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Theory and Modern Applications

Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators

Abstract

In this article, we are concerned with the existence of mild solutions and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators and nonlocal conditions. The existence results are obtained by first defining Green’s function and approximate controllability by specifying a suitable control function. These results are established with the help of Schauder’s fixed point theorem and theory of almost sectorial operators in a Banach space. An example is also presented for the demonstration of obtained results.

1 Introduction

In current times, the rising interest of researchers in fractional calculus reflects the popularity of this branch [13, 513, 15, 19, 3032]. Differential equations of fractional order are widely applicable in the areas of physics, chemistry, electromagnetics, and mechanics. Fractional differential equations (FDEs) have been used widely in the identification of different physical systems, in control theory, in simulating viscoelastic materials, and in the modeling of different complex phenomena [2229, 39, 42]. The concept of exact controllability and approximate controllability of FDEs is an active field of investigation because of its major applications in physical sciences. Under some admissible control inputs, exact controllability steers the system to arbitrary final state, while approximate controllability steers the system to the small neighborhood of arbitrary final state. In the published works, there are numerous articles focussing on the exact or approximate controllability of systems represented by FDEs, neutral FDEs, FDEs with impulsive inclusions, and FDEs with delay functions [4, 18, 40, 44].

In particular, approximate controllability of Hilfer FDEs under different conditions has been discussed widely. Mahmudov et al. [36] investigated the exact controllability of Hilfer FDEs in a Hilbert space under the assumption that a linear system of the given equation is approximate controllable. In 2017, Yang et al. [41] discussed the approximate controllability of Hilfer FDEs with nonlocal conditions in a Banach space with the help of semigroup theory, fixed point techniques, and multivalued analysis. Later on, Debbouche et al. [14] and Du et al. [17] studied the approximate controllability of Hilfer FDEs and semilinear Hilfer FDEs with impulsive control inclusions in Banach spaces, respectively. In 2018, Lv and Yang [34] investigated the approximate controllability of neutral Hilfer FDEs by applying the techniques of stochastic analysis theory and semigroup operator theory in a Hilbert space. In recent works, Lv and Yang [35], with the help of Banach contraction principle, discussed the approximate controllability for a class of Hilfer FDEs of order \(1< \alpha < 2\) and type \(0\le \beta \le 1\). It is noted that, in almost all the problems discussed above, a linear operator generates the strongly continuous semigroup of bounded linear operators.

In [37] Periago and Straub formulated theory to analyze almost sectorial operators. Here authors also mentioned the suitable assumptions required to establish the existence of mild solutions and classical solutions of FDEs with almost sectorial operator. There are numerous works focussing on the existence of mild solutions and analytical solutions of fractional evolution equations with almost sectorial operators [16, 33, 43]. Recently, Jaiswal et al. [21] proved the existence of mild solutions of Hilfer FDEs with almost sectorial operators. We found that in the available literature, approximate controllability of Hilfer FDEs with almost sectorial operators has not been discussed yet. Thus, motivated by the above-discussed works, we consider here the following system of Hilfer FDEs for investigating proposed results:

(1.1)

where \({}^{H}{\mathcal{D}}^{\nu , \mu }_{0^{+}}\) is the Hilfer fractional derivative of order \(0<\nu <1\) and type \(0\le \mu \le 1\). \({\mathcal{A}}:D({\mathcal{A}})\subset {\mathcal{V}}\longrightarrow { \mathcal{V}}\) is an almost sectorial operator in Banach space \({\mathcal{V}}\) and , . The control function takes value in \(\mathcal{L}^{2}(\mathfrak{J}, \mathcal{W})\). \(\mathcal{L}^{2}(\mathfrak{J}, \mathcal{W} )\) denotes the space of admissible control functions for Banach space \(\mathcal{W}\). ϒ is a bounded linear operator from \(\mathcal{W}\) into \(\mathcal{V}\). \(\varPsi :\mathfrak{J}\times \mathcal{V} \times \mathcal{V} \longrightarrow \mathcal{V} \) is a continuous linear mapping, , \(\mathfrak{m}\in \mathbb{N}\), and are real numbers such that . The characteristics functions \(\mathfrak{q}(\mathfrak{t},\mathfrak{s}) : \varDelta \longrightarrow \mathbb{R}\) and \(\varphi : \varDelta \times \mathcal{V} \longrightarrow \mathcal{V}\), are specified in the next section.

