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On stability analysis and existence of positive solutions for a general non-linear fractional differential equations

Abstract

In this article, we deals with the existence and uniqueness of positive solutions of general non-linear fractional differential equations (FDEs) having fractional derivative of different orders involving p-Laplacian operator. Also we investigate the Hyers–Ulam (HU) stability of solutions. For the existence result, we establish the integral form of the FDE by using the Green function and then the existence of a solution is obtained by applying Guo–Krasnoselskii’s fixed point theorem. For our purpose, we also check the properties of the Green function. The uniqueness of the result is established by applying the Banach contraction mapping principle. An example is offered to ensure the validity of our results.

Introduction

Fractional calculus concerns the applications of derivatives and integrals of arbitrary order. During the last few decades, it received great attention because of its various applications in diverse scientific fields. Arbitrary-order models are more flexible than integer-order models. FDEs arise in numerous scientific and engineering fields such as physics, polymer rheology, geophysics, biophysics, aerodynamics, capacitor theory, biology, non-linear oscillation of earthquake, control theory, blood flow phenomena, viscoelasticity, and electrical circuits. For the exhaustive study of its applications, we refer to extensive work [17]. The fundamental differences between exponential decay, the power law, the Mittag-Leffler law and some possible applications in nature are presented in [8, 9].

Nowadays, the existence and uniqueness (EU) of solutions for different type of FDEs is a field of intensive research. Here, we introduce some important and recent work of several researcher about the existence of a positive solution (EPS) of different classes of FDEs. For example, the EU results for Dirichlet and mixed problems of singular FDEs with the Riemann–Liouville sense of fractional derivative were investigated by Agarwal et al. [10, 11]. Baleanu et al. in [12] established the existence of a solution on partially ordered Banach spaces for a non-linear FDEs. Vong studied the singular FDEs involving non-local type boundary conditions in [13] by using fixed point techniques. For more details of EU of different types of FDEs with different types of fractional derivatives, see [1435]. Numerical solutions for the fractional Fisher’s type equations involving the Atangana–Baleanu fractional derivative by methods of spectral collocation are in [26]. As of lately, some authors investigated the FDEs with p-Laplacian operator by diverse types of mathematical techniques. For instance, Khan et al. investigated the existence criterion for solutions for FDEs involving the \(\phi _{p}\)-Laplacian operator in [36]. The EU of results for FDEs with \(\phi _{p}\)-Laplacian operator are analyzed by Chuanzhi Bai in [37] via fixed point theorems. Also we present the Green function’s properties and two examples to illustrate the results. The EPS for FDEs with the \(\phi _{p}\)-Laplacian operator is studied by Tian et al. [38] and EPSs are obtained with the help of a monotone iterative method. For more EU results for FDEs with a p-Laplacian operator one may refer to [3942]

Recently, a great interest has been shown in the study of HU stability of non-linear FDEs with different type of boundary conditions. By HU stability we mean that there exists an exact solution very close to the approximate solution of a FDE and that the error can be calculated. The EU of solutions and HU stability FDEs with p-Laplacian operator and ABC-fractional derivative involving a spatial singularity is derived by Khan et al. in [43] using the well-known Guo–Krasnoselskii theorem. Khan et al. [44] discussed the analytical study of existence and stability results of a singular non-linear FDEs with \(\phi _{p}\)-operator involving fractional integral and differential boundary conditions. The EU and HU stability of solutions for a coupled system of FDEs involving the derivative in Caputo’s sense are proved by Khan et al. [45] using a Leray–Schauder-type fixed point theorem and topological degree theory. Li et al. investigated the HU stability of FDEs in [46] and also presented an example to illustrate their result. Stability and EU of solutions for the fractional order HIV model were introduced by Khan et al. in [47]. Existence and stability of solutions for singular delay FDEs with fractional integral initial conditions by using the Green function and the fixed point theorem were established by the Khan et al. in [48]. For more details of stability analysis, see [4957].

Motivated by the above work, we introduce the EU and HU stability results, for non-liner FDEs involving Caputo fractional derivatives of distinct orders with \(\phi ^{*}_{P}\) Laplacian operator:

$$ { } \textstyle\begin{cases} \mathcal{ {}^{c}D}^{\mathcal{\zeta }}\phi ^{*}_{\mathcal{p}} [ \mathcal{{}^{c}D}^{\mathcal{ \sigma }} (\mathcal{z(t)}-\sum_{i=1}^{m} \lambda _{i}(t) ) ] =-\psi ^{\ast }\mathcal{(t, z(t))}, \quad \mathcal{t\in [0,1]}, \\ \phi ^{*}_{\mathcal{p}} [\mathcal{{}^{c}D}^{\mathcal{ \sigma }} \mathcal{z(t)}-\sum_{i=1}^{m}\lambda _{i}(t) ]|_{t=0} =0,\qquad \mathcal{z}(0)=\sum_{i=1}^{m}\lambda _{i}(0), \\ \mathcal{z}^{\prime }(\mathcal{1})=\sum_{i=1}^{m}\lambda _{i}^{\prime }(1), \qquad \mathcal{z}^{j}(\mathcal{0})=\sum_{i=1}^{m}\lambda _{i}^{j}(0) \quad \text{for } j=2,3,4,\ldots,n-1, \end{cases} $$
(1.1)

where \({}^{c}\mathcal{D}^{\mathcal{\zeta }}\), \({}^{c}\mathcal{D}^{ \mathcal{ \sigma }}\) denotes the derivative of fractional order ζ and σ in Caputo’s sense, respectively, and \({\psi ^{\ast }}\), \(\lambda _{i}(t)\) are continuous functions. The orders \(n-1< \sigma \le n\), \(0<\zeta \le 1 \) where \(n\ge 4\), \(\psi ^{\ast }\in \mathcal{L}[0,1] \) and \(\phi ^{\ast }_{\mathcal{p}}(\mathcal{z})=|\mathcal{z}|^{p-1} \mathcal{z}\) denotes the \(\mathcal{p}\)-Laplacian operator and satisfies \(\frac{1}{\mathcal{p}}+\frac{1}{\mathcal{q}}=1\), \((\phi ^{\ast }_{ \mathcal{p}})^{-1}=\phi ^{\ast }_{\mathcal{q}}\). The rest of article is divided in four parts. Basic definitions and desired lemmas are presented in Sect. 1 and properties of the Green functions are discussed in Sect. 2. The EU results are given in Sect. 3. HU stability is discussed in Sect. 4. In Sect. 5, we introduce an example.

Here, we introduce certain definitions, desired lemmas and theorems, which are essential to find the main result.

Definition 1.1

([1])

For an integrable and real valued continuous function \(\psi ^{\ast } \) defined on \((0,+\infty )\), the Riemann–Liouville integral of fractional order \(\delta \in \mathbb{R}\) is defined as

$$ { } I^{\delta }\mathcal{\psi ^{\ast }(\mathcal{y})} = \frac{1}{\varGamma (\delta )} \int _{0}^{\mathcal{y}}(\mathcal{y-x})^{ \delta -1} \mathcal{\psi ^{\ast }(x)\,dx}, \quad \delta >0. $$

Definition 1.2

([1])

For an n-times continuously differentiable real valued function \(\psi ^{\ast }\) defined on \((0,+\infty )\), the Caputo derivative of fractional order \(\delta \in \mathbb{R}\) (\(\delta >0\)) is defined as

$$ { } \mathcal{ {}^{c}D}^{\delta }\mathcal{\psi ^{\ast }(y)} = \frac{1}{\varGamma (\mathsf{n}-\delta )} \int _{0}^{\mathcal{y}}( \mathcal{y-x})^{\mathsf{n}-\delta -1} \mathcal{\bigl(\psi ^{\ast }\bigr)^{n}(x)\,dx}, \quad \mathsf{n-1}< \delta < \mathsf{n}, \mathsf{n}=[\delta ]+1, $$

where \([\delta ]\) represents the greatest integer and the integral exists on the \((0,+\infty )\) interval.

Lemma 1.1

([2])

Let\(\mathcal{ \sigma }\in \mathcal{(k-1, k]}\)and\(\mathcal{\psi ^{\ast }(t)}\in C^{k-1}\), then

$$ { } \mathcal{I^{ \sigma }D^{ \sigma }\mathcal{\psi ^{\ast }(y)}=\mathcal{\psi ^{\ast }(y)}+a_{0}+a_{1}y+a_{2}y^{2}+a_{3}y^{3}+ \cdots+a_{k-1}y^{k-1},} $$

for the\(a_{j}\in \mathbb{R}\)for\(j=0,1,2,\ldots,k-1\).

