Abstract
This paper studies uniqueness of solutions for a nonlinear Hadamard-type integro-differential equation in the Banach space of absolutely continuous functions based on Babenko’s approach and Banach’s contraction principle. We also include two illustrative examples to demonstrate the use of main theorems.
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1 Introduction
The Hadamard fractional integration and differentiation are based on the nth integral of the form [1, 2]
and the corresponding derivative
where \(\log ( \cdot ) = \log _{e} (\cdot )\), \(0 < a < x < b\), and \(\mu \in R\).
The fractional version of the Hadamard-type integral and derivative are given by
and
where \(n = [\alpha ] + 1\), and \([\alpha ]\) being integral part of α.
When \(0 < \alpha < 1\), the fractional derivative turns out to be
In particular, for \(\alpha = 1\),
which leads to defining the space \({X}_{\mu }[a, b]\) of those Lebesgue measurable functions u on \([a, b]\) for which \(x^{\mu - 1} u(x)\) is absolutely integrable [2]:
Let \(\operatorname{AC}[a, b]\) be the set of absolutely continuous functions on \([a, b]\). Then it follows from [3] that
Obviously,
The latter is a Banach space under its norm. We further define the space
Clearly, \(\lVert u \rVert _{0}\) is a norm in \(\operatorname{AC}_{0}[a, b]\). Indeed, if \(\lVert u \rVert _{0} = 0\) then \(u(x) = u(a) = 0\). To show that \(\operatorname{AC}_{0}[a, b]\) is complete, we assume \(\{u_{n} (x) \}\) is a Cauchy sequence in \(\operatorname{AC}_{0}[a, b]\), then we need to find a function \(u(x)\) such that \(u(x)\) is absolutely continuous and \(u_{n} \rightarrow u\) under its norm. Since \(\{u_{n} (x) \}\) is Cauchy in \(\operatorname{AC}_{0}[a, b]\), we claim that \(u_{n}(a) = 0\) and \(\{u'_{n} (x) \}\) is Cauchy in \(L[a, b]\). Hence, there exists \(g \in L[a, b]\) such that \(u'_{n} \rightarrow g \) in \(L[a, b] \). Define
Then \(u(a) = 0\) and \(u(x)\) is absolutely continuous on \([a, b]\), and
converges to zero. Therefore, \(\operatorname{AC}_{0}[a, b]\) is a Banach space.
Lemma 1.1
If \(\alpha > 0\), \(\mu \in R\), and \(0 < a < b < \infty \), then the operator \(\mathcal{J}^{\alpha }_{ a + , \mu }\) is bounded in \(\operatorname{AC}_{0}[a, b]\) and
where \(C_{\mu }\) is the maximum value of the function
on \([a, b] \times [a, b]\).
Proof
Let \(u \in \operatorname{AC}_{0}[a, b]\). Then
and
by changing the order of integration. Using
we imply that
This completes the proof of Lemma 1.1. □
Kilbas showed the following lemma in reference [2], which is soon to be used.
Lemma 1.2
-
(i)
If \(\alpha >0\), \(\beta > 0\), \(\mu \in R\), and \(u \in X_{\mu }[a, b]\), then the semigroup property holds
$$ \mathcal{J}^{\alpha }_{ a + , \mu } \mathcal{J}^{\beta }_{ a + , \mu } u = \mathcal{J}^{\alpha + \beta }_{ a + , \mu }u. $$ -
(ii)
If \(0 < \alpha < 1\) and \(u \in \operatorname{AC}[a, b]\), then
$$ \mathcal{J}^{\alpha }_{a + , \mu } \mathcal{D}^{\alpha }_{a + , \mu } u = u. $$
Let \(u \in \operatorname{AC}[a, b]\) and \(0 < \beta < 1\). It follows from Lemma 1.2 that
if \(\alpha \geq \beta \).