2 Preliminaries

Let us consider \(\mathcal{V}\) and \(\mathcal{W} \) as real Banach spaces with respective norms \(\lVert \cdot \rVert _{\mathcal{V}}\) and \(\lVert \cdot \rVert _{\mathcal{W}}\). \(\mathcal{C}(\mathfrak{J}, \mathcal{V})\) denotes the Banach space formed by all continuous functions from \(\mathfrak{J}\) into \(\mathcal{V}\) with corresponding norm function . \(\mathcal{L}^{2}(\mathfrak{J}, \mathcal{W} )\) denotes the Banach space of all \(\mathcal{W}\)-valued Bochner square integrable functions defined on \(\mathfrak{J}\) w.r.t. norm function

Definition 2.1

([20])

Hilfer fractional derivative of a continuously differentiable function f of order \(0 < \nu <1\) and \(0\le \mu \le 1\) is defined as

$$ {\mathcal{D}}^{\nu , \mu }_{0^{+}}{ f}({\mathfrak{t}})= { \mathcal{I}}^{ \nu (1-\mu )}_{0^{+}}\frac{d}{d\mathfrak{t}}{ \mathcal{I}}^{(1-\nu ) (1- \mu )}_{\mathcal{0}^{+}}{ f}(\mathfrak{t})={ \mathcal{I}}^{\nu (1-\mu )}_{0^{+}}{ \mathcal{D}}^{\nu + \mu -\nu \mu }_{0^{+}}{f}( \mathfrak{t}), $$

where \({\mathcal{I}}^{\nu (1-\mu )}_{0^{+}}\) is a Riemann–Liouville fractional integral and \({\mathcal{D}}^{\nu +\mu -\nu \mu }_{0^{+}}\) is a Riemann–Liouville fractional derivative.

Definition 2.2

([38])

For \(\alpha >0\), the Riemann–Liouville fractional integral of a continuously differentiable f of order α is defined as follows:

$$ {\mathcal{I}}^{\alpha }_{0^{+}} {f}({\mathfrak{t}})= \frac{1}{\Gamma (\alpha )} \int _{0}^{\mathfrak{t}} (\mathfrak{t}- \mathfrak{s})^{\alpha -1}f(\mathfrak{s})\,d\mathfrak{s}. $$

Almost sectorial operator

([21, 37])

Let \(0< \beta < \pi \) and \(-1<\gamma <0\). We define \(S^{o}_{\beta } = \lbrace \omega \in \mathbb{C}\setminus \lbrace 0 \rbrace \colon \lvert \operatorname{arg} \omega \rvert < \beta \rbrace\) and \(S_{\beta } =\bar{S^{o}_{\beta }}= \lbrace \omega \in \mathbb{C} \setminus \lbrace 0 \rbrace \colon \lvert \operatorname{arg} \omega \rvert \le \beta \rbrace \cup \lbrace 0 \rbrace \). A closed linear operator \({\mathcal{A}}:D({\mathcal{A}})\subset {\mathcal{V}}\longrightarrow { \mathcal{V}}\) is called an almost sectorial operator if the following hold:

  1. 1.

    \(\sigma (\mathcal{A})\) is contained in \(S_{\omega }\).

  2. 2.

