\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 01, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/01\hfil Fractional regularization of operator equations] {Singular regularization of operator equations in $L_1$ spaces via fractional differential equations} \author[G. L. Karakostas, I. K. Purnaras \hfil EJDE-2016/01\hfilneg] {George L. Karakostas, Ioannis K. Purnaras} \address{George L. Karakostas \newline Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece} \email{gkarako@uoi.gr} \address{Ioannis K. Purnaras \newline Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece} \email{ipurnara@uoi.gr} \thanks{Submitted June 8, 2015. Published January 4, 2016.} \subjclass[2010]{34K35, 34A08, 47045, 65J20} \keywords{Causal operator equations; fractional differential equations; \hfill\break\indent regularization; Banach space} \begin{abstract} An abstract causal operator equation $y=Ay$ defined on a space of the form $L_1([0,\tau],X)$, with $X$ a Banach space, is regularized by the fractional differential equation $$ \varepsilon(D_0^{\alpha}y_{\varepsilon})(t) =-y_{\varepsilon}(t)+(Ay_{\varepsilon})(t), \quad t\in[0,\tau], $$ where $D_0^{\alpha}$ denotes the (left) Riemann-Liouville derivative of order $\alpha\in(0,1)$. The main procedure lies on properties of the Mittag-Leffler function combined with some facts from convolution theory. Our results complete relative ones that have appeared in the literature; see, e.g. \cite{cord} in which regularization via ordinary differential equations is used. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Regularization employs several techniques in order to approximate solutions of ill-posed problems such as \begin{equation}\label{ia} My = f, \end{equation} where $M$ is an operator acting on a space $X$ and taking values in another space $Y$. Basically, the problem is characterized as an ill-posed problem, if either solutions do not exist for some $f$, or uniqueness of solutions is not guaranteed, or continuous dependence on data does not hold. The latter is equivalent to saying that there is no continuous inverse of $M$. In order to solve an ill- posed problem (approximately), we should regularize it, namely, replace this problem by a suitable family of well-posed problems whose solutions approximate (in some sense) the solution of the ill-posed problem which we look for. However, it is not true that such a process may produce an approximation of the solutions of the original equation for all situations. To see it, we borrow an example from the literature (e.g., \cite{lin, mar}) adopted to our situation, as follows: Consider the $2\times 2$ matrix-operator $M$ and the function $f$ given by \[ M:=\begin{bmatrix} \frac{d}{dt}& -1\\ 1& 0 \end{bmatrix}\quad \text{and}\quad f(t):=\begin{bmatrix} 0\\ p(t) \end{bmatrix}, \] where $p$ is a differentiable function on $[0,1]$, say. The exact solution of the operator equation \eqref{ia} in the space $C^{1}([0,1],\mathbb{R})\times C([0,1],\mathbb{R})$ is given by $$ x(t)=p(t),\quad y(t)=p'(t),\quad t\in[0,1]. $$ Take a small number $\varepsilon$ and let \[ f_{\varepsilon}(t):=f(t)+\begin{bmatrix} 0\\ {\varepsilon}\sin(t/\varepsilon^2) \end{bmatrix} \] be a small perturbation of $f$. Then we obtain the exact solution $$ x_{\varepsilon}(t)=p(t)+{\varepsilon}\sin(t/\varepsilon^2),\quad y_{\varepsilon}(t)=p'(t)+\frac{1}{\varepsilon}\cos(t/\varepsilon^2). $$ Hence the quantity \[ \begin{bmatrix} x_{\varepsilon}(t)\\ y_{\varepsilon}(t) \end{bmatrix} -\begin{bmatrix} x(t)\\ y(t) \end{bmatrix}= \begin{bmatrix} {\varepsilon}\sin(t/\varepsilon^2)\\ \frac{1}{\varepsilon}\cos(t/\varepsilon^2) \end{bmatrix} \] becomes large enough if the number $\varepsilon$ tends to 0. This means that the solution changes a lot after a small change in the right side of equation. In case that $M$ is a compact linear operator between two Hilbert spaces, a regularizing form should consist of the equation \begin{equation}\label{i3} (M^*M+\varepsilon)x_{\varepsilon}=M^*f, \end{equation} where $M^*$ is the adjoint of $M$, see \cite{cwg}. In \cite{he} the regularization \eqref{i3} has its right side $M^*f_{\delta}$, where $f_{\delta}$ is a (noisy) approximation of $f$. The works \cite{neu, amw} refer to \emph{Tikhonov-regularization}, i.e. regularization of minimazing problems. According to such problems, an equation of the form \begin{equation}\label{i1} \int_a^bk(t,s)x(s)ds=f(t) \end{equation} is replaced by the equation $$ \int_a^bk(t,s)x_{\varepsilon}(s)ds+\varepsilon x_{\varepsilon}(t)=f(t), $$ or the equation $$ \int_a^bk(t,s)x_{\varepsilon}(s)ds+\varepsilon x_{\varepsilon}(t)=f_{\delta}(t), $$ and then one looks for the convergence of the net $x_{\varepsilon}$. Here a noisy $f_{\delta}$ replaces $f$, for small $\delta$; see, e.g., the interesting survey presented in \cite{lamm}. Approximation of the kernel $k$ of \eqref{i1} is used by other authors, see, e.g., \cite{men}. Approximation of both the perturbation and the operator applies elsewhere, \cite{gg}. Some authors, as, e.g. \cite{bll}, dealing with the Volterra equation \begin{equation}\label{i2} \int_0^tk(t,s)x(s)ds=f(t)\,, \end{equation} apply the so called method of \emph{the simplified (or Lavrentiev) regularization}, consisting of an approximation of the perturbation $f$ and the \emph{local regularization}, realized by an approximate equation of the form $$ \int_t^{t+\varepsilon}k(t+\varepsilon,s)x(s)ds +\int_0^tk(t+\varepsilon,s)x(s)ds=f(t+\varepsilon), $$ where $\varepsilon$ is a parameter tending to 0. In \cite{sav} another approach is applied to \eqref{i1} by taking an approximation of both the kernel $k$ and the output $f$. For a more general setting see, also, \cite{sav1}. Regularization of abstract equations of the form \eqref{ia} can be realized by approximating the output $f$, as, e.g. in \cite{hls} and for Fredholm integral equations, as, e.g., in \cite{ww}. Regularization of the Hammerstein's type equation $x+BAx =f$, is achieved, (see, e.g., \cite{pop}) by replacing it with the equation $x_{\varepsilon}+(B+\varepsilon J)(A+\varepsilon J )x_{\varepsilon}=f_{\delta}$, where $\varepsilon, \delta$ are positive reals tending to 0 and the functions $f, f_{\delta}$ are such that $\|f-f_{\delta}\|\leq\delta$. Here $A$ and $B$ are operators, and $x, f$ are elements in a given Banach space $X$, with $x$ being the unknown element in $X$. In case that the operator $M$ has the form $My=Ay-y+f$, the problem \eqref{ia} leads to the fixed point problem \begin{equation}\label{i4} y=Ay. \end{equation} It is known (see, e.g., \cite[p. 89]{cord}) that a continuous compact operator $A$ (in the sense of Krasnoselskii) defined on a locally convex Hausdorff space has a fixed point. Regularization theory of such an equation (especially), when $A$ is a monotone or a non-expansive operator defined in a Hilbert or (even in a) Banach space, forms a large field, and most of the authors make use of variation techniques, see, e.g. \cite{a3, a2, a1, kar4, a4} and the references therein. In case \eqref{i4} refers to a space of functions $y:[0,1]\to\mathbb{R}$, say, namely we have \begin{equation}\label{i5} y(t)=(Ay)(t), \quad t\in[0,1], \end{equation} regularization is achieved by a differential equation of the form \begin{equation}\label{i6} \varepsilon\frac{d}{dt}y(t)+y(t)-(Ay)(t)=0. \end{equation} This is done elsewhere (see, e.g., the book \cite[p. 140]{cord}, and the references therein), when $y$ has to be a continuous function, say, $y\in C([0,T],\mathbb{R})$. Similar things occur for a neutral differential equation discussed in \cite{gh}. An immediate consequence of this approach is that, in this case, a solution of \eqref{i5} is approximated by a sequence $(y_{\varepsilon_n})$ of real-valued functions having continuous first order derivatives. For fractional differential equations a few results, analogous to above, are known. We should refer to the problem $$ D_0^{\alpha}(x-x(0)-\varepsilon)=f(t,x)+\varepsilon, \quad x(0)=x_0+\varepsilon, $$ discussed in \cite{lak}, where conditions are given so that, as $\varepsilon$ tends to 0, the maximal solution $\eta(t;\varepsilon)$ tends to the maximal solution $\eta(t)$ of the problem $$ D_0^{\alpha}(x-x(0))=f(t,x), \quad x(0)=x_0, $$ uniformly on any compact interval $[0,t_1]$ of the domain of $\eta$. In this work we assume that $A$ is defined on an $L_1$-space of $X$-valued functions, where $X$ is a Banach space, and we regularize \eqref{i5} by an equation involving continuous functions with Lebesgue-integrable first order derivatives. To succeed in such an approach we work in $L_1$-spaces and use the fractional equation \begin{equation}\label{e01} \varepsilon(D_0^{\alpha}y_{\varepsilon})(t) =-y_{\varepsilon}(t)+(Ay_{\varepsilon})(t), \quad\text{a.a.