Theorem 1.2

([58, 59], Guo–Krasnoselskii theorem)

Consider\({\varOmega }^{\ast }\)to be a Banach space and let a cone\(\mathcal{K^{\ast }}\in \varOmega ^{\ast } \). Assume that\(\mathcal{A^{\ast }_{1},A^{\ast }_{2}}\)are two bounded subsets of\(\varOmega ^{\ast }\)such that\(0\in \mathcal{A^{\ast }_{1}}\), \(\mathcal{\overline{A^{\ast }_{1}}} \subset \mathcal{A^{\ast }_{2}}\). Then an operator\(\mathcal{G^{\ast }}:\mathcal{K^{\ast }}\cap ( \mathcal{\overline{A^{\ast }_{2}}} \setminus \mathcal{A^{\ast }_{1}}) \longrightarrow \mathcal{K^{\ast }}\), which is completely continuous and satisfies

$$\begin{aligned}& \bigl(\mathcal{P_{1}^{\ast }}\bigr) \quad \bigl\Vert \mathcal{G^{*} z} \bigr\Vert \le \Vert \mathcal{z} \Vert \quad \textit{if } \mathcal{z}\in \mathcal{K^{\ast }}\cap \mathcal{\partial {A^{\ast }_{1}}} \quad \textit{and} \quad \bigl\Vert \mathcal{G^{\ast }z} \bigr\Vert \ge \Vert \mathcal{z} \Vert \quad \textit{if } \mathcal{z}\in \mathcal{K^{\ast }}\cap \mathcal{\partial {A^{\ast }_{2}}}, \end{aligned}$$

or

$$\begin{aligned}& \bigl(\mathcal{P_{2}^{\ast }}\bigr) \quad \bigl\Vert \mathcal{G^{\ast }z} \bigr\Vert \ge \Vert \mathcal{z} \Vert \quad \textit{if }\mathcal{z}\in \mathcal{K^{\ast }}\cap \mathcal{\partial {A^{\ast }_{1}}} \quad \textit{and} \quad \bigl\Vert \mathcal{G^{\ast }z} \bigr\Vert \le \Vert \mathcal{z} \Vert \quad \textit{if } \mathcal{z}\in \mathcal{K^{\ast }}\cap \mathcal{\partial {A^{\ast }_{2}}}, \end{aligned}$$

has a fixed point in\(\mathcal{K^{\ast }}\cap (\mathcal{\overline{A^{\ast }_{2}}} \setminus \mathcal{A^{\ast }_{1}})\).

Lemma 1.3

([44, 45])

For the\(\mathcal{p}\)-Laplacian operator\(\phi ^{*}_{\mathcal{p}}\), the following conditions hold true:

  1. (1)

    If\(|\gamma _{1}|, |\gamma _{2}|\ge \sigma >0\), \(1< \mathcal{p}\le 2\), \(\gamma _{1}\gamma _{2}>0 \), then

    $$ \begin{aligned}[b] \bigl\vert \phi ^{*}_{\mathcal{p}}(\gamma _{1}) -\phi ^{*}_{\mathcal{p}}( \gamma _{2}) \bigr\vert \le (\mathcal{p}-1)\sigma ^{\mathcal{p}-2} \vert \gamma _{1}- \gamma _{2} \vert . \end{aligned} $$
  2. (2)

    If\(\mathcal{p}> 2\), \(|\gamma _{1}|, |\gamma _{2}|\le \sigma ^{\ast }>0\), then

    $$ \bigl\vert \phi ^{*}_{\mathcal{p}}(\gamma _{1}) -\phi ^{*}_{\mathcal{p}}( \gamma _{2}) \bigr\vert \le (\mathcal{p}-1) \bigl(\sigma ^{\ast }\bigr)^{\mathcal{p}-2} \vert \gamma _{1}-\gamma _{2} \vert . $$

Green function and properties

Theorem 2.1

Consider\(\psi ^{\ast }\in \mathcal{C}[0,1]\)satisfying the FDE with\(\phi ^{*}_{\mathcal{p}}\) (1.1). Then, for\(\zeta \in (0,1]\)and\(\sigma \in (n-1,n]\), the FDEs (1.1) involving the\(\phi ^{*}_{p}\)Laplacian operator has a solution equivalent to

$$ { } \mathcal{z(t)}=\sum_{i=1}^{m} \lambda _{i}(t)+ \int _{0}^{1} \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{ \mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s}, $$
(2.1)

where the Green function\(\mathcal{H}^{ \sigma }(\mathcal{t}, \mathcal{s})\)is defined by

$$ { } \mathcal{H}^{ \sigma }(\mathcal{t}, \mathcal{s})= \textstyle\begin{cases} \frac{-(\mathcal{t}- \mathcal{s})^{ \sigma -1}}{\varGamma ( \sigma )}+ \mathcal{t}\frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}, & 0< \mathcal{s} \le t< 1, \\ \mathcal{t}\frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}, & 0< t\le \mathcal{s}< 1. \end{cases} $$
(2.2)

Proof

Taking the integral operator \(I^{\mathcal{\zeta }}\) on both sides (1.1) and using Lemma 1.1, Eq. (1.1) becomes

$$ { } \phi ^{*}_{\mathcal{p}} \Biggl( \mathcal{{}^{c}D}^{\mathcal{ \sigma }}\Biggl[ \mathcal{z(t)}-\sum _{i=1}^{m}\lambda _{i}(t)\Biggr] \Biggr) =-I^{ \mathcal{\zeta }}\bigl[\psi ^{\ast }\mathcal{\bigl(t, z(t)\bigr)} \bigr]+C_{0}. $$
(2.3)

From the condition \(\phi ^{*}_{\mathcal{p}} (\mathcal{{}^{c}D}^{\mathcal{ \sigma }}[ \mathcal{z(t)}-\sum_{i=1}^{m}\lambda _{i}(t)] ) |_{t=0} =0\), \(\implies C_{0}=0\).

Using the value of \(C_{0}=0\), then (2.3) becomes

$$ { } \phi ^{*}_{\mathcal{p}} \Biggl( \mathcal{{}^{c}D}^{\mathcal{ \sigma }}\Biggl[ \mathcal{z(t)}-\sum _{i=1}^{m}\lambda _{i}(t)\Biggr] \Biggr) =-I^{ \mathcal{\zeta }}\bigl[\psi ^{\ast }\mathcal{\bigl(t, z(t)\bigr)}\bigr]. $$
(2.4)

Applying the \(\mathcal{q}\)-Laplacian operator further on (2.4) we get the form

$$ { } \mathcal{{}^{c}D}^{\mathcal{ \sigma }}\Biggl[ \mathcal{z(t)}-\sum_{i=1}^{m} \lambda _{i}(t)\Biggr]=-\phi ^{*}_{\mathcal{q}} \bigl(I^{\mathcal{\zeta }}\bigl[ \psi ^{\ast }\mathcal{\bigl(t, z(t)\bigr)}\bigr] \bigr). $$
(2.5)

Again taking the integral operator \(I^{\mathcal{ \sigma }}\) to both sides of (2.5) and using Lemma 1.1, then (2.5) becomes

$$ { } \mathcal{z(t)}-\sum_{i=1}^{m} \lambda _{i}(t)=-I^{\mathcal{ \sigma }} \bigl(\phi ^{*}_{\mathcal{q}} \bigl(I^{\mathcal{\zeta }}\bigl[\psi ^{\ast } \mathcal{\bigl(t, z(t)\bigr)} \bigr] \bigr) \bigr)+a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3}+ \cdots+a_{n-1}t^{n-1}, $$
(2.6)

where \(a_{j}\in \mathbb{R}\) for \(j=0,1,2,\ldots,n-1\).

Using the boundary conditions \(\mathcal{z}^{j}(\mathcal{0})=\sum_{i=1}^{m}\lambda _{i}^{j}(0) \) for \(j= 0,2,3,4,\ldots,n-1\), in (2.6), \(\implies a_{j}=0 \) for \(j= 0,2,3,4,\ldots,n-1 \). and \(\mathcal{z}^{\prime }(\mathcal{1})=\sum_{i=1}^{m}\lambda _{i}^{\prime }(1)\), implies that \(a_{1}= [4] I^{\mathcal{ \sigma -1}} (\phi ^{*}_{\mathcal{q}} (I^{ \mathcal{\zeta }}[\psi ^{\ast }\mathcal{(t, z(t))}] ) ) |_{\mathcal{t=1}}\)

Putting the values of the constants \(a_{i}\) in (2.6), we get

$$ { } \mathcal{z(t)}=\sum_{i=1}^{m} \lambda _{i}(t)+ \int _{0}^{1} \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{ \mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s}, $$
(2.7)

where \(\mathcal{H}^{ \sigma }(\mathcal{t}, \mathcal{s})\) is defined in (2.2). □

Lemma 2.2

The Green function\(\mathcal{H}^{ \sigma }(\mathcal{t}, \mathcal{s})\)defined in (2.2) satisfies the following conditions:

\((\mathcal{B_{1}})\):

\(\mathcal{H}^{ \sigma }(\mathcal{t}, \mathcal{s})> 0\)\(\forall \mathcal{s},\mathcal{t}\in (0,1)\);

\((\mathcal{B_{2}})\):

the function\(\mathcal{H}^{ \sigma }(\mathcal{t}, \mathcal{s})\)is increasing and\(\mathcal{H}^{ \sigma }(\mathcal{1}, \mathcal{s})= \max_{t\in [0,1]}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\);

\((\mathcal{B_{3}})\):

\(\mathcal{H}^{ \sigma }(\mathcal{1}, \mathcal{s}) \ge \mathcal{t}^{ \sigma -1}\max_{t\in [0,1]} \mathcal{H}^{ \sigma }(\mathcal{t}, \mathcal{s})\)for\(\mathcal{t},\mathcal{s}\in (0,1)\).

Proof

To prove \((\mathcal{B_{1}})\), we take two cases \(\forall \mathcal{t}, \mathcal{s} \in (0,1)\).

Case 1. For \(\mathcal{s}\le t\). As \(\sigma >3\), then

$$ { } \begin{aligned}[b] \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s}) &= \frac{-(\mathcal{t}- \mathcal{s})^{ \sigma -1}}{\varGamma ( \sigma )}+ \mathcal{t}\frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)} \\ & = \frac{-\mathcal{t}^{ \sigma -1}(1-\frac{ \mathcal{s}}{\mathcal{t}})^{ \sigma -1}}{\varGamma ( \sigma )}+ \mathcal{t}\frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)} \\ & \ge \frac{-\mathcal{t}^{ \sigma -1}(1- \mathcal{s})^{ \sigma -1}}{\varGamma ( \sigma )}+ \mathcal{t}^{ \sigma -1} \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}>0. \end{aligned} $$
(2.8)

Case 2. When \(t\le \mathcal{s}\), we evaluate

$$ { } \begin{aligned}[b] \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})=\mathcal{t} \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}>0. \end{aligned} $$
(2.9)

From (2.8) and (2.9), it is proved that \(\mathcal{H}^{ \sigma }(\mathcal{t}, \mathcal{s})> 0\)\(\forall \mathcal{s},\mathcal{t}\in (0,1)\).

