Abstract
We present the equivalence of some stochastic fixed point iterative algorithms by proving the equivalence between the convergence of random implicit Jungck-Kirk-multistep, random implicit Jungck-Kirk-Noor, random implicit Jungck-Kirk-Ishikawa, and random implicit Jungck-Kirk-Mann iterative algorithms for generalized -contractive-like random operators defined on separable Banach spaces.
1. Introduction
Probabilistic functional analysis is an aspect of mathematics that deals with probabilistic models to solve uncertainties and ambiguities that exist in real-world problems. Random nonlinear analysis is a vital area of probabilistic functional analysis that deals with various classes of random operator equations and related problems and solutions. The development of various random methods has transformed the financial markets. Random fixed point theorems for random contraction mappings on separable complete metric spaces were first proved by Spacek [1] and Hans (see [2, 3]). Random fixed point theory has become the full-fledged research area and various ideas associated with the theory are applied to obtain the solutions to a class of stochastic integral equations (see [4, 5]). Random fixed theorems are well known stochastic generalizations of classical fixed point theorems and are usually needed in the theory of random equations, random matrices, random differential equations, and different classes of random operators emanating in physical systems [6, 7]. The origin and various generalizations of random fixed point theorem exist in the literature; for complete survey, see [3] and several related references therein.
The concept of employing various iterative schemes in approximating fixed points of contractive-like operators is very useful in fixed point theory and applications and other relevant fields like numerical analysis, operation research, and so forth (see [8–15]) This is due to the close relationship that exists between the problem of solving nonlinear equations and that of approximating fixed points of corresponding contractive-like operator.
However, while many researchers have proved useful results on the equivalence of the various iterations, that is, the convergence of any of the iterative methods to the unique fixed point of the contractive operator for single map is equivalent to the convergence of the other iterations (see [16–24]), it is observed that little result is known of the equivalence of implicit schemes for a pair of maps [25]. This work will address these areas.
2. Preliminary
Definition 1. Let be a complete probability measure space and a nonempty subset of a separable Banach space . For two random mappings with and being a nonempty closed convex subset of a separable Banach space , there exists a real number and a monotone increasing function with , and for all , one hasand for and , one has For , ,For and since is a monotone increasing sequence with , then
Remark 2. Observe that the contractive conditions imply one another but not conversely: (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (6).
Definition 3. Let be a measurable space and be a nonempty closed convex sunset of a separable Banach space . A function is said to be measurable if for each Borel set of . A function is called a random operator if is measurable for every . A measurable function is called a random fixed point for the operator if .
Definition 4. Let be a measurable space and be a nonempty closed convex subset of a separable Banach space . A measurable function is called a random coincidence for two random mappings if for all . The maps are said to be random weakly compatible if they commute at their random coincidence; that is, if for every , then or
We define the random explicit Jungck-Kirk-multistep hybrid iterative algorithm as follows.
Let be a measurable space and be a nonempty closed convex subset of a separable Banach space . Let be two random mappings with . Let be an arbitrary measurable mapping for , for .
(i) The random explicit Jungck-Kirk-multistep hybrid iterative algorithm is a sequence defined iteratively bywhere , , for each and are measurable sequences in for each while are fixed integers (for each ).
(ii) The random explicit Jungck-Kirk-Noor hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and are fixed integers.
(iii) The random explicit Jungck-Kirk-Ishikawa hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and are fixed integers.
(iv) The random explicit Jungck-Kirk-Mann hybrid iterative algorithm is a sequence defined iteratively bywhere , and is a measurable sequence in and is a fixed integer.
Next, we define the random implicit Jungck-Kirk-multistep hybrid iterative algorithms as follows.
Let be a measurable space and be a nonempty closed convex subset of a separable Banach space . Let be two random mappings with . Let be an arbitrary measurable mapping for .
(i) The random implicit Jungck-Kirk-multistep hybrid iterative algorithm is a sequence defined iteratively bywhere are fixed integers (for each ) with , , and (for each ) are measurable sequences in ; (11) is called random implicit Jungck-Kirk-multistep hybrid iterative algorithm.
(ii) The random implicit Jungck-Kirk-Noor hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and are fixed integers.
(iii) The random implicit Jungck-Kirk-Ishikawa hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and are fixed integers.
(iv) The random implicit Jungck-Kirk-Mann hybrid iterative algorithm is a sequence defined iteratively bywhere , and are measurable sequences in and is a fixed integer.
Next, we introduce the random Jungck-Kirk-SP multiple iterative algorithm.
Let be a measurable space and be a nonempty closed convex subset of a separable Banach space . Let be two random mappings with . Let be an arbitrary measurable mapping for .
(i) The random Jungck-Kirk-multistep-SP hybrid iterative algorithm is a sequence defined iteratively bywhere are fixed integers (for each ) with , and (for each ) are measurable sequences in .
