Solutions of the Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space

The present paper aims to define three new notions: Θe-contraction, a Hardy–Rogers-type Θ-contraction, and an interpolative Θ-contraction in the framework of extended b-metric space. Further, some fixed point results via these new notions and the study endeavors toward a feasible solution would be suggested for nonlinear Volterra–Fredholm integral equations of certain types, as well as a solution to a nonlinear fractional differential equation of the Caputo type by using the obtained results. It also considers a numerical example to indicate the effectiveness of this new technique.


Introduction
The fixed point theory is derived from the investigation of the solution for certain types of differential equations using the successive approximation method.Indeed, the renowned fixed point theorems of Banach [1] are a reformulation of the successive approximation method that was used by some famous mathematicians, namely Cauchy, Liouville, Picard, Lipschitz, Peano, etc.This fact also indicates that the advances and progress in fixed point theory can be referred back to differential equations and the integral equations.On the other hand, in recent years, fixed point theory has been used very extensively to find solutions of nonlinear fractional differential equations.
Indeed, in the last few decades, fractional calculus and fractional differential and integral equations have been the most interesting research topics, not only in mathematics, but also in physics .We can find a brief historical introduction to fractional derivatives with basic notations, illustrations, and results in [2][3][4].Since the beginning, it has been known that the theory has wide applications not only in nonlinear analysis and computational mathematics, but also in applied sciences, including computer science and economics.The applications of these fixed point theories have been presented in the last century, due to this strong relation of fixed point theory and the applications used in several disciplines.
Remark 1.It is clear that in the case of θ(x, y) = s, for s ≥ 1, the extended b-metric becomes the standard b-metric.As is known well, the b-metric does not need to be continuous.As a result, the extended b-metric is not necessarily continuous either.In this paper, it is presumed that the extended b-metric is continuous.

Definition 2 ([15]
).Let S be a non-empty set endowed with the extended b-metric δ e , and a sequence {x n } in S is said to: (a) converge to x if for any given > 0, there exists N = N( ) ∈ N such that δ e (x n , x) < , for all n ≥ N.
In brief, we write lim n→∞ x n = x.(b) be fundamental (Cauchy) if for every > 0, there exists N = N( ) ∈ N such that δ e (x m , x n ) < , for all m, n ≥ N.
Furthermore, this study defines the completeness of δ e -metric space as follows: (c) If any fundamental (Cauchy) sequence in S is convergent, then we say that (S, δ e ) is complete.
For more interesting examples and basic results in δ e -metric space, we refer to [16][17][18][19][20].For some recent modifications or developments to extended b−metric spaces, the reader may refer to the so-called controlled and double-controlled metric type spaces in [21,22] and for further fixed point investigations in extended b-metric spaces to [23].
With reference to the above facts, the proposed three new concepts are Θ e -contraction, a Hardy-Rogers-type Θ-contraction, and an interpolative Θ-contraction in δ e -metric space, and we prove pertinent fixed point theorems in Section 2. By using the obtained results in Section 2, we propose the solutions of the nonlinear integral equation and fractional differential equation via the fixed point approach, which are presented in Sections 3 and 4. The effectiveness of this approach is illustrated by a numerical experiment in Section 5.