    \(\forall \beta \in (\omega , \pi )\) there exists a constant \(\mathcal{M}_{\beta }>0\) such that \(\lVert \mathcal{R}(\mathfrak{z}, \mathcal{A} ) \rVert _{L( \mathcal{V})}\le {\mathcal{M}}_{\beta } |\mathfrak{z}|^{\gamma }\),

where \(\mathcal{R}(\mathfrak{z}, \mathcal{A})= (\mathfrak{z} I- \mathcal{A} )^{-1}\) is the resolvent operator of \(\mathcal{A} \) for \(\mathfrak{z} \in \rho (\mathcal{A}) \).

Let us define \(\Theta ^{\gamma }_{\omega }\) as a family of almost sectorial operators.

Proposition 2.1

([21])

Let be the compact semigroup defined in [37] and \(\mathcal{A} \in \Theta ^{\gamma }_{\omega }\) for \(-1<\gamma <0\) and \(0<\omega <\frac{\pi }{2}\). Then following holds for :

(i) is analytic and , \(\mathfrak{t} \in S^{o}_{ \frac{\pi }{2}-\omega } \). (ii) \(\forall \mathfrak{t}\), \(\mathfrak{s}\in S^{o}_{ \frac{\pi }{2}-\omega } \). (iii) , \(\mathfrak{t}>0\); where \(c_{0}>0 \) is a constant. (iv) Let , then if \(\theta > 1+\gamma \).

Let us define the operator families and as follows:

where \(\mathcal{M}_{\nu }(\theta )\) is a Wright-type function defined as follows:

$$ \mathcal{M}_{\nu }(\theta ) = \sum_{n \in \mathbb{N}} \frac{(-\theta )^{n-1}}{\Gamma (1-\nu n)(n-1)!},\quad \theta \in \mathbb{C}. $$

Proposition 2.2

([21])

(i) For each fixed \(\mathfrak{t} \in S^{o}_{\frac{\pi }{2}-\omega }\), and are bounded linear operators in \(\mathcal{V}\). Also

where and are constants depending only on ν and γ. (ii) and are continuous in the uniform operator topology for \(\mathfrak{t}>0 \).

Lemma 2.1

([2, 20])

The mild solution for the system of equations Eq. (1.1) is defined as follows:

(2.1)

where and .

Proposition 2.3

([21])

(i) For every fixed \(\mathfrak{t}\in S^{o}_{\frac{\pi }{2}-\omega }\), and are bounded linear operators on \(\mathcal{V}\). For \(\mathfrak{t}>0\),

(ii) and are strongly continuous operators.

Let , . Assume that .

We have

By the operator spectrum theorem, exists bounded and \(D( \mathcal{O})= \mathcal{V}\).

By the Neumann series expression, \(\mathcal{O}\) can be expressed as .

Therefore

By Lemma 2.1 mild solutions of Eq. (1.1) are given by

At ,

Thus the mild solutions of Eq. (1.1) are defined as follows:

Now we introduce the Green’s function \(\mathcal{G}(\mathfrak{t}, \mathfrak{s})\) as follows:

where

The mild solutions of Eq. (1.1) in terms of the Green’s function are expressed as

(2.2)

In addition to the above-mentioned propositions and lemma, we assume here the following assumptions to establish the proposed results:

  1. (A1)

    For each \(\mathfrak{t} \in \mathfrak{J}^{\prime }\), \(\varPsi :\mathfrak{J}^{\prime } \times \mathcal{V} \times \mathcal{V} \longrightarrow \mathcal{V}\) is a Caratheodory function.

  2. (A2)

    There exist \(\psi _{1}\in L^{1}(\mathfrak{J}, \mathbb{R}^{+})\) and a continuous function \(\psi _{2}\) such that, for and \(\mathfrak{t} \in \mathfrak{J}\), .

  3. (A3)

    \(\varphi (\mathfrak{t},\mathfrak{s}, \cdot): \mathcal{V} \longrightarrow \mathcal{V}\) is a Caratheodory function and there exists with

  4. (A4)

    is bounded on with .

