Abstract
The goal of this paper is to study the uniqueness of solutions of several Hadamard-type integral equations and a related coupled system in Banach spaces. The results obtained are new and based on Babenko’s approach and Banach’s contraction principle. We also present several examples for illustration of the main theorems.
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1 Introduction
The Hadamard-type fractional integral of order \(\alpha > 0\) for a function u is defined in [1, 2] as
where \(\log ( \cdot ) = \log _{e} (\cdot )\), \(0 < a < x < b\), and \(\mu \in R\). The corresponding derivative is given by
where \(n = [\alpha ] + 1\), \([\alpha ]\) being the integral part of α. When \(\mu = 0\), they take the forms
respectively. In particular, for \(\alpha = 1\),
which leads to definition of the space \({X}_{\mu }(a, b)\) of Lebesgue-measurable functions u on \([a, b]\) for which \(x^{\mu - 1} u(x)\) is absolutely integrable [2]:
Clearly, for \(a > 0\),
for every \(\mu \in R\). Hence \(X_{\mu }(a, b)\) is a Banach space, since \(L(a, b)\) with the norm
is complete and the norms \(\lVert u \rVert _{X_{\mu }}\) and \(\lVert u \rVert _{L} \) are equivalent.
We need the following lemmas shown by Kilbas [2].
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 \(X\mu (a, b)\), and for \(u \in X\mu (a, b)\),
where
Lemma 1.2
If \(\alpha >0\), \(\beta > 0\), \(\mu \in R\), and \(u \in X_{\mu }(a, b)\), then the semigroup property holds:
There are a lot of studies on fractional differential and integral equations involving Riemann–Liouville or Caputo operators with boundary value problems or initial conditions [3–11]. Li and Sarwar [12] considered the existence of solutions for the following fractional-order initial value problems:
where \(0 < \alpha < 1\), and \({}_{C}D_{0, t}^{\alpha }\) is the Caputo derivative.
Wu et al. [13] studied the existence and uniqueness of solutions by fixed point theory for the following fractional differential equation with nonlinearity depending on fractional derivatives of lower order on an infinite interval:
where \(2 < \alpha \leq 3\), \(D_{0 +}^{\alpha }\), \(D_{0 +}^{\alpha - 1}\), and \(D_{0 +}^{\alpha - 2}\) are all Riemann–Liouville fractional derivatives.
Ahmad and Ntouyas [14] considered a coupled system of Hadamard-type fractional differential equations and integral boundary conditions
where \(\gamma > 0\), \(1 < \sigma _{1} < e\), \(1 < \sigma _{2} < e\), and \(w_{1}, w_{2}: [1, e] \times R \times R \rightarrow R\) are continuous functions satisfying certain conditions. They showed the existence of solutions by Leray–Schauder’s alternative and the uniqueness by Banach’s fixed point theorem, based on the fact that for \(1 < q \leq 2\) and \(z \in C([1, e], R)\), the problem
has a unique solution
where
Let \(g: [a, b] \times R \rightarrow R\) be a continuous function. In this paper, we study the following nonlinear Hadamard-type (μ is arbitrary in R) integral equation in the space \(X_{\mu }(a, b)\):
where \(\alpha _{n} > \alpha _{n - 1} > \cdots > \alpha _{1} > 0\), and \(a_{i}\), \(i = 1, 2,\ldots,n\), are complex numbers, not all zero.
To the best of the author’s knowledge, equation (1) is new in the framework of Hadamard-type integral equations. First, by Babenko’s approach we will construct the solution as a convergent infinite series in \(X_{\mu }(a, b)\) for the integral equation
where \(f \in X_{\mu }(a, b)\). Then we will show that there exists a unique solution for equation (1) using Banach’s contraction principle. Furthermore, we present the solution for the Hadamard-type integral equation
by the Hadamard fractional derivative and show the uniqueness for the coupled system of integral equations
where \(\alpha _{n} > \alpha _{n - 1} > \cdots > \alpha _{1} > 0\), \(\beta _{n} > \beta _{n - 1} > \cdots > \beta _{1} > 0\), and there exist at least one nonzero \(a_{i}\) and one nonzero \(b_{j}\) for some \(1 \leq i, j \leq n\). We also present several examples for illustration of our results.
