Keywords

2010 Mathematics Subject Classification

16.1 Introduction

To model a process with delay, it is not sufficient to employ an ordinary or partial differential equations. An approach to resolve this problem is to use integrodifferential equations. In some fields such as nuclear reactor dynamics and thermoelasticity, we need to reflect the effect of the memory of the systems in the model. If such systems are modeled using partial differential equation, the effect of past history is ignored. Therefore in order to incorporate the memory effect in such systems, an integral term in the partial differential equation is introduced and this leads to a partial integrodifferential equation [7]. In recent years, due to the novel surprising insights and framework of fractional calculus, the fractional partial integrodifferential equations have been scrutinized by several authors. Historically, the origins of fractional calculus can be traced back to the end of the seventeenth century, the time when Newton and Leibniz developed the foundations of differential and integral calculus. It extends the differentiation and integration of integer order to an arbitrary order and concatenates these two operators. To be precise, it consists of integrodifferential operators with the convolution type integrals and power-law type weakly singular kernels. An imperative feature is that fractional derivatives and integrals are non local, since it depends on all of its historical states. This is very effective when the system has a longterm memory and any evaluation point depends on the past values of the function. For example, the use of half derivatives and integrals lead to the formulation of certain electrochemical problems which are more economical and useful than the classical approach in terms of Fick’s law of diffusion. Some of the applications of fractional calculus in interdisciplinary sciences can be found in [30, 34]. During the last few decades, fractional differentiation is drawing huge consideration toward physical and biological behaviors. The reason behind using fractional order differential equation is that it is naturally related to systems with memory which exists in most biological systems and fractional order system response ultimately converges to the integer-order equations. The elementary theory and some applications of fractional differential equations are widely covered in [14, 26, 29] and for the books associated with fractional differential equations, see [18, 21, 24, 28]. The applications of fractional derivatives in reservoir engineering problems are given in [23]. Jesus et al. [19] investigated the fractional model of the electrical impedance for botanical elements according to Bode and polar diagrams. A review of some applications of fractional derivatives in continuum and statistical mechanics is given by Carpinteri and Mainardi [11]. Next, we propose some of the works concerning the solvability of fractional differential equations. For instance, Balachandran et al. [8, 9] studied the existence results for several kinds of fractional integrodifferential equations in a Banach space using a fixed point technique. In [36], Zhang et al. investigated the existence of nonnegative solutions for nonlinear fractional differential equations with nonlocal fractional integrodifferential boundary conditions on an unbounded domain by using the Leray–Schauder nonlinear alternative theorem. The differential transform method was applied to fractional integrodifferential equations in [6] to solve those equations analytically. To know more details about the existence of solutions of integrodifferential equation, see the papers [1, 2, 5, 12, 20] and for fractional partial integrodifferential equations refer [3, 4]. In this paper, we extend the results of [25] to fractional order partial integrodifferential equation of diffusion type with integral kernel.

16.2 Basic Concepts

Now, we present the definitions of some well-known fractional operators that play an important role in fractional calculus. For any \(n-1<\alpha <n\), \(n\in \mathbb {N}\), the Rieman–Liouville fractional integral operator is defined as follows:

Definition 16.2.1

([21]) The partial Riemann–Liouville fractional integral operator of order \(\alpha \) with respect to t of a function f(x, t) is defined by

$$\begin{aligned} I^\alpha f(x,t)=\displaystyle {\frac{1}{\varGamma (\alpha )}\int \limits _0^t\frac{f(x,s)}{(t-s)^{n-\alpha }}\ \mathrm {d}s}. \end{aligned}$$

where \(f(\cdot ,t)\) is an integrable function.

The most popular definition of fractional calculus is Riemann–Liouville fractional derivative definition, which is basic for the Caputo fractional derivative. It is written as follows:

Definition 16.2.2

([21]) The partial Riemann–Liouville fractional derivative of order \(\alpha \) of a function f(x, t) with respect to t is of the form

$$\begin{aligned} \frac{\partial ^\alpha }{\partial t^\alpha } f(x,t)=\displaystyle {\frac{1}{\varGamma (n-\alpha )}\frac{\partial ^n}{\partial t^n}\int \limits _0^t \frac{f(x,s)}{(t-s)^{\alpha -n+1}}\ \mathrm {d}s}. \end{aligned}$$

where the function \(f(\cdot ,t)\) has absolutely continuous derivatives up to order \((n-1)\).

Since the Riemann–Liouville fractional derivative of a constant is a function, to overcome this difficulty, Caputo [10] reformulated the Riemann–Liouville fractional derivative to handle integer order initial conditions, in the following way.

Definition 16.2.3

([21]) The Caputo partial fractional derivative of order \(\alpha \) with respect to t of a function f(x, t) is defined as

$$\begin{aligned} \frac{^C\partial ^\alpha }{\partial t^\alpha } f(x,t)=\displaystyle {\frac{1}{\varGamma (n-\alpha )}} \int \limits _0^t \frac{1}{(t-s)^{\alpha -n+1}}\frac{\partial ^n f(x,s)}{\partial s^n}\ \mathrm {d}s. \end{aligned}$$

where the function \(f(\cdot ,t)\) has absolutely continuous derivatives up to order \((n-1)\).

To know the properties of these operators, see the books [21, 28] and for more facts on the geometric and physical interpretation of fractional derivatives with Riemann–Liouville and Caputo types, see [17, 22]. There has been a significant development in ordinary and partial differential equations involving both Riemann–Liouville and Caputo fractional derivatives in the past few years, for instance, see the papers of Gejji and Jafari [16], Furati and Tatar [15]. The Riemann Liouville and Caputo fractional derivatives are linked by the following relationship:

$$\begin{aligned} \frac{^C\partial ^\alpha }{\partial t^\alpha } f(x,t)=\frac{\partial ^\alpha }{\partial t^\alpha } f(x,t)-\sum \limits _{k=0}^{n-1}\displaystyle {\frac{t^{k-\alpha }}{\varGamma (k+1-\alpha )}}\frac{\partial ^k}{\partial t^k}f(x,0). \end{aligned}$$

Before looking at the existence result of fractional partial integrodifferential equations, we introduce some basic results that are inherently tied to existence theory.

