On the Solutions of Nonlinear Hybrid Fractional Integrodi erential Equations

In the present work we study the existence of solutions for hybrid nonlinear fractional integrodi erential equations. We developed an algorithm by using the operator theoretical techniques in order to obtain the approximate solutions. The main results depend on the Dhage iteration method that were incorporated with the modern hybrid xed point theorems. The approximate solutions were obtained by using Lipschitz conditions and weaker form of mixed partial continuity. Further, we provide some examples to explain the hypotheses and the related results.


Introduction
Nonlinear fractional integrodierential equations are important and widely applied in many areas such as in physics, mechanics, electromagnetics, biology, signal processing, economics and more. There are also many dierent methods to solve these type of equations. In particular, the solutions of existence, uniqueness and other properties for these type of equations have been studied by many authors by using dierent techniques. In 2016 Dhage et al. (see, [1]) introduced and proved algorithms for the existence of nonlinear rst order ordinary integrodierential equations and the approximation of solutions to initial value problems.In this paper, we extend their study into the classes of nonlinear hybrid fractional integrodierential equations. Let J = [t 0 , t 0 + a] be a closed-bounded interval in the real space R for some t 0 , a ∈ R with t 0 ≥ 0 and a > 0. Consider the nonlinear hybrid fractional integrodierential equation (N on − HF IDE) of the initial value problem (IV P ) (D α u)(t) + λu(t) = f t, u(t), t t0 (t − s) α−1 Γ(α) g(s, u(s))ds , (1) where f : J × R × R → R and g : J × R → R continuous functions and λ ∈ R(λ > 0), for all t ∈ J. We mean by the solution of Eq.(1), the function u ∈ C 1 (J, R) that holds Eq.(1), where C 1 (J, R) the space of continuously dierentiable real−valued functions on J.
Denition 2.2 [9,10] Riemann & Liouville fractional dierential operator is dened by where m is an integer and α is a real number.
Throughout this study, E denotes a partially ordered real linear normed space with order relation and the norm . in which the scalar product and the addition by non-negative real numbers are preserved by . The space E is called a regular ( see, [11]) if for any non-increasing (resp., non-decreasing) sequence {u n } in E such that u n → u * as n → ∞, u n u * (resp., u n u * ) for all n ∈ N . Denition 2.3 [12,14] A function T : E → E is said to be partially continuous at point a ∈ E if for every > 0 there exists δ > 0 such that  [13] Let (E, , . ) be a complete regular space partially ordered so that the norm . and the order relation in E are compatible in any compact chain C of E. Then T : E → E is a partially continuous, partially compact operator and increasing. If an element u 0 ∈ E exists such that u 0 T u 0 or T u 0 u 0 , then the equation of the operator T u = u has a solution u * in E, and the sequence {T n u 0 } of successive iterations converges monotonously towards u * . Remark 2.1 The regularity of E in Theorem 2.1 above can be substituted by a stronger continuity condition in the operator T in E (see [13]).
In this work, we will prove the existence and uniqueness solution of the N on − HF IDE (1) by using hybrid xed point theorems where we need the following notion of a D−function. Denition 2.7 An upper semi−continuous and monotone increasing function Ξ : R + → R + is said to be a D−function provided Ξ(0) = 0. Denition 2.8 An operator T : E → E is said to be a partial nonlinear D−contraction if there is a D−function such that for u, v ∈ E, where 0 < Ξ(r) < r, ∀r > 0. In particular, if Ξ(r) = kr, ∀k > 0, then T is called a partial Lipschitz operator with constant k and moreover, if 0 < k < 1, then T is known as partial linear contraction on E.
Theorem 2.2 (E, , . ) is a partially ordered, regular, a complete normalized linear space so that the norm . and the order relation in E are compatible in every compact chain C of E. Now A, B : E → E are two increasing operators such that (a) the operator A is partially nonlinear D−contraction and partially bounded, (b) the operator B is partially compact and partially continuous, and (c) there is an element u 0 ∈ E such that u 0 Au 0 + Bu 0 or u 0 Au 0 + Bu 0 .
Then the equation Au + Bu = u has a solution u * in E and the sequence {u n } of successive iterations introduced by u n+1 = Au n + Bu n , n = 0, 1, ..., converges monotonically to u * .