(ii) The random Jungck-Kirk Noor-SP hybrid iterative algorithm is a sequence defined iteratively bywhere are fixed integers with , and (for each ) are measurable sequences in
Lemma 5. If is a real number such that and is a sequence of positive numbers such that , then, for any sequence of positive numbers satisfying one has
Lemma 6. Let be a normed linear space and be a non-self-random commuting operator on an arbitrary set with values in satisfying (6) such that : Let be a sublinear, monotone increasing function such that and for all . Then, for every and , we have
Proof. Lemma 6 is proved by mathematical induction in [17].
3. Main Results
Theorem 7. Let be a separable Banach space and be two random commuting mappings satisfying (19) such that Assume that and are random weakly compatible. Let be the random common point of (i.e., If defined by as sequences satisfying (14) and (11), respectively, then the following are equivalent:(i)Random implicit Jungck-Kirk-Mann iteration (14) converges strongly to (ii)Random implicit Jungck-Kirk-multistep iteration (11) converges strongly to
Proof. We first prove that (i) ⇒ (ii).
Assume , and then using (14) and (11) and generalized contractive condition (19), we get From (20), Substituting (21) into (20), we get Using (11) and (14) and contractive condition (19), we get Following the method of proof in (21), we can writeSubstituting (24) into (23) and simplifying, we obtain Also, using (11) and (14) and contractive condition (19), we get Substituting (25) and (26) into (22), we get Continuing this process to and simplifying, we get the following: Recall that
Let ; then, Substituting (29) into (28), we get where ,Applying Lemma 5 in (30), it follows that
By assumption , then as , that is,
Next, we show that (ii) (i). Assume ; then, using (11) and (14) and contractive condition (19), we have From (32), Substituting (33) into (32) and simplifying, we obtain Using (11) and (14) and contractive condition (19), we have From (35), Substituting (37) and (38) into (36) yieldsSubstituting (40) into (35), we obtain , and are similarly obtained and substituted into (41) to get Substituting (42) into (34), we have Recall that .
Let ; then, Using (44) in (43), we have where , Using Lemma 5 in (45), it follows that
Since, by assumption, , then as ; that is,
Since and , it is shown that the convergence of random implicit Jungck-Kirk-Mann iteration (14) is equivalent to the convergence of random implicit Jungck-Kirk-multistep iteration (11) when applied to a pair of random weakly compatible generalized -contractive-like conditions (19). This ends the proof.
Since the random implicit Jungck-Kirk-multistep iteration (11) generalizes other random implicit Jungck-Kirk-type iterations (12), (13), and (14), then Theorem 7 leads to the following corollaries.
Corollary 8. Let be a separable Banach space and be two random commuting mappings satisfying (19) such that Assume that and are random weakly compatible. Let be the random common point of (i.e., If defined by and as sequences satisfying (14), (13), and (12), respectively, then the following are equivalent: (a) (i)Random implicit Jungck-Kirk-Mann iteration (14) converges strongly to .(ii)Random implicit Jungck-Kirk-Ishikawa iteration (13) converges strongly to (b) (i)Random implicit Jungck-Kirk-Mann iteration (14) converges strongly to .(ii)Random implicit Jungck-Kirk-Noor iteration (12) converges strongly to
Proof. The proof of Corollary 8 is similar to that of Theorem 7.
Corollary 9. Let be a separable Banach space and be two random commuting mappings satisfying (19) such that Assume that and are random weakly compatible. Let be the random common point of (i.e., If defined by and as sequences satisfying (14), (13), (12), and (11), respectively, then the following are equivalent:(i)Random implicit Jungck-Kirk-Mann iteration (14) converges strongly to .(ii)Random implicit Jungck-Kirk-Ishikawa iteration (13) converges strongly to .(iii)Random implicit Jungck-Kirk-Noor iteration (12) converges strongly to .(iv)Random implicit Jungck-Kirk-multistep iteration (11) converges strongly to
Example 10. Consider the equation , where is a random real function defined on interval by .
Note that can be decomposed as , where the maps and are the self-mappings in defined by and . Clearly, and satisfy the generalized -contractive condition (19). They coincide at and we have . Thus, is a solution to . However, if and are random weakly compatible, then .
Using MATLAB, we show in Table 1 that (11), (12), (13), and (14) are equivalently used to approximate the stochastic common fixed point .
4. Conclusion
In this work, we have investigated the equivalence of convergence of various stochastic implicit Jungck-Kirk-type iterations for generalized random contractive-like operators in separable Banach spaces. We also gave an example to show the equivalence results. We hope that the findings in this paper will help researchers enhance and promote the further study on iterative schemes for two or more maps in well known spaces to carry out a general framework for their applications in real life.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors are grateful to Professor J. O. Olaleru for supervising their Ph.D. thesis. The first author is also grateful to his second Ph.D. supervisor, Professor J. A. Adepoju.