In this section, we establish existence results for mild solutions of (3).
Theorem 3.1
Let
\(F:J\times E\rightarrow P_{ck}(E)\)
be a multifunction and
\(g_{i}:[t_{i},s_{i}]\times E\rightarrow E, i=1,2,\ldots,m\). We assume the following conditions:
-
\((H_{1})\)
:
-
For every
\(x\in E\), \(t\rightarrow F(t,x)\)
is measurable and for a.e. \(t\in J\), \(x\rightarrow F(t,x)\)
is upper semicontinuous.
-
\((H_{2})\)
:
-
There exist a function
\(\varphi\in L^{p}(J,\mathbb {R}^{+}),p>\frac{1}{\alpha}\), and a nondecreasing continuous function
\(\Omega:[0,\infty)\rightarrow[0,\infty]\)
such that, for any
\(x\in E\),
$$ \bigl\Vert F(t,x) \bigr\Vert \leq\varphi(t)\Omega\bigl( \Vert x \Vert \bigr), \quad\textit{a.e. } t\in J. $$
-
\((H_{3})\)
:
-
There exists a function
\(\beta\in L^{p}(J, \mathbb{R}^{+}),p>\frac{1}{\alpha}\), satisfying
$$ 8\eta \Vert \beta \Vert _{L^{p}(J,\mathbb{R}^{+})}< 1, $$
(8)
and for every bounded subset
\(D\subseteq E\),
$$\chi\bigl(F(t,D)\bigr)\leq\beta (t)\chi(D)\quad\textit{for a.e. }t\in J, $$
where
χ
is the Hausdorff measure of noncompactness in
E
and
\(\eta=\frac{b^{\alpha-\frac {1}{p}}}{\Gamma (\alpha)}(\frac{p-1}{\alpha p-1})^{\frac{p-1}{p}}\).
-
\((H_{4})\)
:
-
For every
\(i=1,2,\ldots,m\), \(g_{i}\)
is continuous and completely continuous and there exists a positive constant
\(h_{i}\)
such that
$$ \bigl\Vert g_{i}(t,x) \bigr\Vert \leq h_{i} \Vert x \Vert ,\quad t\in [ t_{i},s_{i}], x\in E. $$
Then problem (3) has a PC-mild solution provided that there is a
\(r>0\)
such that
$$ \Vert x_{0} \Vert +\bar{h}r+2\Omega(r)\eta \Vert \varphi \Vert _{L^{p}(J,\mathbb {R}^{+})}\leq r, $$
(9)
where
\(\bar{h}=\sum_{i=1}^{m}h_{i}\).
Proof
From \((H_{1})\) and \((H_{2})\), \(S_{F(\cdot,x(\cdot))}^{p}\) is non-empty (see [56]). Now we turn problem (3) into a fixed point problem and define a multifunction \(R:\mathrm{PC}(J,E)\rightarrow2^{\mathrm{PC}(J,E)}\) as follows: for \(x\in \mathrm{PC}(J,E)\), \(R(x)\) is the set of all functions \(y\in R(x)\) such that
$$ y(t)=\textstyle\begin{cases} x_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha -1}f(s)\,ds,\quad t\in [ 0,t_{1}], \\ g_{i}(t,x(t_{i}^{-})),\quad t\in(t_{i},s_{i}],i=1,2,\ldots,m, \\ g_{i}(s_{i},x(t_{i}^{-}))-\frac{1}{\Gamma(\alpha)} \int_{0}^{s_{i}}(s_{i}-s)^{\alpha-1}f(s)\,ds \\ \quad{}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s)\,ds,\quad t\in[ s_{i},t_{i+1}],i=1,2,\ldots,m, \end{cases} $$
(10)
where \(f\in S_{F(\cdot,x(\cdot))}^{p}\).
It is easy to see that any fixed point for R is a mild solution for (3). We prove using Lemma 2.10 that R has a fixed point.
Let \(B_{0}=\{x\in \mathrm{PC}(J,E):\Vert x\Vert\leq r\}\). Note \(B_{0}\) is a bounded, closed and convex subset of \(\mathrm{PC}(J,E)\). Now, we claim that \(R(B_{0})\subseteq B_{0}\). To prove this, let \(x\in B_{0}\) and \(y\in R(x)\). Using \((H_{2})\), (9), (10) and Hölder’s inequality we get for \(t\in [0,t_{1}]\),
$$\begin{aligned} \bigl\Vert y(t) \bigr\Vert &\leq \Vert x_{0} \Vert +\frac{\Omega(r)}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}\varphi(s)\,ds \\ &\leq \Vert x_{0} \Vert +\frac{\Omega(r)}{\Gamma(\alpha)} \Vert \varphi \Vert _{L^{p}(J, \mathbb{R}^{+})}\biggl( \int_{0}^{t}(t-s)^{\frac{(\alpha -1)p}{p-1}}\,ds \biggr)^{\frac{p-1}{p}} \\ &\leq \Vert x_{0} \Vert +\frac{\Omega(r)}{\Gamma(\alpha)} \Vert \varphi \Vert _{L^{p}(J,\mathbb{R}^{+})}b^{\alpha-\frac{1}{p}}\biggl(\frac {p-1}{\alpha p-1} \biggr)^{\frac{p-1}{p}} \\ &=\Vert x_{0}\Vert+\Omega(r)\eta \Vert \varphi \Vert _{L^{p}(J,\mathbb{R}^{+})}\leq r. \end{aligned}$$
(11)
If \(t\in(t_{i} s_{i}],=1,2,\ldots,m\), then
$$ \bigl\Vert y(t) \bigr\Vert \leq \bigl\Vert g_{i} \bigl(t,x\bigl(t_{i}^{-}\bigr)\bigr) \bigr\Vert \leq h_{i} \bigl\Vert x\bigl(t_{i}^{-}\bigr) \bigr\Vert \leq h_{i} r\leq r. $$
(12)
Similarly, for \(t\in[ s_{i} t_{i+1}]\), \(i=1,2,\ldots,m\), we get
$$\begin{aligned} &\bigl\Vert y(t) \bigr\Vert \\ &\quad\leq \bigl\Vert g_{i}\bigl(s_{i},x \bigl(t_{i}^{-}\bigr)\bigr) \bigr\Vert +\frac{1}{\Gamma(\alpha)} \int_{0}^{s_{i}}(s_{i}-s)^{\alpha-1} \bigl\Vert f(s) \bigr\Vert \,ds+\frac{1}{\Gamma (\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \bigl\Vert f(s) \bigr\Vert \,ds \\ &\quad\leq h_{i}r+\frac{\Omega(r)}{\Gamma(\alpha)} \Vert \varphi \Vert _{L^{p}(J,\mathbb{R}^{+})} \biggl[\biggl( \int_{0}^{s_{i}}(s_{i}-s)^{\frac{(\alpha -1)p}{p-1}}\,ds \biggr)^{\frac{p-1}{ p}}+\biggl( \int_{0}^{t}(t-s)^{\frac{(\alpha-1)p}{p-1}}\,ds \biggr)^{\frac{p-1}{p}}\biggr] \\ &\quad\leq h_{i} r+2\Omega(r)\eta \Vert \varphi \Vert _{L^{p}(J,\mathbb {R}^{+})} \leq r. \end{aligned}$$
(13)
Therefore \(R(B_{r})\subseteq B_{r}\).