To prove the condition \((\mathcal{B_{2}})\), we assume that \(\forall \mathcal{s}, \mathcal{t}\in (0,1)\).

Case 1. For \(\mathcal{s}\le t\). As \(\sigma >3\), then

$$ { } \begin{aligned}[b] \frac{\partial }{\partial \mathcal{t}} \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s}) &= \frac{-(\mathcal{t}- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}+ \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)} \\ & = \frac{-\mathcal{t}^{ \sigma -2}(1-\frac{ \mathcal{s}}{\mathcal{t}})^{ \sigma -2}}{\varGamma ( \sigma -1)}+ \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)} \\ & \ge \frac{-\mathcal{t}^{ \sigma -2}(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}+ \mathcal{t}^{ \sigma -2} \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}>0. \end{aligned} $$
(2.10)

Case 2. When \(t\le \mathcal{s}\), we find that

$$ { } \begin{aligned}[b] \frac{\partial }{\partial \mathcal{t}} \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})= \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}>0. \end{aligned} $$
(2.11)

From Eqs. (2.10) and (2.11), it is shown that \(\frac{\partial }{\partial \mathcal{t}}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})>0\)\(\forall \mathcal{s}, \mathcal{t}\in (0,1)\), consequently, \(\frac{\partial }{\partial \mathcal{t}}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\) is an increasing function. Thus, we have for \(\mathcal{t}\ge \mathcal{s}\)

$$ \max_{t\in [0,1]}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})= \frac{-(\mathcal{1}- \mathcal{s})^{ \sigma -1}}{\varGamma ( \sigma )}+ \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}= \mathcal{H}^{ \sigma }(\mathcal{1}, \mathcal{s}), $$
(2.12)

and for \(\mathcal{s}\ge \mathcal{t}\)

$$ \max_{t\in [0,1]}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})= \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}= \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}). $$
(2.13)

To prove the condition \((\mathcal{B_{3}})\), we assume that

Case 1. For \(\mathcal{s}\le \mathcal{t}\). As \(\sigma >3\), then

$$ { } \begin{aligned}[b] \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s}) &= \frac{-(\mathcal{t}- \mathcal{s})^{ \sigma -1}}{\varGamma ( \sigma )}+ \mathcal{t}\frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)} \\ & = \frac{-\mathcal{t}^{ \sigma -1}(1-\frac{ \mathcal{s}}{\mathcal{t}})^{ \sigma -1}}{\varGamma ( \sigma )}+ \mathcal{t}\frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)} \\ & \ge \frac{-\mathcal{t}^{ \sigma -1}(1- \mathcal{s})^{ \sigma -1}}{\varGamma ( \sigma )}+ \mathcal{t}^{ \sigma -1} \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)} \\ & =\mathcal{t}^{ \sigma -1} \biggl( \frac{-(1- \mathcal{s})^{ \sigma -1}}{\varGamma ( \sigma )}+ \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)} \biggr)= \mathcal{t}^{ \sigma -1}\mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}). \end{aligned} $$
(2.14)

Case 2. For \(\mathcal{t}\le \mathcal{s}\), we evaluate

$$ { } \begin{aligned}[b] \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})&=\mathcal{t} \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)} \\ & \ge \mathcal{t}^{ \sigma -1} \frac{(1- \mathcal{s})^{ \sigma -2}}{\varGamma ( \sigma -1)}= \mathcal{t}^{ \sigma -1} \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}). \end{aligned} $$
(2.15)

Thus, by Eqs. (2.14) and (2.15), condition \(\mathcal{B_{3}}\) is proved. □

Existence result

Now we prove our existence result by introducing the following conditions.

Let \(\varOmega ^{\ast }=\mathcal{ C[0,1]}\) be the Banach space containing all real valued functions defined on \(\mathcal{[0,1]}\), which are continuous and endowed with the \(\|{\mathcal{z}}\|= \max_{\mathcal{t\in [0,1]}}\{| \mathcal{z(t)}|:\mathcal{z}\in \varOmega ^{\ast }\}\). Suppose that \(\mathcal{K^{\ast }}\in \varOmega ^{\ast } \) is a cone of functions, which are positive and of the type \(\mathcal{K^{\ast }}=\{ \mathcal{z}\in \varOmega ^{\ast }: \mathcal{z(t)} \ge \mathcal{t}^{ \sigma }\|\mathcal{z}\|, \mathcal{t \in [0,1]}\}\). Let \(\mathcal{A}^{\ast }(r)=\{ \mathcal{z}\in \mathcal{K^{\ast }}:\| \mathcal{z}\|< r \}\), \(\partial \mathcal{A}^{\ast }(r)=\{ \mathcal{z}\in \mathcal{K^{\ast }}:\|\mathcal{z}\|=r \}\). By using Theorem 2.1. Equation (1.1) is equivalent to

$$ { } \mathcal{z(t)}=\sum_{i=1}^{m} \lambda _{i}(t)+ \int _{0}^{1} \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{ \mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s}. $$
(3.1)

Let us define an operator \(\mathcal{G}^{\ast }:\mathcal{K^{\ast }}\setminus \{0\}\rightarrow \varOmega ^{\ast }\) associated with problem (1.1), such that

$$ { } \mathcal{G}^{\ast }\mathcal{z(t)}=\sum _{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s}. $$
(3.2)

By using Theorem 2.1, the solution of FDE given by Eq. (1.1) is a fixed point \(\mathcal{z(t)}\) of \(\mathcal{G}^{\ast }\) i.e.,

$$ \mathcal{z(t)}=\mathcal{G}^{\ast }\mathcal{z(t)}. $$
(3.3)

To obtain the existence result we need the following assumptions:

\((\mathcal{R_{1}})\):

\(\mathcal{\psi ^{\ast }({t},z(t))}:[0,1] \times (0,+ \infty ) \longrightarrow \mathbb{R^{+}}\) is a continuous function.

\((\mathcal{R_{2}})\):

\(\lambda _{i}(t):[0,1] \longrightarrow \mathbb{R^{+}}\) are also continuous functions for each \(i=1,2,3,\ldots,m\), with

$$ \sum_{i=1}^{m}\lambda _{i}(t)\| \le \overline{\Delta }< + \infty . $$
\((\mathcal{R_{3}})\):

\(|\mathcal{\psi ^{\ast }({t},z(t))}|\le \phi ^{*}_{\mathcal{p}}( \varLambda _{1}|\mathcal{z(t)}|^{l_{1}}+\varLambda _{2})\)\(\forall \mathcal{t} \in [0,1]\), \(\mathcal{z} \in \varOmega ^{\ast }\) where \(\varLambda _{1}\), \(\varLambda _{2}\) are positive constants and \(l_{1}\in [0,1]\).

\((\mathcal{R_{4}})\):

\(\mathcal{|\psi ^{\ast }(t,z)-\psi ^{\ast }(t,\upsilon )|\le L(|z-\upsilon |)}\)\(\mathcal{\forall L>0}\), \(\mathcal{t \in [0,1]}\), \(\mathcal{L>0}\), \(\mathcal{z,\upsilon \in \varOmega ^{\ast }}\).

Theorem 3.1

Let us assume that conditions\((\mathcal{R_{1}})\)\(( \mathcal{R_{3}})\)are satisfied. Then\(\mathcal{G^{\ast }}\)is a completely continuous operator.

Proof

For any \(\mathcal{z}\in (\overline{\mathcal{A^{\ast }}_{2}(r)}) \setminus \mathcal{A^{\ast }_{1}}(r))\), using Lemma 2.2 and (3.2), we have

$$\begin{aligned}& { } \begin{aligned}[b] \mathcal{G}^{\ast } \mathcal{z(t)}&=\sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \le \sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s}, \end{aligned} \end{aligned}$$
(3.4)
$$\begin{aligned}& { } \begin{aligned}[b] \mathcal{G}^{\ast } \mathcal{z(t)}&=\sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \ge \sum_{i=1}^{m}\lambda _{i}(t)+\mathcal{t}^{ \sigma -1} \int _{0}^{1} \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s})\phi ^{\ast }_{ \mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s}. \end{aligned} \end{aligned}$$
(3.5)

From (3.4) and (3.5), we arrive at

$$ { } \mathcal{G}^{\ast }\mathcal{z(t)}\ge \mathcal{t}^{ \sigma -1} \bigl\Vert \mathcal{G}^{\ast }\mathcal{z(t)} \bigr\Vert , \quad \mathcal{t}\in [0,1]. $$
(3.6)

This implies that \(\mathcal{G}^{\ast }:(\overline{\mathcal{A^{\ast }}_{2}(r)}) \setminus \mathcal{A^{\ast }_{1}}(r))\rightarrow \mathcal{K}^{\ast }\).

