1. Introduction
Fractional differential equations arise in many engineering and
scientific disciplines as the mathematical modeling of systems and
processes in the fields of physics, chemistry, aerodynamics,
electrodynamics of complex medium, polymer rheology, Bode’s analysis
of feedback amplifiers, signal and image processing, capacitor
theory, electrical circuits, electron-analytical chemistry, biology,
ow in porous media, aerodynamics, viscoelasticity, control theory,
fitting of experimental data, and so forth, and involves derivatives
of fractional order. Fractional derivatives provide an excellent
tool for the description of memory and hereditary properties of
various materials and processes (for details, see [1,2,3,4,5,6,7,8,9]). The fractional differential equations under
various conditions have been studied by ([10,11,12,13]), etc. The three point boundary value problem
given by a coupled system of FDE on the interval \((0,1)\) was studied
by Bashir [10]
\begin{equation}
\begin{cases}
D^{\alpha}u(t)=f(t,v(t),D^{p}v(t)),& t\in(0,1),\\
D^{\beta}v(t)=f(t,u(t),D^{q}u(t)),& t\in(0,1),\\
u(0)=0,~~u(1)=au(\xi),~~v(0)=0,~~v(1)=av(\xi),&\end{cases}
\end{equation}
(1)
where \(1< \alpha\), \(\beta0\), \(0< \xi< 1\),
\(\alpha-q\geq1\), \(\beta-p\geq1\), \(a\xi^{\alpha-1}< 1\) and
\(a\xi^{\beta-1}< 1\). \(D\) is the standard Riemann-Liouville fractional
derivative operator and \(f: [0,1]\times
\mathbf{R}^{2}\longrightarrow \mathbf{R}^{2}\).
Infinite systems of ODE’s was first studied by Persidskii
[14] with the aid of classical tools such as successive
approximation and the classical Banach fixed point principle. The
infinite systems of differential equations emerge in study of
various topics of nonlinear analysis. For example semidiscretization
of certain parabolic partial differential equation leads to an
infinite system of ODE [15], while modeling certain physical
phenomenon in theory of neural sets, branching process and mechanics
([16,17]), where the infinite system can be represented
as an ordinary differential equation. Consider the following
infinite system of fractional differential equations [18]
\begin{equation}
\begin{cases}
D^{\alpha}u_{i}(t)=f_{i}(t,u(t)),& t\in(0,T)\\
u_{i}(0)=u^{0}_{i}=0,\quad u_{i}(T)=au_{i}(\xi),&i=1,2,3…\\
1< \alpha< 2,\quad a\xi^{\alpha-1}< T^{\alpha-1},&
\end{cases}
\end{equation}
(2)
where each \(u_{i}(t)\) is a differentiable function of class
\(C^{[\alpha]+1}\). We will denote the sequence
\(\{u_{i}(t)\}^{\infty}_{i=1}=u(t)\),
\(\{u_{i}(0)\}^{\infty}_{i=1}=u_{0}\),
\(\{u_{i}(\xi)\}^{\infty}_{i=1}=u(\xi)\) and
\(\{f_{i}(t,u(t))\}^{\infty}_{i=1}=f(t,u(t))\) which is an element of
some Banach sequence space \((E,\|.\|)\).
Motivated by the above works, the aim of this paper is to establish
some sufficient conditions for the existence of nontrivial solution
for the fractional differential equations (FDE) as follows
\begin{equation}
\label{eq3}
\begin{cases}
D^{\alpha}u(t)=f(t,v(t),D^{\nu}v(t)),& t\in(0,T)\\
u(0)=0,\quad u(T)=au(\xi),&
\end{cases}
\end{equation}
(3)
where \(1< \alpha0\), \(\xi\in (0,T)\); \(\alpha-\mu\geq1\) and
\(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\). \(D\) is the standard
Riemann-Liouville fractional derivative operator and \(f\in
C([0,1]\times\mathbf{R}^{2},\mathbf{R})\).
This paper is organized as follows. In Section 2, we present some
definitions and lemmas that will be used to prove the results. Then,
in Section 3, we present and prove our main results which consists
of existence theorems and corollary for nontrivial solution of the
FDE 3, and we establish some existence criteria of at least one
solution by using the Leray-Schauder nonlinear alternative. Finally,
in Section 4, as an application, we give some examples to
illustrate the results we obtained.
