1 Introduction

It can be said that most physical or engineering phenomena can be modeled with some categories such time-dependent (or time-fractional), fractional differential and some variables partial equations. One can find many published papers on delayed time-fractional problems, fractional differential equations [1–30] and some variables partial fractional problems [31–36]. During the history of mathematics, physics and engineering, we can find many equations which have a special role in progress of these sciences. One of the important frameworks of problems is the Sturm–Liouville differential equation (in brief SLDE) have been in the spotlight of the mathematicians of applied mathematics, engineering and scientists of physics, quantum mechanics, classical mechanics (see, [37, 38] and the references therein). In such a manner, it is important that mathematicians and researchers design complicated and more general abstract mathematical models of procedures in the format of applicable fractional SLDE [33, 39–41]. One can find a variety of recent work about this equation, but the aim of this work is studying partial version of the Sturm–Liouville differential equation.

Let \(\hat{k}=(\hat{k}_{1},\hat{k}_{2})\) where \(\hat{k}_{1},\hat{k}_{2}>0\) and \(\mathcal{J}_{a_{0}}=[0,a_{0}]\) and \(\mathcal{J}_{b_{0}}=[0,b_{0}]\) where \(a_{0},b_{0}>0\). For \(\sigma \in \mathcal{L}^{1}(\mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}, \mathbb{R})=\mathcal{L}^{1}(\mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}})\), the partial left-sided mixed Riemann–Liouville integral (of order k̂) is defined by (see [42])

$$\begin{aligned} \mathcal{I}_{0}^{\hat{k}}\sigma \bigl(p^{\star },q^{\star } \bigr)= \int _{0}^{p^{ \star }} \int _{0}^{q^{\star }} \frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})} \sigma (s,t)\,dt \,ds. \end{aligned}$$

Also the partial derivative in the sense of Caputo (of order k̂) is defined by

$$\begin{aligned} D_{c_{0}}^{\hat{k}}\sigma \bigl(p^{\star },q^{\star } \bigr)&=\mathcal{I}_{0}^{1- \hat{k}}\biggl(\frac{\partial ^{2}}{\partial p^{\star }\partial q^{\star }} \sigma \bigl(p^{\star },q^{\star }\bigr)\biggr) \\ &= \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})} \frac{\partial ^{2}}{\partial s\,\partial t} \sigma (s,t)\,dt\,ds. \end{aligned}$$

Let \((\mathcal{Z}^{\star },d_{\mathcal{Z}^{\star }})\) be a metric space. \(\mathcal{P}_{cl}(\mathcal{Z}^{\star })\) the set of all closed subsets of \(\mathcal{Z}^{\star }\) and \(2^{\mathcal{Z}^{\star }}\) the set of all nonempty subsets of \(\mathcal{Z}^{\star }\). It is well known that the Pompeiu–Hausdorff metric \(PH_{d_{\mathcal{Z}^{\star }}}:\mathcal{P}_{cl}(\mathcal{Z}^{\star }) \times \mathcal{P}_{cl}(\mathcal{Z}^{\star })\to \mathbb{R}\cup \{ \infty \}\) is defined by

$$\begin{aligned} PH_{d_{\mathcal{Z}^{\star }}}\bigl(A_{1}^{d_{\mathcal{Z}^{\star }}}, A_{2}^{d_{ \mathcal{Z}^{\star }}} \bigr)=\max \Bigl\{ \sup_{a_{1}^{\star }\in A_{1}^{d_{ \mathcal{Z}^{\star }}}} d_{\mathcal{Z}^{\star }} \bigl(a_{1}^{\star },A_{2}^{d_{ \mathcal{Z}^{\star }}}\bigr),\sup _{a_{2}^{\star }\in A_{2}^{d_{\mathcal{Z}^{ \star }}}} d_{\mathcal{Z}^{\star }}\bigl(A_{1}^{d_{\mathcal{Z}^{\star }}},a_{2}^{ \star } \bigr)\Bigr\} \end{aligned}$$

for all \(A_{1}^{d_{\mathcal{Z}^{\star }}}, A_{2}^{d_{\mathcal{Z}^{\star }}}\in \mathcal{P}(\mathcal{Z}^{\star })\), where \(d_{\mathcal{Z}^{\star }}(a_{1}^{\star },A_{2}^{d_{\mathcal{Z}^{\star }}})= \inf_{a_{1}^{\star }\in A_{1}^{d_{\mathcal{Z}^{\star }}}} d_{ \mathcal{Z}^{\star }}(a_{1}^{\star },a_{2}^{\star })\) and \(d_{\mathcal{Z}^{\star }}(A_{1}^{d_{\mathcal{Z}^{\star }}}, a_{2}^{\star })= \inf_{a_{2}^{\star }\in A_{2}^{d_{\mathcal{Z}^{\star }}}} d_{ \mathcal{Z}^{\star }}(a_{1}^{\star },a_{2}^{\star })\) [43]. We say that a set-valued mapping \(\Psi: \mathcal{Z}^{\star }\to \mathcal{P}_{cl}(\mathcal{Z}^{\star })\) is called Lipschitzian with Lipschitz constant \(k>0\) whenever \(PH_{d_{\mathcal{Z}^{\star }}}(\Psi (\sigma _{1}),\Psi (\sigma _{1})) \leq kd_{\mathcal{Z}^{\star }}(\sigma _{1},\sigma _{2})\) for all \(\sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }\). If \(0< k<1\), then we ay that Ψ is a contraction [43]. An operator \(\Psi: [0,1]\to \mathcal{P}_{cl}(\mathcal{R})\) is called measurable whenever the function \(t\to d_{\mathcal{Z}^{\star }}(\omega _{0},\Psi (t))=\inf \{|\omega _{0}-y|: y\in \Psi (t) \}\) is measurable for all real constant ω [43, 44]. The following notions were introduced in 2012 [45].

  • \(\Psi = \{\psi | \sum_{n=1}^{\infty }\psi ^{n}(t)<\infty, \forall t>0\}\) where \(\psi: [0,\infty )\times [0,\infty )\to [0,\infty )\).

  • Assume that \(\alpha:\mathcal{Z}^{\star }\times \mathcal{Z}^{\star }\to [0,\infty )\) and \(T:\mathcal{Z}^{\star }\to \mathcal{Z}^{\star }\) are two mappings. Now T is α-admissible whenever for each \(\sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }\) with \(\alpha (\sigma _{1},\sigma _{2})\geq 1\), we get \(\alpha (T\sigma _{1},T\sigma _{2})\geq 1\).

  • T is α-ψ-contractive mapping whence \(\alpha (\sigma _{1},\sigma _{2})d_{\mathcal{Z}^{\star }}(T\sigma _{1},T \sigma _{2})\leq \psi (d_{\mathcal{Z}^{\star }}(\sigma _{1},\sigma _{2}))\) for all \(\sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }\).

Lemma 1

([45])

Assume that the metric space \((\mathcal{Z}^{\star }, d_{\mathcal{Z}^{\star }})\) is complete, T is α-admissible and α-ψ-contractive mapping and there exists \(\sigma _{0}\in \mathcal{Z}^{\star }\) such that \(\alpha (\sigma _{0},T\sigma _{0})\geq 1\). Further, for every convergent sequence \(\{\sigma _{n}\}_{n\geq 1}\subseteq \mathcal{Z}^{\star }\) with \(\sigma _{n}\to \sigma \) and \(\alpha (\sigma _{n},\sigma _{n+1})\geq 1\) for all \(n\geq 1\), we have \(\alpha (\sigma _{n},\sigma )\geq 1\) for all \(n\geq 1\). Then T has a fixed point.

After this, multifunction version of α-ψ-contractive maps introduced in 2013 as follows [46].

  • A multifunction \(F: \mathcal{Z}^{\star } \to CB(\mathcal{Z}^{\star })\) is α-admissible whenever for each \(\sigma _{1} \in \mathcal{Z}^{\star }\) and \(\sigma _{2} \in F\sigma _{1}\) with \(\alpha (\sigma _{1}, \sigma _{2}) \geq 1\), we have \(\alpha (\sigma _{2},w_{0}) \geq 1\), for all \(w_{0} \in F\sigma _{2}\).

  • The metric space \(\mathcal{Z}^{\star }\) possesses the \(C_{\alpha }\)-property if for every convergent sequence \(\{\sigma _{n}\}_{n\geq 1}\subseteq \mathcal{Z}^{\star }\) with \(\sigma _{n}\to \sigma \) and \(\alpha (\sigma _{n},\sigma _{n+1})\geq 1\) for all \(n\geq 1\), there exists a subsequence \(\{\sigma _{n_{j}}\}_{j\geq 1}\) of \(\sigma _{n}\) such that \(\alpha (\sigma _{n_{j}},\sigma )\geq 1\) for all \(j\geq 1\).

  • F is α-ψ-contractive multifunction whenever \(\alpha (\sigma _{1},\sigma _{2})PH_{d_{\mathcal{Z}^{\star }}}(T \sigma _{1},T\sigma _{2})\leq \psi (d_{\mathcal{Z}^{\star }}(\sigma _{1}, \sigma _{2}))\) for all \(\sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }\).

Lemma 2

([46])

Assume that the metric space \((\mathcal{Z}^{\star }, d_{\mathcal{Z}^{\star }})\) is complete, F is α-admissible and α-ψ-contractive multifunction and there exist \(\sigma _{0}\in \mathcal{Z}^{\star }\) and \(\sigma _{1}\in F\sigma _{0}\) such that \(\alpha (\sigma _{0},\sigma _{1})\geq 1\). If \(\mathcal{Z}^{\star }\) possesses the \(C_{\alpha }\)-property, then F has a fixed point.