Next, to show that \(\mathcal{G}^{\ast }\) is a continuous map, we prove that \(\| \mathcal{G}^{\ast }\mathcal{z_{n}(t)}-\mathcal{G}^{\ast } \mathcal{z(t)}\|\longrightarrow {0}\) as \(\mathcal{n}\longrightarrow \infty \); let us address

$$ { } \begin{aligned}[b] &\bigl\Vert \mathcal{G}^{\ast }\mathcal{z_{n}(t)}-\mathcal{G}^{\ast } \mathcal{z(t)} \bigr\Vert \\ &\quad = \Biggl\vert \sum_{i=1}^{m} \lambda _{i}(t)+ \int _{0}^{1} \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{ \mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z_{n}( \epsilon )\bigr)\,d\epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \qquad {}-\sum_{i=1}^{m}\lambda _{i}(t)- \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \Biggr\vert \\ & \quad \le \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s}) \bigr\vert \biggl\vert \phi ^{\ast }_{\mathcal{q}} \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{ \ast }\mathcal{\bigl(\epsilon , z_{n}( \epsilon )\bigr)\,d\epsilon } \biggr) \,\mathcal{d} \mathcal{s} \\ & \qquad {}-\phi ^{\ast }_{\mathcal{q}} \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{ \ast }\mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \biggr\vert . \end{aligned} $$
(3.7)

By continuity of the function \(\psi ^{\ast }\), we have \(\| \mathcal{G}^{\ast }\mathcal{z_{n}(t)}-\mathcal{G}^{\ast } \mathcal{z(t)}\|\longrightarrow {0}\) as \(\mathcal{n}\longrightarrow \infty \). This implies that \(\mathcal{G}^{\ast }\) is a continuous map.

Now, we have to prove \(\mathcal{G^{\ast }}\) is uniformly bounded on \((\overline{\mathcal{A^{\ast }}_{2}(r)}) \setminus \mathcal{A^{\ast }}_{1}(r)\).

By (3.2) and using \((\mathcal{R_{2}}\mathcal{R_{3}})\), for any \(\mathcal{t}\in [0,1]\), we get

$$ { } \begin{aligned}[b] &\bigl\Vert \mathcal{G^{\ast }z} \bigr\Vert \\ &\quad = \Biggl\vert \sum _{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1} \mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{ \mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \Biggr\vert \\ & \quad \le \Biggl\vert \sum_{i=1}^{m}\lambda _{i}(t) \Biggr\vert + \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}) \bigr\vert \phi ^{\ast }_{ \mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\phi ^{*}_{\mathcal{p}}\bigl(\varLambda _{1} \bigl\Vert \mathcal{z(\epsilon )} \bigr\Vert ^{l_{1}}+ \varLambda _{2}\bigr)\,\mathcal{d}\epsilon \biggr)\,\mathcal{d} \mathcal{s} \\ &\quad \le \overline{\Delta }+ \biggl[ \frac{1}{\varGamma (\mathcal{ \sigma +1})}+\frac{1}{\varGamma ( \sigma )} \biggr] \biggl[\frac{1}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q}-1}\bigl( \varLambda _{1} \bigl\Vert \mathcal{z(\epsilon )} \bigr\Vert ^{l_{1}}+ \varLambda _{2}\bigr) \\ &\quad =\overline{\Delta }+ \varTheta \bigl(\varLambda _{1} \bigl\Vert \mathcal{z(\epsilon )} \bigr\Vert ^{l_{1}}+\varLambda _{2}\bigr), \end{aligned} $$
(3.8)

where \(\varTheta = [ \frac{1}{\varGamma (\mathcal{ \sigma +1})}+ \frac{1}{\varGamma ( \sigma )} ] [\frac{1}{\varGamma (\zeta +1)} ]^{\mathcal{q}-1}\).

This proves that \(\mathcal{G^{\ast }}\) is uniformly bounded.

In order to show that the operator \(\mathcal{G^{\ast }}\) is compact, we show the equicontinuity of the operator \(\mathcal{G^{\ast }}\).

For \(0<\mathcal{t_{1}}<\mathcal{t_{2}}<1\), we have

$$ { } \begin{aligned}[b] &\bigl\vert \mathcal{G^{\ast }z}(\mathcal{t_{2}})-\mathcal{G^{\ast }z}( \mathcal{t_{1}}) \bigr\vert \\ &\quad \le \Biggl\vert \sum_{i=1}^{m}\lambda _{i}(t_{2})-\sum_{i=1}^{m} \lambda _{i}(t_{1}) \Biggr\vert \\ & \qquad {}+ \biggl\vert \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t_{2}, \mathcal{s}})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \qquad {}- \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t_{1}, \mathcal{s}}) \phi ^{\ast }_{\mathcal{q}} \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \biggr\vert \\ & \quad \le \Biggl\vert \sum_{i=1}^{m}\lambda _{i}(t_{2})-\sum_{i=1}^{m} \lambda _{i}(t_{1}) \Biggr\vert \\ & \qquad {}+ \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{t_{2}, \mathcal{s}})-\mathcal{H}^{ \sigma }( \mathcal{t_{1}, \mathcal{s}}) \bigr\vert \phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\phi ^{*}_{\mathcal{p}}\bigl(\varLambda _{1} \bigl\Vert \mathcal{z(\epsilon )} \bigr\Vert ^{l_{1}}+ \varLambda _{2}\bigr)\,\mathcal{d}\epsilon \biggr)\,\mathcal{d} \mathcal{s} \\ & \quad \le \biggl( \frac{ \vert \mathcal{t_{2}}^{ \sigma }-\mathcal{t_{1}}^{ \sigma } \vert }{\varGamma ( \sigma +1)}+ \frac{ \vert \mathcal{t_{2}}-\mathcal{t_{1}} \vert }{\varGamma ( \sigma )} \biggr) \biggl[ \frac{1}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q}-1}\bigl( \varLambda _{1} \bigl\Vert \mathcal{z(\epsilon )} \bigr\Vert ^{l_{1}}+\varLambda _{2}\bigr) \\ &\qquad {}\times \bigl\vert \mathcal{G^{*}z}(\mathcal{t_{2}})- \mathcal{G^{*}z}(\mathcal{t_{1}}) \bigr\vert \\ &\quad \longrightarrow {0} \quad \text{as } (\mathcal{t_{2}}- \mathcal{t_{1}}) \longrightarrow {0}. \end{aligned} $$
(3.9)

Thus, \(\mathcal{G^{*}}\) is an equicontinuous operator on \((\overline{\mathcal{A^{\ast }}_{2}(r)}) \setminus \mathcal{A^{\ast }_{1}}(r)\) and by the Arzela–Ascoli theorem \(\mathcal{G^{*}}\) is compact on \((\overline{\mathcal{A^{\ast }}_{2}(r)}) \setminus \mathcal{A^{\ast }_{1}}(r)\). In fact, all the conditions of Theorem 2.1 [58] are satisfied. Thus \(\mathcal{G}^{\ast }:(\overline{\mathcal{A^{\ast }}_{2}(r)}) \setminus \mathcal{A^{\ast }_{1}}(r))\rightarrow \mathcal{K}^{\ast }\) is a completely continuous operator.

Now here, let us determine the hight functions for \(\psi ^{\ast }\mathcal{(t, z(t))}\) for \(r>0\), \(\forall \mathcal{t}\in [0,1]\)

$$ { } \textstyle\begin{cases} \phi ^{*}_{\min}(\mathcal{t},r)= \min_{\mathcal{t}\in [0,1]}\{\psi ^{\ast } \mathcal{(t, z(t))}:\mathcal{t}^{ \sigma -1}r\le \mathcal{z}\le r \} \ge \mathsf{\overline{m}}>-\infty , \\ \phi ^{*}_{\max}(\mathcal{t},r)= \max_{\mathcal{t}\in [0,1]}\{\psi ^{\ast } \mathcal{(t, z(t))}:\mathcal{t}^{ \sigma -1}r\le \mathcal{z}\le r \} \le \mathsf{\overline{M}}< +\infty . \end{cases} $$
(3.10)

 □

Theorem 3.2

Suppose that assumptions\((\mathcal{R_{1}})\)\(( \mathcal{R_{3}})\), are satisfied and\(\exists \mathcal{a},\mathcal{b}\in \mathbb{R^{+}}\)such that any of the following condition is satisfied:

\((\mathcal{S_{1}})\):

\(\mathcal{a}\le \|\sum_{i=1}^{m}\lambda _{i}(t)\|+ \int _{0}^{1} |\mathcal{H}^{ \sigma }(\mathcal{1}, \mathcal{s}) |\phi ^{\ast }_{\mathcal{q}} (\frac{1}{\varGamma (\zeta )}\int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\phi ^{*}_{\min}( \mathcal{\epsilon },\mathcal{a})\,\mathcal{d}\epsilon )\,\mathcal{d} \mathcal{s}< {+\infty }\)and\(\|\sum_{i=1}^{m}\lambda _{i}(t)\|+\int _{0}^{1} | \mathcal{H}^{ \sigma }(\mathcal{1}, \mathcal{s}) |\phi ^{\ast }_{ \mathcal{q}} (\frac{1}{\varGamma (\zeta )}\int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\phi ^{*}_{\max}( \mathcal{\epsilon },\mathcal{b})\,\mathcal{d}\epsilon )\,\mathcal{d} \mathcal{s}\le \mathcal{b}\), or

\((\mathcal{S_{2}})\):

\(\|\sum_{i=1}^{m}\lambda _{i}(t)\|+\int _{0}^{1} |\mathcal{H}^{ \sigma }(\mathcal{1}, \mathcal{s}) |\phi ^{ \ast }_{\mathcal{q}} (\frac{1}{\varGamma (\zeta )}\int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\phi ^{*}_{\max}( \mathcal{\epsilon },\mathcal{a})\,\mathcal{d}\epsilon )\,\mathcal{d} \mathcal{s}< \mathcal{a}\)and\(\mathcal{b}\le \|\sum_{i=1}^{m}\lambda _{i}(t)\|+\int _{0}^{1} |\mathcal{H}^{ \sigma }(\mathcal{1}, \mathcal{s}) |\phi ^{ \ast }_{\mathcal{q}} (\frac{1}{\varGamma (\zeta )}\int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\phi ^{*}_{\min}( \mathcal{\epsilon },\mathcal{b})\,\mathcal{d}\epsilon )\,\mathcal{d} \mathcal{s}< {+\infty }\).

