Abstract
The Sturm–Liouville equation is one of classical famous differential equations which has been studied from different of views in the literature. In this work, we are going to review its fractional hybrid inclusion version. In this way, we investigate two fractional hybrid Sturm–Liouville differential inclusions with multipoint and integral hybrid boundary conditions. Also, we provide two examples to illustrate our main results.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
1 Introduction
Some differential equations such as that of Sturm–Liouville have established important relations between physics, mathematics, and other fields of engineering (see [1, 2]). During the last decades, many researchers have been studying some well-known problems involving differential equations such as Sturm–Lioville boundary value problems from different views (see, for example, [3–16]). It is important that researchers try to investigate distinct versions of famous and applicable differential equations (see, for example, [17–20]). On the other hand, some interesting integro-differential equations have been investigated by researchers. Among these interesting ones are hybrid differential equations (see, for example, [21–35]).
In 2010, Dhage and Lakshmikantham introduced hybrid differential equations [36]. In 2011, Zhao et al. extended Dhage’s work to fractional order and investigated the hybrid fractional differential equations [25]. In 2012, Sun et al. studied a fractional hybrid two point boundary value problem [23]. In 2016, Baleanu et al. reviewed some existence results for the Caputo fractional hybrid inclusion problem
with boundary value conditions \(z(0)=z_{0}^{*}\) and \(z(1)=z_{1}^{*}\), where \(p\in (1, 2]\), \({^{c}}D^{\alpha }\) and \(I^{\gamma }\) denote the Caputo derivative operator of the fractional order α and the Riemann–Liouville integral operator of the fractional order \(\gamma \in \{\alpha _{i},\beta _{j}\}\subset (0,\infty )\) for \(i=1,\ldots,n\) and \(j=1,\ldots,k\), respectively [37]. In 2019, El-Sayed et al. investigated the fractional version of the Sturm–Liouville differential equation with multipoint boundary condition
where \(\alpha \in (0,1]\), \({^{c}}D^{\alpha }\) denotes the Caputo fractional derivative, \(p\in C^{1}(I,\mathbb{R})\), \(q(t)\) and \(h(t)\) are absolutely continuous functions on \(I=[0,T]\) with \(T<\infty \) and \(p(t)\neq 0\) for all \(t\in I\), \(f:\mathbb{R}\to \mathbb{R}\) is defined and differentiable on the interval I, \(0\leq a_{1}< a_{2}<\cdots <a_{m}<c\), \(d\leq b_{1}< b_{2}<\cdots <b_{n}<T\), \(c< d\) and \(\xi _{1},\ldots, \xi _{m}\), \(\eta _{1},\ldots,\eta _{n}\), and ν are some real constants [6]. Since inclusion problems are really strong versions of the usual differential equations, by using and mixing the main ideas of these works, we are going to investigate the fractional hybrid inclusion version of the Sturm–Lioville equation given by
with multipoint hybrid boundary conditions
where \(\alpha \in (0,1]\), \({^{c}}D^{\alpha }\) denotes the Caputo fractional derivative, \(\Psi : [0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})\) is a multivalued map with some properties, \(p,\tilde{p}\in C^{1}(I,\mathbb{R})\), \(\tilde{p}(t)\) is absolutely continuous function on \([0,1]\), \(p(t)\neq 0\) for all \(t\in I\), \(\inf_{t\in I}\vert p(t)\vert =p\), \(\tilde{f}:\mathbb{R}\to \mathbb{R}\) is defined and differentiable on the interval \([0,1]\), \(0\leq a_{1}< a_{2}<\cdots <a_{m}<c\), \(d\leq b_{1}< b_{2}<\cdots <b_{n}<1\), \(c< d\), and \(\xi _{1},\ldots, \xi _{m}\), \(\eta _{1},\ldots,\eta _{n}\), and ν are some real constants with \(\sum_{i=1}^{m}\xi _{i}-\nu \sum_{j=1}^{n}\eta _{j}\neq 0\). Moreover, we review the fractional hybrid Sturm–Liouville differential inclusion
with integral hybrid boundary conditions
where \(\alpha \in (0,1]\), \({^{c}}D^{\alpha }\) denotes the Caputo fractional derivative, \(\Psi : [0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})\) is a multivalued map with some properties, \(f\in C([0,1]\times \mathbb{R}, \mathbb{R})\), \(g\in C([0,1]\times \mathbb{R},\mathbb{R}\setminus \{0\})\), \(\nu \in \mathbb{R}\), \(\varpi ,\upsilon :[0,1]\to \mathbb{R}\) are two increasing functions, the integrals are in the Riemann–Stieltjes sense, and \(0\leq a< c\leq d< e\leq 1\).
2 Preliminaries
We consider the norm \(\Vert u\Vert =\sup_{t\in [0,1]}\vert u(t)\vert \) on the space \(\mathcal{C}_{\mathbb{R}}([0,1])\) and \(\Vert u\Vert _{\mathcal{L}^{1}}=\int _{0}^{1}\vert u(s)\vert \,ds\) on \(\mathcal{L}^{1}[0,1]\). The Riemann–Liouville fractional integral of order α for a function f is defined by \(I^{\alpha }f(t)=\frac{1}{\Gamma (\alpha )}\int _{0}^{t}(t-s)^{\alpha -1}f(s)\,ds\) (\(\alpha >0\)) and the Caputo derivative of order α for a function f is defined by \({^{c}}D^{\alpha }f(t)=I^{n-\alpha }\frac{d^{n}}{dt^{n}}f(t)= \frac{1}{\Gamma (n-\alpha )}\int _{0}^{t} \frac{f^{(n)}(s)}{(t-s)^{\alpha -n+1}}\,ds\), where \(n=[\alpha ]+1\) (see [38, 39]).
Suppose that \((\mathcal{X},\Vert \cdot \Vert _{\mathcal{X}})\) is a normed space. Denote by \(\mathcal{P}(\mathcal{X})\), \(\mathcal{P}_{cl}(\mathcal{X})\), \(\mathcal{P}_{b}(\mathcal{X})\), \(\mathcal{P}_{cp}(\mathcal{X})\), and \(\mathcal{P}_{cv}(\mathcal{X})\) the set of all subsets of \(\mathcal{X}\), the set of all closed subsets of \(\mathcal{X}\), the set of all bounded subsets of \(\mathcal{X}\), the set of all compact subsets of \(\mathcal{X}\) and the set of all convex subsets of \(\mathcal{X}\), respectively. We say that a set-valued map Ψ has convex values whenever the set \(\Psi (z)\) is convex for each element \(z\in \mathcal{X}\). A set-valued map Ψ is called upper semicontinuous (u.s.c.) whenever for each \(z^{*}\in \mathcal{X}\) and open set \(\hat{\mathcal{V}}\) containing \(\Psi (z^{*})\) there exists an open neighborhood \(\hat{\mathcal{U}}_{0}\) of \(z^{*}\) such that \(\Psi (\hat{\mathcal{U}}_{0})\subseteq \hat{\mathcal{V}}\) [40]. An element \(z^{*}\in \mathcal{X}\) is called a fixed point for the multivalued map \(\Psi : \mathcal{X}\to \mathcal{P}(\mathcal{X})\) whenever \(z^{*}\in \Psi (z^{*})\). The set of all fixed points of the multifunction Ψ is denoted by \(\operatorname{Fix}(\Psi )\) [40].