Main Results
Now, we start this section by introducing the concept of Θ e -contraction.Definition 3. A self-mapping T, on an extended b-metric space (S, δ e ), is named a Θ e -contraction if there exists a function θ ∈ Θ such that: θ(δ e (Tx, Ty)) ≤ [θ(δ e (x, y))] r if δ e (Tx, Ty) = 0 for x, y ∈ S, where r ∈ [0, 1) such that lim sup m,n→∞ ω(x n , x m ) < 1 r ; here, x n = T n x 0 for x 0 ∈ S.
Theorem 1.If a self-mapping T, on a completed extended b-metric space (S, δ e ), forms a Θ e -contraction, then T has a unique fixed point in S.
Proof.For an arbitrary point x 0 ∈ S, we construct an iterative sequence {x n } 0 ∞ as follows: x n = T n x 0 for all n ∈ N.
Suppose, if T n * x = T n * +1 x for some n * ∈ N, then T n * x will be a fixed point of T. Therefore, without loss of generality, we can assume that δ e (T n x, T n+1 x) > 0 for all n ∈ N. From Definition 3, we have: Recursively, we find that: Accordingly, we obtain that: Letting n → ∞ in (1), we get θ(δ e (x n , x n+1 )) → 1 as n → ∞.
From (Θ 2 ), we have: lim From (Θ 3 ), there exist q ∈ (0, 1) and ∈ (0, ∞] such that: We presume < ∞ and B = 2 > 0. On account of the limit definition, there exists n 0 ∈ N such that: Then, we derive that: Assume = ∞ and B > 0 (an arbitrary positive number).From the definition of the limit, there exists Subsequently, in all cases, there exist 1 B > 0 and n 0 ∈ N such that: Using Equation (1), we obtain: As n → ∞ in the inequality above, we find: Thus, there exists n 1 ∈ N such that: Let N = max{n 0 , n 1 }.Due to the modified triangle inequality, we derive that: This can be written as, , we have: , which is convergent as n, m → ∞ and 1 q > 1.Thus, the sequence {x n } in S is a Cauchy sequence.Since (S, δ e ) is a complete δ e -metric space, there exists a point η in S such that {x n } converges to η.
One can easily note that T is continuous.Suppose that Tx = Ty.Taking the expression (3) into account, we have: Regarding (Θ 1 ), it implies that, δ e (Tx, Ty) ≤ δ e (x, y) for all distinct x, y ∈ S. From this evaluation, we can get, δ e (x n+1 , Tη) = δ e (Tx n , Tη) ≤ δ e (x n , η) for all n ∈ N.
As n → ∞ in the inequality above, we derive x n+1 → Tη.By the uniqueness of the limit, Tη = η.
Suppose f has another fixed point Now, using the condition (3), we get, Therefore, η = ζ.This claims that T has a unique fixed point in S.
Example 3. Let S = [0, ∞).Define δ e : S × S → [0, ∞) as: and ω : Then, (S, δ e ) is a complete extended b-metric space.Define T : X → X as Tx = x 3 , so that δ e (Tx, Then, all the conditions of Theorem 1 are satisfied so that the mapping T has a unique fixed point "0" in S. If we take ω(x, y) = b > 1 in the above theorem, then we get the below corollary.If we take ω(x, y) = 1 in the above theorem, then we get the below corollary.
Corollary 2. Let T be a self-mapping on a complete metric space (S, d).If there exist ϑ ∈ Θ and r ∈ (0, 1) such that: then T has a unique fixed point in S.
Theorem 2. If a self-mapping T, on a completed extended b-metric space (S, δ e ) forms an HR-Θ-contraction, then T has a unique fixed point in S.
Proof.As in Theorem 1, we construct an iterative sequence {x n } 0 ∞ by starting at an arbitrary point x 0 ∈ S as follows: x n = f n x 0 for all n ∈ N.
Without loss of generality, we suppose that δ e ( f n x, f n+1 x) > 0 for all n ∈ N. Indeed, if f n * x = f n * +1 x for some n * ∈ N, then f n * x will be a fixed point of T.
We prove that lim n→∞ δ e (x n , x n+1 ) = 0. Employing the contraction condition (5), we get, where: )), then the inequality (6) becomes: which is a contradiction (since r < 1).Thus, we have M f ,θ (x n , x n−1 ) = θ(δ e (x n 1 , x n )).It is yielded from (6) that: Iteratively, we find that: After this observation, by following the related lines in the proof of Theorem 2, we conclude that the sequence {x n } in S is a Cauchy sequence.Regarding that (S, δ e ) is a complete δ e -metric space, there exists a point η in S such that {x n } converges to η.
Without lose of generality, we may assume that f n x = η for all n (or, for large enough n.) Assume that δ e (η, Tη) > 0. Employing (5), we get: for all x, y ∈ S, where: By taking n → ∞ in the inequality above, we derive that: That is, f has a fixed point in S. Suppose f has another fixed point ζ such that η = ζ.Then, clearly, δ e (η, ζ) = δ e ( f η, f ζ) = 0. Now, using the condition (7), we get, Thus, f has a unique fixed point in S.
Definition 5. Let (S, δ e ) be a δ e -metric space and f : X → X be a mapping.Then, f is said to be an interpolative-Θ-contraction if there exists a function θ ∈ Θ and non-negative real numbers r 1 , r 2 , r 3 , r 4 with r 1 + r 2 + r 3 + 2r 4 < 1 such that: for all x, y ∈ S.
Theorem 3. Let (S, δ e ) be a complete δ e -metric space such that δ e is a continuous functional and f : X → X be an interpolative-Θ-contraction.Then, f has a unique fixed point in S.
We skip the proof since: Thus, it is sufficient to choose r := r 1 + r 2 + r 3 + 2r 4 < 1 in Theorem 2 to conclude the theorem above.
In Theorem 3, if we take r 2 = 0, r 3 = 0, r 4 = 0, then the above theorem reduces to as below.
Corollary 3. labelJS1-c-4 Let (S, δ e ) be an extended b-metric space and θ ∈ Θ.If a mapping f : X → X satisfies that there exists r 1 ∈ [0, 1) such that: where r ∈ [0, 1) and lim sup , then f has a unique fixed point in S.
In Theorem 3, if we take r 1 = 0, r 2 = 0, r 3 = 0, then the above theorem reduces to as below.