2. Preliminaries
In this section, we introduce some necessary definitions and lemmas
of fractional calculus to facilitate the analysis of the Problem
(3). These details can be found in the recent literature, see
([
3,
7,
19,
20,
21,
22,
23]) and the references therein.
Definition 1. Let \(\alpha> 0\), \(n-1< \alpha< n\), \(n=[\alpha]+1\) and \(u\in C([0,1),
\mathbf{R})\). The Caputo derivative of fractional order \(\alpha\) for
the function \(u\) is defined by
\[^{c}D^{\alpha}u(t)=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}(t-s)^{n-\alpha-1}u^{(n)}(s)ds,\]
where \(\Gamma(\cdot)\) is the Gamma function.
Definition 2. The Riemann-Liouville fractional integral of order \(\alpha>0\) of a
function \(u: (0, \infty)\longrightarrow \mathbf{R}\) is given by
\[I^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}u(s)ds,~~t>0,\]
where \(\Gamma(\cdot)\) is the Gamma function, provided that the right
side is pointwise defined on \((0, \infty).\)
Lemma 1.([23]) Let \(\alpha, \beta>0\) and \(u\in L^{p}(0,1)\), \(1\leq p\leq +\infty\). Then the next
formulas hold;
- (i) \((I^{\beta}I^{\alpha}u)(t)=I^{\alpha+\beta}u(t),\)
- (ii) \((D^{\beta}I^{\alpha}u)(t)=I^{\alpha-\beta}u(t),\)
- (iii) \((D^{\alpha}I^{\alpha}u)(t)=u(t).\)
Lemma 2. Let \(\alpha>0\), \(n-1< \alpha0\). Then, the general solution of the
fractional differential equation \(^{c}D^{\alpha}g(t)=0\) is given by
\[g(t)=c_{0}+c_{1}t+…+c_{n-1}t^{n-1},\]
where \(c_{0}, c_{1},…, c_{n-1}\) are real constants and
\(n=[\alpha]+1.\)
Lemma 3. Assume that \(u\in C[0,1]\cap L^{1}(0,1)\) with a Caputo fractional
derivative of order \(\alpha>0\) that belongs to \(u\in C^{n}[0, 1]\),
then
\[I^{\alpha}~^{c}D^{\alpha}u(t)=u(t)+c_{0}+c_{1}t+…+c_{n-1}t^{n-1},\]
where \(c_{0}, c_{1},…, c_{n-1}\) are real constants and
\(n=[\alpha]+1.\)
Lemma 4. For \(\alpha>0\), the general solution of the fractional
differential equation \(D^{\alpha}u(t)=0\) with \(u\in C[0,1]\cap
L^{1}(0,1)\) is given by
\[u(t)=c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2}+…+c_{n}t^{\alpha-n},\]
where \(c_{i}\in\mathbf{R},~~i=1,2,…,n\). Hence for \(u\in C[0,1]\cap
L^{1}(0,1)\), we have
\[I^{\alpha}D^{\alpha}u(t)=u(t)+c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2}+…+c_{n}t^{\alpha-n}.\]
Lemma 5. Let \(y\in C([0,T])\), \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\). Then FDE
\[
\begin{cases}
D^{\alpha}u(t)=y(t),& t\in(0,T)\\
u(0)=0,\; u(T)=au(\xi), &
\end{cases}
\]
has a unique solution
\begin{eqnarray*}u(t)&=&\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\left[(t-s)^{\alpha-1}-\frac{(t(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]y(s)ds-\frac{1}{(T^{\alpha-1}
-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t}^{T}(t(T-s))^{\alpha-1}y(s)ds
\\&&+\frac{a}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(t(\xi-s))^{\alpha-1}y(s)ds.\end{eqnarray*}
Proof. ([10])The general solution of FDE is
\[u(t)=I^{\alpha}y(t)+c_{1}t^{\alpha-1}+c_{2}t^{\alpha-2},~where~~c_{1}, c_{2}\in\mathbf{R}.\]
Using the boundary conditions, we find that \(c_{2}=0\) and
\[c_{1}=-\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})}\left[\int_{0}^{T}\frac{y(s)ds}{(T-s)^{\alpha-1}\Gamma(\alpha)}-a\int_{0}^{\xi}\frac{y(s)ds}{(\xi-s)^{\alpha-1}\Gamma(\alpha)}\right].\]
Substituting \(c_{1}\) and \(c_{2}\) by their values in \(u(t)\), we
obtain the solution in the statement of the lemma. This completes
the proof.