In this paper, first we investigate the partial fractional Sturm–Liouville differential equation

$$\begin{aligned} \textstyle\begin{cases} D^{\hat{k}}_{c_{0}} (l(p^{\star },q^{\star })\mathcal{D}^{ \hat{\ell }}_{c_{0}}\sigma (p^{\star },q^{\star }) )+o(p^{\star },q^{ \star })\sigma (p^{\star },q^{\star })=h(p^{\star },q^{\star })f(\sigma (p^{ \star },q^{\star })), \\ (p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}} (l(p^{\star },q^{\star })\mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma (p^{ \star },q^{\star }) )_{q^{\star }=0}=\theta _{1}(p^{\star }), \\ (l(p^{\star },q^{\star })\mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma (p^{ \star },q^{\star }) )_{p^{\star }=0}=\theta _{2}(q^{\star }), \\ \sigma (p^{\star },0)=\kappa (p^{\star }) \quad\text{and}\quad \sigma (0,q^{ \star })=\omega (q^{\star }), \end{cases}\displaystyle \end{aligned}$$
(1)

where \(\hat{k},\hat{\ell }\in (0,1]\times (0,1]\), \(D^{\hat{k}}_{c_{0}}\) and \(D^{\hat{\ell }}_{c_{0}}\) denote the Caputo partial fractional derivatives, \(l,o,h\) belong to \(\mathcal{C}(\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}})\) with \(l(p^{\star },q^{\star })\neq 0\) for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\) and \(f:\mathbb{R}\to \mathbb{R}\) is a function. Here, \(\theta _{1}:\mathcal{J}_{a_{0}}\to \mathbb{R}\), \(\theta _{2}:\mathcal{J}_{b_{0}}\to \mathbb{R}\), \(\kappa:\mathcal{J}_{a_{0}}\to \mathbb{R}\) and \(\omega:\mathcal{J}_{b_{0}}\to \mathbb{R}\) are absolutely continuous with \(\theta _{1}(0)=\theta _{2}(0)=\kappa (0)=\omega (0)\). Also, we investigate the partial fractional Sturm–Liouville differential inclusion problem

$$\begin{aligned} \textstyle\begin{cases} D^{\hat{k}}_{c_{0}} (l(p^{\star },q^{\star })\mathcal{D}^{ \hat{\ell }}_{c_{0}}\sigma (p^{\star },q^{\star }) )\in \mathcal{H} (p^{\star }q^{\star },\sigma (p^{\star },q^{\star }) ), (p^{ \star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}},\\ (l(p^{\star },q^{\star })\mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma (p^{ \star },q^{\star }) )_{q^{\star }=0}=\theta _{1}(p^{\star }), \\ (l(p^{\star },q^{\star })\mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma (p^{ \star },q^{\star }) )_{p^{\star }=0}=\theta _{2}(q^{\star }), \\ \sigma (p^{\star },0)=\kappa (p^{\star }) \quad\text{and}\quad \sigma (0,q^{ \star })=\omega (q^{\star }), \end{cases}\displaystyle \end{aligned}$$
(2)

where \(\hat{k},\hat{\ell }\in (0,1]\times (0,1]\), \(D^{\hat{k}}_{c_{0}}\) and \(D^{\hat{\ell }}_{c_{0}}\) denote the Caputo partial fractional derivatives, \(l,o,h\) belong to \(\mathcal{C}(\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}})\) with \(l(p^{\star },q^{\star })\neq 0\) for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\) and \(f:\mathbb{R}\to \mathbb{R}\) is a function. Here, \(\theta _{1}:\mathcal{J}_{a_{0}}\to \mathbb{R}\), \(\theta _{2}:\mathcal{J}_{b_{0}}\to \mathbb{R}\), \(\kappa:\mathcal{J}_{a_{0}}\to \mathbb{R}\) and \(\omega:\mathcal{J}_{b_{0}}\to \mathbb{R}\) are absolutely continuous with \(\theta _{1}(0)=\theta _{2}(0)=\kappa (0)=\omega (0)\). Also, \(\mathcal{H}:\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\times \mathbb{R}\to \mathcal{P}_{cl}(\mathbb{R})\) is an integrable bounded multifunction so that \(\mathcal{H} (.,.,\sigma )\) is measurable for all \(\sigma \in \mathbb{R}\).

2 Main results

Assume that \(\mathcal{Z}^{\star }=\{\sigma | \sigma \in \mathcal{C}(\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}})\}\) and \(\|\sigma \|=\sup_{(p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}}|\sigma (p^{\star },q^{\star })|\), where \(\sigma \in \mathcal{Z}^{\star }\). Then \((\mathcal{Z}^{\star },\|.\|)\) is a Banach space.

Lemma 3

Let \(\hat{k}=(\hat{k}_{1},\hat{k}_{2}),\hat{\ell }=(\hat{\ell }_{1}, \hat{\ell }_{2})\in (0,1]\times (0,1]\) and \(g\in \mathcal{L}^{1}(\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}})\). Consider the problem

$$\begin{aligned} \mathcal{D}^{\hat{k}}_{c_{0}} \bigl(l\bigl(p^{\star }, q^{\star }\bigr) \mathcal{D}^{\hat{\ell }}_{c_{0}} \sigma \bigl(p^{\star },q^{\star }\bigr) \bigr) =g\bigl(p^{ \star },q^{\star } \bigr), \end{aligned}$$
(3)

with boundary conditions

$$\begin{aligned} \textstyle\begin{cases} (l(p^{\star },q^{\star })\mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma (p^{ \star },q^{\star }) )_{y=0}=\theta _{1}(p^{\star }), & \\ (l(p^{\star },q^{\star })\mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma (p^{ \star },q^{\star }) )_{x=0}=\theta _{2}(q^{\star }), \\ \sigma (p^{\star },0)=\kappa (p^{\star }) \quad \textit{and}\quad \sigma (0,q^{ \star })=\omega (q^{\star }). \end{cases}\displaystyle \end{aligned}$$
(4)

Then the function \(\sigma \in \mathcal{C}(\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}})\) is a solution of the problem (3)–(4) whenever

$$\begin{aligned} \sigma \bigl(p^{\star },q^{\star }\bigr)={}& \Theta \bigl(p^{\star },q^{\star }\bigr) \\ &{}+ \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1} (s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}g(\wp _{\bar{1}},\zeta _{\bar{2}})}{\Gamma (\hat{k}_{1}) \Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}, \end{aligned}$$

where

$$\begin{aligned} \Theta \bigl(p^{\star },q^{\star }\bigr)= \kappa \bigl(p^{\star }\bigr)+\omega \bigl(q^{ \star }\bigr)-\kappa (0)+ \mathcal{I}^{\hat{\ell }}_{0} \biggl( \frac{\theta _{1}(p^{\star })+\theta _{2}(q^{\star })-\theta _{1}(0)}{l(p^{\star },q^{\star })} \biggr). \end{aligned}$$

Proof

Note that Eq. (3) can be written as

$$\begin{aligned} \mathcal{I}^{1-\hat{k}}_{0} \biggl( \frac{\partial ^{2}}{\partial p^{\star }\partial q^{\star }} \bigl(l\bigl(p^{ \star },q^{\star }\bigr) \mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma \bigl(p^{\star },q^{ \star } \bigr) \bigr) \biggr)=g\bigl(p^{\star },q^{\star }\bigr). \end{aligned}$$

Operating by \(\mathcal{I}^{\hat{k}}_{0}\) on both sides we get

$$\begin{aligned} \mathcal{I}^{1}_{0} \biggl( \frac{\partial ^{2}}{\partial p^{\star }\partial q^{\star }} \bigl(l\bigl(p^{ \star },q^{\star }\bigr) \mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma \bigl(p^{\star },q^{ \star } \bigr) \bigr) \biggr)=\mathcal{I}^{\hat{k}}_{0}g \bigl(p^{\star },q^{\star }\bigr). \end{aligned}$$

Since

$$\begin{aligned} &\mathcal{I}^{1}_{0} \biggl( \frac{\partial ^{2}}{\partial p^{\star }\partial q^{\star }} \bigl(l\bigl(p^{ \star },q^{\star }\bigr) \mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma \bigl(p^{\star },q^{ \star } \bigr) \bigr) \biggr) \\ &\quad=l\bigl(p^{\star },q^{\star }\bigr)\mathcal{D}^{\hat{\ell }}_{c_{0}} \sigma \bigl(p^{ \star },q^{\star }\bigr) - \bigl(l\bigl(p^{\star },q^{\star } \bigr)\mathcal{D}^{\hat{\ell }}_{c_{0}} \sigma \bigl(p^{\star },q^{\star } \bigr) \bigr)_{q^{\star }=0} \\ &\qquad{}- \bigl(l\bigl(p^{\star },q^{\star }\bigr)\mathcal{D}^{\hat{\ell }}_{c_{0}} \sigma \bigl(p^{ \star },q^{\star }\bigr) \bigr)_{p^{\star }=0} + \bigl(l\bigl(p^{\star },q^{\star }\bigr) \mathcal{D}^{\hat{\ell }}_{c_{0}} \sigma \bigl(p^{\star },q^{\star }\bigr) \bigr)_{p^{ \star }=0, q^{\star }=0} \\ &\quad=l\bigl(p^{\star },q^{\star }\bigr)\mathcal{D}^{\hat{\ell }}_{c_{0}} \sigma \bigl(p^{ \star },q^{\star }\bigr)-\theta _{1} \bigl(p^{\star }\bigr)-\theta _{2}\bigl(q^{\star }\bigr)+ \theta _{1}(0), \end{aligned}$$

we get \(l(p^{\star },q^{\star })\mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma (p^{ \star },q^{\star })=\theta _{1}(p^{\star })+\theta _{2}(q^{\star })- \theta _{1}(0)+\mathcal{I}^{\hat{k}}_{0}g(p^{\star },q^{\star })\). Hence,

$$\begin{aligned} \mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma \bigl(p^{\star },q^{ \star }\bigr)= \frac{\theta _{1}(p^{\star })+\theta _{2}(q^{\star })-\theta _{1}(0)}{l(p^{\star },q^{\star })}+ \frac{1}{l(p^{\star },q^{\star })} \mathcal{I}^{\hat{k}}_{0}g\bigl(p^{ \star },q^{\star } \bigr). \end{aligned}$$

This equation can be written as

$$\begin{aligned} \mathcal{I}^{1-\hat{\ell }}_{0} \biggl( \frac{\partial ^{2}}{\partial p^{\star }\partial q^{\star }}\sigma \bigl(p^{ \star },q^{\star }\bigr) \biggr)= \frac{\theta _{1}(p^{\star })+\theta _{2}(q^{\star })-\theta _{1}(0)}{l(p^{\star },q^{\star })}+ \frac{1}{l(p^{\star },q^{\star })} \mathcal{I}^{\hat{k}}_{0}g \bigl(p^{ \star },q^{\star }\bigr). \end{aligned}$$