Fixed Point
Clearly, (S, δ e ) is a complete extended b-metric space.Consider the nonlinear Volterra-Fredholm integral equation: where x(t) is the unknown solution, κ i (t, s, x(s)); (i = 1, 2) are called smooth functions, and λ i , a, and b are constants.
Let us assume a = 0 for an effortlessness detailed examination of the nonlinear Volterra-Fredholm integral Equation (10).
Suppose that the following conditions hold: 1. the mapping f : for all x ∈ C[0, b] and 0 ≤ s ≤ t ≤ b, is a continuous mapping and 3. κ i for some constant A i satisfies: Then, the nonlinear Volterra-Fredholm Equation ( 10) has a unique solution. Consider: , for all x, y ∈ S. which yields, It follows that f satisfies all the conditions of Corollary 1.Hence, f has a unique fixed point.This yields that there exists a unique solution of the non-linear Volterra-Fredholm integral equation.

Fixed Point Method for Common Solution of Nonlinear Fractional Differential Equations
In this section, by using Corollary 2, we investigate the existence and uniqueness solution for nonlinear fractional differential equations (NFDE), in the sense of the Caputo derivative.Recall that the Caputo fractional derivative of ℘(t) order > 0 is denoted by C D ℘(t), and it is defined as: and [ ] denotes the greatest integer of (i.e., the integral part of ) and ℘ : [0, ∞) → R is a continuous function.
In this section, we shall investigate the existence of uniqueness solutions to a non-linear fractional differential equation.

Numerical Example
In this section, a numerical example is established to indicate the significance of the given results.Let S be a set of all continuous real-valued functions defined on [0,1], i.e., S = C([0, 1], R).Define δ e : S × S → R and ω respectively.Clearly, (S, δ e ) is a complete δ e -metric space.Let f : X → X be the operator defined by: Let ϑ(t) = t cos(t), κ(t, s) = t, and x(s) = sin(x(s)).Then, (5.1) becomes: Suppose the following conditions hold.
As a result, the conclusion is that all axioms of Theorem 1 are satisfied.Consequently, the integral Equation ( 12) has a unique solution.It can be easily checked that x(t) = t is the exact solution of Equation (12).Now, we shall use the iteration method to underline the validity of our approaches: Tables 1-5 shown the examples.Figures 1 and 2 shown the sequence of x n+1 (t) = f x n (t) = tcos(t) + 1 0 t sin(x n (s))ds converges to the exact solution 0.2 and 0.6 respecially.Let x 0 (t) = 0 be an initial solution.

Discussion and Conclusions
Since Jleli and Samet's [2] characterization of the contraction principle, many characterizations of contraction principle-type results have been presented in the literature.In this article, we introduced various topics called Θ e -contraction, a Hardy-Rogers-type Θ-contraction, and an interpolative Θ-contraction in extended b-metric space (simply, δ e -metric space) and proved pertinent fixed point theorems.Thereafter, we proposed a simple solution for the nonlinear integral equation and fractional differential equation using the technique of a fixed point in δ e -metric space.We used an iterative method based on the fixed point approach.We found the approximate solution of Equation (12).The numerical results have verified that the approach employed in this article is valid.
The obtained results are significant since this will build new avenues for working in Θ-contraction (and/or its extensions) and its applications to differential, integral, and functional equations with numerical experiments.

Corollary 1 .
Let T be a self-mapping on a complete b-metric space (S, d).If there exist ϑ ∈ Θ and r ∈ (0, 1) such that: ϑ(d(Tx, Ty)) ≤ [ϑ(d(x, y))] r if d(Tx, Ty) = 0 f or x, y ∈ S, then T has a unique fixed point in S.
Method for the Common Solution of Nonlinear Volterra-Fredholm Integral Equations Let X := C([a, b], R); the set of all continuous real valued functions is defined on [a, b].Define δ e : S × S → R and γ : S × S → [1, ∞) by: