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20 pages, 904 KiB

Open AccessArticle

Analytical and Numerical Approaches via Quadratic Integral Equations

by Jihan Alahmadi, Mohamed A. Abdou and Mohamed A. Abdel-Aty

Axioms 2024, 13(9), 621; https://doi.org/10.3390/axioms13090621 - 12 Sep 2024

Cited by 1 | Viewed by 849

A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space

C ([0, T] \times [0, T]) .

The existence of at least one solution to the QIE is [...] Read more.

A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space

C ([0, T] \times [0, T]) .

The existence of at least one solution to the QIE is discussed in this article. Our evidence depends on a suitable combination of the measures of the noncompactness approach and the fixed-point principle of Darbo. The quadratic integral equation can be used to derive a system of integral equations of the second kind using the quadrature method. With the aid of two different polynomials, Laguerre and Hermite, the system of integral equations is solved using the collocation method. In each numerical approach, the estimation of the error is discussed. Finally, using some examples, the accuracy and scalability of the proposed method are demonstrated along with comparisons. Mathematica 11 was used to obtain all of the results from the techniques that were shown. Full article

(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)

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15 pages, 280 KiB

Open AccessArticle

New Extension of Darbo’s Fixed Point Theorem and Its Application to a System of Weighted-Fractional-Type Integral Equations

by Marija Paunović, Ana Savić, Hemanta Kalita, Sudip Deb and Vahid Parvaneh

Mathematics 2024, 12(13), 2133; https://doi.org/10.3390/math12132133 - 7 Jul 2024

Cited by 1 | Viewed by 1009

Abstract

In this article, we introduce several new extensions of Darbo’s fixed point theorem with newly constructed contraction functions associated with the measure of noncompactness. We apply our new extensions to prove the existence of solutions for a system of weighted fractional integral equations [...] Read more.

In this article, we introduce several new extensions of Darbo’s fixed point theorem with newly constructed contraction functions associated with the measure of noncompactness. We apply our new extensions to prove the existence of solutions for a system of weighted fractional integral equations in Banach space

B C (R_{+})

. At the end, we establish an example to show the applicability of our discovery. Full article

(This article belongs to the Special Issue Soft Computing and Fuzzy Mathematics: New Advances and Applications)

12 pages, 276 KiB

Open AccessArticle

Darbo’s Fixed-Point Theorem: Establishing Existence and Uniqueness Results for Hybrid Caputo–Hadamard Fractional Sequential Differential Equations

by Muhammad Yaseen, Sadia Mumtaz, Reny George, Azhar Hussain and Hossam A. Nabwey

Fractal Fract. 2024, 8(6), 326; https://doi.org/10.3390/fractalfract8060326 - 29 May 2024

Cited by 1 | Viewed by 1125

Abstract

This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo’s fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The [...] Read more.

This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo’s fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo’s fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo’s theorem to analyze the solutions’ existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)

11 pages, 233 KiB

Open AccessArticle

On Darbo- and Sadovskii-Type Fixed Point Theorems in Banach Spaces

by Leszek Olszowy and Tomasz Zając

Symmetry 2024, 16(4), 392; https://doi.org/10.3390/sym16040392 - 27 Mar 2024

Cited by 3 | Viewed by 1220

Abstract

The paper aims to generalize several known Darbo- and Sadovskii-type fixed point theorems. These generalizations weaken the assumptions used so far. In addition, an example of an application is presented. Full article

(This article belongs to the Special Issue Nonlinear Differential and Integral Equations and Their Infinite Systems)

19 pages, 325 KiB

Open AccessArticle

Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations

by Józef Banaś and Justyna Madej

Symmetry 2024, 16(1), 107; https://doi.org/10.3390/sym16010107 - 16 Jan 2024

Cited by 4 | Viewed by 1306

Abstract

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real [...] Read more.

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper. Full article

(This article belongs to the Special Issue Nonlinear Differential and Integral Equations and Their Infinite Systems)

18 pages, 310 KiB

Open AccessArticle

Asymptotic Stability and Dependency of a Class of Hybrid Functional Integral Equations

by Ahmed M. A. El-Sayed, Malak M. S. Ba-Ali and Eman M. A. Hamdallah

Mathematics 2023, 11(18), 3953; https://doi.org/10.3390/math11183953 - 17 Sep 2023

Cited by 2 | Viewed by 1216

Abstract

Here, we discuss the solvability of a class of hybrid functional integral equations by applying Darbo’s fixed point theorem and the technique of the measure of noncompactness (MNC). This study has been located in space BC

(R_{+})

. Furthermore, we [...] Read more.

