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Existence and uniqueness of nonlocal boundary conditions for Hilfer–Hadamard-type fractional differential equations
Advances in Difference Equations volume 2021, Article number: 198 (2021)
Abstract
In this paper, we use some fixed point theorems in Banach space for studying the existence and uniqueness results for Hilfer–Hadamard-type fractional differential equations
on the interval \((1,e]\) with nonlinear boundary conditions
1 Introduction
In this paper, we discuss the existence and uniqueness of the solutions for the n-point nonlinear boundary value problems for Hilfer–Hadamard-type fractional differential equations of the form
where \({}_{\mathrm{H}}D^{\alpha ,\beta }\) is the Hilfer–Hadamard fractional derivative of order \(1<\alpha \leq 2\) and type \(\beta \in [0,1]\), \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) is a continuous function, \(0<\epsilon <1\), \(\zeta _{i}\in (1,e)\), \(\nu _{i},\sigma _{i}\in \mathbb{R}\) for all \(i=1, 2,\dots ,n-2\), \(\zeta _{1}<\zeta _{2}<\cdots <\zeta _{n-2}\), and \({}_{\mathrm{H}}D^{1,1}=t\frac{d}{dt}\).
The fractional differential equations appear as more appropriate models for describing real world problems. Indeed, these problems cannot be described using the classical integer-order differential equations. In the past years, the theory of fractional differential equations has received much attention from the authors and has become an important field of investigation due to existing applications in engineering, biology, chemistry, economics, and numerous branches of physics [20, 27, 33, 40]. For example, the fractional differential equations are applied to describe the abundant phenomena such as flow in nonlinear electric circuits [15, 16, 20], properties of viscoelastic and dielectric materials [20, 21, 32], nonlinear oscillations of an earthquake [28], mechanics [35], aerodynamics, regular variations in thermodynamics [18], etc.
Fractional derivatives can be of several kinds, one of them is the Hadamard fractional derivative innovated by Hadamard in 1892 [17]. It differs from the preceding Riemann–Liouville- and Caputo-type fractional derivatives [33] in the sense that the kernel of the integral contains the logarithmic function of an arbitrary exponent. The properties of Hadamard fractional integral and derivative can be found in [26, 27]. Recently, scholars have studied the Hadamard-, Caputo–Hadamard- and Hilfer–Hadamard-type fractional derivatives by using the fixed point theorems with the boundary value problems and have given results of the existence and uniqueness of solutions, see [1–13, 22–25, 30, 31, 34, 36–39, 41, 43–45] and the references mentioned therein.
In this paper, we find a variety of results for the boundary value problem (1.1) by using traditional fixed point theorems. The first result is Theorem 3.2, which depends on Banach contraction mapping principle and presents the existence and uniqueness result for the solution of problem (1.1). In Theorem 3.3, we prove the second result of the existence and uniqueness through a fixed point theorem and for nonlinear contractions due to Boyd and Wong. In Theorem 3.4, we prove the third existence result by using Krasnoselskii’s fixed point theorem. By using Leray–Schauder type of nonlinear alternative for single-valued maps, we prove the last result of existence, which is Theorem 3.5. Examples are included to illustrative our main results.
2 Preliminaries
In this section, we introduce some notations and definitions of Hilfer–Hadamard-type fractional calculus.
Definition 2.1
(Riemann–Liouville fractional integral, [27, 40])
The Riemann–Liouville fractional integral of order \(\alpha >0\) of a function \(\varphi :[1,\infty )\rightarrow \mathbb{R}\) is defined by
Here, \(\Gamma (\alpha )\) is the Euler’s Gamma function defined by
Definition 2.2
(Riemann–Liouville fractional derivative, [27, 40])
The Riemann–Liouville fractional derivative of order \(\alpha >0\) of a function \(\varphi :[1,\infty )\rightarrow \mathbb{R}\) is defined by
where \([\alpha ]\) is the integer part of α.
