Integral-Type Fractional Equations with a Proportional Riemann–Liouville Derivative

Lately, many researchers have been focusing on the study of various types of fractional problems; we refer the reader to [1–17]. 'e fixed point and the monotone iterative techniques can be very useful tools to prove the existence and uniqueness of a solution to this type of problems; see [1]. In this manuscript, inspired by the work of Jankowski in [1], we investigate the existence and uniqueness of a solution to the following problem:

(i) e following integral is called the proportional Riemann-Liouville fractional integral: (ii) e following derivative is called the proportional Riemann-Liouville fractional derivative: Next, we present the following proposition.
Proposition 1 (see [18]). If α, c ∈ C, where Re(α) > 0 and Re(c) > 0, then for any In Section 2, we prove the existence and uniqueness of a solution to problem (1) using the fixed point technique. In Section 3, we prove the existence and uniqueness of a solution to problem (1) using the monotone iterative method. In the conclusion, we present an open question.
Note that if S has a unique fixed point and that is Sξ(t) � ξ(t), then initial value problem (1) has a unique solution, i. e., it will be enough to show that S is a contraction map. So, let ξ, Y ∈ C 1− α (J, R); we have two cases: Hence, S is a contraction map. erefore, S has a unique fixed point as desired. Case 2: α ∈ ((1/2), 1); in this case, we use ‖ · ‖ * with the positive constant λ > 0 such that 2

Journal of Mathematics
It is not difficult to see the following: Also, recall the Schwarz inequality for integrals: us, S is a contraction map. erefore, S has a unique fixed point as required.

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As an application to eorem 1, consider the following problem: then it is not difficult to see that, by using eorem 1, problem (9) has a unique solution. In closing of this section, the following linear problem is considered: Now, we introduce the following hypothesis.

Hypothesis 1 (H 1 )
(1) L(t) � L, t ∈ J or (2) e function L is nonconstant on J and e following lemma is a consequence of eorem 1.
We would like to bring to the reader's attention that, in [1], in the hypothesis ρ should be as follows: ρ ≡ (T q /Γ(2q))[K + (WLT/2q)] which he used to prove the case where q ∈ (0, (1/2)]. is way, his result will be stronger or he can just change the last equality to the inequality.

Monotone Iterative Method
First of all, we start by introducing the following hypothesis.

Hypothesis 2 (H 2 )
(1) L(t) � L, t ∈ J or (2) e function L is nonconstant, and if L(t) is negative, then there exists L which is nondecreasing, where − L(t) ≤ L(t) on J and for every x ∈ J, we have Now, for our purpose, we prove the following useful lemma.
Proof. Assume that our lemma is false, that is, there exist x, y ∈ [0, a) such that q(x) � 0, q(y) > 0, and q(t) ≤ 0 for t ∈ (0, x]; q(t) > 0 for t ∈ (x, y]. Let x 0 be the first maximal point of q on [x, y]. which leads us to a contradiction given the fact that B ≤ 0. Case 2: assume that L(t) ≤ 0 for all t ∈ J, and consider L to be nondecreasing on J. Now, if we apply I α,ρ on problem (14), we obtain Notice that q(0)(e βt t α− 1 /ρ α− 1 Γ(α)) ≤ 0 which is due to the fact that q(0) ≤ 0. us, 4 Journal of Mathematics Using hypothesis (H 2 ) implies that q(x 0 ) ≤ 0, which leads us to a contradiction, and this concludes our proof.

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We say that y is a lower solution of problem (1) if and we say that y is an upper solution of problem (1) if Next, the following hypothesis is defined.

Hypothesis 3. (H 3 ).
ere exists a function L ∈ C(J, R) where Theorem 2. Assume that x 0 is a lower solution of problem (1) and y 0 is an upper solution of problem (1), where x 0 , y 0 ∈ C 1− α (J, R). Moreover, assume that hypotheses H 1 , H 2 , and H 3 hold; problem (1) Proof. Using Lemmas 1 and 2, the proof is similar to the proof of eorem 2 in [1].
□ Now, we present the following example.

Conclusion
In closing, note that the results of Jankowski [1] are a special case of our work which is by taking ρ � 1. Also, we would like to bring to the reader attention the following open question.
What are the necessary and sufficient conditions for problem (1) to have a unique solution if ρ is not constant , but it is a function of $t$ say g(t), so that the problem involves D α,g(t) ?

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.