Open Access

Existence results for a class of generalized fractional boundary value problems

Contributed equally
Advances in Difference Equations20172017:348

https://doi.org/10.1186/s13662-017-1374-0

Received: 2 June 2017

Accepted: 22 September 2017

Published: 27 October 2017

Abstract

In this paper, we study a class of generalized fractional order three-point boundary value problems that involve fractional derivative defined in terms of weight and scale functions. Using several fixed point theorems, the existence and uniqueness results are obtained.

Keywords

fractional calculusgeneralized fractional derivativefractional boundary value problemfixed point theoremexistence result

1 Introduction

Fractional calculus is the subject of studying fractional integrals and fractional derivatives, which means that the orders of integration and differentiation are not integers but non-integers, and even complex numbers. The history of fractional calculus is more than three hundreds years. However, only in the recent forty years, it was realized that these fractional integrals and derivatives may have many potential applications. Fractional differential equation is a differential equation which involves fractional derivatives, and it has been successfully used to model many real-world phenomena such as heat conduction [1], diffusion process [2], and quantum mechanics [3]. More applications can be seen in [4], Chapter 10.

Fractional boundary value problems (FBVPs) appear in many of these applications. In recent twenty years, considerable work has been done in the field of FBVPs. To verify the existence result of a solution and to study the behavior of the solution of FBVPs have become more and more popular. There are several methods to verify the existence of FBVPs, in which the topological degree method is one of the most effective techniques. By using the fixed point theorems, FBVPs with different types of boundary conditions have been studied. More specifically, in [5], the existence and multiplicity of positive solutions for a nonlinear FBVP with two-point boundary condition are studied. In [6], the existence of solutions for a class of three-point FBVPs involving nonlinear impulsive fractional differential equations is considered. In [7], the existence and uniqueness of solutions for a four-point nonlocal FBVP are derived. In [8], the positive solution of FBVP with integral boundary condition is obtained. In [9], the existence theory of FBVP with anti-periodic boundary condition is discussed. Furthermore, for the existence results of FBVPs with some mixed-type boundary conditions, the readers are referred to [1016] and the references therein.

The literature above only focuses on the FBVP with classical fractional derivatives, i.e., Riemann-Liouville or Caputo derivatives. Fractional derivative also has some limit, since it can be regarded as a convolution between a function and a fractional power kernel. The fractional power kernel puts much weight in the present and less weight in the past, which causes the nice property of fractional derivative called nonlocal property or short memory property. The short memory property is very effective in modeling some physical processes such as diffusion phenomenon in material with memory. However, some real-world phenomena cannot be modeled by such a fractional power kernel properly. For example, an old man and a child have different memory ability. The old man may remember things that happened several decades ago, but forget what happened yesterday. The child has an opposite ability, i.e., he may have no idea about things that happened in his early years, but remember almost everything in the recent week. To model this phenomenon, we need different kernels to weight the function differently. Hence in 2012, a new class of generalized fractional integrals and derivatives defined by using a weight function and a scale function was introduced in [17]. The new fractional operators contain many existing fractional integrals and derivatives as special cases. It is shown that many integral equations can be written and solved in an elegant way using the new operators. Therefore, using different weight functions and scale functions, many fractional problems are significantly generalized. It is also possible that the new generalized fractional integrals and derivatives will bring some interest in the near future, although the theoretical study and applications of them are in the very first stage right now.

Motivated by [18], in this present paper, we consider the following three-point FBVP:
$$ \textstyle\begin{cases} {}^{*}D^{\alpha }_{0+,[z;w]} u(t) = f(t,u(t)),\quad t\in [0,1], 1< \alpha < 2, \\ u(0)=0,\qquad \gamma {u(p)}=u(1), \end{cases} $$
(1)
where \(p\in (0,1)\), and γ is a positive constant. \({}^{*}D^{ \alpha }_{0+,[z;w]} u(t)\) is the generalized fractional derivative of function u with respect to t, and its definition is given in the next section. f is a continuous function satisfying \(\lim_{\vert u\vert \rightarrow +\infty }f ( t,u) = 0\), and \(u\in {X}\), \(f:[0,1]\times {X}\rightarrow {X}\). Here \((X,\Vert \cdot \Vert )\) is a Banach space and \(C=C([0,1],X)\) denotes the Banach space of all continuous functions from \([0,1]\) to X equipped with a topology of uniform convergence with the norm denoted by \(\Vert \cdot \Vert \). Next, we will apply some fixed point theorems to study the existence and uniqueness results of this generalized fractional boundary value problem.

2 Preliminaries

We introduce the generalized fractional integral and derivative directly, and for more details about the classical fractional integral and derivative, such as Riemann-Liouville, Caputo, and Riesz operators, we refer to [19], Chapter 2.

Definition 1

([17])

The left-sided generalized fractional integral of order \(\alpha >0\) of a function \(u(t)\), with respect to a scale function \(z(t)\) and a weight function \(w(t)\), is defined as
$$ \bigl( I^{\alpha }_{a+,[z;w]}u \bigr) (t) = \frac{[w(t)]^{-1}}{\Gamma ( \alpha)} \int^{t}_{a} \frac{w(s)z'(s)u(s)}{[z(t)-z(s)]^{1-\alpha }}\,ds, $$
(2)
provided the integral exists.

Definition 2

([17])

The left-sided generalized derivative of order 1 of a function \(u(t)\), with respect to a scale function \(z(t)\) and a weight function \(w(t)\), is defined as
$$ ( D_{[z,w]}u) (t) = \bigl[w(t) \bigr]^{-1} \biggl[ \biggl( \frac{1}{z'(t)}D _{t} \biggr) \bigl(w(t)u(t) \bigr) \biggr] , $$
(3)
provided the right-hand side of equation is finite.

Definition 3

([17])

The left-sided generalized fractional derivative of order m of a function \(u(t)\), with respect to a scale function \(z(t)\) and a weight function \(w(t)\), is defined as
$$ \bigl( D^{m}_{[z,w]}u \bigr) (t) = \bigl[w(t) \bigr]^{-1} \biggl[ \biggl( \frac{1}{z'(t)}D _{t} \biggr) ^{m} \bigl(w(t)u(t) \bigr) \biggr] , $$
(4)
provided the right-hand side of equation is finite, where m is a positive integer.

