fractal and fractional Article Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces Chenkuan Li 1,* and Hari M. Srivastava 2,3,4,5 1 Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada 2 Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada; [email protected] 3 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 4 Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan 5 Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy * Correspondence: [email protected] Abstract: This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some examples for the illustration of our main theorems. Keywords: Riemann–Liouville fractional integral; Banach’s fixed point theorem; Babenko’s approach; Wright’s generalized Bessel function; multivariate Mittag–Leffler function Citation: Li, C.; Srivastava, H.M. 1. Introduction Uniqueness of Solutions of the Let T > 0. The space L[0, T] is given by Generalized Abel Integral Equations in Banach Spaces. Fractal Fract. 2021, Z T 5, 105. https://doi.org/10.3390/ L[0, T] = u(x) : kuk = ju(x)jdx < ¥ . 0 fractalfract5030105 Clearly, L[0, T] is a Banach space. The product space L[0, T] × L[0, T] (which is also a Academic Editor: Paul Eloe Banach space) is defined as follows: Received: 4 August 2021 L[0, T] × L[0, T] = u(x), v(x) : u(x), v(x) 2 L[0, T] , Accepted: 28 August 2021 Published: 31 August 2021 with the norm given by k(u, v)k = kuk + kvk. Publisher’s Note: MDPI stays neutral a + with regard to jurisdictional claims in The Riemann–Liouville fractional integral I of order a 2 R is defined for the published maps and institutional affil- function u(x) by (see [1,2]): iations. 1 Z x (Iau)(x) = (x − t)a−1u(t)dt. G(a) 0 In particular, we have Copyright: © 2021 by the authors. (I0u)(x) = u(x). Licensee MDPI, Basel, Switzerland. < = ··· This article is an open access article Let 0 5 a0 ai for i 1, 2, , m and a0 5 a. In this paper, we begin to construct an distributed under the terms and explicit solution in L[0, T] to the following Abel’s integral equation by using Babenko’s conditions of the Creative Commons approach and the multivariate Mittag–Leffler function: Attribution (CC BY) license (https:// m creativecommons.org/licenses/by/ a0 ai a I u(x) + ∑ ai I u(x) = I f (x), (1) 4.0/). i=1 Fractal Fract. 2021, 5, 105. https://doi.org/10.3390/fractalfract5030105 https://www.mdpi.com/journal/fractalfract Fractal Fract. 2021, 5, 105 2 of 13 where each ai (i = 1, 2, ··· , m) is a constant and f (x) 2 L[0, T]. We then further investigate the uniqueness of solutions in L[0, T] for the following nonlinear Abel’s integral equation by using Banach’s fixed point theorem: m a0 ai a I u(x) + ∑ ai I u(x) = I g x, u(x) , (2) i=1 where g is a mapping from [0, T] × R to R and satisfies certain conditions. Finally, the sufficient conditions are given for the uniqueness of solutions in the product space L[0, T] × L[0, T] to the associated system given by 8 m a ai a >I 0 u(x) + ai I u(x) = I g1 x, u(x), v(x) > ∑ <> i=1 (3) > m > b0 bi b :>I v(x) + ∑ bi I v(x) = I g2 x, u(x), v(x) , i=1 2 where g1 and g2 are mappings from [0, T] × R to R, 0 5 b0 5 b and b0 < bi for all i = 1, 2, ··· , m. Equations (1)–(3) are new and, to the best of our knowledge, have never been investigated earlier. The single-term (for m = 1) Equation (1) turns out to be a1 u(x) + a1 I u(x) = f (x)(a = a0 = 0), (4) which is the classical Abel’s integral equation of the second kind with the following solution given by Hille and Tamarkin (see, for details [3]; see also [4–6]): Z x a1−1 a1 u(x) = f (x) − a1 (x − t) Ea1,a1 (−a1(x − t) ) f (t)dt, 0 where Ea,b(z) given by ¥ zj E (z) = (a, b > 0), a,b ∑ ( + ) j=0 G aj b is the two-parameter Mittag–Leffler function. The above solution can also be easily deduced by the Laplace transform. Indeed, a1 L(u(x) + a1 I u(x)) = L f (x) = f˜(s) infers that a u˜(s) + 1 u˜(s) = f˜(s). sa1 Hence, we have a ( ) = − 1 ˜( ) u˜ s 1 a f s . a1 + s 1 Using the formula (1.80) from [7] Z ¥ 1 −st a1−1 (− a1 ) = e t Ea1,a1 a1t dt a , 0 a1 + s 1 we arrive at a1−1 a1 u(x) = f (x) − a1x Ea1,a1 (−a1x ) ∗ f (x) Z x a1−1 a1 = f (x) − a1 (x − t) Ea1,a1 (−a1(x − t) ) f (t)dt, 0 Fractal Fract. 