2 Main results
We begin by showing the solution for equation (2) as a convergent series in the space \(X_{\mu }(a, b)\) by Babenko’s approach [15], which is a powerful tool 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 [16, 17], such as solving integral or fractional differential equations with distributions whose Laplace transforms do not exist in the classical sense. Clearly, it is always necessary to show the convergence of the series obtained as solutions. Podlubny [16] also provided interesting applications to solving certain partial differential equations for heat and mass transfer by Babenko’s method. Recently, Li and Plowman [18] and Li [19] studied the generalized Abel’s integral equations of the second kind with variable coefficients by Babenko’s technique.
Theorem 2.1
Let \(f \in X_{\mu }(a, b)\) with \(0 < a < b < \infty \). Then equation (2) has a unique solution in the space \(X_{\mu }(a, b)\),
where \(\alpha _{n} > \cdots > \alpha _{1} > 0\), and \(a_{i}\), \(i = 1, 2,\ldots, n\), are complex numbers, not all zero.
Proof
Equation (2) can be written as
By Babenko’s method we arrive at
using Lemma 1.2 and the multinomial theorem. Note that
It remains to show that series (5) converges in the space \(X_{\mu }(a, b)\). By Lemma 1.1
where
Therefore
where
is the value of the multivariate Mittag-Leffler function \(E_{(\alpha _{1},\ldots, \alpha _{n}, 1)}(z_{1},\ldots, z_{n})\) given in [7] at
Thus \(u \in X_{\mu }(a, b)\), and the series on the right-hand of equation (5) is convergent.
To verify that the series is a solution, we substitute it into the left-hand side of equation (2):
as
by cancelation. Note that all series are absolutely convergent and the term rearrangements are feasible for cancelation.
Clearly, the uniqueness immediately follows from the fact that the integral equation
only has zero solution by Babenko’s method. This completes the proof of Theorem 2.1. □
Let \(\nu > 0\) and \(x \geq 0\). The incomplete gamma function is defined by
From the recurrence relation [20]
we get
Example 1
Let \(0< a < x < b\). Then the Hadamard-type integral equation
has the solution
Indeed, it follows from Lemma 2.4 in [2] that
where \(\mu + w > 0\).
By Theorem 2.1
Applying equation (6), we have
Thus
is a solution in the space \(X_{\mu }(a, b)\).
The following theorem shows the uniqueness of solution of equation (1).
Theorem 2.2
Let \(g: [a, b] \times R \rightarrow R\) be a continuous function and suppose that there exists a constant \(C > 0\) such that for all \(x \in [a, b]\),
Furthermore, suppose that
Then equation (1) has a unique solution in the space \(X_{\mu }(a, b)\) for every \(\mu \in R\).
Proof
Let \(u \in X_{\mu }(a, b)\). Then \(g (x, u(x)) \in X_{\mu }(a, b)\) since
by noting that \(g(x, 0)\) is a continuous function on \([a, b]\). Define the mapping T on \(X_{\mu }(a, b)\) by
In particular, for \(k = 0\),
From the proof of Theorem 2.1 we have
Clearly,
Hence T is a mapping from \(X_{\mu }(a, b)\) to \(X_{\mu }(a, b)\). It remains to prove that T is contractive. We have
Since
we derive
Therefore T is contractive. This completes the proof of Theorem 2.2. □
Example 2
Let \(a = 1\), \(b = e\), and \(\mu \in R\). Then for every \(\mu \in R\), there is a unique solution for the following Hadamard-type integral equation:
Clearly, the function
is a continuous function from \([1, e] \times R\) to R and satisfies
Obviously, \(a_{2} = a_{1} = 1\), and \(\log b/a = 1\). By Theorem 2.2 we need to calculate the value
For \(k \geq 1\) and \(j \geq 0\), we have
Therefore
Then
By Theorem 2.2 equation (7) has a unique solution.