Again operating by \(\mathcal{I}^{\hat{\ell }}_{0}\) on both sides, we obtain

$$\begin{aligned} \mathcal{I}^{1}_{0} \biggl( \frac{\partial ^{2}}{\partial p^{\star }\partial q^{\star }}\sigma \bigl(p^{ \star },q^{\star }\bigr) \biggr)= \mathcal{I}^{\hat{\ell }}_{0} \biggl( \frac{\theta _{1}(p^{\star })+\theta _{2}(q^{\star }) -\theta _{1}(0)}{l(p^{\star },q^{\star })} \biggr)+ \mathcal{I}^{\hat{\ell }}_{0} \biggl(\frac{1}{l(p^{\star },q^{\star })} \mathcal{I}^{\hat{k}}_{0}g\bigl(p^{\star },q^{\star } \bigr) \biggr). \end{aligned}$$

Since

$$\begin{aligned} \mathcal{I}^{1}_{0} \biggl( \frac{\partial ^{2}}{\partial p^{\star }\partial q^{\star }}\sigma \bigl(p^{ \star },q^{\star }\bigr) \biggr)&= \sigma \bigl(p^{\star },q^{\star }\bigr)-\sigma \bigl(p^{ \star },0 \bigr)-\sigma \bigl(0,q^{\star }\bigr)+\sigma (0,0) \\ &=\sigma \bigl(p^{\star },q^{\star }\bigr)-\kappa \bigl(p^{\star }\bigr)-\omega \bigl(q^{\star }\bigr)+ \kappa (0), \end{aligned}$$

we get

$$\begin{aligned} \sigma \bigl(p^{\star },q^{\star }\bigr)={}& \kappa \bigl(p^{\star }\bigr) + \omega \bigl(q^{\star }\bigr)-\kappa (0)+ \mathcal{I}^{\hat{\ell }}_{0} \biggl( \frac{\theta _{1}(p^{\star }) +\theta _{2}(q^{\star })-\theta _{1}(0)}{l(p^{\star },q^{\star })} \biggr) \\ &{} + \mathcal{I}^{\hat{\ell }}_{0} \biggl( \frac{1}{l(p^{\star },q^{\star })} \mathcal{I}^{\hat{k}}_{0}g\bigl(p^{\star },q^{ \star } \bigr) \biggr)=\Theta \bigl(p^{\star },q^{\star }\bigr)+ \mathcal{I}^{ \hat{\ell }}_{0} \biggl(\frac{1}{l(p^{\star },q^{\star })} \mathcal{I}^{ \hat{k}}_{0}g\bigl(p^{\star },q^{\star } \bigr) \biggr). \end{aligned}$$

On the other hand,

$$\begin{aligned} & \mathcal{I}^{\hat{\ell }}_{0} \biggl( \frac{1}{l(p^{\star },q^{\star })} \mathcal{I}^{\hat{k}}_{0}g\bigl(p^{\star },q^{ \star } \bigr) \biggr)\\ &\quad= \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})} \frac{1}{l(s,t)} \mathcal{I}^{\hat{k}}_{0}g(s,t)\,dt\,ds \\ &\quad= \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \frac{1}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})} \bigl(p^{\star }-s\bigr)^{ \hat{k}_{1}-1}\bigl(q^{\star }-t \bigr)^{\hat{k}_{2}-1}\frac{1}{l(s,t)} \\ &\qquad{}\times \biggl( \int _{0}^{s} \int _{0}^{t} \frac{(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}}{\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})}g( \wp _{\bar{1}},\zeta _{\bar{2}})\,d\wp _{\bar{1}} \,d\zeta _{\bar{2}} \biggr)\,dt\,ds \\ &\quad= \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}g(\wp _{\bar{1}},\zeta _{\bar{2}})}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \end{aligned}$$

and so

$$\begin{aligned} \sigma \bigl(p^{\star },q^{\star }\bigr)={}& \Theta \bigl(p^{\star },q^{\star }\bigr) \\ &{}+ \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &{}\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}g(\wp _{\bar{1}},\zeta _{\bar{2}})}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}. \end{aligned}$$

This completes the proof. □

Now we establish and prove our first main theorem.

Theorem 4

Assume that \(\upsilon: \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is a function and \(\Phi: \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\to [0,\infty )\) is a bounded function such that

$$\begin{aligned} \bigl\vert f\bigl(\sigma _{1} \bigl(p^{\star },q^{\star }\bigr)\bigr)-f\bigl(\sigma _{2} \bigl(p^{ \star },q^{\star }\bigr)\bigr) \bigr\vert \leq \Phi \bigl(p^{\star },q^{\star }\bigr) \bigl\vert \sigma _{1} \bigl(p^{ \star },q^{\star }\bigr)-\sigma _{2} \bigl(p^{\star },q^{\star }\bigr) \bigr\vert \end{aligned}$$

for all \(\sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }\) with \(\upsilon (\sigma _{1}(p^{\star },q^{\star }),\sigma _{2}(p^{\star },q^{ \star }))\geq 0\), where \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\). Suppose that, for all \(\sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }\) with \(\upsilon (\sigma _{1}(p^{\star },q^{\star }),\sigma _{2}(p^{\star },q^{ \star }))\geq 0\), we have \(\upsilon (\mathcal{Q}_{\bar{0}}^{\star }\sigma _{1}(p^{\star },q^{ \star }),\mathcal{Q}_{\bar{0}}^{\star }\sigma _{2}(p^{\star },q^{\star })) \geq 0\), where

$$\begin{aligned} &\mathcal{Q}_{\bar{0}}^{\star }\sigma \bigl(p^{\star },q^{\star }\bigr)\\ &\quad= \Theta \bigl(p^{\star },q^{\star } \bigr)\\ &\qquad{} + \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t}\\ &\qquad \frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1} (s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}\mathcal{H}(\wp _{\bar{1}},\zeta _{\bar{2}},\sigma (\wp _{\bar{1}},\zeta _{\bar{2}}))}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}, \end{aligned}$$

\(\Theta (p^{\star },q^{\star })=\kappa (p^{\star })+\omega (q^{\star })- \kappa (0)+\mathcal{I}^{\hat{\ell }}_{0} ( \frac{\theta _{1}(p^{\star })+\theta _{2}(q^{\star })-\theta _{1}(0)}{l(p^{\star },q^{\star })} )\),

$$\begin{aligned} \mathcal{H}\bigl(\wp _{\bar{1}},\zeta _{\bar{2}},\sigma (\wp _{ \bar{1}},\zeta _{\bar{2}})\bigr)=h(\wp _{\bar{1}},\zeta _{\bar{2}})f\bigl( \sigma (\wp _{\bar{1}},\zeta _{\bar{2}})\bigr)-o(\wp _{\bar{1}},\zeta _{ \bar{2}})\sigma (\wp _{\bar{1}},\zeta _{\bar{2}}) \end{aligned}$$

and there exists \(\sigma _{0}\) so that \(\upsilon (\sigma _{0}(p^{\star },q^{\star }),\mathcal{Q}_{\bar{0}}^{ \star }\sigma _{0}(p^{\star },q^{\star }))\geq 0\) whenever \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\). Assume that, for every sequence \(\{\sigma _{n}\}_{n\geq 1}\subseteq \mathcal{Z}^{\star }\) with \(\sigma _{n}\to \sigma \) and \(\upsilon (\sigma _{n}(p^{\star },q^{\star }),\sigma _{n+1}(p^{\star },q^{ \star }))\geq 0\) for all \(n\geq 1\) and \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\), we have \(\upsilon (\sigma _{n}(p^{\star },q^{\star }),\sigma (p^{\star },q^{ \star }))\geq 0\) for all \(n\geq 1\) and \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\). If \(\frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}}(\|h\|\Phi ^{*}+\|o\|)}{l\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)}<1\), then the fractional Sturm–Liouville problem (1) has a solution, where \(\Phi ^{*}=\sup_{(p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}}\Phi (p^{\star },q^{\star })\) and

$$\begin{aligned} l=\inf_{(p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}}l\bigl(p^{\star },q^{\star }\bigr). \end{aligned}$$

Proof

By using Lemma 3, \(\sigma _{0}\) is a solution of the partial fractional Sturm–Liouville problem (1) if and only if \(\sigma _{0}=\mathcal{Q}_{\bar{0}}^{\star }\sigma _{0}\). Let \(\sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }\) and \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\) with

$$\begin{aligned} \upsilon \bigl(\sigma _{1}\bigl(p^{\star },q^{\star }\bigr), \sigma _{2}\bigl(p^{\star },q^{ \star }\bigr)\bigr)\geq 0. \end{aligned}$$

Hence, we get

$$\begin{aligned} & \bigl\vert \mathcal{Q}_{\bar{0}}^{\star } \sigma _{1}\bigl(p^{\star },q^{ \star }\bigr)- \mathcal{Q}_{\bar{0}}^{\star }\sigma _{2} \bigl(p^{\star },q^{\star }\bigr) \bigr\vert \\ &\quad = \biggl\vert \Theta \bigl(p^{\star },q^{\star }\bigr) + \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1} (t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}\mathcal{H}(\wp _{\bar{1}},\zeta _{\bar{2}},\sigma _{1}(\wp _{\bar{1}},\zeta _{\bar{2}}))}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \\ &\qquad{}-\Theta \bigl(p^{\star },q^{\star }\bigr) - \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1} \mathcal{H}(\wp _{\bar{1}},\zeta _{\bar{2}},\sigma _{2}(\wp _{\bar{1}},\zeta _{\bar{2}}))}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \biggr\vert \\ &\quad\leq \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad{}\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1} \vert \mathcal{N}(\wp _{\bar{1}},\zeta _{\bar{2}}) \vert }{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}, \end{aligned}$$

where \(\mathcal{N}(\wp _{\bar{1}},\zeta _{\bar{2}})=\mathcal{H}(\wp _{ \bar{1}},\zeta _{\bar{2}},\sigma _{1}(\wp _{\bar{1}},\zeta _{\bar{2}}))- \mathcal{H}(\wp _{\bar{1}},\zeta _{\bar{2}},\sigma _{2}(\wp _{\bar{1}}, \zeta _{\bar{2}}))\). Since

