Fixed Point Technique Research Articles

Solving fractional integro-differential equations with delay and relaxation impulsive terms by fixed point techniques

This paper presents a systematic approach to investigating the existence of solutions for fractional integro-differential equation systems incorporating delay and relaxation impulsive terms. By employing suitable definitions of fractional derivatives, we establish physically interpretable boundary conditions. To account for abrupt state changes, impulsive conditions are integrated into the model. The system is transformed into an equivalent integral equation, facilitating the application of Banach and Schaefer fixed-point theorems to prove the existence and uniqueness of solutions. The practical applicability of our findings is demonstrated through an illustrative example.

An optimal control for non-autonomous second-order stochastic differential equations with delayed arguments

Abstract Optimal control of non-autonomous second-order stochastic differential equations with delayed arguments is indispensable for managing systems exposed to uncertainty, time-dependent dynamics, and historical influences. These equations underpin a wide range of applications, including finance, engineering, and biology, where it’s imperative to make informed decisions that mitigate risks or maximize returns while considering the inherent randomness, evolving conditions, and the impact of past states. By employing optimal control techniques, we can devise strategies that are resilient to uncertainty, adaptable to changing circumstances, and capable of accounting for the memory effects of previous events. This empowers us to optimize system performance, bolster stability, and attain desired objectives in intricate and dynamic environments. So, the goal of this article is to introduce a novel model of second-order perturbed stochastic differential equations incorporating non-local finite delay and deviated arguments in the setting of Hilbert spaces. Moreover, essential criteria are presented to examine the existence of a mild solution and evaluate the potential for approximate and optimal control of the proposed system. These results have been obtained by using evolution operators, fixed point techniques, random analytic methods, and compact semigroup theory. Further, to support the theoretical results, the optimal controllability of our model was studied by considering the Lagrange problem. Finally, the results were applied to discuss the approximate controllability of a partial differential equation. These models have the potential to advance the understanding and application of optimal control techniques for a wider range of complex systems.

Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability

Abstract Developing a model of fractional differential systems and studying the existence and stability of a solution is considebly one of the most important topics in the field of analysis. Therefore, this manuscript was dedicated to deriving a new type of fractional system that arises from the combination of three sequential fractional derivatives with fractional pantograph equations. Also, the fixed-point technique was used to evaluate the existence and uniqueness of solutions to the supposed hybrid model. Furthermore, stability results for the intended system in the sense of the Mittag-Leffler-Ulam have been investigated. Ultimately, an illustrative example has been highlighted in order to reinforce the theoretical results and suggest applications for this article.

On a Dirichlet problem related to anisotropic fluid flow in bidisperse porous media

This paper is concerned with the study of a Dirichlet boundary value problem for a system of two coupled anisotropic Darcy–Forchheimer–Brinkman equations on a bounded Lipschitz domain in . Using variational methods and fixed point techniques, we obtain a well‐posedness result in ‐based Sobolev spaces for sufficiently small data. As an application, we investigate numerically the lid‐driven flow problem in a square cavity saturated with a bidisperse porous medium and analyze the effect of various physical parameters on the fluid motion.

An existence of the solution for generalized system of fractional q-differential inclusions involving p-Laplacian operator and sequential derivatives

In this paper, we investigate the presence of positive solutions for system of fractional q-differential inclusions involving sequential derivatives with respect to the p-Laplacian operator. By using fixed point technique we obtain a new solution for inclusion or boundary value problems with special integral and derivative conditions. At the end, we give an example to show the effect of the solution of this device. The conclusion is expressed to introduce future works.

New exploration on approximate controllability of nondensely defined Hilfer neutral-type delayed nonlinear differential inclusion system with non-instantaneous impulse

This manuscript aims to investigate the existence and approximate controllability results for integral solution of nondensely defined Hilfer neutral differential inclusion system of Sobolev-type with infinite delay and non-instantaneous impulse in a Hilbert space. By utilizing the semigroup theory of bounded linear operators, fixed point technique, fractional calculus, and the theory of multivalued maps, the principal discussions are demonstrated. The sufficient conditions for the existence and approximate controllability are demonstrated under the consideration that the linear operator A is defined nondensely and satisfies the Hille–Yosida condition. Finally, an example is given to support the validity of the study.