Remark 1
There are algorithms for computation of the Mittag-Leffler function [21]
and its derivative. In particular,
where \(0 < \alpha \leq 1\), \(\beta \in R\), \(\vert \arg z \vert > \pi \alpha \), \(z \neq 0\).
The Mittag-Leffler function is widely used in studying fractional differential equations and fractional calculus. Li [22] studied three classes of fractional oscillators and obtained the solutions of the first class in terms of the Mittag-Leffler function.
Define the product space \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) with the norm
Clearly, \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) is a Banach space.
Now we can extend Theorem 2.2 to the coupled system of the Hadamard-type integral equations given by (4).
Theorem 2.3
Let \(g_{1}, g_{2}: [a, b] \times R \times R \rightarrow R\) be continuous functions and suppose that there exist nonnegative constants \(C_{i}\), \(i = 1, 2, 3, 4\), such that for all \(x \in [a, b]\) and \(u_{i}, v_{i} \in R\), \(i = 1, 2\),
Furthermore, suppose that
Then system (4) has a unique solution in the product space \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) for every \(\mu \in R\).
Proof
Let \(u, v \in X_{\mu }(a, b)\). Then \(g_{1}(x, u(x), v(x)), g_{2}(x, u(x), v(x)) \in X_{\mu }(a, b)\) since
by noting that \(g_{1}(x, 0, 0)\) is a continuous function on \([a, b]\). Furthermore,
for every \(\mu \in R\).
Define the mapping T on \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) by
where
and
Clearly, from the proof of Theorem 2.2 we have
and
Hence
which implies that T maps the Banach space \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) into itself. It remains to show that T is contractive. Indeed,
and
Thus
where \(q < 1\) by assumption. By Banach’s contractive principle system (4) has a unique solution in the space \(X_{\mu }(a, b) \times X_{\mu }(a, b)\). This completes the proof of Theorem 2.3. □
Let \(\operatorname{AC}[a, b]\) be the set of absolutely continuous functions on \([a, b]\), which coincides with the space of primitives of Lebesgue-measurable functions [3]:
Clearly, if \(f \in \operatorname{AC}[a, b]\) with \(0 < a < b < \infty \), then \(x^{\mu } f(x) \in \operatorname{AC}[a, b]\) since \(x^{\mu }\in \operatorname{AC}[a, b]\).
The following results are from Lemma 2.3 and Theorem 5.5(a) in [2].
-
(i)
If \(\alpha > \beta > 0\) and \(\mu \in R\), then for \(u \in X_{\mu }(a, b)\),
$$ \mathcal{D}^{\beta }_{a + , \mu } \mathcal{J}^{\alpha }_{a + , \mu }u = \mathcal{J}^{\alpha - \beta }_{a + , \mu }u. $$ -
(ii)
If \(\alpha > 0\) and \(u \in X_{\mu }(a, b)\), then
$$ \mathcal{D}^{\alpha }_{a + , \mu } \mathcal{J}^{\alpha }_{a + , \mu }u = u. $$
Theorem 2.4
Let \(\alpha _{n} > \cdots > \alpha _{1} > \alpha _{0}\) with \(0 < \alpha _{0} < 1\), and let \(f \in \operatorname{AC}[a, b]\). In addition, let \(a_{i}\), \(i = 1, 2,\ldots, n\), be complex numbers, not all zero. Then equation (3) has a unique solution in the space \(X_{\mu }(a, b)\),
Proof
It follows from Theorem 5.3 in [2] that
where \(f_{0}(x) = x^{\mu }f(x) \in \operatorname{AC}[a, b]\). We first claim that \((\mathcal{D}^{\alpha _{0}}_{a + , \mu }f)(x) \in X_{\mu }(a, b)\). Indeed,
Similarly,
by noting that \(f_{0}'(t) \in L[a, b]\) and
where
For \(u \in X_{\mu }(a, b)\), equation (3) turns out to be
by applying the fractional differential operator \(\mathcal{D}^{\alpha _{0}}_{a + , \mu }\) to both sides. Then by Theorem 2.1 we have
To remove the differential operator \(\mathcal{D}^{\alpha _{0}}_{a + , \mu }\), we compute the Hadamard-type fractional integral of order \(\alpha >0\) for the first term in \((\mathcal{D}^{\alpha _{0}}_{a + , \mu }f)(x)\):