$$\begin{aligned} & \bigl\vert \mathcal{H}\bigl(\wp _{\bar{1}}, \zeta _{\bar{2}},\sigma _{1}( \wp _{\bar{1}},\zeta _{\bar{2}})\bigr)-\mathcal{H}\bigl(\wp _{\bar{1}},\zeta _{ \bar{2}},\sigma _{2}(\wp _{\bar{1}},\zeta _{\bar{2}})\bigr) \bigr\vert \\ &\quad\leq \bigl\vert h(\wp _{\bar{1}},\zeta _{\bar{2}}) \bigl(f\bigl( \sigma _{1}(\wp _{ \bar{1}},\zeta _{\bar{2}})\bigr)-f\bigl( \sigma _{2}(\wp _{\bar{1}},\zeta _{ \bar{2}})\bigr) \bigr)-o( \wp _{\bar{1}},\zeta _{\bar{2}}) \bigl(\sigma _{1}(\wp _{ \bar{1}},\zeta _{\bar{2}})-\sigma _{2}(\wp _{\bar{1}},\zeta _{\bar{2}})\bigr) \bigr\vert \\ &\quad\leq \bigl\vert h(\wp _{\bar{1}},\zeta _{\bar{2}}) \bigr\vert \bigl\vert f\bigl(\sigma _{1}(\wp _{ \bar{1}},\zeta _{\bar{2}})\bigr)-f\bigl(\sigma _{2}(\wp _{\bar{1}},\zeta _{ \bar{2}})\bigr) \bigr\vert + \bigl\vert o(\wp _{\bar{1}},\zeta _{\bar{2}}) \bigr\vert \bigl\vert \sigma _{1}(\wp _{ \bar{1}},\zeta _{\bar{2}})-\sigma _{2}(\wp _{\bar{1}},\zeta _{\bar{2}}) \bigr\vert \\ &\quad\leq \bigl\vert h(\wp _{\bar{1}},\zeta _{\bar{2}}) \bigr\vert \Phi (\wp _{\bar{1}},\zeta _{ \bar{2}}) \bigl\vert \sigma _{1}(\wp _{\bar{1}},\zeta _{\bar{2}})-\sigma _{2}( \wp _{\bar{1}},\zeta _{\bar{2}}) \bigr\vert + \bigl\vert o(\wp _{\bar{1}},\zeta _{\bar{2}}) \bigr\vert \bigl\vert \sigma _{1}(\wp _{\bar{1}},\zeta _{\bar{2}})-\sigma _{2}( \wp _{ \bar{1}},\zeta _{\bar{2}}) \bigr\vert \\ &\quad\leq \bigl( \Vert h \Vert \Phi ^{*}+ \Vert o \Vert \bigr) \Vert \sigma _{1}-\sigma _{2} \Vert , \end{aligned}$$

we have

$$\begin{aligned} \begin{aligned} & \bigl\vert \mathcal{Q}_{\bar{0}}^{\star } \sigma _{1}\bigl(p^{\star },q^{ \star }\bigr)- \mathcal{Q}_{\bar{0}}^{\star }\sigma _{2} \bigl(p^{\star },q^{\star }\bigr) \bigr\vert \\ &\quad\leq \frac{( \Vert h \Vert \Phi ^{*}+ \Vert o \Vert ) \Vert \sigma _{1}-\sigma _{2} \Vert }{l\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})} \\ &\qquad{}\times \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t}\bigl(p^{\star }-s \bigr)^{\hat{k}_{1}-1}\bigl(q^{\star }-t\bigr)^{ \hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}\\ &\qquad{}\times(t-\zeta _{\bar{2}})^{ \hat{\ell }_{2}-1}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}, \end{aligned} \end{aligned}$$
(5)

where

$$\begin{aligned} & \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t}\bigl(p^{\star }-s \bigr)^{\hat{k}_{1}-1}\bigl(q^{\star }-t\bigr)^{ \hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{ \hat{\ell }_{2}-1}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \\ &\quad= \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \frac{s^{\hat{\ell }_{1}}(p^{\star }-s)^{\hat{k}_{1}-1}}{\hat{\ell }_{1}} \frac{t^{\hat{\ell }_{2}}(q^{\star }-t)^{\hat{k}_{2}-1}}{\hat{\ell }_{2}} \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \\ &\quad= \int _{0}^{p^{\star }} \frac{s^{\hat{\ell }_{1}}(p^{\star }-s)^{\hat{k}_{1}-1}}{\hat{\ell }_{1}} \,d\wp _{\bar{1}}\times \int _{0}^{q^{\star }} \frac{t^{\hat{\ell }_{2}}(q^{\star }-t)^{\hat{k}_{2}-1}}{\hat{\ell }_{2}} \,d\zeta _{\bar{2}} \\ &\quad\leq \frac{1}{\hat{\ell }_{1}\hat{\ell }_{2}} \int _{0}^{a_{0}}s^{ \hat{\ell }_{1}}(a_{0}-s)^{\hat{k}_{1}-1} \,ds\times \int _{0}^{b_{0}}t^{ \hat{\ell }_{2}}(b_{0}-t)^{\hat{k}_{2}-1} \,dt. \end{aligned}$$

Put \(s=a_{0}u\) and \(t=b_{0}v\). Thus, we obtain

$$\begin{aligned} & \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t}\bigl(p^{\star }-s \bigr)^{\hat{k}_{1}-1}\bigl(q^{\star }-t\bigr)^{ \hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{ \hat{\ell }_{2}-1}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \\ &\quad\leq \frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}}}{\hat{\ell }_{1}\hat{\ell }_{2}} \int _{0}^{1}u^{\hat{\ell }_{1}}(1-u)^{\hat{k}_{1}-1}\,du \times \int _{0}^{1}v^{\hat{\ell }_{2}}(1-v)^{\hat{k}_{2}-1} \,dv. \end{aligned}$$

On the other hand,

$$\begin{aligned} \mathbf{B}(\hat{\ell }_{1}+1,\hat{k}_{1})= \int _{0}^{1}u^{\hat{\ell }_{1}}(1-u)^{ \hat{k}_{1}-1} \,du= \frac{\Gamma (\hat{\ell }_{1}+1)\Gamma (\hat{k}_{1})}{\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)} \end{aligned}$$

and

$$\begin{aligned} \mathbf{B}(\hat{\ell }_{2}+1,\hat{k}_{2})= \int _{0}^{1}v^{\hat{\ell }_{2}}(1-v)^{ \hat{k}_{2}-1} \,dv= \frac{\Gamma (\hat{\ell }_{2}+1)\Gamma (\hat{k}_{2})}{\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)}. \end{aligned}$$

Hence,

$$\begin{aligned} & \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t}\bigl(p^{\star }-s \bigr)^{\hat{k}_{1}-1}\bigl(q^{\star }-t\bigr)^{ \hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{ \hat{\ell }_{2}-1}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \\ &\quad\leq \frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}}\Gamma (\hat{\ell }_{1}+1)\Gamma (\hat{\ell }_{2}+1)\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})}{\hat{\ell }_{1}\hat{\ell }_{2} \Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)}. \end{aligned}$$

By using (5), we derive

$$\begin{aligned} & \bigl\vert \mathcal{Q}_{\bar{0}}^{\star } \sigma _{1}\bigl(p^{\star },q^{ \star }\bigr)- \mathcal{Q}_{\bar{0}}^{\star }\sigma _{2} \bigl(p^{\star },q^{\star }\bigr) \bigr\vert \\ &\quad\leq \frac{( \Vert h \Vert \Phi ^{*}+ \Vert o \Vert ) \Vert \sigma _{1}-\sigma _{2} \Vert }{l\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})} \\ &\qquad{}\times \frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}}\Gamma (\hat{\ell }_{1}+1)\Gamma (\hat{\ell }_{2}+1)\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})}{\hat{\ell }_{1}\hat{\ell }_{2}\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)} \\ &\quad= \frac{a_{0}^{\hat{k}_{1}+\hat{\ell } _{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}}( \Vert h \Vert \Phi ^{*}+ \Vert o \Vert )}{l\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)} \Vert \sigma _{1}-\sigma _{2} \Vert , \end{aligned}$$

which means \(\|\mathcal{Q}_{\bar{0}}^{\star }\sigma _{1}-\mathcal{Q}_{\bar{0}}^{ \star }\sigma _{2}\|\leq \frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}}(\|h\|\Phi ^{*}+\|o\|)}{l\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)} \|\sigma _{1}-\sigma _{2}\|\). Define \(\psi: [0,\infty )\to [0,\infty )\) and \(\alpha: \mathcal{Z}^{\star }\times \mathcal{Z}^{\star }\to [0,\infty )\) by \(\psi (t)= \frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}}(\|h\|\Phi ^{*} +\|o\|)}{l\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)}t\) and

$$\begin{aligned} \alpha (\sigma _{1},\sigma _{2})=\textstyle\begin{cases} 1, & \upsilon (\sigma _{1}(p^{\star },q^{\star }),\sigma _{2}(p^{\star },q^{ \star }))\geq 0 \text{ with } (p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}, \\ 0, & \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

It is clear that \(\psi \in \Psi \). If \(\alpha (\sigma _{1},\sigma _{2})\geq 1\), then \(\upsilon (\sigma _{1}(p^{\star },q^{\star }),\sigma _{2}(p^{\star },q^{ \star }))\geq 0\). From the hypotheses, \(\upsilon (\mathcal{Q}_{\bar{0}}^{\star }\sigma _{1}(p^{\star },q^{ \star }),\mathcal{Q}_{\bar{0}}^{\star }\sigma _{2}(p^{\star },q^{\star })) \geq 0\) and so \(\alpha (\mathcal{Q}_{\bar{0}}^{\star }\sigma _{1},\mathcal{Q}_{ \bar{0}}^{\star }\sigma _{2})\geq 1\). Thus, \(\mathcal{Q}_{\bar{0}}^{\star }\) is an α-admissible mapping. Also, there exists \(\sigma _{0}\in \mathcal{Z}^{\star }\) such that \(\alpha (\sigma _{0},\mathcal{Q}_{\bar{0}}^{\star }\sigma _{0})\geq 1\). For every sequence \(\{\sigma _{n}\}_{n\geq 1}\subseteq \mathcal{Z}^{\star }\) with \(\sigma _{n}\to \sigma \) and \(\alpha (\sigma _{n},\sigma _{n+1})\geq 1\) for all \(n\geq 1\), we have \(\alpha (\sigma _{n},\sigma )\geq 1\) for all \(n\geq 1\). Assume that \(\alpha (\sigma _{1},\sigma _{2})=0\). Then \(\alpha (\sigma _{1},\sigma _{2})\|\mathcal{Q}_{\bar{0}}^{\star } \sigma _{1}-\mathcal{Q}_{\bar{0}}^{\star }\sigma _{2}\|=0\leq \psi (\| \sigma _{1}-\sigma _{2}\|)\) and so

$$\begin{aligned} \alpha (\sigma _{1},\sigma _{2}) \bigl\Vert \mathcal{Q}_{\bar{0}}^{\star } \sigma _{1}- \mathcal{Q}_{\bar{0}}^{\star }\sigma _{2} \bigr\Vert \leq \psi \bigl( \Vert \sigma _{1}-\sigma _{2} \Vert \bigr) \end{aligned}$$

for all \(\sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }\). Thus all conditions of Lemma 1 hold and so \(\mathcal{Q}_{\bar{0}}^{\star }\) has a fixed point which is a solution for the partial fractional Sturm–Liouville problem (1). □