Define the integral operator \(F: E\rightarrow E\), by
\begin{eqnarray*}Fu(t)&=&\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\left[(t-s)^{\alpha-1}-\frac{(t(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]f(s,v(s),D^{\nu}v(s))ds\\
&&-\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t}^{T}(t(T-s))^{\alpha-1}f(s,v(s),D^{\nu}v(s))ds\\
&&+\frac{a}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(t(\xi-s))^{\alpha-1}f(s,v(s),D^{\nu}v(s))ds.\end{eqnarray*}
By Lemma 5, the FDE (3) has a solution if and only if the
operator \(F\) has a fixed point in \(E\). So we only need to seek a
fixed point of \(F\) in \(E\). By Ascoli-Arzela theorem, we can prove
that \(F\) is a completely continuous operator. Now we cite the
Leray-Schauder nonlinear alternative.
Lemma 6. Let \(E\) be a Banach space and
\(\Omega\) be a bounded open subset of \(E\), \(0\in\Omega\).
\(F:\overline{\Omega}\rightarrow E\) be a completely continuous
operator. Then, either
- (i) there exists \(u\in \partial \Omega\) and \(\lambda>1\) such that
\(F(u)=\lambda u\), or
- (ii) there exists a fixed point \(u^{\ast}\in \overline {\Omega}\)
of \(F\).
3. Main results
In this section, we prove the existence of a nontrivial solution for
the FDE (3). Let \(E=C([0,T])\) with the norm
\(\|v\|=\max_{t\in[0,T]}\{|v(t)|,|D^{\nu}v(t)|\}\) for any \(v\in E\),
\(f\in C([0,T]\times\mathbf{R}^{2},\mathbf{R}).\)
Theorem 1. Suppose that \(f(t,0,0)\neq 0\), \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\), and there exist nonnegative functions \(k,h,l \in L^{1}[0,T]\) such that
\[|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in[0,T]\times \mathbf{R}^{2},\]
and
\[\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds< 1.\]
Then the FDE (3) has at least one nontrivial solution \(u^{\ast}\in
C([0,T]).\)
Proof. Let
\[A=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds,\]
and
\[B=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}l(s)ds+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}l(s)ds,\]
then \(A0\),
and, as \(l(t)\geq |f(t,0,0)|\), a.e., and \(t\in [0,T]\), so \(B>0\).
Let \(C=B(1-A)^{-1}\) and \(\Omega=\{(u, v)\in E^{2}: \|(u,
v)\|_{E^{2}}1\) such that \(Fu=\lambda u\), then
\begin{eqnarray*}\lambda C&=&\lambda \|u\|=\|Fu\|=\max_{0\leq t\leq T}|(Fu)(t)|\\
&\leq&\frac{1}{\Gamma(\alpha)}\max_{t\in[0,T]}\int_{0}^{t}\left|(t-s)^{\alpha-1}-\frac{(t(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right||f(s,v(s),D^{\nu}v(s))|ds\\
&&+\max_{t\in[0,T]}\frac{1}{|T^{\alpha-1}-a\xi^{\alpha-1}|\Gamma(\alpha)}\int_{t}^{T}(t(T-s))^{\alpha-1}|f(s,v(s),D^{\nu}v(s))|ds\\
&&+\max_{t\in[0,T]}\frac{a}{|T^{\alpha-1}-a\xi^{\alpha-1}|\Gamma(\alpha)}\int_{0}^{\xi}(t(\xi-s))^{\alpha-1}|f(s,v(s),D^{\nu}v(s))|ds\\
&\leq&\frac{1}{\Gamma(\alpha)}\int_{0}^{T}\left[(T-s)^{\alpha-1}+\frac{(T(T-s))^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})}\right]|f(s,v(s),D^{\nu}v(s))|ds\\
&&+\frac{a}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(T(\xi-s))^{\alpha-1}|f(s,v(s),D^{\nu}v(s))|ds\\
&\leq&\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)|v(s)|+h(s)|D^{\nu}v(s)|+l(s))ds\\
&&+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)|v(s)|+h(s)|D^{\nu}v(s)|+l(s))ds\\
&\leq&\left[\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))\|v\|ds\right.\\
&&\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))\|v\|ds\right]\\
&&+\left[\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}l(s)ds\right.\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}l(s)ds\right]\\
&=&A\|v\|+B.\end{eqnarray*}
Therefore, \(\lambda \leq A+\frac{B}{C}=A+\frac{B}{B(1-A)^{-1}}=A+(1-A)=1.\)
This contradicts \(\lambda>1\). By Lemma 6, \(F\) has a fixed point
\(u^{\ast}\in\overline{\Omega}\). In view of \(f(t,0,0)\neq0\), the FDE
(3) has a nontrivial solution \(u^{\ast}\in E\).