Let \((\mathcal{X},d_{\mathcal{X}})\) be a metric space. For each \(A_{1}, A_{2}\in \mathcal{P}(\mathcal{X})\), the Pompeiu–Hausdorff metric \(PH_{d}:\mathcal{P}(\mathcal{X})\times \mathcal{P}(\mathcal{X})\to \mathbb{R}\cup \{\infty \}\) is defined by
where \(d_{\mathcal{X}}(a_{1},A_{2})=\inf_{a_{1}\in A_{1}} d_{\mathcal{X}}(a_{1},a_{2})\) and \(d_{\mathcal{X}}(A_{1},a_{2})=\inf_{a_{2}\in A_{2}} d_{\mathcal{X}}(a_{1},a_{2})\) [40]. A multivalued function \(\Psi : \mathcal{X}\to \mathcal{P}_{cl}(\mathcal{X})\) is said to be Lipschitz with Lipschitz constant \(k>0\) whenever \(PH_{d}(\Psi (z_{1}),\Psi (z_{1}))\leq kd_{\mathcal{X}}(z_{1},z_{2})\) holds for all \(z_{1},z_{2}\in \mathcal{X}\). A Lipschitz map Ψ is called a contraction whenever \(0< k<1\) [40]. A set-valued operator \(\Psi : [0,1]\to \mathcal{P}_{cl}(\mathcal{R})\) is called measurable whenever the function \(t\to d_{\mathcal{X}}(\omega ,\Psi (t))=\inf \{\vert \omega -y\vert : y\in \Psi (t) \}\) is measurable for any real constant ω [40, 41]. The graph of a set-valued function \(\Psi : \mathcal{X}\to \mathcal{P}_{cl}(\Omega )\) is defined by \(\operatorname{Graph} (\Psi )= \{(z, \omega )\in \mathcal{X}\times \Omega : \omega \in \Psi (z)\}\) [40]. We say that the graph of Ψ is closed whenever for each sequence \(\{z_{n}\}\) in \(\mathcal{X}\) and \(\{\omega _{n}\}\) in Ω with \(z_{n}\to z_{0}\), \(\omega _{n}\to \omega _{0}\) and \(\omega _{n}\in \Psi (z_{n})\), we have \(\omega _{0}\in \Psi (z_{0})\) [41].
A multifunction Ψ is said to be a completely continuous operator whenever the set \(\Psi (\mathcal{W})\) is relatively compact for all \(\mathcal{W}\in \mathcal{P}_{b}(\mathcal{X})\). If the multifunction \(\Psi : \mathcal{X}\to \mathcal{P}_{cl}(\Omega )\)) is upper semicontinuous, then \(\operatorname{Graph} (\Psi )\) is a subset of the product space \(\mathcal{X}\times \Omega \) with the closedness property. Conversely, if the set-valued mapping Ψ is completely continuous and has a closed graph, then Ψ is upper semicontinuous (see [40], Proposition 2.1). A set-valued map \(\Psi : [0,1]\times \mathcal{R}\to \mathcal{P}(\mathbb{R})\) is said to be a Caratheodory multifunction whenever \(t\to \Psi (t,z)\) is a measurable mapping for all \(z\in \mathbb{R}\) and \(z\to \Psi (t,z)\) is an upper semicontinuous mapping for almost all \(t\in [0,1]\) (see [40, 41]). Also, a Caratheodory multifunction \(\Psi : [0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})\) is said to be \(\mathcal{L}^{1}\)-Caratheodory whenever for each constant \(\mu >0\) there exists function \(\phi _{\mu }\in \mathcal{L}^{1}([0,1],\mathcal{R})\) such that
for all \(\vert z\vert \leq \mu \) and for almost all \(t\in [0,1]\) (see [40, 41]). The set of selections of a multifunction Ψ at a point \(z\in \mathcal{C}_{\mathbb{R}}([0,1])\) is defined by \((\mathcal{SEL})_{\Psi ,z}:=\{y\in \mathcal{L}^{1}([0,1],\mathcal{R}): y(t)\in \Psi (t,z) \}\) for almost all \(t\in [0,1]\) (see [40, 41]). Let Ψ be a set-valued map. It is known that \((\mathcal{SEL})_{\Psi ,z}\neq \emptyset \) for all \(z\in \mathcal{C}_{\mathbb{R}}([0,1])\) whenever \(\dim \mathcal{X}< \infty \) [40]. We need the following results.
Theorem 1
([42])
Suppose that \(\mathcal{X}\)is a separable Banach space, \(\Psi :[0,1]\times \mathcal{X}\to \mathcal{P}_{cp,cv}(\mathcal{X})\)is an \(\mathcal{L}^{1}\)-Carathéodory multifunction and \(\hat{\Upsilon }:\mathcal{L}^{1}([0,1],\mathcal{X})\to C([0,1], \mathcal{X})\)is a linear continuous mapping. Then
is an operator in the product space \(C([0,1],\mathcal{X})\times C([0,1],\mathcal{X})\)with the action \(z\to (\hat{\Upsilon }\circ (\mathcal{SEL})_{\Psi })(z)=\hat{\Upsilon }(( \mathcal{SEL})_{\Psi ,z})\)having the closed graph property.
Theorem 2
(Theorem 4.8 of [43])
Let \(V_{\zeta }(0)\)and \(\bar{V}_{\zeta }(0)\)denote respectively the open and closed balls centered at the origin 0 of radius \(\zeta >0\)in a Banach algebra \(\mathcal{X}\)and let \(\hat{\mathcal{B}}_{1},\hat{\mathcal{B}}_{3}:\bar{V}_{\zeta }(0)\to \mathcal{X}\)and \(\hat{\mathcal{B}}_{2}:\bar{V}_{\zeta }(0)\to \mathcal{P}_{cp,cv}( \mathcal{X})\)be three operators such that
-
(i)
\(\hat{\mathcal{B}}_{1}\)and \(\hat{\mathcal{B}}_{3}\)are single-valued Lipschitz with the Lipschitz constants \(\ell _{1}^{*}\)and \(\ell _{2}^{*}\), respectively;
-
(ii)
\(\hat{\mathcal{B}}_{2}\)is u.s.c. and compact;
-
(iii)
\(\ell _{1}^{*}\hat{M}+\ell _{2}^{*}<\frac{1}{2}\)where \(\hat{M}=\Vert \hat{\mathcal{B}}_{2}(\bar{V}_{\zeta }(0))\Vert =\sup \{\Vert \hat{\mathcal{B}}_{2}z\Vert : z\in \bar{V}_{\zeta }(0)\}\).
Then either
-
(a)
the operator inclusion \(z\in \hat{\mathcal{B}}_{1}z\hat{\mathcal{B}}_{2}z+\hat{\mathcal{B}}_{3}z\)has a solution, or
-
(b)
there exists \(z\in \mathcal{X}\)with \(\Vert z\Vert =\zeta \)such that \(\mu z\in \hat{\mathcal{B}}_{1}z\hat{\mathcal{B}}_{2}z+ \hat{\mathcal{B}}_{3}z\)for some \(\mu >1\).
Theorem 3
(Theorem 4.13 of [43])
Let \(\mathcal{X}\)be a Banach algebra. Let \(\hat{\mathcal{B}}_{1},\hat{\mathcal{B}}_{3}:\mathcal{X}\to \mathcal{X}\)be two single-valued operators and \(\hat{\mathcal{B}}_{2}:\mathcal{X}\to \mathcal{P}_{cp,cv}(\mathcal{X})\)be a multivalued map such that
-
(i)
\(\hat{\mathcal{B}}_{1}\)and \(\hat{\mathcal{B}}_{3}\)are single-valued Lipschitz with the Lipschitz constants \(\ell _{1}^{*}\)and \(\ell _{2}^{*}\), respectively;
-
(ii)
\(\hat{\mathcal{B}}_{2}\)is u.s.c. and compact;
-
(iii)
\(\ell _{1}^{*}\hat{M}+\ell _{2}^{*}<\frac{1}{2}\)where \(\hat{M}=\Vert \hat{\mathcal{B}}_{2}(\bar{V}_{\zeta }(0))\Vert =\sup \{\Vert \hat{\mathcal{B}}_{2}z\Vert : z\in \mathcal{X}\}\).
