Definition 1
If \(g \in C([a,b])\) and \(q > 0\), then the Riemann-Liouville fractional integral is defined by
where \(\Gamma(\cdot)\) is the gamma function.
Definition 2
Let \(q \geq0\) and \(n = [q]+1\). If \(g \in \mathrm {AC}^{n}[a,b]\), then the Caputo fractional derivative of order q of g defined by
exists almost everywhere on \([a,b]\) (\([q]\) is the integer part of q).
Lemma 3
Let \(h \in C(I)\). Then the linear boundary value problem (LBVP)
has a unique solution
where
Moreover, the solution operator \(S:\mathrm {AC}(I)\rightarrow \mathrm {AC}^{(n-1)}(I)\) is a completely continuous linear operator.
Proof
We may deduce equation (2) equivalent to an integral equation
Since \(u^{(j)}(0)=0\), we deduce that \(c_{j}=0\), \(j=0,1,\ldots,n-2\). Therefore, taking the derivatives of equation (5) gives
and we have
Because of the integral boundary condition \(u^{(n-1)}(0)=\rho I^{ \gamma}_{0^{+}}u(1)\), we have
Substituting the values of \(c_{j}\), \(c_{n-1}\), \(j=0,1,\ldots,n-2\), into (5), we obtain
which can be written as
From expression (3) we easily see that \(S:\mathrm {AC}(I)\rightarrow \mathrm {AC}^{(n-1)}(I)\) is a completely continuous linear operator. This completes the proof. □
Lemma 4
Let \(h \in C^{+}(I)\). Then the unique solution \(u=Sh\) of LBVP (2) has the following properties:
Proof
By expression (3) of the solution of LBVP (2) we easily see that \(u(t)\geq0\). Next, we show that \(^{C}D^{ \beta _{i}}_{0^{+}}u(t)\geq0\), \(i=1,2,\ldots,n-1\).
From (3) we have
where
and \(G_{s}^{(i)}(s,r)\) is the ith-order partial derivative of \(G(s,r)\) to s, which is given by
Consequently, (6) becomes
From (7) we see that
Combining (8) and this inequality, we have
and the proof is completed. □
Now, by expression (3) of the solution to LBVP (2) we easily see that problem (1) is equivalent to the integral equation
Therefore, the solution of problem (1) is equivalent to the fixed point of operator T. Next, we give a comparison theorem.
Lemma 5
Comparison result
If \(u(t)\in \mathrm {AC}^{n}(I)\) satisfies
then \(u(t)\geq0\), \(t\in I\).
Proof
By Lemma 3 we know that LBVP (2) has a unique solution \(u(t)=\int^{1}_{0}G(t,s)h(s)\,ds\). From (4) it is easy to verify that Green’s function \(G(t,s)\geq0\), \(t,s \in I\). Let \(h(t)\in C^{+}(I)\). Then \(u(t)\geq0\), \(t\in I\). □
According to the comparison result of Lemma 5, we give the definition of upper solution and lower solutions.
Definition 6
If \(v\in \mathrm {AC}^{n}(I)\) satisfies
then we call v a lower solution of problem (1). If \(w\in \mathrm {AC}^{n}(I)\) satisfies
then we call w an upper solution of problem (1).
3 Main results
Theorem 7
Let v, w be lower solution and upper solutions of problem (1) such that \(v(t)\leq w(t)\) for all \(t\in I\). Assume that the nonlinear term \(f:[0,1]\times\mathbb {R}^{n}\rightarrow\mathbb{R}\) is continuous and satisfies the following assumption:
-
(H)
For all \(t\in I\), \(x_{0}, y_{0}\in[v,w]\) and \(x_{i},y_{i}\in [^{C}D^{ \beta_{i}}_{0^{+}}v,^{C}D^{ \beta_{i}}_{0^{+}}w]\), \(i=1,2,\ldots,n-1\), such that \(x_{0}\geq y_{0}\), \(x_{i}\geq y_{i}\), we have
$$f(t,x_{0},x_{1},x_{2},\ldots,x_{n-1}) \geq f(t,y_{0},y_{1},y_{2},\ldots ,y_{n-1}). $$
Then problem (1) has a minimum solution \(\underline{u}\) and maximum solution u̅ between v and w.
Proof
Denote
Then \(D\subset \mathrm {AC}^{n-1}(I)\) is a nonempty, convex, and closed set. Define the operator \(F: D\rightarrow \mathrm {AC}(I)\) as follows:
From the continuity of f we easily see that \(F: D\rightarrow \mathrm {AC}(I)\) is a continuous operator that maps bounded sets into bounded sets. By Lemma 3 we know that the composite mapping \(S\circ F: D\rightarrow \mathrm {AC}^{n-1}(I)\) is a completely continuous operator. Therefore, by (9), for every \(u\in D\), we have \(Tu=(S\circ F)(u)\), and \(T: D\rightarrow \mathrm {AC}^{n-1}(I)\) is a completely continuous operator. Then the solution of problem (1) is equivalent to the fixed point of operator T defined by (9). We the proof in three steps.