Now, we prove that the operator \(F\) is completely continuous, we
have \(B_{C}=\{v \in E: ||v||\leq C\}\) is a bounded closed
convex set of \(E\). We shall prove that \(F(B_{C})\) is
relatively compact. The proof will be done is some steps.
- (i) Let \(v\in B_{C}\), we have
\(|Fu(t)|\leq A\|v\|+B.\) Consequently \(F(B_{C})\) is uniformly bounded.
- (ii) Let us prove that \(F(B_{C})\) is equicontinuous. Let
\(t_{1}, t_{2}\in[0, T], \;\text{with}\;t_{1}< t_{2},\;\text{and}\; v\in B_{C}\), we have
\begin{eqnarray*}\bigg|Fu(t_{1})-Fu(t_{2})\bigg|&=&\bigg|\frac{1}{\Gamma(\alpha)} \int_{0}^{t_{1}}\left[(t_{1}-s)^{\alpha-1}-\frac{(t_{1}(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]
f(s,v(s), D^{\nu}v(s))ds \\
&&-\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{1}}^{T} (t_{1}(T-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\
&&+\frac{a}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi} (t_{1}(\xi-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\
&&-\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{2}}\left[(t_{2}-s)^{\alpha-1}-\frac{(t_{2}(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]f(s,v(s), D^{\nu}v(s))ds \\
&&+\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{2}}^{T} (t_{2}(T-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\
&&-\frac{a}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi} (t_{2}(\xi-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\bigg|\\
&=&\bigg|\frac{1}{\Gamma(\alpha)}
\int_{0}^{t_{1}}\left[(t_{1}-s)^{\alpha-1}-\frac{(t_{1}(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]
f(s,v(s), D^{\nu}v(s))ds \\
&&-\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{1}}^{T} (t_{1}(T-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\
&&-\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{2}}\left[(t_{2}-s)^{\alpha-1}-\frac{(t_{2}(T-s))^{\alpha-1}}{(T^{\alpha-1}-a\xi^{\alpha-1})}\right]f(s,v(s), D^{\nu}v(s))ds \\
&&+\frac{1}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{2}}^{T} (t_{2}(T-s))^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\\
&&+\frac{a(t_{1}^{\alpha-1}-t_{2}^{\alpha-1})}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1} f(s,v(s), D^{\nu}v(s))ds\bigg|\\
&\leq&\frac{1}{\Gamma(\alpha)} \int_{0}^{t_{1}}\left[(t_{1}-s)^{\alpha-1}+\frac{(t_{1}(T-s))^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})}\right]
|f(s,v(s), D^{\nu}v(s))|ds \\
&&+\frac{1}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{1}}^{T} (t_{1}(T-s))^{\alpha-1} |f(s,v(s), D^{\nu}v(s))|ds\\
&&+\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{2}}\left[(t_{2}-s)^{\alpha-1}+\frac{(t_{2}(T-s))^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})}\right]|f(s,v(s), D^{\nu}v(s))|ds\\
&&+\frac{1}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{2}}^{T} (t_{2}(T-s))^{\alpha-1} |f(s,v(s), D^{\nu}v(s))|ds\\
&&+\frac{a|t_{1}^{\alpha-1}-t_{2}^{\alpha-1}|}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1} |f(s,v(s), D^{\nu}v(s))|ds.\end{eqnarray*}
Therefore,
\begin{eqnarray*}&&\bigg|Fu(t_{1})-Fu(t_{2})\bigg|\leq\frac{1}{\Gamma(\alpha)}
\int_{t_{2}}^{t_{1}}\left[((t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha-1})\right.\left.+\frac{[(t_{1}(T-s))^{\alpha-1}-(t_{2}(T-s))^{\alpha-1}]}
