Let \(\mathit{PC}(J, \mathbb{R})\) = {\(x: J\rightarrow\mathbb{R}: x(t)\) is continuous everywhere except for some \(t_{k}\) at which \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) exist and \(x(t_{k}^{-})=x(t_{k})\), \(k=1,2,\ldots,m\)}. Observe that \(\mathit{PC}(J, \mathbb{R})\) is a Banach space equipped with the norm \(\|x\|_{\mathit{PC}}=\sup\{|x(t)| : t\in J\}\).
In view of Lemma 2.5, we define an operator \(\mathcal{A}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\) by
$$\begin{aligned} \mathcal{A}x(t) =&-\frac{1}{2}\sum _{i=1}^{m} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}f \bigl(t_{i},x(t_{i})\bigr)+\varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}}x(t_{i}) \bigr) \bigr] \\ &{}- \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}f\bigl(t_{i},x(t_{i}) \bigr)+\varphi _{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}x(t_{i}) \bigr) \bigr\} \\ &{}- \frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}f \bigl(T,x(T)\bigr)+ \biggl(t-\frac{T}{2} \biggr) \Biggl[-\frac{1}{2} \sum_{i=1}^{m} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} f\bigl(t_{i},x(t_{i})\bigr) \\ &{}+\varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\} -\frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha _{m}-1}f \bigl(T,x(T)\bigr) \Biggr] \\ &{}+\sum_{i=1}^{k} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}}f \bigl(t_{i},x(t_{i})\bigr)+\varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}}x(t_{i}) \bigr) \bigr] \\ &{}+ \sum_{i=1}^{k}(t-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}f\bigl(t_{i},x(t_{i}) \bigr)+\varphi_{i}^{*} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}x(t_{i}) \bigr) \bigr\} \\ &{}+ {}_{t_{k}}I_{q_{k}}^{\alpha_{k}}f\bigl(t,x(t)\bigr), \end{aligned}$$
(3.1)
where
$$ {}_{a}I_{q}^{p} f\bigl(u,x(u)\bigr)= \frac{1}{\Gamma_{q}(p)} \int_{a}^{u}{}_{a}\bigl(u-{}_{a} \Phi _{q}(s)\bigr)_{q}^{(p-1)}f\bigl(s,x(s)\bigr)\, {}_{a}d_{q}s, $$
\(p\in\{\alpha_{0},\ldots,\alpha_{m},\alpha_{0}-1,\ldots,\alpha _{m}-1,\beta_{0},\ldots,\beta_{m-1},\gamma_{0},\ldots,\gamma_{m-1}\}\), \(q\in\{q_{0},\ldots,q_{m}\}\), \(a\in\{t_{0},\ldots,t_{m}\}\), and \(u\in \{t,t_{1},t_{2},\ldots,t_{m},T\}\).
For computational convenience, we set
$$\begin{aligned}& \Omega_{1}= \frac{3}{2}\sum _{i=1}^{m+1}\frac{(t_{i}-t_{i-1})^{\alpha _{i-1}}}{\Gamma_{q_{i-1}}(\alpha_{i-1}+1)} +\frac{3}{2}\sum _{i=1}^{m}\frac{(T-t_{i})(t_{i}-t_{i-1})^{\alpha_{i-1}-1}}{ \Gamma_{q_{i-1}}(\alpha_{i-1})} \\& \hphantom{\Omega_{1}={}}{}+\frac{T}{4}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})}, \end{aligned}$$
(3.2)
$$\begin{aligned}& \Omega_{2}= \frac{3}{2}m M_{1}+ \frac{3}{2}M_{2}\sum_{i=1}^{m}(T-t_{i})+ \frac{T}{4}m M_{2}. \end{aligned}$$
(3.3)
Now we present our first existence result for the problem (1.1), which is based on the Schauder fixed point theorem.
