Abstract
In this paper using topological degree we study the existence of nontrivial solutions for a fractional differential equation involving integral boundary conditions. Here, the nonlinear term may be sign-changing and may also depend on the derivatives of the unknown function.
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1 Introduction
We study the existence of nontrivial solutions for the following integral boundary value problem involving Riemann–Liouville fractional derivatives:
where \(\alpha ,\beta \in (2,3]\) are two real numbers, \(D_{0+}^{\alpha }\), \(D_{0+}^{\beta }\) are the Riemann–Liouville fractional derivatives, and \(f\in C( [0,1]\times \mathbb{R}^{3}, \mathbb{R})\) (\(\mathbb{R}=(- \infty ,+\infty )\)). Moreover, the functions g, h are defined on \([0,1]\) and satisfy the condition:
-
(H0)
\(g,h\ge 0\) with \(\int _{0}^{1} g(t)t^{\alpha -2}\,dt\in [0,1)\), and \(\int _{0}^{1} h(t) t^{\beta -2} \,dt\in [0,1)\).
Fractional-order problems arise naturally in engineering and scientific disciplines such as physics, biophysics, chemistry, control theory, signal and image processing, and aerodynamics; we refer the reader to [1,2,3]. For example, in [4, 5] the authors introduced a fractional-order model of infection of CD4+ T-cells, and the system takes the following form:
where \(D^{\alpha _{i}}\) are fractional derivatives, \(i=1,2,3\). Many results on the existence and multiplicity of solutions (or positive solutions) of nonlinear fractional differential equations can be found for example in [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] and the references therein. In [6,7,8,9,10,11, 19, 20, 26], the authors used the fixed point index theory to study the existence of (positive) solutions for various boundary value problems of fractional differential equations, for example, Bai in [6] obtained positive solutions for the nonlocal fractional-order differential equation boundary value problem
where \(1<\alpha \le 2\), \(0<\beta \eta ^{\alpha -1}<1\), \(0<\eta <1\), f is continuous on \([0,1]\times \mathbb{R}^{+}\) and satisfies the conditions
and
where \(\lambda _{1}\) is the first eigenvalue corresponding of the relevant linear operator. These conditions can also be found in some integer-order differential equations; we refer the reader to [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52] and the references therein. Integral boundary conditions arise in thermal conduction problems, semiconductor problems and hydrodynamic problems (see [11]) and we refer the reader to [10, 11, 18, 19, 26, 28,29,30, 33, 34, 39, 43, 50,51,52] and the references therein. In [26] the authors studied the integral boundary value problem of the nonlinear Hadamard fractional differential equation
where \(\alpha \in (2,3]\), \(\beta \in (1,2]\) and \(D^{\alpha }\), \(D^{\beta }\) are Hadamard fractional derivatives.
Many papers in the literature also considered sign-changing nonlinearity problems; see [4, 5, 7,8,9, 13,14,15,16,17,18,19, 22,23,24,25,26,27,28,29,30, 34, 35, 38,39,40,41,42,43,44,45,46,47,48,49,50,51,52] and the references therein. In [15], the authors studied the fractional differential equation with a singular decreasing nonlinearity and a p-Laplacian operator:
Using a double iterative technique, they showed that the above problem has a unique positive solution, and from an iterative technique, they established an appropriate sequence, which converges uniformly to the unique positive solution.
In this paper we use topological degree theory to consider the existence of nontrivial solutions for (1.1). The novelty is twofold: (1) the nonlinearity depends on the unknown function u and its integer-order, fractional-order derivatives \(u'\), \(-D_{0+}^{\alpha }u\), (2) the nonlinearity can be unbounded on \([0,1]\times \mathbb{R}^{3}\), which improves results on semipositone problems (i.e., boundedness from below); see [5, 21, 42, 43, 45].
2 Preliminaries
We present some definitions and notations from fractional calculus theory involving Riemann–Liouville fractional derivatives; for details see the books [1,2,3].
Definition 2.1
The Riemann–Liouville fractional derivative of order \(\alpha > 0\) of a continuous function \(f : (0,+\infty )\to (-\infty ,+\infty ) \) is given by
where \(n =[\alpha ] + 1\), \([\alpha ]\) denotes the integer part of number α, provided that the right side is pointwise defined on \((0,+\infty )\).
Definition 2.2
The Riemann–Liouville fractional integral of order \(\alpha > 0\) of a function \(f : (0,+\infty )\to (-\infty , +\infty )\) is given by
provided that the right side is pointwise defined on \((0,+\infty )\).