Then Eq. (1.1) has a positive solution\(\mathcal{z}\in \mathcal{K^{\ast }}\)and\(\mathcal{a}\le \|\mathcal{z}\| \le \mathcal{b}\).

Proof

Firstly, we are considering the case \((\mathcal{S_{1}})\). If \(\mathcal{z}\in \partial \mathcal{A^{\ast }}(\mathcal{a}) \) then \(\|\mathcal{z}\|=\mathcal{a}\) and \(\forall \mathcal{t}\in [0,1]\), \(\mathcal{t}^{ \sigma -1}\mathcal{a} \le \mathcal{z}\le \mathcal{a}\). Using (3.10), \(\phi ^{*}_{\min}(\mathcal{t},a)\le \psi ^{\ast } \mathcal{(t, z(t))}\), we write

$$ { } \begin{aligned}[b] \bigl\Vert \mathcal{G^{\ast }z} \bigr\Vert &=\max_{\mathcal{t}\in [0,1]} \Biggl\vert \sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \Biggr\vert \\ & \ge \Biggl\Vert \sum_{i=1}^{m}\lambda _{i}(t) \Biggr\Vert +\mathcal{t}^{ \sigma -1} \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}) \bigr\vert \phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \ge \Biggl\Vert \sum_{i=1}^{m}\lambda _{i}(t) \Biggr\Vert +\mathcal{t}^{ \sigma -1} \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}) \bigr\vert \phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\phi ^{*}_{\min}(\mathcal{\epsilon },a)\,d \epsilon \biggr)\,\mathcal{d} \mathcal{s} \ge \mathcal{a} \\ & = \Vert \mathcal{z} \Vert . \end{aligned} $$
(3.11)

Now, for all \(\mathcal{t}\in [0,1]\), \(\mathcal{t}^{ \sigma -1}\mathcal{b}\le \mathcal{z}\le \mathcal{b}\).

If \(\mathcal{z}\in \partial \mathcal{A^{\ast }}(\mathcal{b}) \) then \(\|\mathcal{z}\|=\mathcal{b}\) and, using (3.10), we have \(\phi ^{*}_{\max}(\mathcal{t},b)\ge \psi ^{\ast } \mathcal{(t, z(t))}\); we find

$$ { } \begin{aligned}[b] \bigl\Vert \mathcal{G^{\ast }z} \bigr\Vert &=\max_{\mathcal{t}\in [0,1]} \Biggl\vert \sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \Biggr\vert \\ & \le \Biggl\Vert \sum_{i=1}^{m}\lambda _{i}(t) \Biggr\Vert + \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}) \bigr\vert \phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \le \Biggl\Vert \sum_{i=1}^{m}\lambda _{i}(t) \Biggr\Vert +\mathcal{t}^{ \sigma -1} \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}) \bigr\vert \phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\phi ^{*}_{\min}(\mathcal{\epsilon },a)\,d \epsilon \biggr)\,\mathcal{d} \mathcal{s} \ge \mathcal{a} \\ & = \Vert \mathcal{z} \Vert . \end{aligned} $$
(3.12)

Using Lemma 1.2, \(\mathcal{z}\in (\overline{\mathcal{A^{\ast }}(\mathcal{b})}) \setminus \mathcal{A^{\ast }}(\mathcal{a})\) is a fixed point of \(\mathcal{G^{*}}\). By using Lemma 2.2 and Theorem 2.1, for \(\mathcal{t}\in (0,1)\) and \(\mathcal{a}\le \|\mathcal{z}\|\le \mathcal{b}\), we have \(\mathcal{z(t)} \ge \mathcal{t}^{ \sigma -1}\|\mathcal{z(t)}\|\ge \mathcal{at}^{ \sigma -1} > 0 \). Therefore \(\mathcal{z(t)} \) is positive solution. It obeys

$$\begin{aligned}& \begin{aligned}[b] \frac{\partial }{\partial \mathcal{t}}\mathcal{z(t)}&= \frac{\partial }{\partial \mathcal{t}}\mathcal{G^{*}}\mathcal{z(t)} \\ &=\frac{\partial }{\partial \mathcal{t}}\sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\frac{\partial }{\partial \mathcal{t}} \mathcal{H}^{ \sigma }(\mathcal{t}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & >0. \end{aligned} \end{aligned}$$
(3.13)

 □

Uniqueness result

Theorem 3.3

Let us assume that assumptions\(\mathcal{(R_{1})}\), \(\mathcal{(R_{2})}\)and\(\mathcal{(R_{4})}\)are satisfied. Then there exists a unique solution for Eq. (1.1) on\(\mathcal{[0,1]}\), if

$$ { } \Delta ^{\ast }=\mathcal{L}(\mathcal{q-1}) \biggl[ \frac{\mathsf{\overline{M}}}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q-2}} \biggl[ \frac{1}{\varGamma (\mathcal{ \sigma +1})}+ \frac{1}{\varGamma ( \sigma )} \biggr] \biggl[\frac{1}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q}-1}\le 1. $$
(3.14)

Proof

We prove the uniqueness result for \(\mathcal{p}\ge 2 \).

By (3.10) and, for all \(\mathcal{t}\in [0,1]\),

$$ { } \begin{aligned}[b] I^{\mathcal{\zeta }}\bigl[\psi ^{\ast }\mathcal{\bigl(t, z(t)\bigr)}\bigr]&= \frac{1}{\varGamma (\zeta )} \int _{0}^{\mathcal{t}}(\mathcal{t}- \mathcal{s})^{\zeta -1} \psi ^{\ast } \mathcal{\bigl(\mathcal{s}, z(\mathcal{s})\bigr)\,\mathcal{d} \mathcal{s}}\\ &\le \frac{1}{\varGamma (\zeta )} \int _{0}^{\mathcal{t}}(\mathcal{t}- \mathcal{s})^{\zeta -1} \mathsf{\overline{M}}\,\mathcal{d}\mathcal{s} \le \frac{\mathsf{\overline{M}}}{\varGamma (\zeta +1)}. \end{aligned} $$
(3.15)

For each \(\mathcal{z}\in (\overline{\mathcal{A^{\ast }}({r})}) \setminus \mathcal{A^{\ast }}({r})\) and using (3.15) we have

$$ \begin{aligned}[b] &\bigl\Vert \mathcal{G^{*}z}- \mathcal{G^{*}\upsilon } \bigr\Vert \\ &\quad = \max_{\mathcal{t}\in [0,1]} \Biggl\vert \sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \qquad {}-\sum_{i=1}^{m}\lambda _{i}(t)- \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , \upsilon (\epsilon )\bigr)\,d\epsilon } \biggr) \,\mathcal{d} \mathcal{s} \Biggr\vert \\ & \quad \le \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}) \bigr\vert \biggl\vert \phi ^{\ast }_{\mathcal{q}} \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \qquad {}-\phi ^{\ast }_{\mathcal{q}} \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{ \ast }\mathcal{\bigl(\epsilon , \upsilon (\epsilon )\bigr)\,d\epsilon } \biggr) \,\mathcal{d} \mathcal{s} \biggr\vert \\ & \quad \le (\mathcal{q-1}) \biggl[ \frac{\mathsf{\overline{M}}}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q-2}} \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}) \bigr\vert \biggl\vert \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \qquad {}- \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , \upsilon (\epsilon )\bigr)\,d\epsilon } \biggr) \,\mathcal{d} \mathcal{s} \biggr\vert \\ & \quad \le \mathcal{L}(\mathcal{q-1}) \biggl[ \frac{\mathsf{\overline{M}}}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q-2}} \biggl[ \frac{1}{\varGamma (\mathcal{ \sigma +1})}+ \frac{1}{\varGamma ( \sigma )} \biggr] \biggl[\frac{1}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q}-1}\mathcal{ \Vert z-\upsilon \Vert } \\ & \quad =\Delta ^{\ast } \quad \forall \mathcal{t \in [0,1]}, \end{aligned} $$

but in (3.14) we assumed that \(\Delta ^{\ast }<1\). This proves that \(\mathcal{G^{*}}\) is a contraction map. Hence by the Banach contraction mapping principle there exists a unique fixed point for operator \(\mathcal{G^{*}}\). Hence, there exists a unique solution of Eq. (1.1) on \(\mathcal{[0,1]}\). □

Hyers–Ulam stability

Here, we analyze the HU stability of (1.1). We define the HU stability as follows.

Definition 4.1

([60])

The integral equation (3.1) is said to be HU stable if there exists a non-negative constant \(\Delta ^{\ast }\), for some fixed non-negative constant \(\gamma ^{\ast }>0\) satisfying the following:

If

$$ { } \begin{aligned}[b] \Biggl\vert \mathcal{z}( \mathcal{t}) - \sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \Biggr\vert \le \gamma ^{*} , \end{aligned} $$
(4.1)

then there exists a function \(\upsilon (t)\), which is continuous and satisfies

$$ { } \mathcal{\upsilon }(\mathcal{t}) = \sum _{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , \upsilon (\epsilon )\bigr)\,d\epsilon } \biggr) \,\mathcal{d} \mathcal{s}, $$
(4.2)

with

$$ \bigl\vert \mathcal{z(t)-\upsilon (t)} \bigr\vert \le \Delta ^{\ast }\gamma ^{\ast }. $$
(4.3)

Theorem 4.1

The FDE (1.1) with\(\phi ^{\ast }_{\mathcal{p}}\)operator is HU stable for\(\mathcal{p}>2\)provided that\(\mathcal{(R_{1})}\), \(\mathcal{(R_{2})}\)and\(\mathcal{(R_{4})}\)are satisfied.