Definition 4

([17])

The left-sided Caputo type generalized fractional derivative of order \(\alpha >0\) of a function \(u(t)\), with respect to a scale function \(z(t)\) and a weight function \(w(t)\), is defined as
$$ \bigl( {}^{*}D^{\alpha }_{a+,[z;w]}u \bigr) (t) = I^{m-\alpha }_{a+,[z;w]} \bigl( D^{m}_{[z,w]}u \bigr) (t), $$
(5)
provided the right-hand side of equation is finite, where \(m-1<\alpha <m\), and m is a positive integer. Specifically, for \(1<\alpha <2\),
$$ \bigl( {}^{*}D^{\alpha }_{a+,[z;w]}u \bigr) (t) = \frac{[w(t)]^{-1}}{ \Gamma (2-\alpha)} \int^{t}_{a} \frac{[z(t)-z(s)]^{1-\alpha }}{z'(s)} \cdot \frac{d^{2} [ w(s)u(s) ] }{ds^{2}}\,ds. $$
(6)
Moreover, for \(z(t)=t\) and \(w(t)=1\), the generalized fractional derivative reduces to the Caputo fractional derivative as
$$\bigl( {}^{c}D^{\alpha }_{a+}u \bigr) (t) = \frac{1}{\Gamma (2-\alpha)} \int^{t}_{a} (t-s)^{1-\alpha }u''(s) \,ds. $$

In the above definitions, we only present the ‘left-sided’ sense of generalized fractional integrals and derivatives. The ‘right-sided’ sense of generalized fractional integrals and derivatives and their properties are discussed in [17]. We will not repeat them here since the derivative we use in this paper is defined in the left-sided sense. For simplicity, in what follows, we remove the term ‘\([z;w]\)’ from the subscript in equation (6).

Remark 1

To be more specific, we assume that the weight function is positive and the scale function \(z ( t) \) is monotone increasing over \([0,1]\). Moreover, both \(w(t)\) and \(z(t)\) are continuously differentiable.

Remark 2

Indeed, the generalized fractional derivatives have extended the classical Caputo fractional derivative. For example,
$$f ( x) = \textstyle\begin{cases} -1, & x\in [-1,0), \\ 1, & x\in (0,1], \end{cases} $$
is discontinuous, then \(f\notin {AC} ( [-1,1] ) \), and hence it cannot have Caputo fractional derivative according to [19], Equations (2.4.17)-(2.4.18), p.92. Nevertheless, when we take \(\alpha \in (0,1)\), \(w ( x) = \vert x\vert \), and \(z ( x) = x\) in \(x\in [-1,0)\cup (0,1]\), then
$$\begin{aligned} \bigl( {}^{*} D^{\alpha }_{(-1)+} f \bigr) ( x) =& \frac{1}{\vert x\vert \Gamma ( 1-\alpha) } \int_{-1}^{x}\frac{s'}{ [ x-s ] ^{\alpha }}\,ds \\ =& \frac{ ( x+1) ^{1-\alpha }}{\vert x\vert \Gamma ( 2-\alpha) },\quad x\neq {0}. \end{aligned}$$
Let us consider the following generalized fractional boundary value problem:
$$ \textstyle\begin{cases} {}^{*} D^{\alpha }_{0+} u(t) = \sigma (t),\quad t\in [0,1], 1< \alpha < 2, \\ u(0)=0, \qquad \gamma {u(p)}=u(1), \end{cases} $$
(7)
where σ is a sufficiently smooth function, \(p\in (0,1)\).

To solve problem (7), we have the following lemma.

Lemma 1

Assume that \(z(t)\) is strictly monotone increasing and \(w(t)\) is positive on \([0,1]\), and
$$ L := \gamma \frac{z(p)}{w(p)} - \frac{z(1)}{w(1)} + \frac{z(0)}{w(1)} - \gamma \frac{z(0)}{w(p)} \neq 0, $$
(8)
then the solution of problem (7) is given as
$$\begin{aligned} u(t)=\frac{C_{0}}{w(t)} + \frac{C_{1}z(t)}{w(t)} + \frac{[w(t)]^{-1}}{ \Gamma (\alpha)} \int^{t}_{0} \frac{w(s)z'(s)\sigma (s)\,ds}{[z(t)-z(s)]^{1- \alpha }}, \end{aligned}$$
(9)
where
$$C_{0} = \frac{z(0)}{L\cdot \Gamma (\alpha)} \biggl\{ \frac{\gamma }{w(p)} \int^{p}_{0}\frac{w(s)z'(s)\sigma (s)\,ds}{[z(p)-z(s)]^{1- \alpha }} - \frac{1}{w(1)} \int^{1}_{0}\frac{w(s)z'(s)\sigma (s)\,ds}{[z(1)-z(s)]^{1- \alpha }} \biggr\} $$
and
$$C_{1} = \frac{1}{L\cdot \Gamma (\alpha)} \biggl\{ \frac{1}{w(1)} \int ^{1}_{0}\frac{w(s)z'(s)\sigma (s)\,ds}{[z(1)-z(s)]^{1-\alpha }} - \frac{ \gamma }{w(p)} \int^{p}_{0}\frac{w(s)z'(s)\sigma (s)\,ds}{[z(p)-z(s)]^{1- \alpha }} \biggr\} . $$

Proof

According to equation (6), we have
$$\begin{aligned} {}^{*} D^{\alpha }_{0+} u(t) =& \frac{[w(t)]^{-1}}{\Gamma (2-\alpha)} \int^{t}_{a} \frac{[z(t)-z(s)]^{1- \alpha }}{z'(s)}\cdot \frac{d^{2} [ w(s)u(s) ] }{ds^{2}}\,ds \\ =& \frac{[w(t)]^{-1}}{\Gamma (2-\alpha)} \int^{z(t)}_{z(0)} \bigl[z(t)- \xi \bigr]^{1-\alpha } \frac{d^{2} [ w(z^{-1}(\xi))u(z^{-1}(\xi)) ] }{d \xi^{2}} \bigl[z' \bigl(z^{-1}(\xi) \bigr) \bigr]^{2}\,d\xi \\ =& \frac{1}{w(t)} {^{c}D}^{\alpha }_{z(0)+} \bigl[ w \bigl(z^{-1}(\phi) \bigr)u \bigl(z ^{-1}(\phi) \bigr) \bigr] , \end{aligned}$$
(10)
where \(\phi =z(t)\in [z(0),z(1)]\) and \(\xi =z(s)\in [z(0),z(t)]\). We apply the mean value theorem to move \([z'(z^{-1}(\xi))]^{2}\) outside as a constant, i.e., \([z'(z^{-1}(\xi^{*}))]^{2}\), \(\xi^{*}\in [z(0),z(1)]\), which is absorbed by the weight \(\frac{1}{w(t)}\). Hence, problem (7) is transformed to
$$ \textstyle\begin{cases} {^{c}D}^{\alpha }_{z(0)+} [ w(z^{-1}(\phi))u(z^{-1}(\phi)) ] = w(z^{-1}(\phi))\sigma (z^{-1}(\phi)), \\ u(z^{-1}(z(0)))=0,\qquad \gamma {u(z^{-1}(z(p)))}=u(z^{-1}(z(1))). \end{cases} $$
(11)
Finally, it suffices to verify that (11) is solvable under assumptions in Remark 1.
According to [20], Section 5.5, the general solution of equation (11) is
$$\begin{aligned} w \bigl(z^{-1}(\phi) \bigr)u \bigl(z^{-1}(\phi) \bigr) =& C_{0} + C_{1}z(t) + \frac{1}{\Gamma (\alpha)} \int^{z(t)}_{z(0)} \frac{w(z^{-1}( \xi))\sigma (z^{-1}(\xi))}{[z(t)-\xi ]^{1-\alpha }}\,d\xi, \end{aligned}$$
(12)
which implies
$$\begin{aligned} u(t) = C_{0}\frac{1}{w(t)} + C_{1} \frac{z(t)}{w(t)} + \frac{[w(t)]^{-1}}{ \Gamma (\alpha)} \int^{t}_{0} \frac{w(s)z'(s)\sigma (s)}{[z(t)-z(s)]^{1- \alpha }}\,ds. \end{aligned}$$
(13)
Imposing the initial and boundary conditions on equation (13) gives
$$ C_{0}\frac{1}{w(0)} + C_{1} \frac{z(0)}{w(0)} = 0, $$
(14)
and
$$\begin{aligned}& \gamma \biggl[ C_{0}\frac{1}{w(p)} + C_{1}\frac{z(p)}{w(p)} + \frac{[w(p)]^{-1}}{ \Gamma (\alpha)} \int^{p}_{0} \frac{w(s)z'(s)\sigma (s)}{[z(p)-z(s)]^{1- \alpha }}\,ds \biggr] \\& \quad = C_{0}\frac{1}{w(1)} + C_{1}\frac{z(1)}{w(1)} + \frac{[w(1)]^{-1}}{ \Gamma (\alpha)} \int^{1}_{0} \frac{w(s)z'(s)\sigma (s)}{[z(1)-z(s)]^{1- \alpha }}\,ds. \end{aligned}$$
(15)