Example 1

Consider the partial fractional Sturm–Liouville equation

$$\begin{aligned} \textstyle\begin{cases} D_{c_{0}}^{(\frac{999}{1000},\frac{1999}{2000})} (100e^{- \sqrt[4]{p^{\star }}}\cosh q^{\star }D_{c_{0}}^{(\frac{89}{90}, \frac{79}{80})}\sigma (p^{\star },q^{\star }) ) + \frac{e^{-p^{\star }-q^{\star ^{3}}}}{300(1+p^{\star ^{2}}+p^{\star ^{2}})} \sigma (p^{\star }, q^{\star }) \\ \quad =\frac{1}{600}e^{\frac{p^{\star ^{2}}}{1+p^{\star ^{2}}}}\sigma (p^{ \star }, q^{\star }), \quad (p^{\star },q^{\star })\in [0,1]\times [0,1], \\ (100e^{-\sqrt[4]{p^{\star }}}\cosh q^{\star }\mathcal{D}^{( \frac{89}{90},\frac{79}{80})}_{c_{0}}\sigma (p^{\star },q^{\star }) )_{q^{\star }=0}=\frac{1}{100}p^{\star ^{2}}, \\ (100e^{-\sqrt[4]{p^{\star }}}\cosh q^{\star }\mathcal{D}^{( \frac{89}{90},\frac{79}{80})}_{c_{0}}\sigma (p^{\star },q^{\star }) )_{q^{\star }=0}=\frac{1}{400}q^{\star ^{3}},\\ \sigma (p^{\star },0)=\frac{1}{500}p^{\star } \quad\text{and}\quad \sigma (0,q^{ \star })=\frac{1}{350}q^{\star ^{2}}. \end{cases}\displaystyle \end{aligned}$$
(6)

Put \(\hat{k}=(\hat{k}_{1},\hat{k}_{2})=(\frac{999}{1000}, \frac{1999}{2000})\), \(\hat{\ell }=(\hat{\ell }_{1},\hat{\ell }_{2})=(\frac{89}{99}, \frac{79}{80})\), \(a_{0}=1\), \(b_{0}=1\), \(\theta _{1}(p^{\star })=\frac{1}{100}p^{\star ^{2}}\), \(\theta _{2}(q^{\star })=\frac{1}{400}q^{\star ^{3}}\), \(\kappa (p^{\star })=\frac{1}{500}p^{\star }\), \(\omega (q^{\star })=\frac{1}{350}q^{\star ^{2}}\), \(l(p^{\star },q^{\star })=100e^{-\sqrt[4]{p^{\star }}}\cosh q^{\star }\), \(o(p^{\star },q^{\star })= \frac{e^{-p^{\star }-q^{\star ^{3}}}}{300(1+p^{\star ^{2}}+q^{\star ^{2}})}\), \(h(p^{\star },q^{\star })=\frac{1}{600}e^{ \frac{p^{\star ^{2}}}{1+q^{\star ^{2}}}}\). The diagrams are plotted in Figs. 1 and 2, and obviously they satisfy the conditions of the partial Sturm–Liouville differential problem. Put \(f(r)=r\), \(\upsilon (r_{1},r_{2})=3\) whenever \(|r_{1}|\leq 1\) and \(|r_{2}|\leq 1\) and \(\upsilon (r_{1},r_{2})=-1\) otherwise, and

$$\begin{aligned} &\mathcal{Q}_{\bar{0}}^{\star }\sigma \bigl(p^{\star },q^{\star }\bigr)\\ &\quad= \Theta \bigl(p^{\star },q^{\star } \bigr) + \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\frac{-1}{1000}}(q^{\star }-t)^{\frac{-1}{2000}} (s-\wp _{\bar{1}})^{\frac{-1}{90}}(t-\zeta _{\bar{2}})^{\frac{-1}{80}}\mathcal{H}(\wp _{\bar{1}},\zeta _{\bar{2}},\sigma (\wp _{\bar{1}},\zeta _{\bar{2}}))}{\Gamma (\frac{999}{1000})\Gamma (\frac{1999}{2000}) \Gamma (\frac{89}{99})\Gamma (\frac{79}{80})100e^{-\sqrt[4]{p^{\star }}}\frac{\cosh q^{\star } }{50}}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}, \end{aligned}$$

where

$$\begin{aligned} \Theta \bigl(p^{\star },q^{\star }\bigr)={}& \frac{1}{500}p^{\star }+ \frac{1}{350}p^{\star ^{2}} \\ &{}+ \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \frac{(p^{\star }-s)^{-\frac{1}{90}} (q^{\star }-t)^{-\frac{1}{80}}(\frac{1}{100}s^{2}+\frac{1}{400}t^{3})}{100e^{-\sqrt[4]{p^{\star }}}\cosh q^{\star } \Gamma (\frac{89}{90})\Gamma (\frac{79}{80})}\,dt\,ds. \end{aligned}$$

Note that \(\mathcal{H}(\wp _{\bar{1}},\zeta _{\bar{2}},\sigma (\wp _{\bar{1}}, \zeta _{\bar{2}}))=h(\wp _{\bar{1}},\zeta _{\bar{2}})f(\sigma (\wp _{ \bar{1}},\zeta _{\bar{2}}))-o(\wp _{\bar{1}},\zeta _{\bar{2}})\sigma ( \wp _{\bar{1}},\zeta _{\bar{2}})\). Now, assume that \(\upsilon (\sigma _{1}(p^{\star },q^{\star }),\sigma _{2}(p^{\star },q^{ \star }))\geq 0\). Then we have \(|\sigma _{1}(p^{\star },q^{\star })|\leq 1\) and \(|\sigma _{2}(p^{\star },q^{\star })|\leq 1\). Suppose that \(|\sigma _{1}(p^{\star },q^{\star })|\leq 1\). Then we get

$$\begin{aligned} & \bigl\vert \mathcal{H}\bigl(\wp _{\bar{1}}, \zeta _{\bar{2}},\sigma ( \wp _{\bar{1}},\zeta _{\bar{2}})\bigr) \bigr\vert \\ &\quad= \biggl\vert \frac{1}{600}e^{ \frac{\wp _{\bar{1}}^{2}}{1+\zeta _{\bar{2}}^{2}}}f\bigl(\sigma (\wp _{ \bar{1}},\zeta _{\bar{2}})\bigr)- \frac{e^{-\wp _{\bar{1}}-\zeta _{\bar{2}}^{3}}}{300(1+\wp _{\bar{1}}^{2}+\zeta _{\bar{2}}^{2})} \sigma _{1}(\wp _{\bar{1}},\zeta _{\bar{2}}) \biggr\vert \\ &\quad\leq \biggl\vert \frac{1}{600}e^{ \frac{\wp _{\bar{1}}^{2}}{1+\zeta _{\bar{2}}^{2}}} \biggr\vert \bigl\vert \sigma _{1}(\wp _{ \bar{1}},\zeta _{\bar{2}}) \bigr\vert + \biggl\vert \frac{e^{-p^{\star }-p^{\star ^{3}}}}{300(1+p^{\star ^{2}}+p^{\star ^{2}})} \biggr\vert \bigl\vert \sigma _{1}(\wp _{\bar{1}},\zeta _{\bar{2}}) \bigr\vert \\ &\quad\leq \frac{e}{600}+\frac{1}{300}=\frac{e+2}{600} \end{aligned}$$

and so

$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \int _{0}^{s} \int _{0}^{t} \\ &\quad\frac{(p^{\star }-s)^{\frac{-1}{1000}}(q^{\star }-t)^{\frac{-1}{2000}} (s-\wp _{\bar{1}})^{\frac{-1}{90}}(t-\zeta _{\bar{2}})^{\frac{-1}{80}} \vert \mathcal{H}(\wp _{\bar{1}},\zeta _{\bar{2}},\sigma _{1}(\wp _{\bar{1}},\zeta _{\bar{2}})) \vert }{\Gamma (\frac{999}{1000})\Gamma (\frac{1999}{2000}) \Gamma (\frac{89}{90})\Gamma (\frac{79}{80})100e^{-\sqrt[4]{p^{\star }}}\cosh q^{\star }}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \\ &\qquad\leq 0.0000285064 \int _{0}^{p^{\star }} \int _{0}^{q^{ \star }} \int _{0}^{s} \int _{0}^{t}\bigl(p^{\star }-s \bigr)^{\frac{-1}{1000}}\bigl(q^{ \star }-t\bigr)^{\frac{-1}{2000}} (s-\wp _{\bar{1}})^{\frac{-1}{90}}\\ &\qquad\quad{}\times(t- \zeta _{\bar{2}})^{\frac{-1}{80}}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{ \bar{1}} \\ &\qquad\leq 0.0000285064 \times \frac{\Gamma (\frac{89}{90}+1)\Gamma (\frac{79}{80}+1)\Gamma (\frac{999}{1000})\Gamma (\frac{1999}{2000})}{\frac{89}{90}\frac{79}{80} \Gamma (\frac{999}{1000}+\frac{89}{90}+1)\Gamma (\frac{1999}{2000}+\frac{79}{80}+1)}=0.0000296489. \end{aligned}$$

Also,

$$\begin{aligned} &\bigl\vert \Theta \bigl(p^{\star },q^{\star } \bigr) \bigr\vert \\ &\quad\leq \biggl\vert \frac{1}{500}p^{ \star }+ \frac{1}{350}q^{\star ^{2}} \biggr\vert + \biggl\vert \int _{0}^{p^{\star }} \int _{0}^{q^{ \star }} \frac{(p^{\star }-s)^{-\frac{1}{90}} (q^{\star }-t)^{-\frac{1}{80}}(\frac{1}{100}p^{\star ^{2}}+\frac{1}{400}p^{\star ^{3}})}{100e^{-\sqrt[4]{p^{\star }}}\cosh q^{\star } \Gamma (\frac{89}{90})\Gamma (\frac{79}{80})}\,dt\,ds \biggr\vert \\ &\quad\leq \frac{1}{500}+\frac{1}{350}+0.0000453519 \int _{0}^{1} \int _{0}^{1}(1-s)^{- \frac{1}{90}}(1-t)^{-\frac{1}{80}} \,dt\,ds \\ &\quad= \frac{1}{500}+\frac{1}{350}+0.0000453519\times 1.0240364102=0.0053797753. \end{aligned}$$

