Positive solutions for a system of fractional integral boundary value problem

In this paper, we shall discuss the existence of positive solutions for the 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, i.e., \eta (t) has bounded variation. By virtue of some inequalities associated with Green’s function, without the assumption of the nonnegativity of f_i, we utilize the fixed point index theory to establish our main results. In addition, 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.

MSC: 34B10, 34B18, 34A34, 45G15, 45M20.

In this paper, we study the existence of positive solutions for the 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])

The Riemann-Liouville fractional derivative of order \alpha >0 of a continuous function f:(0,+\mathrm{\infty })\to \mathbf{R} is given by

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])

The Riemann-Liouville fractional integral of order \alpha >0 of a function f:(0,+\mathrm{\infty })\to \mathbf{R} is given by

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])

Let \alpha >0. If we assume u\in C(0,1)\cap L(0,1), then the fractional differential equation D_{0+}^{\alpha }u(t)=0 has a unique solution

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

Lemma 2.2 (see [11])

Assume that u\in C(0,1)\cap L(0,1) with a fractional derivative of order \alpha >0 that belongs to C(0,1)\cap L(0,1). Then

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

In what follows, we need to consider the following fractional integral boundary value problem:

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}}^{+}.

Lemma 2.3 Let (H0), (H1) hold. Then problem (2.1) is equivalent to

where

and

Proof By Lemmas 2.1 and 2.2, we can reduce the equation of problem (2.1) to an equivalent integral equation

where c_i (i=1,2,3) are fixed constants. By u(0)=0, there is c_3=0. Thus,

Differentiating (2.5), we have

By (2.6) and u^{\prime }(0)=0, we have c_2=0. Then

From u(1)={\int }_0^{1}u(t)\mathrm{d}\eta (t), we arrive at

and thus

Therefore, we obtain by (2.7)

where H(t,s) is defined by (2.3). From (2.8), we have

and by (H0) we find

Combining (2.8) and (2.10), we see

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

Lemma 2.4 (see [[10], Lemma 3.2])

For any (t,s)\in [0,1]\times [0,1], let k(t):=t^{\alpha -1}(1-t)+{\kappa }_0^{-1}{t}^{\alpha -1}{\int }_0^1{t}^{\alpha -1}(1-t)\mathrm{d}\eta (t), \phi (t):=\frac{t{(1-t)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}, \mu =(\alpha -1)(1+\eta (1){\kappa }_0^{-1})>0. Then the following two inequalities are satisfied:

1. (i)

   k(t)\phi (s)\le G(t,s)\le \mu \phi (s),

2. (ii)

   H(t,s)\le {\mathrm{\Gamma }}^{-1}(\alpha )(\alpha -1)t^{\alpha -1}(1-t).

Proof (i) For s\le t, we have 1-s\ge 1-t, then

On the other hand, for t\le s, since \alpha >2, we have

Consequently,

Moreover, for s\le t, note that {(t-s)}^{\alpha -2}\le {(t-ts)}^{\alpha -2}, {(1-s)}^{\alpha -2}\ge {(1-s)}^{\alpha -1}, and t^{\alpha -2}\ge t^{\alpha -1}, then we find

On the other hand, for t\le s, we have

Therefore, we get

1. (ii)

   If t\le s, since \alpha >2, we have {(1-s)}^{\alpha -1}\le {(1-t)}^{\alpha -1}\le (1-t) and

   H(t,s)\le {\mathrm{\Gamma }}^{-1}(\alpha )(\alpha -1)t^{\alpha -1}{(1-s)}^{\alpha -1}\le {\mathrm{\Gamma }}^{-1}(\alpha )(\alpha -1)t^{\alpha -1}(1-t).

For s\le t, we have 1-s\ge 1-t, then by (2.12) we get

This completes the proof. □

Lemma 2.5 Let {\kappa }_1:=\frac{\alpha \mathrm{\Gamma }(\alpha +1)}{\mathrm{\Gamma }(2\alpha +2)}+\frac{{\kappa }_0^{-1}\mathrm{\Gamma }(\alpha +1){\int }_0^1{t}^{\alpha -1}(1-t)\mathrm{d}\eta (t)}{\mathrm{\Gamma }(2\alpha +1)} and {\kappa }_2:=\frac{(\alpha -1)(1+\eta (1){\kappa }_0^{-1})}{\mathrm{\Gamma }(\alpha +2)}. Then the following inequality holds:

Proof By (i) of Lemma 2.4, we have

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

Let

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.