Lemma 16.2.1

((Leray–Schauder fixed point theorem) [25]) If U is a closed bounded convex subset of a Banach space X and \(T:U\rightarrow U\) is completely continuous, then T has at least a fixed point in U.

Lemma 16.2.2

((Arzela–Ascoli Theorem) [25]) Assume that K is a compact set in \(\mathbb {R}^n,\ n\ge 1\), then a set \(S\subset C(K)\) is relatively compact in C(K) if and only if the functions in S are uniformly bounded and equicontinuous on K.

Lemma 16.2.3

((Green’s Identity) [13]) Let \(\varOmega \) be a bounded domain in \(\mathbb {R}^m\) with smooth boundary \(\partial \varOmega \). Then, for any \(u,v\in C^2(\varOmega )\),

$$\begin{aligned} \int _\varOmega v\varDelta u\ \mathrm {d}x=\int _{\partial \varOmega } v\frac{\partial u}{\partial n}\ \mathrm {d}s-\int _\varOmega \nabla u\cdot \nabla v\ \mathrm {d}x, \end{aligned}$$

where n is the outward unit normal to the boundary \(\partial \varOmega \) and \(\mathrm {d}s\) is the element of arc length. For the special case \(v=1\),

$$\begin{aligned} \int _\varOmega \varDelta u\ \mathrm {d}x=\int _{\partial \varOmega } \frac{\partial u}{\partial n}\ \mathrm {d}s. \end{aligned}$$
(16.1)

This is called Green’s first identity.

16.3 Fractional Partial Differential Equations with Kernel

Let \(\varOmega \) be a bounded subset of an m-dimensional space with smooth boundary and let \(J=[0,T]\). Consider the fractional partial integrodifferential equation of the form

$$\begin{aligned} \displaystyle {\frac{^C\partial ^\alpha u}{\partial t^\alpha }}&=a(t)\varDelta u(x,t)+\int _0^t h(t-s)\varDelta u(x,s)\ \mathrm {d}s+f\big (t,u(x,t)\big )\nonumber \\ {}&\quad +\int _0^t g(t,s,u(x,s))\ \mathrm {ds},\ t\in J, \end{aligned}$$
(16.2)

with the initial condition

$$\begin{aligned} u(x,0)&= u_0(x),\;\;\;\;\;\;\;\;\;\; x\in \;\varOmega , \end{aligned}$$

where \(0<\alpha < 1\), \(h:J\rightarrow \mathbb {R}\) is a positive kernel and \(f:J\times \mathbb {R}\rightarrow \mathbb {R}\) is a nonlinear function. This (16.2) is a special case of integrodifferential equation of motion of fractional Maxwell fluid with zero pressure. This type of equation appears in the investigation of viscoelastic property. This equation gets attention from the fact that the fractional derivatives are used to depict the viscoelasticity phenomena with little amount of constraints. A viscoelastic fractional order mathematical model of a human root dentin is proposed by Petrovic et al. in [27]. Fractional partial integrodifferential equation has also been applied in the study of signal processing, turbulence, plasma physics, and in many other fields, for instance, see [31,32,33]. The integral equation corresponding to (16.2) can be written as

$$\begin{aligned} u(x,t)&=u_0(x)+\frac{1}{\varGamma (\alpha )}\!\int _0^t\!(t-s)^{\alpha -1}a(s)\varDelta u(x,s)\ \mathrm {d}s\nonumber \\ {}&\quad +\frac{1}{\varGamma (\alpha )}\!\int _0^t\!(t-s)^{\alpha -1}\Big (\!\int _0^s\!h(s-\tau )\varDelta u(x,\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\nonumber \\&\quad +\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}f\big (s,u(x,s)\big )\ \mathrm {d}s\nonumber \\ {}&\quad +\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\bigg (\int _0^sg(s,\tau ,u(x,\tau ))\ \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$
(16.3)

Next, we present some hypotheses which will be used to prove our main result.

  1. (H1)

    a(t) is continuous on J and \(a(t)\in L^{1/\beta }(0,t)\), for all \(t\in J\) and some \(\beta \in (0,\alpha )\). That is, \(\bigg ( \displaystyle \int _0^t (a(s))^{\frac{1}{\beta }}\ \mathrm {d}s\bigg )^\beta \le C_1.\)

  2. (H2)

    f(t, u) is continuous with respect to u, Lebesgue measurable with respect to t and satisfies

    $$\begin{aligned} \frac{\int \limits _\varOmega \phi (x)f(t,u)\ \mathrm {d}x}{\int \limits _\varOmega \phi (x)\ \mathrm {d}x}\le f\left( t,\frac{\int \limits _\varOmega \phi (x)u(x,t)\ \mathrm {d}x}{\int \limits _\varOmega \phi (x)\ \mathrm {d}x} \right) , \end{aligned}$$

    for some function \(\phi (x)\).

  3. (H3)

    There exists an integrable function \(m_1(t): J\rightarrow [0,\infty )\) such that

    $$\begin{aligned} \parallel f(t,u)\parallel \le m_1(t)\Vert u\Vert , \end{aligned}$$

    where \(m_1(t)\in L^{1/\beta }(0,t)\), for all \(t\in J\) and \(\displaystyle \Big (\int _0^t(m_1(s))^{\frac{1}{\beta }}\ \mathrm {d}s\Big )^\beta \le C_2\), for \(\beta \) as in (H1) and \(C_2\ge 0\).

