Hybrid Fixed Point Theorem Research Articles

Hybrid fixed point theorems of graphic contractions with applications

In this work, a family of hybrid contractions, termed Jaggi-type hybrid (ℵ-τ)-contractive mapping is proposed in metric space equipped with a graph and new conditions under which the mapping is a Picard operator are studied. The novel ideas proposed in this manuscript are exemplified to display the validity of the presented results and to show how they differ from the existing ones. Additionally, some corollaries which reduce our proposed notion to some recently announced concepts in the existing findings are indicated and examined. Finally, we study Ulam-type stability for the fixed point equations with hybrid contractions.

On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations

Abstract Recently, fractals and fractional calculus have received much attention from researchers of various fields of science and engineering. Because the said area has been found applicable in modeling various real-world processes and phenomena. Hybrid differential equations (HDEs) play significant roles in mathematical modeling of various processes because the aforesaid equations incorporate different dynamical systems as specific cases. For instance, it is possible to model and describe non-homogeneous physical phenomena on using the said equations. Therefore, this research work is concerned with studying a class of nonlinear hybrid fractal–fractional differential equations. We develop the existence result for the qualitative study using a hybrid fixed point theorem. For the mentioned goal, a fixed point theory for the product of two operators is applied to deduce appropriate conditions for the existence of exactly one solution. Additionally, the stability result based on Ulam–Hyers is also deduced. The said stability results play an important role in numerical investigations. In addition, a numerical method based on Euler procedure is utilized to approximate the solution of the proposed problems. Various computational test problems are given to demonstrate the results. Also, using various fractal–fractional order values, several graphical presentations are given for the examples. The concerned analysis will help in investigating many real-world problems modeled using HDEs with fractal–fractional orders in the near future.

AN ALGORITHMIC APPROACH TO LOCAL SOLUTION OF THE NONLINEAR HIGHER ORDER ORDINARY HYBRID DIFFERENTIAL EQUATIONS

In this paper, we introduce a new notion of local or neighborhood solutions and establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear higher order ordinary hybrid differential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2023). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Our main abstract results of this paper are also illustrated with a couple of numerical examples. Finally, we compare our existence and uniqueness results with those existing in the literature via other operator theoretic methods from nonlinear functional analysis. We claim that the method and the results of this paper are new to the literature.

USING LEARNING ABILITIES OF COMPUTATION IN COUNTING

In this paper, we introduce a new notion of local or neighborhood solutions and establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear higher order ordinary hybrid differential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2023). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Our main abstract results of this paper are also illustrated with a couple of numerical examples. Finally, we compare our existence and uniqueness results with those existing in the literature via other operator theoretic methods from nonlinear functional analysis. We claim that the method and the results of this paper are new to the literature.

Approximating local solution of IVPs of nonlinear first order ordinary hybrid integrodifferential equations

In this paper, we prove a couple of approximation results for local existence and uniqueness of the solution of a IVP of nonlinear first order ordinary hybrid integrodifferential equations by using the Dhage monotone iteration method based on a hybrid fixed point theorem of Dhage (2022) and Dhage {et al.} (2022). An approximation result for the Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Finally, our main abstract results are also illustrated with a couple of numerical examples.

Approximation results for PBVPs of nonlinear first order ordinary functional differential equations in a closed subset of the Banach space

In this paper we prove the approximation results for existence and uniqueness of the solution of PBVPs of nonlinear first order ordinary functional differential equations in a closed subset of the Banach space. We employ the Dhage monotone iteration method based on a recent hybrid fixed point theorem of Dhage (2022) and Dhage {\em et al.} (2022) for the main results of this paper. Finally an example is indicated to illustrate the abstract ideas involed in the approximation results.

Analysis of Implicit Solutions for a Coupled System of Hybrid Fractional Order Differential Equations with Hybrid Integral Boundary Conditions in Banach Algebras

This paper investigates the existence and uniqueness of implicit solutions in a coupled symmetry system of hybrid fractional order differential equations, along with hybrid integral boundary conditions in Banach Algebras. The methodology centers on a hybrid fixed-point theorem that involves mixed Lipschitz and Carathéodory conditions, serving to establish the existence of solutions. Moreover, it derives sufficient conditions for solution uniqueness and establishes the Hyers–Ulam types of solution stability. This study contributes valuable insights into complex hybrid fractional order systems and their practical implications.

