## Inequality for integro-differential system

Reading a paper about homogenization of stochastic coefficients [A. Gloria et al., Invent. math. (2015)], I found the following lemma that gives an estimate of the solution for a given ODE system. The lemma is:

Let $$1\le p,\gamma<\infty$$ and $$a(t),b(t)\ge 0$$. Suppose that there exists $$C_1 <\infty$$ such that for all $$t\ge0$$, $$\begin{gather} \tag{1}\label{eq1} a(t) \le C_1 \left( (t+1)^{-\gamma} + \int_{0}^{t} (t-s+1)^{-\gamma} b(s) \,ds \right), \ b(t)^p \le C_1\left(-\frac{d}{dt}a(t)^p\right). \end{gather}$$ Then there exists $$C_2<\infty$$ depending only on $$C_1,p$$ and $$\gamma$$ such that $$$$\tag{2}\label{eq2} a(t) \le C_2 (t+1)^{-\gamma}.$$$$

My question is: Is it possible to prove a similar result replacing \eqref{eq1} by $$\begin{equation*} a(t) \le C_1 \left( t^{-\gamma} + \int_{0}^{t} (t-s)^{-\gamma} b(s) \,ds \right)? \end{equation*}$$ How would \eqref{eq2} change?

## Coupled partial differential and integro-differential equation

I have derived two equations of the following type $$\dfrac{\partial A}{\partial x}=a\dfrac{\partial B}{\partial t}-b\dfrac{\partial^3 B}{\partial x^2 \partial t}$$ and $$\dfrac{\partial B}{\partial x}=\int _0^l e^{-\lambda|x-x’|}\dfrac{\partial A(x’)}{\partial t} dx’$$ Where $$A$$ and $$B$$ are functions of $$x$$ and $$t$$, $$x$$ and $$x’$$ are any point between $$0$$ and $$l$$ and $$a, b, \lambda$$ are constants.
Is it possible to transform these two equations into a single partial differential equation for $$B$$?

## Conversion of Integro-differential equation into integral equation

I am stuck how equation (72) can come by just integrating equation (71) 3 times from 0 to x how to integrate eq 71 to convert it into eq 72