Now we can solve equations (14) and (15) to get \(C_{0}\) and \(C_{1}\). Since \(L:= \gamma \frac{z(p)}{w(p)} - \frac{z(1)}{w(1)} + \frac{z(0)}{w(1)} - \gamma \frac{z(0)}{w(p)}\) is the determinant of the coefficient matrix of equations (14) and (15), when \(L\neq 0\), equations (14) and (15) have a unique nonzero solution. This completes the proof. □

The following theorems play important roles in studying the existence and uniqueness of fractional boundary value problems.

Theorem 1

(Contraction mapping principle, see [21])

Let E be a Banach space, \({D}\subset {E}\) be closed, and \(F:{D}\rightarrow {D}\) be a strict contraction, i.e., \(\vert Fx-Fy\vert \leq k\vert x-y\vert \) for some \(k\in (0,1)\) and all \(x,y\in {D}\). Then F has a unique fixed point \(x^{*}\). Furthermore, the successive approximations \(x_{n+1}=Fx_{n}=F^{n}x_{0}\), starting at any \(x_{0}\in {D}\), converge to \(x^{*}\) and satisfy \(\vert x_{n}-x^{*}\vert \leq (1-k)^{-1}k^{n}\vert Fx_{0}-x_{0}\vert \).

Theorem 2

(Arzelà-Ascoli, see [21])

If a sequence \(\{x_{n}\} ^{\infty }_{n=0}\) in a compact subset of X is uniformly bounded and equicontinuous, then it has a uniformly convergent subsequence.

Theorem 3

([22])

Let X be a Banach space. Assume that Ω is an open bounded subset of X with \(\theta \in \Omega \), and let \(T:\bar{\Omega } \rightarrow {X}\) be a completely continuous operator such that
$$\Vert Tu\Vert \leq \Vert u\Vert ,\quad \forall {u}\in \partial \Omega. $$
Then T has a fixed point in Ω̄.

Theorem 4

(Krasnosel’skii, see [22])

Let M be a closed convex and nonempty subset of a Banach space X. Let A and B be two operators such that:
(H1): 

\(Ax+By\in {M}\), wherever \(x,y\in {M}\);

(H2): 

A is compact and continuous; and

(H3): 

B is a contraction mapping.

Then there exists \(z^{*}\in {M}\) such that \(z^{*}=Az^{*}+Bz^{*}\).

3 Main results

In this section, we present some existence results of boundary value problem (1). Let \(C=C ( [0,1],R ) \) denote the Banach space of all continuous functions mapping \([0,1]\) to R equipped with the norm defined by \(\Vert u\Vert =\sup_{0\leq t\leq 1}\{\vert u(t)\vert \}\).

Define the operator \(T:C\rightarrow {C}\) as
$$\begin{aligned} (Tu) (t) =& \frac{L^{-1}z(0)}{w(t)\Gamma (\alpha)} \biggl[ \int^{p} _{0}\frac{\gamma w(s)z'(s)f(s,u(s))\,ds}{w(p)[z(p)-z(s)]^{1-\alpha }} \\ &{} - \int^{1}_{0}\frac{w(s)z'(s)f(s,u(s))\,ds}{w(1)[z(1)-z(s)]^{1- \alpha }} \biggr] \\ &{} + \frac{L^{-1}z(t)}{w(t)\Gamma (\alpha)} \biggl[ \int^{1}_{0}\frac{w(s)z'(s)f(s,u(s))\,ds}{w(1)[z(1)-z(s)]^{1- \alpha }} \\ & {}- \int^{p}_{0}\frac{\gamma w(s)z'(s)f(s,u(s))\,ds}{w(p)[z(p)-z(s)]^{1- \alpha }} \biggr] \\ &{} + \frac{[w(t)]^{-1}}{\Gamma (\alpha)} \int^{t}_{0} \frac{w(s)z'(s)f(s,u(s))}{[z(t)-z(s)]^{1- \alpha }}\,ds, \end{aligned}$$
(16)
where \(0< p<1\), \(0< t<1\).

If the operator \(T:C\rightarrow {C}\) defined by equation (16) has a fixed point, then the fixed point coincides with the solution of fractional boundary problem (1). In what follows, we prove the complete continuity property of operator T.

Lemma 2

The operator \(T:C\rightarrow {C}\) defined by equation (16) is completely continuous.