Next, for every \(n\geq1\), set \(B_{n}=\overline{\operatorname{conv}}R(B_{n-1})\). Note that \(B_{n}\) is a non-empty, closed and convex subset of \(\mathrm{PC}(J,E)\). Moreover, \(B_{1}=\overline{\operatorname{conv}}R(B_{0})\subseteq B_{0}\). Also \(B_{2}=\overline{\operatorname{conv}} R(B_{1})\)
\(\subseteq\overline{\operatorname{conv}}R(B_{0})\subseteq B_{1}\). By induction, the sequence \((B_{n})\), \(n\geq1\) is a decreasing sequence of non-empty, closed and bounded subsets of \(\mathrm{PC}(J,E)\). Set \(B=\bigcap_{n=1}^{ \infty}B_{n}\).
Now, by arguing as in Steps 1, 3 and 6 in the proof of Theorem 3.1 in [41], we see that the values of R are closed, the set \(Z_{\mid\overline{J_{i}}}\) is equicontinuous for every \(i=0,1,2,\ldots,m\), and the graph of the multivalued function \(R_{|B}:B \rightarrow2^{B}\) is closed; here \(Z=R(B_{0})\) and
$$ Z_{\mid\overline{J_{i}}}=\bigl\{ y^{\ast}\in C(\overline {J_{i}},E):y^{\ast }(t)=y(t),t \in J_{i}, y^{\ast }(t_{i})=y\bigl(t_{i}^{+} \bigr),y\in Z\bigr\} . $$
We now show that the subset \(B=\bigcap_{n=1}^{\infty}B_{n}\) is non-empty and compact in \(\mathrm{PC}(J,E)\). From the generalized Cantor intersection theorem [43], it is enough to show that
$$ \lim_{n\rightarrow\infty}\chi_{\mathrm{PC}}(B_{n})=0, $$
(14)
where \(\chi_{\mathrm{PC}}\) is the Hausdorff measure of noncompactness on \(\mathrm{PC}(J,E)\) defined in Section 2.
Let \(n\geq1\) be a fixed natural number and \(\varepsilon>0\). In view of [58], p.125, there exists a sequence \((y_{k})\), \(k\geq1\) in \(R(B_{n-1})\) such that
$$ \chi_{\mathrm{PC}}(B_{n})=\chi_{\mathrm{PC}}R(B_{n-1}) \leq2\chi_{\mathrm{PC}}\{y_{k}:k\geq 1\}+\varepsilon. $$
From the definition of \(\chi_{\mathrm{PC}}\), the above inequality becomes
$$ \chi_{\mathrm{PC}}(B_{n})\leq2\max _{J= 0,1,\ldots,m}\chi _{i}(S_{\mid \overline{J_{i}}})+\varepsilon, $$
(15)
where \(S=\{y_{k}:k\geq1\}\) and \(\chi_{i}\) is the Hausdorff measure of noncompactness on \(C(\overline{J_{i}},E)\). Since \(B_{n\mid\overline {J_{i}}}\), \(J=0,1,\ldots, m\), is equicontinuous,
$$ \chi_{i}(S_{\mid\overline{J_{i}}})=\sup_{t\in\overline {J_{i}}}\chi \bigl(S(t)\bigr), $$
where χ is the Hausdorff measure of noncompactness on E. Therefore, using the nonsingularity of χ, (15) becomes
$$\begin{aligned} \chi_{\mathrm{PC}}(B_{n}) &\leq 2\max _{i=0,1,\ldots,m}\Bigl[\sup_{t\in \overline{J_{i}}}\chi\bigl(S(t)\bigr) \Bigr]+\epsilon \\ &= 2\sup_{t\in J}\chi\bigl(S(t)\bigr)+\varepsilon \\ &= 2\sup_{t\in J}\chi\bigl\{ y_{k}(t):k\geq1\bigr\} + \varepsilon. \end{aligned}$$
(16)
Now, since \(y_{k}\in R(B_{n-1})\), \(k\geq1\) there is \(x_{k}\in B_{n-1}\) such that \(y_{k}\in R(x_{k})\), \(k\geq1\). Recalling the definition of R for every \(k\geq1\) there is a \(f_{k}\in S_{F(\cdot,x_{k}(\cdot ))}^{p}\) such that for every \(t\in J\),