Definition 2.3
(Hadamard fractional integral, [27])
The Hadamard fractional integral of order \(\alpha \in \mathbb{R}^{+}\) for a function \(\varphi :[1,\infty )\rightarrow \mathbb{R}\) is defined as
where \(\log (\cdot )=\log _{e}(\cdot )\).
Definition 2.4
(Hadamard fractional derivative, [27])
The Hadamard fractional derivative of order α applied to the function \(\varphi :[1,\infty )\rightarrow \mathbb{R}\) is defined as
where \(\delta ^{n}=(t\frac{d}{dt})^{n}\) and \([\alpha ]\) denotes the integer part of the real number α.
Definition 2.5
(Caputo–Hadamard fractional derivative, [17])
The Caputo–Hadamard fractional derivative of order α applied to the function \(\varphi \in AC_{\delta }^{n}[a,b]\) is defined as
where \(\varphi \in AC_{\delta }^{n}[a,b]= \{\varphi :[a,b]\rightarrow \mathbb{C}:\delta ^{(n-1)}\varphi \in AC[a,b],\delta =t\frac{d}{dt}\}\).
Definition 2.6
(Hilfer fractional derivative, [20, 22])
Let \(n-1<\alpha <n\), \(0\leq \beta \leq 1\), \(\varphi \in L^{1}(a,b)\). The Hilfer fractional derivative \(D^{\alpha ,\beta }\) of order α and type β of φ is defined as
where \(I^{(\cdot )}\) and \(D^{(\cdot )}\) are the Riemann–Liouville fractional integral and derivative defined by Definitions 2.1 and 2.2, respectively.
In particular, if \(0<\alpha <1\), then
Proposition 2.7
Let \(0<\alpha <1\), \(0\leq \beta \leq 1\), \(\gamma =\alpha +\beta -\alpha \beta \), and \(\varphi \in L^{1}(a,b)\). If \(D^{\gamma }\varphi \) exists and is in \(L^{1}(a,b)\), then
Definition 2.8
(Hilfer–Hadamard fractional derivative, [12, 21])
Let \(0<\alpha <1\), \(0\leq \beta \leq 1\), \(\varphi \in L^{1}(a,b)\). The Hilfer–Hadamard fractional derivative \({}_{\mathrm{H}}D^{\alpha ,\beta }\) of order α and type β of φ is defined as
where \({}_{\mathrm{H}}I^{(\cdot )}\) and \({}_{\mathrm{H}}D^{(\cdot )}\) are the Hadamard fractional integral and derivative defined by Definitions 2.3 and 2.4, respectively.
Theorem 2.9
Let \(\Re (\alpha )>0\), \(n=[\Re (\alpha )]+1\), and \(0< a< b<\infty \). If \(\varphi \in L^{1}(a,b)\) and \(({}_{\mathrm{H}}I_{a+}^{n-\alpha }\varphi )(t)\in AC_{\delta }^{n}[a,b]\), then
Theorem 2.10
([17])
Let \(\varphi (t)\in AC_{\delta }^{n}[a,b]\) or \(\varphi (t)\in C_{\delta }^{n}[a,b]\), and \(\alpha \in \mathbb{C}\), then
Definition 2.11
([45])
Let E be a Banach space and let \(F :E\rightarrow E \) be a mapping. Then F is said to be a nonlinear contraction if there exists a continuous nondecreasing function \(\psi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) such that \(\psi (0) = 0\) and \(\psi (\phi ) < \phi \) for all \(\phi > 0\) with the property
Lemma 2.12
([14])
Let E be a Banach space and let \(F :E\rightarrow E \) be a nonlinear contraction. Then, F has a unique fixed point in E.
Theorem 2.13
(Krasnoselskii’s fixed point theorem, [29])
Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that
-
(a)
\(Ax+By\in M\), whenever \(x, y\in M\);
-
(b)
A is compact and continuous;
-
(c)
B is a contraction mapping.
Then, there exists \(z\in M\) such that \(z=Az+Bz\).
Theorem 2.14
(Nonlinear alternative for single-valued maps, [19, 42])
Let E be a Banach space, C a closed, convex subset of E, U an open subset of C, and \(0\in U\). Suppose that \(F : \overline{U}\rightarrow C\) is a continuous, compact (i.e., \(F(\overline{U})\) is a relatively compact subset of C) map. Then, either
-
(i)
F has a fixed point in U̅ or
-
(ii)
there is a \(u\in \partial U\) (the boundary of U in C) and \(\bar{\lambda }\in (0,1)\) with \(u =\bar{\lambda } F(u)\).
Definition 2.15
(Hilfer–Hadamard fractional derivative, [38])
Let \(n-1<\alpha<n\), \(0\leq\beta\leq 1\), \(\varphi\in L^{1}(a,b)\). The Hilfer–Hadamard fractional derivative \({}_{\mathrm{H}}D^{\alpha ,\beta }\) of order α and type β of φ is defined as
where \({}_{\mathrm{H}}I^{(\cdot )}\) and \({}_{\mathrm{H}}D^{(\cdot )}\) are the Hadamard fractional integral and derivative defined by Definitions 2.3 and 2.4, respectively.
Lemma 2.16
([38])
Let \(\Re (\alpha )>0\), \(0\leq \beta \leq 1\), \(\gamma =\alpha +n \beta -\alpha \beta \), \(n-1<\gamma \leq n\), \(n=[\Re (\alpha )]+1\), and \(0< a< b<\infty \). If \(\varphi \in L^{1}(a,b)\) and \(({}_{\mathrm{H}}I_{a+}^{n-\gamma }\varphi )(t)\in AC_{\delta }^{n}[a,b]\), then
From this lemma, we notice that if \(\beta =0\) then the equation reduces to the equation in Theorem 2.9, and if the \(\beta =1\) then the equation reduces to the equation in Theorem 2.10.
3 Main results
Lemma 3.1
For \(1<\alpha \leq 2\), \(0\leq \beta \leq 1\), \(\gamma =\alpha +2\beta -\alpha \beta \), \(\gamma \in (1,2]\), and \(\varphi \in C([1,e],\mathbb{R})\), the problem
has a unique solution given by
where
Proof
In view of Lemma 2.16, the solution of the Hilfer–Hadamard differential equation (3.1) can be written as
and
The boundary condition \(x(1+\epsilon )=\sum_{i=1}^{n-2}\nu _{i}x(\zeta _{i})\) gives
where
In view of the boundary condition \({}_{\mathrm{H}}D^{1,1}x(e)=\sum_{i=1}^{n-2}\sigma _{i} \, {}_{\mathrm{H}}D^{1,1}x(\zeta _{i})\) and from equations (3.3) and (3.4), we have
where
By using (3.5) in equation (3.4), we have
where
By substituting the value of \(c_{1}\) into (3.5), we have
Now, substituting the values of \(c_{0}\) and \(c_{1}\) in (3.2), we obtain the solution of problem (3.1). □
Next, we present the existence and uniqueness of solutions for Hilfer–Hadamard-type fractional differential equation (1.1). For that, suppose that
is a Banach space of all continuous functions from \([1,e]\) into \(\mathbb{R}\) equipped with the norm \(\| x\|=\sup_{t\in J}|x(t)|\). From Lemma 3.1, we get an operator \(\rho :K\rightarrow K\) defined as
It must be noticed that problem (1.1) has a solution if and only if operator ρ has a fixed point. The results of existence and uniqueness are based on the Banach contraction mapping principle.
Theorem 3.2
Let \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) be a continuous function satisfying the assumption
- (\(Q_{1}\)):
-
there exists a constant \(C > 0 \) such that \(| f(t,x)-f(t,y)|\leq C| x-y|\) for each \(t\in J\) and \(x,y\in \mathbb{R}\). If Φ is such that \(C\Phi <1\), where
$$\begin{aligned} \Phi =& \Biggl\{ \frac{1}{\Gamma (\alpha +1)}+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert + ( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert \Gamma (\alpha +1)} \\ &{}\times \Biggl[\bigl(\log (1+\epsilon )\bigr)^{\alpha }+\sum _{i=1}^{n-2} \vert \nu _{i} \vert \bigl(\log ( \zeta _{i})\bigr)^{\alpha } \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert \Gamma (\alpha )} \Biggl[1+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \bigl(\log (\zeta _{i}) \bigr)^{\alpha -1} \Biggr] \Biggr\} , \end{aligned}$$(3.8)then the boundary value problem (1.1) has a unique solution on J.
Proof
We are using Banach contraction mapping principle to transform the boundary value problem (1.1) into a fixed point problem \(x=\rho x\), where the operator ρ is defined by (3.7). We will show that ρ has a fixed point, which is a unique solution of problem (1.1).
We put \(\sup |f(\tau ,0)|= p <\infty \) and choose
Now, assume that \(B_{r}=\{x\in K:| x|\leq r\}\). We will show that \(\rho B_{r}\subset B_{r}\).
For any \(x\in B_{r}\), we have
Thus, we have shown that \(\rho B_{r}\subset B_{r}\).
Now, for \(x,y\in K\) and \(t\in J\), we have
Therefore, it has been shown that \(\|(\rho x)(t)-(\rho y)(t)\|\leq C \Phi \| x-y\|\), where \(C \Phi <1\). Hence, ρ is a contraction. Thus, by Banach contraction mapping principle, problem (1.1) has a unique solution. □
Theorem 3.3
Let \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) be a continuous function satisfying the assumption
- (\(Q_{2}\)):
-
\(|f(t,x)-f(t,y) |\leq \varphi (t) (|x-y|/(P^{*}+|x-y|) )\), \(t\in J\), \(x,y\geq 0\), where \(\varphi :J\rightarrow \mathbb{R}^{+}\) is continuous and a constant \(P^{*}\) is defined by
$$\begin{aligned} P^{*} =& {}_{\mathrm{H}}I^{\alpha } \varphi (e)+ \frac{( \vert \gamma -1 \vert ) \vert \delta _{1} \vert +( \vert \gamma -2 \vert ) \vert \delta _{2} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha } \varphi (1+\epsilon )+ \sum_{i=1}^{n-2} \vert \nu _{i} \vert \, {}_{\mathrm{H}}I^{ \alpha } \varphi (\zeta _{i}) \Biggr] \\ &{} + \frac{ \vert \mu _{2} \vert + \vert \mu _{1} \vert }{ \vert \lambda \vert } \Biggl[ {}_{\mathrm{H}}I^{\alpha -1} \varphi (e)+ \sum_{i=1}^{n-2} \vert \sigma _{i} \vert \, {}_{\mathrm{H}}I^{\alpha -1}\varphi ( \zeta _{i}) \Biggr]. \end{aligned}$$(3.12)
Then, the boundary value problem (1.1) has a unique solution on J.
Proof
We have the operator \(\rho :K\rightarrow K\) defined by (3.7) and by applying Definition 2.11, we can define a continuous nondecreasing function \(\Psi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) by
where the function Ψ satisfies \(\Psi (0) = 0\) and \(\Psi (\phi )<\phi \) for all \(\phi > 0\).
For any \(x,y \in K\) and for each \(t\in J\), we have
which implies that \(\|\rho x-\rho y\|\leq \Psi (\|x-y\|)\). Then, the operator ρ is a nonlinear contraction. Thus, by Lemma 2.12 (Banach contraction mapping principle) the operator ρ has a unique fixed point, which is the unique solution of problem (1.1). □
Next, we will give the existence result by using Theorem 2.13 (Krasnoselskii’s fixed point theorem).
Theorem 3.4
Let \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) be a continuous function satisfying the assumption \((Q_{1})\). In addition, assume that
If
then the boundary value problem (1.1) has at least one solution on J.
Proof
We put \(\sup_{t\in J}|g(t)|=\|g\|\) and choose a suitable constant r̂ such that
where Φ is defined by (3.8). Moreover, we consider the operators \(\mathscr{F}\) and \(\mathscr{G}\) on \(B_{\hat{r}}=\{x\in K:\|x\|\leq \hat{r}\}\) defined as
For any \(x,y\in B_{\hat{r}}\), we have
which implies that \(\mathscr{F}x+\mathscr{G}x\in B_{\hat{r}}\). It follows from assumption \((Q_{1})\), together with (3.15), that \(\mathscr{G}\) is a contraction. Furthermore, it is easy to show that the operator \(\mathscr{F}\) is continuous. Moreover,
Hence, \(\mathscr{F}\) is uniformly bounded on \(B_{\hat{r}}\).
Next, we prove that the operator \(\mathscr{F}\) is compact. For that, we put \(\sup_{(t,x)\in J\times B_{\hat{r}}}|f(t,x)|=\bar{p}<\infty \).
Consequently, for \(t_{1},t_{2}\in J\), we get
which is independent of x and tends to zero as \(t_{2}\rightarrow t_{1}\). Thus, \(\mathscr{F}\) is equicontinuous. Hence, \(\mathscr{F}\) is relatively compact on \(B_{\hat{r}}\). Therefore, by the Arzelà–Ascoli theorem, \(\mathscr{F}\) is compact on \(B_{\hat{r}}\). Thus, by Theorem 2.13, the boundary value problem (1.1) has at least one solution on J. □
Now, the final existence result is based on Theorem 2.14 (nonlinear alternative for single-valued maps).
Theorem 3.5
Let \(f:J\times \mathbb{R}\rightarrow \mathbb{R}\) be a continuous function, and assume that:
- (\(Q_{4}\)):
-
there exists a continuous nondecreasing function \(\vartheta : \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\backslash \{0 \}\) such that
$$ \bigl\vert f(t,x) \bigr\vert \leq q(t)\vartheta \bigl( \vert x \vert \bigr)\quad \textit{for each } (t,x)\in J \times \mathbb{R}, $$(3.20)where \(q\in C([1,e],\mathbb{R}^{+})\) is a function;
- (\(Q_{5}\)):
-
there exists a constant \(L>0\) such that
$$ \frac{L}{ \Vert q \Vert \vartheta (L)\Phi }>1, $$(3.21)where Φ is defined by (3.8). Then, the boundary value problem (1.1) has at least one solution on J.
Proof
We have the operator ρ defined by (3.7). Firstly, we will show that ρ maps bounded sets (balls) into bounded sets in K. For that, let r̄ be a positive number, and \(B_{\bar{r}}=\{x\in K :\|x\|\leq \bar{r}\}\) be a bounded ball in K, where K is defined by (3.6). For \(t\in J\), we have
which implies that \(\|\rho x\|\leq C_{1}\).
Now, we will show that ρ maps bounded sets into equicontinuous sets of K. For that, let \(\sup_{(t,x)\in J\times B_{\bar{r}}}|f(t,x)|=p^{\star }<\infty \), where \(\omega _{1}, \omega _{2}\in J\), with \(\omega _{1}<\omega _{2}\) and \(x\in B_{\bar{r}}\). Hence, we have
Clearly, as \(\omega _{2}\rightarrow \omega _{1}\), the right-hand side of the latter inequality tends to zero, which happens independently of \(x\in B_{\bar{r}}\). Thus, by the Arzelà–Ascoli theorem, it follows that \(\rho : K\rightarrow K\) is completely continuous.
Finally, let x be a solution. So, for \(t\in J\), following similar computations as in the first step, we have
Thus, we have
In view of \((Q_{5})\), there exists L such that \(\| x\|\neq L\). Let us set
Note that the operator \(\rho :\overline{V}\rightarrow K\) is continuous and completely continuous. From the choice of V, there is no \(x\in \partial V\) such that \(x=\bar{\lambda }\rho x\) for some \(\bar{\lambda }\in (0,1)\). Thus, by Theorem 2.14, the operator ρ has a fixed point in V̅, which is a solution of the boundary value problem (1.1). □
4 Example
Example 4.1
Consider the following boundary value problem for Hilfer–Hadamard-type fractional differential equation:
Here, \(\alpha =3/2\), \(\beta =1/2\), \(\gamma =7/4\), \(\nu _{1}=1/2\), \(\nu _{2}=-3/4 \), \(\sigma _{1}=2/3\), \(\sigma _{2}=4/3\), \(\zeta _{1}=3/2\), \(\zeta _{2}=7/4\), \(\epsilon =0.3\), \(1+\epsilon =1.3\), and
Clearly,
Hence, (\(Q_{1}\)) is satisfied with \(C=\frac{3}{64e}\). We can show that
Therefore, by Theorem 3.2, the boundary value problem (4.1) has a unique solution on J.
Example 4.2
Consider the following boundary value problem for Hilfer–Hadamard-type fractional differential equation:
Here, \(\alpha =3/2\), \(\beta =2/3\), \(\gamma =11/6\), \(\nu _{1}=2\), \(\nu _{2}=-1/2\), \(\nu _{3}=5/3\), \(\sigma _{1}=-1\), \(\sigma _{2}=3\), \(\sigma _{3}=-11/3\), \(\zeta _{1}=4/3\), \(\zeta _{2}=2\), \(\zeta _{2}=9/7\), \(\epsilon =0.5\), \(1+ \epsilon =1.5\), and
Clearly,
We choose \(q(t)=1+\log t\) and \(\vartheta (|x|)=(| x(t)|+1)/12\). Then, we can show that
Now, by (\(Q_{5}\)) we have
Hence, \(L>1.320578171\). Therefore, by Theorem 3.5, the boundary value problem (4.2) has at least one solution on J.
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References
Abdeljawad, T., Agarwal, R.P., Karapinar, E., Kumari, P.S.: Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space. Symmetry 11(5), 686 (2019)
Afshari, H., Atapour, M., Karapinar, E.: A discussion on a generalized Geraghty multi-valued mappings and applications. Adv. Differ. Equ. 2020, 356 (2020). https://doi.org/10.1186/s13662-020-02819-2
Afshari, H., Jarad, F., Abdeljawad, T.: On a new fixed point theorem with an application on a coupled system of fractional differential equations. Adv. Differ. Equ. 2020, 461 (2020). https://doi.org/10.1186/s13662-020-02926-0
Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional differential equations via coupled fixed point. Electron. J. Differ. Equ. 2015, 286 (2015)
Afshari, H., Karapinar, E.: A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces. Adv. Differ. Equ. 2020, 616 (2020). https://doi.org/10.1186/s13662-020-03076-z
Ahmad, B., Ntouyas, S.K.: An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions. Abstr. Appl. Anal. 2014, Article ID 705809 (2014)
Ahmad, B., Ntouyas, S.K.: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17(2), 348–360 (2014)
Ahmad, B., Ntouyas, S.K.: On Hadamard fractional integro-differential boundary value problems. J. Appl. Math. Comput. 47, 119–131 (2015)
Ahmad, B., Ntouyas, S.K., Alsaedi, A.: A study of nonlinear fractional differential equations of arbitrary order with Riemann–Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013, Article ID 320415 (2013)
Ahmad, B., Ntouyas, S.K., Alsaedi, A.: New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions. Bound. Value Probl. 2013, 275 (2013)
Alqahtani, B., Aydi, H., Karapinar, E., Rakocevic, V.: A solution for Volterra fractional integral equations by hybrid contractions. Mathematics 7(8), 694 (2019)
Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal., Theory Methods Appl. 72(2), 916–924 (2010)
Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal., Theory Methods Appl. 71(7–8), 2391–2396 (2009)
Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)
Caponetto, R., Dongola, G., Fortuna, I., Petras, I.: Fractional Order Systems. Modeling and Control Applications. World Scientific Series on Nonlinear Science Series A, vol. 72. World Scientific, Singapore (2010)
Corduneanu, C.: Integral Equations and Stability of Feedback Systems. Academic Press, San Diego (1973)