  4. (H4)

    g(t, s, u) is continuous with respect to u, Lebesgue measurable with respect to t and also satisfies the inequality

    $$\begin{aligned} \frac{\int \limits _\varOmega \phi (x)g(t,s,u)\ \mathrm {d}x}{\int \limits _\varOmega \phi (x)\ \mathrm {d}x}\le g\left( t,s,\frac{\int \limits _\varOmega \phi (x)u(x,t)\ \mathrm {d}x}{\int \limits _\varOmega \phi (x)\ \mathrm {d}x}\right) . \end{aligned}$$
  5. (H5)

    There exists an integrable function \(m_2(t,s):J\times J\rightarrow [0,\infty )\), such that

    $$\begin{aligned} \parallel g(t,s,u)\parallel \le m_2(t,s)\Vert u\Vert , \end{aligned}$$

    and for a nonnegative integer \(C_3\),

    $$\displaystyle \Big (\int _0^t\bigg (\int _0^s m_2(s,\tau )\ \mathrm {d}\tau \bigg )^{\frac{1}{\beta }}\ \mathrm {d}s\Big )^\beta \le C_3.$$
  6. (H6)

    The integral kernel satisfies

    $$\begin{aligned}\Big (\int _0^t\Big ( \int _0^s h(s-\tau )\ \mathrm {d}\tau \Big )^{\frac{1}{\beta }} \Big )^\beta \le C_4,\end{aligned}$$

    where \(C_4\ge 0\).

16.3.1 Dirichlet Boundary Condition

This section is consecrated to the existence of solution of (16.2) with Dirichlet boundary condition

$$\begin{aligned} u(x,t)= 0, \;\;\;\;\;\;\;(x,t)\ \in \partial \varOmega \times J, \end{aligned}$$
(16.4)

where \(\partial \varOmega \) is the boundary of \(\varOmega \). In order to achieve the required result, consider the following eigenvalue problem:

$$\begin{aligned} \left. \begin{array}{rcll} \varDelta u+\lambda u&{}=&{}0, &{} (x,t)\in \varOmega \times J,\\ u&{}=&{}0, &{} (x,t)\in \partial \varOmega \times J, \end{array} \right\} \end{aligned}$$
(16.5)

where \(\lambda \) is a constant not depending on the variables x and t. The theory of eigenvalue problems is well known [35]. Thus for \(x\in \varOmega \), the smallest eigenvalue \(\lambda _1\) of the problem (16.5) is positive and the corresponding eigenfunction is \(\phi (x)\ge 0\). Now, we define the function U(t) as

$$\begin{aligned} U(t)=\frac{\int \limits _\varOmega u(x,t)\phi (x)\ \mathrm {d}x}{\int \limits _\varOmega \phi (x)\ \mathrm {d}x}. \end{aligned}$$
(16.6)

The main theorem is as follows:

Theorem 16.3.1

Assume that there exists a \(\beta \in (0,\alpha )\) for some \(0<\alpha <1\) such that (H1)–(H3) and (H6) hold. For any constant \(b> 0\), suppose that

$$\begin{aligned} r_1=\min \left\{ T,\left[ \frac{\varGamma (\alpha )b}{(\Vert U(0)\Vert +b)(\lambda _1(C_1+C_4)+C_2+C_3)}\left( \frac{\alpha -\beta }{1-\beta }\right) ^{1-\beta }\right] ^{\frac{1}{\alpha -\beta }}\right\} . \end{aligned}$$

Then there exists at least one solution for the initial value problem (16.2) on \(\varOmega \times [0,r_1]\).

Proof

Our first aim is to prove that the initial value problem (16.2) has a solution if and only if the equation

$$\begin{aligned} U(t)&=U(0)-\frac{\lambda _1}{\varGamma (\alpha )}\int _0^t (t-s)^{\alpha -1}a(s)U(s)\ \mathrm {d}s\nonumber \\&-\displaystyle {\frac{\lambda _1}{\varGamma (\alpha )}}\int _0^t (t-s)^{\alpha -1}\Big (\int _0^s h(s-\tau )U(\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\nonumber \\&+\frac{1}{\varGamma (\alpha )}\int _0^t (t-s)^{\alpha -1} f(s,U(s))\ \mathrm {d}s\nonumber \\&+\frac{1}{\varGamma (\alpha )}\int _0^t (t-s)^{\alpha -1}\bigg (\int _0^sg\big (s,\tau ,U(\tau )\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s \end{aligned}$$
(16.7)

has a solution.

Step 1. We start the proof by assuming u(x, t) to be a solution of (16.3). On integrating both sides of (16.3) with respect to \(x\in \varOmega \), we get

$$\begin{aligned} \int _\varOmega u(x,t)\ \mathrm {d}x=&\int _\varOmega u_0(x)\ \mathrm {d}x+\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _\varOmega \int _0^t (t-s)^{\alpha -1}a(s)\varDelta u(x,s)\ \mathrm {d}s\ \mathrm {d}x\nonumber \\ {}&+\frac{1}{\varGamma (\alpha )}\int _\varOmega \int _0^t (t-s)^{\alpha -1}\Big (\int _0^s h(s-\tau )\varDelta u(x,\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\ \mathrm {d}x\nonumber \\ {}&+\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _\varOmega \int _0^t (t-s)^{\alpha -1}f(s,u(x,s))\ \mathrm {d}s\ \mathrm {d}x\nonumber \\&+\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _\varOmega \int _0^t (t-s)^{\alpha -1}\bigg (\int _0^sg\big (s,\tau ,u(x,\tau )\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s\ \mathrm {d}x. \end{aligned}$$
(16.8)