Theorem 3.1
Assume that
- (H1):
-
there exist continuous functions
\(a(t)\), \(b(t)\), and nonnegative constants
\(M_{1}\), \(M_{2}\)
such that
$$ \bigl\vert f(t,x)\bigr\vert \leq a(t)+b(t)\vert x\vert , \quad (t,x)\in J\times\mathbb{R} $$
(3.4)
with
\(\sup_{t\in J}|a(t)|=a_{1}\), \(\sup_{t\in J}|b(t)|=b_{1}\), and
$$ \bigl\vert \varphi_{k}(x)\bigr\vert \leq M_{1}, \qquad \bigl\vert \varphi_{k}^{*}(x)\bigr\vert \leq M_{2},\quad \forall x\in\mathbb{R}, k=1,2,\ldots,m. $$
(3.5)
Then the anti-periodic boundary value problem (1.1) has at least one solution on
J
if
$$ b_{1}\Omega_{1}< 1. $$
(3.6)
Proof
Let us define a closed ball \(B_{R}=\{x\in \mathit{PC}(J,\mathbb{R}) : \|x\|_{\mathit{PC}}\leq R\}\) with
$$ R>\frac{a_{1}\Omega_{1}+\Omega_{2}}{1-b_{1}\Omega_{1}}, $$
where \(a_{1}\), \(b_{1}\) are defined in (H1) and \(\Omega_{1}\), \(\Omega_{2}\) are, respectively, given by (3.2) and (3.3). Clearly \(B_{R}\) is a bounded, closed, and convex subset of \(\mathit{PC}(J,\mathbb{R})\). Now we show that the operator \(\mathcal{A}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\) defined by (3.1) has a fixed point in the following two steps.
Step 1. \(\mathcal{A}:B_{R}\to B_{R}\).
For any \(x\in B_{R}\), using (2.12), we have
$$\begin{aligned} \bigl\vert \mathcal{A}x(t)\bigr\vert \leq&\frac{1}{2}\sum _{i=1}^{m} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}} \bigl\vert f\bigl(t_{i},x(t_{i})\bigr)\bigr\vert +\bigl\vert \varphi_{i}\bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}}x(t_{i}) \bigr)\bigr\vert \bigr] \\ &{}+ \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}\bigl\vert f \bigl(t_{i},x(t_{i})\bigr)\bigr\vert +\bigl\vert \varphi_{i}^{*}\bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr)\bigr\vert \bigr\} \\ &{}+ \frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}}\bigl\vert f\bigl(T,x(T)\bigr)\bigr\vert +\frac {T}{2} \Biggl[ \frac{1}{2}\sum_{i=1}^{m} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} \bigl\vert f \bigl(t_{i},x(t_{i})\bigr)\bigr\vert \\ &{}+\bigl\vert \varphi_{i}^{*}\bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr)\bigr\vert \bigr\} +\frac {1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}-1} \bigl\vert f\bigl(T,x(T)\bigr)\bigr\vert \Biggr] \\ &{}+\sum_{i=1}^{k}\bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}} \bigl\vert f\bigl(t_{i},x(t_{i})\bigr)\bigr\vert +\bigl\vert \varphi_{i} \bigl({}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}}x(t_{i}) \bigr)\bigr\vert \bigr] \\ &{}+ \sum_{i=1}^{k}(t-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}\bigl\vert f \bigl(t_{i},x(t_{i})\bigr)\bigr\vert +\bigl\vert \varphi_{i}^{*}\bigl({}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}}x(t_{i}) \bigr) \bigr\vert \bigr\} \\ &{}+ {}_{t_{k}}I_{q_{k}}^{\alpha_{k}}\bigl\vert f \bigl(t,x(t)\bigr)\bigr\vert \\ \leq&\frac{1}{2}\sum_{i=1}^{m} \bigl[\bigl(a_{1}+b_{1}\|x\| _{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}1(t_{i}) +M_{1}\bigr] \\ &{}+ \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ \bigl(a_{1}+b_{1}\|x\| _{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}1(t_{i}) +M_{2} \bigr\} \\ &{}+ \frac{1}{2}\bigl(a_{1}+b_{1}\|x \|_{\mathit{PC}}\bigr){}_{t_{m}}I_{q_{m}}^{\alpha _{m}}1(T) \\ &{}+\frac{T}{2} \Biggl[\frac{1}{2}\sum _{i=1}^{m} \bigl\{ \bigl(a_{1}+b_{1} \|x\|_{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} 1(t_{i})+M_{2} \bigr\} \\ &{}+\frac{1}{2}\bigl(a_{1}+b_{1}\|x \|_{\mathit{PC}}\bigr){}_{t_{m}}I_{q_{m}}^{\alpha _{m}-1}1(T) \Biggr] \\ &{}+\sum_{i=1}^{m}\bigl[\bigl(a_{1}+b_{1} \|x\| _{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}1(t_{i})+M_{1} \bigr] \\ &{}+ \sum_{i=1}^{m}(T-t_{i}) \bigl\{ \bigl(a_{1}+b_{1}\|x\| _{\mathit{PC}}\bigr){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}1(t_{i})+M_{2} \bigr\} \\ &{}+ \bigl(a_{1}+b_{1}\|x\|_{\mathit{PC}}\bigr){}_{t_{m}}I_{q_{m}}^{\alpha_{m}}1(T) \\ =&\frac{3}{2}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha _{i-1}}}{\Gamma_{q_{i-1}}(\alpha_{i-1}+1)}\bigl(a_{1}+b_{1}\|x\|_{\mathit{PC}}\bigr)+ \frac {3}{2}mM_{1} \\ &{}+ \frac{3}{2}\sum_{i=1}^{m}(T-t_{i}) \biggl\{ \frac {(t_{i}-t_{i-1})^{\alpha_{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha _{i-1})}\bigl(a_{1}+b_{1}\|x\|_{\mathit{PC}}\bigr) +M_{2} \biggr\} \\ &{}+ \frac{T}{4}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})}\bigl(a_{1}+b_{1}\|x\|_{\mathit{PC}}\bigr)+ \frac {T}{4}M_{2}m \\ =&a_{1}\Omega_{1}+\Omega_{2}+b_{1} \|x\|_{\mathit{PC}}\Omega_{1} \leq R, \end{aligned}$$
which implies \(\|\mathcal{A}x\|_{\mathit{PC}}\leq R\). Therefore, \(\mathcal{A}:B_{R}\to B_{R}\).
Step 2. The operator
\(\mathcal{A}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\)
is completely continuous on
\(B_{R}\).
Let \(\sup_{(t,x)\in J\times B_{R}}|f(t,x)|=F_{1}\). For any \(\tau_{1}, \tau_{2}\in J_{k}\), \(k=0,1,\ldots,m\), with \(\tau_{1}<\tau_{2}\), we have
$$\begin{aligned} \bigl\vert \mathcal{A}x(\tau_{2})-\mathcal{A}x(\tau_{1}) \bigr\vert \leq&\vert \tau_{2}-\tau _{1}\vert \Biggl[ \frac{1}{2}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})} F_{1}+\frac{mM_{2}}{2} \Biggr] \\ &{}+\vert \tau_{2}-\tau_{1}\vert \sum _{i=1}^{k} \biggl[\frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})}F_{1}+M_{2} \biggr] \\ &{}+\frac{F_{1}}{\Gamma_{q_{k}}(\alpha_{k})}\biggl\vert \int_{t_{k}}^{\tau _{2}}{}_{t_{k}}(\tau_{2}-{}_{t_{k}} \Phi_{q_{k}})_{q_{k}}^{(\alpha _{k}-1)}\, {}_{t_{k}}d_{q_{k}}s \\ &{}- \int_{t_{k}}^{\tau_{1}}{}_{t_{k}}(\tau_{1}-{}_{t_{k}} \Phi _{q_{k}})_{q_{k}}^{(\alpha_{k}-1)}\, {}_{t_{k}}d_{q_{k}}s \biggr\vert , \end{aligned}$$
which is independent of x and tends to zero as \(\tau_{2}-\tau_{1}\to 0\). Therefore \(\mathcal{A}\) is equicontinuous. Thus \(\mathcal{A}B_{R}\) is relatively compact as \(\mathcal{A}B_{R}\subset B_{R}\) is uniformly bounded. In view of the continuity of f, \(\varphi_{k}\) and \(\varphi_{k}^{*}\), \(k=1,2,\ldots,m\), it is clear that the operator \(\mathcal{A}\) is continuous. Hence the operator \(\mathcal{A}:\mathit{PC}(J,\mathbb{R})\to \mathit{PC}(J,\mathbb{R})\) is completely continuous on \(B_{R}\). Applying the Schauder fixed point theorem, we deduce that the operator \(\mathcal{A}\) has at least one fixed point in \(B_{R}\). This shows that the problem (1.1) has at least one solution on J. □
In the next existence result, we make use of Leray-Schauder’s nonlinear alternative.