}\quad t\in[0,\tau]:=I_{\tau}, \end{equation} for $\varepsilon$ tending to 0. Here, $D_0^{\alpha}y_{\varepsilon}$ is the (left) Riemann-Liouville derivative of $f$ of order $\alpha$. A central role to our approach is played by some facts from convolution theory, as well as the Mittag-Leffler function. It is known that the relation of the latter with the fractional calculus, is analogous of that of the exponential function with standard calculus. See, for instance, \cite[subsection 3.2]{jum}. We investigate when, for some $\tau\in(0,T]$, there is a sequence of solutions of the fractional differential equation \eqref{e01} converging in the sense of $L_1$-norm on $[0,\tau]$ to solutions of equation \eqref{i5}, when the parameter $\varepsilon$ approaches 0. \section{Preliminaries} \subsection{Fractional calculus} Throughout this paper we shall work on a real Banach space $X$ endowed with a norm $\|\cdot\|_X$, and on the space $L_1^T:=L_1([0,T],X)$, for some $T>0$ fixed, with norm $$ \|y\|_1^{\tau}:=\int_0^{\tau}\|y(s)\|_Xds. $$ Several books in the literature present surveys on the classical fractional calculus. Two exhaustive such books are the ones by Podlubny \cite{po} and Miller and Ross \cite{miro}. We recall some basic definitions and results adopted for our purposes, namely we consider the meaning of fractional derivative and integral on an $X$-valued function defined on the interval $[0,T]$. Let $\Gamma$ be the Euler Gamma function. It is well known (see, e.g., \cite{wiki}) that on the positive real axis the function $\Gamma$ admits a local minimum $0.885603..$. at $x_{\rm min}=1.461632144..$. and it is increasing for $x>x_{\rm min}$. Later on we shall use the monotonicity of $\Gamma$ on the interval $[2,+\infty)$. For $u\in L_1^T$ and $\alpha\in(0,1)$, the (left) fractional Riemann-Liouville derivative of $f$ of order $\alpha$, is defined by $$ (D_0^{\alpha}u)(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt} \int_0^t(t-s)^{-\alpha}u(s)ds, $$ where the integral is in the Bohner sense. As in \cite{po}, [pp. 59-73, and relation (2.122)], we can see that the first composition formula with integer order $n$ derivative holds\footnote{The relation holds even for $\alpha<0.$}: \begin{equation}\label{10} D_0^{\alpha}(u^{(n)})(t)=D_0^{\alpha+n}u(t) -\sum_{j=0}^{n-1}\frac{u^{(j)}(0)t^j}{\Gamma(j+1)}. \end{equation} Now consider the problem \begin{equation}\label{eq1} (D_0^{\alpha}u)(t)=f(t), \quad \text{a.a. } t\in[0,T], \quad (D_0^{\alpha-1}u)(t)\Big|_{t=0}=b, \end{equation} where $b\in X$. Although the following result can be implied from arguments borrowed from the literature (see, e.g., \cite{po} Theorem 3.1, p. 122 and relation (3.7) in p. 123), we shall give our proof for two reasons: First we want this work to be complete. Second, the functions used here take values in the abstract Banach space $X$ and not in $\mathbb{R}$, as it is used elsewhere (and in \cite[Theorem 3.1]{po}). Let $B$ be the (real) Betta function, namely the function defined for $\rho, \sigma>0$ by $$ B(\rho,\sigma)=\int_0^1(1-\theta)^{\rho-1}\theta^{\sigma-1}d\theta $$ This is connected with the Gamma function by the relation $$ B(\rho,\sigma)=\frac{\Gamma(\rho)\Gamma(\sigma)}{\Gamma(\rho+\sigma)}. $$ \begin{lemma}\label{l2} The function $y$ defined by $$ y(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}b +\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}f(s)ds,\quad\text{a.a. } t\in[0,T], $$ is the only solution of the problem \eqref{eq1}. \end{lemma} \begin{proof} We show that $y$ satisfies the problem \eqref{eq1}. We have \begin{equation}\begin{aligned} (D_0^{\alpha}y)(t) &=\frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)}\frac{d}{dt} \int_0^t{(t-s)^{-\alpha}}s^{\alpha-1}dsb\\ &\quad +\frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)} \frac{d}{dt}\int_0^t{(t-s)^{-\alpha}}\int_0^s(s-r)^{\alpha-1}f(r)\,dr\,ds\\ &=\frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)}\frac{d}{dt}B(1-\alpha,\alpha)b\\ &\quad +\frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)} \frac{d}{dt}\int_0^t{(t-s)^{-\alpha}}\int_r^t(s-r)^{\alpha-1}f(r)dsdr\\ &=\frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)} \frac{d}{dt}\int_0^tf(r)drB(1-\alpha,\alpha)\\ &=\frac{d}{dt}\int_0^tf(r)dr=f(t), \quad\text{a.e.