Making the change of variable
we get
where B denotes the beta function. Hence
The second term in \((\mathcal{D}^{\alpha _{0}}_{a + , \mu }f)(x)\) is
Therefore the solution immediately follows by substituting equations (9) and (10) into equation (8). This completes the proof of Theorem 2.4. □
Remark 2
It seems impossible to deal with the case \(\alpha _{0} \geq 1\) along the same lines as \(\mathcal{D}^{\alpha _{0}}_{a + , \mu }f \notin X_{\mu }(a, b)\) for \(f \in \operatorname{AC}[a, b]\). Furthermore, \(\mathcal{D}^{\alpha _{0}}_{a + , \mu }\) is not a bounded operator on \(\operatorname{AC}[a, b]\). The single-term Hadamard-type integral equation
was studied in [2] with the necessary and sufficient conditions given in Theorem 3.1.
3 Conclusions
Using Babenko’s approach and Banach’s contraction principle, we have derived the uniqueness of solution for several Hadamard-type integral equations and related coupled system. The results obtained are new in the present configuration of integral equations.
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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)
Li, C.: Several results of fractional derivatives in \(D'(R_{+})\). Fract. Calc. Appl. Anal. 18, 192–207 (2015)
Li, C., Clarkson, K.: Babenko’s approach to Abel’s integral equations. Mathematics (2018). https://doi.org/10.3390/math6030032
Li, C., Huang, J.: Remarks on the linear fractional integro-differential equation with variable coefficients in distribution. Fract. Differ. Calc. 10, 57–77 (2020)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Marasi, H., Piri, H., Aydi, H.: Existence and multiplicity of solutions for nonlinear fractional differential equations. J. Nonlinear Sci. Appl. 9, 4639–4646 (2016)
Zhang, L., Ahmad, B., Wang, G., Agarwal, R.P.: Nonlinear fractional integrodifferential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 249, 51–56 (2013)
Ahmad, B., Nieto, J.J.: Existence results for nonlinear boundary value problems of fractional integro differential equations with integral boundary conditions. Bound. Value Probl. 2009, Article ID 708576 (2009)
Ahmad, B., Ntouyas, S.K.: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 15, 362–382 (2012)
Li, C., Sarwar, S.: Existence and continuation of solutions for Caputo type fractional differential equations. Electron. J. Differ. Equ. 2016, Article ID 207 (2016)
Wu, J., Zhang, X., Liu, L., Wu, Y., Cui, Y.: Convergence analysis of iterative scheme and error estimation of positive solution for a fractional differential equation. Math. Model. Anal. 23, 611–626 (2018)
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)
Babenko, Y.I.: Heat and Mass Transfer. Khimiya, Leningrad (1986) (in Russian)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Li, C., Li, C.P., Clarkson, K.: Several results of fractional differential and integral equations in distribution. Mathematics 6(6), 97 (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)
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products. Academic Press, New York (1980)
Gorenflo, R., Loutchko, J., Luchko, Y.: Computation of the Mittag-Leffler function \(E_{\alpha , \beta }(z)\) and its derivative. Fract. Calc. Appl. Anal. 5, 491–518 (2002)
Li, M.: Three classes of fractional oscillators. Symmetry 10, 40 (2018). https://doi.org/10.3390/sym10020040
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Li, C. Uniqueness of the Hadamard-type integral equations. Adv Differ Equ 2021, 40 (2021). https://doi.org/10.1186/s13662-020-03205-8
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DOI: https://doi.org/10.1186/s13662-020-03205-8