Lemma 3.1
(Nonlinear alternative for single valued maps) [19]
Let
E
be a Banach space, C
a closed, convex subset of
E, U
an open subset of
C, and
\(0\in U\). Suppose that
\(F:\overline{U}\to C\)
is a continuous, compact (that is, \(F(\overline{U})\)
is a relatively compact subset of
C) map. Then either
-
(i)
F
has a fixed point in
U̅, or
-
(ii)
there is a
\(u\in\partial U\) (the boundary of
U
in
C) and
\(\theta\in(0,1)\)
with
\(u=\theta F(u)\).
In the sequel, we set
$$\begin{aligned}& \Omega_{3}=\frac{3}{2}\sum _{i=1}^{m}\frac{(t_{i}-t_{i-1})^{\beta _{i-1}}}{\Gamma_{q_{i-1}}(\beta_{i-1}+1)}, \end{aligned}$$
(3.7)
$$\begin{aligned}& \Omega_{4}=\frac{3}{2}\sum _{i=1}^{m}\frac {(T-t_{i})(t_{i}-t_{i-1})^{\gamma_{i-1}}}{\Gamma_{q_{i-1}}(\gamma_{i-1}+1)} +\frac{T}{4}\sum _{i=1}^{m}\frac{(t_{i}-t_{i-1})^{\gamma _{i-1}}}{\Gamma_{q_{i-1}}(\gamma_{i-1}+1)}. \end{aligned}$$
(3.8)
Theorem 3.2
Assume that
- (H2):
-
there exist a continuous nondecreasing function
\(\psi:[0,\infty)\to(0,\infty)\), a continuous function
\(p:J\to \mathbb{R}^{+}\)
with
\(p^{*}=\sup_{t\in J}|p(t)|\)
and constants
\(M_{3},M_{4}>0\)
such that
$$ \bigl\vert f(t,x)\bigr\vert \leq p(t)\psi\bigl(\vert x\vert \bigr), \quad \forall(t,x)\in J\times \mathbb{R} $$
(3.9)
and
$$ \bigl\vert \varphi_{k}(x)\bigr\vert \leq M_{3}\vert x \vert ,\qquad \bigl\vert \varphi_{k}^{*}(x)\bigr\vert \leq M_{4}\vert x\vert ,\quad \forall x\in R, k=1,\ldots,m; $$
(3.10)
- (H3):
-
there exists a constant
\(N>0\)
such that
$$ \frac{(1-M_{3}\Omega_{3}-M_{4}\Omega_{4})N}{p^{*}\psi(N)\Omega_{1}}> 1,\qquad M_{3}\Omega_{3}+M_{4} \Omega_{4}< 1, $$
(3.11)
where
\(\Omega_{3}\), \(\Omega_{4}\)
are, respectively, given by (3.7) and (3.8).
Then the problem (1.1) has at least one solution
J.
Proof
We shall show that the operator \(\mathcal{A}\) defined by (2.5) has a fixed point. To accomplish this, for a positive number ρ, let \(B_{\rho} = \{x \in \mathit{PC}(J,\mathbb{R}): \|x\|_{\mathit{PC}} \le \rho\}\) denote a closed ball in \(\mathit{PC}(J,\mathbb{R})\). Then for \(x\in B_{\rho}\), \(t\in J\), and using (2.12), we have
$$\begin{aligned} \bigl\vert \mathcal{A}x(t)\bigr\vert \leq&\frac{1}{2}\sum _{i=1}^{m} \bigl[p^{*}\psi(\rho ){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}1(t_{i}) +\rho M_{3} {}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}}1(t_{i}) \bigr] \\ &{}+ \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ p^{*}\psi(\rho ){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}1(t_{i}) +\rho M_{4} {}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}1(t_{i}) \bigr\} \\ &{}+ \frac{1}{2}p^{*}\psi(\rho){}_{t_{m}}I_{q_{m}}^{\alpha _{m}}1(T)+ \frac{T}{2} \Biggl[\frac{1}{2}\sum_{i=1}^{m} \bigl\{ p^{*}\psi (\rho){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} 1(t_{i}) \\ &{}+\rho M_{4} {}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}1(t_{i}) \bigr\} +\frac {1}{2}p^{*}\psi(\rho){}_{t_{m}}I_{q_{m}}^{\alpha_{m}-1}1(T) \Biggr] \\ &{}+\sum_{i=1}^{m} \bigl[p^{*}\psi( \rho){}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}}1(t_{i})+\rho M_{3} {}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}}1(t_{i}) \bigr] \\ &{}+ \sum_{i=1}^{m}(T-t_{i}) \bigl\{ p^{*}\psi(\rho ){}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}1(t_{i})+ \rho M_{4} {}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}}1(t_{i}) \bigr\} \\ &{}+ p^{*}\psi(\rho){}_{t_{m}}I_{q_{m}}^{\alpha_{m}}1(T) \\ =&p^{*}\psi(\rho)\Omega_{1}+\rho M_{3}\Omega_{3}+ \rho M_{4} \Omega_{4}:=K, \end{aligned}$$
which implies that \(\|\mathcal{A}x\|_{\mathit{PC}}\leq K\).