}, \end{aligned} \end{equation} where, in the integration, we used the substitution $s=:(1-\theta)r+\theta t, \quad \theta\in[0,1]$. Similarly we obtain $$ (D_0^{\alpha-1}y)(t)\Big|_{t=0}=\frac{d}{dt}(t) \Big|_{t=0}b+\frac{d}{dt}\int_0^t(t-r)f(r)dr\Big|_{t=0}=b. $$ The inverse is implied by an application \cite[Theorem 3.1, p.122]{po}. \end{proof} \subsection{The Mittag-Leffler function} \label{p1} The Mittag-Leffler function of order $\alpha (>0)$ is defined on the complex plane by $$ E_{\alpha}(z):=\sum_0^{\infty}\frac{z^j}{\Gamma(ja+1)}\,. $$ From a result of Feller referred by Pollard \cite{poll}, we know that there is a nondecreasing and bounded function $F_{\alpha}$ such that \begin{equation}\label{r2} E_{\alpha}(-x)=\int_0^{+\infty}e^{-xs}dF_{\alpha}(s), \quad x\geq 0. \end{equation} It follows that this function is positive, non-increasing, it tends to 0 as $x\to+\infty$ and since $E_{\alpha}(0)=1$, the quantity $E_{\alpha}(-x)$ is not greater than 1. More properties of this function and of some generalizations of it can be found in \cite{po}. \section{Main results} Let $A: L_1^T\to L_1^T$ be a causal operator, namely, it satisfies $(Ax)(t)=(Ay)(t)$, whenever $x(s)=y(s)$, for a.a. $ s\in[0,t]$, (for the continuous case see, e.g., \cite{kar3}, \cite{new} and the references therein). This characteristic guarantees that, for any $\tau\in(0,T]$, the operator $A$ maps the ball $$ B_{\tau}^r:=\{y\in L_1^{\tau}: \|y\|_1^{\tau}0$ it holds $$ A(\overline{B_{\tau}^r})\subseteq \overline{B_{\tau}^r}, $$ the following Schauder's fixed point theorem applies and ensures the existence of a fixed point of $A$ in $\overline{B_{\tau}^r}$. \begin{theorem}[{\cite[p. 89]{cord}}] \label{t1} Let $E$ be a real Banach space and $K\subset E$ a closed, bounded and convex set. If $C:K\to K$ is a continuous compact operator, then $C$ has at least one fixed point. \end{theorem} Now, for any fixed $\varepsilon>0$ and small enough, say $\varepsilon<1$, consider the fractional differential equation \begin{equation}\label{e1} \varepsilon(D_0^{\alpha}y)(t)=-y(t)+(Ay)(t), \quad \text{a.a. }\ t\in[0,T], \end{equation} where the derivative $D_0^{\alpha}y$ is in the sense of Riemann-Liouville and $\alpha\in(0,1)$. Let $b$ be a (nonzero) real number and consider the initial value problem $$ (D_0^{\alpha}y)(t)=-\frac{1}{\varepsilon}y(t)+\frac{1}{\varepsilon}(Ay)(t), \quad (D_0^{\alpha-1}y)(t)\Big|_{t=0}=b. $$ According to Lemma \ref{l2}, a function $y$ is a solution of the problem, if and only if it satisfies the equation \begin{equation}\label{a1} y(t)=\frac{b}{\Gamma(\alpha)}t^{\alpha-1} -\frac{1}{\varepsilon\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}y(s)ds +\frac{1}{\varepsilon\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}(Ay)(s)ds. \end{equation} Our main result in this work is given in the following theorem: \begin{theorem}\label{t2} If $A$ is a causal, compact and continuous operator on $L_1^T$, then, there exists a certain $\tau\in(0,T]$, such that, for any sequence $(\varepsilon_n)$ converging to 0, there is a sequence of solutions $(y_{n})$ of equation \eqref{a1} converging in the $L_1^{\tau}$-sense to a solution $y$ of equation $$ y(t)=(Ay)(t), \quad\text{a.a } t\in [0,\tau]. $$ \end{theorem} The proof of the above theorem will be given in the last section. It is noteworthy that the theorem has several interesting consequences, as the following one. \begin{corollary} \label{coro3.3} Let $k$ be a positive integer, $W$ a continuous and causal operator defined on the $C^{k}([0,T],X)$-space and let $\alpha\in(0,1)$. Then, there exists a certain $\tau\in(0,T]$ such that, for any sequence $(\varepsilon_n)$ converging to 0, there is a sequence of solutions $(x_{n})$ of the problem \begin{gather}\label{es1} \varepsilon(D_0^{k+\alpha}x)(t)=-x^{(k)}(t)+(Wx)(t), \quad\text{a.a. } t\in[0,\tau], \\ x^{(j)}(0)=0,\quad j=0, 1, \dots, k-1, \quad (D_0^{k+\alpha-1}x)(t)\Big|_{t=0}=b, \nonumber \end{gather} converging, in the sup-norm $\|\cdot\|^{\tau}_{\infty}$ sense, to a solution of the problem \begin{gather*} x^{(k)}(t)=(Wx)(t)\\ x^{(j)}(0)=0,\quad j=0, 1, \dots, k-1. \end{gather*} \end{corollary} \begin{proof} Set $y=x^{(k)}$. Then, due to \eqref{10}, we have $$ (D_{0}^{\alpha}y)(t)=(D_0^{k+\alpha}x)(t)\quad\text{and}\quad (D_{0}^{\alpha-1}y)(t)\big|_{t=0}=(D_0^{k+\alpha-1}x)(t)\Big|_{t=0}=b $$ and, moreover, $$ x(t)= \int_0^t\frac{(t-s)^{k-1}}{(k-1)!}y(s)ds=:(Uy)(t). $$ Thus problem \eqref{es1} is transformed into problem \eqref{i5}, where $Au:=W\circ U(u)$, with $A$ continuous, compact and, obviously, causal. Take any sequence $(\varepsilon_n)$ converging to 0. Then applying the results above, we obtain the existence of a sequence of solutions $y_n$ of \eqref{e1} satisfying $(D_{0}^{\alpha-1}y_n)(t)\big|_{t=0}=b$ and converging in the $L_1^{\tau}$-sense to a solution of equation $y=Ay$. We set $$ x_n:=Uy_n \quad\text{and}\quad x:=Uy. $$ Then, evidently, $x_n$ satisfies the problem \eqref{es1} and $$ x^{(k)}(t)=y(t)=(Ay)(t)=W(Uy)(t)=Wx(t), $$ for a.a. $t\in[0,\tau]$ and $x^{(j)}(0)=0$, $j=0, 1,\dots, k-1$. Finally, we observe that $$ \|x_n-x\|^{\tau}_{\infty} =\sup_{t\in[0,\tau]}\big\|\int_0^t\frac{(t-s)^{k-1}}{(k-1)!