Step 1: \(T: D\rightarrow D\) is an increasing operator.
For \(u\in D\), suppose that \(x=Tu=(S\circ F)(u)\). Letting \(h=F(u)\), we know that \(x=Sh\) is a solution of LBVP (2). Then \(x\in \mathrm {AC}^{n}(I)\) satisfies
Thus, using the definition of upper and lower solutions and condition (H), we have
Then, by Lemma 5 we have
Further, we have
Similarly,
From Lemma 5 we have
Namely,
Hence,
This implies that \(T: D\rightarrow D\).
For every \(u_{1}, u_{2}\in D\), and
Assume that \(x_{1}=Tu_{1}\) and \(x_{2}=Tu_{2}\), this implies that \(x_{1}\) and \(x_{2}\) satisfy (11), respectively. Then, from condition (H) we have
By Lemma 5 we have
namely,
Therefore, T is an increasing operator.
Step 2: Problem (1) has solutions between v and w.
Define two iterative sequences \(\{v_{n}\}\) and \(\{w_{n}\}\) starting from \(v_{0}=v\) and \(w_{0}=w\), respectively, by the following procedure
This implies that \(\{v_{n}\}\), \(\{w_{n}\}\) satisfy the following monotonous conditions
where \(i=1,\ldots,n-1\). Namely, \(\{v_{n}\}\), \(\{^{C}D^{ \beta_{i}}_{0^{+}}v_{n}\}\) are increasing sequences in \([v,w]\), \([^{C}D^{ \beta_{i}}_{0^{+}}v,^{C}D^{ \beta _{i}}_{0^{+}}w]\), and \(\{w_{n}\}\), \(\{^{C}D^{ \beta_{i}}_{0^{+}}w_{n}\}\) are decreasing sequences in \([v,w]\), \([^{C}D^{ \beta _{i}}_{0^{+}}v,^{C}D^{ \beta_{i}}_{0^{+}}w]\), respectively. By the compactness of T we easily see that \(\{v_{n}\},\{w_{n}\}\subset T(D)\) are relatively compact in \(\mathrm {AC}^{n-1}(I)\), which means that they have at least one uniformly convergent subsequence, respectively. From the monotonicity of \(\{v_{n}\}\), \(\{w_{n}\}\) we obtain that \(\{v_{n}\}\), \(\{ w_{n}\}\) are convergent in \(\mathrm {AC}^{n-1}(I)\), which implies that there exist \(\underline{u},\overline{u}\in \mathrm {AC}^{n-1}\) such that \(v_{n}\rightarrow\underline{u}\), \(w_{n}\rightarrow\overline{u}\). Since D is a convex closed set, we also obtain \(\underline{u},\overline {u}\in D\). Further, by the continuity of T we know that \(\underline {u}=T\underline{u}\), \(\overline{u}=T\overline{u}\). Therefore, \(\underline{u}\) and u̅ are solutions of problem (1).
Step 3: We show that \(\underline{u}\) and u̅ are minimum and maximum solutions between v and w, respectively.
Suppose that \(u\in D\) is an arbitrary solution of problem (1). Then u satisfies
Applying to T to (16), we have
Further, we have
Letting \(n\rightarrow\infty\), we obtain
Thus, we see that \(\underline{u}\), u̅ are minimum and maximum solutions between v and w, respectively. The proof is complete. □
By the proof procedure of Theorem 7, we have the following result.
Corollary 8
Let v, w be lower and upper solutions of problem (1) such that \(v(t)\leq w(t)\) for \(t\in I\). Assume that the nonlinear term \(f:[0,1]\times\mathbb{R}^{n}\rightarrow\mathbb{R}\) is continuous and satisfies assumption (H). Then using the linear iterative equation starting from \(u_{0}=v\) and \(u_{0}=w\), respectively,
we define iterative sequences \(\{v_{n}\}\), \(\{w_{n}\}\). By this procedure we can obtain
uniformly for every \(t\in I\), where \(\underline{u}\), u̅ are minimum and maximum solutions between v and w, \(i=1,2,\ldots ,n-1\).
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Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions.
Research supported by NNSFs of China (11501455, 11661071), Key Project of Gansu Provincial National Science Foundation (1606RJZA015) and Project of NWNU-LKQN-14-6.
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Gao, Y., Chen, P. Existence of solutions for a class of nonlinear higher-order fractional differential equation with fractional nonlocal boundary condition. Adv Differ Equ 2016, 314 (2016). https://doi.org/10.1186/s13662-016-1034-9
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DOI: https://doi.org/10.1186/s13662-016-1034-9