Lemma 2.3
(see [5, Lemma 2.3])
Let \(\alpha >0\), then, for \(u, D_{0+}^{\alpha }u\in C(0,1) \cap L(0,1)\), we have
where N is the smallest integer greater than or equal to α.
Lemma 2.4
Suppose that (H0) holds. If let \(-D^{\alpha }_{0+}u:=v\), then the fractional boundary value problem
can be transformed into its equivalent Hammerstein integral equation, which takes the form
where
Proof
From Lemma 2.3 we have
Note that \(u(0)=u'(0)=0\), and we obtain \(c_{2}=c_{3}=0\). Then we have
Therefore, from the condition \(u'(1)=\int _{0}^{1} g(t)u'(t)\,dt\), we have the equation
Solving this, we have
As a result, we obtain
This completes the proof. □
For convenience, let
Lemma 2.5
Suppose that (H0) holds. If α, β, f are as in (1.1), then the fractional boundary value problem (1.1) is equivalent to the Hammerstein integral equation
where
Proof
Substituting \(-D^{\alpha }_{0+}u=v\) into (1.1), we have
Using \(\widetilde{f}(t)\) to replace \(f(t,\cdot ,\cdot ,\cdot )\), and from Lemma 2.3 we obtain
Note that \(v(0)=v'(0)=0\) implies \(c_{2}=c_{3}=0\). Hence,
From the condition \(v'(1)=\int _{0}^{1} h(t) v'(t)\,dt\), we get
Solving this equation, we obtain
Consequently, we have
This completes the proof. □
Lemma 2.6
The functions \(G_{i}\), \(H_{1}\) (\(i=1,2\)) satisfy the properties:
-
(i)
\(G_{i}(t,s),H_{1}(t,s)\ge 0\) for \(t,s\in [0,1]\times [0,1]\),
-
(ii)
\(t^{\beta -1}\phi _{\beta }(s)\le H_{1}(t,s)\le \phi _{\beta }(s) \) for \(t,s\in [0,1]\times [0,1]\), where \(\phi _{\beta }(s)=\frac{s(1-s)^{ \beta -2}}{\varGamma (\beta )}+\frac{\int _{0}^{1} h(t)\widetilde{H}_{2}(t,s) \,dt}{1-\int _{0}^{1} h(t) t^{\beta -2} \,dt}\), \(s \in [0,1]\).
Proof
We only prove (ii). From [20] we have
Combining this with (2.5), we easily obtain the inequalities in (ii). This completes the proof. □
Let \(E:=C[0,1]\), \(\|v\|:=\max_{t\in [0,1]}|v(t)|\) (here \(v\in E\)), \(P:=\{v\in E:v(t)\geqslant 0,\forall t\in [0,1]\}\). Then \((E,\|\cdot \|)\) is a real Banach space, and P is a cone on E. Now, we define an operator \(A: E\to E\) as follows:
for all \(v\in E\). Moreover, we note that the continuity of \(G_{1}\), \(G _{2}\), \(H_{1}\) and f implies that \(A:E\rightarrow E\) is a completely continuous operator. Note that the existence of solutions of (2.6) is equivalent to that of fixed points of A, and then from (1.1) and (2.6) (\(-D^{\alpha }_{0+}u=v\)), we see that if there exists a \(\overline{y}\in E\) such that \(A\overline{y}= \overline{y}\), then y̅ is a solution for (1.1). Therefore, in what follows we study the existence of fixed points of A. For this purpose we need to define a linear operator \(L_{a,b,c}: E\to E \) as follows:
where \(H_{a,b,c}(t,s):=aH_{3}(t,s)+bH_{2}(t,s)+cH_{1}(t,s)\), \(\forall t,s\in [0,1]\), with \(a, b, c\geqslant 0\) and \(a^{2}+b^{2}+c^{2}\neq 0\); here
Moreover, we know that the continuity of \(H_{i}\) (\(i=1,2,3\)) implies that \(L_{a,b,c}\) is a completely continuous operator and \(L_{a,b,c}(P) \subset P\). Let \(r(L_{a,b,c})\) denote the spectral radius of \(L_{a,b,c}\), and from Gelfand’s theorem we see that \(r(L_{a,b,c})>0\) (the proof is standard; see [20, Lemma 5]).
Let \(P_{0}=\{v\in P: v(t)\ge t^{\beta -1}\|v\|, \forall t\in [0,1]\}\). Then if we define an operator \((A_{1}v)(t)=\int _{0} ^{1} H_{1}(t,s)v(s)\,ds\), where \(H_{1}\) is defined by (2.5), and from Lemma 2.6(ii) we have
Indeed, if \(v\in P\), Lemma 2.6(ii) implies that
and
Lemma 2.7
(see [53, Theorem 19.3])
Let P be a reproducing cone in a real Banach space E and let \(L : E \to E\) be a compact linear operator with \(L(P) \subset P\). Let \(r(L)\) be the spectral radius of L. If \(r(L)>0\), then there exists \(\varphi \in P\setminus \{0\}\) such that \(L\varphi =r(L)\varphi \).