}[y_n(s)-y(s)]ds \big\|_X \leq\frac{\tau^{k-1}}{(k-1)!}\|y_n-y\|_1^{\tau}. $$ The right-hand side tends to zero. The proof is complete. \end{proof} \section{Auxiliary Lemmas} Before giving the proof of Theorem \ref{t2}, we need some auxiliary facts concerning the series \begin{equation}\label{s1} \sum_{j=1}^{\infty}\frac{(-1)^{j-1}s^{j\alpha-1}}{\varepsilon^j\Gamma(j\alpha)}, \quad s>0. \end{equation} \begin{lemma}\label{lem1} The series \eqref{s1} converges absolutely and uniformly on compact subsets of $[0,+\infty)$ to a function $k(s;\varepsilon),\quad s>0$, which is continuous and positive. \end{lemma} \begin{proof} Define the sets $$ Q_1:=\{j\in\mathbb{Z}: \alpha\leq j\alpha<1\},\quad Q_k:=\{j\in\mathbb{Z}: k\leq j\alpha0 $$ is an $L_1^T$ function, for any $T>0$. Now, by using the fact that $(s+1)^{\alpha}>1>\varepsilon$ and the monotonicity of the function $\Gamma$ on the interval $[2,+\infty)$, we obtain \begin{align*} \sum_{j=1}^{\infty}\frac{(s+1)^{j\alpha-1}}{\varepsilon^j\Gamma(j\alpha)} & \leq \Lambda(s)+ \sum_{k=3}^{\infty}\sum_{j\in Q_k} \frac{1}{s+1}\frac{(\frac{(s+1)^{\alpha}}{\varepsilon})^j}{\Gamma(k)}\\ &\leq\Lambda(s)+\sum_{k=3}^{\infty}\sum_{j\in Q_k}\frac{1}{s+1} \frac{(\frac{(s+1)^{\alpha}}{\varepsilon})^{\frac{k+1}{\alpha}}}{\Gamma(k)}\\ &=\Lambda(s)+\sum_{k=3}^{\infty}\sum_{j\in Q_k}\varepsilon^{\frac{1}{\alpha}} \frac{(\frac{(s+1)}{\varepsilon^{1/\alpha}})^k}{\Gamma(k)}\\ &\leq \Lambda(s)+\mu\sum_{k=3}^{\infty}\varepsilon^{\frac{1}{\alpha}} \frac{(\frac{(s+1)}{\varepsilon^{1/\alpha}})^k}{(k-1)!}\\ &=\Lambda(s)+\mu (s+1)\sum_{k=3}^{\infty} \frac{(\frac{(s+1)}{\varepsilon^{1/\alpha}})^{k-1}}{(k-1)!}\\ &=\Lambda(s)-\mu (s+1)(1+\frac{(s+1)}{\varepsilon^{1/\alpha}}) +\mu (s+1)\exp({\frac{(s+1)}{\varepsilon^{1/\alpha}}}). \end{align*} The right-hand side defines an $L_1^T$ function, for any $T>0$. Obviously, this proves the first part of the lemma. It remains to show that the function $k(\cdot;\varepsilon)$ is positive. Indeed, by the previous arguments, we can apply the Lebesgue Dominated Convergence Theorem and get, for fixed $\theta\in[0,t]$, that \begin{equation}\label{e5} \begin{aligned} \int_{t-\theta}^tk(s;\varepsilon)ds &=\int_0^{\theta}k(t-s;\varepsilon)ds =\int_0^{\theta}\sum_{j=1}^{\infty} \frac{(-1)^{j-1}(t-s)^{j\alpha-1}}{\varepsilon^j\Gamma(j\alpha)}ds\\ &=\sum_{j=1}^{\infty}\frac{(-1)^{j}(t-\theta)^{j\alpha}} {\varepsilon^j\Gamma(j\alpha+1)}-\sum_{j=1}^{\infty} \frac{(-1)^{j}t^{j\alpha}}{\varepsilon^j\Gamma(j\alpha+1)}\\ &=\sum_{j=0}^{\infty}\frac{(-1)^{j}(t-\theta)^{j\alpha}} {\varepsilon^j\Gamma(j\alpha+1)}-\sum_{j=0}^{\infty} \frac{(-1)^{j}t^{j\alpha}}{\varepsilon^j\Gamma(j\alpha+1)}\\ &=E_{\alpha}(\frac{-(t-\theta)^{\alpha}}{\varepsilon})-E_{\alpha} (\frac{-t^{\alpha}}{\varepsilon}). \end{aligned} \end{equation} By using \eqref{r2}, relation \eqref{e5} gives $$ \int_0^{\theta}\sum_{j=1}^{\infty} \frac{(-1)^{j-1}(t-s)^{j\alpha-1}}{\varepsilon^j \Gamma(j\alpha)}ds=\int_0^{+\infty}(e^{-(t-\theta)s} -e^{-ts})dF_{\alpha}(s)\geq 0. $$ From the properties of $E_{\alpha}$ which we mentioned in Subsection \ref{p1}, it follows that the quantity $E_{\alpha}(\frac{-t^{\alpha}}{\varepsilon})$ is positive and less than 1 and it tends to zero monotonically when $t$ tends to $+\infty$. The latter implies that \begin{equation}\label{e15} \lim_{x\to+\infty}E_{\alpha}(-x)=0, \end{equation} namely, \begin{gather}\label{e7} 00$. Extend $u$ from $[0,T]$ to $\mathbb{R}$ by setting $\bar{u}(s)=0$, if $s\notin[0,\tau]$ and $\bar{u}(s)=u(s)$, $s\in[0,T]$. Then $\bar{u}$ is an element of $L_1(\mathbb{R},X)$ and, so it satisfies $\lim_{s\to0}\|\bar{u}(\cdot-s)-\bar{u}(\cdot)\|_1^T=0$, (see. e.g. \cite[Thm 1.4.2 p. 298]{cc}). This means that there is an $s_0>0$ such that $$ \|\bar{u}(\cdot-s)-\bar{u}(\cdot)\|_1^T\leq \eta, \quad 0\leq s\leq s_0. $$ Take any $\delta\in(0,s_0]$. By \eqref{e10}, there is some $\varepsilon_{\delta}>0$, such that for all $\varepsilon\in(0,\varepsilon_{\delta}]$ it holds $$ \big|\int_0^tk(t-s;\varepsilon)ds-1\big|<\eta, \quad t\in[{\delta},T]. $$ Hence, we have $$ \big\|\int_0^tk(t-s;\varepsilon)u(t)ds-u(t)\big\|_X\leq\eta\|u(t)\|_X, \quad t\in [{\delta},T], $$ or \begin{equation}\label{e13} \big\|\int_0^t\big[k(s;\varepsilon)u(s)-\frac{1}{t}u(t)\big]ds\big\|_X \leq\eta\|u(t)\|_X, \quad t\in [{\delta},T]. \end{equation} Taking into account Lemma \ref{lem1} (i.e. that $k$ is positive), we observe that \begin{equation}\label{16} \begin{aligned} &\int_{\delta}^T\big\|\int_0^t\big[k(t-s;\varepsilon)u(s)ds-u(t)\big]\big\|_Xdt\\ &=\int_{\delta}^T\big\|\int_0^t\big[k(s;\varepsilon)\bar{u}(t-s) -\frac{1}{t}\bar{u}(t)\big]ds\big\|_Xdt\\ &\leq \int_{\delta}^T\big\|\int_0^t\big[k(s;\varepsilon)\bar{u}(t-s)ds -\int_0^tk(s;\varepsilon)\bar{u}(t)ds\big]\big\|_Xdt\\ &\quad +\int_{\delta}^T\big\|\int_0^t\Big(k(s;\varepsilon)\bar{u}(t) -\frac{1}{t}\bar{u}(t)\Big)ds\big\|_Xdt\\ &\leq \int_{\delta}^T\big\|\int_0^tk(s;\varepsilon) [\bar{u}(t-s)-\bar{u}(t)]ds\big\|_Xdt+\eta\int_{\delta}^T\|\bar{u}(t)\|_Xdt\\ &\leq \int_{\delta}^T\int_0^{\delta}k(s;\varepsilon) \|\bar{u}(t-s)-\bar{u}(t)\|_X\,ds\,dt\\ &\quad +\int_{\delta}^T\int_{\delta}^tk(s;\varepsilon)\|\bar{u}(t-s) -\bar{u}(t)\|_X\,ds\,dt+\eta\|u\|_1^T. \end{aligned} \end{equation} We estimate the right-hand side of relation \eqref{16}. We have \begin{align*} &\int_{\delta}^T\int_0^{\delta}k(s;\varepsilon)\|\bar{u}(t-s) -\bar{u}(t)\|_X\,ds\,dt\\ &=\int_0^{\delta}k(s;\varepsilon)\int_{\delta}^T\|\bar{u}(t-s)-\bar{u}(t)\|_X\,dt\,ds\\ &\leq\int_0^{\delta}k(s;\varepsilon)\|\bar{u}(\cdot-s)-\bar{u}(\cdot)\|_1^Tds \\ &\leq \eta\int_0^{\delta}k(s;\varepsilon)ds. \end{align*} Also \begin{align*} &\int_{\delta}^T\int_{\delta}^tk(s;\varepsilon)\|\bar{u}(t-s) -\bar{u}(t)\|_X\,ds\,dt\\ &= \int_{\delta}^Tk(s;\varepsilon)\int_{s}^T\|\bar{u}(t-s)-\bar{u}(t)\|_X\,dt\,ds\\ &\leq \int_{\delta}^T\int_0^T\big(k(s;\varepsilon)(\|\bar{u}(t-s)\|_X+\|\bar{u}(t)\|_X\big)\,dt\,ds\\ &\leq 2\|u\|_1^T\int_{\delta}^Tk(s;\varepsilon)ds. \end{align*} Hence, \eqref{e16} becomes \begin{align*} &\int_{\delta}^T\big\|\int_0^t\big[k(t-s;\varepsilon)u(s) -\frac{1}{t}u(t)\big]ds\big\|_Xdt\\ &\leq \eta\int_0^{\delta}k(s;\varepsilon)ds+2\|u\|_1^T\int_{\delta}^T k(s;\varepsilon)ds+\eta\|u\|_1^T. \end{align*} Now, in view of \eqref{e10} and \eqref{e11} as $\varepsilon$ tends to 0, the right-hand side tends to $\eta(1+\|u\|_1^T)$. Since $\delta$ is arbitrary and small, we obtain $$ \int_0^T\big\|\int_0^t\big[k(t-s;\varepsilon)u(s) -\frac{1}{t}u(t)\big]ds\big\|_Xdt\leq\eta(1+\|u\|_1^T). $$ The fact that $\eta$ is arbitrary completes the proof of relation \eqref{e12}. \end{proof} \section{Proof of theorem \ref{t2}} To simplify notation, we set $$ \phi(t):=\frac{t^{\alpha-1}}{\Gamma(\alpha)}b, \quad t\in(0,T] $$ and observe that $\phi$ is an element of $L_1^T$, for all $T>0$. Also, consider the operator $$ (L_{\varepsilon}u)(t):=\frac{1}{\varepsilon\Gamma(\alpha)} \int_0^t(t-s)^{\alpha-1}u(s)ds, \quad u\in L_1^T. $$ Then relation \eqref{a1} takes the form $$ y(t)=\phi(t)-(L_{\varepsilon}y)(t)+(L_{\varepsilon}Ay)(t) $$ which, by iteration, for each $n=1, 2, \dots$, gives \begin{equation}\label{e3} y(t)=\sum_{j=0}^{n-1}(-1)^{j}(L_{\varepsilon}^{(j)}\phi)(t) +(-1)^{n}(L_{\varepsilon}^{n}y)(t) +\sum_{j=1}^n(-1)^{j-1}(L_{\varepsilon}^{(j)}Ay)(t). \end{equation} Let $u\in L_1^T$. We observe that $$ (L_{\varepsilon}^{(2)}u)(t)=\frac{1}{\varepsilon^2\Gamma(2\alpha)} \int_0^t(t-s)^{2\alpha-1}u(s)ds. $$ By induction we obtain $$ (L_{\varepsilon}^{(j)}u)(t) =\frac{1}{\varepsilon^j\Gamma(j\alpha)} \int_0^t(t-s)^{j\alpha-1}u(s)ds, \quad j=1, 2, \dots. $$ Then we have \begin{align*} \|L_{\varepsilon}^{(j)}u\|_1^T &=\int_0^T\big\|\frac{1}{\varepsilon^j\Gamma(j\alpha)} \int_0^t(t-s)^{j\alpha-1}u(s)ds\big\|_Xdt\\ &\leq\int_0^T\frac{1}{\varepsilon^j\Gamma(j\alpha)} \int_s^t(t-s)^{j\alpha-1}\|u(s)\|_X\,dt\,ds\\ &\leq \frac{T^{j\alpha}}{\varepsilon^j\Gamma(j\alpha+1)}\|u\|_1^T. \end{align*} Since by definition $$ \sum_0^{+\infty}\frac{T^{j\alpha}}{\varepsilon^j\Gamma(j\alpha+1)} =E_{\alpha}(\frac{T^{\alpha}}{\varepsilon}), $$ where $E_{\alpha}$ is the Mittag-Leffler function, it follows that both series in \eqref{e3} converge, yet $$ \lim_j L_{\varepsilon}^{(j)}u=0. $$ So the right side of \eqref{e3} converges to $$ \sum_{j=0}^{\infty}(-1)^{j}(L_{\varepsilon}^{(j)}\phi)(t) +\sum_{j=1}^{\infty}(-1)^{j-1}(L_{\varepsilon}^{(j)}Au)(t)=:Su(t) $$ and, therefore, we obtain \begin{equation}\label{ea8} \begin{aligned} Su(t)-\phi(t)&=\sum_{j=1}^{\infty}(-1)^{j-1}\big(L_{\varepsilon}^{(j)} (Au-\phi)\big)(t)dt\\ &=\sum_{j=1}^{\infty}(-1)^{j-1}\frac{1}{\varepsilon^j\Gamma(j\alpha)} \int_0^t(t-s)^{j\alpha-1}(Au(s)-\phi(s))ds\\ &=\int_0^t\sum_{j=1}^{\infty}\frac{(-1)^{j-1}(t-s)^{j\alpha-1}} {\varepsilon^j\Gamma(j\alpha)}(Au(s)-\phi(s))ds\\ &=\int_0^tk(t-s;\varepsilon)(Au(s)-\phi(s))ds, \end{aligned} \end{equation} where $$ k(s;\varepsilon):=\sum_{j=1}^{\infty}\frac{(-1)^{j-1}s^{j\alpha-1}} {\varepsilon^j\Gamma(j\alpha)}. $$ The interchange of integration and summation is permitted because of Lemma \ref{lem1}. From \eqref{ea8} and the