To show that the operator \(\mathcal{A}\) maps bounded sets into equicontinuous sets of \(\mathit{PC}(J,\mathbb{R})\), we take \(\tau_{1}, \tau_{2}\in J_{k}\) for some \(k\in\{0, 1,2,\ldots, m\}\) with \(\tau_{1}<\tau_{2}\) and \(x\in B_{\rho}\). Then we have
$$\begin{aligned}& \bigl\vert \mathcal{A}x(\tau_{2})-\mathcal{A}x(\tau_{1}) \bigr\vert \\& \quad \leq \vert \tau_{2}-\tau_{1}\vert \Biggl[ \frac{p^{*}\psi(\rho)}{2}\sum_{i=1}^{m+1} \frac{(t_{i}-t_{i-1})^{\alpha_{i-1}-1}}{\Gamma _{q_{i-1}}(\alpha_{i-1})} +\frac{\rho M_{4}}{2}\sum_{i=1}^{m} \frac{(t_{i}-t_{i-1})^{\gamma _{i-1}}}{\Gamma_{q_{i-1}}(\gamma_{i-1}+1)} \Biggr] \\& \qquad {}+\vert \tau_{2}-\tau_{1}\vert \sum _{i=1}^{k} \biggl[\frac{(t_{i}-t_{i-1})^{\alpha _{i-1}-1}}{\Gamma_{q_{i-1}}(\alpha_{i-1})}p^{*}\psi(\rho)+ \rho M_{4}\frac{(t_{i}-t_{i-1})^{\gamma_{i-1}}}{\Gamma_{q_{i-1}}(\gamma _{i-1}+1)} \biggr] \\& \qquad {}+\frac{p^{*}\psi(\rho)}{\Gamma_{q_{k}}(\alpha_{k})}\biggl\vert \int _{t_{k}}^{\tau_{2}}{}_{t_{k}}( \tau_{2}-{}_{t_{k}}\Phi_{q_{k}})_{q_{k}}^{(\alpha _{k}-1)} \, {}_{t_{k}}d_{q_{k}}s - \int_{t_{k}}^{\tau_{1}}{}_{t_{k}}(\tau_{1}-{}_{t_{k}} \Phi _{q_{k}})_{q_{k}}^{(\alpha_{k}-1)}\, {}_{t_{k}}d_{q_{k}}s \biggr\vert , \end{aligned}$$
which tends to zero independent of x as \(\tau_{1}\rightarrow \tau_{2}\). Thus, by the Arzelá-Ascoli theorem, the operator \(\mathcal{A}: \mathit{PC}(J,\mathbb{R})\rightarrow \mathit{PC}(J,\mathbb{R})\) is completely continuous.
Finally, for \(\lambda\in(0,1)\), let \(x=\lambda\mathcal{A}x\). Then, as in the first step, we can get
$$ \|x\|_{\mathit{PC}}\leq p^{*}\psi\bigl(\|x\|_{\mathit{PC}}\bigr)\Omega_{1}+ \|x\|_{\mathit{PC}} M_{3}\Omega_{3}+\|x\|_{\mathit{PC}} M_{4} \Omega_{4}, $$
which can alternatively be written as
$$ \frac{(1-M_{3}\Omega_{3}-M_{4}\Omega_{4})\|x\|_{\mathit{PC}}}{p^{*}\psi(\Vert x\Vert _{\mathit{PC}})\Omega_{1}}\leq 1. $$
In view of (H3), there exists N such that \(\|x\|_{\mathit{PC}}\neq N\). We define \(\mathcal{U} = \{x \in \mathit{PC}(J,\mathbb{R}) : \|x\|_{\mathit{PC}} < N\}\). Note that the operator \(\mathcal{A} :\overline{\mathcal{U}} \to \mathit{PC}\) is continuous and completely continuous. From the choice of \(\mathcal{U}\), there is no \(x \in\partial\mathcal{U}\) such that \(x=\lambda {\mathcal{A}}x\) for some \(\lambda\in(0,1)\). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that \({\mathcal{A}}\) has a fixed point \(x \in \overline{\mathcal{U}}\) which is a solution of the problem (1.1) on J. This completes the proof. □
In the last theorem, we apply Banach’s contraction principle to establish the uniqueness of solutions for the problem (1.1).