2021, 5, 105 3 of 13 where f ∗ y denotes the Laplace convolution given by Z x (f ∗ y)(x) = f(x − t)y(t)dt. 0 On the other hand, Babenko’s approach is a potentially powerful tool for solving dif- ferential, integral and integro-differential equations by treating integral operators like vari- ables. The method itself is similar to the Laplace transform method while dealing with such equations with constant coefficients, but it can be used in other cases as well, such as han- dling integral equations with variable coefficients (see [8,9]). To demonstrate this method, we are going to solve Equation (4) with Babenko’s approach. Clearly, Equation (4) becomes a1 (1 + a1 I )u(x) = f (x). Therefore, we get ¥ a1 −1 k a1 k u(x) = (1 + a1 I ) f (x) = ∑ (−1) (a1 I ) f (x) k=0 ¥ k+1 a1 k+1 = f (x) + ∑ (−1) (a1 I ) f (x) k=0 ¥ 1 Z x = f (x) − a (−a )k (x − t)a1k+a1−1 f (t)dt 1 ∑ 1 ( + ) k=0 G a1k a1 0 Z x ¥ 1 = f (x) − a (x − t)a1−1 (−a )k (x − t)a1k f (t)dt 1 ∑ 1 ( + ) 0 k=0 G a1k a1 Z x a1−1 a1 = f (x) − a1 (x − t) Ea1,a1 (−a1(x − t) ) f (t)dt. 0 We now recall Wright’s generalized Bessel Function j(b, d; z) defined as follows: ¥ zj j(b, d; z) = (b, d > 0). ∑ ( + ) j=0 j! G bj d We also define m Sm(x; z1, ··· , zm; b1, ··· , bm) = (h1 ∗ h2 ∗ · · · ∗ hm)(x), where mk−1 bk hk = hk(x) = x f bk, mk; zkx (x, bk > 0; zk 2 R), m m = ∑ mk, mk > 0. k=1 Let m Gm(x; g1, ··· , gm; b1, ··· , bm) Z ¥ −t m = e Sm(x; g1t, ··· , ymt; b1, ··· , bm)dt, 0 and m−a0 wm(x) = Gm (x; −a1, ··· , −am; a1 − a0, ··· , am − a0)(m > a0). Fractal Fract. 2021, 5, 105 4 of 13 In 2013, Pskhu [10] constructed an explicit solution for the following Abel’s integral equation which is a special case of the Equation (1): m a0 ai a0 I u(x) + ∑ ai I u(x) = I f (x)( f 2 L[0, T]), i=1 as follows: m u(x) = D0,x( f ∗ wm)(x)(m > a0), where the solution u(x) is independent of the parameter m and 8 1 Z x > f(t)(x − t)−m−1 dt = I−mf(x)(m < 0) > G(−m) > 0 > m < D0,xf(x) := f(x)(m = 0) > > > n > d n−m : fI f(x)g (n − 1 < m n; n 2 N), dxn 5 N being the set of positive integers. We would also like to add that Gorenflo and Luchko [11] established an explicit solution to the following generalized Abel integral equation of the second kind, which was based on a modification of the Mikusi´nskioperational calculus and the Mittag–Leffler function of several variables (see, for details, [12]): m ajm u(x) − ∑ aj I u(x) = f (x)(aj > 0; m = 1; m > 0; x > 0), j=1 which is also a special case of Equation (1). There are many analytic and numerical studies on Abel’s integral equation and its variants in distribution, as well as the existence and uniqueness of the corresponding solu- tions by using fixed point theorems [7,8,13–15]. For example, Brunner et al. [16] considered numerical solutions of Abel’s integral equation of the second kind: Z x u(x) = f (x) + (x − t)−a kx, t, u(t)dt (x 2 [0, T]), 0 where 0 < a < 1 and f 2 C[0, T], and the kernel k is continuous on S × R, with S = f(t, s) : 0 5 s 5 t 5 Tg, and satisfies the Lipschitz conditions in the third argument. The multivariate Mittag–Leffler function was studied by (among others) Hadid and Luchko [17] for solving linear fractional differential equations with constant coefficients by applying the operational calculus: E (z ··· z ) (a1,··· ,am),b 1, , m ¥ k1 km k z ··· zm = ∑ ∑ 1 , k1, ··· , km G(a1k1 + ··· + amkm + b) k=0 k1+···+km=k where ai > 0 (i = 1, 2, ··· , m) and b > 0.