fact that $k$ is positive, we obtain \begin{equation} \label{e9} \begin{aligned} \|Su-\phi\|_1^T&=\int_0^T\|Su(t)-\phi(t)\|_Xdt\\ &\leq \int_0^T\int_0^tk(t-s;\varepsilon)\|Au(s)-\phi(s)\|_X\,ds\,dt\\ &=\int_0^T\int_s^Tk(t-s;\varepsilon)\|Au(s)-\phi(s)\|_X\,dt\,ds\\ &=\int_0^T\Big[1-E_{\alpha}\Big(\frac{-(T-s)^{\alpha}}{\varepsilon}\Big) \Big]\|Au(s)-\phi(s)\|_Xds\\ &\leq \|Au-\phi\|_1^T. \end{aligned} \end{equation} We claim that, for any $R>0$, there exists $\tau\in(0,T]$, such that in the space $L_1^{\tau}$, it holds $$ S(\overline{B(\phi, R)})\subseteq \overline{B(\phi,R)}. $$ By \eqref{e9}, to show this fact, it is sufficient to prove that there is a $\tau\in(0,T]$, such that in the space $L_1^{\tau}$, it holds \begin{equation}\label{e17} A(\overline{B(\phi, R)})\subseteq \overline{B(\phi,R)}. \end{equation} Let $\overline{B(\phi,R)}$ be the closed ball $\{u\in L_1^{\tau}: \quad \|u-\phi\|_1^T\leq R\}$. Fix any $\zeta\in(0,\frac{R}{2}]$. Since the set $A(\overline{B(\phi,R)})$ has compact closure, there is a finite $\zeta$-dense subset of it, say, $Au_1, Au_2, \dots, Au_k\in A(\overline{B(\phi,R)})$. Also, we can find $\tau\in(0,T]$ such that $$ \|Au_j-\phi\|_1^{\tau}=\int_0^{\tau}\|(Au_j)(t)-\phi(t)\|_Xdt\leq\zeta, \quad j=1, 2, \dots, k. $$ Take any $u\in \overline{B(\phi,R)}$. Then $Au\in A(\overline{B(\phi,R)})$ and, thus, $\|Au-Au_j\|_1^{\tau}\leq \zeta$, for some $j$. Hence, $$ \|Au-\phi\|_1^{\tau}\leq \|Au-Au_j\|_1^{\tau} +\|Au_j-\phi\|_1^{\tau}\leq 2\zeta\leq R. $$ Therefore \eqref{e17} is true. Because of the previous facts, the fixed point Theorem \ref{t1} applies and we conclude that there is $y_{\varepsilon}\in \overline{B([0,\tau],R)}$, such that $$ y_{\varepsilon}(t)=(Sy_{\varepsilon})(t) =\sum_{j=0}^{\infty}(-1)^{j}(L_{\varepsilon}^{(j)}\phi)(t) +\sum_{j=1}^{\infty}(-1)^{j-1}(L_{\varepsilon}^{(j)}Ay_{\varepsilon})(t), \quad t\in[0,\tau], $$ or, by \eqref{ea8}, $$ y_{\varepsilon}(t)-\phi(t)=\int_0^tk(t-s;\varepsilon)(Ay_{\varepsilon}(s) -\phi(s))ds, \quad t\in[0, \tau]. $$ Next, we take any sequence $\varepsilon_n$ tending to 0, and denote by $y_n$ the solution $ y_{\varepsilon_n}$. Hence we have \begin{equation}\label{e19} y_{n}(t)-\phi(t)=\int_0^tk(t-s;\varepsilon_n)(Ay_{n}(s)-\phi(s))ds, \quad t\in[0, \tau]. \end{equation} By the relative compactness of the set $A(\overline{(B(\phi,R)})$, we can assume that the sequence $(Ay_{n})$ converges to some $y\in L_1^{\tau}$. Then, for almost all $t\in[0,\tau]$, from \eqref{e19} we obtain $$ y_n(t)-y(t)=\int_0^tk(t-s;\varepsilon_n)(Ay_n(s)-\phi(s))ds-(y(t)-\phi(t)) $$ and, therefore, it follows that \begin{align*} \|y_n-y\|_1^{\tau} &=\int_0^{\tau}\big\|\Big(\int_0^tk(t-s;\varepsilon_n) \big[Ay_n(s)-\phi(s)\big]ds\Big)-(y(t)-\phi(t))\big\|_Xdt\\ &\leq \int_0^{\tau}\int_0^tk(t-s;\varepsilon_n)\|Ay_n(s)-y(s)\|_X\,ds\,dt\\ &\quad +\int_0^{\tau}\big\|\int_0^tk(t-s;\varepsilon_n)(y(s)-\phi(s))ds -(y(t)-\phi(t))\big\|_Xdt. \end{align*} For the first integral on the right side we have \begin{align*} &\int_0^{\tau}\int_0^tk(s;\varepsilon_n)\big\|(Ay_n)(t-s) -y(t-s)\big\|_X \,ds\,dt \\ &= \int_0^{\tau}\int_s^{\tau}k(s;\varepsilon_n) \big\|(Ay_n)(t-s)-y(t-s)\big\|_X\,dt\,ds \\ &\leq\int_0^{\tau}k(s;\varepsilon_n)\int_s^{\tau}\big\|(Ay_n)(t-s)-y(t-s) \big\|_X\,dt\,ds\\ &=\int_0^{\tau}k(s;\varepsilon_n)\int_0^{\tau-s}\big\|(Ay_n)(\xi)-y(\xi) \big\|_Xd\xi ds\\ &\leq \int_0^{\tau}k(s;\varepsilon_n)ds\|Ay_n-y\|_1^{\tau}, \end{align*} which tends to $0$. Also, the sequence $$ \int_0^{\tau}\big\|\int_0^tk(t-s;\varepsilon_n)(y(s) -\phi(s))ds-(y(t)-\phi(t))\big\|_Xdt $$ tends to $0$, because of \eqref{e12}. Hence, we have $\lim y_n=y$ and, by the continuity of $A$, it follows that $y=\lim Ay_n=Ay$. The proof is complete. \begin{thebibliography}{00} \bibitem{a3} Ya. I. Alber; Recurrence relations and variational inequalities, \emph{Soviet Math. Dokl.}, \textbf{27} (1983), 511-517. \bibitem{a2} Yakov Alber, Simeon Reich, David Shoikhet; Iterative approximations of null points of uniformly accretive operators with estimates of the convergence rate, \emph{Commun. Appl. Nonlinear Anal.} \textbf{3} (8-9) (2002), 1107--1124. \bibitem{bll} Cara D. Brooks, Patricia K. Lamm, Xiaoyue Luo; Local regularization of nonlinear Volterra equations of Hammerstein type, \emph{Integral Equations Appl.}, 09/2010; 22(2010). DOI: 10.1216/JIE-2010-22-3-393. \bibitem{a1} F. E. Browder, W. V. Petryshyn; Construction of fixed points of nonlinear mappings in Hilbert space, \emph{J. Math. Anal. Appl.