Existence of Solutions and Uniqueness
The equivalent integral form of N on − HF ID (1) is considered in the function space C(J, R) of continuous real−valued functions introduced on J. We dene the order relation and a norm . in C(J, R) by and Obviously, C(J, R) is a Banach space with respect to previous supremum norm and is also partially ordered with respect to the previous partially order relation . Further, it is also clear that Banach space of partially ordered C(J, R) is regular and is a lattice, so each pair of elements in the space has an upper and a lower bound in the space.
The following lemma regarding the compatibility of sets in C(J, R) followed by an application theorem of the Arzela-Ascoli. Lemma 3.3 Let (C(J, R), , . ) be a Banach space of partially ordered with the order relation and the norm . introduced by (4) and (5), respectively. Then, and . are compatible in each partially compact subset of C(J, R).
We can see the proof of the lemma in [1]. Before to show our result, we need the following denition: for every t in J. Likewise, an upper solution y ∈ C 1 (J, R) to the N on − HF IDE(1) is introduced on J by inverting the previous inequalities.
likewise, an upper solution y in C 1 (J, R) to the N on − HF IDE (19) is introduced on J by inverting the above inequalities.

Existence theorem
In the following hypothesis relating to our further discussion: The following lemma is important in our next work and its proof is clear by a direct verication (by using the denition Riemann & Liouville dierential operator).

Lemma 3.4 Let
if and only if it is a solution of the folloeing nonlinear integral equation Theorem 3.5 Let the conditions (A 1 − A 4 ) be held. Then the N on − HF IDE (4) has a solution u * on J and the sequence {u n } ∞ n=1 the successive approximations, dened by g(ξ, u n (ξ))dξ ds (8) for all t in R, converges monotonically to u * .
Proof. From Lemma 3.4, the N on − Hf IDE (4) is equivalent to the nonlinear integral equation Set E = C(J, R). Then, by Lemma 3.3 it follows that each compact chain in E has the property of compatibility with respect to the order relation and the norm . in E. We introduce the operator T by Through a series of steps, we must prove that the operator T fullls all the conditions of Theorem 2.1. Step for all t in J. This proves that T is a increasing operator on E.
Step II: Let T be partially continuous on E. {u n } is a chain points sequences C in E such that u n → u, ∀n ∈ N . Then, by the controlled convergence theorem, for every t in J. This proves that {T u n } converges to T u point-wise on J. Therefore, we prove that {T u n } is an equi-continuous sequence of functions in E. Let t 1 , t 2 in J with t 1 < t 2 . Then Step III: Let T be a operator of partially compact on E and C is an arbitrary chain in E. Then we prove that T (C) is a equi-continuous an uniformly bounded set in E. Firstly, we prove that T (C) is uniformly bounded. We put u ∈ C is arbitrary. Then, = r for every t in J. We are taking the supremum over t, we get T u ≤ r for every u ∈ C. Thus T (C) ≺ E is a uniformly bounded. Therefore, we will prove that T (C) is an equicontinuous set in E. We put t 1 , t 2 ∈ J is arbitrary with t 1 < t 2 . Then Since the functions t → (t − s) α−1 and t → p(t) are uniformly continuous on compact J = [t 0 , t 0 + a], we have that |T u(t 2 ) − T u(t 1 )| → 0 as t 2 → t 1 , uniformly for every u ∈ C. This proves that T (C) is an equi-continuous set in E. Thus T (C) ≺ E is compact and consequently T is a operator of partially compact on E → E.
Step IV : z fullls the operator inequality z ≤ T z.
from the supposition (A 4 ) Achieves, y is a lower solution of N on − HF IDE (4) introduced on J. Thus and for every t in J. The integrating (12) from t 0 to t, we have We applied the theorem 3.5, we get λ = 1, c = 1, g(t, u) = sinh u and f (t, u, v) = tanh u + tanh v.
Obviously, the functions f and g are continuous on J × R, and f achieves (A 1 ) with M f = 2. Furthermore, g(t, u) is increasing in u for any t in J, and f (t, u, v) is increasing in u and v for any t in J, thus conditions (A 2 ) and (A 3 ) are achieved. Finally, the N on − HF IDE (15) has a lower solution z dened by Hence, all the assumption of Theorem 2.1 are achieved, and thus N on − HF IDE (15) has a solution u * introduced on J, and the sequence {u n }, introduced by sinh u n (ξ)dξ ds for all t ∈ J and where Γ( 3