{(T^{\alpha-1}+a\xi^{\alpha-1})}\right]\\&&\times|f(s,v(s), D^{\nu}v(s))|ds +\frac{1}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{t_{1}}^{t_{2}} [(t_{1}(T-s))^{\alpha-1}-(t_{2}(T-s))^{\alpha-1}]\\&&\times |f(s,v(s), D^{\nu}v(s))|ds+\frac{a|t_{1}^{\alpha-1}-t_{2}^{\alpha-1}|}{(T^{\alpha-1}-a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1} |f(s,v(s), D^{\nu}v(s))|ds.\end{eqnarray*}
Letting \(t_{1}\rightarrow t_{2}\), then \(|Fu(t_{1})-Fu(t_{2})|\) tends
to \(0\). Consequently \(F( B_{C})\) is equicontinuous. From
Ascoli-Arzela theorem, we deduce that \(F\) is a completely
continuous. This completes the proof.
Theorem 2. Suppose that \(f(t,0,0)\neq0\), \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\), and there exist
nonnegative functions \(k,h,l\in L^{1}[0,T]\) such that
\(|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in [0,T]\times\mathbf{R}^{2}.\)
If one of the following conditions is fulfilled;
-
There exists a constant \(p>1\) such that
\[\int_{0}^{1}(k(s)+h(s))^{p}ds< \left[\frac{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}
{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}\right]^{p},~\frac{1}{p}+\frac{1}{q}=1,\]
- \(k(s)+h(s)\) satisfies
\[k(s)+h(s)\leq\frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}},\quad a.e.~~~s\in [0,T],\]
\[meas\left\{s\in[0,T] : k(s)+h(s)0.\]
Then the FDE (3) has at least one nontrivial solution \(u^{\ast}\in
E.\)
Proof.
Let \(A\) be defined as in the proof of Theorem 1.
To prove Theorem 2, we only need to prove that \(A< 1\). Since
\(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\), we have
\[A=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds
+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds.\]
-
Using the Hölder inequality, we have
\begin{align*}A&\leq\left[\int_{0}^{1}(k(s)+h(s))^{p}ds\right]^{1/p}\left\{\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\left[\int_{0}^{T}\left((T-s)^{\alpha-1}\right)^{q}ds\right]^{1/q}\right.\\
&\;\;\;\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\left[\int_{0}^{\xi}\left((\xi-s)^{\alpha-1}\right)^{q}ds\right]^{1/q}\right\}\\
&\leq\left[\int_{0}^{1}(k(s)+h(s))^{p}ds\right]^{1/p}\left\{\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\left[\frac{T^{1+q(\alpha-1)}}{(1+q(\alpha-1))}\right]^{1/q}\right.\\
&\;\;\;\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\left[\frac{\xi^{1+q(\alpha-1)}}{1+q(\alpha-1)}\right]^{1/q}\right\}\\
&\leq\left[\int_{0}^{1}(k(s)+h(s))^{p}ds\right]^{1/p}\left[\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}\right.
\left.+\frac{aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}\right]\\
&\leq\frac{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}\times
\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}\\
&=1.
\end{align*}
-
In this case, we have
\begin{align*}A&\leq\frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}}\left[\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}ds\right.\\
&\;\;\;\left.+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}ds\right]
\end{align*}
\begin{align*}&\leq\frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}}\left[\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}}{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}
+\frac{aT^{\alpha-1}\xi^{\alpha}}{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}\right]\\
&\leq\frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}}
.\frac{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}}{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}=1.\end{align*}
This completes the proof.