Combining (16.6) and assumptions (H2) and (H6), (16.8) we get

$$\begin{aligned} U(t)&\le U(0)+\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _0^t (t-s)^{\alpha -1}a(s)\varDelta U(s)\ \mathrm {d}s\nonumber \\&+\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _0^t (t-s)^{\alpha -1}\Big (\int _0^s h(s-\tau )\varDelta U(\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\nonumber \\&+\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}f(s,U(s))\ \mathrm {ds}\nonumber \\&+\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\bigg (\int _0^sg\big (s,\tau ,U(\tau )\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$
(16.9)

Let \(K=\{U:U\in C(J,\ \mathbb {R}),\ \parallel U(t)-U(0)\parallel \le b\}\) and define an operator \(T:C(J,\ \mathbb {R})\rightarrow C(J,\ \mathbb {R})\) by

$$\begin{aligned} TU(t)=&\ U(0)+\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _0^t (t-s)^{\alpha -1}a(s)\varDelta U(s)\ \mathrm {d}s\nonumber \\ +&\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _0^t (t-s)^{\alpha -1}\Big (\int _0^s h(s-\tau )\varDelta U(\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\nonumber \\ +&\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}f(s,U(s))\ \mathrm {ds}\nonumber \\&+\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\bigg (\int _0^sg\big (s,\tau ,U(\tau )\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$
(16.10)

Clearly, \(U(0)\in K\). This means that K is nonempty. From our construction of K, we can say that K is closed and bounded. Now, for any \(U_1,U_2\in K\) and for any \(a_1,a_2\ge 0\) such that \(a_1+a_2=1\),

$$\begin{aligned} \parallel a_1U_1(t)+a_2U_2(t)-U(0)\parallel\le & {} a_1\parallel U_1(t)-U(0)\parallel +\,a_2\parallel U_2(t)-U(0)\parallel \\\le & {} a_1b+a_2b=b. \end{aligned}$$

Thus \(a_1U_1+a_2U_2\in K\). Therefore K is a nonempty closed convex set. Next, we move on to verify that T maps K into itself.

$$\begin{aligned} \parallel TU(t)-TU(0)\parallel&\le \ \frac{\lambda _1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \int _0^t(t-s)^{\alpha -1}\Vert a(s)\Vert \ \mathrm {d}s\nonumber \\&+\frac{\lambda _1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \int _0^t(t-s)^{\alpha -1}\Big (\int _0^s h(s-\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\nonumber \\&+\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\Vert f(s,U(s))\Vert \ \mathrm {ds}\nonumber \\&+\frac{1}{\varGamma (\alpha )} \int _0^t(t-s)^{\alpha -1}\bigg (\int _0^s\Vert g\big (s,\tau ,U(\tau )\big )\Vert \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$

Making use of Holder’s inequality and the assumptions, for any \(U\in K\), we can establish

$$\begin{aligned} \parallel TU(t)-&TU(0)\parallel \, \le \,\frac{\lambda _1C_1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \left( \int _0^t\left( (t-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }}\ \mathrm {d}s\right) ^{1-\beta }\\ +&\,\frac{\lambda _1C_4}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \left( \int _0^t\left( (t-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }}\ \mathrm {d}s\right) ^{1-\beta }\\ +&\,\frac{1}{\varGamma (\alpha )}\int _0^t m_1(s)(t-s)^{\alpha -1}\Vert U(s)\Vert \ \mathrm {d}s\\ +&\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\bigg (\int _0^s m_2(s,\tau )\Vert U(s)\Vert \ \mathrm {d}\tau \bigg )\ \mathrm {d}s \end{aligned}$$
$$\begin{aligned} \le&\,\frac{\left( \Vert U(0)\Vert +b\right) \lambda _1C_1}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_1^{\alpha -\beta }+\frac{\left( \Vert U(0)\Vert +b\right) \lambda _1C_4}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_1^{\alpha -\beta }\\ +&\,\frac{\left( \Vert U(0)\Vert +b\right) C_2}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_1^{\alpha -\beta }+\frac{\left( \Vert U(0)\Vert +b\right) C_3}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_1^{\alpha -\beta }\\ =&\,\frac{\left( \Vert U(0)\Vert +b\right) (\lambda _1(C_1+C_4)+C_2+C_3)}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_1^{\alpha -\beta }\\ \le&\, b, \ \ \ \ \ t\in [\,0,r_1\,]. \end{aligned}$$

Now, define a sequence \(\{U_k(t)\}\) in K such that

$$\begin{aligned} U_0(t)=U(0)\;\; \mathrm {and}\; \; U_{k+1}(t)=U_k(t),\;\; k=0,1,2,\ldots \end{aligned}$$

Since K is closed, there exists a subsequence \(\{U_{k_i}(t)\}\) of \(U_k(t)\) and \(\widetilde{U}(t)\in K\) such that

$$\begin{aligned} \lim _{k_i\rightarrow \infty }U_{k_i}(t)=\widetilde{U}(t). \end{aligned}$$
(16.11)

Then, Lebesgue’s dominated convergence theorem yields

$$\begin{aligned} \widetilde{U}(t)&=\widetilde{U}(0)-\frac{\lambda _1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}a(s)\widetilde{U}(s)\ \mathrm {d}s\\&-\frac{\lambda _1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\Big ( \int _0^sh(s-\tau )\widetilde{U}(\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\nonumber \\&+\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1} f(s,\widetilde{U}(s))\ \mathrm {d}s\\&+\frac{1}{\varGamma (\alpha )}\int _0^t\!(t-s)^{\alpha -1}\bigg (\int _0^s g\big (s,\tau ,\widetilde{U}(s)\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$

Next, we claim that T is continuous.

