Iterating Fixed Point via Generalized Mann’s Iteration in Convex b-Metric Spaces with Application

Abstract

This manuscript investigates fixed point of single-valued Hardy-Roger’s type -contraction globally as well as locally in a convex -metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom (F3) of the -contraction is removed, and thus the mapping is relaxed. An important approach used in the article is, though a subset closed ball of a complete convex -metric space is not necessarily complete, the convergence of the Cauchy sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations.

1. Introduction and Preliminaries

Various authors generalized metric space into many interesting generalizations (see [1–8]). Among them, Czerwik [9] introduced the idea of b-metric and proved fixed point theorems in it. It was further extended to partial b-metric and dislocated b-metric spaces in the past years [10, 11]. Chen et al. [12] introduced convex b-metric space and established various fixed point theorems.

On the contrary, different authors extended Banach contraction to its generalizations (see [2–17]). Wardowski [18] introduced the idea of -contraction which was later followed by many authors who delivered interesting results of -contraction. One of them was presented by Cosentino et al. [19] who expanded -contraction in -contraction of Hardy-Roger’s type.

In this article, we discuss -contraction in the frame of convex b-metric space using generalized Mann’s iteration algorithm. However, we have modified definition of -contraction by eliminating two of its conditions and define -Hardy-Roger’s contraction on closed ball in convex b-metric space. A detailed example explains the proved results.

Some fundamental definitions related to our work are given below.

Definition 1 (see [20]). Let be a real number. We denote by the family of all functions.
with the following properties:(F1) is strictly increasing.(F2)for each sequence , if and only if .(F3)for each sequence , , there exists such that .(F4)for each sequence of positive numbers such that for all and some , then for all n ∈ N.

Definition 2 (see [20]). Let be a b-metric space where . A multivalued mapping is called an -contraction of Nadler type if there exist and such that for all with .
Note that, in our theorems, we will consider as the class of functions satisfying only (F1) and (F2) which modifies the definition of -contraction.

Definition 3 (see [9]). Assume that with . If satisfies the following axioms, for each .(1) if and only if .(2).(3).Then the pair is known as b-metric space with .

Definition 4 (see [9]). Suppose is a sequence in Then(1) is convergent to a point if (2) is Cauchy if (3)The space is complete if every Cauchy sequence is convergent to a point .

Definition 5 (see [12]). Assume that and is a mapping. Let with a continuous mapping such thatfor each and . is said to be a convex structure on .

Definition 6 (see [12]). Assume that is a convex structure on a b-metric space . Then is known as convex b-metric space.
Suppose that is convex b-metric space with a self-mapping . Then for and , a generalize Mann’s iteration sequence is defined as

Definition 7 (see [12]). Let be a metric space. A self-mapping on is called an -contraction of Hardy-Roger’s type if there exist and such thatfor all with , where , , and .
In the above definition, is the class of functions that satisfies (F1)-(F3).

2. Fixed Point Results of -Hardy-Roger’s Contractions on Closed Ball

This section examines -Hardy-Roger’s contraction for fixed point while executing the contraction only on closed ball rather than the whole convex b-metric. Also, an example is given to explain the proved theorem.

Note that, in our theorems, we will consider as the class of functions satisfying only (F1) and (F2) which modifies the definition of -contraction and .

Definition 8. Let is a convex b-metric space, is some element in , and ; then the set is called a closed ball in
Not that, is a nonempty subset containing at least one element . Also, a subset of a complete b-metric space may not be complete. Hence a Cauchy sequence in the subset closed ball of a complete b-metric space may not converge in closed ball.

Definition 9. Assume that , be a convex b-metric space with , for , and be a closed ball in . Then is known as -Hardy-Roger’s contraction on closed ball if for with , the following hold:for every .

Theorem 1. Suppose is a complete convex b-metric space with generalized Mann’s iteration having . Suppose is an -Hardy-Roger’s contraction on closed ball, and for all , and the following holds:Then has a unique fixed point in .

Proof. First we prove that the iteration belongs to the closed ball. To do so, we will use mathematical induction. Sincethat is, , therefore . Suppose . Now for , asThen by (4),and this implies thatUsing (F1), we writeand this implies thatthat is, for all ,Note that . Consequently,and hence for all . Thus by (7), for all . By triangular inequalityHence . Therefore, we conclude that for all . Now from (4), (7), and (12), we writeExerting limit , we obtainConsequently by (F2), we acquireHence, . It remains to prove that the sequence is a Cauchy sequence. Suppose on the contrary that is not Cauchy. Then we can find a positive real number and two subsequences and of while with as the smallest natural index such thatUsing the above inequalities, we can writeNow,Since, by (4),and therefore,using (F1) and (20), we writeExerting limit , we obtainand hence, by hypothesis, we getThis shows thatwhich is a contradiction. Hence, is a Cauchy sequence. The completeness of assures the existence of an element in such thatHowever, this is not clear whether the convergent point lies inside the closed ball or not. Therefore, before proceeding to the proof of fixed point, we first prove that . By triangular inequality, we writeexerting limit , we get . That is, . Next, we prove that is the fixed point of :Astherefore, using above equation and (29),and utilizing (F1), we obtainClearly,and therefore, by (F2), we getand consequently, by (F2) again, we obtainthat is, , Hence is the fixed point of . It remains to prove that is the only fixed point . Suppose on the contrary that be another fixed point of . Thenwhich is a contradiction because , is strictly increasing, and by hypothesis, . Hence, . That is, .