Therefore,

$$\begin{aligned} & \bigl\vert \mathcal{Q}_{\bar{0}}^{\star } \sigma _{1}\bigl(p^{\star },q^{ \star }\bigr) \bigr\vert \\ &\quad\leq \bigl\vert \Theta \bigl(p^{\star },q^{\star }\bigr) \bigr\vert + \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\frac{-1}{1000}}(q^{\star }-t)^{\frac{-1}{2000}} (s-\wp _{\bar{1}})^{\frac{-1}{90}}(t-\zeta _{\bar{2}})^{\frac{-1}{80}} \vert \mathcal{H}(\wp _{\bar{1}},\zeta _{\bar{2}},\sigma _{1}(\wp _{\bar{1}},\zeta _{\bar{2}})) \vert }{\Gamma (\frac{999}{1000})\Gamma (\frac{1999}{2000}) \Gamma (\frac{89}{99})\Gamma (\frac{79}{80})100e^{-\sqrt[4]{p^{\star }}}\frac{\cosh q^{\star } }{50}}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \\ &\quad\leq 0.0053797753+0.0000296489=0.0054094242\leq 1. \end{aligned}$$

Similarly, we obtain \(|\mathcal{Q}_{\bar{0}}^{\star }\sigma _{1}(p^{\star },q^{\star })|\leq 1\) and \(\upsilon (\mathcal{Q}_{\bar{0}}^{\star }\sigma _{1}(p^{\star },q^{ \star }),\mathcal{Q}_{\bar{0}}^{\star }\sigma _{2}(p^{\star },q^{\star })) \geq 0\). Assume that \(\{\sigma _{n}\}_{n\geq 1}\subseteq \mathcal{Z}^{\star }\) is a sequence such that \(\sigma _{n}\to \sigma \) and

$$\begin{aligned} \upsilon \bigl(\sigma _{n}\bigl(p^{\star },q^{\star } \bigr),\sigma _{n+1}\bigl(p^{\star },q^{ \star }\bigr)\bigr) \geq 0 \end{aligned}$$

for all \(n\geq 1\) and \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\). Then we have \(|\sigma _{n}(p^{\star },q^{\star })|\leq 1\) and \(|\sigma _{n+1}(p^{\star },q^{\star })|\leq 1\) for all \(n\geq 1\) and \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\). Since \(\sigma _{n}\to \sigma \), \(|\sigma (p^{\star },q^{\star })|=\lim_{n\to \infty } |\sigma _{n}(p^{ \star },q^{\star })|\leq 1\), we obtain \(\upsilon (\mathcal{Q}_{\bar{0}}^{\star }\sigma _{n}(p^{\star },q^{ \star }),\mathcal{Q}_{\bar{0}}^{\star }\sigma (p^{\star },q^{\star })) \geq 0\) for all \(n\geq 1\) and \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\). Since \(|0|\leq 1\) and \(|\mathcal{Q}_{\bar{0}}^{\star } 0|\leq 1\), we get \(\upsilon (0,\mathcal{Q}_{\bar{0}}^{\star } 0)\geq 1\). Note that \(\Phi ^{*}=1\),

$$\begin{aligned} &l=\inf_{(p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}}l\bigl(p^{\star },q^{\star }\bigr)=\inf _{(p^{\star },q^{\star }) \in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}}100e^{- \sqrt[4]{p^{\star }}}\cosh q^{\star }=100e, \\ &\Vert o \Vert =\sup_{(p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}} o\bigl(p^{\star },q^{\star } \bigr) =\sup_{(p^{\star },q^{ \star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}} \frac{e^{-p^{\star }-q^{\star ^{3}}}}{300(1+p^{\star ^{2}}+q^{\star ^{2}})}= \frac{1}{300}, \end{aligned}$$

and \(\|h\|=\sup_{(p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}}\frac{1}{600}e^{ \frac{p^{\star ^{2}}}{1+q^{\star ^{2}}}}=\frac{e}{600}\). Hence,

$$\begin{aligned} \frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}}( \Vert h \Vert \Phi ^{*}+ \Vert o \Vert )}{l\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)} &= \frac{\frac{e}{600}+\frac{1}{300}}{100e\Gamma (\frac{999}{1000}+\frac{89}{90}+1)\Gamma (\frac{1999}{2000}+\frac{79}{80}+1)}\\ &=0.0000074014 \leq 1. \end{aligned}$$

Now by using Theorem 4, the problem (6) has a solution.

Figure 1
figure 1

Plots of functions \(h(p^{*},q^{*}),l(p^{*},q^{*}),o(p^{*},q^{*})\) in \([0,1]\)

Full size image
Figure 2
figure 2

Plots of functions \(\theta _{1}, \theta _{2}, \kappa, \omega \) in \([0,1]\)

Full size image

Definition 5

We say that a function \(\sigma \in \mathcal{C}(\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}})\) is a solution for the partial fractional Sturm–Liouville differential inclusion problem problem (2) whenever there is a function v in \(\mathcal{ L}^{1}(\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}, \mathbb{R})\) such that \(v(p^{\star },q^{\star })\in \mathcal{H} (p^{\star },q^{\star }, \sigma (p^{\star },q^{\star }) )\) for almost all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\),

$$\begin{aligned} \textstyle\begin{cases} (l(p^{\star },q^{\star })\mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma (p^{ \star }, q^{\star }) )_{q^{\star }=0}=\theta _{1}(p^{\star }), \\ (l(p^{\star },q^{\star })\mathcal{D}^{\hat{\ell }}_{c_{0}}\sigma (p^{ \star },q^{\star }) )_{p^{\star }=0}=\theta _{2}(q^{\star }), \\ \sigma (p^{\star },0)=\kappa (p^{\star }) \quad \text{and}\quad \sigma (0,q^{ \star })=\omega (q^{\star }), \end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned} &\sigma \bigl(p^{\star },q^{\star }\bigr)\\ &\quad= \Theta \bigl(p^{\star },q^{\star }\bigr) \\ &\qquad{}+ \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}v(\wp _{\bar{1}},\zeta _{\bar{2}})}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}, \end{aligned}$$

for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\).

For given \(\sigma \in \mathcal{Z}^{\star }\), define the set

$$\begin{aligned} \mathcal{S}_{\mathcal{H},\sigma }= \bigl\{ v\in \mathcal{L}^{1}( \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}})| v\in \mathcal{H} \bigl(p^{\star },q^{\star },\sigma \bigl(p^{\star },q^{\star }\bigr) \bigr)\text{ on } \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}} \bigr\} . \end{aligned}$$

Assume that

\((H1)\):

\(\mathcal{H}:\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\times \mathbb{R}\to \mathcal{P}_{cl}(\mathbb{R})\) is an integrable bounded multifunction so that

\(\mathcal{H} (\cdot,\cdot,\sigma )\) is measurable for all \(\sigma \in \mathbb{R}\).

\((H2)\):

There exists \(\rho \in \mathcal{C}(\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}, \mathbb{R}^{+})\) so that

$$\begin{aligned} PH_{d_{\mathcal{Z}^{\star }}}\bigl(\mathcal{H} \bigl(p^{\star },q^{ \star },r_{1} \bigr),\mathcal{H} \bigl(p^{\star },q^{\star },r_{2} \bigr)\bigr) \leq \rho \bigl(p^{\star },q^{\star }\bigr)\psi \bigl( \vert r_{1}-r_{2} \vert \bigr) \end{aligned}$$

for all \(r_{1},r_{2}\in \mathbb{R}\), where \(\psi \in \Psi \), \(\frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}}\|\rho \|_{\infty }}{l\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)} \leq 1\) and

$$\begin{aligned} l=\inf_{(p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}} l\bigl(p^{\star },q^{\star }\bigr). \end{aligned}$$
\((H3)\):

Define \(N:\mathcal{Z}^{\star }\to 2^{\mathcal{Z}^{\star }}\) by

$$\begin{aligned} N(\sigma )={}&\bigl\{ h\in \mathcal{Z}^{\star }| \text{ there exists } v \in S_{\mathcal{H},\sigma } \text{ so that } h\bigl(p^{\star },q^{\star }\bigr)=w \bigl(p^{ \star },q^{\star }\bigr) \\ &\text{for all } \bigl(p^{\star },q^{\star } \bigr)\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\bigr\} , \end{aligned}$$

where

$$\begin{aligned} &w\bigl(p^{\star },q^{\star }\bigr)\\ &\quad=\Theta \bigl(p^{\star },q^{\star }\bigr) + \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}v(\wp _{\bar{1}},\zeta _{\bar{2}})}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}. \end{aligned}$$
\((H4)\):

Suppose that \(\upsilon: \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) be a real valued function and for every convergent sequence \(\{\sigma _{n}\}_{n\geq 1}\subseteq \mathcal{Z}^{\star }\) with \(\sigma _{n}\to \sigma \) and \(\upsilon (\sigma _{n}(p^{\star },q^{\star }),\sigma (p^{\star },q^{ \star }))\geq 0\) for all n and \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\), there exists a subsequence \(\{\sigma _{n_{j}}\}_{j\geq 1}\) of \(\sigma _{n}\) so that \(\upsilon (\sigma _{n_{j}}(p^{\star },q^{\star }),\sigma (p^{\star },q^{ \star }))\geq 0\) for all n and \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\). Assume that, for every \(\sigma \in \mathcal{Z}^{\star }\) and \(h\in N(\sigma )\) with \(\upsilon (\sigma (p^{\star },q^{\star }),h(p^{\star },q^{\star }))\geq 0\) for each \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\), there exists \(w\in N(\sigma )\) so that \(\upsilon (h(p^{\star },q^{\star }),w(p^{\star },q^{\star }))\geq 0\) for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\). Suppose that there exists \(\sigma _{0}\in \mathcal{Z}^{\star }\) and \(h\in N(\sigma _{0})\) so that \(\upsilon (\sigma _{0}(p^{\star },q^{\star }),h(p^{\star },q^{\star })) \geq 0\) for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\).

Theorem 6

Assume that \((H1)\)–\((H4)\) hold. Then the partial fractional Sturm–Liouville problem (2) has a solution.