Step 2. Let \(\{U_m(t)\}\) be a converging sequence in K to U(t). Then, for any \(\varepsilon >0\), let

$$\begin{aligned} \Vert U_m(t)-U(t)\Vert \le \frac{\varGamma (\alpha )\varepsilon }{4\lambda _1Cr_1^{\alpha -\beta }}\left( \frac{\alpha -\beta }{1-\beta }\right) ^{1-\beta }, \end{aligned}$$
(16.12)

where \(C=\max \{C_1,C_4\}\). By assumption (H2) and (H4),

$$\begin{aligned} f(t,U_m(t))\longrightarrow f(t,U(t))\ \ \ \ \ \text{ and }\ \ \ \ \ g(t,s,U_m(t)) \longrightarrow g(t,s,U(t)) \end{aligned}$$

for each \(t\in [0,r_1]\). Therefore, for any \(\varepsilon >0\), we can take

$$\begin{aligned} \big \Vert f(t,U_m(t))-f(t,U(t))\big \Vert&\le \ \frac{\alpha \varGamma (\alpha )\varepsilon }{4r_1^\alpha }\ \left( \frac{\alpha -\beta }{1-\beta }\right) ^{1-\beta },\end{aligned}$$
(16.13)
$$\begin{aligned} \big \Vert g(t,s,U_m(t))-g(t,s,U(t))\big \Vert&\le \frac{\varGamma (\alpha )\varepsilon }{4Tr_1^\alpha }\ \left( \frac{\alpha -\beta }{1-\beta }\right) ^{1-\beta }. \end{aligned}$$
(16.14)

Employing (16.12) and (16.13) and simplifying, we have

$$\begin{aligned} \Vert TU_m(t)-TU(t)\Vert \,\le&\ \frac{\lambda _1C_1}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta }r_1^{\alpha -\beta }\Vert U_m(t)-U(t)\Vert \\ +&\frac{\lambda _1C_4}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta }r_1^{\alpha -\beta }\Vert U_m(t)-U(t)\Vert \\ +&\frac{r_1^\alpha }{\alpha \varGamma (\alpha )}\big \Vert f\left( s,U_m(s)\right) -f\left( s,U(s)\right) \big \Vert \\ +&\,\frac{r_1^\alpha }{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } \int _0^s\big \Vert g(t,s,U_m(t))-g(t,s,U(t))\big \Vert \ \mathrm {d}s\\ \le&\ \varepsilon . \end{aligned}$$

Since \(\varepsilon \) can be arbitrarily small, taking limit \(m\rightarrow \infty \) implies T is continuous.

Step 3. Moreover, for \(U\in K\),

$$\begin{aligned} \parallel TU(t)\parallel\le & {} \Vert U(0)\Vert +\frac{\lambda _1(C_1+C_4)+C_2+C_3}{\varGamma (\alpha )}(\Vert U(0)\Vert +b)\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_1^{\alpha -\beta }\\\le & {} \Vert U(0)\Vert +b. \end{aligned}$$

Hence, TK is uniformly bounded and so T is completely continuous. At this point, it remains to show that T maps K into an equicontinuous family.

Step 4. Now, let \(U\in K\) and \(t_1,t_2\in J\). Then, if \(0<t_1<t_2\le r_1\), by the assumptions (H1)–(H6), we obtain

$$\begin{aligned} \parallel TU(t_1)&-TU(t_2)\parallel \ \\&\le \ \frac{\lambda _1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \int _0^{t_1}\left( (t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}\right) \Vert a(s)\Vert \ \mathrm {d}s\\ {}&+\frac{\lambda _1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \int _{t_1}^{t_2} (t_2-s)^{\alpha -1}\Vert a(s)\Vert \ \mathrm {d}s\\&+\frac{\lambda _1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \int _0^{t_1}\left( (t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}\right) \Big (\int _0^sh(s-\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\\&+\frac{\lambda _1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \int _{t_1}^{t_2} (t_2-s)^{\alpha -1}\Big (\int _0^sh(s-\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\\&+ \frac{1}{\varGamma (\alpha )}\int _0^{t_1}\Big ((t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}\Big )\big \Vert f(s,U(s))\big \Vert \ \mathrm {d}s\\ {}&+\frac{1}{\varGamma (\alpha )}\int _{t_1}^{t_2} (t_2-s)^{\alpha -1}\big \Vert f(s,U(s))\big \Vert \ \mathrm {d}s\\&+\frac{1}{\varGamma (\alpha )}\bigg \Vert \!\int _0^{t_1}\!\!\!\Big ((t_2-s)^{\alpha -1}\!-(t_1-s)^{\alpha -1}\Big )\bigg (\!\int _0^s\!\! g\big (s,\tau ,U(\tau )\ \mathrm {d}\tau \big )\bigg )\ \mathrm {d}s\bigg \Vert \\&+\frac{1}{\varGamma (\alpha )}\bigg \Vert \int _{t_1}^{t_2} (t_2-s)^{\alpha -1}\bigg (\int _0^s g\big (s,\tau ,U(\tau )\ \mathrm {d}\tau \big )\bigg )\ \mathrm {d}s\bigg \Vert \\&\le \ \frac{\lambda _1C_1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \Bigg (\int _0^{t_1}\left( (t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }}\ \mathrm {d}s\Bigg )^{1-\beta }\\&+\frac{\lambda _1C_1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \Bigg (\int _{t_1}^{t_2} ((t_2-s)^{\alpha -1})^{\frac{1}{1-\beta }}\ \mathrm {d}s\Bigg )^{1-\beta } \end{aligned}$$
$$\begin{aligned}&+\frac{\lambda _1C_4}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \Bigg (\int _0^{t_1}\left( (t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }}\ \mathrm {d}s\Bigg )^{1-\beta }\\&+\frac{\lambda _1C_4}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \Bigg (\int _{t_1}^{t_2} ((t_2-s)^{\alpha -1})^{\frac{1}{1-\beta }}\ \mathrm {d}s\!\Bigg )^{1-\beta }\\&+ \frac{\left( \Vert U(0)\Vert +b\right) }{\varGamma (\alpha )}\Bigg (\!\!\int _0^{t_1}\!\Big ((t_2-s)^{\alpha -1}\!-(t_1-s)^{\alpha -1}\Big )^{\frac{1}{1-\beta }}\ \mathrm {d}s\!\Bigg )^{\!1-\beta }\!\!\left( \!\int _0^t\!\left( m_1(s)\right) ^{\frac{1}{\beta }} \mathrm {d}s\!\right) ^{\!\beta }\\&+ \frac{1}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \Bigg (\int _{t_1}^{t_2} \big ((t_2-s)^{\alpha -1}\big )^{\frac{1}{1-\beta }}\ \mathrm {d}s\Bigg )^{1-\beta }\left( \int _0^t\left( m_1(s)\right) ^{\frac{1}{\beta }} \mathrm {d}s\right) ^\beta \\&+\frac{C_3}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \left( \int _0^{t_1}\left( (t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }}\ \mathrm {d}s\right) ^{1-\beta }\\&+\frac{C_3}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \left( \int _{t_1}^{t_2}((t_2-s)^{\alpha -1})^{\frac{1}{1-\beta }}\ \mathrm {d}s\right) ^{1-\beta }\\&\le \ \frac{\lambda _1(C_1+C_4)+C_2+C_3}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \Bigg (\!\!\int _0^{t_1}\!\!\!\big ((t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}\big )^{\frac{1}{\!1-\beta }}\ \mathrm {d}s\!\Bigg )^{\!1-\beta }\\&+ \frac{\lambda _1(C_1+C_4)+C_2+C_3}{\varGamma (\alpha )}\left( \Vert U(0)\Vert +b\right) \left( \int _{t_1}^{t_2}\big ((t_2-s)^{\alpha -1}\big )^{\frac{1}{1-\beta }}\ \mathrm {d}s\right) ^{1-\beta }. \end{aligned}$$

