A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative

We present a new method to investigate some fractional integro-differential equations involving the Caputo-Fabrizio derivation and we prove the existence of approximate solutions for these problems. We provide three examples to illustrate our main results. By checking those, one gets the possibility of using some discontinuous mappings as coefficients in the fractional integro-differential equations.


Introduction
The fractional calculus has an old history and several fractional derivations where defined but the most utilized are Caputo  . Thus, the fractional Caputo-Fabrizio derivative of order α for the function u is given by CF D α u(t) =  -α t  exp(-α -α (ts))u (s) ds, where t ≥  and  < α <  []. If n ≥  and α ∈ [, ], then the fractional derivative CF D α+n of order n + α is defined by CF D α+n u := CF D α (D n u(t)) []. We need the following results.

Lemma . ([]) Let  < α < . Then the unique solution for the problem
To discuss the existence of solutions for most fractional differential equations in analytic methods, the well-known fixed point results such as the Banach contraction principle is used. In fact, the existence of solutions and the existence of fixed points are equivalent. As is well known, there are many fractional differential equations which have no exact solutions. Thus, the researchers utilize numerical methods usually for obtaining an approximation of the exact solutions. We say that u is an approximate solution for fractional integro-differential equation whenever we could obtain a sequence of functions {u n } n≥ with u n → u. We use this notion when we could not obtain the exact solution u. This appears usually when you want to investigate the fractional integro-differential equation in a non-complete metric space.
In this manuscript, we prove the existence of approximate solutions analytically for some fractional integro-differential equations involving the Caputo-Fabrizio derivative. In fact, the approximate solution of an equation is equivalent to the approximate fixed point of an appropriate operator. This says that by using numerical methods, one can obtain approximations of the unknown exact solution. We will not check the estimates of the exact solution in our examples because our aim is to show the existence of approximate solutions within the analytical method.
Here, we provide some basic needed notions. Let (X, d) be a metric space, F a selfmap on X, α : X × X → [, ∞) a mapping and ε a positive number. We say that F is α-admissible whenever α(x, y) ≥  implies α(Fx, Fy) ≥  []. An element x  ∈ X is called ε-fixed point of F whenever d(Fx  , x  ) ≤ ε. We say that F has the approximate fixed point property whenever F has an ε-fixed point for all ε >  []. Some mappings have approximate fixed points, while they have no fixed points []. Denote by R the set of all continuous mappings g : ∞)  and μ ≥  and also g(x  , x  , x  , , x  ) ≤ g(y  , y  , y  , , y  ) and g(x  , x  , x  , x  , ) ≤ g(y  , y  , y  , y  , ) whenever x  , . . . , x  , y  , . . . , y  ∈ [, ∞) with x i < y i for i = , , ,  []. We say that F is a generalized α-contractive mapping whenever there Then F has an approximate fixed point.

Main results
Now, we are ready to state and prove our main results.
Lemma . Suppose that u, v ∈ H  (, ) and there exists a real number K such that . Also by checking the proof of the last result, one can prove the next lemma.
where a α and b α are given in Lemma .. This completes the proof.
Theorem . Let η(t) ∈ L ∞ (I) and f : I × R  → R be a continuous function such that for all t ∈ I and x, y, w, x , y , w ∈ R. Then the problem () with the boundary condition has an approximate solution whenever  = η * ( + γ  + λ  ) < .
where a α and b α are given in Lemma .. Note that defined by g(t  , t  , t  , t  , t  ) =  t  and α(x, y) =  for all x, y ∈ H  . One can easily check that g ∈ R and F is a generalized α-contraction. By using Theorem ., F has an approximate fixed point which is an approximate solution for the problem ().
Note that H  with the sup norm is not Banach. Thus, we used a new method for investigation of the problem. Now, we investigate the fractional integro-differential problem with boundary condition u() = c, where μ ≥  and α, β, γ , θ , δ ∈ (, ) and c ∈ R.
Theorem . Let η(t) ∈ L ∞ (I) and f : [, ] × R  → R be a continuous function such that for all t ∈ I and x, y, w, x , y , w , u  u  , v  , v  ∈ R. Then the problem () with the boundary condition has an approximate solution whenever  < , where where a α and b α are given in Lemma .. By using Lemmas . and ., we obtain  (t  + t  ) and α(x, y) =  for all x, y ∈ H  . One can easily see that g ∈ R and F is a generalized α-contractive map. By using Theorem ., F has an approximate fixed point which is an approximate solution for the problem ().
Let k and h be bounded functions on I = [, ] and s an integrable bounded function on I with M  = sup t∈I |k(t)|, M  = sup t∈I |s(t)| < ∞ and M  = sup t∈I |h(t)| < ∞. Now, we investigate the fractional integro-differential problem for all t ∈ I and x, y, w, v, x , y , w , v ∈ R. Then the problem () has an approximate solution whenever  < , where Define the map F : for all t ∈ I, where a α and b α introduced in Lemma .. By using Lemma ., we get for all t ∈ I and u, v ∈ H  . Also, we have } and α(x, y) =  for all x, y ∈ H  . One can check that g ∈ R and F is a generalized αcontraction. By using Theorem ., F has an approximate fixed point which is an approximate solution for the problem ().