Nonlocal Hybrid Integro-Differential Equations Involving Atangana–Baleanu Fractional Operators

In this study, we develop a theory for the nonlocal hybrid boundary value problem for the fractional integro-differential equations featuring Atangana–Baleanu derivatives. The corresponding hybrid fractional integral equation is presented. Then, we establish the existence results using Dhage’s hybrid fixed point theorem for a sum of three operators. We also offer additional exceptional cases and similar outcomes. In order to demonstrate and verify the results, we provide an example as an application.

A new sequential proportional fractional derivative of hybrid di?erential equations with nonlocal hybrid condition

In this paper, we study the existence of solutions for a new problem of hybrid differential equations with nonlocal integro multi point boundary conditions by using the proportional fractional derivative. The presented results are obtained by using hybrid fixed point theorems for three Dhage operators. The application of theoretical conclusions is demonstrated through an example.

Hybrid Fixed Point Theorems of Fuzzy Soft Set-Valued Maps with Applications in Integral Inclusions and Decision Making

A lot of work has been completed in efforts to extend the notions of soft set and fuzzy soft set and their applications to other domains. However, neither of the two concepts has been examined in the study of functional equations in b-metric spaces. Given this background information, this paper proposes the idea of b-hybrid fuzzy soft contraction in b-metric space and investigates new criteria for the existence of fixed points for such mappings. The significance of the obtained principal result lies in the fact that the contractive inequalities can be specialized in various ways, depending on the choice of the parameters, thereby making it possible to unify, deduce, and refine several corresponding results. To motivate further studies in the directions studied herein, two applications regarding decision-making problems and an existence theorem of integral inclusion are considered, using fuzzy soft set-valued maps.

AN ALGORITHMIC APPROACH TO LOCAL SOLUTION OF THE NONLINEAR SECOND ORDER ORDINARY HYBRID INTEGRODIFFERENTIAL EQUATIONS

In this paper, we establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear second order ordinary hybrid integrodifferential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2022) and Dhage et al. (2022). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Finally, our main abstract results are also illustrated with a couple of numerical examples.

On the solvability of a system of Caputo-Hadamard fractional hybrid differential equations subject to some hybrid boundary conditions

We give some existence and regularity results for a system of a new class of hybrid Caputo-Hadamard fractional differential equations under hybrid boundary conditions. The technique of investigation is essentially based on the use of a well known hybrid fixed point theorem.

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

Pantograph, a device in which an electric current is collected from overhead contact wires, is introduced to increase the speed of trains or trams. The work aims to study the stability properties of the nonlinear fractional order generalized pantograph equation with discrete time, using the Hilfer operator. Hybrid fixed point theorem is considered to study the existence of solutions, and the uniqueness of the solution is proved using Banach contraction theorem. Stability results in the sense of Ulam and Hyers, and its generalized form of stability for the considered initial value problem are established and we depict numerical simulations to demonstrate the impact of the fractional order on stability.

A new tripled system of hybrid differential equations with φ-Caputo derivatives

In this paper, we study the existence of solutions for a new tripled system of hybrid differential equations with nonlocal integro multi point boundary conditions by using the φ-Caputo derivatives. The presented results are obtained by using hybrid fixed point theorems for three Dhage operators. An illustrative example is presented for the applicability of the theoretical results.

A SCHAUDER TYPE HYBRID FIXED POINT THEOREM IN A PARTIALLY ORDERED METRIC SPACE WITH APPLICATIONS TO NONLINEAR FUNCTIONAL INTEGRAL EQUATIONS

In this paper we prove some hybrid fixed point theorems for the monotonic nondecreasing mappings in a partially ordered metric space which includes the Schauder type fixed point theorems in an ordered Banach space proved by Dhage (2013,2014) and Dhage et al. (2022) in a partially ordered Banach space as the special cases. As an application, we discuss a nonlinear functional integral equation of Fredholm type and a nonlinear functional boundary value problem for proving the existence and approximation of solution by constructing the algorithms via Dhage monotone iteration method under some natural condtions.