Proof

Let \(\Omega \subset {C}\) be a bounded set, then for any \(t\in [0,1]\) and \(u\in \Omega \), since \(f(t,u)\) is continuous on \([0,1]\times {\mathbb{R}}\), there exists a positive constant \(L_{1}\) such that \(\vert f(t,u)\vert \leq {L_{1}}\). Thus one can have
$$\begin{aligned}& \bigl\vert (Tu) (t) \bigr\vert \\& \quad \leq \frac{\vert L^{-1}\vert z(0)}{w(t)\Gamma (\alpha)} \biggl[ \int^{p}_{0}\frac{ \gamma w(s)z'(s)\vert f(s,u(s))\vert \,ds}{w(p)[z(p)-z(s)]^{1-\alpha }} \\& \qquad {} + \int^{1}_{0}\frac{w(s)z'(s)\vert f(s,u(s))\vert \,ds}{w(1)[z(1)-z(s)]^{1- \alpha }} \biggr] + \frac{\vert L^{-1}\vert z(t)}{w(t)\Gamma (\alpha)} \biggl[ \int^{1}_{0} \frac{w(s)z'(s)\vert f(s,u(s))\vert \,ds}{w(1)[z(1)-z(s)]^{1-\alpha }} \\& \qquad {} + \int^{p}_{0}\frac{\gamma w(s)z'(s)\vert f(s,u(s))\vert \,ds}{w(p)[z(p)-z(s)]^{1- \alpha }} \biggr] + \frac{[w(t)]^{-1}}{\Gamma (\alpha)} \int^{t}_{0} \frac{w(s)z'(s)\vert f(s,u(s))\vert }{[z(t)-z(s)]^{1-\alpha }}\,ds \\& \quad \leq \frac{z(0)L_{1}w_{\max }}{w_{\min }\Gamma (\alpha)\vert L\vert } \biggl[ \int^{p}_{0}\frac{\gamma z'(s)\,ds}{w(p)[z(p)-z(s)]^{1-\alpha }} \\& \qquad {} + \int^{1}_{0}\frac{z'(s)\,ds}{w(1)[z(1)-z(s)]^{1-\alpha }} \biggr] + \frac{z(t)L_{1}w_{\max }}{w_{\min }\Gamma (\alpha)\vert L\vert } \biggl[ \int^{1}_{0}\frac{z'(s)\,ds}{w(1)[z(1)-z(s)]^{1-\alpha }} \\& \qquad {} + \int^{p}_{0}\frac{\gamma z'(s)\,ds}{w(p)[z(p)-z(s)]^{1- \alpha }} \biggr] + \frac{L_{1}w_{\max }}{w_{\min }\Gamma (\alpha)} \int^{t}_{0} \frac{z'(s)}{[z(t)-z(s)]^{1- \alpha }}\,ds \\& \quad = \frac{z(0)L_{1}w_{\max }}{w_{\min }\Gamma (\alpha +1)\vert L\vert } \biggl\{ \frac{ \gamma }{w(p)} \bigl[ z(p)-z(0) \bigr] ^{\alpha } + \frac{1}{w(1)} \bigl[ z(1)-z(0) \bigr] ^{\alpha } \biggr\} \\& \qquad {} + \frac{z(t)L_{1}w_{\max }}{w_{\min }\Gamma (\alpha +1)\vert L\vert } \biggl\{ \frac{1}{w(1)} \bigl[ z(1)-z(0) \bigr] ^{\alpha } + \frac{\gamma }{w(p)} \bigl[ z(p)-z(0) \bigr] ^{\alpha } \biggr\} \\& \qquad {} + \frac{L_{1}w_{\max }}{w_{\min }\Gamma (\alpha +1)} \bigl[ z(t)-z(0) \bigr] ^{\alpha } \\& \quad \leq \frac{z(0)L_{1}w_{\max }}{w_{\min }\Gamma (\alpha +1)\vert L\vert } \biggl\{ \frac{\gamma }{w(p)} \bigl[ z(p)-z(0) \bigr] ^{\alpha } + \frac{1}{w(1)} \bigl[ z(1)-z(0) \bigr] ^{\alpha } \biggr\} \\& \qquad {} + \frac{z(1)L_{1}w_{\max }}{w_{\min }\Gamma (\alpha +1)\vert L\vert } \biggl\{ \frac{1}{w(1)} \bigl[ z(1)-z(0) \bigr] ^{\alpha } \\& \qquad {} + \frac{\gamma }{w(p)} \bigl[ z(p)-z(0) \bigr] ^{\alpha } \biggr\} + \frac{L_{1}w_{\max }}{w_{\min }\Gamma (\alpha +1)} \bigl[ z(1)-z(0) \bigr] ^{\alpha } \\& \quad :=L_{2}, \end{aligned}$$
(17)
where \(w_{\max }=\max_{0\leq t\leq 1}\{w(t)\}\), \(w_{\min }= \min_{0\leq t\leq 1}\{w(t)\}\), and \(L_{2}\) is a positive constant. Equation (17) implies that \(\Vert Tu\Vert \leq {L_{2}}\). Moreover, for the derivative of T, we have
$$\begin{aligned} \bigl\vert (Tu)'(t) \bigr\vert =& \biggl\vert \frac{-L^{-1}z(0)w'(t)}{w^{2}(t) \Gamma (\alpha)}L_{31} + \frac{L^{-1}[z'(t)w(t)-z(t)w'(t)]}{w ^{2}(t)\Gamma (\alpha)}L_{32} \\ & {}- \frac{[w'(t)]^{-1}}{w^{2}(t)\Gamma (\alpha)} \int^{t}_{0} \frac{w(s)z'(s)f(s,u(s))}{[z(t)-z(s)]^{1- \alpha }}\,ds \\ &{} + \frac{[w(t)]^{-1}}{\Gamma (\alpha -1)} \int^{t}_{0} \frac{w(s)z'(s)z'(t)\vert f(s,u(s))\vert }{[z(t)-z(s)]^{2- \alpha }}\,ds \biggr\vert \\ \leq &\frac{\vert L^{-1}\vert z(0)w'_{\max }}{w^{2}_{\min }\Gamma (\alpha)} \vert L_{31}\vert + \frac{\vert L^{-1}\vert (z'_{\max }w_{\max }+z_{\max }w'_{ \max })}{w^{2}_{\min }\Gamma (\alpha)} \vert L_{32}\vert \\ &{} + \frac{w'_{\max }w_{\max }L_{1}}{w^{2}_{\min }\Gamma (\alpha)} \bigl[ z(1)-z(0) \bigr] ^{\alpha } \\ &{} + \frac{w_{\max }z'_{\max }z'_{\max }L_{1}}{w_{\min }\Gamma ( \alpha -1)} \bigl[ z(1)-z(0) \bigr] ^{\alpha -1} \\ :=&L_{3}, \end{aligned}$$
(18)
where
$$\begin{aligned}& L_{31} = \int^{p}_{0}\frac{\gamma w(s)z'(s)f(s,u(s))\,ds}{w(p)[z(p)-z(s)]^{1- \alpha }} - \int^{1}_{0}\frac{w(s)z'(s)f(s,u(s))\,ds}{w(1)[z(1)-z(s)]^{1- \alpha }}, \\& L_{32} = \int^{1}_{0}\frac{w(s)z'(s)f(s,u(s))\,ds}{w(1)[z(1)-z(s)]^{1- \alpha }} - \int^{p}_{0}\frac{\gamma w(s)z'(s)f(s,u(s))\,ds}{w(p)[z(p)-z(s)]^{1- \alpha }}, \end{aligned}$$
are constants, i.e., \(L_{31}+L_{32}=0\), and \(w'_{\max }\), \(z'_{\max }\) indicate the maximum values of the derivative of functions \(w(t)\), \(z(t)\) on \([0,1]\), respectively.
Therefore, for all \(0\leq t_{1}\leq t_{2}\leq 1\),
$$ \bigl\vert (Tu) (t_{2})-(Tu) (t_{1}) \bigr\vert \leq \int^{t_{2}}_{t_{1}} \bigl\vert (Tu)'(s) \bigr\vert \,ds \leq L_{3}(t_{2}-t_{1}), $$
(19)
which implies that the operator T is equicontinuous on \([0,1]\). Hence, by the Arzelà-Ascoli theorem, the operator \(T:C\rightarrow {C}\) is completely continuous. □