Theorem 3.3
Assume that there exist a function
\(\mathcal{W}(t)\in C(J,\mathbb{R}^{+})\)
with
\(W=\sup_{t\in J}|\mathcal{W}(t)|\)
and positive constants
\(M_{5}\), \(M_{6}\)
such that
$$ \bigl\vert f(t,x)-f(t,y)\bigr\vert \leq\mathcal{W}(t)\vert x-y\vert , \quad \forall(t,x)\in J\times\mathbb{R} $$
(3.12)
and
$$ \begin{aligned} &\bigl\vert \varphi_{k}(x)-\varphi_{k}(y)\bigr\vert \leq M_{5}\vert x-y\vert , \\ &\bigl\vert \varphi_{k}^{*}(x)- \varphi_{k}^{*}(y)\bigr\vert \leq M_{6}\vert x-y\vert , \quad x,y\in\mathbb{R}, \end{aligned} $$
(3.13)
for
\(k=1,2,\ldots,m\). If
$$ W\Omega_{1}+M_{5}\Omega_{3}+M_{6} \Omega_{4} < 1, $$
(3.14)
then the problem (1.1) has a unique solution on
J.
Proof
For any \(x,y\in \mathit{PC}(J,\mathbb{R})\), we have
$$\begin{aligned}& \bigl\vert \mathcal{A}x(t)-\mathcal{A}y(t)\bigr\vert \\& \quad \leq \frac{1}{2}\sum_{i=1}^{m} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}}\bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert +M_{5}{}_{t_{i-1}}I_{q_{i-1}}^{\beta_{i-1}} \vert x-y\vert (t_{i}) \bigr] \\& \qquad {} + \frac{1}{2}\sum_{i=1}^{m}(T-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1}\bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert +M_{6}{}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}} \vert x-y\vert (t_{i}) \bigr\} \\& \qquad {} + \frac{1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}} \bigl\vert f(T,x)-f(T,y)\bigr\vert +\frac {T}{2} \Biggl[ \frac{1}{2}\sum_{i=1}^{m} \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha_{i-1}-1} \bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert \\& \qquad {} +M_{6}{}_{t_{i-1}}I_{q_{i-1}}^{\gamma_{i-1}} \vert x-y\vert (t_{i}) \bigr\} +\frac {1}{2}{}_{t_{m}}I_{q_{m}}^{\alpha_{m}-1} \bigl\vert f(T,x)-f(T,y)\bigr\vert \Biggr] \\& \qquad {} +\sum_{i=1}^{k} \bigl[{}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}}\bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert +M_{5}{}_{t_{i-1}}I_{q_{i-1}}^{\beta _{i-1}} \vert x-y\vert (t_{i}) \bigr] \\& \qquad {} + \sum_{i=1}^{k}(t-t_{i}) \bigl\{ {}_{t_{i-1}}I_{q_{i-1}}^{\alpha _{i-1}-1}\bigl\vert f(t_{i},x)-f(t_{i},y)\bigr\vert +M_{6}{}_{t_{i-1}}I_{q_{i-1}}^{\gamma _{i-1}} \vert x-y\vert (t_{i}) \bigr\} \\& \qquad {} + {}_{t_{k}}I_{q_{k}}^{\alpha_{k}}\bigl\vert f(t,x)-f(t,y)\bigr\vert \\& \quad \leq (W\Omega_{1}+M_{5}\Omega_{3}+M_{6} \Omega_{4})\|x-y\|_{\mathit{PC}}, \end{aligned}$$
which yields
$$ \|\mathcal{A}x-\mathcal{A}y\|_{\mathit{PC}}\leq(W\Omega_{1}+M_{5} \Omega _{3}+M_{6}\Omega_{4})\|x-y \|_{\mathit{PC}}. $$
By (3.14), we conclude that \(\mathcal{A}\) is a contraction. Thus, by Banach’s contraction mapping principle, the problem (1.1) has a unique solution on J. This completes the proof. □