Positive solutions for a system of fractional integral boundary value problem

where $\alpha \in (2,3]$ is a real number$,{\mathbf{D}}_{0+}^{\alpha }$ is the standard Riemann-Liouville fractional derivative of order α and $f_i\in C([0,1]\times {\mathbf{R}}^{+}\times {\mathbf{R}}^{+},\mathbf{R})$, $i=1,2$. ${\int }_0^1{u}_i(t)\mathrm{d}\eta (t)$ denotes the Riemann-Stieltjes integral, η is right continuous on $[0,1)$, left continuous at $t=1$, and nondecreasing on $[0,1]$, with $\eta (0)=0$.

The subject of multi-point nonlocal boundary value problems, initiated by Il’in and Moiseev [1], has been addressed by many authors. The multi-point boundary conditions appear in certain problems of thermodynamics, elasticity, and wave propagation; see [2] and the references therein. For example, the vibrations of a guy wire of a uniform cross-section and composed of N parts of different densities can be set up as a multi-point boundary value problem (see [3]); many problems in the theory of elastic stability can be handled by the method of multi-point problems (see [4]). On the other hand, we all know that the Riemann-Stieltjes integral, as in the form of ${\int }_0^{1}u(s)\mathrm{d}\eta (s)$, where η is of bounded variation, that is, dη can be a signed measure, includes as special cases the multi-point boundary value problems and integral boundary value problems. That is why many authors are particularly interested in Riemann-Stieltjes integral boundary value problems.

Meanwhile, we also note that fractional differential equation’s modeling capabilities in engineering, science, economy, and other fields, have resulted in a rapid development of the theory of fractional differential equations in the last few decades; see the recent books [5–9]. This may explain the reason why the last few decades have witnessed an overgrowing interest in the research of such problems, with many papers in this direction published. Recently, there are some papers dealing with the existence of solutions (or positive solutions) of nonlinear fractional differential equation by the use of techniques of nonlinear analysis (fixed-point theorems, Leray-Schauder theory, upper and lower solution method, etc.); for example, see [10–18] and the references therein.

However, to the best our knowledge, there are only a few papers dealing with systems with fractional boundary value problems. In [13] and [18], Bai and Su considered respectively the existence of solutions for systems of fractional differential equations, and obtained some excellent results. Motivated by the works mentioned above, in this paper, we shall discuss the existence of positive solutions for the system of fractional integral boundary value problem (1.1). It is interesting that a square function and its inverse function are used to characterize coupling behaviors of $f_i$, so that $f_i$ are allowed to grow superlinearly and sublinearly.

We first offer some definitions and fundamental facts of fractional calculus theory, which can be found in [5–9].

Definition 2.1 (see [7, 8], [[6], pp.36-37])

where $n=[\alpha ]+1$, $[\alpha ]$ denotes the integer part of number α, provided that the right-hand side is pointwise defined on $(0,+\mathrm{\infty })$.

Definition 2.2 (see [[6], Definition 2.1])

provided that the right-hand side is pointwise defined on $(0,+\mathrm{\infty })$.

From the definition of the Riemann-Liouville derivative, we can obtain the following statement.

Lemma 2.1 (see [11])

where N is the smallest integer greater than or equal to α.

Lemma 2.2 (see [11])

where N is the smallest integer greater than or equal to α.

then we present Green’s function for (2.1), and study the properties of Green’s function. In our paper, we always assume that the following two conditions are satisfied:

(H0) ${\kappa }_0:=1-{\int }_0^1{t}^{\alpha -1}\mathrm{d}\eta (t)>0$.

(H1) $h\in C([0,1]\times {\mathbf{R}}^{+},\mathbf{R})$ is bounded from below, i.e., there is a positive constant M such that $h(t,u)\ge -M$, $\mathrm{\forall }(t,u)\in [0,1]\times {\mathbf{R}}^{+}$.

where $G(t,s)$ is determined by (2.2). This completes the proof. □

Lemma 2.4 (see [[10], Lemma 3.2])

This completes the proof. □

and we easily obtain (2.13), as claimed. This completes the proof. □

Then $(E,\parallel \cdot \parallel )$ is a real Banach space and P is a cone on E.

The norm on $E\times E$ is defined by $\parallel (u,v)\parallel :=\parallel u\parallel +\parallel v\parallel$, $(u,v)\in E\times E$. Note that $E\times E$ is a real Banach space under the above norm, and $P\times P$ is a positive cone on $E\times E$.

where $G(t,s)$ is defined by (2.2).

i.e., $u_{\ast }(t)+w(t)$ satisfies (2.15). Therefore, (i) holds, as claimed. Similarly, it is easy to prove that (ii) is also satisfied. This completes the proof. □

where $G(t,s)$ is determined by (2.2). Clearly, the continuity and nonnegativity of G and F imply that $T:P\to P$ is a completely continuous operator.