Therefore, from Lemma 2.7 we see that there exists \(\varphi _{a,b,c} \in P\setminus \{0\}\) such that
In what follows, we prove that
Indeed, from (2.10) we have
Using Lemma 2.6(ii) and the definitions of \(H_{i}\) (\(i=1,2,3\)), we have
and
Therefore, (2.11) is true.
Lemma 2.8
(see [54])
Let E be a Banach space and Ω a bounded open set in E. Suppose that \(A: \varOmega \to E\) is a continuous compact operator. If there exists \(u_{0}\in E\setminus \{0\}\) such that
then the topological degree \(\deg (I-A,\varOmega ,0)=0\).
Lemma 2.9
(see [54])
Let E be a Banach space and Ω a bounded open set in E with \(0\in \varOmega \). Suppose that \(A: \varOmega \to E\) is a continuous compact operator. If
then the topological degree \(\deg (I-A,\varOmega ,0)=1\).
3 Main results
Let \(\alpha _{i}, \beta _{i}, \gamma _{i}\geqslant 0\) (\(i=1,2\)) with \(\alpha _{1}^{2}+\beta _{1}^{2}+\gamma _{1}^{2} \neq 0\), \(\alpha _{2}^{2}+\beta _{2}^{2}+\gamma _{2}^{2} \neq 0\), and \(r^{-1}(L_{\alpha _{i},\beta _{i},\gamma _{i}})= \lambda _{\alpha _{i},\beta _{i},\gamma _{i}}\) for \(i=1,2\). Now, we list our assumptions for f as follows:
-
(H1)
\(f\in C([0,1]\times \mathbb{R}^{3},\mathbb{R})\).
-
(H2)
There exist two nonnegative functions \(b(t),c(t)\in C[0,1]\) with \(c(t)\not \equiv 0\) and a function \(K(x_{1},x_{2},x_{3})\in C[ \mathbb{R}^{3}, \mathbb{R}^{+}]\) such that
$$ f(t,x_{1},x_{2},x_{3})\geqslant -b(t)-c(t)K(x_{1},x_{2},x_{3}), \quad \forall x_{i}\in \mathbb{R}, t\in [0,1], i=1,2,3. $$ -
(H3)
\(\lim_{\alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+\gamma _{1}|x_{3}|\to +\infty } \frac{K(x_{1},x_{2},x_{3})}{\alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+ \gamma _{1}|x_{3}|}=0\).
-
(H4)
\(\liminf_{\alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+\gamma _{1}|x_{3}|\to + \infty }\frac{f(t,x_{1},x_{2},x_{3})}{\alpha _{1}|x_{1}|+\beta _{1}|x _{2}|+\gamma _{1}|x_{3}|}>\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}\), uniformly for \(t\in [0,1]\).
-
(H5)
\(\limsup_{\alpha _{2}|x_{1}|+\beta _{2}|x_{2}|+\gamma _{2}|x_{3}|\to 0}\frac{|f(t,x _{1},x_{2},x_{3})|}{\alpha _{2}|x_{1}|+\beta _{2}|x_{2}|+\gamma _{2}|x _{3}|}<\lambda _{\alpha _{2},\beta _{2},\gamma _{2}}\), uniformly for \(t\in [0,1]\).
We now present our main result.
Theorem 3.1
Suppose that (H0)–(H5) hold. Then (1.1) has at least one nontrivial solution.
Proof
From (H4) there exist \(\varepsilon _{0}>0\) and \(X_{0}>0\) such that
For any given ε with \(\varepsilon _{0} - \|c\|\varepsilon >0\), and from (H3) there exists \(X_{1}>X_{0}\) such that
It follows from (H2), (3.1), (3.2) that
Let \(C_{X_{1}}= (\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}+\varepsilon _{0}-\varepsilon \|c\| )X_{1}+\max_{0\leqslant t\leqslant 1, \alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+\gamma _{1}|x_{3}|\leqslant X_{1}}|f(t,x_{1},x_{2},x_{3})|\), \(K^{*}=\max_{\alpha _{1}|x_{1}|+\beta _{1}|x_{2}|+\gamma _{1}|x_{3}|\leqslant X_{1}}K(x_{1},x_{2},x_{3})\). Then it easy to see that
Note that ε can be chosen arbitrarily small, and we let
where \(M_{\alpha _{1},\beta _{1},\gamma _{1}}=\int _{0}^{1} \phi _{h}(s) (\alpha _{1}\int _{0}^{1}G_{1}(s,\tau )\,d\tau +\beta _{1}\int _{0} ^{1}G_{2}(s,\tau )\,d\tau +\gamma _{1} ) \,ds\), and \(\phi _{h}(s)=\frac{(1-s)^{ \beta -2}}{\varGamma (\beta )}+\frac{\int _{0}^{1} h(t)\widetilde{H}_{2}(t,s) \,dt}{1-\int _{0}^{1} h(t) t^{\beta -2} \,dt}\), \(s \in [0,1]\).