  5. (A5)

    There exists \(\kappa \in L^{1}(\mathfrak{J}, \mathbb{R}^{+})\) such that , \(\forall \mathfrak{t} \in \mathfrak{J}\) and .

3 Existence result

For and , we introduce

with the norm defined as .

Let . Then iff and .

We define

Define an operator \(\mathcal{P} : \mathcal{B}_{\delta }(\mathfrak{J}) \longrightarrow \mathcal{B}_{\delta }(\mathfrak{J}) \) such that

where is defined as

Let us denote , , \(\hat{\mathfrak{m}}= \sup_{\mathfrak{t}\in \mathfrak{J}^{\prime } } \lbrace \kappa (\mathfrak{t}) \rbrace \), and .

Lemma 3.1

Let \(\mathcal{A} \in \Theta ^{\gamma }_{\omega }\) for \(-1<\gamma <0\) and \(0<\omega <\frac{\pi }{2}\). Assume that assumptions (A1)(A5) and Proposition 2.3hold. Then the operator is bounded and continuous if .

Proof

For , we have

If possible, suppose .

This implies

Applying \(\lim_{\delta \longrightarrow \infty }\) on both sides, we have

which is a contradiction.

So our supposition is wrong. Therefore . This implies the boundedness of operator \(\mathcal{P}\).

Now, to show that \(\mathcal{P}: \mathcal{B}_{\delta }(\mathfrak{J}) \longrightarrow \mathcal{B}_{\delta }(\mathfrak{J})\) is a continuous operator, let and with , \(n \in \mathbb{N}\).

By the continuity of function Ψ w.r.t. to the second and third variable

For \(\mathfrak{t} \in \mathfrak{J}\),

Since

by the Lebesgue dominated convergence theorem and the continuity of function Ψ,

This completes the proof of Lemma 3.1. □

Lemma 3.2

Let \(\mathcal{A} \in \Theta ^{\gamma }_{\omega }\) for \(-1<\gamma <0\) and \(0<\omega <\frac{\pi }{2}\). If assumptions (A1)(A5) and Proposition 2.3are satisfied, the operator is equicontinuous for \(\mathfrak{t} \in \mathfrak{J}\).

Proof

For and ,

Here,

The strong continuity of operator yields as \(\mathfrak{t}_{1}\longrightarrow \mathfrak{t}_{2}\).

Following Lemma 3.1 and assumptions (A1)–(A5), it is easy to see that

exists and is bounded. Thus as \(\mathfrak{t}_{2} \longrightarrow \mathfrak{t}_{1}\).

After following the given assumptions and performing some steps of calculation, we have

Clearly, as \(\mathfrak{t}_{2} \longrightarrow \mathfrak{t}_{1} \).

So, we have

This proves the equicontinuity of operator . □

Theorem 3.1

Let \(-1<\gamma <0\), \(0<\omega <\frac{\pi }{2}\), and . The system of equations Eq. (1.1) has at least one mild solution in if assumptions (A1)(A5) hold along with Lemma 3.1and Lemma 3.2.

Proof

The mild solution of Eq. (1.1) is equivalent to the fixed point of operator \(\mathcal{P} : \mathcal{B}_{\delta }(\mathfrak{J}) \longrightarrow \mathcal{B}_{\delta }(\mathfrak{J}) \). Here we prove that the operator \(\mathcal{P}\) has at least one fixed point.

In the following, we show that the operator \(\mathcal{P}\) is relatively compact in \(\mathcal{V}\) for every \(\mathfrak{t} \in \mathfrak{J}\).

We prove this by showing that there is a relatively compact set arbitrarily close to the set in \(\mathcal{V}\) for .