Lemma 2.7 Put $P_1:=\{u\in P:u(t)\ge {\mu }^{-1}k(t)\parallel u\parallel \mathit{\text{ for }}t\in [0,1]\}$. Then $T(P)\subset P_1$, where μ and T are defined by Lemma  2.4 and (2.17), respectively.

This completes the proof. □

In this paper, we assume that $f_i$ ($i=1,2$) satisfy the following condition:

(H2) $f_i(t,x,y)\in C([0,1]\times {\mathbf{R}}^{+}\times {\mathbf{R}}^{+},\mathbf{R})$ and there is a positive constant M such that $f_i(t,x,y)\ge -M$, $\mathrm{\forall }(t,x,y)\in [0,1]\times {\mathbf{R}}^{+}\times {\mathbf{R}}^{+}$.

the function ${\tilde{f}}_i(t,x,y)=f_i(t,x,y)+M$, ${\tilde{f}}_i\in C([0,1]\times {\mathbf{R}}^{+}\times {\mathbf{R}}^{+},{\mathbf{R}}^{+})$ and $w(t)$ is denoted by (2.16). By Lemma 2.6, we know if $(u_1,u_2)$ is a solution of (2.18) and $u_i(t)\ge w(t)$, $t\in [0,1]$, then ($u_1^{\ast }=u_1-w$, $u_2^{\ast }=u_2-w$) is a positive solution of (1.1).

It is obvious that $A_i(i=1,2):P\times P\to P$, $A:P\times P\to P\times P$ are completely continuous operators. Clearly, $(u_1-w,u_2-w)\in P\times P$ is a positive solution of (1.1) if and only if $(u_1,u_2)\in (P\times P)\setminus \{0\}$ is a fixed point of A and $u_i\ge w$, $i=1,2$.

The following two lemmas play some important roles in our proofs involving fixed point index.

Lemma 2.8 ([19])

Let $\mathrm{\Omega }\subset E$ be a bounded open set, and let $A:\overline{\mathrm{\Omega }}\cap P\to P$ be a completely continuous operator. If there exists $v_0\in P\setminus \{0\}$ such that $v-Av\ne \lambda v_0$ for all $v\in \partial \mathrm{\Omega }\cap P$ and $\lambda \ge 0$, then $i(A,\mathrm{\Omega }\cap P,P)=0$.

Lemma 2.9 ([19])

Let $\mathrm{\Omega }\subset E$ be a bounded open set with $0\in \mathrm{\Omega }$. Suppose that $A:\overline{\mathrm{\Omega }}\cap P\to P$ is a completely continuous operator. If $v\ne \lambda Av$ for all $v\in \partial \mathrm{\Omega }\cap P$ and $0\le \lambda \le 1$, then $i(A,\mathrm{\Omega }\cap P,P)=1$.

We list the assumptions on $F_i$ ($i=1,2$) in this section.

We adopt the convention in the sequel that $c_1,c_2,\dots$ stand for different positive constants. We denote $B_{\rho }:=\{u\in E:\parallel u\parallel <\rho \}$ for $\rho >0$ in the sequel.

Theorem 3.1 Suppose that (H2)-(H4) hold, (1.1) has at least a positive solution.

Therefore, $\parallel u_i\parallel \ge M\mu {\mathrm{\Gamma }}^{-1}(\alpha )(\alpha -1)$ leads to $u_i(t)\ge w(t)$, $t\in [0,1]$.

By the concavity of , we have by (3.4)

$\begin{array}{rl}\sqrt{v(s)}& \ge \sqrt{v(s)+c_1}-\sqrt{c_1}\\ \ge \sqrt{{\int }_0^{1}G(s,\tau ){\xi }_2{(u(\tau )-w(\tau ))}^2\mathrm{d}\tau }-\sqrt{c_1}\\ \ge {\int }_0^1\sqrt{G(s,\tau ){\xi }_2{(u(\tau )-w(\tau ))}^2}\mathrm{d}\tau -\sqrt{c_1}\\ ={\int }_0^1\sqrt{\mu {\mathrm{\Gamma }}^{-1}(\alpha ){\xi }_2{\mu }^{-1}\mathrm{\Gamma }(\alpha )G(s,\tau )}(u(\tau )-w(\tau ))\mathrm{d}\tau -\sqrt{c_1}\\ \ge {\int }_0^1\sqrt{\mu {\mathrm{\Gamma }}^{-1}(\alpha ){\xi }_2}{\mu }^{-1}\mathrm{\Gamma }(\alpha )G(s,\tau )(u(\tau )-w(\tau ))\mathrm{d}\tau -\sqrt{c_1}\\ \ge {\mu }^{-\frac{1}{2}}{\mathrm{\Gamma }}^{\frac{1}{2}}(\alpha ){\xi }_2^{\frac{1}{2}}{\int }_0^{1}G(s,\tau )u(\tau )\mathrm{d}\tau -c_3.\end{array}$

(3.5)