Let \(0 < \alpha _{0} < \alpha _{ 1} < \cdots < \alpha _{n} < 1\) and \(0 \leq \beta _{n + 1} < \cdots < \beta _{m} \in R\), where \(n = 0, 1, \ldots \) and \(m > n\). In this paper, we show the uniqueness of solutions for the following new nonlinear Hadamard-type integro-differential equation for all \(\mu \in R\) in the space \(\operatorname{AC}_{0}[a, b]\):
by Banach’s contraction principle and Babenko’s approach [4], with two applicable examples presented to illustrate the main results. It seems impossible to obtain these by any existing integral transforms or analytic local model methods. Babenko’s approach treats integral operators like variables in solving differential and integral equations. The method itself is close to the Laplace transform method in the ordinary sense, but it can be used in more cases [5, 6], such as dealing with integral or fractional differential equations with distributions whose Laplace transforms do not exist in the classical sense. Furthermore, it works well on certain differential or integral equations whose solutions cannot be achieved by the local model. Clearly, it is always necessary to show convergence of the series obtained as solutions. Recently, Li studied the generalized Abel’s integral equations of the first [7] and second kind with variable coefficients by Babenko’s technique [8–10].
It is well known that fractional calculus [3, 11, 12] has been an emergent tool which uses fractional differential and integral equations to develop more sophisticated mathematical models that can accurately describe complex systems. There are many definitions of fractional derivatives available in the literature, such as the Riemann–Liouville derivative which played an important role in the development of the theory of fractional analysis. However, the commonly used is the Hadamard fractional derivative (with \(\mu = 0\)) given by Hadamard in [13]. Butzer et al. [14–16] studied various properties of the Hadamard-type derivative which is more generalized than the Hadamard fractional derivative. In particular, Hadamard fractional differential equations with boundary value problems or initial conditions have been investigated by researchers using fixed point theories [17, 18]. In 2014, Thiramanus et al. [19] studied the existence and uniqueness of solutions for a fractional boundary value problem involving Hadamard differential equations of order \(q \in (1, 2]\) and nonlocal fractional integral boundary conditions by fixed point theories. In 2018, Matar [20] obtained the solution of the linear equations with the initial conditions (three terms on the left-hand side at most and a given function on the right) by the parameter technique, and then investigated the existence problems of the corresponding nonlinear types of Hadamard equations using fixed point theorems. Very recently, Ding et al. [21] applied the fixed point index and nonnegative matrices to study the existence of positive solutions for a system of Hadamard-type fractional differential equations with semipositone nonlinearities. In 1967, Caputo [22] introduced another type of fractional derivative which has an advantage over R-L derivative in differential equations since it does not require to define fractional order initial conditions. Jarad et al. [23] defined the Caputo-type modification of the Hadamard fractional derivatives which preserve physically interpretable initial conditions similar to the ones in Caputo fractional derivatives. Gambo et al. [24] further presented the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo–Hadamard setting with several new results. Adjabi et al. [25] studied Cauchy problems for a differential equation with a left Caputo–Hadamard fractional derivative in spaces of continuously differentiable functions.
There are new studies on fixed point theorems for different operators on metric spaces [26–28], as well as their applications in differential and integral equations, existence and uniqueness of solutions for equations [29–31]. Palve et al. [32] recently constructed the existence and uniqueness of solutions for the fractional implicit differential equation with boundary condition of the form
where \({{}_{H}}D^{\alpha , \beta }_{1+}\) is the Hilfer–Hadamard type fractional derivative of order α and type β given by
and \(c_{1}, c_{2}, c_{3} \in R\) with \(c_{1} + c_{2} \neq 0\) and \(c_{2} \neq 0\). Li [33] obtained uniqueness of solutions for the coupled system of integral equations
on the product space \(X_{\mu }[a, b] \times X_{\mu }[a, b]\) (it is a Banach space), based on Babenko’s approach and Banach’s contraction principle.
2 Main results
Theorem 2.1
Assume that \(a_{i}\) and \(b_{j}\) for \(i = 0, 1, \ldots , n -1\) and \(j = n +1, \ldots , m\) are arbitrary complex numbers, and \(g \in \operatorname{AC}_{0}[a, b]\). In addition, we let \(0 < \alpha _{0} < \alpha _{ 1} < \cdots < \alpha _{n} < 1\) and \(0 \leq \beta _{n + 1} < \cdots < \beta _{m} \in R\), where \(n = 0, 1, \ldots \) . Then equation
has a unique solution
in the space \(\operatorname{AC}_{0}[a, b]\).
