Fixed-Point Theorems for Rational Interpolative-Type Operators with Applications

In the current manuscript, two fixed-point theorems for Dass-Gupta and Gupta-Saxena rational interpolative-type operators are studied in the setting of metric spaces. For the authenticity of the presented work, examples and applications to the existence of a solution to the Caputo-Fabrizio fractional derivative and Caputo-Fabrizio fractal-fractional derivative are also discussed.


Introduction and Preliminaries
In [1], Banach proposed his famous postulation, the Banach contraction principle. Banach proposed that a continuous self-operator in the setting of complete metric space possesses a unique fixed point. The contraction condition was generalized in several directions. One of the generalizations was supposed by Kannan [2,3] in which he dropped the continuity assumption of the operator. As a part of the generalization of the Banach contraction principle, Dass-Gupta [4] and Gupta-Saxena [5] introduced the innovation of rational contraction. In recent times, Karapinar [6] converted the classical Kannan [2] contraction to an interpolative Kannan contraction in order to maximize the rate of convergence of an operator to a unique fixed point. However, Karapinar and Agarwal [7] found a little gap in the article [6] about the assumption of the fixed point being unique. They provided a counterexample to verify that the fixed point need not be unique and invalidate the assumption of a unique fixed point.
Invented more than a century ago, fractional calculus has attracted numerous physicists, mathematicians, engineers, and researchers in the field of biological sciences because of its extraordinary involvement in these fields of sciences. Despite the long debate, fractional calculus is still not mature enough, and scientists from every field need to do a lot more in this particular branch of mathematics to solve some of its complicated problems. Fractional calculus is mainly composed of fractional integral equation, fractional differential equations, and fractional integrodifferential equations. From the very first day, scientists are finding ways to solve these types of integral equations cited in the manuscripts which can provide enough insight on these problems and their solution [8][9][10][11][12]; among all the other techniques, one technique is a fixed-point theory; several articles can be found on this topic (see, for instance, [13][14][15][16][17] and references therein).
In the fixed-point theory, researchers mostly try to check the existence of a solution to the problem in an underlying set. Because of its complicated nature, it is sometimes impossible to find the exact solution to fractional-type equations in some cases; therefore, frequently, scientists attempt to find the nature of the solution rather than an analytic or exact solution. The particular type of differential equations, namely, Caputo-Fabrizio fractional and Caputo-Fabrizio fractal-fractional differential equations [18][19][20], can also be used in many problems of the aforesaid fields like heat transfer problem, Fisher reaction diffusion equation, mass spring damper system, modeling of steady heat flow, etc.
Motivated by the work done in [6,7,21], we have converted the famous rational contractions in [4,5] to interpolation rational contractions with examples given in each case to verify each theorem. Consequently, the enormous amount of applications of fractional differential equation has led us to contribute the existence of a solution to a couple of fractional differential equations, i.e., the Caputo-Fabrizio fractional differential equation and Caputo-Fabrizio fractal-fractional differential equation. The analysis in the current study indicates that whenever the aforementioned fractional differential equations satisfy certain conditions under specific circumstances, then a contraction theorem guarantees the existence of at least one solution.
Theorem 1 (see [1]). Consider ðM, pÞ to be a complete metric space. Let T : M ⟶ M be a continuous self-operator if there exists a constant δ ∈ ½0, 1Þ , such that ∀f, g ∈ M, then T possesses a unique fixed point.
Afterwards, a question was raised: what happens to the fixed point when the operator is not continuous? Ultimately, the answer to the question was given by Kannan in [2]. He proposed the following theorem as a modified version of Theorem 1.
Theorem 2 (see [2]). Consider ðM, pÞ to be a complete metric space. Let T : M ⟶ M be as a self-operator if there exists a constant δ ∈ ½0, 1/2Þ , such that ∀f, g ∈ M, then T possesses a unique fixed point.
Theorem 4 (see [5]). Consider ðM, pÞ to be a complete metric space. Let T : M ⟶ M be a self-operator if there exists con- ð4Þ ∀ distinct f, g ∈ M, then T possesses a unique fixed point.

Main Results
This section provides the extension of the famous rationaltype contractions to interpolative rational contraction. Before proceeding to the first result of the current section, consider the definition which can later be used in the proof of the first main result of this section.
It can be analyzed that Definition 5 is the conversion of the Dass-Gupta rational contraction in Theorem 3 to an interpolation Dass-Gupta rational contraction.

Theorem 6.
In the setting of a complete metric space, the operator for interpolative Dass and Gupta rational-type contraction defined in Definition 5 possesses a fixed point.