Clearly, T maps K into an equicontinuous family of functions and it is noted that T is completely continuous by Ascoli–Arzela theorem. Then, applying Leray–Schauder fixed point theorem, we achieve that T has a fixed point in K which is a solution of (16.2).

16.3.2 Neumann Boundary Condition

Next, our aim is to show the existence of solutions of (16.2) with Neumann boundary condition instead of Dirichlet boundary condition. That is,

$$\begin{aligned} \displaystyle {\frac{\partial u(x,t)}{\partial n}}&=\;\ 0, \;\;\;\;\;\;\;(x,t) \in \partial \varOmega \times J, \end{aligned}$$
(16.15)

where n is an outward unit normal. Now, we define the function V(t) by

$$\begin{aligned} V(t)=\frac{\int \limits _\varOmega u(x,t)\ \mathrm {d}x}{\int \limits _\varOmega \mathrm {d}x}. \end{aligned}$$
(16.16)

The following theorem asserts the existence of solution of (16.2) with Neumann boundary conditions (16.15).

Theorem 16.3.2

Assume that there exists a \(\beta \in (0,\alpha )\) for some \(0<\alpha <1\) such that (H2) and (H3) hold. For any constant \(b> 0\), suppose that

$$\begin{aligned} r_2=\min \left\{ T,\left[ \frac{\varGamma (\alpha )b}{(\Vert V(0)\Vert +b)C_2}\left( \frac{\alpha -\beta }{1-\beta }\right) ^{1-\beta }\right] ^{\frac{1}{\alpha -\beta }}\right\} . \end{aligned}$$

Then, there exists at least one solution for the initial value problem (16.2) on \(\varOmega \times [0,r_2]\).

Proof

In order to prove the existence of solutions of (16.2), it is enough to show that the equation

$$\begin{aligned} V(t)&=V(0)+\displaystyle \frac{1}{\varGamma (\alpha )}\int _0^t (t-s)^{\alpha -1} f(s,V(s))\ \mathrm {ds}\nonumber \\&+\frac{1}{\varGamma (\alpha )}\int _0^t (t-s)^{\alpha -1}\bigg (\int _0^sg\big (s,\tau ,V(\tau )\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s \end{aligned}$$
(16.17)

has a solution.

Step 1. Assume u(x, t) to be a solution of (16.2). Then, it follows that u(x, t) is a solution of (16.3). Now, integrating both sides of Eq. (16.3) with respect to \(x\in \varOmega \), we are led to

$$\begin{aligned} \int _\varOmega u(x,t)\ \mathrm {d}x=&\int _\varOmega u_0(x)\ \mathrm {d}x+\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _\varOmega \int _0^t (t-s)^{\alpha -1}a(s)\varDelta u(x,s)\ \mathrm {d}s\ \mathrm {d}x\nonumber \\ {}&+\frac{1}{\varGamma (\alpha )}\int _\varOmega \int _0^t (t-s)^{\alpha -1}\Big (\int _0^s h(s-\tau )\varDelta u(x,\tau )\ \mathrm {d}\tau \Big )\ \mathrm {d}s\ \mathrm {d}x\nonumber \\ {}&+\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _\varOmega \int _0^t (t-s)^{\alpha -1}f(s,u(x,s))\ \mathrm {d}s\ \mathrm {d}x\nonumber \\&+\displaystyle {\frac{1}{\varGamma (\alpha )}}\int _\varOmega \int _0^t (t-s)^{\alpha -1}\bigg (\int _0^sg\big (s,\tau ,u(x,\tau )\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s\ \mathrm {d}x. \end{aligned}$$
(16.18)

Combining Green’s identity and the Neumann boundary condition, the assumption (H2), (16.18) can be written as

$$\begin{aligned} V(t)\le&\, V(0) +\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}f(s,V(s))\ \mathrm {ds}\nonumber \\ +&\,\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\bigg (\int _0^sg\big (s,\tau ,V(\tau )\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$
(16.19)