Gambo, Y.Y., Jarad, F., Baleanu, D., Abdeljawad, T.: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014(1), 10 (2014)
Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Vienna (1997)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Hilfer, R.: Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284(1–2), 399–408 (2002)
Hilfer, R.: Threefold introduction to fractional derivatives. In: Anomalous Transport: Foundations and Applications, pp. 17–73 (2008)
Karapinar, E., Binh, H.D., Luc, N.H., Can, N.H.: On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems. Adv. Differ. Equ. 2021(1), 70 (2021)
Karapinar, E., Fulga, A., Rashid, M., Shahid, L., Aydi, H.: Large contractions on quasi-metric spaces with an application to nonlinear fractional differential equations. Mathematics 7(5), 444 (2019)
Keyantuo, V., Lizama, C., Warma, M.: Asymptotic behavior of fractional order semilinear evolution equations. Differ. Integral Equ. 26(7/8), 757–780 (2013)
Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191–1204 (2001)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Kiryakova, V.S.: Generalized Fractional Calculus and Applications. CRC Press, Boca Raton (1993)
Krasnosel’skii, M.A.: Two remarks on the method of successive approximations. Usp. Mat. Nauk 10(1), 123–127 (1955)
Li, C.-G., Kostic, M., Li, M., Piskarev, S.: On a class of time-fractional differential equations. Fract. Calc. Appl. Anal. 15(4), 639–668 (2012)
Lizama, C.: Solutions of two-term fractional order differential equations with nonlocal initial conditions. Electron. J. Qual. Theory Differ. Equ. 2012, 82 (2012)
Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2010)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Qassim, M.D., Furati, K.M., Tatar, N.-E.: On a differential equation involving Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2012, Article ID 391062 (2012)
Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3581 (1997)
Salamooni, A.Y.A., Pawar, D.D.: Unique positive solution for nonlinear Caputo-type fractional q-difference equations with nonlocal and Stieltjes integral boundary conditions. Fract. Differ. Calc. 9(2), 295–307 (2019)
Salamooni, A.Y.A., Pawar, D.D.: Existence and uniqueness of generalised fractional Cauchy-type problem. Univers. J. Math. Appl. 3(3), 121–128 (2020)
Salamooni, A.Y.A., Pawar, D.D.: Existence and uniqueness of boundary value problems for Hilfer–Hadamard-type fractional differential equations. Ganita 70(2), 1–16 (2020)
Salamooni, A.Y.A., Pawar, D.D.: Existence and stability results for Hilfer–Katugampola-type fractional implicit differential equations with nonlocal conditions. J. Nonlinear Sci. Appl. 14(3), 124–138 (2021)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York (1993). Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987)
Sevinik Adigüzel, R., Aksoy, Ü., Karapinar, E., Erhan, I.M.: On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6652
Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980)
Tariboon, J., Ntouyas, S.K., Sudsutad, W.: Nonlocal Hadamard fractional integral conditions for nonlinear Riemann–Liouville fractional differential equations. Bound. Value Probl. 2014(1), 253 (2014)
Thabet, S.T.M., Dhakne, M.B.: On boundary value problems of higher order abstract fractional integro-differential equations. Int. J. Nonlinear Anal. Appl. 7(2), 165–184 (2016)
Thiramanus, P., Ntouyas, S.K., Tariboon, J.: Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions. Abstr. Appl. Anal. 2014, Article ID 902054 (2014)
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Salamooni, A.Y.A., Pawar, D.D. Existence and uniqueness of nonlocal boundary conditions for Hilfer–Hadamard-type fractional differential equations. Adv Differ Equ 2021, 198 (2021). https://doi.org/10.1186/s13662-021-03358-0
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DOI: https://doi.org/10.1186/s13662-021-03358-0
MSC
- 34A08
- 35R11
Keywords
- Existence
- Uniqueness
- Nonlinear boundary value problems
- Hilfer–Hadamard type
- Fractional differential equation and fractional calculus