Uniqueness theorem
In this section, we investigate a uniqueness theorem for the N on − Hf IDE (4) by using the weaker partially Lipschitz condition.
Theorem 3.6 A suppose that conditions (A 4 ) − (A 6 ) achieve. Then the N on − Hf IDE (4) contain of an unique solution u * introduced on J, and the sequence {u n } of successive approximations introduced by the Eq. (8) converges monotonically to u * .
Proof. Set E = C(J, R). Obviously, E is a lattice with respect to the order relation and thus upper and lower bounds there is for all pair of elements in E. We introduce the operator T by (10). Then, the N on − Hf IDE (4) is equivalent to the operator equation (11). We must prove that T fullls all the conditions of Theorem 2.1.
Obviously, T is a increasing operator from E → E. We want to prove that the operator T is a partially nonlinear D−contraction on E, thus let u, v ∈ E with u ≥ v. Then, by (A 5 ) and for every t ∈ J, where ξ(r) = a α Γ(α+1) ξ 1 (r) + ξ 2 (Lar) < r, r > 0.
Taking the supremum over t, we get for every u, v ∈ E, with u ≥ v. Consequently, T is a partially nonlinear D−contraction in E. In addition, as in the proof of Theorem 3.5, we can show that the function z given in condition (A 4 ) achieves the inequality of the operator z ≤ T z in J. Now, we apply direction of Theorem 2.1 gives that the N on − Hf IDE (4) has a unique solution u * , and the sequence {u n } of successive approximations introduced by Eq.(10) converges monotonically to u * .  where g : J × R → R is the function introduced by Obviously, the functions f and g are continuous on J × R × R and J × R, consecutively. The function f achieves (A 1 ) with M f = π 2 and it is easy to prove that g achieves (A 5 ) with L = 1. Further, f (t, u, v) is increasing in u and v for any t in J. To prove that f fullls (A 6 ) on J × R × R, let u 1 , u 2 , v 1 , v 2 ∈ R be such that u 1 ≥ v 1 and u 2 ≥ v 2 . Then, for all t ∈ J and for some u 1 > η 1 > v 1 and u 2 > η 2 > v 2 , where Ξ 1 and Ξ 2 are D−functions introduced by , and Ξ 2 (r) = r 1 + η 2 2 for 0 < η 1 , η 2 < r. Furthermore, In the end, the N on − HF IDE (4) has a lower solution z(t) = for every t in J, converges monotonically to u * .

The rst type linear perturbations
At times, it is conceivable that the non-linearity of those involved in N on − HF IDE (1) does not fulll either the supposition of theorem 3.5 or the supposition of theorem 3.6. In spite of, from incising the functions f 1 and f 2 of f in the form f = f 1 + f 2 fulll the conditions of Theorems (3.5 and 3.6).
Consecutively. in Dhage's terminology [1], the resulting equation is said to be the hybrid integrodierential equation with the rst type linear disturbance. The objective of this section is to get an Following similar arguments to those used in the proofs of Theorems 3.5 and 3.6, we can show that operator A is a nonlinear Dcontraction and partially bounded and B is a partially compact operator and partially continuous in E. From the direct application of Theorem 2.2 gives that the operator equation u ≤ Au + Bu has a solution u * . Thus, N on − HF IDE (19) has a sequence {u n } ∞ n=1 , and the solution u * introduced by (20) monotonic converges to u * . This completes the proof. The inference of Theorem 3.7 residues true if we substitute (A 7 ) with (B 7 ). The N on − HF IDE (19) has a upper solution y ∈ C 1 (J, R). u(0)=1.
We applied the Theorem 3.7, we get λ = 1, c = 1, f 1 (t, u, v) = tan −1 u, f 2 (t, u, v) = tanh v and g(t, v) = sinh v. Therefore the function f 1 fullls (A 1 ) with M f1 = π 2 and also fullls (A 6 ) with Ξ 1 (r) = r 1+ξ 2 , 0 < ξ < r and Ξ 2 (r) = 0. Now f 2 fullls (A 1 ) with M f2 = 1 and it is increasing in v, thus (A 2 ) fullls. Likewise, g fullls (A 3 ). In the end, z(t) = 3(e −t − 1), for every t ∈ J, is a lower solution of the N on − HF IDE (25) on J, and thus (A 7 ) is fullled. Next, by Theorem 3.7, the N on−HF IDE (25) possesses a solution u * on J, and the sequence {u n } ∞ n=1 , introduced by for t ∈ J, converges monotonically to x * . Remark 3. 4 We observe that if the N on−HF IDE ( (1) or (19)) have a upper y solution, in addition to a lower solution z such that y ≥ z, then the congruous solutions u * and u * of the N on−HF IDE( (1) or(19)) fullls u * ≤ u * , and these are the maximum and minimum solutions in the vector segment [z, y] of the Banach space E = C(J, R). In fact, the order relation introduced by Eq. (5) is equivalent to the order relation introduced by the order etcher K = {u ∈ c(J, R) | u(t) ≥ 0, f or every t ∈ J} that is a closed set in C(J, R).