Kyungpook Mathematical Journal 2023; 63(1): 45-60
Published online March 31, 2023
Copyright © Kyungpook Mathematical Journal.
-quasi-Geraghty Contractive Mappings and Application to Perturbed Volterra and Hypergeometric Operators
Olalekan Taofeek Wahab
Department of Mathematics and Statistics, Kwara State University, Malete, P. M. B. 1530 Ilorin, Nigeria
e-mail : firstname.lastname@example.org
Received: November 30, 2021; Revised: April 25, 2022; Accepted: May 3, 2022
In this paper we suggest an enhanced Geraghty-type contractive mapping for examining the existence properties of classical nonlinear operators with or without prior degenerates. The nonlinear operators are proved to exist with the imposition of the Geraghty-type condition in a non-empty closed subset of complete metric spaces. To showcase some efficacies of the Geraghty-type condition, convergent rate and stability are deduced. The results are used to study some asymptotic properties of perturbed integral and hypergeometric operators. The results also extend and generalize some existing Geraghty-type conditions.
Application of Banach's contraction map  in the area of applied and social sciences has birthed many general concepts by abstracting some common properties of Banach's condition. Two of these general concepts appear in [11, 20]. The Banach-type map is reformulated by:
in partially ordered set
In , a generalized
was introduced and proved in the framework of partially ordered metric spaces. Another recent extension was proved in  for the set of all functions
at some distinct points
Lemma 2.1. Suppose that
Since T is a λ-pseudocontraction map (II), the above quotient does not exceed 1. Let
Next, assume that
Now assume that
But, by hypothesis of the lemma, it follows that
This violates the latter condition. Hence,
Observe that for
Motivated by Lemma 2.1, let
By resolving in terms of
The inequalities (2.4) and (2.5) shall be formalised in the sequel. Before then, the following class of test functions are defined.
Definition 2.2. For
Definition 2.3. Let
: is lower semi-continuous and non-decreasing function;
: if and only if t=0; and
: is subadditive.
Motivated by the above classes of functions, Lemma 2.1, conditions (2.4) and (2.5), the
Definition 2.4. Let
By comparing condition (2.4) and (2.6), the role of function
Definition 2.5. Let
Condition (2.7) is a special case of condition (2.6) if
3. Main Results
In this section, the existence properties and stability of the nonlinear self-operator
Theorem 3.1. Let
By the condition (2.6), there gives
This further implies that
More so, since
This implies that
By taking limit of inequality (3.1) as
This contradicts the hypothesis. Hence,
Next is to prove that
Using (3.2) and triangle inequality in (3.3), we have the following:
Taking limit as
Thus, by using (2.6), (3.2), (3.4) and triangle inequality, there results
Applying limit as
Next is to prove that
By (3.2) and convergence of
Next, assume that
By the properties on
Hence a contradiction. Therefore,
Remark 3.2. If
Example 3.3. Let
It is easily verified that the fixed nodes are (1,1) and (2,1). Then, the map
Corollary 3.4. Let
Corollary 3.5. Let
This follows from Corollary 3.4.
Corollary 3.6. Let
Immediate from Corollary 3.4.
Corollary 3.7. Let
The estimate of an operator satisfying (2.6) is presented as follow:
Theorem 3.8. Let
This further implies
Remark 3.9. If
Suppose, by Theorem 3.1, that
Theorem 3.10. Let
On the other hand, suppose
This further implies
Both (3.7) and (3.8) give that
Therefore, the operator satisfying (2.6) is stable.
The existence properties of some non-degenerate nonlinear operators given by the solutions of differential equations (DEs), namely, perturbed Volterra and hypergeometric operators are studied in this section with the imposition of the
4.1. Application I
Here, Theorem 3.1. is employed to study the existence theorem of solutions for the perturbed integral equations of Volterra-like in complete metric spaces. This is facilitated by the collection of results in [5, 10, 8, 9, 15].
Clearly, the pair
Now, consider the Volterra equation
is a perturbed operator,
The purpose of equation (4.3) is to study some asymptotic properties of solutions of the perturbed equation. To ensure that the solutions of (4.2) and (4.3) are the same, the condition imposed on (4.2) is also on (4.3).
This condition leads to the admissibility of the pair
I. The pair
is admissible with respect to (4.2);
II. For each
, there corresponds such that for .
III. There exist positive function
and such that
IV. For all
and , there give such that
, there exists such that , for and .
