Existence results of certain nonlinear polynomial and integral equations via (cid:122) -contractive operators

: In this manuscript, the concept of (cid:122) -contraction is applied to extend the notion of Jaggi-Suzuki-type hybrid contraction in the framework of G -metric space, which is termed Jaggi-Suzuki-type hybrid (cid:122) -( G - α - φ )-contraction, and invariant point results which cannot be inferred from their cognate ones in metric space are established. The results obtained herein provide a new direction and are generalizations of several well-known results in ﬁxed point theory. An illustrative, comparative example is constructed to give credence to the results obtained. Furthermore, su ﬃ cient conditions for the existence and uniqueness of solutions of certain nonlinear polynomial and integral equations are established. For the purpose of future research, an open problem is highlighted regarding discretized population balance model whose solution may be investigated from the techniques proposed herein.


Introduction
The Banach contraction principle (BCP) [1], as one of the most acclaimed tools in nonlinear analysis, took on a noteworthy role and opened up a gateway in metric fixed point theory.Over the years, researchers have worked tremendously to extend this principle towards different wide-ranging directions.One of such generalizations satisfying the contractive inequality of rational type was presented by Jaggi [2].Refer to Cho [3] for some extensive advancements of fixed point results of contraction mappings.
Owing to the versatility of metric fixed point (FP) theory, a lot of attention has been given to generalizing the metric space (MS) and worthy of note are those given by Gähler [4] in 1963 (called 2-MS) and Dhage [5] in 1992 (called D-MS).The concept of generalized MS (or simply G-MS) was first proposed by Mustafa and Sims [6] in 2006 as an extension of MSs, where each triplet of elements of an abstract set is given a non-negative real number.This was aimed at addressing some flaws in the idea of generalized MS given by Dhage and to properly depict the geometry of three points along the perimeter of a triangle.Consequently, Mustafa et al. [7] established some FP results of Hardy-Rogers type for mappings satisfying certain constraints on a complete G-MS.However, according to Jleli et al. [8] as well as Samet et al. [9], several invariant point outcomes in the framework of G-MS can be inferred from some earlier findings in (quasi)-MS.In essence, an equivalent FP result in symmetric spaces if the contraction constraint of the invariant point theorems in G-MS can be reduced to two variables from three variables.Although, as observed by [10,11], this is pertinent only if the contractive conditions in the result can be reduced to two variables.Consult Jiddah et al. [12] for a comprehensive survey on the developments of FP results in G-MS.
In a bid to generalize and enrich the BCP, Suzuki [13] in 2008 refined the results of Edelstein and Banach; hence introducing a family of mappings known as the Suzuki-type generalized non-expansive mappings.The existence and uniqueness of FPs for this class of mappings were discussed in the setup of a compact MS.Also, in 2012 Wardowski [14] pioneered a new class of contractions known as -contraction and proved an invariant point result.Piri and Kumam [15] in 2014 extended the result of Wardowski by enforcing less stringent additional constraints on a self map defined on a complete MS.In 2015, Alsulami et al. [16] presented the concept of generalized -Suzuki type contraction in the setting of b-MS and investigated the existence of FPs of such a family of mappings.Minak et al. [17] discussed some new FP results for a variant of -contraction including Ćirić-type generalized -contraction on a complete MS.A new form of applying the Hardy-Roger type contraction in G-MSs was analysed by Singh et al. [18] and this enhanced the main work of [15].Aydi et al. [19] pioneered a modification of -contraction via α-admissible mappings and discussed the subsistence and uniqueness of such mappings.For some important trends in -contraction type FP results, refer to Joshi and Jain [20], Fabiano et al. [21].Karapinar and Fulga [22] pioneered a new hybrid type contraction that combines interpolativetype contractions and Jaggi-type contractions in the framework of a complete symmetric space.The notion of Jaggi-Suzuki-type hybrid contraction mappings was initiated by Noorwali and Yes ¸ilkaya [23] with the aim of putting together Suzuki-type contractions, Jaggi hybrid type contractions and hence investigating some theorems that guarantees the existence and uniqueness of FPs of such contractions.Jiddah et al. [24] unified the work of [23] in the setting of G-MS and proved some invariant point results via (G-α-φ)-contraction.Based on our surveyed literature, we noticed that a hybrid form of Jaggi and Suzuki invariant point results has not been adequately examined, notwithstanding the fact that hybrid FP theorems have enormous applications in various areas of nonlinear analysis.Motivated by this background orientation, we propose an innovative approach called Jaggi-Suzuki-type hybrid -(G-α-φ)-contraction, and via the aid of some supplementary functions, certain FP results are discussed in the set-up of a complete G-MS.A comparative, non-trivial example is given to authenticate the applicability of our results and its enhancement over previous findings.It is important to highlight that the principal ideas presented herein cannot be shrunk to any known results.It has been determined, with the help of some obtained consequences, that the notion put forth here is an extension of several existing FP results in the area of -contraction in the setting of G-MS.Moreover, one of the corollaries acquired is applied to establish the existence and uniqueness of solutions of certain nonlinear polynomial and integral equations.
The paper is organized as follows: In Section 1, the overview and introduction of relevant literature are highlighted.The fundamental results and mathematical notations employed in this work are outlined in Section 2. In Section 3, the key findings and some special cases of the obtained FP results are discussed.The criteria for the existence of a solution to a polynomial equation is analyzed in Section 4. Using one of the acquired consequences herein, the existence and uniqueness of solutions of certain nonlinear integral equations are investigated in Section 5.An open problem on a discretized population model is highlighted in Section 6.In Section 7, deductions, recommendations and conclusions are given.