Proof

For the HU stability of the problem (1.1), we prove that for the integral equation (3.1), with assumptions \(\mathcal{(R_{1})}\), \(\mathcal{(R_{2})}\) and \(\mathcal{(R_{4})}\). we have

$$ \begin{aligned}[b] &\bigl\vert \mathcal{z(t)}- \mathcal{\upsilon (t)} \bigr\vert \\ &\quad = \max_{\mathcal{t}\in [0,1]} \Biggl\vert \sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \qquad {}-\sum_{i=1}^{m}\lambda _{i}(t)- \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , \upsilon (\epsilon )\bigr)\,d\epsilon } \biggr) \,\mathcal{d} \mathcal{s} \Biggr\vert \\ & \quad \le \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}) \bigr\vert \biggl\vert \phi ^{\ast }_{\mathcal{q}} \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \qquad {}-\phi ^{\ast }_{\mathcal{q}} \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{ \ast }\mathcal{\bigl(\epsilon , \upsilon (\epsilon )\bigr)\,d\epsilon } \biggr) \,\mathcal{d} \mathcal{s} \biggr\vert \\ &\quad \le (\mathcal{q-1}) \biggl[ \frac{\mathsf{\overline{M}}}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q-2}} \int _{0}^{1} \bigl\vert \mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s}) \bigr\vert \biggl\vert \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \qquad {}- \biggl(\frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , \upsilon (\epsilon )\bigr)\,d\epsilon } \biggr) \,\mathcal{d} \mathcal{s} \biggr\vert \\ & \quad \le \mathcal{L}(\mathcal{q-1}) \biggl[ \frac{\mathsf{\overline{M}}}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q-2}} \biggl[ \frac{1}{\varGamma (\mathcal{ \sigma +1})}+ \frac{1}{\varGamma ( \sigma )} \biggr] \biggl[\frac{1}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q}-1}\mathcal{ \Vert z-\upsilon \Vert } \\ & \quad =\Delta ^{\ast } \quad \forall \mathcal{t \in [0,1]}. \end{aligned} $$
(4.4)

Hence using (4.4) Eq. (3.1) is HU stable. As a result, the FDE (1.1) is HU stable. □

Example

Here, we present some examples to illustrate our results.

Example 5.1

Let us take the following FDE:

$$ { } \textstyle\begin{cases} \mathcal{ {}^{c}D}^{\mathcal{\zeta }}\phi ^{*}_{\mathcal{p}} [ \mathcal{{}^{c}D}^{\mathcal{ \sigma }} (\mathcal{z(t)}-\sum_{i=1}^{m} \lambda _{i}(t) ) ] =-\psi ^{\ast }\mathcal{(t, z(t))}, \quad \mathcal{t\in [0,1]}, \\ \phi ^{*}_{\mathcal{p}} [\mathcal{{}^{c}D}^{\mathcal{ \sigma }} \mathcal{z(t)}-\sum_{i=1}^{m}\lambda _{i}(t) ]|_{t=0} =0, \qquad \mathcal{z}(0)=\sum_{i=1}^{m}\lambda _{i}(0), \\ \mathcal{z}^{\prime }(\mathcal{1})=\sum_{i=1}^{m}\lambda _{i}^{\prime }(1), \qquad \mathcal{z}^{j}(\mathcal{0})=\sum_{i=1}^{m}\lambda _{i}^{j}(0) \quad \text{for } j=2,3,4,\ldots,n-1, \end{cases} $$
(5.1)

where \(\mathcal{\zeta }=0.5\), \(\mathcal{ \sigma }=3.6\), \(\mathcal{p}={5}\), \(\mathcal{q}=1.25\), \(m=3\). We have\(\sum_{i=1}^{m}\lambda _{i}(t)=\frac{1}{\mathcal{t}^{2}+100+i}\)\(\forall \mathcal{t}\in [0,1]\), \(\psi ^{\ast }\mathcal{(t},\mathcal{z(t)})=\frac{1}{{t^{2}}+20} [\mathcal{|z|^{5}}+ \frac{1}{ (1+7\mathcal{|z|^{\frac{5}{39}}} )} ]\).

Let us consider

$$\begin{aligned}& \begin{aligned}[b] \mathsf{\overline{M}}&=\phi ^{*}_{\max}( \mathcal{t},r)= \max_{\mathcal{t}\in [0,1]} \biggl\{ \frac{1}{{t^{2}}+20} \biggl[ \mathcal{ \vert z \vert ^{5}}+ \frac{1}{ (1+7\mathcal{ \vert z \vert ^{\frac{5}{39}}} )} \biggr]: \mathcal{t}^{\frac{13}{5}}r\le \mathcal{z}\le r \biggr\} \\ & \le \max_{\mathcal{t}\in [0,1]} \biggl( \frac{1}{{t^{2}}+20} \biggl[ \vert r \vert ^{5}+ \frac{1}{ (1+7 \vert \mathcal{t}^{\frac{1}{3}}r^{\frac{5}{39}} \vert } \biggr] \biggr) \\ &\le 0.1 \quad \forall \mathcal{t}\in [0,1] , r=b=1, \end{aligned} \\& \begin{aligned}[b] \phi ^{*}_{\min}(\mathcal{t},r)&= \min _{\mathcal{t}\in [0,1]} \biggl\{ \frac{1}{{t^{2}}+20} \biggl[\mathcal{ \vert z \vert ^{5}}+ \frac{1}{ (1+7\mathcal{ \vert z \vert ^{\frac{5}{39}}} )} \biggr]: \mathcal{t}^{\frac{13}{5}}r \le \mathcal{z}\le r \biggr\} \\ & \ge \min_{\mathcal{t}\in [0,1]} \biggl( \frac{1}{{t^{2}}+20} \biggl[ \bigl\vert \mathcal{t}^{13}r^{5} \bigr\vert + \frac{1}{ (1+7 \vert r^{\frac{5}{39}} \vert } \biggr] \biggr) \\ &\ge 0.0122503 \quad \forall \mathcal{t}\in [0,1] , r=a=\frac{1}{1000}. \end{aligned} \end{aligned}$$

Then

$$ \begin{aligned}[b] &\max_{\mathcal{t}\in [0,1]} \Biggl(\sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \Biggr) \\ &\quad \le 0.115663+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\phi ^{*}_{\max}(\mathcal{t},\mathcal{b}) \mathcal{d\epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ &\quad \le 0.115663+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\phi ^{*}_{\max}(\mathcal{t},\mathcal{1}) \mathcal{d\epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ & \quad \le 0.115663+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\phi ^{*}_{\max}(\mathcal{t},\mathcal{1}) \mathcal{d\epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ &\quad \le 0.115663+\mathsf{\overline{M}}^{\mathcal{q}-1} \biggl[ \frac{1}{\varGamma (\mathcal{ \sigma +1})}+ \frac{1}{\varGamma ( \sigma )} \biggr] \biggl[\frac{1}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q}-1} \\ &\quad \le 0.314901< 1. \end{aligned} $$
(5.2)

Also we have

$$\begin{aligned} &\min_{\mathcal{t}\in [0,1]} \Biggl(\sum_{i=1}^{m}\lambda _{i}(t)+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{t}, \mathcal{s})\phi ^{ \ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}-\epsilon )^{\zeta -1}\psi ^{\ast } \mathcal{\bigl(\epsilon , z(\epsilon )\bigr)\,d \epsilon } \biggr)\,\mathcal{d} \mathcal{s} \Biggr) \\ &\quad \ge 0.029414+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\phi ^{*}_{\min}(\mathcal{t},\mathcal{a}) \mathcal{d\epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ &\quad \ge 0.029414+ \int _{0}^{1}\mathcal{H}^{ \sigma }( \mathcal{1}, \mathcal{s})\phi ^{\ast }_{\mathcal{q}} \biggl( \frac{1}{\varGamma (\zeta )} \int _{0}^{ \mathcal{s}}( \mathcal{s}- \epsilon )^{\zeta -1}\phi ^{*}_{\min}\biggl(\mathcal{t}, \frac{1}{1000}\biggr) \,\mathcal{d\epsilon } \biggr)\,\mathcal{d} \mathcal{s} \\ &\quad \ge 0.029414>\frac{1}{1000}. \end{aligned}$$
(5.3)

Using Theorem 3.2, Eq. (5.1) has a solution \(\mathcal{z}^{*}\) which satisfies \(\frac{1}{1000}\le \|\mathcal{z}\|\le 1\).