Proof.
Taking an arbitrary f 0 ∈ ðM, pÞ and constructing an iterative sequence ff n g n∈ℕ as f n = Tðf n−1 Þ = T n ðf 0 Þ. If f n 0 = f n 0 +1 for any n 0 > 0, then f n 0 is a fixed point for T, which completes the proof. Consequently, taking f n ≠ f n+1 for each n ≥ 0 and by replacing f by f n and g by f n−1 in (5), it is deduced that With straightforward calculation, it can be analyzed that Hence, it can be observed that the sequence fpðf n , f n−1 Þg is a sequence of nonnegative terms which is nonincreasing. As a consequence, there is a nonnegative constant L such 2 Journal of Function Spaces that lim n→∞ pðf n , f n−1 Þ = L. It is presumed that L > 0. Indeed, from (7), it is obvious that Letting n ⟶ ∞ in (8), it can be concluded that L = 0. As a proceeding step, it is proven that the given sequence is a Cauchy sequence. Using the triangle inequality Letting n ⟶ ∞ in (9), it is deduced that the sequence ff n g n∈ℕ is a Cauchy sequence. As stated, ðM, pÞ is a complete metric space; such an assumption guarantees the existence of a number f ∈ M such that lim n→∞ pðf n , fÞ = 0. At last, it is proven that f is the fixed point of T of the sequence ff n g n∈ℕ . Suppose that f ≠ Tf; therefore, pðf, TfÞ > 0. Recall that f ≠ Tf and for each n ≥ 0 and by letting f = f n and g = f in (5), it is determined that By lettingn ⟶ ∞in ((10)), it is analyzed thatpðTf, fÞ = 0; thus,f = Tfwhich is, subsequently, a contradiction. Therefore, f = Tf.
The proof is complete.
Next, consider the extension of the Gupta-Saxena-type rational contraction to interpolative contraction. Before proceeding to the theorem, consider the definition. ð15Þ ∀ distinct f, g ∈ M/FixðMÞ. Then, T is known as an interpolative Gupta-Saxena rational-type contraction.
It can be analyzed that Definition 8 is the conversion of Gupta-Saxena rational contraction in Theorem 4 to an interpolation Gupta-Saxena rational contraction.

Theorem 9.
In the setting of a complete metric space, the operator for interpolative Gupta-Saxena rational-type contraction defined in Definition 8 possesses a fixed point.
Proof. By taking an arbitrary f 0 ∈ ðM, pÞ and constructing an iterative sequence ff n g n∈ℕ as f n = Tðf n−1 Þ = T n ðf 0 Þ. If f n 0 = f n 0 +1 for any n 0 > 0, then f 0 is a fixed point for T, which completes the proof. Consequently, taking f n ≠ f n+1 for each n ≥ 0 and by replacing f by f n and g by f n−1 in (15), it is deduced that 3 Journal of Function Spaces With straightforward calculation, it can be analyzed that Hence, it can be observed that the sequence fpðf n , f n−1 Þg is a sequence of nonnegative terms which is nonincreasing. As a consequence, there is a nonnegative constant L such that lim n→∞ pðf n , f n−1 Þ = L. It is presumed that L > 0. Indeed, from (17), it is obvious that Letting n ⟶ ∞ in (18), it can be concluded that L = 0. As a proceeding step, it is proven that the given sequence is a Cauchy sequence. Using the triangle inequality Letting n ⟶ ∞ in (19), it is deduced that the sequence ff n g n∈ℕ is a Cauchy sequence. As stated, ðM, pÞ is a complete metric space; such an assumption guarantees the existence of a number f 0 ∈ M such that lim n→∞ pðf n , f 0 Þ = 0. At last, it is proven that f 0 is the fixed point of T of the sequence ff n g n∈ℕ . Suppose that f ≠ Tf; therefore, pðf, TfÞ > 0. Recall that f ≠ Tf and for each n ≥ 0 and by letting f = f n and g = f in (15), it is determined that By letting n ⟶ ∞ in (20), it is analyzed that pðTf n , fÞ = 0; thus,f = Tf, which is, subsequently, a contradiction. Therefore, f = Tf.
The proof is complete.

Applications
The current section provides the existence of a solution to the Caputo-Fabrizio fractional derivative and the Caputo-Fabrizio fractal-fractional derivative. The very first result of this section is the existence of the solution to the Caputo-Fabrizio fractional derivative.
To begin the proof, consider Using inequality (26), the above inequality can be written as So, where δ = 2ð2 − γÞNðγÞ/κð1 − γÞ. Consequently, all the propositions of Theorem 6 are satisfied. Therefore, the equation (24) possesses a solution.
Next, we discuss the existence of a solution to the Caputo-Fabrizio fractal-fractional derivative Consider the Caputo-Fabrizio fractal-fractional derivative of order γ as where NðγÞ is a normalization function satisfying N ð0Þ = Nð1Þ = 1.