Proof
Applying the operator \(\mathcal{J}^{\alpha _{n }}_{a + , \mu }\) to both sides of equation (2), we get
Using Lemma 1.2,
by noting that \(0 < \alpha _{0} < \alpha _{ 1} < \cdots < \alpha _{n} < 1\). Hence,
By Babenko’s method we come to
using Lemma 1.2 and the multinomial theorem. Clearly, \(u(a) = 0\) since \(\alpha _{n } > 0\) and
It remains to show that the series converges in the space \(\operatorname{AC}_{0}[a, b]\) and is absolutely continuous on \([a, b]\). By Lemma 1.1,
where
Therefore,
where
is the value at
of the multivariate Mittag-Leffler function \(E_{(\alpha _{n} - \alpha _{n - 1}, \ldots , \alpha _{n} + \beta _{m}, \alpha _{n} + 1)}(z_{1}, \ldots , z_{m})\) given in [12]. Thus, the series on the right-hand side of equation (3) is convergent. To see \(u(x)\) is absolutely continuous,
as the function inside of the outer integral
uniformly converges with respect to t and belongs to \(L[a, b]\) from Lemma 1.1 and the multivariate Mittag-Leffler function used above. Thus, \(u(x)\) is absolutely continuous on \([a, b]\). To verify that the obtained series is a solution, we substitute it into the left-hand side of equation (2):
by the cancelation. Note that all series are absolutely convergent and the term rearrangements are feasible for the cancelation.
Indeed,
The rest terms cancel each other similarly.
Clearly, the uniqueness follows immediately from the fact that the integro-differential equation
only has solution zero by Babenko’s method. This completes the proof of Theorem 2.1. □
Remark 1
-
(i)
It follows from Theorem 5.3 in [2] that for \(0 < \alpha < 1\)
$$ \bigl(\mathcal{D}^{\alpha }_{a + , \mu }u\bigr) (x) = \frac{x^{- \mu }}{\Gamma (1 - \alpha )} \biggl[u_{0}(a) \biggl(\log \frac{x}{a} \biggr)^{- \alpha } + \int _{a}^{x} \biggl(\log \frac{x}{t} \biggr)^{- \alpha } u_{0}'(t) \,dt \biggr], $$where \(u_{0}(x) = x^{\mu }u(x) \in \operatorname{AC}[a, b]\). Hence, for \(u \in \operatorname{AC}_{0}[a, b]\),
$$\begin{aligned}& \bigl(\mathcal{D}^{\alpha }_{a + , \mu }u\bigr) (x) = \frac{x^{- \mu }}{\Gamma (1 - \alpha )} \int _{a}^{x} \biggl(\log \frac{x}{t} \biggr)^{- \alpha } u_{0}'(t) \,dt,\quad \mbox{and} \\& \bigl(\mathcal{D}^{\alpha }_{a + , \mu }u\bigr) (a) = 0. \end{aligned}$$ -
(ii)
A solution of equation (2) in the space \(\operatorname{AC}_{0}[a, b]\) is said to be stable if \(\forall \epsilon > 0\) \(\exists \delta > 0\), such that \(\lVert u \rVert _{0} < \epsilon \) if \(\lVert g \rVert _{0} < \delta \). Using the inequality
$$\begin{aligned} \lVert u \rVert _{0} \leq & C_{\mu }E_{(\alpha _{n} - \alpha _{n - 1}, \ldots , \alpha _{n} + \beta _{m}, \alpha _{n} + 1)} \\ &{}\cdot \biggl( \vert a_{n - 1} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{n} - \alpha _{n - 1}}, \ldots , \vert b_{m} \vert \biggl(\log \frac{b}{a} \biggr)^{ \alpha _{n} + \beta _{m}} \biggr) \lVert g \rVert _{0}, \end{aligned}$$(4)we imply that the solution u is stable.
-
(iii)
The multivariate Mittag-Leffler function was initially introduced by Hadid and Luchko [34], who used it for solving linear fractional differential equations with constant coefficients by the operational method. Suthar et al. [35] studied some properties of generalized multivariate Mittag-Leffler function and established two theorems giving the image of this function under certain integral operators. Haubold et al. [36] presented a good survey of the Mittag-Leffler function, generalized Mittag-Leffler functions, Mittag-Leffler type functions, their interesting and useful properties, and applications in certain areas of physical and applied sciences. The Mittag-Leffler function plays an important role in the investigations of the fractional generalization of the kinetic equation, random walks, Lévy flights, superdiffusive transport and in the study of complex models.