Now, an operator \(P:C(J,\ \mathbb {R})\rightarrow C(J,\ \mathbb {R})\) is defined by

$$\begin{aligned} PV(t)=&\, V(0) +\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}f(s,V(s))\ \mathrm {ds}\nonumber \\ +&\,\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\bigg (\int _0^sg\big (s,\tau ,V(\tau )\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$
(16.20)

Next, we have to prove that the operator P maps K into itself. From the above equation, we observe that

$$\begin{aligned} \parallel PV(t)-PV(0)\parallel&\le \frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\Vert f(s,V(s))\Vert \ \mathrm {ds}\\&+\frac{1}{\varGamma (\alpha )} \int _0^t(t-s)^{\alpha -1}\bigg (\int _0^s\Vert g\big (s,\tau ,V(\tau )\big )\Vert \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$

Then, by using the Holder inequality and the assumptions (H2) and (H3), we obtain

$$\begin{aligned} \parallel PV(t)&-PV(0)\parallel \le \frac{1}{\varGamma (\alpha )}\int _0^t (t-s)^{\alpha -1}\big \Vert f(s,V(s))\big \Vert \ \mathrm {d}s\\&+\frac{1}{\varGamma (\alpha )}\int _0^t (t-s)^{\alpha -1}\bigg (\int _0^s\Vert g(s,\tau ,V(\tau ))\Vert \ \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$
$$\begin{aligned}&\le \frac{1}{\varGamma (\alpha )}\int _0^t m_1(s)(t-s)^{\alpha -1}\left( \Vert V(s)\Vert \right) \mathrm {d}s\\&+\frac{1}{\varGamma (\alpha )}\int _0^t(t-s)^{\alpha -1}\bigg (\int _0^s m_2(s,\tau )\Vert V(s)\Vert \ \mathrm {d}\tau \bigg )\ \mathrm {d}s\\&\le \frac{1}{\varGamma (\alpha )}\left( \Vert V(0)\Vert +b\right) \left( \int _0^t\left( (t-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }}\ \mathrm {d}s\right) ^{1-\beta }\left( \int _0^t\left( m_1(s)\right) ^{\frac{1}{\beta }}\ \mathrm {d}s\right) ^\beta \\&+\frac{1}{\varGamma (\alpha )}\left( \Vert V(0)\Vert +b\right) \left( \!\int _0^t\!\left( (t-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }} \mathrm {d}s\right) ^{1-\beta }\!\left( \!\int _0^t\!\left( \int _0^s\!m_2(s,\tau )\ \mathrm {d}\tau \right) ^{\frac{1}{\beta }}\! \mathrm {d}s\!\right) ^\beta \\&\le \frac{\left( \Vert V(0)\Vert +b\right) C_2}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_2^{\alpha -\beta }+\frac{\left( \Vert V(0)\Vert +b\right) C_3}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_2^{\alpha -\beta }\\&=\frac{\left( \Vert V(0)\Vert +b\right) (C_2+C_3)}{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_2^{\alpha -\beta }\\&\le b, \ \ \ \ t\in [\,0,r_2\,]. \end{aligned}$$

Since K is closed, we next define a sequence \(\{V_k(t)\}\) in K which has a subsequence \(\{V_{k_i}(t)\}\) such that

$$\begin{aligned} \lim _{k_i\rightarrow \infty }V_{k_i}(t)=\widetilde{V}(t). \end{aligned}$$
(16.21)

Thus, by Lebesgue’s dominated convergence, we obtain

$$\begin{aligned} \widetilde{V}(t)&=\widetilde{V}(0)+\frac{1}{\varGamma (\alpha )}\int _0^t\!(t-s)^{\alpha -1} f\big (s,\widetilde{V}(s)\big )\ \mathrm {d}s\\&+\frac{1}{\varGamma (\alpha )}\int _0^t\!(t-s)^{\alpha -1}\bigg (\int _0^s g\big (s,\tau ,\widetilde{V}(s)\big )\ \mathrm {d}\tau \bigg )\ \mathrm {d}s. \end{aligned}$$

Now, we intend to show that P is continuous.

Step 2. Let \(\{V_m(t)\}\) be a converging sequence in K to V(t). Therefore, for any \(\varepsilon >0\) and for each \(t\in [\,0,r_2\,]\), let

$$\begin{aligned} \big \Vert f\big (t,V_m(t)\big )-f\big (t,V(t)\big )\big \Vert&\le \displaystyle \frac{\alpha \varGamma (\alpha )\varepsilon }{2r_2^\alpha }\ \left( \frac{\alpha -\beta }{1-\beta }\right) ^{1-\beta },\end{aligned}$$
(16.22)
$$\begin{aligned} \big \Vert g(t,s,V_m(t))-g(t,s,V(t))\big \Vert&\le \frac{\varGamma (\alpha )\varepsilon }{2Tr_2^\alpha }\ \left( \frac{\alpha -\beta }{1-\beta }\right) ^{1-\beta }. \end{aligned}$$
(16.23)

Making use of (16.13) and then simplifying, we have

$$\begin{aligned} \Vert PV_m(t)-PV(t)\Vert \le&\ \frac{r_2^\alpha }{\alpha \varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta }\big \Vert f\left( s,V_m(s)\right) -f\left( s,V(s)\right) \big \Vert \\ +&\,\frac{r_2^\alpha }{\varGamma (\alpha )}\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } \int _0^s\big \Vert g(t,s,V_m(t))-g(t,s,V(t))\big \Vert \ \mathrm {d}s\\ \le&\ \varepsilon . \end{aligned}$$

Taking limit \(m\rightarrow \infty \), for sufficiently small \(\varepsilon \), P is continuous.

