Analysis of implicit type of a generalized fractional differential equations with nonlinear integral boundary conditions

The given paper describes the implicit fractional differential equation with nonlinear integral boundary conditions in the frame of Caputo-Katugampola fractional derivative. We obtain an analogous integral equation of the given problem and prove the existence and uniqueness results of such a problem using the Banach and Krasnoselskii fixed point theorems. To show the effectiveness of the acquired results, convenient examples are presented.


Introduction
The subject of fractional differential equations (FDEs) has lately developed as a motivating area of research. In reality, fractional derivatives kinds supply an excellent tool for the description of the memory and hereditary properties of different materials and operations. More researchers have established that FDEs show in many research scopes, such as physics, chemical technology, biotechnology, population dynamics, and economics. There has been considerable growth in FDEs involving many fractional derivatives such as Caputo, Riemann-Liouville, Hilfer, and Hadamard have been investigated and advanced by employing various tools from the nonlinear analysis. See the monographs of Podlubny [1], Kilbas et al. [2], Malinowska et al. [3], and some articles, for e.g., [4,5,6,7,8,9,10,11,12,13,14,15,16,17] and the references cited therein. The forward evolution of the FDEs have been caught much interest lately as study some results on the existence and uniqueness of solutions of various types of FDEs under different conditions have been studied by many authors by employing fixed point techniques, see [18,19,20,21,22,23,24,25,26] and many other references.
In recent years [27] the researcher inserted a new fractional integral, which generalizes the Riemann-Liouville and Hadamard integrals into one form. For more properties of this new operator and similar operators, can be seen in [28,29]. The identical fractional derivatives were established in [3,30,31] which named Katugampola fractional operators.
The existence and uniqueness results of FDEs involving Caputo-Katugampola deriva- have been discussed using the Peano and Picard-Lindelö theorems in [32].
In [33], the author presented a new type of fractional operator as a generalization of the Caputo and Caputo-Hadamard fractional derivative operators. Also, he applied the Gronwall inequality type to obtain the uniqueness theorem of the problem (1.1)-(1.2).
In [34], the authors established the existence, uniqueness, and Ulam-Hyers stability results of boundary value problems (BVP) for an implicit FDEs with anti-periodic condition involving Caputo-Katugampola type by applying some fixed point theorems and generalized Gronwall inequality.
Recently, there are some papers dealing with the qualitative properties of solutions of nonlinear FDEs by using techniques of nonlinear functional analysis see for e.g. [35,36,37,38,39,40,41].
The aim of this study is to extend and generalize some reported results in the literature through investigation in the existence and uniqueness of solutions for the given problem (1.3)-(1.4), in which the studied problem with nonlinear integral boundary condition is more general. Our analysis is depends on Banach's and Krasnoselskii's fixed point theorems [42].
The remainder of the paper is displayed as follows: In Section 2, we recall some essential definitions and properties which will be useful throughout this article and we proving some axiom lemmas which play a key role in the sequel. Section 3 contains certain sufficient conditions to corroborate the existence and uniqueness results of the

Preliminaries
For the sake of convenience of the readers, we present some background materials from fractional calculus theory and nonlinear analysis [27,30,33,34,42,43] to facilitate the analysis of our problem ( [27] The Katugampola fractional integral of order σ > 0 with ρ > 0 is defined by if the integral exists, where, Γ (·) is a gamma function.
Theorem 2.9. [42] (Krasnoselskii's fixed point theorem) Let X be a Banach space, let Ω be a bounded closed convex subset of X and let F 1 , F 2 : Ω → Ω a be mapping such that F 1 x + F 2 y ∈ Ω for every pair x, y ∈ Ω. If F 1 is contraction and F 2 is completely continuous, then there exists z ∈ Ω such that F 1 z + F 2 z = z.

Existence and uniqueness theorems for (1.3)-(1.4)
In this section, we give the results on the existence, the uniqueness of solution for problem (1.3)-(1.4) depending on Theorems 2.8, 2.9. The next lemma plays a necessary role in analysis our results.
if ω(ϑ) satisfies the following fractional integral equation Proof. Applying I σ;ρ a + on both sides of (3.1, and employing Lemma (2.7), we get where c 0 , c 1 ∈ R. Take the limit of the equations (3.4) as ϑ → a, ϑ → T respectively, it follows from the integral boundary conditions (3.2) that and The converse follows by Lemmas 2.5 and 2.6. The proof is completed.
As result of Lemma 3.1, we have the following Lemma:  (H 1 ) There exists a constant 0 < L g < 1 such that: (H 2 ) There exists a constant 0 < L h < 1 such that: (H 3 ) The following inequaility holds
Since G ω (·) is continuous on J, the operator CK D σ;ρ a + Fω(ϑ) is continuous on J, that is Second, we apply the Theorem 2.8 to show that F has a fixed point. In fact, it sufficient to prove that F is contraction. Let ω 1 , ω 2 ∈ C(J, R) and ϑ ∈ J. Then By (H 1 ), we get and using (H 2 ), we obtain 14) The relations (3.13), (3.14) and (3.12) give Since Υ < 1, the operator F is contraction. Consequently, Theorem 2.8 shows that the problem (1. Proof. Consider the operator F defined by (3.7). Set the ball B r 0 := {ω ∈ C(J, R) : ω r 0 }, with r 0 Θ 1−Υ , where Θ and Υ is defined as in Theorem 3.3. Moreover, we define the operators F 1 and F 2 on B r 0 by Clearly, for any ω ∈ C(J, R), The proof will be divided into several stages as follows: Applying the same arguments in (3.10) and (3.11), we obtain For each ω 1 , ω 2 ∈ B r 0 and ϑ ∈ J, we have This proves that F 1 ω 1 + F 2 ω 2 ∈ B r 0 for every ω 1 , ω 2 ∈ B r 0 . Stage 2 F 1 is a contration mapping on B r 0 .
Since F is contraction mapping as in Theorem 3.3, then F 1 is a contraction map too. Stage 3. Here, we shall prove that operator F 2 is completely continuous on B r 0 . Obviously, F 2 is continuous due to the continuity of G ω (·). Next, it is not difficult to conclude that due to definitions of Υ and r 0 . This verifies that F 2 is uniformly bounded on B r 0 Finally, we show that F 2 maps bounded sets into equicontinuous sets of C(J, R).
Let ϑ 1 , ϑ 2 ∈ J, with ϑ 1 < ϑ 2 and for any ω ∈ B r 0 . Then we have As ϑ 1 −→ ϑ 2 the right-hand side of the above inequality is not dependent on ω and tends to zero. Therefore, This proves that F 2 is equicontinuous on B r 0 . An application of Arzela-Ascoli Theorem shows that F 2 is relatively compact on B r 0 . Hence all the assumptions of the Theorem 2.9 are satisfied. Thus, we deduce that the problem (1.3)-(1.4) has at least one solution on J.

Conclusions
In this article, we have studied a type of a nonlinear IFDE with the nonlinear integral boundary condition involving a Caputo-Katugampola fractional derivative. We have also established sufficient conditions ensuring existence, and uniqueness of solutions for a proposed proplem by applying some fixed point theorems. We confident the obtained results here will have a favorable impact on the evolution of more applications in applied sciences and engineering.