$$\begin{aligned} & \chi\bigl\{ y_{k}(t):k\geq1\bigr\} \\ &\quad \leq\textstyle\begin{cases} \frac{1}{\Gamma(\alpha)}\chi\{\int_{0}^{t}(t-s)^{\alpha -1}f_{k}(s)\,ds: k\geq1\}\quad \text{if }t\in[0,t_{1}], \\ \chi\{g_{i}(t,x_{k}(t_{i}^{-})):k\geq1\} \quad\text{if }t\in(t_{i}, s_{i}], i=1,2,\ldots,m, \\ \chi\{g_{i}(s_{i},x_{k}(t_{i}^{-})):k\geq1\} \\ \quad{}+\frac{1}{\Gamma(\alpha)}\chi\{\int_{0}^{s_{i}}(s_{i}-s)^{\alpha -1}f_{k}(s)\,ds:k\geq1\} \\ \quad{}+\frac{1}{\Gamma(\alpha)}\chi\{\int_{0}^{t}(t-s)^{\alpha -1}f_{k}(s)\,ds:k\geq1\}\quad\text{if }t\in[ s_{i}, t_{i+1}],i=1,2,\ldots,m. \end{cases}\displaystyle \end{aligned}$$
(17)
Note that, by the complete continuity of \(g_{i}\), \(i=1,2,\ldots,m\), we get
$$ \chi\bigl\{ g_{i}\bigl(t,x_{k} \bigl(t_{i}^{-}\bigr)\bigr):k\geq1\bigr\} =\chi \bigl\{ g_{i}\bigl(s_{i},x_{k}\bigl(t_{i}^{-} \bigr)\bigr):k\geq1\bigr\} =0. $$
(18)
Next, we observe that from \((H_{3})\), that for a.e. \(t\in J\)
$$\begin{aligned} \chi\bigl\{ f_{k}(t):k\geq1\bigr\} &\leq \chi\bigl\{ F \bigl(s,x_{k}(t)\bigr):k\geq1\bigr\} \\ &\leq \beta(t)\chi\bigl\{ x_{k}(t):k\geq1\bigr\} \\ &\leq \beta(t)\chi\bigl(B_{n-1}(t)\bigr) \\ &\leq \beta(t)\chi_{\mathrm{PC}}(B_{n-1}):=\gamma(t). \end{aligned}$$
(19)
Furthermore, by \((H_{2})\), for any \(k\geq1\) and for almost \(t\in J\), \(\Vert f_{k}(t) \Vert \leq\varphi(t)\Omega(r)\). Consequently, \(f_{k}\in L^{p}(J,E),k\geq1\). Note that \(\gamma\in L^{p}(J,\mathbb{R} ^{+})\). Then, by virtue of [59], Lemma 4(iii), there exists a compact \(K_{\epsilon }\subseteq E\), a measurable set \(J_{\epsilon}\subset J\), with measure less than ϵ, and a sequence of functions \(\{z_{k}^{\epsilon}\} \subset L^{p}(J,E)\) such that, for all \(s\in J,\{z_{k}^{\epsilon}(s):k\geq 1\}\subseteq K\) and
$$ \bigl\Vert f_{k}(s)-z_{k}^{\epsilon}(s) \bigr\Vert < 2\gamma(s)+\epsilon\quad\text{for every } k\geq1\text{ and every }s\in J-J_{\epsilon}. $$
(20)
Using (19) and Hölder’s inequality, we get \(k\geq1\)
$$\begin{aligned} & \biggl\Vert \int_{[0,t_{1}]-J_{\epsilon}}(t-s)^{\alpha -1}\bigl(f_{k}(s)-z_{k}^{\epsilon }(s) \bigr)\,ds \biggr\Vert \\ &\quad\leq \int_{[0,t_{1}]-J_{\epsilon}}(t-s)^{\alpha -1} \bigl\Vert f_{k}(s)-z_{k}^{\epsilon}(s) \bigr\Vert \,ds \\ &\quad\leq \bigl\Vert f_{k}-z_{k}^{\epsilon} \bigr\Vert _{L^{p}([0,t_{1}]-J_{\epsilon}, E)}\biggl( \int_{[0,t_{1}]- J_{\epsilon}}(t-s)^{\frac{(\alpha -1)p}{p-1}}\,ds\biggr)^{\frac{ p-1}{p}} \\ &\quad \leq \bigl(2 \Vert \gamma \Vert _{L^{p}([0,t_{1}]-J_{\epsilon}, \mathbb{R} ^{+})}+\epsilon b^{p} \bigr)\eta\Gamma(\alpha). \end{aligned}$$
(21)
From Hölder’s inequality we get for any \(k\geq1\),
$$\begin{aligned} & \biggl\Vert \int_{J_{\epsilon}}(t-s)^{\alpha-1}f_{k}(s)\,ds \biggr\Vert \\ &\quad \leq \Omega(r) \int_{J_{\epsilon}}(t-s)^{\alpha-1}\varphi (s)\,ds \\ &\quad\leq \Omega(r) \Vert \varphi \Vert _{L^{p}(J_{\epsilon},\mathbb {R}^{+})}\biggl( \int_{J_{\epsilon}}(t-s)^{\frac{(\alpha -1)p}{p-1}}\,ds\biggr)^{\frac{p-1}{p} } \\ &\quad \leq\Omega(r) \Vert \varphi \Vert _{L^{p}(J_{\epsilon},\mathbb{R} ^{+})}\eta \Gamma(\alpha). \end{aligned}$$
(22)
From (21) and (22) we have, for \(t\in[0,t_{1}]\),