Then either
-
(a)
the operator inclusion \(z\in \hat{\mathcal{B}}_{1}z\hat{\mathcal{B}}_{2}z+\hat{\mathcal{B}}_{3}z\)has a solution, or
-
(b)
the set \(\sum^{*}=\{z\in \mathcal{X}: \mu z\in \hat{\mathcal{B}}_{1}z \hat{\mathcal{B}}_{2}z+\hat{\mathcal{B}}_{3}z, \mu >1\}\)is unbounded.
3 Main results
Now, we investigate the fractional Sturm–Liouville differential inclusion (1)–(2).
Lemma 4
Let \(y\in \mathcal{L}^{1}([0,1],\mathbb{R})\). A function z is a solution for the fractional hybrid Sturm–Liouville differential equation
with multipoint hybrid boundary conditions
if and only if z is a solution for the integral equation
where \(\mathcal{H}=\frac{1}{\sum_{i=1}^{m}\xi _{i}-\nu \sum_{j=1}^{n}\eta _{j}}\).
Proof
First assume that z is a solution for the hybrid fractional equation (5). Note that equation (5) can be written as
Then, \(I^{1}(\frac{d}{dt}[p(t) (\frac{z(t)}{g(t,z(t))} )'-\tilde{p}(t)\tilde{f}(z(t)])=I^{\alpha }y(t)\) and so
Since \((\frac{z(t)}{g(t,z(t))} )'_{t=0}= ( \frac{\tilde{p}(t)\tilde{f}(z(t))}{p(t)} )_{t=0}\), one has \(p(t) (\frac{z(t)}{g(t,z(t))} )'=\tilde{p}(t) \tilde{f}(z(t))+I^{\alpha }y(t)\), and so
By integrating from 0 to t, we get
where \(\ell =\frac{z(0)}{g(0,z(0))}\). Now, we can write
and
Now by subtracting (10) from (11) and utilizing \(\sum_{i=1}^{m}\xi _{i}(\frac{z(a_{i})}{g(t,z(a_{i}))} ) =\nu \sum_{j=1}^{n}\eta _{j} (\frac{z(b_{j})}{g(t,z(b_{j}))} )\), we conclude that
where \(\mathcal{H}=\frac{1}{\sum_{i=1}^{m}\xi _{i}-\nu \sum_{j=1}^{n}\eta _{j}}\). Now by substituting the value of ℓ in (9), we get
Conversely, to complete the equivalence between integral equation (7) and problem (5)–(6), by using (8), we obtain
and \((\frac{z(t)}{g(t,z(t))} )'_{t=0}= ( \frac{\tilde{p}(t)}{p(t)}\tilde{f}(z(t)) )_{t=0} \). Also by using simple computations and (7), we obtain \(\sum_{i=1}^{m}\xi _{i}(\frac{z(a_{i})}{g(a_{i},z(a_{i}))} )=\nu \sum_{j=1}^{n}\eta _{j} (\frac{z(b_{j})}{g(b_{j},z(b_{j}))} )\). This completes the proof. □
Definition 5
We say that an absolutely continuous function \(z:[0,1]\to \mathbb{R}\) is a solution for the fractional hybrid Sturm–Liouville differential inclusion (1)–(2) whenever there is an integrable function \(y\in \mathcal{L}^{1}([0,1],\mathbb{R})\) with \(y(t)\in \Psi (t,z(t))\) for almost all \(t\in [0,1]\),
and
for all \(t\in [0,1]\).
Theorem 6
Assume that
- \((\mathcal{Q}_{1})\):
-
there exists a bounded mapping \(\chi : [0,1]\to \mathbb{R}^{+}\)such that \(\vert g(t,x)-g(t,y)\vert \leq \chi (t)\vert x-y\vert \)for all \((t,x,y)\in [0,1]\times \mathbb{R} \times \mathbb{R}\);
- \((\mathcal{Q}_{2})\):
-
the function \(\tilde{f}:\mathbb{R}\to \mathbb{R}\)is differentiable on \([0,1]\), and \(\frac{\partial \tilde{f}}{\partial u}\)is bounded on \([0,1]\)with \(\frac{\partial \tilde{f}}{\partial u}\leq \tilde{\mathcal{K}}\);
- \((\mathcal{Q}_{3})\):
-
the set-valued map \(\Psi : [0,1]\times \mathbb{R}\to \mathcal{P}_{cp,cv}(\mathbb{R})\)has \(\mathcal{L}^{1}\)-Caratheodory property;
- \((\mathcal{Q}_{4})\):
-
there exists a positive mapping \(\sigma \in \mathcal{C}([0,1],\mathbb{R}^{+})\)such that
$$ \bigl\Vert \Psi (t,x) \bigr\Vert =\sup \bigl\{ \vert y \vert : y\in \Psi (t,x) \bigr\} \leq \sigma (t) $$for all \(x\in \mathbb{R}\)and almost all \(t\in [0,1]\);
- \((\mathcal{Q}_{5})\):
-
there exists \(\zeta >0\)such that \(\zeta >\bar{\Delta }(\chi ^{*}\zeta +g_{0})(\Vert \tilde{p}\Vert ( \widetilde{\mathcal{K}}\zeta +\tilde{f_{0}})+ \frac{{\Vert \sigma \Vert }}{\Gamma (\alpha +2)})\)and
$$\begin{aligned} \begin{aligned} \biggl( \frac{\chi ^{*}{ \Vert \sigma \Vert }}{\Gamma (\alpha +2)} + \Vert \tilde{p} \Vert \bigl(2\widetilde{\mathcal{K}}\chi ^{*}\zeta +\chi ^{*} \tilde{f_{0}}+\widetilde{\mathcal{K}}g_{0} \bigr) \biggr) \bar{\Delta }< \frac{1}{2}, \end{aligned} \end{aligned}$$
where \(\bar{\Delta }=\frac{1}{p} (\vert \mathcal{H}\vert (\sum_{i=1}^{m}\vert \xi _{i}\vert +\vert \nu \vert \sum_{j=1}^{n}\vert \eta _{j}\vert )+1 )\), \(\chi ^{*}=\sup_{t\in [0,1]}\chi (t)\), \(g_{0}=\sup_{t\in [0,1]}g(t,0)\), and \(\tilde{f_{0}}=\tilde{f}(0)\). Then the fractional hybrid Sturm–Liouville inclusion problem (1)–(2) has at least one solution.