Let \(\nu > 0\) and \(x \geq 0\). The incomplete gamma function is defined by
From the recurrence relation [37]
we get
Example 1
Let \(0< a < x < b\). Then the Hadamard-type integro-differential equation
has the solution
in the space \(\operatorname{AC}_{0}[a, b]\). Indeed, it follows from Lemma 2.4 in [2] that
where \(\mu + w > 0\).
By Theorem 2.1,
Applying equation (5),
Thus,
is the solution in the space \(\operatorname{AC}_{0}[a, b]\).
The following theorem shows the uniqueness of equation (1).
Theorem 2.2
Assume that \(f: [a, b] \times R \rightarrow R\) is a continuous function, and there exists a constant C such that
for all \(x \in [a, b]\) and \(y_{1}, y_{2} \in R\). Furthermore,
Then equation (1) has a unique solution in the space \(\operatorname{AC}_{0}[a, b]\) for every \(\mu \in R\).
Proof
Let \(u \in \operatorname{AC}_{0}[a, b]\). Then
as \(u'(\tau ) \in L[a, b]\) and \(f(\tau , u'(\tau )) \in L[a, b]\). Clearly,
Define a mapping T on \(\operatorname{AC}_{0}[a, b]\) by
Using inequality (4), we claim that
Furthermore, \(T(u)\) is absolutely continuous on \([a, b]\) from the proof of Theorem 2.1. Hence, T is a mapping from \(\operatorname{AC}_{0}[a, b]\) to \(\operatorname{AC}_{0}[a, b]\). It remains to prove that T is contractive. Indeed,
Since
we derive
Therefore T is contractive. This completes the proof of Theorem 2.2. □
Example 2
Let \(a = 1\), \(b = e\) and \(\mu = 2\). Then there is a unique solution for the following nonlinear Hadamard-type integro-differential equation:
where the constant C is to be determined.
Clearly, \(C_{2} = e^{2}\) is the maximum value of the function \((\frac{t}{x} )^{2}\) over the interval \([1, e] \times [1, e]\), and the function
is a continuous function from \([1, e] \times R\) to R and satisfies
Obviously \(\log b/a = 1\). By Theorem 2.2, we need to calculate the value
For \(k \geq 1\),
Therefore,
Then, choose a positive C such that
By Theorem 2.2, equation (6) has a unique solution. We note that the series \(\sum_{k = 0}^{\infty }\frac{3^{k}}{k!} \) converges.
3 Conclusions
Using Babenko’s approach and Banach’s contraction principle, we have derived the uniqueness of solutions for the new nonlinear Hadamard-type integro-differential equation for all \(\mu \in R\):
in the Banach space \(\operatorname{AC}_{0}[a, b]\), with two examples given to illustrate the main theorems. The results obtained are fresh in the present studies, and they cannot be achieved via any existing integral transforms or local model methods to the best knowledge of the author.