}, \textbf{20} (1967), 197-228. \bibitem{cord} C. Corduneanu; \emph{Integral Equations and Applications}, Cambridge Univ. Press, New York, 1991. \bibitem{cc} Misha Cotlar, Roberto Cignoli; \emph{An Introduction to Functional Analysis}, American Elsevier Publ. Co. New York, 1974. \bibitem{he} Heinz W. Engl; On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems, \emph{Journal of Approx. Theory} 49 (1987), 55-63. \bibitem{hls} Markus Haltmeier, Antonio Leit\~ao, Otmar Scherzer; Kaczmarz methods for regularizing nonlinear ill-posed equations I: Convergence analysis, \emph{Inverse Problems and Imaging} 1(2007), 289-298. \bibitem{gg} A. L. Gaponenko, Yu L. Gaponenko; A method of regularization for operator equations of the first kind, \emph{Zh. $v\tilde{y}chisl$. Mat. mat. Fiz.}, 16 (1976), 577-584. \bibitem{cwg} C. W. Groetsch; Integral equations of the first kind, inverse problems and regularization: a crash course, \emph{Journal of Physics: Conference Series} 73 (2007) 1-32. \bibitem{gh} Nicola Guglielmi, Ernst Hairer; Regularization of neutral delay differential equations with several delays, \emph{J. Dynam. Differential Equations} 7, (2012), 1-26. \bibitem{jum} O. Jumarie; Modified Riemann-Liouville Derivative and Fractional Taylor Series of Nondifferentiable Functions Further Results, \emph{Comput. Math. Appl.} 51 (2006) 1367-1376. \bibitem{kar3} George L. Karakostas; Causal operators and Topological Dynamics, \emph{Ann. Matematica Pura ed Appl.} Vol. CXXXI, 1982, 1-27. \bibitem{kar4} George L. Karakostas; Strong approximation of the solutions of a system of operator equations in Hilbert spaces, \emph{J. Difference Equ. Appl.} 12 (2006), 619-632. \bibitem{lak} V. Lakshmikantham, A. S. Vatsala; General uniqueness and monotone iterative technique for fractional differential equations, \emph{Appl. Math. Lett.} 21 (2008), 828-834. \bibitem{lamm} Patricia K. Lamm; A Survey of Regularization Methods for First-Kind Volterra Equations, Mathematics Dept., Michigan State University, E. Lansing, MI 48824-1027 USA, http://www.mth.msu.edu/∼lamm (May 19, 2015). \bibitem{lin} Ping Lin; Regularization methods for differential equations and their numerical solution, Ph. D. Thesis, The University of British Columbia, 1995. \bibitem{mar} R. M\"arz; \emph{Numerical methods for differential-algebraic equations., Part I: Characterizing DAEe}, Preprint No. 91-32/I, Humboldt Universit\"at zu Berlin, 1991. \bibitem{men} Abdelaziz Mennouni; A regularization procedure for solving some singular integral equations of the second kind, \emph{Internat. J. Difference Equations} 8 (2013), 71-76. \bibitem{miro} Kenneth S. Miller, Bertram Ross; \emph{An Introduction to the Fractional Calculus and Fractional Differential Equations}, John Wiley and Sons, Inc. New York, 1993. \bibitem{neu} A. Neubauer; Tikhonov-Regularization of ill-Posed Linear Operator Equations on Closed Convex Sets, \emph{J. Approx. Theory} 53(1988), 304-320. \bibitem{amw} Abdul-Majid Wazwaz; Solving Schl\"omilch's integral equation by the regularization-Adomian method, \emph{Rom. Journ. Phys.}, 60 (2015), 56-72. \bibitem{new} L. W. Neustadt; On the solutions of certain integral like operator equations. Existence, uniqueness and dependence theorem, \emph{Arch. Rat. Mech. Anal.}, 38 (1970), 131-160. \bibitem{po} Igor Podlubny; \emph{Fractional Differential Equation}, Mathematics in Science and Engineering, Vol. 118, Acad. Press, 1999. \bibitem{poll} Harry Pollard; The completely monotonic character of the Mittag-Leffler function $E_{\alpha}(x)$, \emph{Bull. Amer. Math. Soc.} Vol. 54, (12), (1948), 1115-1116. \bibitem{pop} E. Prempeh, I. Owusu-Mensay, K. Piesie-Frimbong; On the regularization of Hammerstein's type operator equations, \emph{Aust. J. Math. Anal. Appl.}, 11 (2014), 1-10. \bibitem{sav} T. I. Savelova; Optimal regularization of equations of the convolution type with random noise in the kernel and right-hand side, \emph{U.S.U.R. Comput. Math. Phys.} 18(1978), 1-7. \bibitem{sav1} T. I. Savelova; Regularization of non-linear integral equations of the convolution type, \emph{U.S.U.R. Comput. Math. Phys.} 19(1979), 20-27. \bibitem{a4} Ishikawa Shiro; Fixed points by a new iteration method, \emph{Proc. Amer. Math. Soc.}, \textbf{44}(1) (1974), 147-150. \bibitem{ww} Jin Wen, Ting Wei; Regularized solution to the Fredholm integral equation of the first kind with noisy data, \emph{J. Appl. Math. and Informatics} 29(2011), 23-37. \bibitem{wiki} Wikipedia, http://en.wikipedia.org/wiki/Particular\_values\_of\_the\_Gamma\_function \#Other\_ constants (May 26, 2015). \end{thebibliography} \end{document}