$$\begin{aligned} &\chi\biggl(\biggl\{ \int_{0}^{t}(t-s)^{\alpha-1}f_{k}(s)\,ds :k\geq 1\biggr\} \biggr) \\ &\quad \leq \chi\biggl(\biggl\{ \int_{[0,t_{1}]-J_{\epsilon}}(t-s)^{\alpha -1}f_{k}(s)\,ds:k\geq 1 \biggr\} \biggr) \\ &\qquad{}+\chi\biggl(\biggl\{ \int_{J_{\epsilon}}(t-s)^{\alpha-1}f_{k}(s)\,ds :k\geq1 \biggr\} \biggr) \\ &\quad \leq\chi\biggl(\biggl\{ \int_{[0,t_{1}]-J_{\epsilon}}(t-s)^{\alpha -1}\bigl(f_{k}(s) -z_{k}^{\epsilon}(s)\bigr)\,ds:k\geq1\biggr\} \biggr) \\ &\qquad{}+\chi\biggl(\biggl\{ \int_{[0,t_{1}]-J_{\epsilon}}(t-s)^{\alpha -1}z_{k}^{\epsilon}(s)\,ds :k\geq1\biggr\} \biggr) \\ &\qquad{}+\chi\biggl(\biggl\{ \int_{J_{\epsilon}}(t-s)^{\alpha-1}f_{k}(s) \,ds:k\geq1 \biggr\} \biggr) \\ &\quad\leq \bigl(2 \Vert \beta \Vert _{L^{p}([0,t_{1}]-J_{\epsilon},\mathbb{R} ^{+})}\chi_{\mathrm{PC}}(B_{n-1})+ \epsilon b^{p}\bigr)\eta\Gamma(\alpha)+ \Omega(r) \Vert \varphi \Vert _{L^{p}(J_{\epsilon},\mathbb{R} ^{+})}\eta\Gamma(\alpha). \end{aligned}$$
(23)
Taking into account that ε is arbitrary, the inequality (23) gives us for all \(t\in[0,t_{1}]\),
$$ \chi\biggl(\biggl\{ \int_{0}^{t}(t-s)^{\alpha-1}f_{k}(s)\,ds:k \geq1\biggr\} \biggr)\leq2 \Vert \beta \Vert _{L^{p}([0,t_{1}],\mathbb{R} ^{+})} \chi_{\mathrm{PC}}(B_{n-1})\eta\Gamma(\alpha). $$
(24)
Similarly, we can show that if \(t\in[s_{i},t_{i+1}],i=1,2,\ldots ,m\), then
$$ \chi\biggl\{ \int_{0}^{s_{i}}(s_{i}-s)^{\alpha-1}f_{k}(s)\,ds:k \geq 1\biggr\} \leq2 \Vert \beta \Vert _{L^{p}([s_{i},t_{i+1}], \mathbb{R} ^{+})}\chi_{\mathrm{PC}}(B_{n-1}) \eta\Gamma(\alpha) $$
(25)
and
$$ \chi\biggl\{ \int_{0}^{t}(t-s)^{\alpha-1}f_{k}(s)\,ds:k \geq1\biggr\} \leq2 \Vert \beta \Vert _{L^{p}([s_{i},t_{i+1}],\mathbb{R} ^{+})} \chi_{\mathrm{PC}}(B_{n-1}) \eta\Gamma(\alpha). $$
(26)
From (17), (18), (24), (25) and (26) for every \(t\in J\),
$$ \chi\bigl\{ y_{k}(t):k\geq1\bigr\} \leq8 \Vert \beta \Vert _{L^{\frac{1}{q}}(J, \mathbb{R} ^{+})}\chi_{\mathrm{PC}}(B_{n-1})\eta. $$
(27)
Now (27), (16) and the fact that ε is arbitrary, yields
$$ \chi_{\mathrm{PC}}(B_{n})\leq8 \Vert \beta \Vert _{L^{\frac{1}{q}}(J, \mathbb{R} ^{+})}\chi_{\mathrm{PC}}(B_{n-1})\eta, $$
so
$$ 0\leq\chi_{\mathrm{PC}}(B_{n})\leq\bigl(8\eta \Vert \beta \Vert _{L^{p}(J, \mathbb{R} ^{+})}\bigr)^{n-1}\chi_{\mathrm{PC}}(B_{1}),\quad \forall n\geq1. $$
Since this inequality is true for every \(n\in\mathbb{N}\), from (8) and by passing to the limit as \(n\rightarrow+\infty \), we obtain (14).
From the generalized Cantor intersection property the set \(B=\bigcap_{n=1}^{ \infty}B_{n}\) is a non-empty and compact subset of \(\mathrm{PC}(J,E)\). Moreover, since every \(B_{n}\) is bounded, closed and convex, B is also bounded closed and convex. We now claim that \(R(B)\subseteq B\). Indeed, \(R(B)\subseteq R(B_{n})\subseteq\overline{\operatorname{conv}}R(B_{n})=B_{n+1}\), for every \(n\geq1\). Therefore, \(R(B)\subset\bigcap_{n=2}^{\infty}B_{n}\). Also \(B_{n}\subset B_{1}\) for every \(n\geq1\), so \(R(B)\subset \bigcap_{n=2}^{\infty}B_{n}=\bigcap_{n=1}^{\infty}B_{n}=B\).
Therefore, the multivalued \(R_{|B}:B\rightarrow2^{B} \) is a closed compact map with non-empty convex compact values, and hence \(u.s.c.\) From Lemma 2.10, there is a \(x\in B\) such that \(x\in R(x)\). Thus x is a PC-mild solution for (3). □
In the following theorem we prove that the set of PC-mild solutions of (3) is compact.