Example 5.2

Let us take the following FDE:

$$ { } \textstyle\begin{cases} \mathcal{ {}^{c}D}^{\mathcal{\zeta }}\phi ^{*}_{\mathcal{p}} [ \mathcal{{}^{c}D}^{\mathcal{ \sigma }} (\mathcal{z(t)}-\sum_{i=1}^{m} \lambda _{i}(t) ) ] =-\psi ^{\ast }\mathcal{(t, z(t))}, \quad \mathcal{t\in [0,1]}, \\ \phi ^{*}_{\mathcal{p}} [\mathcal{{}^{c}D}^{\mathcal{ \sigma }} \mathcal{z(t)}-\sum_{i=1}^{m}\lambda _{i}(t) ]|_{t=0} =0, \qquad \mathcal{z}(0)=\sum_{i=1}^{m}\lambda _{i}(0), \\ \mathcal{z}^{\prime }(\mathcal{1})=\sum_{i=1}^{m}\lambda _{i}^{\prime }(1), \qquad \mathcal{z}^{j}(\mathcal{0})=\sum_{i=1}^{m}\lambda _{i}^{j}(0) \quad \text{for } j=2,3,4,\ldots,n-1, \end{cases} $$
(5.4)

where \(\mathcal{\zeta }=0.5\), \(\mathcal{ \sigma }=3.5\), \(\mathcal{p}={5}\), \(\mathcal{q}=1.25\), \(m=3\). We have \(\sum_{i=1}^{m}\lambda _{i}(t)=\frac{1}{t+100+i}\)\(\forall \mathcal{t}\in [0,1]\). \(\psi ^{\ast }\mathcal{(t},\mathcal{z(t)})=\frac{1}{{t^{2}}+20} [\frac{\mathcal{|z|}}{ (1+\mathcal{|z|} )} ]\), which satisfies the assumption \(\mathcal{(R_{4})}\) and where \(\mathcal{L}=\frac{1}{20}\), that is,

$$ \mathcal{\bigl|\psi ^{\ast }\bigl({t},z(\mathcal{t})\bigr)-\psi ^{\ast }( \mathcal{t},\upsilon ({t})}\bigr| \le \frac{1}{20}\mathcal{ \vert z-\upsilon \vert }. $$

Consider

$$\begin{aligned} \mathsf{\overline{M}}&=\phi ^{*}_{\max}(\mathcal{t},r)= \max _{\mathcal{t}\in [0,1]} \biggl\{ \frac{1}{{t^{2}}+20} \biggl[ \frac{\mathcal{ \vert z \vert }}{ (1+\mathcal{ \vert z \vert } )} \biggr]:\mathcal{t}^{ \frac{5}{2}}r\le \mathcal{z}\le r \biggr\} \\ & \le \max_{\mathcal{t}\in [0,1]} \biggl( \frac{1}{{t^{2}}+20} \biggl[ \frac{ \vert r \vert }{ (1+\mathcal{|t^{\frac{5}{2}}}r| )} \biggr] \biggr) \\ &\le 0.05 \quad \forall \mathcal{t}\in [0,1] , r=b=1. \end{aligned}$$

Then

$$ \Delta ^{\ast }=\mathcal{L}(\mathcal{q-1}) \biggl[ \frac{\mathsf{\overline{M}}}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q-2}} \biggl[ \frac{1}{\varGamma (\mathcal{ \sigma +1})}+ \frac{1}{\varGamma ( \sigma )} \biggr] \biggl[\frac{1}{\varGamma (\zeta +1)} \biggr]^{\mathcal{q}-1}\approx 0.04306\le 1. $$
(5.5)

Hence there exists a unique solution of Eq. (5.4) on \(\mathcal{[0,1]}\) by Theorem 3.2. We can easily check all the conditions of Theorem 4.1 are also satisfied. As a result, Eq. (5.4) is HU stable.

Conclusion

In this investigation, the existence results for general FDEs (1.1) involving a \(\phi ^{*}_{p}\)-Laplacian operator is established by using Guo–Krasnoselskii’s fixed point theorem [58]. The uniqueness results are proved by using the Banach contraction mapping principle and HU stability is also evaluated. The properties of the Green function also proved. The validity of our result is illustrated by examples. Also, one can study the multiple solutions, periodic solutions and controllability for the proposed general non-linear FDEs.

References

  1. 1.

    Podlubny, I.: Fractional Differential Equations, vol. 198. Academic Press, San Diego (1998). https://doi.org/10.2307/2653160

    Google Scholar 

  2. 2.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Amsterdam (2006)

    Google Scholar 

  3. 3.

    Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus, vol. 4. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-4020-6042-7

    Google Scholar 

  4. 4.

    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2010). https://doi.org/10.1142/p614

    Google Scholar 

  5. 5.

    Yang, X.J.: General Fractional Derivatives: Theory, Methods and Applications. CRC Press, New York (2019)

    Google Scholar 

  6. 6.

    Gómez-Aguilar, J.F., Atangana, A.: Fractional Hunter–Saxton equation involving partial operators with bi-order in Riemann–Liouville and Liouville–Caputo sense. Eur. Phys. J. Plus 132(2), 1–18 (2017). https://doi.org/10.1140/epjp/i2017-11371-6

    Article  Google Scholar 

  7. 7.

    Yang, X.J., Gao, F., Ju, Y., Zhou, H.E.: Fundamental solutions of the general fractional-order diffusion equations. Math. Methods Appl. Sci. 41, 9312–9320 (2018). https://doi.org/10.1002/mma.5341

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Atangana, A., Gómez-Aguilar, J.F.: Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 133, 1–23 (2018). https://doi.org/10.1140/epjp/i2018-12021-3

    Article  Google Scholar 

  9. 9.

    Atangana, A., Hammouch, Z.: Fractional calculus with power law: the cradle of our ancestors. Eur. Phys. J. Plus 134(9), 1–15 (2019). https://doi.org/10.1140/epjp/i2019-12777-8

    Article  Google Scholar 

  10. 10.

    Agarwal, R.P., O’Regan, D., Stanek, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371(1), 57–68 (2010). https://doi.org/10.1016/j.jmaa.2010.04.034

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Agarwal, R.P., O’Regan, D., Stanek, S.: Positive solutions for mixed problems of singular fractional differential equations. Math. Nachr. 285(1), 27–41 (2012). https://doi.org/10.1002/mana.201000043

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Baleanu, D., Agarwal, R.P., Mohammadi, H., Rezapour, S.: Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces. Bound. Value Probl. 2013, 112 (2013). http://www.boundaryvalueproblems.com/content/2013/1/112

    MathSciNet  Article  Google Scholar 

  13. 13.

    Vong, S.: Positive solutions of singular fractional differential equations with integral boundary conditions. Math. Comput. Model. 57(5–6), 1053–1059 (2013). https://doi.org/10.1016/j.mcm.2012.06.024

    MathSciNet  Article  Google Scholar 

  14. 14.

    Abdeljawad, T., Alzabut, J.: On Riemann–Liouville fractional q-difference equations and their application to retarded logistic type model. Math. Methods Appl. Sci. 41(18), 8953–8962 (2018). https://doi.org/10.1002/mma.4743

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Liu, J.G., Yang, X.J., Feng, Y.Y.: On integrability of the time fractional nonlinear heat conduction equation. J. Geom. Phys. 144, 190–198 (2019). https://doi.org/10.1016/j.geomphys.2019.06.004

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Liu, J.G., Yang, X.J., Feng, Y.Y.: Analytical solutions of some integral fractional differential-difference equations. Mod. Phys. Lett. B 34(1), 02050009 (2020). https://doi.org/10.1142/S0217984920500098

    MathSciNet  Article  Google Scholar 

  17. 17.

    Abdeljawad, T., Baleanu, D., Jarad, F.: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives. J. Math. Phys. 49, 083507 (2008). https://doi.org/10.1063/1.2970709

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Yang, X.J., Feng, Y.Y., Cattani, C., Inc, M.: Fundamental solutions of anomalous diffusion equations with the decay exponential kernel. Math. Methods Appl. Sci. 42, 4054–4060 (2019). https://doi.org/10.1002/mma.5634

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Saad, K.M., Gómez-Aguilar, J.F.: Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel. Phys. A, Stat. Mech. Appl. 509, 703–716 (2018). https://doi.org/10.1016/j.physa.2018.05.137

    MathSciNet  Article  Google Scholar 

  20. 20.

    Yang, X.J., Gao, F.: A new technology for solving diffusion and heat equations. Therm. Sci. 21(1A), 133–140 (2017)

    Article  Google Scholar 

  21. 21.

    Maraaba, T.A., Jarad, F., Baleanu, D.: On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives. Sci. China Ser. A, Math. 51, 1775–1786 (2008). https://doi.org/10.1007/s11425-008-0068-1

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Atangana, A., Gómez-Aguilar, J.F.: Numerical approximation of Riemann–Liouville definition of fractional derivative: from Riemann–Liouville to Atangana–Baleanu. Numer. Methods Partial Differ. Equ. 34(5), 1–22 (2017). https://doi.org/10.1002/num.22195

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Morales-Delgado, V.F., Gómez-Aguilar, J.F., Taneco-Hernandez, M.A.: Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation. Rev. Mex. Fis. 65(1), 82–88 (2018)

    MathSciNet  Google Scholar 

  24. 24.

    Atangana, A., Gómez-Aguilar, J.F.: Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 114, 516–535 (2018). https://doi.org/10.1016/j.chaos.2018.07.033

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H., Reyes, J.M., Sosa, I.O.: Series solution for the time-fractional coupled mkdv equation using the homotopy analysis method. Math. Probl. Eng. 2016, Article ID 7047126 (2016). https://doi.org/10.1155/2016/7047126

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Saad, K.M., Khader, M.M., Gómez-Aguilar, J.F., Baleanu, D.: Numerical solutions of the fractional Fisher’s type equations with Atangana–Baleanu fractional derivative by using spectral collocation methods. Chaos, Interdiscip. J. Nonlinear Sci. 29(2), 1–13 (2019). https://doi.org/10.1063/1.5086771

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Yang, X.J.: New general fractional-order rheological models with kernels of Mittag-Leffler functions. Rom. Rep. Phys. 69(4), 1–15 (2017)

    Google Scholar 

  28. 28.

    Yang, X.J., Machado, J.A., Nieto, J.J.: A new family of the local fractional PDEs. Fundam. Inform. 151, 63–75 (2017). https://doi.org/10.3233/FI-2017-1479

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Bhatter, S., Mathur, A., Kumar, D., Singh, J.: A new analysis of fractional Drinfeld–Sokolov–Wilson model with exponential memory. Phys. A, Stat. Mech. Appl. 537, 122578 (2020). https://doi.org/10.1016/j.physa.2019.122578

    MathSciNet  Article  Google Scholar 

  30. 30.

    Ravichandran, C., Logeswari, K., Jarad, F.: New results on existence in the framework of Atangana–Baleanu derivative for fractional integro-differential equations. Chaos Solitons Fractals 125, 194–200 (2019). https://doi.org/10.1016/j.chaos.2019.05.014

    MathSciNet  Article  Google Scholar 

  31. 31.