Now we prove that
where \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\) is the positive eigenfunction of \(L_{\alpha _{1},\beta _{1},\gamma _{1}}\) corresponding to the eigenvalue \(\lambda _{\alpha _{1},\beta _{1},\gamma _{1}}\), and then \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}= \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}L_{\alpha _{1},\beta _{1}, \gamma _{1}}\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\) and \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\in P_{0}\) by (2.11).
Suppose (3.5) is not true. Then there exists \(v_{0}\in \partial B_{R}\) and \(\mu _{0}>0\) such that
Let
Then we have
Consequently, we have
Note that \(\widetilde{v}\in P_{0}\), and then from (2.9), \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\in P_{0}\), and
we have
using the fact that
Therefore, (2.10), (3.4) and (3.7) enable us to obtain
From the definition of operator \(L_{\alpha _{1},\beta _{1},\gamma _{1}}\), we get
From (3.10), we get \(v_{0}(t)+\widetilde{v}(t)\geqslant t^{ \beta -1}\|v_{0}+\widetilde{v}\|\geqslant t^{\beta -1}(\|v_{0}\|-\| \widetilde{v}\|)\), \(t\in [0,1]\), and hence from (3.8) we have
Combining (3.11), (3.12) and (3.13), and we obtain
Therefore, using (3.6) and (3.14) we have
Define
It is easy to see that \(\mu ^{*}\geqslant \mu _{0}\) and \(v_{0}+ \widetilde{v}\geqslant \mu ^{*}\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\). From \(\varphi _{\alpha _{1},\beta _{1},\gamma _{1}}= \lambda _{\alpha _{1},\beta _{1},\gamma _{1}}L_{\alpha _{1},\beta _{1}, \gamma _{1}} \varphi _{\alpha _{1},\beta _{1},\gamma _{1}}\), we obtain
Hence
which contradicts the definition of \(\mu ^{*}\). Therefore, (3.5) holds, and from Lemma 2.8 we obtain
From (H5) there exist \(0<\varepsilon _{1}< \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}\) and \(0< r< R\) such that
for all \(x_{i}\in \mathbb{R}\), \(i=1, 2,3\), \(t\in [0,1]\) with \(0\leqslant \alpha _{2}|x_{1}|+\beta _{2}|x_{2}|+\gamma _{2}|x_{3}|< r\). Consequently, we obtain
Now for this r, we prove that
Assume the contrary. Then there exist \(v_{0}\in \partial B_{r}\) and \(\lambda _{0}\geqslant 1\) such that \(Av_{0}=\lambda _{0}v_{0}\). Let \(\omega (t)=|v_{0}(t)|\). Then \(\omega \in \partial B_{r}\cap P\) and
By induction, we have \(\omega \leqslant ( \lambda _{\alpha _{2},\beta _{2},\gamma _{2}}-\varepsilon _{1})^{n}L^{n} _{\alpha _{2},\beta _{2},\gamma _{2}}\omega \), for \(n=1,2,\ldots \) . As a result, we have
and thus
Therefore, by Gelfand’s theorem, we have
This contradicts
Thus (3.16) holds and from Lemma 2.9 we have
Now (3.15) and (3.17) imply that
Therefore the operator A has at least one fixed point in \(B_{R} \setminus \overline{B}_{r}\). Equivalently, (1.1) has at least one nontrivial solution. This completes the proof. □
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Research supported by the National Natural Science Foundation of China (Grant No. 11601048), Natural Science Foundation of Chongqing (Grant No. cstc2016jcyjA0181), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201800533), Natural Science Foundation of Chongqing Normal University (Grant No. 16XYY24).
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Fu, Z., Bai, S., O’Regan, D. et al. Nontrivial solutions for an integral boundary value problem involving Riemann–Liouville fractional derivatives. J Inequal Appl 2019, 104 (2019). https://doi.org/10.1186/s13660-019-2058-y
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DOI: https://doi.org/10.1186/s13660-019-2058-y