Generalized form of fixed-point theorems in generalized Banach algebra relative to the weak topology with an application

In this paper, a general hybrid fixed point theorem for the contractive mappings in generalized Banach spaces is proved via measure of weak non-compactness and it is further applied to fractional integral equations for proving the existence results for the solutions under mixed Lipschitz and weakly sequentially continuous conditions. Finally, an example is given to illustrate the result.

Analysis of fractional hybrid differential equations with impulses in partially ordered Banach algebras

In this paper, we investigate a class of fractional hybrid differential equations with impulses, which can be seen as nonlinear differential equations with a quadratic perturbation of second type and a linear perturbation in partially ordered Banach algebras. We deduce the existence and approximation of a mild solution for the initial value problems of this system by applying Dhage iteration principles and related hybrid fixed point theorems. Compared with previous works, we generalize the results to fractional order and extend some existing conclusions for the first time. Meantime, we take into consideration the effect of impulses. Our results indicate the influence of fractional order for nonlinear hybrid differential equations and improve some known results, which have wider applications as well. A numerical example is included to illustrate the effectiveness of the proposed results.

Analytical Study of Two Nonlinear Coupled Hybrid Systems Involving Generalized Hilfer Fractional Operators

In this research paper, we dedicate our interest to an investigation of the sufficient conditions for the existence of solutions of two new types of a coupled systems of hybrid fractional differential equations involving ϕ-Hilfer fractional derivatives. The existence results are established in the weighted space of functions using Dhage’s hybrid fixed point theorem for three operators in a Banach algebra and Dhage’s helpful generalization of Krasnoselskii fixed- point theorem. Finally, simulated examples are provided to demonstrate the obtained results.

EXISTENCE THEORY TO A CLASS OF FRACTIONAL ORDER HYBRID DIFFERENTIAL EQUATIONS

In this paper, we develop the theory of fractional order hybrid differential equations involving Riemann–Liouville differential operators of order [Formula: see text]. We study the existence theory to a class of boundary value problems for fractional order hybrid differential equations. The sum of three operators is used to prove the key results for a couple of hybrid fixed point theorems. We obtain sufficient conditions for the existence and uniqueness of positive solutions. Moreover, examples are also presented to show the significance of the results.

Approximate controllability of hybrid Hilfer fractional differential inclusions with non-instantaneous impulses

• Approximate controllability associated to hybrid fractional differential inclusions with impulses conditions has not yet been addressed by any researcher until now. • The Wright function is used to find the solution of the studied problem and it is defined only when 0 ≤ σ ≤ 1 . • As we cannot apply the Wright function, observing that our problem deals with the special case 0 ≤ σ ≤ 2 . We make use of the general fractional resolvent operator and the Laplace transform to find the solution. • We make use of some facts about fractional calculus and set-valued maps, properties of resolvent operators, and a hybrid fixed point the- orem for three operators of Schaefer type. • First we prove the existence of a mild solution. • Then the approximate controllability of a problem governed by hybrid fractional differential inclusions under Hilfer derivative of order 1 ≤ σ ≤ 2 and type 0 ≤ ζ ≤ 1 , supplemented with non-instantaneous impulses in a weighted space is proved. • An example, illustrating the abstract theory, is given. This paper deals with the approximate controllability of a class of non-instantaneous impulsive hybrid systems for fractional differential inclusions under Hilfer derivative of order 1 < σ < 2 and type 0 ≤ ζ ≤ 1 , on weighted spaces. As an alternative to the Wright function which is defined only when 0 < σ < 1 , we make use of a family of general fractional resolvent operators to give a proper form of the mild solution. This latter is consequently formulated by Laplace transform, improving and extending important results on this topic. Based on known facts about fractional calculus and set-valued maps, properties of the resolvent operator, and a hybrid fixed point theorem for three operators of Schaefer type, the existence result and the approximate controllability of our system is achieved. An example is given to demonstrate the effectiveness of our result.