Define the operator by

Since and are compact for \(\mathfrak{t}>0\), the set is compact for \(\mathfrak{t} \in \mathfrak{J}^{\prime }\). In the following

Clearly,

We have shown that the set is arbitrarily close to the relatively compact set . This implies is relatively compact in \(\mathcal{V}\). Also, by Lemma 3.1 and Lemma 3.2, the operator \(\mathcal{P} : \mathcal{B}_{\delta }(\mathfrak{J}^{\prime }) \longrightarrow \mathcal{B}_{\delta }(\mathfrak{J}^{\prime }) \) is bounded, continuous, and equicontinuous in \(\mathcal{V}\). So by the Arzela–Ascoli theorem, is a compact operator. Hence, by Schauder’s fixed point theorem, \(\mathcal{P}\) has at least one fixed point .

Let . Then is a mild solution of Eq. (1.1). This completes the proof. □

4 Approximate controllability

In this section, we discuss the approximate controllability of Eq. (1.1).

The system of equations Eq. (1.1) is said to be approximate controllable on if, for every desired final state and \(\epsilon >0\), there exists a control function such that the mild solution of Eq. (1.1) satisfies

Following this, we first introduce the following two operators:

(4.1)

where \(\varUpsilon ^{*}\), \(\mathcal{O}^{*}\), , and characterize the adjoint operators of ϒ, \(\mathcal{O}\), , and respectively,

Theorem 4.1

Let \(\mathcal{A} \in \Theta ^{\gamma }_{\omega }\) for \(-1<\gamma <0\) and \(0<\omega <\frac{\pi }{2}\). Assume that assumptions (A1)(A5) and Proposition 2.3hold. The system of equations Eq. (1.1) is approximate controllable on \(\mathfrak{J}\) if as \(\lambda \longrightarrow 0^{+}\) in the strong operator topology.

Proof

By Theorem 3.1, the system of equations Eq. (1.1) has at least one mild solution given by

(4.2)

where we define the control function as

(4.3)

with

(4.4)

At ,

(4.5)

which implies that the sequence is bounded in the Hilbert space . Therefore, there exists a subsequence of the sequence converging weakly to some point .

Let us write

(4.6)

Hence, by Eqs. (4.4) and (4.6), we have

(4.7)

By the compactness of operators and for \(\mathfrak{t}>0\), one gets the compactness of Green’s function \(\mathcal{G}(\mathfrak{t}, \mathfrak{s})\) for \(\mathfrak{t}, \mathfrak{s}>0\), which implies that

(4.8)

Thus from Eq. (4.7) and Eq. (4.8) we get

(4.9)

Equation (4.5) implies that

Hence the approximate controllability of Eq. (1.1). □

5 Applications

Here we investigate the proposed results for the following Cauchy problem:

(5.1)

in the Banach space , \(0<\eta <1\), where \(\nu = \frac{1}{4}\), \(\mu = \frac{1}{2}\), \(\wp >0\) and , , are such that . On substituting , Eq. (5.1) reduces to Eq. (1.1) with and . It follows from article [37] that there exist constants \(\rho , \epsilon >0\) such that \(\mathcal{A}+ \rho \in \Theta ^{\frac{\eta }{2}-1}_{\frac{\pi }{2}- \epsilon }(\mathcal{V})\). The compactness of semigroup \(\lbrace \mathfrak{T}(\mathfrak{t}) \rbrace \) follows from (Lemma 4.66) [42]. Since and is embedded in , the compactness of resolvent operators follows for every \(\eta >0\).

The bounded linear operator \(\varUpsilon :\mathcal{W}= \mathcal{V}\longrightarrow \mathcal{V}\) is defined as .

Following this discussion and the definition of function Ψ and bounded operator ϒ, it is easy to verify that assumptions (A1)–(A5) hold with

Hence the existence and approximate controllability of Eq. (5.1) follow from Theorem 3.1 and Theorem 4.1 respectively.

6 Conclusion

In this paper, we discussed the approximate controllability of Hilfer fractional differential equations with almost sectorial operators. We first prove the existence of mild solutions for similar equations by applying fixed point theory. We will try to investigate the exact controllability and stability of a similar problem in our future research work.