    Panda, S.K., Abdeljawad, T., Ravichandran, C.: Novel fixed point approach to Atangana–Baleanu fractional and \(L^{p}\)-Fredholm integral equations. Alex. Eng. J. 1–12 (2020). https://doi.org/10.1016/j.aej.2019.12.027

    Article  Google Scholar 

  32. 32.

    Yokus, A., Gülbahar, S.: Numerical solutions with linearization techniques of the fractional Harry Dym equation. Appl. Math. Nonlinear Sci. 4(1), 35–42 (2019). https://doi.org/10.2478/AMNS.2019.1.00004

    MathSciNet  Article  Google Scholar 

  33. 33.

    Kumar, D., Singh, J., Baleanu, D.: A new numerical algorithm for fractional Fitzhugh–Nagumo equation arising in transmission of nerve impulses. Nonlinear Dyn. 91, 307–317 (2018). https://doi.org/10.1007/s11071-017-3870-x

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Cattani, C., Ciancio, A.: Existence theorem for hybrid competition model. In: BSG Proceedings 18. The Int. Conf. of Diff. Geom. and Dynamical Systems, vol. 18, pp. 32–38 (2011)

    Google Scholar 

  35. 35.

    Uçar, S., Uçar, E., Özdemir, N., Hammouch, Z.: Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative. Chaos Solitons Fractals 118, 300–306 (2019). https://doi.org/10.1016/j.chaos.2018.12.003

    MathSciNet  Article  Google Scholar 

  36. 36.

    Jafari, H., Baleanu, D., Khan, H., Khan, R.A., Khan, A.: Existence criterion for the solutions of fractional order p-Laplacian boundary value problems. Bound. Value Probl. 2015(1), 164 (2015). https://doi.org/10.1186/s13661-015-0425-2

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Bai, C.: Existence and uniqueness of solutions for fractional boundary value problems with p-Laplacian operator. Adv. Differ. Equ. 2018(4), 1 (2018). https://doi.org/10.1186/s13662-017-1460-3

    MathSciNet  Article  Google Scholar 

  38. 38.

    Yan, R., Sun, S., Lu, H., Zhao, Y.: Existence of solutions for fractional differential equations with integral boundary conditions. Adv. Differ. Equ. 2014(1), 1 (2014). https://doi.org/10.1186/1687-1847-2014-25

    MathSciNet  Article  Google Scholar 

  39. 39.

    Li, Y.: Existence of positive solutions for fractional differential equation involving integral boundary conditions with p-Laplacian operator. Adv. Differ. Equ. 2017, 135 (2017) 1–11. https://doi.org/10.1186/s13662-017-1172-8

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Chen, T., Liu, W., Hu, Z.: A boundary value problem for fractional differential equation with p-Laplacian operator at resonance. Nonlinear Anal. 75, 3210–3217 (2012). https://doi.org/10.1016/j.na.2011.12.020

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Tan, J.J., Li, M.: Solutions of fractional differential equations with p-Laplacian operator in Banach spaces. Bound. Value Probl. 2018(15), 1 (2018). https://doi.org/10.1186/s13661-018-0930-1

    MathSciNet  Article  Google Scholar 

  42. 42.

    Lu, H., Han, Z., Sun, S., Liu, J.: Existence on positive solutions for boundary value problems of nonlinear fractional differential equations with p-Laplacian. Adv. Differ. Equ. 2013, 30 (2013) 1–16. https://doi.org/10.1186/1687-1847-2013-30

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Khan, H., Jarad, F., Abdeljawad, T., Khan, A.: A singular ABC-fractional differential equation with p-Laplacian operator. Chaos Solitons Fractals 129, 56–61 (2019). https://doi.org/10.1016/j.chaos.2019.08.017

    MathSciNet  Article  Google Scholar 

  44. 44.

    Khan, H., Chen, W., Sun, H.: Analysis of positive solution and Hyers–Ulam stability for a class of singular fractional differential equations with p-Laplacian in Banach space. Math. Methods Appl. Sci. 41, 3430–3440 (2018). https://doi.org/10.1002/mma.4835

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Khan, H., Li, Y., Suna, H., Khan, A.: Existence of solution and Hyers–Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator. Bound. Value Probl. 2017, 157 (2017) 1–16. https://doi.org/10.1186/s13661-017-0878-6

    Article  MATH  Google Scholar 

  46. 46.

    Wang, J., Li, X.: Ulam–Hyers stability of fractional Langevin equations. Appl. Math. Comput. 258, 72–83 (2015). https://doi.org/10.1016/j.amc.2015.01.111

    MathSciNet  Article  MATH  Google Scholar 

  47. 47.

    Khan, A., Gómez-Aguilar, J.F., Khan, T.S., Khan, H.: Stability analysis and numerical solutions of fractional order HIV/AIDS model. Chaos Solitons Fractals 122, 119–128 (2019). https://doi.org/10.1016/j.chaos.2019.03.022

    MathSciNet  Article  Google Scholar 

  48. 48.

    Khan, H., Tunc, C., Khan, A.: Stability results and existence theorems for nonlinear delay-fractional differential equations with \(\phi ^{*}_{p}\)-operator. J. Appl. Anal. Comput. 10, 584–597 (2020). https://doi.org/10.11948/20180322

    Article  Google Scholar 

  49. 49.

    Khan, H., Tunc, C., Khan, A.: Green function’s properties and existence theorem for nonlinear delay-fractional differential equations. Discrete Contin. Dyn. Syst., Ser. S 13, 1–13 (2020). https://doi.org/10.3934/dcdss.2020139

    MathSciNet  Article  Google Scholar 

  50. 50.

    Rizwan, R., Zada, A., Wang, X.: Stability analysis of nonlinear implicit fractional Langevin equation with non instantaneous impulses. Adv. Differ. Equ. 2019, 85 (2019) 1–31. https://doi.org/10.1186/s13662-019-1955-1

    Article  MATH  Google Scholar 

  51. 51.

    Khan, H., Gómez-Aguilar, J.F., Khan, A., Khan, T.S.: Stability analysis for fractional order advection–reaction diffusion system. Phys. A, Stat. Mech. Appl. 521, 737–751 (2019). https://doi.org/10.1016/j.physa.2019.01.102

    MathSciNet  Article  Google Scholar 

  52. 52.

    Zada, A., Ali, W., Farina, S.: Hyers–Ulam stability of non linear differential equations with fractional integrable impulses. Math. Methods Appl. Sci. 40(15), 5502–5514 (2017). https://doi.org/10.1002/mma.4405

    MathSciNet  Article  MATH  Google Scholar 

  53. 53.

    Khan, A., Syam, M.I., Zada, A., Khan, H.: Stability analysis of nonlinear fractional differential equations with Caputo and Riemann–Liouville derivatives. Eur. Phys. J. Plus 264, 1–9 (2018). https://doi.org/10.1140/epjp/i2018-12119-6

    Article  Google Scholar 

  54. 54.

    Khan, H., Gómez-Aguilar, J.F., Khan, A., Khan, T.S.: Stability analysis for fractional order advection-reaction diffusion system. Phys. A, Stat. Mech. Appl. 521, 737–751 (2019). https://doi.org/10.1016/j.physa.2019.01.102

    MathSciNet  Article  Google Scholar 

  55. 55.

    Khan, H., Abdeljawad, T., Aslam, M., Khan, R.A., Khan, A.: Existence of positive solution and Hyers Ulam stability for a nonlinear singular-delay-fractional differential equation. Adv. Differ. Equ. 2019, 104 (2019) 1–13. https://doi.org/10.1186/s13662-019-2054-z

    MathSciNet  Article  MATH  Google Scholar 

  56. 56.

    Khan, A., Khan, H., Gómez-Aguilar, J.F., Abdeljawad, T.: Existence and Hyers Ulam stability for a nonlinear singular fractional differential equation with Mittag-Leffler kernel. Chaos Solitons Fractals 127, 422–427 (2019). https://doi.org/10.1016/j.chaos.2019.07.026

    MathSciNet  Article  Google Scholar 

  57. 57.

    Khan, H., Khan, A., Abdeljawad, T., Alkhazzan, A.: Existence results in Banach space for a non linear impulsive system. Adv. Differ. Equ. 2019(18), 1 (2019). https://doi.org/10.1186/s13662-019-1965-z

    Article  MATH  Google Scholar 

  58. 58.

    Krasnoselsky, M.A.: Two remarks on the method of successive approximation. Usp. Mat. Nauk 10, 123–127 (1955) http://mi.mathnet.ru/eng/umn7954

    MathSciNet  Google Scholar 

  59. 59.

    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones, vol. 5. Academic Press, San Diego (2014)

    Google Scholar 

  60. 60.

    Hyers, D.H.: On the stability of the linear functional equations. Proc. Natl. Acad. Sci. USA 27(4), 222–224 (1941). https://doi.org/10.1073/pnas.27.4.222

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The first author acknowledges with gratitude the Council of Scientific and Industrial Research (CSIR)-New Delhi, India, for supporting this research work under grant no. 09/1051(0031)/2019-EMR-1 and the Department of Mathematics and Statistics, Central University of Punjab, Bathinda, India.

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Devi, A., Kumar, A., Baleanu, D. et al. On stability analysis and existence of positive solutions for a general non-linear fractional differential equations. Adv Differ Equ 2020, 300 (2020). https://doi.org/10.1186/s13662-020-02729-3

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MSC

  • 26A33
  • 34BB2
  • 45ND5

Keywords

  • Hyers–Ulam stability
  • p-Laplacian operator
  • Caputo fractional derivative
  • Guo–Krasnoselskii’s fixed point theorem
  • EU of positive solutions