Further results on existence of positive solutions of generalized fractional boundary value problems

*Correspondence: hojat.afshari@yahoo.com 1Department of Mathematics, Faculty of Basic Science, University of Bonab, Bonab, Iran Full list of author information is available at the end of the article Abstract This paper studies two classes of boundary value problems within the generalized Caputo fractional operators. By applying the fixed point result of α-φ-Geraghty contractive type mappings, we derive new results on the existence and uniqueness of the proposed problems. Illustrative examples are constructed to demonstrate the advantage of our results. The theorems reported not only provide a new approach but also generalize existing results in the literature.

To the best of our observation, the investigation of positive solutions to fractional BVP has not been studied within κ-Ca and κ-RL FOs yet. Moreover, the FP technique based on α-ψ-GC has never been applied to such problems. Inspired by the above results and motivated by the recent evolutions in κ-fractional calculus, in this paper, we apply the FP technique of α-ψ-GC type mappings to investigate the existence of positive solutions for the following fractional BVPs: and where 1 < ı ≤ 2, and C D ı,κ 0+ is κ-FD of order ı in the sense the κ-Ca operator, and g : × R → R + is a continuous function. Throughout the article = [0, 1] and R + = [0, ∞).
We claim that our approach is new and the reported results are different form existing ones in the literature.
The remaining parts of the paper are outlined as follows: Some preliminary facts needed for the proofs of the main results are recalled in Sect. 2. In Sect. 3, we prove the existence of positive solutions for problems (1) and (2) by the aid of the FP result of α-ψ-GC type mappings. Examples are given in Sect. 4 to check the applicability of the theoretical findings. We end the paper by a conclusion.

Preliminaries
Definition 2.1 ([32]) Let ι > 0 and κ be an increasing function, having a continuous derivative κ on (a, b). The left-sided κ-RL fractional integral of a function h with respect to κ on [a, b] is defined by provided that I ι,κ a + exists. Note that when κ( ) = , we obtain the known classical RL fractional integral.   Let be set of all increasing and continuous functions φ : R + → R + satisfying the property: φ(c ) ≤ cφ( ) ≤ c for c > 1 and φ(0) = 0. We denote by F the family of all nondecreasing functions λ : R + → [0, 1 r 2 ) for some r ≥ 1. We say that T is α-admissible if, for , ς ∈ M, we have

Main results
Let M = C( , R + ) and d : M × M → R + be given by Then, (M, d) is a complete b-MS with r = 2.
Then the problem (4) has at least one solution.
Proof By Lemma 2.8, ∈ C( ) is a solution of (6) if and only if is a solution of the integral equation By Lemma 2.8, for 0 < κ < ϑ < 1 we have For 0 < ϑ < κ < 1, the same estimates can be proved in analogous way to the previous one. So we will omit it. Using (i), we get Thus, So for , w ∈ C( ) with μ( (ϑ), w(ϑ)) ≥ 0, we have So, we conclude that O is a α-φ-GC type mapping. From (iii), we get for , w ∈ C( ). Thus, O is α-admissible. From (ii), there exists 0 ∈ C( ) with α( 0 , O 0 ) ≥ 1. By (iv) and Theorem 2.7, we find * with * = O * , that is, a positive solution of (4).
Then (4) has at least one solution.
Then (6) has at least one solution.

Conclusion
In recent years and with the explosive growth of studies of derivatives of fractional order, there have appeared tremendous numbers of papers that reported their results by using the classical FDs and FP theorems. Meanwhile, interested researchers have raised the question of the possibility of introducing a different approach that covers all classical cases.
In this paper, we provided an affirmative answer to this inquiry by investigating the notion of existence of solutions for BVPs defined within κ-generalized FD and with the help of the FP technique based on α-φ-GC type mappings. The results reported in this paper generalize existing results in the literature. Two examples are presented as particular cases for our proposed theorems. It is proved that the results obtained are consistent with our theoretical findings.
We believe that the investigation of this problem in terms of a general approach will provide an effective platform for the study of BVPs via generalized FOs.