Availability of data and materials

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References

  1. Alkhazzan, A., Jiang, P., Baleanu, D., Khan, H., Khan, A.: Stability and existence results for a class of nonlinear fractional differential equations with singularity. Math. Methods Appl. Sci. 41(18), 9321–9334 (2018). https://doi.org/10.1002/mma.5263

    Article  MathSciNet  MATH  Google Scholar 

  2. Bedi, P., Kumar, A., Abdeljawad, T., Khan, A.: Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations. Adv. Differ. Equ. 2020, 155 (2020). https://doi.org/10.1186/s13662-020-02615-y

    Article  MathSciNet  Google Scholar 

  3. Bedi, P., Kumar, A., Abdeljawad, T., Khan, A.: S-asymptotically ω-periodic mild solutions and stability analysis of Hilfer fractional evolution equations. Evol. Equ. Control Theory (2020). https://doi.org/10.3934/eect.2020089

    Article  Google Scholar 

  4. Chang, Y.K., Pereira, A., Ponce, R.: Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators. Fract. Calc. Appl. Anal. 20(4), 963–987 (2017)

    Article  MathSciNet  Google Scholar 

  5. Chen, P., Zhang, X.: Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations. Evol. Equ. Control Theory (2019). https://doi.org/10.3934/eect.2020076

    Article  Google Scholar 

  6. Chen, P., Zhang, X., Li, Y.: Approximation technique for fractional evolution equations with nonlocal integral conditions. Mediterr. J. Math. 14(6), 226 (2017). https://doi.org/10.1007/s00009-017-1029-0

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, P., Zhang, X., Li, Y.: A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Commun. Pure Appl. Anal. 17(5), 1975–1992 (2018)

    Article  MathSciNet  Google Scholar 

  8. Chen, P., Zhang, X., Li, Y.: Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families. J. Fixed Point Theory Appl. 21(3), 84 (2019). https://doi.org/10.1007/s11784-019-0719-6

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, P., Zhang, X., Li, Y.: Non-autonomous evolution equations of parabolic type with non-instantaneous impulses. Mediterr. J. Math. 16(5), 118 (2019). https://doi.org/10.1007/s00009-019-1384-0

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, P., Zhang, X., Li, Y.: Fractional non-autonomous evolution equation with nonlocal conditions. J. Pseudo-Differ. Oper. Appl. 10(4), 955–973 (2019). https://doi.org/10.1007/s11868-018-0257-9

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, P., Zhang, X., Li, Y.: Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators. Fract. Calc. Appl. Anal. 23(1), 268–291 (2020). https://doi.org/10.1515/fca-2020-0011

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, P., Zhang, X., Li, Y.: Approximate controllability of non-autonomous evolution system with nonlocal conditions. J. Dyn. Control Syst. 26(1), 1–16 (2020). https://doi.org/10.1007/s10883-018-9423-x

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, P., Zhang, X., Li, Y.: Cauchy problem for fractional non-autonomous evolution equations. Banach J. Math. Anal. 14(2), 559–584 (2020). https://doi.org/10.1007/s43037-019-00008-2

    Article  MathSciNet  MATH  Google Scholar 

  14. Debbouche, A., Antonov, V.: Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces. Chaos Solitons Fractals 102, 140–148 (2017). https://doi.org/10.1016/j.chaos.2017.03.023

    Article  MathSciNet  MATH  Google Scholar 

  15. Devi, A., Kumar, A., Abdeljawad, T., Khan, A.: Existence and stability analysis of solutions for fractional Langevin equation with nonlocal integral and anti-periodic type boundary conditions. Fractals (2020). https://doi.org/10.1142/S0218348X2040006X