Discussion on the existence and controllability of Hilfer fractional stochastic differential equation with non-dense domain

This article studies the approximate controllability for a class of fractional control system with semigroup operators governed by differential equations with Hilfer fractional derivatives of order α ∈ ( 0 , 1 ) and type β ∈ [ 0 , 1 ] in Hilbert spaces. First, we establish a set of sufficient conditions for the approximate controllability for Hilfer fractional stochastic differential equation with non-dense domain in Hilbert spaces. The existence results are obtained by using the fractional calculus, semi-group theory, Wiener process and fixed point technique to prove our main results. Further, we extend the result to study the approximate controllability concept. An example is provided to illustrate the application of the obtained theory.

Modified Inertial Krasnosel'skii-Mann Type Method for Solving Fixed PointProblems in Real Uniformly Convex Banach Spaces

We present an altered version of the inertial Krasnosel'skii-Mann algorithm and demonstrate convergence outcomes for mappings that are asymptotically nonexpansive within real, uniformly convex Banach spaces. To achieve our results, we skillfully construct the inequality in equation \eqref{6} and apply it accordingly. Our findings support and broadly generalize a number of significant findings from the literature. We demonstrate, as an application, the generation of maximal monotone operators' zeros via fixed point methods in Hilbert spaces. Additionally, we solve convex minimization issues using our fixed-point techniques.

On the approximate controllability of second-order stochastic differential systems with hemivariational inequalities with damping

In this article, we investigate the approximate controllability of damping second-order stochastic differential systems with hemivariational inequalities. First, we show the existence of a mild solution to the suggested second-order differential system with the notions of generalized Clarke's subdifferential type. Next, by adopting the fixed point technique of condensing multi-valued maps, the conditions for generalized Clarke's subdifferential and the strongly continuous cosine family is employed for the existence of a mild solution. Furthermore, the approximate controllability results for the given system are proven. In the end, an illustration is presented to highlight our primary findings.

On the solvability of some systems of integro-differential equations with and without a drift

We prove the existence of solutions for some integro-differential systems containing equations with and without drift terms in the H 2 spaces by virtue of the fixed point technique when the elliptic equations contain second order differential operators with and without Fredholm property, on the whole real line or on a finite interval with periodic boundary conditions. Let us emphasise that the study of the system case is more complicated than of the scalar situation and requires to overcome more cumbersome technicalities.

On the application of some fixed-point techniques to Fredholm integral equations of the second kind

It is known that the global convergence of the method of successive approximations is obtained by means of the Banach contraction principle. In this paper, we study the global convergence of the method by means of a technique that uses auxiliary points and, as a consequence of this study, we obtain fixed-point type results on closed balls. We apply the study to nonlinear Fredholm integral equations of the second kind.

Global existence and stability results for a time-fractional diffusion equation with variable exponents

This paper aims to study the existence and stability results concerning a fractional partial differential equation with variable exponent source functions. The local existence result for α∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1)$$\\end{document} is established with the help of the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-resolvent kernel and the Schauder-fixed point theorem. The non-continuation theorem is proved by the fixed point technique and accordingly the global existence of solution is achieved. The uniqueness of the solution is obtained using the contraction principle and the stability results are discussed by means of Ulam-Hyers and generalized Ulam-Hyers-Rassias stability concepts via the Picard operator. Examples are provided to illustrate the results.