Proof
Let \(\mathcal{X}=\mathcal{C}_{\mathbb{R}}([0,1]) \), and let \(V_{\zeta }(0):=\{z\in \mathcal{X}: \Vert z\Vert <\zeta \}\) and \(\bar{V}_{\zeta }(0):=\{z\in \mathcal{X}: \Vert z\Vert \leq \zeta \}\) be the open and closed balls centered at the origin 0 of radius ζ, respectively. Consider the operator \(\hat{\mathcal{K}}:\bar{V}_{\zeta }(0)\to \mathcal{P}(\mathcal{X})\) defined by
where
It is easy to check that fixed point of the set-valued map \(\hat{\mathcal{K}}\) is solution for the fractional hybrid Sturm–Liouville inclusion problem (1)–(2). Define the maps \(\hat{\mathcal{B}}_{1}, \hat{\mathcal{B}}_{3}:\bar{V}_{\zeta }(0)\to \mathcal{X}\) and the set-valued-map \(\hat{\mathcal{B}}_{2}:\bar{V}_{\zeta }(0)\to \mathcal{P}(\mathcal{X})\) by \((\hat{\mathcal{B}}_{1}z)(t)=g(t,z(t))\), \((\hat{\mathcal{B}}_{3}z)(t)=g(t,z(t))(\vartheta z)(t)\),
where
and
Note that \(\hat{\mathcal{K}}(z)=\hat{\mathcal{B}}_{1}z\hat{\mathcal{B}}_{2}z+ \hat{\mathcal{B}}_{3}z\) for all \(z\in \bar{V}_{\zeta }(0)\). We show that the operators \(\hat{\mathcal{B}}_{1}\), \(\hat{\mathcal{B}}_{2}\), and \(\hat{\mathcal{B}}_{3}\) satisfy the conditions of Theorem 2. First, we prove that the set-valued map \(\hat{\mathcal{B}}_{2}\) is convex-valued. Let \(\varphi _{1},\varphi _{2}\in \hat{\mathcal{B}}_{2}z\). Choose \(y_{1},y_{2}\in (\mathcal{SEL})_{\Psi , z}\) such that
for \(k=1,2\). Let \(\lambda \in (0,1)\). Then, we have
for almost all \(t\in [0,1]\). Since Ψ is convex-valued, \((\mathcal{SEL})_{\Psi , z}\) is convex, that is,
Hence, \(\lambda \varphi _{1}(t)+(1-\lambda )\varphi _{2}(t)\in \hat{\mathcal{B}}_{2}z\), and so \(\hat{\mathcal{B}}_{2}z\) is a convex set for all \(z\in \mathcal{X}\). Now, we show that the operator \(\hat{\mathcal{B}}_{2}\) is completely continuous and upper semicontinuous on \(\mathcal{X}\). To establish the complete continuity of the operator \(\hat{\mathcal{B}}_{2}\), we should prove that \(\hat{\mathcal{B}}_{2}(\mathcal{X})\) is an equicontinuous and uniformly bounded set. To do this, first we prove that \(\hat{\mathcal{B}}_{2}\) maps all bounded sets into bounded subsets of \({\mathcal{X}}\). Let \(\hat{\mathcal{V}}\) be a bounded subset of \(\bar{V}_{\zeta }(0)\). Choose \(0<\kappa ^{*}\leq \zeta \) such that \(\Vert z\Vert \leq \kappa ^{*}\) for all \(z\in \hat{\mathcal{V}}\). For each \(z\in \hat{\mathcal{V}}\) and \(\varphi \in \hat{\mathcal{B}}_{2}({\hat{\mathcal{V}}})\), there exists \(y\in (\mathcal{SEL})_{\Psi , z}\) such that
Hence,
Since \(\vert \mathcal{H}\vert \vert \nu \vert \sum_{j=1}^{n}\vert \eta _{j}\vert \int _{0}^{b_{j}} \int _{0}^{s} \frac{(s-\tau )^{\alpha -1}\sigma (\tau )}{\vert p(s)\vert \Gamma (\alpha )}\,d \tau \,ds \leq \vert \mathcal{H}\vert \vert \nu \vert \sum_{j=1}^{n}\vert \eta _{j}\vert \frac{{\Vert \sigma \Vert }}{p\Gamma (\alpha +2)}\),
and \(\int _{0}^{t}\int _{0}^{s} \frac{(s-\tau )^{\alpha -1} \sigma (\tau )}{\vert p(s)\vert \Gamma (\alpha )}\,d \tau \,ds\leq \frac{{\Vert \sigma \Vert }}{p\Gamma (\alpha +2)}\), we get
and so \(\Vert \varphi \Vert \leq \frac{{\Vert \sigma \Vert }}{\Gamma (\alpha +2)} \bar{\Delta }\). Thus, \(\hat{\mathcal{B}}_{2}(\hat{\mathcal{V}})\) is a uniformly bounded. Now, we prove that the operator \(\hat{\mathcal{B}}_{2}\) maps bounded sets onto equicontinuous sets. Let \(z\in \hat{\mathcal{V}} \) and \(\varphi \in \hat{\mathcal{B}}_{2}z\). Choose \(y\in (\mathcal{SEL})_{\Psi , z}\) such that
for all \(t\in [0,1]\). For each \(t_{1}\), \(t_{2}\) with \(t_{1}< t_{2}\), we have
Since the right-hand side of the above inequality tends to zero as \(t_{1}\to t_{2}\), by using the Arzela–Ascoli theorem, the operator \(\hat{\mathcal{B}}_{2}:\bar{V}_{\zeta }(0)\to \mathcal{P}(\mathcal{X})\) is completely continuous. Here, we show that \(\hat{\mathcal{B}}_{2}\) has a closed graph, and this implies that \(\hat{\mathcal{B}}_{2}\) is upper semicontinuous. For this aim, suppose that \(z_{n}\in \hat{\mathcal{V}} \) and \(\varphi _{n}\in \hat{\mathcal{B}}_{2}z_{n}\) with \(z_{n}\to z^{*}\) and \(\varphi _{n}\to \varphi ^{*}\). We show that \(\varphi ^{*}\in \hat{\mathcal{B}}_{2}z^{*}\). For each \(\varphi _{n}\in \hat{\mathcal{B}}_{2}z_{n}\), choose \(y_{n}\in (\mathcal{SEL})_{\Psi , z_{n}}\) such that
It is sufficient to be prove that there exists a function \(y^{*}\in (\mathcal{SEL})_{\Psi , z^{*}}\) such that
for all \(t\in [0,1]\). Consider the continuous linear operator \(\hat{\Upsilon }:\mathcal{L}^{1}([0,1],\mathbb{R})\to \mathcal{X}\) defined by
for all \(t\in [0,1]\). Then, we get
Hence by using Theorem 1, we conclude that the operator \(\hat{\Upsilon }\circ (\mathcal{SEL})_{\Psi }\) has a closed graph. In fact, since \(\varphi _{n}\in \hat{\Upsilon }((\mathcal{SEL})_{\Psi ,z_{n}})\) and \(z_{n}\to z^{*}\), there exists \(y^{*}\in (\mathcal{SEL})_{\Psi ,z^{*}}\) such that
for all \(t\in [0,1]\). Hence, \(\varphi ^{*}\in \hat{\mathcal{B}}_{2}z^{*}\). This means that the graph of \(\hat{\mathcal{B}}_{2}\) is closed. Thus, \(\hat{\mathcal{B}}_{2}\) is upper semicontinuous. Furthermore, by using the assumptions, we know that the operator Ψ has compact values. Hence, \(\hat{\mathcal{B}}_{2}\) is a compact and upper semicontinuous operator. Now, we show that \(\hat{\mathcal{B}}_{1}\) and \(\hat{\mathcal{B}}_{3}\) are Lipschitz. Let \(z_{1},z_{2}\in \bar{V}_{\zeta }(0)\). By using \((\mathcal{Q}_{1})\), we get
and so \(\Vert \hat{\mathcal{B}}_{1}z_{1}-\hat{\mathcal{B}}_{1}z_{2}\Vert \leq \chi ^{*} \Vert z_{1}-z_{2}\Vert \). Hence, \(\hat{\mathcal{B}}_{1}\) is Lipschitz with Lipschitz constant \(\ell _{1}^{*}=\chi ^{*}\). Let \(z\in \bar{V}_{\zeta }(0)\). By using \((\mathcal{Q}_{2})\), we get \(\vert \tilde{f}(z(s))\vert \leq \widetilde{\mathcal{K}}\Vert z\Vert +\tilde{f_{0}} \leq \widetilde{\mathcal{K}}\zeta +\tilde{f_{0}}\). Similarly, one can show that \(\vert g(s,z(s))\vert \leq \chi ^{*}\zeta +g_{0}\). Thus, we obtain
Let \(z_{1},z_{2}\in \bar{V}_{\zeta }(0)\). Similarly, we get
On the other hand, we have
Thus,
and so \(\Vert \hat{\mathcal{B}}_{3}z_{1}-\hat{\mathcal{B}}_{3}z_{2}\Vert \leq \Vert \tilde{p}\Vert (2\widetilde{\mathcal{K}}\chi ^{*}\zeta +\chi ^{*} \tilde{f_{0}}+\widetilde{\mathcal{K}}g_{0} )\bar{\Delta }\Vert z_{1}-z_{2} \Vert \). Hence, \(\hat{\mathcal{B}}_{3}\) is Lipschitz with Lipschitz constant \(\ell _{2}^{*}=\Vert \tilde{p}\Vert (2\widetilde{\mathcal{K}}\chi ^{*} \zeta +\chi ^{*}\tilde{f_{0}}+\widetilde{\mathcal{K}}g_{0} ) \bar{\Delta }\). Note that
and
Thus, the assumptions of Theorem 2 hold for \(\hat{\mathcal{B}}_{1}\), \(\hat{\mathcal{B}}_{2}\), and \(\hat{\mathcal{B}}_{3}\). Hence, one of the conditions \((a)\) or \((b)\) holds. We show that condition (b) is impossible. Let \(z\in \mathcal{X}\) with \(\Vert z\Vert =\zeta \) be such that \(\mu z\in \hat{\mathcal{B}}_{1} z \hat{\mathcal{B}}_{2} z+ \hat{\mathcal{B}}_{3} z\) for some \(\mu >1\). Choose \(y\in (\mathcal{SEL})_{\Psi , z}\) such that
Hence,
and so \(\zeta \leq \bar{\Delta }(\chi ^{*}\zeta +g_{0})[\Vert \tilde{p}\Vert ( \widetilde{\mathcal{K}}\zeta +\tilde{f_{0}})+ \frac{{\Vert \sigma \Vert }}{\Gamma (\alpha +2)}]\), which is a contradiction. Hence, condition (b) is impossible, and so the fractional hybrid Sturm–Liouville inclusion problem (1)–(2) has at least one solution. □
Now, we investigate the fractional hybrid Sturm–Liouville inclusion problem (3)–(4).