Preliminaries
In this section, we will present some fundamental mathematical notations and results that will be adopted throughout this manuscript.Every set Θ is assumed to be non-empty.We represent by N, R + and R as the set of natural numbers, the set of non-negative real numbers and the set of real numbers, respectively.
Consistent with Mustafa and Sims [6], the following definitions and results will be needed in the sequel.i, j→∞ G(x, x i , x j ) = 0; in other words, given > 0, we can find i 0 ∈ N such that G(x, x i , x j ) < , ∀i, j ≥ i 0 .x is referred to as the limit of the sequence {x i }.
The following Figure 2 represents the topological notion of convergence in G-metric spaces.We can observe from the 3D representation that the limiting value of G(x, x i , x j ) (as i, j −→ ∞) is zero.
The subsequent finding was demonstrated by Mustafa [25] in the set-up of G-MS.for all x, y, z ∈ Θ where 0 ≤ η < 1, then γ has a unique FP (say u, i.e., γu = u), and γ is G-continuous at u.
Another intriguing generalization of the BCP presented by Suzuki [26] is given hereunder: Theorem 2.7.[26] Let (Θ, ρ) be a compact MS and γ be a mapping defined on Θ. Assume that for x, y ∈ Θ.Then, γ has a unique FP.
Wardowski [14] presented a variation of the Banach FP theorem as follows: Theorem 2.10.Let (Θ, ρ) be a complete MS and γ : Θ −→ Θ be an -contraction.Then, γ has a unique FP x ∈ Θ and for every x 0 ∈ Θ a sequence {γ i x 0 } i∈N is convergent to x.
Lemma 2.11.[28] If φ ∈ Φ, then the following hold: [29] presented the subsequent definitions in the framework of MSs.Definition 2.12.[29] Given two mappings α : The above definitions were modified and presented in the setting of G-MS by Jiddah et al. [24] as follows: Jiddah, et al. [24] presented the following definition of Jaggi-Suzuki-type hybrid contraction (G-αφ)-contraction in G-MS.