New L-fuzzy fixed point techniques for studying integral inclusions

The survey of the available literature shows that a lot of important invariant point problems of Banach and Heilpern types have been examined in both metric and quasimetric spaces. However, a handful of the existing results employed the recent approaches of interpolative contractions. Therefore, based on the new idea of interpolation techniques in fixed point theory, this article studies new notions of L-fuzzy contractions and investigates conditions for the existence of L-fuzzy fixed points for such mappings. On the fact that fixed points of point-to-point mappings satisfying interpolative-type contraction are not always unique, whence making the concepts more fitted for invariant point results of crisp set-valued maps, new multi-valued analogues of the key findings put forward in this work are derived. Comparative illustrations, which indicate the preeminence of the results presented herein, are constructed. From application viewpoint, one of the theorems so obtained is employed to introduce new solvability conditions of Fredholm-type integral inclusions.

Existence and stability results for a Langevin system with Caputo–Hadamard fractional operators

This paper is devoted to analyzing a new model of a coupled Langevin system with fractional operators under nonlocal antiperiodic integral boundary conditions. This model involves nonlinear Langevin fractional equations with Caputo–Hadamard and Caputo fractional operators. Also, the existence and uniqueness of solutions to the suggested model have been investigated by the fixed-point technique. Moreover, the Hyers–Ulam stability of the solutions has been discussed. Finally, we provide an illustrative example to support the theoretical results.

The well-posedness of semilinear fractional dissipative equations on [formula omitted]

In this paper, we study the well-posedness of time-space fractional dissipative equations. Combining the Hörmander-multipliers theory and asymptotic property of the Mittag-Leffler functions, we give a useful method to estimate the solution operator which is independent of the C0-semigroup generated by the fractional Laplace operator (−Δ)β2 and the Mainardi's Wright type function. Furthermore, we discuss the global/local well-posedness in some appropriate time-space Lebesgue space and Besov spaces. We also obtain several results about blow-up alternative and asymptotic behavior. The approaches are based on the Hörmander-multipliers theory, Gagliardo-Nirenberg inequalities, fixed point techniques, Sobolev embedding and some methods in harmonic analysis.

Modeling the dynamics of the Hepatitis B virus via a variable-order discrete system

We investigate the dynamics of the hepatitis B virus by integrating variable-order calculus and discrete analysis. Specifically, we utilize the Caputo variable-order difference operator in this study. To establish the existence and uniqueness results of the model, we employ a fixed-point technique. Furthermore, we prove that the model exhibits bounded and positive solutions. Additionally, we explore the local stability of the proposed model by determining the basic reproduction number. Finally, we present several numerical simulations to illustrate the richness of our results.

Periodic Boundary Value problem for the Dynamical system with neutral integro-differential equation on time scales

Examining the results of existence and Ulam stability in a fractional dynamic system, including a neutral integro-differential equation that incorporates periodic boundary conditions on time scales and the Caputo fractional nabla derivative, is the main objective of the study. Standard fixed-point techniques are applied to obtain the results. Additionally, controllability and Ulam stability results are included. The application of the theoretical results is illustrated with a suitable example.

Fuzzy fixed point techniques for analyzing differential inclusions

Fuzzy fixed point techniques for analyzing differential inclusions

New investigation on controllability of sobolev-type Volterra-Fredholm functional integro-differential equation with non-local condition

In this article, we analyse the controllability criteria on sobolev-type Volterra-Fredholm functional Integro-differential Equations (SVFIDE) via Atangana–Baleanu Caputo (ABC) derivative. The primary outcomes was established by utlising the concepts on the measure of noncompactness, semigroup theory, the contraction principle and Darbo fixed point technique with nonlocal condition. An illustrative example to demonstrate the analytical result is provided at the end.

Existence of common fuzzy fixed points via fuzzy F-contractions in b-metric spaces

The main goal of this study is to establish common fuzzy fixed points in the context of complete b-metric spaces for a pair of fuzzy mappings that satisfy F-contractions. To strengthen the validity of the derived results, non-trivial examples are provided to substantiate the conclusions. Moreover, prior discoveries have been drawn as logical extensions from pertinent literature. Our findings are further reinforced and integrated by the numerous implications that this technique has in the literature. Using fixed point techniques to approximate the solutions of differential and integral equations is very useful. Specifically, in order to enhance the validity of our findings, the existence result of the system of non-linear Fredholm integral equations of second-kind is incorporated as an application.