Example 1. Suppose and , then . Let be defined byand choose a mapping defined asDemonstrate as for all . Choose , , and fix . Observe that whenever . Now, consider , then we have two cases:(i)When , we get(ii)When . We have the following subcases:(a)If both , then obviously , and hence(b)If only one of and is in , say is in , then obviously , and hence, The same can be done for in and not in .(c)If both , then obviously , and henceFrom all the possible cases, it is clear that is a convex b-metric space with . Next, fix and . Observe thatAlsothat is, . Now checking for the contractive condition, we again have two cases: Case 1: if both , then that is, Thus, the inequality (4) is satisfied for and for all . Case 2: if both , i.e., say and , thenHence the inequality (4) does not hold for all the elements of . Hence the contractive condition (4) holds true only for . Observe that satisfies (F1) and (F2). MoreoverSimilarlyTherefore, and . Letting , we get and . That is, is the fixed point of . For uniqueness, suppose on contrary that is another fixed point of then , say . Hencewhich is a contradiction. Therefore, is the only fixed point of .
Taking in Theorem 1, we obtain the following results.

Corollary 1. Assume that , is a complete convex b-metric space with and is a generalized Mann’s iteration having for all . Suppose be a closed ball in and for with , . If is a mapping such that for , the following holds:

Then has a unique fixed point in .

Taking in Theorem 1, we obtain the following results.

Corollary 2. Assume that , is a complete convex b-metric space with and is a generalized Mann’s iteration having for all . Suppose be a closed ball in and for with , . If is a mapping such that for , the following holds:Then has a unique fixed point in .

3. Fixed Point Results of Globally Contractive Mappings on Convex b-Metric Space

In this section, fixed point is instigated for F-contraction of Hardy-Roger’s type while the contractive condition holds true on the whole b-metric space rather than locally only on closed ball.

Theorem 2. Suppose , is a complete convex b-metric space with and is a generalized Mann’s iteration having for all . If for with and such that the following hold:for every . Then has a unique fixed point in .

Proof. We know by hypothesisBy (53),and this implies thatusing (F1), we writeand this implies thatwhere . Or we can say that for all ,Note that . Now from (53), (54), and (59), we writeexerting limit , we obtainconsequently by (F2), we acquireHence,The Cauchyness of the sequence can be proved by following the same steps as in equation (59). Afterwards, on contrary to the previous of the theorem, we do not need to prove convergence as every Cauchy sequence in complete b-metric space is Convergent. Hence there exist an element in such that . Furthermore, can be proved as the unique fixed point of in the similar way as done in Theorem 1.

Corollary 3. Suppose , is a complete convex b-metric space with and is a generalized Mann’s iteration having for all . If for with and , such that the following hold:or every . Then has a unique fixed point in .

4. Application

In Physics and Engineering, many problems are modelled in to second-order linear differential equations, i.e., system of an object with mass attached to a vertical spring, an inductor, a capacitor connected in series, and an electric circuit with a resistor. The differential equation discussed in this section shows the engineering problem of activation of spring affected by external forces. The differential equation will be reformulated to fixed point problems and the existence of the fixed point will assure the existence of the solution of differential equation (see [1, 17]).

Suppose the boundary value problem for second-order differential equation bewhere is a continuous function. Equation (65) can be equivalently written aswhere is the green function stated as

With formulated in terms of and in (66). Suppose is the collection of all continuous functions from into . Definefor , and as

Consider is defined byfor all and . Note that the existence of fixed point of guarantees the existence of solution of equation (66).

Before starting our main theorem, we first prove the following important lemma.

Lemma 1. Suppose , is defined by (69) and is stated as for all , then is a convex b-metric space.

Proof. For , , and ,Hence is convex b-metric space with .

Theorem 3. Suppose the integral equation (66) and let the following condition holds:(1) is an increasing function.(2)There exist such as where , , and .(3)For with and the norm defined above, the following hold:for . Then the equation (69) has a solution.

Proof. By (70),As , thenSimilarly, we obtainNoworBy hypothesis, we writeTherefore,and hence,and forTherefore, all the conditions of the Theorem 2 are satisfied, so the integral equation (66) has a unique solution. Consequently, the differential equation (65) has a unique solution.

5. Conclusion

This paper has introduced -Hardy-Roger’s contraction on closed ball in convex b-metric space, and generalized Mann’s iteration theorem algorithm is used to find the fixed point. The existence of the limit point inside the closed ball is established despite of the fact that the subset closed ball in a complete b-metric space is not complete. The convergence of the chosen sequence is insured inside the closed ball without completeness of the ball. Hence, a new approach is used to investigate fixed point for -Hardy-Roger’s contractions (and can extend similarly to others contractions). The results are followed by an interesting example in . Furthermore, an application of our results in finding a unique solution to second-order differential equation with boundary valued is described along with an important lemma. The paper furthered the research already done on the topic of -contractions and fixed point theory. In future, these results will be formulated in the structure of Hilbert spaces and orthogonal partial b-metric spaces, and its application in convex minimization and fractional differential equations will be investigated in continuation to the work already done in [21–25].

Data Availability

No data are used in this research.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All the authors contributed equally to this research work.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia, under grant no. KEP-66-130-38. The authors, therefore, acknowledge with thanks DSR technical and financial support.