Lemma 7
Let \(y\in \mathcal{L}^{1}([0,1],\mathbb{R})\). A function z is a solution for the fractional hybrid Sturm–Liouville differential equation
with integral hybrid boundary conditions
if and only if z is a solution for the integral equation
where \(R=\frac{1}{\varpi (c)-\varpi (a)-\nu (\upsilon (e))-\upsilon (d))}\)with \(\varpi (c)-\varpi (a)-\nu (\upsilon (e))-\upsilon (d))\neq 0\).
Proof
First, assume that z is a solution for the fractional hybrid equation (12). Note that
Hence, \(I^{1}(\frac{d}{dt}[p(t) (\frac{z(t)-f(t,z(t))}{g(t,z(t))} )'])=I^{\alpha }y(t)\), and so
Since \((\frac{z(t)-f(t,z(t))}{g(t,z(t))} )'_{t=0}=0\), we get
By integrating from 0 to t, we obtain
where \(\ell ^{*}=\frac{z(0)-f(0,z(0))}{g(0,z(0))}\). Thus,
and \(\nu \int _{d}^{e} (\frac{z(\theta )-f(\theta ,z(\theta ))}{g(\theta ,z(\theta ))} )\,d \upsilon (\theta ) -\ell ^{*}\nu \int _{d}^{e}\,d\upsilon (\theta )= \nu \int _{d}^{e}\int _{0}^{\theta }\frac{1}{p(s)}I^{\alpha }y(s)\,ds\,d \upsilon (\theta )\). Hence,
and
Since \(\int _{a}^{c} ( \frac{z(\theta )-f(\theta ,z(\theta ))}{g(\theta ,z(\theta ))} )\,d \varpi (\theta )=\nu \int _{d}^{e} ( \frac{z(\theta )-f(\theta ,z(\theta ))}{g(\theta ,z(\theta ))} )\,d \upsilon (\theta )\), we have
Now by substituting the value of \(\ell ^{*}\) in (16), we obtain
For the converse part, from (15) we get \({^{c}}D^{\alpha } (p(t) (\frac{z(t)-f(t,z(t))}{g(t,z(t))} )' )={^{c}}D^{\alpha }I^{\alpha }y(t)=y(t)\) and \((\frac{z(t)-f(t,z(t))}{g(t,z(t))} )'_{t=0}=0\). Also by using some simple computations and (14), we obtain \(\int _{a}^{c} ( \frac{z(\theta )-f(\theta ,z(\theta ))}{g(\theta ,z(\theta ))} )\,d \varpi (\theta )=\nu \int _{d}^{e} ( \frac{z(\theta )-f(\theta ,z(\theta ))}{g(\theta ,z(\theta ))} )\,d \upsilon (\theta )\). □
Definition 8
We say that an absolutely continuous function \(z:[0,1]\to \mathbb{R}\) is a solution for the fractional hybrid Sturm–Liouville inclusion problem (3)–(4) whenever there exists \(y\in \mathcal{L}^{1}([0,1],\mathbb{R})\) such that \(y(t)\in \Psi (t,z(t))\) for almost all \(t\in [0,1]\), \((\frac{z(t)-f(t,z(t))}{g(t,z(t))} )'_{t=0}=0\), \(\int _{a}^{c} ( \frac{z(\theta )-f(\theta ,z(\theta ))}{g(\theta ,z(\theta ))} )\,d \varpi (\theta )=\nu \int _{d}^{e} ( \frac{z(\theta )-f(\theta ,z(\theta ))}{g(\theta ,z(\theta ))} )\,d \upsilon (\theta )\), and
for all \(t\in [0,1]\).
Theorem 9
Assume that
- \((\mathcal{Q}^{*}_{1})\):
-
there exists a bounded mapping \(\chi _{1}: [0,1]\to \mathbb{R}^{+}\)such that \(\vert g(t,x)-g(t,y)\vert \leq \chi _{1}(t)\vert x-y\vert \)for all \((t,x,y)\in [0,1]\times \mathbb{R} \times \mathbb{R}\);
- \((\mathcal{Q}^{*}_{2})\):
-
there exists a bounded map \(\chi _{2}: [0,1]\to \mathbb{R}^{+}\)such that \(\vert f(t,x)-f(t,y)\vert \leq \chi _{2}(t)\vert x-y\vert \)for all \((t,x,y)\in [0,1]\times \mathbb{R} \times \mathbb{R}\);
- \((\mathcal{Q}^{*}_{3})\):
-
the set-valued map \(\Psi : [0,1]\times \mathbb{R}\to \mathcal{P}_{cp,cv}(\mathbb{R})\)has \(\mathcal{L}^{1}\)-Caratheodory property;
- \((\mathcal{Q}^{*}_{4})\):
-
there exists a positive map \(\sigma \in \mathcal{C}([0,1],\mathbb{R}^{+})\)such that
$$ \bigl\Vert \Psi (t,x) \bigr\Vert =\sup \bigl\{ \vert y \vert : y\in \Psi (t,x) \bigr\} \leq \sigma (t) $$for all \(x\in \mathbb{R}\)and almost all \(t\in [0,1]\);
- \((\mathcal{Q}^{*}_{5})\):
-
the strict inequality \(\chi _{1}^{*}\Lambda ^{*}+\chi _{2}^{*}<\frac{1}{2}\)holds, where \(\chi _{1}^{*}=\sup_{t\in [0,1]}\chi _{1}(t)\), \(\chi _{2}^{*}= \sup_{t\in [0,1]}\chi _{2}(t)\), and \(\Lambda ^{*}=\frac{{\Vert \sigma \Vert }}{p\Gamma (\alpha +1)} (\vert R\vert ( \varpi (c)-\varpi (a)+\vert \nu \vert (\upsilon (e)-\upsilon (d)) )+1 )\).