Theorem 3.2
If the function Ω in
\((H2)\)
is of the form
\(\Omega(r)=r+1\), then under the assumptions of Theorem
3.1
the set of solutions of (3) is a non-empty compact subset in
\(\mathrm{PC}(J,E)\)
provided that
$$ \bar{h}+2\eta \Vert \varphi \Vert _{L^{p}(J, \mathbb{R} ^{+})}< 1. $$
(28)
Proof
From Theorem 3.1 the set of solutions of (3) is non-empty. Indeed, let \(R:\mathrm{PC}(J,E)\rightarrow2^{\mathrm{PC}(J,E)}\) be defined as in Theorem 3.1 and we take r in (9) as
$$ r=\frac{ \Vert x_{0} \Vert +2\eta \Vert \varphi \Vert _{L^{p}(J,\mathbb{R} ^{+})}}{1-(\bar{h}+2\eta \Vert \varphi \Vert _{L^{p}(J,\mathbb{R} ^{+})})}. $$
Note r is well defined because of (28). From Theorem 3.1 we know that the problem (3) has a PC-mild solution in B. Now, by arguing as in the proof of Theorem 3.1, there is a non-empty convex compact subset B such that \(R_{|B}:B\rightarrow2^{B}\) is a closed compact map with non-empty convex compact values. Then \(R_{|B}\) is \(\gamma _{\mathrm{PC}}\)-condensing on every bounded subset of B. From Lemma 2.11, in order to show that the set of solutions of (3) is compact, it suffices to prove that the set of fixed points of R is bounded. Let x be a fixed point of R. Then there is an integrable selection f for \(F(\cdot,x(\cdot))\) such that
$$ x(t)=\textstyle\begin{cases} x_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha -1}f(s)\,ds,\quad t\in [ 0,t_{1}], \\ g_{i}(t,x(t_{i}^{-})),\quad t\in(t_{i},s_{i}],i=1,2,\ldots,m, \\ g_{i}(s_{i},x(t_{i}^{-}))+\frac{1}{\Gamma(\alpha)}\int_{s_{i}}^{t}(t-s)^{ \alpha-1}f(s)\,ds \\ \quad{}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s)\,ds,\quad t\in [ s_{i},t_{i+1}],i=1,2,\ldots,m. \end{cases} $$
We argue as in Theorem 3.1 and we get
$$ \Vert x \Vert _{\mathrm{PC}(J,E)}\leq \Vert x_{0} \Vert +\bar{h} \Vert x \Vert _{\mathrm{PC}(J,E)}+2\bigl( \Vert x \Vert _{\mathrm{PC}(J,E)}+1 \bigr)\eta \Vert \varphi \Vert _{L^{p}(J,\mathbb{R}^{+})}. $$
From (28), we obtain
$$ \Vert x \Vert _{\mathrm{PC}(J,E)}\leq\frac{ \Vert x_{0} \Vert +2\eta \Vert \varphi \Vert _{L^{p}(J,\mathbb{R} ^{+})}}{1-(\bar{h}+2\eta \Vert \varphi \Vert _{L^{p}(J,\mathbb{R}^{+})})}=r. $$
Finally, from Lemma 2.11, the proof is complete. □
In the following theorem we will show that if we use the Bielecki PCB-norm, then we can establish an existence result for (3) without assuming a condition similar to (9).
Theorem 3.3
Let
\(F:J\times E\rightarrow P_{ck}(E)\)
and
\(g_{i}:[t_{i},s_{i}]\times E\rightarrow E, i=1,2,\ldots,m\). Assume the following assumptions hold:
-
\((H_{5})\)
:
-
For every
\(x\in E,t \longrightarrow F(t,x)\)
is measurable.
-
\((H_{6})\)
:
-
There is a
\(\varsigma\in L^{p}(J, \mathbb{R}^{+}),p>\frac{1}{\alpha}\)
such that
-
(i)
For every
\(x,y\in E\),
$$ h\bigl(F(t,x),F(t,y)\bigr)\leq\varsigma(t) \Vert x-y \Vert ,\quad\textit{for a.e. } t\in J, $$
where
\(h: P_{ck}(E)\times P_{ck}(E)\to \mathbb {R}^{+}\)
is the Hausdorff distance.
-
(ii)
For every
\(x\in E\),
$$ \sup\bigl\{ \Vert x \Vert : x\in F(t,0)\bigr\} \leq\varsigma(t),\quad \textit{for a.e. } t\in J. $$
-
\((H_{7})\)
:
-
For all
\(i=1,2,\ldots,m\), there is a positive constant
\(\xi_{i}\)
such that, for every
\(x,y\in E\)
and every
\(t\in[ t_{i},s_{i}]\), we have
$$ \bigl\Vert g_{i}(t,x)-g_{i}(t,y)\bigr\Vert \leq\xi_{i} \Vert x-y \Vert . $$
Then (3) has a PC-mild solution.
Proof
From \((H_{5})\) and \((H_{6})\), for any \(x\in \mathrm{PC}(J,E)\), the set \(S_{F(\cdot,x(\cdot))}^{p}\) is non-empty. Consider the multifunction map \(R:\mathrm{PC}(J,E)\rightarrow2^{\mathrm{PC}(J,E)}\) as follows: for \(x\in \mathrm{PC}(J,E)\), \(R(x)\) is the set of all functions \(y\in R(x)\) such that (10) holds. It is easy to see that any fixed point for R is a PC-mild solution for (3). We now show that R satisfies the assumptions of Lemma 2.12.