Step 3. Moreover, for \(V\in K\),

$$\begin{aligned} \parallel PV(t)\parallel\le & {} \Vert V(0)\Vert +\frac{(C_2+C_3)}{\varGamma (\alpha )}(\Vert V(0)\Vert +b)\left( \frac{1-\beta }{\alpha -\beta }\right) ^{1-\beta } r_2^{\alpha -\beta }\\\le & {} \Vert V(0)\Vert +b. \end{aligned}$$

Hence, PK is uniformly bounded. Now, it remains to show that P maps K into an equicontinuous family.

Step 4. Now, let \(V\in K\) and \(t_1,t_2\in J\). Then, if \(0<t_1<t_2\le r_2\), by the assumptions (H2) and (H3), we obtain

$$\begin{aligned} \parallel PV(t_1)&-PV(t_2)\parallel \ \\&\le \ \frac{1}{\varGamma (\alpha )}\int _0^{t_1}\Big ((t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}\Big )\big \Vert f(s,V(s))\big \Vert \ \mathrm {d}s\\ {}&+\frac{1}{\varGamma (\alpha )}\int _{t_1}^{t_2} (t_2-s)^{\alpha -1}\big \Vert f(s,V(s))\big \Vert \ \mathrm {d}s\\&+\frac{1}{\varGamma (\alpha )}\bigg \Vert \!\int _0^{t_1}\!\!\Big ((t_2-s)^{\alpha -1}\!-(t_1-s)^{\alpha -1}\!\Big )\bigg (\!\int _0^s\!\! g\big (s,\tau ,V(\tau )\ \mathrm {d}\tau \!\big )\bigg )\ \mathrm {d}s\bigg \Vert \\&+\frac{1}{\varGamma (\alpha )}\bigg \Vert \int _{t_1}^{t_2} (t_2-s)^{\alpha -1}\bigg (\int _0^s g\big (s,\tau ,V(\tau )\ \mathrm {d}\tau \big )\bigg )\ \mathrm {d}s\bigg \Vert \\&\le \, \frac{\left( \Vert V(0)\Vert +b\right) }{\varGamma (\alpha )}\Bigg (\!\!\int _0^{t_1}\!\!\Big ((t_2-s)^{\alpha -1}\!-(t_1-s)^{\alpha -1}\Big )^{\frac{1}{1-\beta }}\ \mathrm {d}s\!\Bigg )^{\!1-\beta }\!\!\left( \!\int _0^t\!\!\left( m_1(s)\right) ^{\frac{1}{\beta }} \mathrm {d}s\!\right) ^{\!\beta }\\&+ \frac{\left( \Vert V(0)\Vert +b\right) }{\varGamma (\alpha )}\Bigg (\int _{t_1}^{t_2} ((t_2-s)^{\alpha -1})^{\frac{1}{1-\beta }}\ \mathrm {d}s\Bigg )^{1-\beta }\left( \int _0^t\left( m_1(s)\right) ^{\frac{1}{\beta }} \mathrm {d}s\right) ^\beta \\&+\frac{1}{\varGamma (\alpha )}\left( \Vert V(0)\Vert +b\right) \left( \!\int _0^{t_1}\!\left( (t-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }} \mathrm {d}s\right) ^{1-\beta }\!\left( \!\int _0^{t_1}\!\left( \int _0^s\!m_2(s,\tau )\ \mathrm {d}\tau \right) ^{\frac{1}{\beta }}\! \mathrm {d}s\!\right) ^\beta \\&+\frac{1}{\varGamma (\alpha )}\left( \Vert V(0)\Vert +b\right) \left( \!\int _{t_1}^{t_2}\!\left( (t-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }} \mathrm {d}s\right) ^{1-\beta }\!\left( \!\int _{t_1}^{t_2}\!\left( \int _0^s\!m_2(s,\tau )\ \mathrm {d}\tau \right) ^{\frac{1}{\beta }}\! \mathrm {d}s\!\right) ^\beta \\&\le \ \frac{C_2}{\varGamma (\alpha )}\left( \Vert V(0)\Vert +b\right) \Bigg (\int _0^{t_1}((t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1})^{\frac{1}{1-\beta }}\ \mathrm {d}s\Bigg )^{1-\beta }\\&+ \frac{C_2}{\varGamma (\alpha )}\left( \Vert V(0)\Vert +b\right) \left( \int _{t_1}^{t_2}((t_2-s)^{\alpha -1})^{\frac{1}{1-\beta }}\ \mathrm {d}s\right) ^{1-\beta }\\&+\frac{C_3}{\varGamma (\alpha )}\left( \Vert V(0)\Vert +b\right) \left( \int _0^{t_1}\!\!\left( (t_2-s)^{\alpha -1}-(t_1-s)^{\alpha -1}\right) ^{\frac{1}{1-\beta }}\ \mathrm {d}s\!\right) ^{\!1-\beta }\\&+\frac{C_3}{\varGamma (\alpha )}\left( \Vert V(0)\Vert +b\right) \left( \int _{t_1}^{t_2}((t_2-s)^{\alpha -1})^{\frac{1}{1-\beta }}\ \mathrm {d}s\right) ^{1-\beta }. \end{aligned}$$

Thus, P maps K into an equicontinuous family of functions. Then, as in the previous case, from Leray–Schauder fixed point theorem, we conclude that P has a fixed point in K which is a solution of (16.2).

Conclusion

In this chapter, we consider fractional integrodifferential equation describing the motion of fractional Maxwell fluid with zero pressure. This equation gets attention from the fact that the fractional derivatives are used to depict the viscoelasticity phenomena with little amount of constraints. Since this equation has a positive kernel with diffusion term, this is different from the integrodifferential equation considered in [3]. Further, our equation is not a particular case of the equation discussed in [3].