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References
Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38, 1191–1204 (2001). https://doi.org/10.1016/j.bulsci.2011.12.004
Kilbas, A.A.: Hadamard-type integral equations and fractional calculus operators. Oper. Theory, Adv. Appl. 142, 175–188 (2003)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York (1993)
Babenkos, Y.I.: Heat and Mass Transfer. Khimiya, Leningrad (1986) (in Russian)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Li, C., Clarkson, K.: Babenko’s approach to Abel’s integral equations. Mathematics (2018). https://doi.org/10.3390/math6030032
Li, C., Li, C.P., Clarkson, K.: Several results of fractional differential and integral equations in distribution. Mathematics (2018). https://doi.org/10.3390/math6060097
Li, C., Plowman, H.: Solutions of the generalized Abel’s integral equations of the second kind with variable coefficients. Axioms (2019). https://doi.org/10.3390/axioms8040137
Li, C.: The generalized Abel’s integral equations on \(R^{n}\) with variable coefficients. Fract. Differ. Calc. 10, 129–140 (2020)
Li, C., Huang, J.: Remarks on the linear fractional integro-differential equation with variable coefficients in distribution. Fract. Differ. Calc. 10, 57–77 (2020)
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, New York (1997)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Hadamard, J.: Essai sur l’etude des fonctions donnees par leur developpment de Taylor. J. Math. Pures Appl. 4, 101–186 (1892)
Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387–400 (2002)
Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1–27 (2002)
Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270, 1–15 (2002)
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, 348–360 (2014)
Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities. Springer, Heidelberg (2017)
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). https://doi.org/10.1155/2014/902054
Matar, M.M.: Solution of sequential Hadamard fractional differential equations by variation of parameter technique. Abstr. Appl. Anal. 2018, Article ID 9605353 (2018). https://doi.org/10.1155/2018/9605353
Ding, Y., Jiang, J., O’Regan, D., Xu, J.: Positive solutions for a system of Hadamard-type fractional differential equations with semipositone nonlinearities. Complexity 2020, Article ID 9742418 (2020). https://doi.org/10.1155/2020/9742418
Caputo, M.: Linear model of dissipation whose Q is almost frequency independent. II. Geophys. J. Int. 13, 529–539 (1967)
Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 142 (2012). http://www.advancesindifferenceequations.com/content/2012/1/142
Gambo, Y.Y., Jarad, F., Baleanu, D., Abdeljawad, T.: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014, 10 (2014). http://www.advancesindifferenceequations.com/content/2014/1/10
Adjabi, Y., Jarad, F., Baleanu, D., Abdeljawad, T.: On Cauchy problems with Caputo Hadamard fractional derivatives. J. Comput. Anal. Appl. 21, 661–681 (2016)
Khan, A., Khan, H., Li, T., Akça, H., Khan, T.S.: Common fixed point theorems for weakly compatible self-mappings sustaining integral type contractions. Int. J. Appl. Math. Stat. 57, 43–55 (2018)
Khan, A., Abdeljawad, T., Shatanawi, W., Khan, H.: Fixed point theorems for quadruple self-mappings satisfying integral type inequalities. Filomat 34, 905–917 (2020)
Khan, A., Khan, H., Baleanu, D., Karapinar, E., Khan, T.S.: Fixed points of weakly compatible mappings satisfying a generalized common limit range property. J. Nonlinear Sci. Appl. 10, 5690–5700 (2017)
Abdo, M.S., Abdeljawad, T., Ali, S.M., Shah, K., Jarad, F.: Existence of positive solutions for weighted fractional order differential equations. Chaos Solitons Fractals 141, 110341 (2020)
Abdo, M.S.: Further results on the existence of solutions for generalized fractional quadratic functional integral equations. J. Math. Anal. Model. 1, 33–46 (2020). https://doi.org/10.48185/jmam.v1i1.2
Abdo, M.S., Panchal, S.K., Wahash, H.A.: Ulam–Hyers–Mittag-Leffler stability for a Ψ-Hilfer problem with fractional order and infinite delay. Res. Appl. Math. 7, 100115 (2020). https://doi.org/10.1016/j.rinam.2020.100115
Palve, L.A., Abdo, M.S., Panchal, S.K.: Some existence and stability results of Hilfer–Hadamard fractional implicit differential fractional equation in a weighted space. arXiv preprint (2019). arXiv:1910.08369
Li, C.: Uniqueness of the Hadamard-type integral equations. Adv. Differ. Equ. 2021, 40 (2021). https://doi.org/10.1186/s13662-020-03205-8
Hadid, S.B., Luchko, Y.: An operational method for solving fractional differential equations of an arbitrary real order. Panam. Math. J. 6, 57–73 (1996)
Suthar, D.L., Andualem, M., Debalkie, B.: A study on generalized multivariable Mittag-Leffler function via generalized fractional calculus operators. J. Math. 2019, Article ID 9864737 (2019). https://doi.org/10.1155/2019/9864737
Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. 2011, Article ID 298628 (2011)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products. Academic Press, New York (1980)
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The author is grateful to the three reviewers for their careful reading of the paper with productive comments and suggestions.
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This work is supported by NSERC (Canada 2019-03907).
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Li, C. On the nonlinear Hadamard-type integro-differential equation. Fixed Point Theory Algorithms Sci Eng 2021, 7 (2021). https://doi.org/10.1186/s13663-021-00693-5
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DOI: https://doi.org/10.1186/s13663-021-00693-5