Main results
In this section, the approach of Jaggi-Suzuki-type hybrid -(G-α-φ)-contraction is introduced in the set-up of G-MS and conditions for the existence of invariant points for such class of operators are investigated.
Next, we present our main results as follows.
Then, γ possesses an FP in Θ.
Proof.By assumption (ii), we have α(x, γx, γ 2 x) ≥ 1.Let x 0 ∈ Θ be an arbitrary but fixed, and consider a sequence {x i } i∈N in Θ defined by x i = γ i x 0 for all i ∈ N. If x i 0 +1 = x i 0 for some i 0 ∈ N, then x i 0 is an FP of γ.Suppose on the contrary that x i x i−1 for all i ∈ N. Since γ is triangular (G-α)-orbital admissible, then α(x 0 , γx 0 , γ Proceeding in like manner, we obtain Owing to the fact that γ is a Jaggi-Suzuki-type hybrid -(G-α-φ)-contraction, then from (3.1), Together with (3.6) and on account of (F1), (3.7) becomes Next, we analyse the following cases of (3.2).Case 1: For κ > 0, we have Therefore, (3.8) becomes ).Using the nondecreasing property of φ, then (3.9) yields which is a contradiction.Therefore, for all i ∈ N. Hence, (3.9) becomes By letting Λ i = G(x i , x i+1 , x i+2 ), we can deduce from (3.10) that

Condition (F2) yields lim
i→∞ As a result of (F3), we can find 0 < η < 1 such that By (3.11), the following holds for all i ∈ N: Observe that from (3.15), we can find i 1 ∈ N such that iΛ i η ≤ 1 for all i ≥ i 1 .It follows that Consider i, j ∈ N such that j > i ≥ i 1 and by rectangle inequality, we have Since the series converges, it follows that the sequence {x i } i∈N is Cauchy in (Θ, G).Hence, by the completeness of (Θ, G), we can find x * ∈ Θ such that {x i } is G-convergent to x * .In other words, lim i→∞ G(x i , x i , x * ) = 0.
We demonstrate that x * is an FP of γ.By Hypothesis (iii) of Theorem 3.2, we obtain and so we get γx * = x * .Hence, we can conclude that x * is an FP of γ.
In a similar way, suppose that γ 3 is continuous, we have To prove that γx * = x * , suppose on the contrary that γx * x * .Then, by (3.1) we have Hence, where which is a contradiction.It follows that γx * = x * .Case 2: For κ = 0, we attain which is a contradiction.Therefore, Furthermore, by (3.8) we have By concurrent method as in the Case of κ > 0, we can show that {x i } in (Θ, G) is G-Cauchy and consequently, by the completeness of (Θ, G) we can find a point x * ∈ Θ such that lim i→∞ x i = x * .Similarly, under the hypothesis that γ is continuous and by the uniqueness of the limit, we obtain γx * = x * .That is, x * is an FP of γ.Also, if γ 3 is continuous as in Case 1, we have γ 3 x * = x * .Suppose on the contrary that γx * x * .Then where = G(x * , γx * , γ 2 x * ).
Proof.Let x 0 ∈ Θ be arbitrary but fixed such that α(x 0 , γx 0 , γ 2 x 0 ) ≥ 1. Define a sequence {x i } i∈N in Θ by x i+1 = γx i for all i ≥ 0. If x i 0 = γx i 0 +1 for some i 0 ∈ N, then x i 0 is an FP of γ.Assume that x i x i+1 for all i ∈ N. Since γ is α-admissible, we have Inductively, we have α( Since γ is a Suzuki-type -(G-α-φ)-contraction and using (3.19), it follows that for all i ≥ 1, we have By letting Λ i = G(x i , x i+1 , x i+2 ) in (3.20) and since φ is nondecreasing, we can infer inductively that Hence, Condition (F2) yields lim From (F3), we can find η ∈ (0, 1) such that By (3.21), the following holds for all i ∈ N: Letting i → ∞ in (3.22), we have lim Note that from (3.23), we can find i 1 ∈ N such that iΛ i η ≤ 1, for all i ≥ i 1 .We have For i, j ∈ N such that j > i ≥ i 1 and by rectangle inequality, we have Since the series Due to the completeness of (Θ, G), there exists x * ∈ Θ such that {x i } i∈N is G-convergent to x * .From the continuity of γ, we see that {γx i } is G-convergent to γx * .By the uniqueness of the limit, we have In what follows, we construct an illustration in the form of an example to substantiate the assumptions of Theorems 3.2 and 3.3.

