Contributions of the fixed point technique to solve the 2D Volterra integral equations, Riemann–Liouville fractional integrals, and Atangana–Baleanu integral operators

In this manuscript, some fixed point results for generalized contractive type mappings under mild conditions in the setting of double controlled metric spaces (in short, η ג -metric spaces) are obtained. Moreover, some related consequences dealing with a common fixed point concept and nontrivial examples to support our results are presented. Ultimately, we use the theoretical results to discuss the existence and uniqueness of solutions of 2D Volterra integral equations, Riemann–Liouville integrals and Atangana–Baleanu integral operators are given.


Introduction
Fixed point theorems have numerous applications in mathematics and are applied in diverse fields as biology [1], chemistry [2], economics [3,4], engineering [5], game theory [6], and physics [7,8]. In the last years, boundary value problems of nonlinear fractional differential equations with a variety of boundary conditions have been studied by various researchers, see [9][10][11][12][13][14][15][16][17][18][19][20]. Fractional differential equations appear naturally in diverse fields of science and engineering. They constitute an important field of research. It should be noted that most papers dealing with the existence of solutions of nonlinear initial value problems of fractional differential equations mainly use the techniques of nonlinear analysis such as fixed point techniques, stability, the Leray-Schauder result, etc. Relatively, fractional calculus and fractional differential/integral equations are very fresh topics for the researchers. For instance, in [21] the authors resolved some fractional differential equations with multiple delays in relation to chaos neuron models by using fixed point results of Lou [22] and E. de Pascale and L. de Pascale [23]. Amann [24] used a fixed point technique when studying some nonlinear eigenvalue problems in ordered Banach spaces. Liu et al. [25] gave applications of mixed monotone operators with superlinear nonlinearity via fixed point theory. On the other hand, by using fixed point theorems, the existence and uniqueness of solutions to differential/integral equations involving fractional operators were studied by a huge number of researchers. For further related results, see for example [26][27][28][29][30]. Definition 1.1 ([31]) The usual form of the RL-fractional integral operator of order τ is where τ > 0, and the function is defined on L 1 (R + ).
In this manuscript, we investigate some fixed point results via a class of contractive type mappings involving mild conditions in the setting of η ν ‫ג‬ -metric spaces. Also, some nontrivial examples are introduced. Finally, as applications, theoretical results are involved to discuss the existence and uniqueness of a solution of 2D Volterra integral equations, Riemann-Liouville integrals, and Atangana-Baleanu integral operators.

Main results
We begin this section with the following definition.
From inequality (2.1), one writes This leads to Iteratively, By taking = σ a b -ρ-μ < 1, one gets For all m, n ∈ N with m > n, we get It follows by (2.2) and the ratio test that the real number sequence { n } exists, and so { n } is Cauchy. Note that the ratio test is applied to the term Relation (2.6) implies that the sequence {ς n } is Cauchy. The completeness of (℘, η ν ‫ג‬ ) yields that there exists some ∈ ℘ such that Applying the triangle inequality, we have By (2.6) and (2.7) in (2.8), we get at the limit Now, we shall show that = . Using the method of contradiction, i.e., let η ν ‫ג‬ ( , ) > 0. Then Letting n → ∞ in the above inequality and with the help of (2.6), (2.7), and (2.9), we conclude that η ν ‫ג‬ ( , ) = 0. It is a contradiction, that is, = . Likewise, we can show that = . This means that is a cfp of and . For uniqueness, suppose that α is another cfp of and such that = α, then by condition (2.1) we obtain that This implies that (a bσ )η ν ‫ג‬ ( , α) ≤ 0, a contradiction. So it should be η ν ‫ג‬ (μ, α) = 0, i.e., μ = α. Hence μ is the unique cfp of and . This ends the proof. Theorem 2.5 reduces to the following corollary if we consider that the two mappings and are equal.
If we take a = b = 1, μ = ρ = 0, and = in Theorem 2.5, we get the following main result.

Corollary 2.7
Let (℘, η ν ‫ג‬ ) be a complete η ν ‫ג‬ -metric space, and let the mapping : ℘ → ℘ satisfy for all ς, υ ∈ ℘, where σ is a nonnegative real number with σ < 1. Choose ς n = n ς • for ς • ∈ ℘, then has a unique fixed point, provided that the following assumptions are satisfied: Another direction to obtain a cfp of and is by considering a strong contractive condition in the following theorem.

Solving the 2D Volterra integral equations
There are many advantages to studying equations of the form (3.1). The authors [37] showed that problem (3.1) arises from the transformation of certain Volterra integral equations of the first kind, with applications, for example, in analysis of Cauchy problems for certain partial differential equations (e.g., the telegraph equation) and in radiation transfer problems. Moreover, the Darboux problem can also be reduced to equation (3.1), as shown in [38].
Applying the double conditions of our theorem, we get Hence the requirements of Corollary 2.7 are fulfilled, therefore we observe that has a unique fixed point, and so problem (3.1) has a unique solution.

A unique solution of Riemann-Liouville fractional integrals
A bunch of scientists have tackled Riemann-Liouville integral equations and recently identified a new technique, so-called 'fixed point approach' . This novel approach has promised the existence of a solution of Riemann-Liouville integral equations. For more details, see [39,40]. Along the same lines, here we study the existence and uniqueness of a solution of Riemann-Liouville (RL) fractional integral in the form of is the set of all continuous functions from [0, 1] onto R) and κ, ∈ [0, 1] which is the fractional integral. Define the distance η ν Then (℘, η ν ‫ג‬ ) is an η ν ‫ג‬ -metric space. Now, we shall show that integral (4.1) has a unique solution under the following condition: where σ ∈ (0, 1) and κ = .
Define also the operator : ℘ → ℘ by Thus, the existence of a unique solution of problem (4.1) is equivalent to finding a unique fixed point of the integral operator (4.2). Assume that Thus, all the assumptions of Corollary 2.7 are verified, so has a unique fixed point, i.e., the Riemann-Liouville fractional integral equation has a unique solution.

Existence of a unique solution of Atangana-Baleanu fractional operator
In 2016, Atangana and Baleanu [41] developed more general definitions of a fractional derivative and an integral operator targeting nonlocal and nonsingular kernel. This operator takes the form (5.1). This study examines the connections between nanofluids, the dynamics of ions over the membrane, material mechanics, and predictor-corrector algorithms [42][43][44][45][46]. This new impression offers the opportunity to elaborate on the new findings/new insights and creative approaches for contextualizing the new topics in various aspects.

Conclusion and discussions
In this manuscript, we considered a double controlled metric space (in short, η ν ‫ג‬ -metric space). Via this space, some novel theoretical results involving fixed point techniques under various suitable assumptions have been established. To confirm our consequences, nontrivial examples have been presented. Finally, short and simple proofs have been obtained to find the existence and uniqueness of solutions of 2D Volterra integral equations, Riemann-Liouville integrals, and Atangana-Baleanu integral operators. In addition, the applications in this manuscript are listed as follows: • A fixed point technique to solve the 2D Volterra integral equation (3.1). This problem is considered without kernels because it is caused by a time-fractional telegraph equation. It is exciting to clarify a few points in this direction: In telegraph's equation characterizing the variation of voltage along with an electrical cable as a function of time and position κκ + ( + ) κ + = 2 ζ ζ , (6.1) which consists of a resistor of resistance R, a coil of inductance L, a resistor of conductance κ, or a capacitor of capacitance , where 2 = 1 LL , = κ , = R L . If R L = κ (or R = Lκ) a constant velocity of propagation would result and the attenuation would be minimized, this result was discussed by the physicist Oliver Heaviside in 1893.