    Article  Google Scholar 

  16. Ding, X.L., Ahmad, B.: Analytical solutions to fractional evolution equations with almost sectorial operators. Adv. Differ. Equ. 2016(1), 203 (2016). https://doi.org/10.1186/s13662-016-0927-y

    Article  MathSciNet  MATH  Google Scholar 

  17. Du, J., Jiang, W., Niazi, A.U.K.: Approximate controllability of impulsive Hilfer fractional differential inclusions. J. Nonlinear Sci. Appl. 10(2), 595–611 (2017)

    Article  MathSciNet  Google Scholar 

  18. Fu, X.: Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evol. Equ. Control Theory 6(4), 517–534 (2017). https://doi.org/10.3934/eect.2017026

    Article  MathSciNet  MATH  Google Scholar 

  19. Gomez-Aguilar, J.F., Cordova-Fraga, T., Abdeljawad, T., Khan, A., Khan, H.: Analysis of fractal-fractional malaria transmission model. Fractals (2020). https://doi.org/10.1142/S0218348X20400411

    Article  Google Scholar 

  20. Gu, H., Trujillo, J.J.: Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257, 344–354 (2015). https://doi.org/10.1016/j.amc.2014.10.083

    Article  MathSciNet  MATH  Google Scholar 

  21. Jaiswal, A., Bahuguna, D.: Hilfer fractional differential equations with almost sectorial operators. Differ. Equ. Dyn. Syst. (2020). https://doi.org/10.1007/s12591-020-00514-y

    Article  Google Scholar 

  22. Khan, A., Khan, T.S., Syam, M.I., Khan, H.: Analytical solutions of time-fractional wave equation by double Laplace transform method. Eur. Phys. J. Plus 134(4), 163 (2019). https://doi.org/10.1140/epjp/i2019-12499-y

    Article  Google Scholar 

  23. Khan, A., Shah, K., Li, Y., Khan, T.S.: Ulam type stability for a coupled system of boundary value problems of nonlinear fractional differential equations. J. Funct. Spaces (2017). https://doi.org/10.1155/2017/3046013

    Article  MathSciNet  MATH  Google Scholar 

  24. 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 133(7), 264 (2018). https://doi.org/10.1140/epjp/i2018-12119-6

    Article  Google Scholar 

  25. Khan, H., Alipour, M., Khan, R.A., Tajadodi, H., Khan, A.: On approximate solution of fractional order logistic equations by operational matrices of Bernstein polynomials. J. Math. Comput. Sci. 14, 222–232 (2015)

    Article  Google Scholar 

  26. Khan, H., Chen, W., Khan, A., Khan, T.S., Al-Madlal, Q.M.: Hyers–Ulam stability and existence criteria for coupled fractional differential equations involving p-Laplacian operator. Adv. Differ. Equ. 2018(1), 455 (2018). https://doi.org/10.1186/s13662-018-1899-x

    Article  MathSciNet  MATH  Google Scholar 

  27. Khan, H., Gomez Aguilar, J.F., Abdeljawad, T., Khan, A.: Existence results and stability criteria for ABC-fuzzy-Volterra integro-differential equation. Fractals (2020). https://doi.org/10.1142/S0218348X20400484

    Article  Google Scholar 

  28. Khan, H., Gómez-Aguilar, J.F., Alkhazzan, A., Khan, A.: A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler law. Math. Methods Appl. Sci. 43(6), 3786–3806 (2020). https://doi.org/10.1002/mma.6155

    Article  MATH  Google Scholar 

  29. Khan, H., Tunç, C., Alkhazan, A., Ameen, B., Khan, A.: A generalization of Minkowski’s inequality by Hahn integral operator. J. Taibah Univ. Sci. 12(5), 506–513 (2018). https://doi.org/10.1080/16583655.2018.1493859

    Article  Google Scholar 

  30. Khan, Z.A.: Integral inequality of Gronwall type with an application. J. Math. Comput. Sci. 5(1), 34–41 (2015)

    Google Scholar 

  31. Khan, Z.A.: Solvability for a class of integral inequalities with maxima on the theory of time scales and their applications. Bound. Value Probl. 2019, 146 (2019). https://doi.org/10.1186/s13661-019-1259-0

    Article  MathSciNet  Google Scholar 

  32. Khan, Z.A.: Analysis on some powered integral inequalities with retarded argument and application. J. Taibah Univ. Sci. 14(1), 488–495 (2020). https://doi.org/10.1080/16583655.2020.1747218

    Article  Google Scholar 

  33. Li, F.: Mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay. Adv. Differ. Equ. 2013(1), 327 (2013). https://doi.org/10.1186/1687-1847-2013-327

    Article  MathSciNet  MATH  Google Scholar 

  34. Lv, J., Yang, X.: Approximate controllability of Hilfer fractional neutral stochastic differential equations. Dyn. Syst. Appl. 27(4), 691–713 (2018). https://doi.org/10.12732/dsa.v27i4.1

    Article  MathSciNet  Google Scholar 

  35. Lv, J., Yang, X.: Approximate controllability of Hilfer fractional differential equations. Math. Methods Appl. Sci. 43(1), 242–254 (2020). https://doi.org/10.1002/mma.5862

    Article  MathSciNet  MATH  Google Scholar 

  36. Mahmudov, N.I., McKibben, M.A.: On the approximate controllability of fractional evolution equations with generalized Riemann–Liouville fractional derivative. J. Funct. Spaces 2015, Article ID 263823 (2015). https://doi.org/10.1155/2015/263823

    Article  MathSciNet  MATH  Google Scholar 

  37. Periago, F., Straub, B.: A functional calculus for almost sectorial operators and applications to abstract evolution equations. J. Evol. Equ. 2(1), 41–68 (2002). https://doi.org/10.1007/s00028-002-8079-9

    Article  MathSciNet  MATH  Google Scholar 

  38. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  39. Ullah, M., Sarwar, M., Khan, H., Abdeljawad, T., Khan, A.: Near-coincidence point results in metric interval space and hyperspace via simulation functions. Adv. Differ. Equ. 2020(1), 291 (2020). https://doi.org/10.1186/s13662-020-02734-6

    Article  MathSciNet  Google Scholar 

  40. Wang, J., Fečkan, M., Zhou, Y.: Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evol. Equ. Control Theory 6(3), 471 (2017)

    Article  MathSciNet  Google Scholar 

  41. Yang, M., Wang, Q.R.: Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. Math. Methods Appl. Sci. 40(4), 1126–1138 (2017). https://doi.org/10.1002/mma.4040

    Article  MathSciNet  MATH  Google Scholar 

  42. Yong, Z., Jinrong, W., Lu, Z.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2016)

    MATH  Google Scholar 

  43. Zhang, L., Zhou, Y.: Fractional Cauchy problems with almost sectorial operators. Appl. Math. Comput. 257, 145–157 (2015). https://doi.org/10.1016/j.amc.2014.07.024

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhou, Y., Vijayakumar, V., Ravichandran, C., Murugesu, R.: Controllability results for fractional order neutral functional differential inclusions with infinite delay. Fixed Point Theory 18(2), 773–798 (2017). https://doi.org/10.24193/fpt-ro.2017.2.62

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author (Pallavi Bedi) acknowledges the support of Council of Scientific and Industrial Research (CSIR)-New Delhi, India, and the third author (Thabet Abdeljawad) acknowledges the support of Prince Sultan University for funding this work through research group in Applied Mathematics (NAMAM) group number RG-DES-201701-17.

Funding

Prince Sultan University for funding this work through research group in Applied Mathematics (NAMAM) group number RG-DES-201701-17.

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Bedi, P., Kumar, A., Abdeljawad, T. et al. Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators. Adv Differ Equ 2020, 615 (2020). https://doi.org/10.1186/s13662-020-03074-1

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