Proof

We prove that the multifunction \(N:\mathcal{Z}^{\star }\to 2^{\mathcal{Z}^{\star }}\) has a fixed point which provides a solution for the partial fractional Sturm–Liouville problem (2). Note that the multifunction \((p^{\star },q^{\star })\to \mathcal{H} (p^{\star },q^{\star },\sigma (p^{ \star },q^{\star }) )\) has a measurable selection. Since it has closed and has measurable values for aa \(\sigma \in \mathcal{Z}^{\star }\), \(S_{\mathcal{H},\sigma }\) is nonempty for every \(\sigma \in \mathcal{Z}^{\star }\). We prove that \(N(\sigma )\) is closed subset of \(\mathcal{Z}^{\star }\). For this aim assume that \(\sigma \in \mathcal{ Z}^{*}\) and \(\{h_{n}\}\subset N(\sigma )\) is a sequence with \(h_{n}\to h\). For each n, choose \(v_{n}\in S_{\mathcal{H},\sigma }\) such that

$$\begin{aligned} &h_{n}\bigl(p^{\star },q^{\star } \bigr)\\ &\quad=\Theta \bigl(p^{\star },q^{\star }\bigr)+ \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}v_{n}(\wp _{\bar{1}},\zeta _{\bar{2}})}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}, \end{aligned}$$

for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\). On the other hand, \(\mathcal{H}\) has compact values. Thus, we may assume that \(\{v_{n}\}\) converges to some \(v\in \mathcal{L}^{1}(\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}})\). Hence,

$$\begin{aligned} &h\bigl(p^{\star },q^{\star }\bigr)\\ &\quad=\Theta \bigl(p^{\star },q^{\star }\bigr)+ \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}v(\wp _{\bar{1}},\zeta _{\bar{2}})}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \end{aligned}$$

and so \(h\in N(\sigma )\). Since \(\mathcal{H}\) is a compact map, \(N(\sigma )\) is a bounded set for al \(\sigma \in \mathcal{Z}^{\star }\). Now, define the function \(\alpha:\mathcal{Z}^{\star }\times \mathcal{Z}^{\star }\to \mathbb{R}^{+}\) by \(\alpha (\sigma _{1},\sigma _{2})\geq 1\) whenever \(\upsilon (\sigma _{1}(p^{\star },q^{\star }),\sigma _{2}(p^{\star },q^{ \star }))\geq 0\) for almost all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\) and \(\alpha (\sigma _{1},\sigma _{2})=0\) otherwise. We show that N is α-ψ-contractive. Let \(\sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }\) and \(h_{1}\in N(\sigma _{2})\). Choose \(v_{1}\in S_{\mathcal{H},\sigma _{2}}\) such that

$$\begin{aligned} &h_{1}\bigl(p^{\star },q^{\star } \bigr)\\ &\quad=\Theta \bigl(p^{\star },q^{\star }\bigr)+ \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}v_{1}(\wp _{\bar{1}},\zeta _{\bar{2}})}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \end{aligned}$$

for almost all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\). Then we get

$$\begin{aligned} & PH_{d_{\mathcal{Z}^{\star }}}\bigl(\mathcal{H} \bigl(p^{\star },q^{ \star },\sigma _{1}\bigl(p^{\star },q^{\star } \bigr) \bigr),\mathcal{H} \bigl(p^{\star },q^{ \star },\sigma _{2}\bigl(p^{\star },q^{\star }\bigr) \bigr)\bigr)\\ &\quad\leq \rho \bigl(p^{\star },q^{ \star }\bigr)\psi \bigl( \bigl\vert \sigma _{1}\bigl(p^{\star },q^{\star }\bigr)-\sigma _{2} \bigl(p^{ \star },q^{\star }\bigr) \bigr\vert \bigr), \end{aligned}$$

for almost all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\) with \(\upsilon (\sigma _{1}(p^{\star },q^{\star }),\sigma _{2}(p^{\star },q^{ \star }))\geq 0\). Now, choose \(w\in \mathcal{H} (p^{\star },q^{\star },\sigma _{1}(p^{\star },q^{ \star }) )\) so that \(|v_{1}(t)-w|\leq \rho (p^{\star },q^{\star })\psi (|\sigma _{1}(p^{ \star },q^{\star })-\sigma _{2}(p^{\star },q^{\star })| )\) for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\). Now, define \(\mathcal{U}:\mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\to \mathcal{P}(\mathbb{R})\) by

$$\begin{aligned} \mathcal{U}\bigl(p^{\star },q^{\star }\bigr)= \bigl\{ w\in \mathbb{R}| \bigl\vert v_{1}\bigl(p^{\star },q^{\star } \bigr)-w \bigr\vert \leq \rho \bigl(p^{\star },q^{\star }\bigr) \psi \bigl( \bigl\vert \sigma _{1}\bigl(p^{\star },q^{\star } \bigr)-\sigma _{2}\bigl(p^{\star },q^{ \star }\bigr) \bigr\vert \bigr) \bigr\} . \end{aligned}$$

Since \(v_{1}\) and \(\omega _{0}^{*}=\rho (p^{\star },q^{\star })\psi (|\sigma _{1}(p^{ \star },q^{\star })-\sigma _{2}(p^{\star },q^{\star })| )\) are measurable, the multifunction \(\mathcal{U}(\cdot,\cdot)\cap \mathcal{H} (\cdot,\cdot,\sigma (\cdot,\cdot) )\) is measurable. Choose \(v_{2}\in \mathcal{H} (p^{\star },q^{\star },\sigma _{1}(p^{\star },q^{ \star }) )\) such that

$$\begin{aligned} \begin{aligned} \bigl\vert v_{1} \bigl(p^{\star },q^{\star }\bigr)-v_{2}\bigl(p^{\star },q^{\star } \bigr) \bigr\vert & \leq \rho \bigl(p^{\star },q^{\star }\bigr)\psi \bigl( \bigl\vert \sigma _{1}\bigl(p^{\star },q^{ \star } \bigr)-\sigma _{2}\bigl(p^{\star },q^{\star }\bigr) \bigr\vert \bigr) \\ &\leq \Vert \rho \Vert _{\infty }\psi \bigl( \Vert \sigma _{1}-\sigma _{2} \Vert \bigr) \end{aligned} \end{aligned}$$
(7)

for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\). Now, choose the element \(h_{2}\in N(\sigma _{1})\) defined by

$$\begin{aligned} &h_{2}\bigl(p^{\star },q^{\star } \bigr)\\ &\quad=\Theta \bigl(p^{\star },q^{\star }\bigr)+ \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}v_{2}(\wp _{\bar{1}},\zeta _{\bar{2}})}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \end{aligned}$$

for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\). By using (7), we get

$$\begin{aligned} & \bigl\vert h_{2}\bigl(p^{\star },q^{\star } \bigr)-h_{1}\bigl(p^{\star },q^{\star }\bigr) \bigr\vert \\ &\quad\leq \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1} \vert v_{2}(\wp _{\bar{1}},\zeta _{\bar{2}})-v_{1}(\wp _{\bar{1}},\zeta _{\bar{2}}) \vert }{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}} \\ &\quad\leq \Vert \rho \Vert _{\infty } \int _{0}^{p^{\star }} \int _{0}^{q^{ \star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{\hat{k}_{1}-1}(q^{\star }-t)^{\hat{k}_{2}-1}(s-\wp _{\bar{1}})^{\hat{\ell }_{1}-1}(t-\zeta _{\bar{2}})^{\hat{\ell }_{2}-1}}{\Gamma (\hat{k}_{1})\Gamma (\hat{k}_{2})\Gamma (\hat{\ell }_{1})\Gamma (\hat{\ell }_{2})l(s,t)}\,dt \,ds \,d\zeta _{\bar{2}} \,d\wp _{\bar{1}}\psi \bigl( \Vert \sigma _{1}-\sigma _{2} \Vert \bigr) \\ &\quad\leq \frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}} \Vert \rho \Vert _{\infty }}{l\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)} \psi \bigl( \Vert \sigma _{1}-\sigma _{2} \Vert \bigr)\leq \psi \bigl( \Vert \sigma _{1}- \sigma _{2} \Vert \bigr) \end{aligned}$$

and so \(\|h_{1}-h_{2}\|\leq \psi (\|\sigma _{1}-\sigma _{2}\|)\). Note that

$$\begin{aligned} \alpha (\sigma _{1},\sigma _{2})PH_{d_{\mathcal{Z}^{\star }}}\bigl(N( \sigma _{1}),N(\sigma _{2})\bigr)\leq \psi \bigl( \Vert \sigma _{1}-\sigma _{2} \Vert \bigr) \quad\text{for all } \sigma _{1},\sigma _{2}\in \mathcal{Z}^{\star }. \end{aligned}$$

Thus, N is α-ψ-contraction. Let \(\sigma _{1}\in \mathcal{Z}^{\star }\) and \(\sigma _{2}\in N(\sigma _{1})\) be such that \(\alpha (\sigma _{1},\sigma _{2})\geq 1\). Then \(\upsilon (\sigma _{1}(p^{\star },q^{\star }),\sigma _{2}(p^{\star },q^{ \star }))\geq 0\) for all \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\). Hence, there exists \(w\in N(\sigma _{2})\) such that \(\upsilon (\sigma _{1}(p^{\star },q^{\star }), w(p^{\star },q^{\star })) \geq 0\). This implies that \(\alpha (\sigma _{1}, w)\geq 1\). Thus, N is α-admissible. Now by using Lemma 2, N has a fixed point which is a solution of the partial fractional Sturm–Liouville problem (2). □

Example 2

Consider the partial fractional Sturm–Liouville inclusion problem

$$\begin{aligned} \textstyle\begin{cases} D_{c_{0}}^{(\frac{1}{2},\frac{1}{3})} (13e^{p^{\star }q^{\star }}D_{c_{0}}^{( \frac{1}{7},\frac{1}{8})} \sigma (p^{\star },q^{\star }) )\in [\sigma (p^{\star },q^{\star }),\sigma (p^{\star },q^{\star }) + \frac{e^{-p^{\star ^{2}}-q^{\star ^{2}}} \vert \sigma (p^{\star },q^{\star }) \vert }{4(1+ \vert \sigma (p^{\star },p^{\star ^{2}}) \vert )} ], \\ (p^{\star },q^{\star })\in [0,1]\times [0,1], &\\ (13e^{p^{\star }q^{\star }}\mathcal{D}^{(\frac{1}{7},\frac{1}{8})}_{c_{0}} \sigma (p^{\star },q^{\star }) )_{q^{\star }=0}=\frac{3}{40}p^{ \star ^{2}}, & \\ (13e^{p^{\star }q^{\star }}\mathcal{D}^{(\frac{1}{7},\frac{1}{8})}_{c_{0}} \sigma (p^{\star },q^{\star }) )_{p^{\star }=0}=\frac{1}{25}q^{ \star ^{3}}, &\\ \sigma (p^{\star },0)=\frac{7}{50}p^{\star ^{2}} \quad\text{and}\quad \sigma (0,q^{\star })=\frac{8}{35}q^{\star ^{3}}. \end{cases}\displaystyle \end{aligned}$$
(8)