Corollary 1. Suppose \(f(t,0,0)\neq0\), \((1+a)T^{\alpha-1}\neq0\), and there exist nonnegative
functions \(k, h, l\in L^{1}[0,T]\) such that
\(|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in[0,T]\times \mathbf{R}^{2}.\)
If one of following conditions is fulfilled;
-
There exists a constant \(p>1\) such that
\[\int_{0}^{1}(k(s)+h(s))^{p}ds< \left[\frac{(1+a)T^{\alpha-1}\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}{2(1+a)T^{\alpha-1}T^{(1+q(\alpha-1))/q}}\right]^{p},~\frac{1}{p}+\frac{1}{q}=1.\]
- \(k(s)+h(s)\) satisfies
\[k(s)+h(s)\leq\frac{\alpha\Gamma(\alpha)}{2T^{\alpha}},\quad a.e.~~~s\in [0,T],\]
\[meas\left\{s\in[0,T] : k(s)+h(s)0.\]
Then, the FDE (3) has at least one nontrivial solution \(u^{\ast}\in
E.\)
Proof. In this case, we have
\begin{align*}A&=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds
+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds\\
&\leq\frac{2T^{\alpha-1}+aT^{\alpha-1}}{(T^{\alpha-1}+aT^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds
+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+aT^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds\\
&=\frac{2(1+a)T^{\alpha-1}}{(1+a)T^{\alpha-1}\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds.\end{align*}
Proof of this Corollary 1 is similar to the proof Theorem
2. This completes the proof.
4. Applications
In order to illustrate the above results, we consider some examples.
Example 1. Consider the following system of FDE
\begin{equation}
\label{eq4}\left\{
\begin{array}{ll}
D^{3/2}u(t)=\frac{t}{207}v(t)+\frac{t+2}{100}D^{5/4}v(t)+t^{2}-1,\quad t\in(0,T)\\
\quad\\
u(0)=0,\quad u(T)=2 u(T/2).
\end{array}\right.
\end{equation}
(4)
Set \(\alpha=3/2\), \(a=2\), \(\xi=T/2\), and
\[f(t,x,y)=\frac{t}{207}x(t)+\frac{t+2}{100}y(t)+t^{2}-1,\]
\[k(t)=\frac{t}{100},\quad h(t)=\frac{t+2}{100},\quad l(t)=t^{2}.\]
It is easy to prove that \(k, h, l\in L^{1}[0,T]\) are nonnegative
functions, and
\[|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in [0,T]\times\mathbf{R}^{2},\]
and
\[T^{\alpha-1}+a\xi^{\alpha-1}=(1+\frac{2}{2^{1/2}})T^{1/2}\neq0.\]
Moreover, we have
\[A=\frac{2T^{\alpha-1}+a\xi^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{T}(T-s)^{\alpha-1}(k(s)+h(s))ds+\frac{aT^{\alpha-1}}{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)}\int_{0}^{\xi}(\xi-s)^{\alpha-1}(k(s)+h(s))ds,\]
\[A\approx 13.10^{-3}.T^{3/2}+4.10^{-3}.T^{5/2}< 1.\]
Hence, by Theorem 1, the FDE (4) has at least one nontrivial
solution \(u^{\ast}\) in \(E.\)
Example 2. Consider the following system of FDE
\begin{equation}
\label{eq5}\left\{
\begin{array}{ll}
D^{1/2}u(t)=\frac{\sqrt[3]{1+t^{5}}}{20}v(t)\sin v(t)+\frac{\sqrt[3]{1+t^{5}}}{5}D^{3/4}v(t)+\cos t-e^{t},\quad t\in(0,T)\\
\quad\\
u(0)=0,\quad u(T)=4 u(T/3).
\end{array}\right.
\end{equation}
(5)
Set \(\alpha=1/2\), \(a=4\), \(\xi=T/3\), and
\[f(t,x,y)=\frac{\sqrt[3]{1+t^{5}}}{20}x(t)\sin x(t)+\frac{\sqrt[3]{1+t^{5}}}{5}y(t)+\cos t-e^{t},\]
\[k(t)=\sqrt[3]{1+t^{5}}/10,\quad h(t)=\sqrt[3]{1+t^{5}}/4,\quad l(t)=\cos t+e^{t}.\]
It is easy to prove that \(k, h, l\in L^{1}[0,T]\) are nonnegative
functions, and
\[|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x,y)\in [0,T]\times\mathbf{R}^{2},\]
and
\[T^{\alpha-1}+a\xi^{\alpha-1}=(1+\frac{4}{3^{-1/2}})T^{-1/2}\neq0.\]
Let \(p=3,~q=3/2\), such that \(\frac{1}{p}+\frac{1}{q}=1,\) then
\[\int_{0}^{1}(k(s)+h(s))^{p}ds=\frac{2401}{48000}\approx 0.05.\]
Moreover, we have
\[\left[\frac{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}
{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}\right]^{p}\approx0.51.T^{-1/2}.\]
Therefore,
\[\int_{0}^{1}(k(s)+h(s))^{p}ds< \left[\frac{(T^{\alpha-1}+a\xi^{\alpha-1})\Gamma(\alpha)(1+q(\alpha-1))^{1/q}}
{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{(1+q(\alpha-1))/q}+aT^{\alpha-1}\xi^{(1+q(\alpha-1))/q}}\right]^{p}.\]
Hence, by Theorem 2(1), the FDE (5) has at least one
nontrivial solution \(u^{\ast}\) in \(E.\)
Example 3. Consider the following system of FDE
\begin{equation}
\label{eq6}\left\{
\begin{array}{ll}
D^{3/2}u(t)=\frac{\sqrt{t}}{2(\frac{1}{2}+v(t))}e^{|v^{2}(t)-1|}\cos v(t)+\frac{(1+t^{2})}{9+e^{t}}D^{7/3}v(t)+e^{-t}-\sin t,\quad t\in(0,T)\\
\quad\\
u(0)=0,\quad u(T)=3 u(T/4).
\end{array}\right.
\end{equation}
(6)
Set \(\alpha=3/2\), \(a=3\), \(\xi=T/4\), and
\[f(t,x,y)=\frac{\sqrt{t}}{2(\frac{1}{2}+x(t))}e^{|x^{2}(t)-1|}\cos x(t)+\frac{(1+t^{2})}{9+e^{t}}y(t)+e^{-t}-\sin t,\]
\[k(t)=\frac{\sqrt{t}}{2},\quad h(t)=\frac{(1+t^{2})}{3},\quad l(t)=e^{-t}+\sin t.\]
It is easy to prove that \(k, h, l\in L^{1}[0,T]\) are nonnegative
functions, and
\[|f(t,x,y)|\leq k(t)|x|+h(t)|y|+l(t),\quad a.e.~~(t,x)\in [0,T]\times\mathbf{R}^{2},\]
and
\[T^{\alpha-1}+a\xi^{\alpha-1}=(1+\frac{3}{4^{1/2}})T^{1/2}\neq0.\]
Moreover, we have
\[\frac{\alpha\Gamma(\alpha)(T^{\alpha-1}+a\xi^{\alpha-1})}{(2T^{\alpha-1}+a\xi^{\alpha-1})T^{\alpha}+aT^{\alpha-1}\xi^{\alpha}}=\frac{15\sqrt{\pi}}{31}T^{-3/2}.\]
Therefore,
\[k(s)+h(s)=\frac{\sqrt{s}}{2}+\frac{(1+s^{2})}{3}< \frac{15\sqrt{\pi}}{31}T^{-3/2},\quad s\in[0,T],\]
\[meas\{s\in[0,T] : k(s)+h(s)0.\]
Hence, by Theorem 2(2), the FDE (6) has at least one
nontrivial solution \(u^{\ast}\) in \(E.\)
The authors want to thank the anonymous referee for
the throughout reading of the manuscript and several suggestions
that help us improve the presentation of the paper.
All authors contributed equally to the writing of this paper. All authors read and approved the
final manuscript.