Note from \((H_{6})\), for every \(n\geq1\), and for \(t\in J\),
$$\begin{aligned} \bigl\Vert F(t,x) \bigr\Vert &= h\bigl(F\bigl(t,x(t)\bigr),\{0\}\bigr) \\ &\leq h\bigl(F\bigl(t,x(t)\bigr),F(t,0)\bigr)+h\bigl(F(t,0),\{0\}\bigr) \\ &\leq \varsigma(t) \bigl\Vert x(t) \bigr\Vert +\varsigma(t) \leq \varsigma(t) \bigl(1+ \Vert x \Vert _{\mathrm{PC}(J,E)}\bigr). \end{aligned}$$
Arguing as in Step 1 in the proof of Theorem 3.3 in [41], we see that the values of R are closed.
We now show that R is contraction. Let \(x_{1},x_{2}\in \mathrm{PC}(J,E)\) and \(y_{1}\in R(x_{1})\). Then there is a \(f\in S_{F(\cdot,x_{1}(\cdot ))}^{p}\) such that
$$ y_{1}(t)=\textstyle\begin{cases} x_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha -1}f(s)\,ds,\quad t\in [0,t_{1}], \\ g_{i}(t,g_{i-1}(s_{i-1},x_{1}(t_{i-1}^{-}))-\frac{1}{\Gamma (\alpha)} \int_{0}^{s_{i-1}}(s_{i-1}-s)^{\alpha-1}f(s)\,ds\\ \quad{}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{i}}(t_{i}-s)^{\alpha -1}f(s)\,ds),\quad t\in (t_{i},s_{i}],=1,2,\ldots,m, \\ g_{i}(s_{i},x_{1}(t_{i}^{-}))-\frac{1}{\Gamma(\alpha)} \int_{0}^{s_{i}}(s_{i}-s)^{\alpha-1}f(s)\,ds\\ \quad{}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s)\,ds,\quad t\in[ s_{i},t_{i+1}],i=1,2,\ldots,m. \end{cases} $$
(29)
Next, since \(F(t,x_{2}(t))\) is compact for any \(t\in J\), the set \(U(t)=\{x\in F(t,x_{2}(t)):d(f(t),x)=d(f(t),G(t))\}\) is non-empty. According to Lemma 2.9, there is a measurable function \(h:J\rightarrow E\) such that \(h(t)\)
\(\in F(t,x_{2}(t))\), a.e. and
$$ \bigl\Vert f(t)-h(t) \bigr\Vert =d\bigl(f(t),F\bigl(t,x_{2}(t) \bigr)\bigr),\quad\text{for a.e. } t\in J. $$
Observe that from \((H_{6})\)(i), we have
$$ h\bigl(F\bigl(t,x_{2}(t)\bigr),F\bigl(t,x_{1}(t)\bigr) \bigr)\leq\varsigma(t) \bigl\Vert x_{1}(t)-x_{2}(t) \bigr\Vert ,\quad\text{for a.e. } t\in J. $$
Then \(h\in S_{F(\cdot,x_{2}(\cdot))}^{p}\) with
$$ \bigl\Vert h(t)-f(t) \bigr\Vert \leq\varsigma(t) \bigl( \bigl\Vert x_{2}(t)-x_{1}(t) \bigr\Vert \bigr),\quad\text{for a.e. }t\in J. $$
(30)
Let
$$ y_{2}(t)=\textstyle\begin{cases} x_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha -1}h(s)\,ds,\quad t\in [0,t_{1}], \\ g_{i}(t,g_{i-1}(s_{i-1},x_{2}(t_{i-1}^{-}))-\frac{1}{\Gamma (\alpha)} \int_{0}^{t_{i-1}}(t_{i-1}-s)^{\alpha-1}h(s)\,ds \\ \quad{}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{i}}(t_{i}-s)^{\alpha -1}h(s)\,ds),\quad t\in (t_{i},s_{i}],=1,2,\ldots,m, \\ g_{i}(s_{i},x_{2}(t_{i}^{-}))-\frac{1}{\Gamma(\alpha)} \int_{0}^{s_{i}}(s_{i}-s)^{\alpha-1}h(s)\,ds\\ \quad{}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}h(s)\,ds,\quad t\in[ s_{i},t_{i+1}],i=1,2,\ldots,m. \end{cases} $$
(31)
Now \(y_{2}\in R(x_{2})\) and if \(t\in[0,t_{1}]\) we get from (30), (31) and Hölder’s inequality
$$\begin{aligned} &e^{-Lt} \bigl\Vert y_{2}(t)-y_{1}(t) \bigr\Vert \\ &\quad\leq \frac{e^{-Lt}}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \bigl\Vert h(s)-f(s) \bigr\Vert \,ds \\ &\quad\leq \frac{e^{-Lt}}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha -1} \varsigma(s)e^{Ls}\bigl(e^{-Ls} \bigl\Vert x_{1}(s)-x_{2}(s) \bigr\Vert \bigr)\,ds \\ &\quad\leq \frac{e^{-Lt}}{\Gamma(\alpha)} \Vert x_{1}-x_{2} \Vert _{\mathrm{PCB}(J, E)} \int_{0}^{t}(t-s)^{\alpha-1}e^{Ls} \varsigma(s)\,ds \\ &\quad\leq \frac{1}{\Gamma(\alpha)} \Vert x_{1}-x_{2} \Vert _{\mathrm{PCB}(J, E)} \int_{0}^{t}(t-s)^{\alpha-1}e^{-L(t-s)} \varsigma(s)\,ds \\ &\quad\leq \frac{t^{\alpha-\frac{1}{p}}}{\Gamma(\alpha)}\biggl(\frac {p-1}{\alpha p-1 }\biggr)^{\frac{p-1}{p}} \Vert x_{1}-x_{2} \Vert _{\mathrm{PCB}(J, E)} \biggl( \int_{0}^{t}\bigl(e^{-L(t-s)}\varsigma(s) \bigr)^{p}\,ds\biggr)^{\frac{1}{p}}. \end{aligned}$$
(32)
Also, if \(t\in[ s_{i} t_{i+1}]\), \(i=1,2,\ldots,m\), we get
$$\begin{aligned} & \bigl\Vert y_{2}(t)-y_{1}(t) \bigr\Vert \\ &\quad\leq \bigl\Vert g_{i}\bigl(t,x_{1} \bigl(t_{i}^{-}\bigr)\bigr)-g_{i} \bigl(t,x_{2}\bigl(t_{i}^{-}\bigr)\bigr) \bigr\Vert \\ &\qquad{}+\frac{1}{\Gamma(\alpha)} \int_{0}^{s_{i}}(s_{i}-s)^{\alpha-1} \bigl\Vert h(s)-f(s) \bigr\Vert \,ds \\ &\qquad{}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \bigl\Vert h(s)-f(s) \bigr\Vert \,ds \\ &\quad\leq\xi_{i} \bigl\Vert x_{1}\bigl(t_{i}^{-} \bigr)-x_{2}\bigl(t_{i}^{-}\bigr) \bigr\Vert \\ &\qquad{}+\frac{1}{\Gamma(\alpha)} \Vert x_{1}-x_{2} \Vert _{\mathrm{PCB}(J, E)} \int_{0}^{s_{i}}(s_{i}-s)^{\alpha-1}e^{Ls} \varsigma(s)\,ds \\ &\qquad{}+\frac{1}{\Gamma(\alpha)} \Vert x_{1}-x_{2} \Vert _{\mathrm{PCB}(J, E)} \int_{0}^{t}(t-s)^{\alpha-1}e^{Ls} \varsigma(s)\,ds. \end{aligned}$$
Then
$$\begin{aligned} &e^{-Lt} \bigl\Vert y_{2}(t)-y_{1}(t) \bigr\Vert \\ &\quad\leq \biggl[e^{-Lt}e^{Lt_{i}}\xi_{i}+ \frac{s_{i}^{\alpha-\frac {1}{p}}}{\Gamma (\alpha)}\biggl(\frac{p-1}{\alpha p-1}\biggr)^{\frac{p-1}{p}}\biggl( \int_{0}^{s_{i}}\bigl(e^{-L(t-s)}\varsigma(s) \bigr)^{p}\,ds\biggr)^{\frac{1}{p}} \\ &\qquad{} +\frac{t^{\alpha-\frac{1}{p}}}{\Gamma(\alpha)}\biggl(\frac {p-1}{\alpha p-1}\biggr)^{ \frac{p-1}{p}}\biggl( \int_{0}^{t}\bigl(e^{-L(t-s)}\varsigma (s)\bigr)^{p}\,ds\biggr)^{\frac{1}{p} }\biggr] \Vert x_{1}-x_{2} \Vert _{\mathrm{PCB}(J, E)} \\ &\quad\leq \Vert x_{1}-x_{2} \Vert _{\mathrm{PCB}(J, E)} \biggl[e^{L(t_{i}-t)}\xi_{i} \\ &\qquad{}+2\frac{t^{\alpha-\frac{1}{p}}}{\Gamma(\alpha )}\biggl(\frac{p-1}{\alpha p-1} \biggr)^{\frac{p-1}{p}}\biggl( \int_{0}^{t}\bigl(e^{-L(t-s)}\varsigma (s) \bigr)^{p}\,ds\biggr)^{\frac{1}{p} }\biggr] \\ &\quad\leq \Vert x_{1}-x_{2} \Vert _{\mathrm{PCB}(J,E)} \biggl[e^{L(t_{i}-s_{i})}\xi_{i} +\frac{2 t^{\alpha-\frac{1}{p}}}{\Gamma(\alpha )}\biggl(\frac{p-1}{\alpha p-1} \biggr)^{\frac{p-1}{p}}\biggl( \int_{0}^{t}\bigl(e^{-L(t-s)}\varsigma (s) \bigr)^{p}\,ds\biggr)^{\frac{1}{p} }\biggr]. \end{aligned}$$
(33)
Similarly, if \(t\in(t_{i} s_{i}]\), \(i=1,2,\ldots,m\), we get from (28), (29), (30) and \((H_{7})\),
$$\begin{aligned} \bigl\Vert y_{2}(t)-y_{1}(t) \bigr\Vert \leq{}& \xi_{i}\biggl(\xi _{i-1} \bigl\Vert x_{1} \bigl(t_{i-1}^{-}\bigr)-x_{2}\bigl(t_{i-1}^{-} \bigr) \bigr\Vert \\ &{}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t_{i-1}}(t_{i-1}-s)^{\alpha -1} \bigl\Vert f(s)-h(s) \bigr\Vert \,ds \\ &{}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \bigl\Vert f(s)-h(s) \bigr\Vert \,ds\biggr) \\ \leq{}&\xi_{i}\biggl(\xi_{i-1} \bigl\Vert x_{1} \bigl(t_{i}^{-}\bigr)-x_{2}\bigl(t_{i}^{-} \bigr) \bigr\Vert \\ &{}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t_{i-1}}(t_{i-1}-s)^{\alpha -1} \varsigma(s)\bigl\Vert x_{2}(s)-x_{1}(s)\bigr\Vert \,ds \\ &{}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}\varsigma (s)\bigl\Vert x_{2}(s)-x_{1}(s)\bigr\Vert \,ds\biggr). \end{aligned}$$
Then, for \(t\in(t_{i},s_{i}], i=1,2,\ldots,m\),
$$\begin{aligned} &e^{-Lt} \bigl\Vert y_{2}(t)-y_{1}(t) \bigr\Vert \\ &\quad\leq e^{-Lt}\xi_{i}\xi_{i-1} \bigl\Vert x_{1}\bigl(t_{i-1}^{-}\bigr)-x_{2} \bigl(t_{i-1}^{-}\bigr) \bigr\Vert \\ &\qquad{}+\frac{\xi_{i}}{\Gamma(\alpha)}\Vert x_{2}-x_{1} \Vert_{\mathrm{PCB}(J,E)} \int _{0}^{t_{i-1}}(t_{i-1}-s)^{\alpha -1}e^{-L(t-s)} \varsigma(s)\,ds \\ &\qquad{}+\frac{\xi_{i}}{\Gamma(\alpha)}\Vert x_{2}-x_{1} \Vert_{\mathrm{PCB}(J,E)} \int_{0}^{t_{i}}(t_{i}-s)^{\alpha -1}e^{-L(t-s)} \varsigma(s)\,ds \\ &\quad\leq\Vert x_{2}-x_{1}\Vert_{\mathrm{PCB}(J,E)} \xi_{i} \biggl[\xi _{i-1}e^{-Lt}e^{Lt_{i-1}} \\ &\qquad{}+\biggl(\frac{t_{i-1}^{\alpha-\frac{1}{p}}}{\Gamma(\alpha )}\biggl(\frac{p-1}{\alpha p-1}\biggr)^{\frac{p-1}{p}}+ \frac{t_{i}^{\alpha-\frac{1}{p}}}{\Gamma (\alpha)}\biggl( \frac{p-1}{\alpha p-1}\biggr)^{\frac{p-1}{p}}\biggr) \biggl( \int _{0}^{t}\bigl(e^{-L(t-s)}\varsigma (s) \bigr)^{p}\,ds\biggr)^{\frac{1}{p}}\biggr]. \end{aligned}$$
Therefore,
$$\begin{aligned} e^{-Lt} \bigl\Vert y_{2}(t)-y_{1}(t) \bigr\Vert \leq{}&\Vert x_{2}-x_{1}\Vert_{\mathrm{PCB}(J,E)} \xi_{i}\biggl[ \xi _{i-1}e^{-L(t_{i}-t_{i-1})} \\ &{}+\frac{2 t^{\alpha-\frac {1}{p}}}{\Gamma(\alpha)}\biggl(\frac{p-1}{\alpha p-1} \biggr)^{\frac{p-1}{p}}\biggl( \int_{0}^{t}\bigl(e^{-L(t-s)}\varsigma (s) \bigr)^{p}\,ds\biggr)^{\frac{1}{p} }\biggr]. \end{aligned}$$
(34)
From (32), (33), (34), we get
$$ \bigl\Vert R(x_{2})-R(x_{1}) \bigr\Vert _{\mathrm{PCB}(J, E)}< K \Vert x_{1}-x_{2} \Vert _{\mathrm{PCB}(J,E)}, $$
where
$$\begin{aligned} K ={}&\max\biggl\{ \max_{1\leq i\leq m}\xi_{i}\biggl[\xi _{i-1}e^{-L(t_{i}-t_{i-1})} \\ &{}+\frac{2}{\Gamma(\alpha)}\biggl(\frac{p-1}{\alpha p-1}\biggr)^{\frac{p-1}{p} } \max _{t\in J}t^{\alpha-\frac{1}{p}}\biggl( \int _{0}^{t}\bigl(e^{-L(t-s)}\varsigma (s) \bigr)^{p}\,ds\biggr)^{\frac{1}{p}}\biggr], \\ &\max_{1\leq i\leq m}e^{-L(s_{i}-t_{i})}\xi_{i}+ \frac{2}{\Gamma (\alpha)}\biggl( \frac{p-1}{\alpha p-1}\biggr)^{\frac{p-1}{p}} \max _{t\in J}t^{\alpha -\frac{1}{p} }\biggl( \int_{0}^{t}\bigl(e^{-L(t-s)}\varsigma(s) \bigr)^{p}\,ds\biggr)^{\frac {1}{p}}\biggr\} . \end{aligned}$$
We can choose a sufficiently large L such that \(K<1\) and so R is contraction. Thus, from Lemma 2.12, R has a fixed point which is a mild solution for (3). □
Remark 3.4
If we take \(\varsigma(\cdot)=M_{F}\), where \(M_{F}\) is a positive real number, in Theorem 3.3, then we can choose
$$\begin{aligned} K ={}&\max\biggl\{ \max_{1\leq i\leq m}\xi_{i} \xi _{i-1}e^{-L(t_{i}-t_{i-1})}+ \frac{2\xi_{i}}{\Gamma(\alpha)}\biggl(\frac{p-1}{\alpha p-1} \biggr)^{\frac {p-1}{p}}\frac{M_{F} }{(Lp)^{\frac{1}{p}}}\max_{t\in J}t^{\alpha-\frac{1}{p}}, \\ &\max_{1\leq i\leq m}e^{-L(s_{i}-t_{i})}\xi_{i}+ \frac{2}{\Gamma (\alpha)}\biggl( \frac{p-1}{\alpha p-1}\biggr)^{\frac{p-1}{p}} \frac{M_{F}}{(Lp)^{\frac{1}{p}}} \max_{t\in J}t^{\alpha-\frac{1}{p}}\biggr\} , \end{aligned}$$
where we use the fact that
$$ \int_{0}^{t}e^{-L(t-s)p}\,ds=\frac{1-e^{-Ltp}}{Lp} \leq\frac {1}{Lp}, \quad t\in J. $$