Then the fractional hybrid Sturm–Liouville inclusion problem (3)–(4) has a solution.
Proof
Let \(\mathcal{X}=\mathcal{C}_{\mathbb{R}}([0,1])\). Consider the operator \(\hat{\mathcal{K}}:\mathcal{X}\to \mathcal{P}(\mathcal{X})\) defined by
where
Note that each fixed point of the set-valued map \(\hat{\mathcal{K}}\) is a solution for the fractional hybrid Sturm–Liouville inclusion problem (3)–(4). Define the single-valued maps \(\hat{\mathcal{B}}_{1},\hat{\mathcal{B}}_{3}:\mathcal{X}\to \mathcal{X}\) by \((\hat{\mathcal{B}}_{1}z)(t)=g(t,z(t))\) and \((\hat{\mathcal{B}}_{3}z)(t)=f(t,z(t))\), and the set-valued map \(\hat{\mathcal{B}}_{2}:\mathcal{X}\to \mathcal{P}(\mathcal{X})\) by
where
Note that \(\hat{\mathcal{K}}(z)=\hat{\mathcal{B}}_{1}z\hat{\mathcal{B}}_{2}z+ \hat{\mathcal{B}}_{3}z\). We prove that the operators \(\hat{\mathcal{B}}_{1}\), \(\hat{\mathcal{B}}_{2}\), and \(\hat{\mathcal{B}}_{3}\) satisfy the conditions of Theorem 3. Now, we prove that the set-valued map \(\hat{\mathcal{B}}_{2}\) is convex-valued. Let \(z_{1},z_{2}\in \hat{\mathcal{B}}_{2}z\). Choose \(y_{1},y_{2}\in (\mathcal{SEL})_{\Psi , z}\) such that
for \(j=1,2\). Let \(\lambda \in (0,1)\). Then, we have
for almost all \(t\in [0,1]\). Since Ψ is convex-valued, \((\mathcal{SEL})_{\Psi , z}\) is convex. This implies that \(\lambda y_{1}(\tau )+(1-\lambda )y_{2}(\tau ))\in (\mathcal{SEL})_{ \Psi ,z}\), and so \(\hat{\mathcal{B}}_{2}z\) is convex set for all \(z\in \mathcal{X}\). Now, we show that the operator \(\hat{\mathcal{B}}_{2}\) is completely continuous and upper semicontinuous on \(\mathcal{X}\). To establish the complete continuity of the operator \(\hat{\mathcal{B}}_{2}\), we should prove that \(\hat{\mathcal{B}}_{2}(\mathcal{X})\) is an equicontinuous and uniformly bounded set. To do this, it is sufficient to prove that \(\hat{\mathcal{B}}_{2}\) maps all bounded sets into bounded subsets of \({\mathcal{X}}\). Assume that \(\mathcal{V}\) is a bounded subset of \(\mathcal{X}\). Choose \(\kappa ^{*}>0\) such that \(\Vert z\Vert <\kappa ^{*}\) for all \(z\in \mathcal{V}\). For every \(z\in { \mathcal{V}}\) and \(\varphi \in \hat{\mathcal{B}}_{2}({\mathcal{V}})\), there exists \(y\in (\mathcal{SEL})_{\Psi , z}\) such that
Hence,
Since \(\vert \nu \vert \vert R\vert \int _{d}^{e}\int _{0}^{\theta } \int _{0}^{s} \frac{(s-\tau )^{\alpha -1} \vert \sigma (\tau )\vert }{\vert p(s)\vert \Gamma (\alpha )}\,d \tau \,ds \,d\upsilon (\theta )\leq \frac{\vert R\vert {\Vert \sigma \Vert }\vert \nu \vert (\upsilon (e)-\upsilon (d))}{p\Gamma (\alpha +1)}\),
and \(\int _{0}^{t} \int _{0}^{s} \frac{(s-\tau )^{\alpha -1} \vert \sigma (\tau )\vert }{\vert p(s)\vert \Gamma (\alpha )}\,d \tau \,ds\leq \frac{{\Vert \sigma \Vert }}{p\Gamma (\alpha +1)}\), we get
Hence, \(\Vert \varphi \Vert \leq \Lambda ^{*}\). This means \(\hat{\mathcal{B}}_{2}(\mathcal{V})\) is a uniformly bounded set. Here, we show that the operator \(\hat{\mathcal{B}}_{2}\) maps bounded sets onto equicontinuous sets. Let \(z\in \mathcal{V} \) and \(\varphi \in \hat{\mathcal{B}}_{2}z\). Choose \(y\in (\mathcal{SEL})_{\Psi , z}\) such that
for all \(t\in [0,1]\). For each \(t_{1}\), \(t_{2}\) with \(t_{1}< t_{2}\), we can write
Note that the right-hand side of the inequality tends to zero as \(t_{1}\to t_{2}\). Now by using the Arzela–Ascoli theorem, the operator \(\hat{\mathcal{B}}_{2}:\mathcal{X}\to \mathcal{P}(\mathcal{X})\) is completely continuous. We show that \(\hat{\mathcal{B}}_{2}\) has a closed graph, and this implies that \(\hat{\mathcal{B}}_{2}\) is upper semicontinuous. For this, suppose that \(z_{n}\in \mathcal{V} \) and \(\varphi _{n}\in \hat{\mathcal{B}}_{2}z_{n}\) with \(z_{n}\to z^{*}\) and \(\varphi _{n}\to \varphi ^{*}\). We show that \(\varphi ^{*}\in \hat{\mathcal{B}}_{2}z^{*}\). For each \(\varphi _{n}\in \hat{\mathcal{B}}_{2}z_{n}\), choose \(y_{n}\in (\mathcal{SEL})_{\Psi , z_{n}}\) such that
It is sufficient to prove that there exists a function \(y^{*}\in (\mathcal{SEL})_{\Psi , z^{*}}\) such that
for all \(t\in [0,1]\). Consider the continuous linear operator \(\hat{\Upsilon }:\mathcal{L}^{1}([0,1],\mathbb{R})\to \mathcal{X}\) defined by
for all \(t\in [0,1]\). Then, we have
Now by using Theorem 1, we conclude that the operator \(\hat{\Upsilon }\circ (\mathcal{SEL})_{\Psi }\) has a closed graph. In fact, since \(\varphi _{n}\in \hat{\Upsilon }((\mathcal{SEL})_{\Psi ,z_{n}})\) and \(z_{n}\to z^{*}\), there exists \(y^{*}\in (\mathcal{SEL})_{\Psi ,z^{*}}\) such that
for all \(t\in [0,1]\). Hence, \(\varphi ^{*}\in \hat{\mathcal{B}}_{2}z^{*}\), and so the graph of \(\hat{\mathcal{B}}_{2}\) is closed. Thus, \(\hat{\mathcal{B}}_{2}\) is upper semicontinuous. Furthermore, by using the assumptions, we know that the operator Ψ has compact values. Hence, \(\hat{\mathcal{B}}_{2}\) is a compact and upper semicontinuous operator. By a similar method as in proof of Theorem 6, we can prove that \(\hat{\mathcal{B}}_{1}\) and \(\hat{\mathcal{B}}_{3}\) are Lipschitz operators on \(\mathcal{X}\) with Lipschitz constants \(\chi _{1}^{*}\) and \(\chi _{2}^{*}\), respectively. Now by using assumption \((\mathcal{Q}^{*}_{5})\), we get \(\ell ^{*}_{1}\hat{M}^{*}+\ell ^{*}_{2}=\chi _{1}^{*}\Vert \hat{\mathcal{B}}_{2}(\mathcal{X})\Vert +\chi _{2}^{*}=\chi _{1}^{*}\sup \{\vert \hat{\mathcal{B}}_{2}z\vert : z\in \mathcal{X}\}+\chi _{2}^{*} \leq \chi _{1}^{*}\Lambda ^{*}+\chi _{2}^{*}<\frac{1}{2}\). Thus, the assumptions of Theorem 3 hold for \(\hat{\mathcal{B}}_{1}\), \(\hat{\mathcal{B}}_{2}\), and \(\hat{\mathcal{B}}_{3}\), and so one of the conditions \((a)\) or \((b)\) holds. We show that condition (b) is impossible. According to the definition of ∑∗, let z be an arbitrary element of ∑∗. Thus, \(\mu z\in \hat{\mathcal{B}}_{1} z \hat{\mathcal{B}}_{2} z+ \hat{\mathcal{B}}_{3} z\) for some \(\mu >1\). Now, choose \(y\in (\mathcal{SEL})_{\Psi , z}\) such that
Hence,
Since \(\vert g(t,z(t))\vert \leq \chi _{1}^{*}\Vert z\Vert +\mathcal{G}^{*}\) and \(\vert f(t,z(t))\vert \leq \chi _{2}^{*}\Vert z\Vert +\mathcal{F}^{*}\), where \(\mathcal{G}^{*}=\sup_{t\in [0,1]}g(t,0)\) and \(\mathcal{F}^{*}=\sup_{t\in [0,1]}f(t,0)\), we deduce that
By taking the supremum over \(t\in [0,1]\) in the above inequality, we find a constant \(\mathcal{M}>0\) such that \(\Vert z\Vert \leq \mathcal{M}:=\frac{\mathcal{G}^{*}\Lambda ^{*}+\mathcal{F}^{*}}{1-\chi _{1}^{*}\Lambda ^{*}-\chi _{2}^{*}}\). Hence, ∑∗ is bounded. Thus, condition (b) is impossible. Now by using Theorem 3, the fractional hybrid Sturm–Liouville inclusion problem (3)–(4) has at least one solution. □
Now, we provide two examples to illustrate our main results.