By Lemma 2.3, we can obtain that system (1.1) is equivalent to the system of nonlinear Hammerstein integral equations

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

Lemma 2.6 (i) If u_{\ast }(t) is a positive solution of (2.1), then u_{\ast }(t)+w(t) is a positive solution of the following differential equation:

where

the function \tilde{h}(t,x)=h(t,x)+M, \tilde{h}:[0,1]\times {\mathbf{R}}^{+}\to {\mathbf{R}}^{+} is continuous,

1. (ii)

   If u(t) is a solution of (2.15) and u(t)\ge w(t), t\in [0,1], then u_{\ast }(t)=u(t)-w(t) is a positive solution of (2.1).

Proof If u_{\ast }(t) is a positive solution of (2.1), then we obtain

By a simple computation, we easily get u_{\ast }(0)+w(0)=u_{\ast }^{\prime }(0)+w^{\prime }(0)=0, u_{\ast }(1)+w(1)={\int }_0^1(u_{\ast }(t)+w(t))\mathrm{d}\eta (t) and

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. □

By Lemma 2.3, we obtain that (2.15) is equivalent to the integral equation

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.

Proof By (i) of Lemma 2.4, we easily find

and thus

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}}^{+}.

By (H2) and Lemma 2.6, (2.14) is turned into the following integral equation:

where

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).

Define the operator A as follows:

where

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.

(H3) There are c>0 and {\xi }_i>0, i=1,2, satisfying {\xi }_1{\xi }_2^{\frac{1}{2}}{\mu }^{-\frac{1}{2}}{\mathrm{\Gamma }}^{\frac{1}{2}}(\alpha ){\kappa }_1^2>1 such that

(H4) There exists Q(t):[0,1]\to (-\mathrm{\infty },+\mathrm{\infty }) such that

(H5) There are c>0 and {\xi }_i>0, i=3,4, satisfying such that

(H6) There exist Q:[0,1]\to (-\mathrm{\infty },+\mathrm{\infty }), \theta \in (0,\frac{1}{2}), and t_0\in [\theta ,1-\theta ] such that

We adopt the convention in the sequel that c_1,c_2,\dots stand for different positive constants. We denote for \rho >0 in the sequel.

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

Proof By Lemma 2.6, it suffices to find a fixed point (u_1,u_2) of A satisfying u_i(t)\ge w(t), t\in [0,1]. By Lemma 2.7, for any u_i\in P and t\in [0,1], noting (ii) of Lemma 2.4, together with

we have

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].

In what follows, we first show that there exists an adequately big positive number R>M\mu {\mathrm{\Gamma }}^{-1}(\alpha )(\alpha -1) such that the following claim holds:

where \psi \in P is a given function. Indeed, if the claim is false, there exist (u,v)\in \partial B_R\cap (P\times P) and \lambda \ge 0 such that (u,v)=A(u,v)+\lambda (\psi ,\psi ), then u\ge A_1(u,v) and v\ge A_2(u,v). In view of (H3) and the definition of A_i (i=1,2), we get

and

By the concavity of , we have by (3.4)

Combining (3.3) and (3.5), we easily find

Multiply the both sides of the above by \phi (t) and integrate over [0,1] and use Lemma 2.5 to obtain

and thus

Noting Lemma 2.7, we obtain

Hence,

On the other hand, noting (3.3), together with the concavity of , we arrive at

Multiply the both sides of the above by \phi (t) and integrate over [0,1] and use Lemma 2.5, Lemma 2.7 to obtain

Consequently,

Taking R>max\{{\mathcal{N}}_1,M\mu {\mathrm{\Gamma }}^{-1}(\alpha )(\alpha -1),{[\frac{{\mathrm{\Gamma }}^{-1}(\alpha +2)({\mathcal{N}}_1+c_2)}{{\mu }^{-1}{\xi }_1{\kappa }_1^2}]}^2\}, which contradicts (u,v)\in \partial B_R\cap (P\times P). As a result, (3.2) is true. Lemma 2.8 implies

On the other hand, by (H4), we have, for i=1,2,