**1. Introduction**

Fixed point theorems are useful tools in nonlinear analysis, the theory of differential equations, and many other related areas of mathematics. One of the most applicable method for various investigations is Banach's contraction principle [1]. Many researchers generalized and extended this theorem to different directions. For example, Boyd and Wong [2] elongated the main result of Banach and they replaced the constant in the contractive condition by an appropriate function. Recently, Samet et al. in [3] defined *α*-admissible and *<sup>α</sup>*-*ψ*-contractive type mappings and studied some of their properties in the framework of complete metric spaces. Later on, Salimi et al. in [4] introduced and investigated the twisted (*<sup>α</sup>*,*β*)-admissible mappings. Many extensions of the notion of *<sup>α</sup>*-*ψ*-contractive type mappings have been developed, see, for example, [5–9] and the references therein.

In 2012, Wardowski ([10]) defined *ϑ*-contraction in the setting of metric space. Wardowski et al. [11] also presented the concept of *ϑ*-weak contraction and generalized the conception of *ϑ*-contraction. Kaddouri et al. [12] extended the notion of *ϑ*-contraction and gave applications of their results to integral inclusions. Arshad et al. in [13] instigated the rational *ϑ*-contraction and obtained some fixed points results in a metric space. Concerning *ϑ*-contractions, we mention the researchers in [14–22].

In all these investigations, the underlying space was complete metric space. There were some open problems for fixed point theorems in ordered metric spaces and cyclic representations of *ϑ*-contraction. To solve the first problem, we define (*<sup>α</sup>*,*β*)-type *ϑ*-contraction with the help of control functions *α* and *β*. With this new notion, we not only generalize the main theorem of Wardowski [10] but also derive the results for ordered metric spaces by these control functions. We also introduce (*<sup>α</sup>*,*β*)-type rational *ϑ*-contraction which extend the notion of *ϑ*-contraction. Moreover, a cyclic (*<sup>α</sup>*-*ϑ*) contraction and cyclic ordered (*<sup>α</sup>*-*ϑ*) contraction are also introduced to solve the second problem.

To illustrate some of the applications of the fixed point theorems studied in this paper, we use the Caputo fractional differential equation. Note that nonlinear fractional differential equations play a very useful role in modeling in various fields of science, such as physics, engineering, bio-physics, fluid mechanics, chemistry, and biology [23,24]. In this paper, based on the proved fixed point theorems, we provide some new sufficient conditions for the existence of the solutions of an integral boundary value problem for a scalar nonlinear Caputo fractional differential equations with fractional order in (1, <sup>2</sup>). We also compare the obtained existence results with known ones in the literature.