Put \(\hat{k}=(\hat{k}_{1},\hat{k}_{2})=(\frac{1}{2},\frac{1}{3})\), \(\hat{\ell }=(\hat{\ell }_{1},\hat{\ell }_{2})=(\frac{1}{7},\frac{1}{8})\), \(a=1\), \(b=1\), \(\theta _{1}(p^{\star })=\frac{3}{40}p^{\star ^{2}}\), \(\theta _{2}(q^{\star })=\frac{1}{25}q^{\star ^{3}}\), \(\kappa (p^{\star })=\frac{7}{50}p^{\star ^{2}}\), \(\omega (q^{\star })=\frac{8}{35}q^{\star ^{3}}\) and \(l(p^{\star },q^{\star })=13e^{p^{\star }q^{\star }}\). The plotted diagrams in Figs. 3 and 4 show that the conditions of the partial Sturm–Liouville differential inclusion problem hold. Also, put \(\mathcal{H} (p^{\star },q^{\star },r )= [r,r+ \frac{e^{-p^{\star ^{2}}-q^{\star ^{2}}}|r|}{4(1+|r|)} ]\), \(\upsilon (r_{1},r_{2})=1\) whenever \(r_{1}\geq 0\) and \(r_{2}\geq 0\) and \(\upsilon (r_{1},r_{2})=-1\) otherwise,

$$\begin{aligned} N(\sigma )={}&\bigl\{ h\in \mathcal{Z}^{\star }| \text{ there exists } v \in S_{\mathcal{H},\sigma } \text{ so that } h\bigl(p^{\star },q^{\star }\bigr)=w \bigl(p^{ \star },q^{\star }\bigr) \\ &\text{for all } \bigl(p^{\star },q^{\star } \bigr)\in \mathcal{J}_{a_{0}} \times \mathcal{J}_{b_{0}}\bigr\} \end{aligned}$$

where

$$\begin{aligned} &w\bigl(p^{\star },q^{\star }\bigr)\\ &\quad=\Theta \bigl(p^{\star },q^{\star }\bigr) \\ &\qquad{}+ \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \frac{(p^{\star }-s)^{-\frac{1}{2}}(q^{\star }-t)^{-\frac{2}{3}}(s-\wp _{\bar{1}})^{-\frac{6}{7}}(t-\zeta _{\bar{2}})^{-\frac{7}{8}}v(\wp _{\bar{1}},\zeta _{\bar{2}})}{13e^{st}\Gamma (\frac{1}{2}) \Gamma (\frac{1}{3})\Gamma (\frac{1}{7})\Gamma (\frac{1}{8})}\,dt \,ds \,d \zeta _{\bar{2}} \,d\wp _{\bar{1}} \end{aligned}$$

and \(\Theta (p^{\star },q^{\star })=\frac{7}{50}p^{\star ^{2}}+\frac{8}{35}p^{ \star ^{3}}+\int _{0}^{p^{\star }}\int _{0}^{q^{\star }} \frac{(p^{\star }-s)^{-\frac{1}{90}} (q^{\star }-t)^{-\frac{1}{80}} (\frac{3}{40}s^{2}+\frac{1}{25}t^{3} )}{13e^{st} \Gamma (\frac{89}{90})\Gamma (\frac{79}{80})}\,dt\,ds\). Assume that \(\sigma \in \mathcal{Z}^{\star }\) and \(h\in N(\sigma )\) with \(\upsilon (\sigma (p^{\star },q^{\star }),h(p^{\star },q^{\star }))\geq 0\) for all \((p^{\star },q^{\star })\in [0,1]\times [0,1]\). Then we have \(\sigma (p^{\star },q^{\star })\geq 0\) and \(h(p^{\star },q^{\star })\geq 0\) for all \((p^{\star },q^{\star })\in [0,1]\times [0,1]\). Since \(\sigma (p^{\star },q^{\star })\geq 0\), we get

$$\begin{aligned} \mathcal{H} \bigl(p^{\star },q^{\star },\sigma \bigl(p^{\star },q^{\star } \bigr) \bigr)= \biggl[\sigma \bigl(p^{\star },q^{\star }\bigr), \sigma \bigl(p^{\star },q^{\star }\bigr)+ \frac{e^{-p^{\star ^{2}}-q^{\star ^{2}}} \vert \sigma (p^{\star },q^{\star }) \vert }{4(1+ \vert \sigma (p^{\star },q^{\star }) \vert )} \biggr]\subseteq [0,\infty ). \end{aligned}$$

Choose \(v(p^{\star },q^{\star })\in \mathcal{H} (p^{\star },q^{\star }, \sigma (p^{\star },q^{\star }) )\) so that \(v(p^{\star },q^{\star })\geq 0\) for all \((p^{\star },q^{\star })\) in \([0,1]\times [0,1]\). Since \(\Theta (p^{\star },q^{\star })\geq 0\), we get

$$\begin{aligned} &w\bigl(p^{\star },q^{\star }\bigr)\\ &\quad:=\Theta \bigl(p^{\star },q^{\star }\bigr) + \int _{0}^{p^{\star }} \int _{0}^{q^{\star }} \int _{0}^{s} \int _{0}^{t} \\ &\qquad\frac{(p^{\star }-s)^{-\frac{1}{2}}(q^{\star }-t)^{-\frac{2}{3}}(s-\wp _{\bar{1}})^{-\frac{6}{7}}(t-\zeta _{\bar{2}})^{-\frac{7}{8}}v(\wp _{\bar{1}},\zeta _{\bar{2}})}{13e^{st}\Gamma (\frac{1}{2}) \Gamma (\frac{1}{3})\Gamma (\frac{1}{7})\Gamma (\frac{1}{8})}\,dt \,ds \,d \zeta _{\bar{2}} \,d\wp _{\bar{1}} \geq 0 \end{aligned}$$

and so \(w(p^{\star },q^{\star })\geq 0\). Thus, \(\upsilon (h(p^{\star },q^{\star }),w(p^{\star },q^{\star }))\geq 0\). Note that \(\upsilon (0,h(p^{\star },q^{\star }))\geq 0\) for \(h\in N(\sigma )\) and also for each convergent sequence \(\{\sigma _{n}\}_{n\geq 1}\subseteq \mathcal{Z}^{\star }\) with \(\sigma _{n}\to \sigma \) and \(\upsilon (\sigma _{n}(p^{\star },q^{\star }),\sigma (p^{\star },q^{ \star }))\geq 0\) for all n and \((p^{\star },q^{\star })\in \mathcal{J}_{a_{0}}\times \mathcal{J}_{b_{0}}\), there exists a subsequence \(\{\sigma _{n_{j}}\}_{j\geq 1}\) of \(\{\sigma _{n}\}_{n\geq 1}\) such that \(\upsilon (\sigma _{n_{j}}(p^{\star },q^{\star }),\sigma (p^{\star },q^{ \star }))\geq 0\). Thus,

$$\begin{aligned} &PH_{d_{\mathcal{Z}^{\star }}}\bigl(\mathcal{H} \bigl(p^{\star },q^{ \star },r_{1} \bigr),\mathcal{H} \bigl(p^{\star },q^{\star },r_{2} \bigr)\bigr)\\ &\quad \leq \frac{e^{-p^{\star ^{2}}-q^{\star ^{2}}}}{4} \biggl\vert \frac{ \vert r_{1} \vert }{1+ \vert r_{1} \vert }-\frac{ \vert r_{2} \vert }{1+ \vert r_{2} \vert } \biggr\vert \\ &\quad=\frac{e^{-p^{\star ^{2}}-q^{\star ^{2}}}}{4} \bigl\vert \vert r_{1} \vert - \vert r_{2} \vert \bigr\vert \leq \frac{e^{-p^{\star ^{2}}-q^{\star ^{2}}}}{4} \vert r_{1}-r_{2} \vert . \end{aligned}$$

If \(\rho (p^{\star },q^{\star })=e^{-p^{\star ^{2}}-q^{\star ^{2}}}\) and \(\psi (t)=\frac{1}{4}t\), then

$$\begin{aligned} PH_{d_{\mathcal{Z}^{\star }}}\bigl(\mathcal{H} \bigl(p^{\star },q^{ \star },r_{1} \bigr),\mathcal{H} \bigl(p^{\star },q^{\star },r_{2} \bigr)\bigr) \leq \Phi \bigl(p^{\star },q^{\star }\bigr) \psi \bigl( \vert r_{1}-r_{2} \vert \bigr) \end{aligned}$$

and \(\|\rho \|_{\infty }=1\). Put \(l(p^{\star },q^{\star })=13e^{p^{\star }q^{\star }}\). Then \(l=13\) and so

$$\begin{aligned} \frac{a_{0}^{\hat{k}_{1}+\hat{\ell }_{1}}b_{0}^{\hat{k}_{2}+\hat{\ell }_{2}} \Vert \rho \Vert _{\infty }}{l\Gamma (\hat{k}_{1}+\hat{\ell }_{1}+1)\Gamma (\hat{k}_{2}+\hat{\ell }_{2}+1)} = \frac{1}{13\Gamma (\frac{1}{2}+\frac{1}{7}+1)\Gamma (\frac{1}{3}+\frac{1}{8}+1)}=0.0966114627 \leq 1. \end{aligned}$$

Now by using Theorem 6, the problem (8) has a solution.

Figure 3
figure 3

Plot of function \(l(p^{*},q^{*})\) in \([0,1]\)

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Figure 4
figure 4

Plots of functions \(\theta _{1}, \theta _{2}, \kappa, \omega \) in \([0,1]\)

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3 Conclusion

In this work, we studied a partial fractional version of the Sturm–Liouville differential equation by using the Caputo derivative. Also, we reviewed inclusion version of the problem. First, by using the technique of α-ψ-contractive mappings, we investigated the existence of solutions for the partial fractional Sturm–Liouville equation. We presented an illustrated example to clear more the result. Secondly, we have investigated the partial fractional Sturm–Liouville inclusion problem by using the technique of α-ψ-contractive multifunctions. We provided an illustrated an example for explaining the second result. In this way, we provided some related figures for the examples.