Remark 3

The absolute value of \(L_{31}\) (or \(L_{32}\)) has the following upper-bound estimation:
$$\begin{aligned} \vert L_{31}\vert =& \biggl\vert \int^{p}_{0}\frac{\gamma w(s)z'(s)f(s,u(s))\,ds}{w(p)[z(p)-z(s)]^{1- \alpha }} - \int^{1}_{0}\frac{w(s)z'(s)f(s,u(s))\,ds}{w(1)[z(1)-z(s)]^{1- \alpha }} \biggr\vert \\ \leq & \biggl\vert \int^{p}_{0}\frac{\gamma w(s)z'(s)f(s,u(s))\,ds}{w(p)[z(p)-z(s)]^{1- \alpha }} \biggr\vert + \biggl\vert \int^{1}_{0}\frac{w(s)z'(s)f(s,u(s))\,ds}{w(1)[z(1)-z(s)]^{1- \alpha }} \biggr\vert \\ \leq & \frac{\gamma L_{1}w_{\max }}{w(p)} \int^{p}_{0}\frac{z'(s)\,ds}{[z(p)-z(s)]^{1- \alpha }} + \frac{L_{1}w_{\max }}{w(1)} \int^{1}_{0}\frac{z'(s)\,ds}{[z(1)-z(s)]^{1- \alpha }} \\ =& \frac{L_{1}w_{\max }}{\alpha } \biggl[ \frac{\gamma }{w(p)} \bigl[z(p)-z(0) \bigr]^{ \alpha } + \frac{1}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr] . \end{aligned}$$
Denote
$$L_{4} = {L_{1}w_{\max }} \biggl[ \frac{\gamma }{w(p)} \bigl[z(p)-z(0) \bigr]^{ \alpha } + \frac{1}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr] , $$
we have
$$\alpha \vert L_{31}\vert \leq L_{4}. $$

Theorem 5

Assume that \(f:[0,1]\times {\mathbb{R}}\rightarrow {\mathbb{R}}\) and \(\lim_{u\rightarrow {0}}f(t,u)=0\). Then the boundary value problem (1) has at least one solution.

Proof

Since \(\lim_{u\rightarrow {0}}f(t,u)=0\), then there exist constants \(d>0\) and \(d_{1}>0\) such that for all \(0<\vert u\vert <d\), we have \(\vert f(t,u)\vert \leq d_{1}\vert u\vert \), where \(d_{1}\) is such that
$$\begin{aligned}& \max_{t\in [0,1]} \biggl\{ \biggl( \frac{\vert L^{-1}\vert z(0)}{w(t)\Gamma (\alpha +1)} + \frac{\vert L^{-1}\vert z(t)}{w(t)\Gamma (\alpha +1)} \biggr) \biggl[ \frac{ \gamma w_{\max }}{w(p)} \bigl[ z(p)-z(0) \bigr] ^{\alpha } \\& \quad {} + \frac{w_{\max }}{w(1)} \bigl[ z(1)-z(0) \bigr] ^{\alpha } \biggr] + \frac{w_{\max }}{w(t)\Gamma (\alpha +1)}\bigl[z(t)-z(0)\bigr]^{\alpha } \biggr\} \cdot d_{1} \leq 1. \end{aligned}$$
(20)
Define \(\Omega_{1}= \{ u\in {C}:\vert u\vert \leq d \} \). Choose \(u_{0}\in {C}\) such that \(\vert u_{0}\vert =d\), which means that
$$u_{0}\in \partial {\Omega_{1}}. $$
By Lemma 2, the operator T is completely continuous, and by equation (20), we have
$$\begin{aligned} \bigl\vert (Tu_{0}) (t) \bigr\vert \leq & \max _{t\in [0,1]} \biggl\{ \biggl( \frac{\vert L^{-1}\vert z(0)}{w(t) \Gamma (\alpha +1)} + \frac{\vert L^{-1}\vert z(t)}{w(t)\Gamma (\alpha +1)} \biggr) \\ & {}\times \biggl[ \frac{\gamma w_{\max }}{w(p)} \bigl[z(p)-z(0) \bigr]^{\alpha } + \frac{w _{\max }}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr] \\ &{} + \frac{w_{\max }}{w(t)\Gamma (\alpha +1)} \bigl[z(t)-z(0) \bigr]^{\alpha } \biggr\} \cdot d_{1}\vert u_{0}\vert \\ \leq & \vert u_{0}\vert . \end{aligned}$$
(21)

Therefore, by Theorem 3, the operator T has at least one fixed point, which implies that the boundary value problem (1) has at least one solution. □

Theorem 6

Let \(f:[0,1]\times {X}\rightarrow {X}\) be a jointly continuous function satisfying the Lipschitz condition
$$\bigl\Vert f(t,u_{1})-f(t,u_{2}) \bigr\Vert \leq L_{5} \Vert u_{1}-u_{2}\Vert ,\quad \forall t \in [0,1], u_{1},u_{2}\in {X}. $$
Then the boundary value problem (1) has a unique solution provided \(\Delta <1\), where
$$\begin{aligned} \Delta =& \frac{L_{5}}{\mu }\cdot \frac{w_{\max }}{w_{\min }\Gamma ( \alpha +1)} \biggl\{ \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)}\bigl[z(p)-z(0)\bigr]^{ \alpha } \\ &{} + \frac{1}{w(1)}\bigl[z(1)-z(0)\bigr]^{\alpha } \biggr] + \bigl[z(1)-z(0)\bigr]^{ \alpha } \biggr\} , \end{aligned}$$
and \(\Delta \leq \mu <1\).