, there exists such that
With respect to the above, result concerning the exponential decay of equation (4.3) is presented as follow:
Theorem 4.1. Suppose that all hypotheses (I-VI) are fulfilled with
Remark 4.2. Result concerning boundedness could be proved for the perturbed equation (4.3) in
4.2. Application II
Here, consider the generalized hypergeometric function denoted and defined by
Practical Example 4.3. Let
Then, the hypergeometric operator
On other hand, if
Observe that the similitude of the latter is in manifolds. Let
5. Concluding Remarks
This study discussed the existence properties, stability and convergent rate of the operator satisfying one of the
respectively, will be investigated in future studies.
- Y. I. Alber and S. Gurre-Delabriere. Principles of weakly contractive maps in Hilbert spaces. In: I. Gohberg and Yu. Lyubich, eds. New Results in Operator Theory, in: Advances and Appl. Basel: Birkhuser; 1997:7-22.
- H. Afshari, H. Aydi and E. Karapınar,
On generalized α-ψ-Geraghty contractions on b-metric spaces, Georgian Math. J., 27(1)(2020), 9-21.
- S. Banach,
Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3(1922), 133-181.
- V. Berinde. Iterative approximation of fixed points. Volume(1912) of Lecture Notes in Mathematics. Springer; 2007.
- P. Borisut, P. Kumam, V. Gupta and N. Mani,
Generalized (ψ,α,β)-weak contractions for initial value problems, Mathematics, 7(3)(2019), 266.
- P. Chaipunya1, Y. J. Cho and P. Kumam,
Geraghty-type theorems in modular metric spaces with an application to partial differential equation, Adv. Difference Equ., 2012(2012).
- S. H. Cho, J. S. Bae and E. Karapınar,
Fixed point theorems for α-Geraghty contraction contraction type maps in metric spaces, Fixed Point Theory Appl., 2013(2013).
- C. Corduneanu,
Some perturbation problems in the theory of integral equations, Math. Systems Theory, 1(1967), 143-155.
- M. Dobritoiu,
The existence and uniqueness of the solution of a nonlinear Fredholm-Volterra integral equation with modified argument via Geraghty contractions, Mathematics, 9(2021), 29-30.
- H. Faraji, D. Savić and S. Radenović,
Fixed point theorems for Geraghty contraction type mappings in b-metric spaces and applications, MDPI: axioms, 8(2019), 34.
- M. Geraghty,
On contractive mappings, Proc. Am. Math. Soc., 40(1973), 604-608.
- M. E. Gordji, M. Ramezani, Y. J. Cho and S. Pirbavafa,
A generalization of Geraghtys theorem in partially ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory Appl., 2012(2012).
- E. Karapnar,
α-ψ-Geraghty contraction type mappings and some related fixed point results, Filomat, 28(1)(2014), 37-48.
- J. Martnez-Moreno, W. Sintunavarat and Y. J. Cho,
Common fixed point theorems for Geraghtys type contraction mappings using the monotone property with two metrics, Fixed Point Theory Appl., 2015(2015).
- E. Messina, Y. S. Raffoul and A. Vecchio,
Analysis of perturbed Volterra integral equations on time scales, Mathematics, 8(2020).
- G. A. Okeke and D. Francis,
Fixed point theorems for Geraghty-type mappings applied to solving nonlinear Volterra-Fredholm integral equations in modular G-metric spaces, Arab J. Math. Sci., 27(2)(2021), 214-234.
- M. O. Osilike,
Some stability results for fixed point iteration procedures, Journal of the Nigeria Society, 14(1995), 17-29.
- M. Pǎcurar and R. V. Pǎcurar,
Approximate fixed point theorems for weak contractions on metric spaces, Carpathian J. Math., 23(2007), 149-155.
- O. Popescu,
Some new fixed point theorems for α-Geraghty contraction type maps in metric spaces, Fixed Point Theory Appl., 2014(2014).
- E. Rakotch,
A note on contractive mappings, Proc. Amer. Math. Soc., 13(1962), 459-465.
- S. Reich,
Kannan's fixed point theorem, Boll. Un. Mat. Ital., 4(4)(1971), 1-11.
- O. T. Wahab and S. A. Musa,
On general class of nonlinear contractive maps and their performance estimates, Aust. J. Maths. Anal. Appl., 18(2)(2021), 292-298.
- O. T. Wahab and K. Rauf,
On faster implicit hybrid Kirk-multistep schemes for contractive-type operators, Int. J. Anal., 2016(2016).
- T. Zamfirescu,
Fix point theorems in metric spaces, Arch. Math., 23(1)(1972), 292-298.