Example 1
Consider the fractional hybrid Sturm–Liouville inclusion problem
with multipoint hybrid boundary conditions
Put \(\alpha =\frac{4}{5}\), \(\zeta =1\), \(\xi _{i}=\frac{1}{2^{i}}\) (\(i=1,2\)), \(\eta _{j}=\frac{1}{(-3)^{j}}\) (\(j=1,2,3\)), \(\nu =-\frac{1}{200}\), \(p(t)=101\sqrt{1+t}\), \(\tilde{p}(t)=e^{-\cos \pi t}\), \(\tilde{f}(t)=\tan ^{-1}(z(t)+1)\), \(g(t,z(t))=e^{-t^{2}-t-1}+\frac{2e^{-\sin \pi t}\vert z(t)\vert }{\vert z(t)\vert +1}\), and
Then, \(\vert g(t,z_{1})-g(t,z_{2})\vert \leq 2e^{-\sin \pi t} \vert z_{1}-z_{2}\vert \) for all \(z_{1},z_{2}\in \mathbb{R}\) and \(t\in [0,1]\). If \(\chi (t)=2e^{\sin \pi t}\), then \(\chi ^{*}=2\) and \(g_{0}=e^{-\frac{7}{4}}\). Also,
for all \(x\in \mathbb{R}\) and almost all \(t\in [0,1]\). Put \(\sigma (t)= \frac{\ln (3)}{4\pi }(1-e^{-4\pi t})\). Then, \(\Vert \sigma \Vert =\frac{\ln (3)}{4\pi }(1-e^{-4\pi })\), \(p=101\), \(\Vert \tilde{p}\Vert =1\), \(\vert \frac{\partial \tilde{f}(z)}{\partial z}\vert \leq 1=\tilde{\mathcal{K}}\), \(\tilde{f}_{0}=\frac{\pi }{4}\), and \(\bar{\Delta }=0.0198816623\). Hence,
and \((\frac{\chi ^{*}\Vert \sigma \Vert }{\Gamma (\alpha +2)} +\Vert \tilde{p}\Vert (2\widetilde{\mathcal{K}}\chi ^{*}\zeta +\chi ^{*}\tilde{f_{0}}+ \widetilde{\mathcal{K}}g_{0} ) )\bar{\Delta }\approx 0.1162851566< \frac{1}{2}\). Then by using Theorem 6, the fractional hybrid inclusion problem (17)–(18) has at least one solution.
Example 2
Consider the fractional hybrid Sturm–Liouville inclusion problem
with integral hybrid boundary conditions
Put \(\alpha =\frac{99}{100}\), \(\nu =-\frac{1}{111}\), \(\tilde{\varepsilon }=1\), \(p(t)=150\sqrt{4+t}\), \(\varpi (\theta )=4\theta +1\), \(\upsilon (\theta )=6\theta +1\),
\(f(t,x)=\frac{1}{60}e^{-\frac{t^{2}}{1+t^{2}}}(\frac{1}{40}z(t)+3)\), and \(\Psi (t,x)= [0, \frac{\cos \frac{\pi }{2} t\sin \frac{\pi \vert x\vert }{2(1+\vert x\vert )}}{1+\sin \frac{\pi \vert x\vert }{2(1+\vert x\vert )}} ]\). Then \(p=300\), \(\varpi (\frac{1}{4})-\varpi (0)=1\), \(\upsilon (1)-\upsilon (\frac{1}{3})=4\), \(\vert g(t,z_{1})-g(t,z_{2})\vert \leq e^{-\ln ^{2}(t+1)}\vert z_{1}-z_{2}\vert \), and
for all \(z_{1},z_{2}\in \mathbb{R}\) and \(t\in [0,1]\). If \(\chi _{1}(t)=e^{-\ln ^{2}(t+1)}\) and \(\chi _{2}(t)=\frac{1}{2400}e^{-\frac{t^{2}}{1+t^{2}}}\), then \(\chi _{1}^{*}=1\) and \(\chi _{1}^{*}=\frac{1}{2400}\). Also,
for all \(x\in \mathbb{R}\) and almost all \(t\in [0,1]\). Put \(\sigma (t)= \frac{2}{\pi }\cos ^{2} \frac{\pi }{2} t\). Then, \(\Vert \sigma \Vert =\frac{2}{\pi }\) and so
Now by using Theorem 9, the hybrid inclusion problem (19)–(20) has at least one solution.
4 Conclusion
Many natural processes are modeled by some types of fractional differential equations. This diversity factor in studying complicated fractional integro-differential equations increases our ability for exact modeling of more phenomena. We know that inclusion problems are real generalizations for differential equations and some economic phenomena could be model by inclusions. Thus, it is important that we study different inclusion problems, especially those related to well-known differential equations such as Sturm–Liouville. In this work, we review fractional hybrid inclusion version of the Sturm–Liouville equation. In this way, we investigate two fractional hybrid Sturm–Liouville differential inclusions with multipoint and integral hybrid boundary conditions. Also, we provide two examples to illustrate our main results.
References
Joannopoulos, J.D., Johnson, S.G., Winnn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton (2008)
Teschl, G.: Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators. Am. Math. Soc., New York (2014)
Al-Mdallal, Q.M.: An efficient method for solving fractional Sturm–Liouville problems. Chaos Solitons Fractals 40(1), 183–189 (2009)
Ashrafyan, Y.: A new kind of uniqueness theorems for inverse Sturm–Liouville problems. Bound. Value Probl. 2017, 79 (2017). https://doi.org/10.1186/s13661-017-0813-x
Bensidhoum, F.Z., Dib, H.: On some regular fractional Sturm–Liouville problems with generalized Dirichlet conditions. J. Integral Equ. Appl. 28(4), 459–480 (2016)
El-Sayed, A.M.A., Gaafar, F.M.: Existence and uniqueness of solution for Sturm–Liouville fractional differential equation with multi-point boundary condition via Caputo derivative. Adv. Differ. Equ. 2019, 46 (2019)
Erturk, V.S.: Computing eigenelements of Sturm–Liouville problems of fractional order via fractional differential transform method. Math. Comput. Appl. 16(3), 712–720 (2011)
Hassana, A.A.: Green’s function solution of non-homogeneous regular Sturm–Liouville problem. J. Appl. Comput. Math. 6(3), 1000362 (2017)
Kiataramkul, C., Ntouyas, S.K., Tariboon, J., Kijjathanakorn, A.: Generalized Sturm–Liouville and Langevin equations via Hadamard fractional derivatives with anti-periodic boundary conditions. Bound. Value Probl. 2016, 217 (2016)
Li, Y., Sun, S., Han, Z., Lu, H.: Existence of positive solutions for Sturm–Liouville boundary value problems on the half-line. Abstr. Appl. Anal., 2013, Article ID 301560 (2013)
Lian, H., Ge, W.: Existence of positive solutions for Sturm–Liouville boundary value problems on the half-line. J. Math. Anal. Appl. 321, 781–792 (2006)
Liu, Y., He, T., Shi, H.: Three positive solutions of Sturm–Liouville boundary value problems for fractional differential equations. Differ. Equ. Appl. 5(1), 127–152 (2013)
Sun, F., Li, K., Qi, J., Liao, B.: Non-real eigenvalues of nonlocal indefinite Sturm–Liouville problems. Bound. Value Probl. 2019, 176 (2019)
Xu, J., Abernathy, Z.: On the solvability of nonlinear Sturm–Liouville problems. J. Math. Anal. Appl. 387, 310–319 (2012)
Yang, Z.: Positive solutions for singular Sturm–Liouville boundary value problems on the half line. Electron. J. Differ. Equ. 2010(171), 1 (2010)
Zhao, J., Ge, W.: Existence results of a kind of Sturm–Liouville type singular boundary value problem with non-linear boundary conditions. J. Inequal. Appl., 2012, 197 (2012)
Baleanu, D., Aydogan, S.M., Mohammadi, H., Rezapour, S.: On modelling of epidemic childhood diseases with the Caputo–Fabrizio derivative by using the Laplace Adomian decomposition method. Alex. Eng. J. (2020). https://doi.org/10.1016/j.aej.2020.05.007
Tuan, N.H., Mohammadi, H., Rezapour, S.: A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Solitons Fractals 140, 110107 (2020). https://doi.org/10.1016/j.chaos.2020.110107
Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 2020, 64 (2020). https://doi.org/10.1186/s13661-020-01361-0
Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: Analysis of the human liver model with Caputo–Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020)
Ahmad, B., Ntouyas, S.K., Tariboon, J.: On hybrid Caputo fractional integro-differential inclusions with nonlocal conditions. J. Nonlinear Sci. Appl. 9, 4235–4246 (2016)
Charandabi, Z.Z., Rezapour, S., Ettefagh, M.: On a fractional hybrid version of the Sturm–Liouville equation. Adv. Differ. Equ. 2020, 301 (2020). https://doi.org/10.1186/s13662-020-02765-z
Sun, S., Zhao, Y., Han, Z., Li, Y.: The existence of solutions for boundary value problem of fractional hybrid differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 4961–4967 (2012)
Ullah, Z., Ali, A., Khan, R.A., Iqbal, M.: Existence results to a class of hybrid fractional differential equations. Matrix Sci. Math. 2(1), 13–17 (2018)
Zhao, Y., Sun, S., Han, Z., Li, Q.: Theory of fractional hybrid differential equations. Comput. Math. Appl. 62(3), 1312–1324 (2011). https://doi.org/10.1016/j.camwa.2011.03.041
Baleanu, D., Etemad, S., Rezapour, S.: On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators. Alex. Eng. J. (2020). https://doi.org/10.1016/j.aej.2020.04.053
Baleanu, D., Rezapour, S., Mohammadi, H.: Some existence results on nonlinear fractional differential equations. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 371, 20120144 (2013). https://doi.org/10.1098/rsta.2012.0144
Ghorbanian, R., Hedayati, V., Postolache, M., Rezapour, S.: Attractivity for a k-dimensional system of fractional functional differential equations and global attractivity for a k-dimensional system of nonlinear fractional differential equations. J. Inequal. Appl., 2014, 319 (2014) https://doi.org/10.1186/1029-242X-2014-319
Baleanu, D., Rezapour, S., Etemad, S., Alsaedi, A.: On a time-fractional partial integro-differential equation via three-point boundary value conditions. Math. Probl. Eng., 2015, Article ID 896871 (2015). https://doi.org/10.1155/2015/785738
Agarwal, R.P., Baleanu, D., Hedayati, V., Rezapour, S.: Two fractional derivative inclusion problems via integral boundary conditions. Appl. Math. Comput. 257, 205–212 (2015). https://doi.org/10.1016/j.amc.2014.10.082
Alsaedi, A., Baleanu, D., Etemad, S., Rezapour, S.: On coupled systems of time-fractional differential problems by using a new fractional derivative. J. Funct. Spaces, 2016, Article ID 4626940 (2016). https://doi.org/10.1155/2016/4626940
Aydogan, S.M., Nazemi, Z., Rezapour, S.: Positive solutions for a sum-type singular fractional integro-differential equation with m-point boundary conditions. Univ. Politech. Buchar. Sci. Bull. Ser. A 79(1), 89–98 (2017)
Etemad, S., Rezapour, S., Samei, M.E.: On a fractional Caputo–Hadamard inclusion problem with sum boundary value conditions by using approximate endpoint property. Math. Model. Appl. Sci. (2020). https://doi.org/10.1002/mma.6644
Rezapour, S., Samei, M.E.: On the existence of solutions for a multi-singular pointwise defined fractional q-integro-differential equation. Bound. Value Probl. 2020, 38 (2020). https://doi.org/10.1186/s13661-020-01342-3
Samei, M.E., Rezapour, S.: On a system of fractional q-differential inclusions via sum of two multi-term functions on a time scale. Bound. Value Probl. 2020, 135 (2020). https://doi.org/10.1186/s13661-020-01433-1
Dhage, B.C., Lakshmikantham, V.: Basic results on hybrid differential equation. Nonlinear Anal. Hybrid Syst. 4, 414–424 (2010)
Baleanu, D., Hedayati, V., Rezapour, S., Al Qurashi, M.M.: On two fractional differential inclusions. SpringerPlus 5(1), 882 (2016). https://doi.org/10.1186/s40064-016-2564-z
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Samko, G., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York (1993)
Deimling, K.: Multi-Valued Differential Equations. de Gruyter, Berlin (1992)
Aubin, J., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Springer, Berlin (1984). https://doi.org/10.1007/978-3-642-69512-4
Lasota, A., Opial, Z.: An application of the Kakutani–Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 13, 781–786 (1965)
Dhage, B.C.: On solvability of operator inclusion \(x\in ax bx+cx\) in Banach algebras and differential inclusions. Commun. Appl. Anal. 14, 567–596 (2010)
Acknowledgements
The first author was supported by Sarab Branch, Islamic Azad University. The second author was supported by Azarbaijan Shahid Madani University. The authors express their gratitude to dear unknown referees for their helpful suggestions which improved the final version of this paper.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Authors’ information
Zohreh Zeinalabedini Charandabi: z.z.charandabi@iausa.ac.ir.
Author information
Authors and Affiliations
Contributions
The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The author declares that he/she has no competing interests.
Consent for publication
Not applicable.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Charandabi, Z.Z., Rezapour, S. Fractional hybrid inclusion version of the Sturm–Liouville equation. Adv Differ Equ 2020, 546 (2020). https://doi.org/10.1186/s13662-020-03011-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-020-03011-2