Application to j th degree polynomial
One of the fundamental roles of fixed point theory is in the solutions of polynomial equations.In this context, many useful results have been developed (e.g., see [30] and the citations in there).Going in this frame, we present the following application: It is easy to show that (Θ, G) is a complete G-MS.To reformulate (4.1) as an FP problem, consider a mapping γ : Θ −→ Θ defined by γx = x j + 1 ( j 5 − 1)x j + j 5 , and let : R + −→ R + be defined by (µ) = ln µ, µ > 0, then ∈ F .We will now show that γ is a Jaggi-Suzuki-type hybrid -(G-α-φ)-contraction.

Applications to solution of nonlinear integral equation
Younis et al. [31] applied their proposed techniques to analyse and examine the existence of solutions for some class of integral equations.Motivated by this, we study new criteria for the existence of a solution to the following integral equation: where are continuous functions and x(µ) is an unknown function.Let Θ = C([q 1 , q 2 ], R) be the set of all real-valued continuous functions defined on [q 1 , q 2 ] and let G : Θ × Θ × Θ −→ R + be endowed with the metric defined by It can be easily verified that (Θ, G) is a complete G-MS.Define a mapping γ : Θ → Θ by then x(µ) is an FP of γ if and only if it is a solution to (5.1).Under the assumptions of the following theorem, we study the existence of solution of the integral equation (5.1).Theorem 5.1.Assume that the following conditions hold: (i) there exists a constant η ∈ (0, 1) such that for some constant ϑ > 0.
Then, the integral equation (5.1) has a solution in Θ.
Consequently, all the assumptions of Theorems 3.2 and 3.3 are satisfied and hence, the integral equation (5.6) has a solution in Θ = C([0, 1], R).

Open problem
The following open problem is proposed as a direction for further consideration.A model representing a discretized population balance for continuous systems at a steady state can be given by the integral equation: Using any of the results proposed herein, we are yet to ascertain whether or not it is possible to obtain the existence conditions for the solution of (6.1).
Remark 3. (i) Several consequences of our principal results can be obtained by particularizing some of the parameters in Definition 3.1.(ii) None of the results proposed in this work can be written in the form of G(x, y, y) or G(x, x, y).
Therefore, they cannot be inferred from their analogues in MS.

Conclusions
An interesting extension of the BCP was presented by Wardowski [14], called -contraction and this has attracted the interest of researchers over the years.In this manuscript, a new family of contractions called Jaggi-Suzuki-type hybrid -(G-α-φ)-contraction was introduced in the setting of G-MS and some FP results that cannot be derived from their corresponding results in MS were discussed.To demonstrate the applicability of the proposed results and to show that they are in fact generalizations of some existing concepts in the literature, a comparative, non-trivial example was provided.We noted that via variants of -contractive operators, more than a handful of existing results can be deduced from our proposed hybrid -(G-α-φ)-contraction.A few of these special cases have been pointed out as corollaries.From an application viewpoint, one of the corollaries gotten was applied to analyse the sufficient criteria for the existence of solutions to a class of nonlinear integral equations.The totally abstract nature of the mathematical framework, analysis and results presented in this work places restriction on its applicability.The proposed family of contractions can be extended in other well-known spaces and new invariant point results can be obtained.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Figure 1
Figure 1 hereunder is a 3D visualisation of G-metric spaces.

Figure 1 .
Figure 1.The visualisation of G-metric transformation in G-metric spaces.

Figure 2 .
Figure 2. The visualisation of a convergent sequence in G-metric spaces.