Proof

First of all, we verify that T maps a bounded ball into itself. Denote \(L_{6}= \sup_{t\in [0,1]}\vert f(t,0)\vert \), and select
$$\begin{aligned} r \geq& \frac{L_{6}}{1-\mu }\cdot \frac{w_{\max }}{w_{\min }\Gamma ( \alpha +1)} \biggl\{ \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)}\bigl[z(p)-z(0) \bigr]^{ \alpha } \\ &{} + \frac{1}{w(1)}\bigl[z(1)-z(0)\bigr]^{\alpha } \biggr] + \bigl[z(1)-z(0)\bigr]^{ \alpha } \biggr\} , \end{aligned}$$
(22)
where μ is a real number satisfying \(\Delta \leq \mu <1\). We define a closed ball as \(B_{r}= \{ u\in {C}:\Vert u\Vert \leq {r} \} \), then
$$\begin{aligned}& \bigl\Vert (Tu) (t) \bigr\Vert \\& \quad \leq \frac{\vert L^{-1}\vert z(0)}{w(t)\Gamma (\alpha)} \biggl\{ \int^{p} _{0}\frac{\gamma w(s)z'(s)\vert f(s,u(s))\vert }{w(p)[z(p)-z(s)]^{1-\alpha }}\,ds \\& \qquad {} + \int^{1}_{0} \frac{w(s)z'(s)\vert f(s,u(s))\vert }{w(1)[z(1)-z(s)]^{1-\alpha }}\,ds \biggr\} \\& \qquad {} + \frac{\vert L^{-1}\vert z(t)}{w(t)\Gamma (\alpha)} \biggl\{ \int^{1}_{0}\frac{w(s)z'(s)\vert f(s,u(s))\vert }{w(1)[z(1)-z(s)]^{1- \alpha }}\,ds \\& \qquad {} + \int^{p}_{0}\frac{\gamma w(s)z'(s)\vert f(s,u(s))\vert }{w(p)[z(p)-z(s)]^{1- \alpha }}\,ds \biggr\} \\& \qquad {} + \frac{1}{w(t)\Gamma (\alpha)} \int^{t}_{0}\frac{w(s)z'(s)\vert f(s,u(s))\vert }{[z(t)-z(s)]^{1- \alpha }}\,ds \\& \quad \leq \frac{\vert L^{-1}\vert z(0)}{w(t)\Gamma (\alpha)} \biggl\{ \int^{p} _{0}\frac{\gamma w(s)z'(s) \{ \vert f(s,u(s))-f(s,0)\vert +\vert f(s,0)\vert \} }{w(p)[z(p)-z(s)]^{1- \alpha }}\,ds \\& \qquad {} + \int^{1}_{0}\frac{w(s)z'(s) \{ \vert f(s,u(s))-f(s,0)\vert +\vert f(s,0)\vert \} }{w(1)[z(1)-z(s)]^{1- \alpha }}\,ds \biggr\} \\& \qquad {} + \frac{\vert L^{-1}\vert z(t)}{w(t)\Gamma (\alpha)} \biggl\{ \int^{1}_{0}\frac{w(s)z'(s) \{ \vert f(s,u(s))-f(s,0)\vert +\vert f(s,0)\vert \} }{w(1)[z(1)-z(s)]^{1-\alpha }}\,ds \\& \qquad {} + \int^{p}_{0}\frac{\gamma w(s)z'(s) \{ \vert f(s,u(s))-f(s,0)\vert +\vert f(s,0)\vert \} }{w(p)[z(p)-z(s)]^{1- \alpha }}\,ds \biggr\} \\& \qquad {} + \frac{1}{w(t)\Gamma (\alpha)} \int^{t}_{0}\frac{w(s)z'(s) \{ \vert f(s,u(s))-f(s,0)\vert +\vert f(s,0)\vert \} }{[z(t)-z(s)]^{1- \alpha }}\,ds \\& \quad \leq \frac{(L_{5}r+L_{6})\vert L^{-1}\vert z(0)}{w(t)\Gamma (\alpha +1)} \biggl\{ \frac{\gamma w_{\max }}{w(p)} \bigl[z(p)-z(0) \bigr]^{\alpha } + \frac{w _{\max }}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr\} \\& \qquad {} + \frac{(L_{5}r+L_{6})\vert L^{-1}\vert z(t)}{w(t)\Gamma (\alpha +1)} \biggl\{ \frac{\gamma w_{\max }}{w(p)} \bigl[z(p)-z(0) \bigr]^{\alpha } + \frac{w_{ \max }}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr\} \\& \qquad {} + \frac{(L_{5}r+L_{6})w_{\max }}{w_{\min }\Gamma (\alpha +1)} \bigl[z(t)-z(0) \bigr]^{ \alpha } \\& \quad \leq \frac{(L_{5}r+L_{6})w_{\max }}{w_{\min }\Gamma (\alpha +1)} \biggl\{ \bigl[z(1)-z(0) \bigr]^{\alpha } \\& \qquad {} + \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)} \bigl[z(p)-z(0) \bigr]^{ \alpha } + \frac{1}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr] \biggr\} \\& \quad \leq ( \mu \Delta +1-\mu) r \\& \quad \leq r, \end{aligned}$$
which implies that \(T(B_{r})\subset {B_{r}}\). Next, for any \(u_{1},u_{2}\in {C}\) and for each \(t\in [0,1]\), one can obtain
$$\begin{aligned}& \bigl\Vert (Tu_{1}) (t)-(Tu_{2}) (t) \bigr\Vert \\& \quad \leq \frac{\vert L^{-1}\vert z(0)}{w(t)\Gamma (\alpha)} \biggl\{ \int^{p} _{0}\frac{\gamma w(s)z'(s)\Vert f(s,u_{1}(s))-f(s,u_{2}(s))\Vert }{w(p)[z(p)-z(s)]^{1- \alpha }}\,ds \\& \qquad {} + \int^{1}_{0}\frac{w(s)z'(s)\Vert f(s,u_{1}(s))-f(s,u_{2}(s))\Vert }{w(1)[z(1)-z(s)]^{1- \alpha }}\,ds \biggr\} \\& \qquad {} + \frac{\vert L^{-1}\vert z(t)}{w(t)\Gamma (\alpha)} \biggl\{ \int^{1}_{0}\frac{w(s)z'(s) \Vert f(s,u_{1}(s))-f(s,u_{2}(s))\Vert }{w(1)[z(1)-z(s)]^{1-\alpha }}\,ds \\& \qquad {} + \int^{p}_{0}\frac{\gamma w(s)z'(s)\Vert f(s,u_{1}(s))-f(s,u_{2}(s))\Vert }{w(p)[z(p)-z(s)]^{1- \alpha }}\,ds \biggr\} \\& \qquad {} + \frac{1}{w(t)\Gamma (\alpha)} \int^{t}_{0}\frac{w(s)z'(s)\Vert f(s,u _{1}(s))-f(s,u_{2}(s))\Vert }{[z(t)-z(s)]^{1-\alpha }}\,ds \\& \quad \leq \frac{L_{5}\Vert u_{1}-u_{2}\Vert w_{\max }}{w_{\min }\Gamma (\alpha +1)} \biggl\{ \bigl[z(1)-z(0) \bigr]^{\alpha } \\& \qquad {} + \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)} \bigl[z(p)-z(0) \bigr]^{ \alpha } + \frac{1}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr] \biggr\} \\& \quad \leq \mu \Delta \Vert u_{1}-u_{2}\Vert \\& \quad \leq \Vert u_{1}-u_{2}\Vert , \end{aligned}$$
which implies that T is a contraction as \(\mu \Delta <1\). Therefore, by the contraction mapping principle (i.e., Banach fixed point theorem), the boundary value problem (1) has a unique solution. □

Theorem 7

Assume that \(f:[0,1]\times {X}\rightarrow {X}\) is a jointly continuous function and further:
(H1): 

\(\vert f(t,u_{1})-f(t,u_{2})\vert \leq L_{5} \vert u_{1}-u_{2}\vert \), \(u_{1},u_{2}\in {X}\),

(H2): 

\(\vert f(t,u)\vert \leq \lambda (t)\), \(\forall (t,u)\in [0,1] \times {X}\), and \(\lambda \in L^{1}([0,1],R^{+})\).

If
$$\begin{aligned}& \frac{w_{\max }L_{5}}{w_{\min }\Gamma (\alpha +1)} \biggl\{ \bigl[z(1)-z(0)\bigr]^{ \alpha } \\& \quad {} + \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)}\bigl[z(p)-z(0) \bigr]^{\alpha } + \frac{1}{w(1)}\bigl[z(1)-z(0)\bigr]^{\alpha } \biggr] \biggr\} < 1, \end{aligned}$$
then the boundary value problem (1) has at least one solution.

Proof

Let
$$\begin{aligned} r \geq & \Vert \lambda \Vert _{L^{1}[0,1]}\times \frac{w_{\max }}{w_{\min } \Gamma (\alpha +1)} \biggl\{ \bigl[z(1)-z(0) \bigr]^{\alpha } \\ &{} + \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)} \bigl[z(p)-z(0) \bigr]^{ \alpha } + \frac{1}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr] \biggr\} , \end{aligned}$$
and consider \(B_{r}= \{ u\in {X}:\Vert u\Vert \leq {r} \} \). We define the operators
$$\begin{aligned}& (\Phi {u}) (t) = \frac{[w(t)]^{-1}}{\Gamma (\alpha)} \int^{t}_{0} \frac{w(s)z'(s)f(s,u(s))}{[z(t)-z(s)]^{1- \alpha }}\,ds, \\& (\Psi {u}) (t) = \biggl[ \frac{L^{-1}z(0)}{w(t)\Gamma (\alpha)} - \frac{L ^{-1}z(t)}{w(t)\Gamma (\alpha)} \biggr] \\& \hphantom{(\Psi {u}) (t) =}{}\times \biggl[ \int^{1}_{0}\frac{w(s)z'(s)f(s,u(s))\,ds}{w(1)[z(1)-z(s)]^{1- \alpha }} - \int^{p}_{0}\frac{\gamma w(s)z'(s)f(s,u(s))\,ds}{w(p)[z(p)-z(s)]^{1- \alpha }} \biggr] . \end{aligned}$$
For \(u_{1},u_{2}\in {B_{r}}\), simple calculation yields
$$\begin{aligned}& \bigl\Vert (\Phi {u_{1}}) (t) + (\Psi {u_{2}}) (t) \bigr\Vert \\& \quad \leq \Vert \lambda \Vert _{L^{1}[0,1]}\times \frac{w_{\max }}{w_{\min } \Gamma (\alpha +1)} \biggl\{ \bigl[z(1)-z(0) \bigr]^{\alpha } \\& \qquad {} + \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)} \bigl[z(p)-z(0) \bigr]^{ \alpha } + \frac{1}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr] \biggr\} \\& \quad \leq r. \end{aligned}$$
Therefore, \((\Phi {u_{1}})(t) + (\Psi {u_{2}})(t) \in B_{r}\). Moreover, by (H1), it is easy to verify that Ψ is a contraction mapping for
$$\begin{aligned}& \frac{w_{\max }L_{5}}{w_{\min }\Gamma (\alpha +1)} \biggl\{ \bigl[z(1)-z(0) \bigr]^{ \alpha } + \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)} \bigl[z(p)-z(0) \bigr]^{ \alpha } \\& \quad {} + \frac{1}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr] \biggr\} < 1. \end{aligned}$$
Since the weight function \(w(t)\), scale function \(z(t)\) and f are continuous, Φ is also continuous. Furthermore, Φ is uniformly bounded in \(B_{r}\) since
$$\begin{aligned} \Vert \Phi {u}\Vert \leq & \Vert \lambda \Vert _{L^{1}} \frac{[w(t)]^{-1}}{\Gamma ( \alpha)} \int^{t}_{0} \frac{w(s)z'(s)}{[z(t)-z(s)]^{1-\alpha }}\,ds \\ \leq & \Vert \lambda \Vert _{L^{1}}\cdot \frac{w_{\max }}{w_{\min }\Gamma ( \alpha +1)}\cdot \bigl[z(t)-z(0) \bigr]^{\alpha } \\ \leq &\frac{w_{\max }}{w_{\min }\Gamma (\alpha +1)}\cdot \bigl[z(1)-z(0) \bigr]^{ \alpha }\cdot \Vert \lambda \Vert _{L^{1}}. \end{aligned}$$
Next, we prove the compactness of the operator Φ. Let \(E=[0,1]\times {B_{r}}\), and denote \(f_{\max }=\sup_{(t,u)\in {E}}\vert f(t,u)\vert \), then
$$\bigl\vert (\Phi {u}) (t_{2}) - (\Psi {u}) (t_{1}) \bigr\vert \leq \frac{w_{ \max }f_{\max }}{w_{\min }\Gamma (\alpha +1)} \bigl\{ \bigl[z(t_{2})-z(0) \bigr]^{ \alpha } + \bigl[z(t_{1})-z(0) \bigr]^{\alpha } \bigr\} , $$
which is independent of u. Thus, Φ is equicontinuous. Since Φ maps bounded subsets into relatively compact subsets, one can deduce that \(\Phi (C_{bs})\) is relatively compact in X for every t, where \(C_{bs}\) is a bounded subset of C. Therefore, Φ is relatively compact on \(B_{r}\), and hence, by the Arzelà-Ascoli theorem, Φ is compact on \(B_{r}\) and conditions (H1) and (H2) are satisfied. Consequently, by Theorem 4, the boundary value problem (1) has at least one solution. □

4 Examples

We present three examples to demonstrate the main results discussed in the last section.

Example 1

Consider the generalized fractional boundary value problem
$$ \textstyle\begin{cases} {}^{*}D^{\alpha }_{0+} u(t) = f(t,u(t)),\quad t\in [0,1], 1< \alpha < 2, \\ u(0)=0, \qquad \gamma {u(p)}=u(1), \end{cases} $$
(23)
where \(\alpha =1.85\), \(\gamma =1/2\), \(p=3/4\), \(w(t)=e^{t}\), \(z(t)=t^{2}\), and \(f(t,u)=u^{2}\sin (0.5\pi {t})\). Since \(\lim_{u\rightarrow {0}}\frac{f(t,u)}{u}=0\) and taking \(0<\vert u\vert <1/5\) such that \(\vert f(t,u)\vert \leq \vert u^{2}\vert \leq \frac{1}{5}\vert u\vert \) and
$$\begin{aligned}& \max_{t\in [0,1]} \biggl\{ \biggl( \frac{L^{-1}z(0)}{w(t)\Gamma (\alpha +1)} + \frac{L^{-1}z(t)}{w(t)\Gamma (\alpha +1)} \biggr) \biggl[ \frac{ \gamma w_{\max }}{w(p)}\bigl[z(p)-z(0) \bigr]^{\alpha } \\& \qquad {} + \frac{w_{\max }}{w(1)}\bigl[z(1)-z(0)\bigr]^{\alpha } \biggr] + \frac{w _{\max }}{w(t)\Gamma (\alpha +1)}\bigl[z(t)-z(0)\bigr]^{\alpha } \biggr\} \cdot d_{1} \\& \quad < 4.5246\times \frac{1}{5}< 1. \end{aligned}$$
Therefore, by Theorem 5, the boundary value problem (23) has at least one solution.

Example 2

Consider the generalized fractional boundary value problem
$$ \textstyle\begin{cases} {}^{*}D^{\alpha }_{0+} u(t) = f(t,u(t)),\quad t\in [0,1], 1< \alpha < 2, \\ u(0)=0,\qquad \gamma {u(p)}=u(1), \end{cases} $$
(24)
where \(\alpha =1.92\), \(\gamma =1/3\), \(p=2/5\), \(w(t)=t^{3}+1\), \(z(t)=t^{1.5}\), and \(f(t,u)=\frac{\Vert u\Vert }{(\Vert u\Vert +1)(t+5)^{3}}\) such that
$$\bigl\Vert f(t,u_{1})-f(t,u_{2}) \bigr\Vert \leq \frac{1}{125}\Vert u_{1}-u_{2}\Vert , $$
and
$$\begin{aligned} \Delta =& \frac{L_{5}}{\mu }\cdot \frac{w_{\max }}{w_{\min }\Gamma ( \alpha +1)} \times \biggl\{ \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)} \bigl[z(p)-z(0) \bigr]^{ \alpha } \\ &{} + \frac{1}{w(1)} \bigl[z(1)-z(0) \bigr]^{\alpha } \biggr] + \bigl[z(1)-z(0) \bigr]^{ \alpha } \biggr\} = 0.0052 < 1, \end{aligned}$$
where we take \(\mu =0.5\). Therefore, by Theorem 6, the boundary value problem (23) has a unique solution.

Example 3

Consider the generalized fractional boundary value problem
$$ \textstyle\begin{cases} {}^{*}D^{\alpha }_{0+} u(t) = f(t,u(t)),\quad t\in [0,1], 1< \alpha < 2, \\ u(0)=0,\qquad \gamma {u(p)}=u(1), \end{cases} $$
(25)
where \(\alpha =1.73\), \(\gamma =1/4\), \(p=1/2\), \(w(t)=e^{t}+1\), \(z(t)=t^{1.2}\), and \(f(t,u)=\frac{\Vert u\Vert }{(\Vert u\Vert +1)(t+3)^{4}}\) such that
$$\begin{aligned} \bigl\Vert f(t,u_{1})-f(t,u_{2}) \bigr\Vert \leq \frac{1}{81}\Vert u_{1}-u_{2}\Vert , \end{aligned}$$
and \(\vert f(t,u)\vert \leq \lambda (t)=\frac{1}{(t+3)^{4}}\in {L^{1}([0,1],R ^{+})}\), and
$$\begin{aligned}& \frac{w_{\max }L_{5}}{w_{\min }\Gamma (\alpha +1)} \biggl\{ \bigl[z(1)-z(0)\bigr]^{ \alpha } + \frac{z(0)+z(1)}{\vert L\vert } \biggl[ \frac{\gamma }{w(p)}\bigl[z(p)-z(0)\bigr]^{ \alpha } \\& \quad {} + \frac{1}{w(1)}\bigl[z(1)-z(0)\bigr]^{\alpha } \biggr] \biggr\} =0.3310 < 1. \end{aligned}$$
Therefore, by Theorem 7, the boundary value problem (23) has at least one solution.

5 Conclusion remark

The existence results of generalized fractional boundary value problem are discussed in this paper by using several fixed point theorems. The generalized fractional derivative is defined upon a weight function and a scale function, which contains many fractional derivatives in the literature as special cases. Hence, the boundary value problems studied in this paper are more general, and it is important to develop certain methods for investigating the existence results of them. In fact, equation (10) provides us with an effective transform, under which the generalized FBVP can be regarded as a regular FBVP defined in a general time scale \(z(t)\) and weighted by a weight function \(w(t)\). We hope that our work will bring much attention into this field in the near future.

Notes

Declarations

Acknowledgements

This work was partly supported by the NSFC (No. 11501581, No. 51134003, No. 51174236), the Research Project (No. 502042032) of Central South University, the Project funded by China Postdoctoral Science Foundation (No. 2015M570683), the Project supported by the National Basic Research Development Program of China (No. 2011CB606306), the project supported by the National Key Laboratory Open Program of Porous Metal Material of China (No. PMM-SKL-4-2012).

Authors’ contributions

The authors have made the same contribution. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interest.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, Central South University

References

  1. Povstenko, Y: Fractional heat conduction equation and associated thermal stress. J. Therm. Stresses 28, 83-102 (2004) View ArticleMathSciNetGoogle Scholar
  2. Gorenflo, R, Mainardi, F: Some recent advances in theory and simulation of fractional diffusion processes. J. Comput. Appl. Math. 229, 400-415 (2009) View ArticleMATHMathSciNetGoogle Scholar
  3. Laskin, N: Fractional quantum mechanics. Phys. Rev. E 62, 3135-3145 (2008) View ArticleGoogle Scholar
  4. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  5. Bai, Z, Lv, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005) View ArticleMathSciNetGoogle Scholar
  6. Tian, Y, Bai, Z: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 59, 2601-2609 (2010) View ArticleMATHMathSciNetGoogle Scholar
  7. Ahmad, B, Sivasundaram, S: On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl. Math. Comput. 217, 480-487 (2010) View ArticleMATHMathSciNetGoogle Scholar
  8. Cabada, A, Wang, G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403-411 (2012) View ArticleMATHMathSciNetGoogle Scholar
  9. Agrawal, R, Ahmad, B: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, 1200-1214 (2011) View ArticleMATHMathSciNetGoogle Scholar
  10. Dong, X, Bai, Z, Zhang, X: Positive solutions to boundary value problems of p-Laplacian with fractional derivative. Bound. Value Probl. 2017, 5 (2017) View ArticleMATHMathSciNetGoogle Scholar
  11. Benchohra, M, Hamani, S, Ntouyas, S: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71, 2391-2396 (2009) View ArticleMATHMathSciNetGoogle Scholar
  12. Agrawal, R, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973-1033 (2010) View ArticleMATHMathSciNetGoogle Scholar
  13. Qiao, Y, Zhou, Z: Existence of solutions for a class of fractional differential equations with integral and anti-periodic boundary conditions. Bound. Value Probl. 2017, 11 (2017) View ArticleMATHMathSciNetGoogle Scholar
  14. Wang, J, Zhou, Y, Lin, Z: On a new class of impulsive fractional differential equations. Appl. Math. Comput. 242, 649-657 (2014) View ArticleMATHMathSciNetGoogle Scholar
  15. Xu, Y: Fractional boundary value problems with integral and anti-periodic boundary conditions. Bull. Malays. Math. Sci. Soc. 39, 571-587 (2016) View ArticleMATHMathSciNetGoogle Scholar
  16. Zhang, K: On a sign-changing solution for some fractional differential equations. Bound. Value Probl. 2017, 59 (2017) View ArticleMATHMathSciNetGoogle Scholar
  17. Agrawal, O: Some generalized fractional calculus operators and their applications in integral equations. Fract. Calc. Appl. Anal. 15, 700-711 (2012) View ArticleMATHMathSciNetGoogle Scholar
  18. Zima, M: Fixed point theorem of Leggett-Williams type and its application. J. Math. Anal. Appl. 299, 254-260 (2004) View ArticleMATHMathSciNetGoogle Scholar
  19. Kilbas, A, Srivastava, H, Trujillo, J: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
  20. Lakshmikantham, V, Leela, S, Devi, J: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009) MATHGoogle Scholar
  21. Deimling, K: Nonlinear Functional Analysis. Springer, Berlin (1985) View ArticleMATHGoogle Scholar
  22. Smart, D: Fixed Point Theorems. Cambridge University Press, Cambridge (1980) MATHGoogle Scholar

Copyright

© The Author(s) 2017