Here, we discuss the solvability of a class of hybrid functional integral equations by applying Darbo’s fixed point theorem and the technique of the measure of noncompactness (MNC). This study has been located in space BC

(R_{+})

. Furthermore, we prove the asymptotic stability of the solution of our problem on

R_{+} .

We introduce the idea of asymptotic dependency of the solutions on some parameters for that class. Moreover, general discussion, examples, and remarks are demonstrated. Full article

15 pages, 298 KiB

Open AccessArticle

New Aspects on the Solvability of a Multidimensional Functional Integral Equation with Multivalued Feedback Control

by Ahmed M. A. El-Sayed, Hind H. G. Hashem and Shorouk M. Al-Issa

Axioms 2023, 12(7), 653; https://doi.org/10.3390/axioms12070653 - 30 Jun 2023

Cited by 5 | Viewed by 1101

Abstract

The current study demonstrates the existence of solutions to a multidimensional functional integral equation with multivalued feedback. We seek solutions for the multidimensional functional problem that is defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated [...] Read more.

The current study demonstrates the existence of solutions to a multidimensional functional integral equation with multivalued feedback. We seek solutions for the multidimensional functional problem that is defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in

B C (R_{+}) .

Also, some sufficient conditions are investigated to demonstrate the asymptotic stability of the solutions to that multidimensional functional equation. Additionally, we give an example and some particular cases to illustrate our outcomes. Full article

(This article belongs to the Section Mathematical Analysis)

16 pages, 328 KiB

Open AccessArticle

Investigating Asymptotic Stability for Hybrid Cubic Integral Inclusion with Fractal Feedback Control on the Real Half-Axis

by Hind H. G. Hashem, Ahmed M. A. El-Sayed and Shorouk M. Al-Issa

Fractal Fract. 2023, 7(6), 449; https://doi.org/10.3390/fractalfract7060449 - 31 May 2023

Cited by 8 | Viewed by 1818

Abstract

In this paper, we discuss the existence of solutions for a hybrid cubic delayed integral inclusion with fractal feedback control. We are seeking solutions for these hybrid cubic delayed integral inclusions that are defined, continuous, and bounded on the semi-infinite interval. Our proof [...] Read more.

In this paper, we discuss the existence of solutions for a hybrid cubic delayed integral inclusion with fractal feedback control. We are seeking solutions for these hybrid cubic delayed integral inclusions that are defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in

B C (R_{+}) .

In addition, some sufficient conditions are investigated to demonstrate the asymptotic stability of the solutions of that integral inclusion. Finally, some cases analyzed are in the presence and absence of the control variable, and two examples are provided in order to indicate the validity of the assumptions. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

17 pages, 323 KiB

Open AccessArticle

Approximate Controllability of Neutral Functional Integro-Differential Equations with State-Dependent Delay and Non-Instantaneous Impulses

by Abdelhamid Bensalem, Abdelkrim Salim, Mouffak Benchohra and Michal Fečkan

Mathematics 2023, 11(7), 1667; https://doi.org/10.3390/math11071667 - 30 Mar 2023

Cited by 9 | Viewed by 1522

Abstract

In this manuscript, we investigate the issue of approximate controllability for a certain class of abstract neutral integro-differential equations having non-instantaneous impulsions and being subject to state-dependent delay. Our methodology relies on the utilization of resolvent operators in conjunction with Darbo’s fixed point [...] Read more.

In this manuscript, we investigate the issue of approximate controllability for a certain class of abstract neutral integro-differential equations having non-instantaneous impulsions and being subject to state-dependent delay. Our methodology relies on the utilization of resolvent operators in conjunction with Darbo’s fixed point theorem. To exemplify the practical implications of our findings, we provide an illustration. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

17 pages, 325 KiB

Open AccessArticle

An Infinite System of Fractional Sturm–Liouville Operator with Measure of Noncompactness Technique in Banach Space

by Ahmed Salem, Hunida Malaikah and Eid Sayed Kamel

Mathematics 2023, 11(6), 1444; https://doi.org/10.3390/math11061444 - 16 Mar 2023

Cited by 1 | Viewed by 1221

Abstract

In the current contribution, an appropriate quantity connected to the space of all convergent sequences is provided and shown to be a measure of noncompactness in a Banach space. Through the application of the fixed point theorems of Darbo and Meir–Keeler, this amount [...] Read more.

In the current contribution, an appropriate quantity connected to the space of all convergent sequences is provided and shown to be a measure of noncompactness in a Banach space. Through the application of the fixed point theorems of Darbo and Meir–Keeler, this amount is used to discuss whether a solution to an infinite system of fractional Sturm–Liouville operators exists. We offer a numerical example as an application of the key finding in the study. Full article

(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)

16 pages, 345 KiB

Open AccessArticle

On the Existence Results for a Mixed Hybrid Fractional Differential Equations of Sequential Type

by Meraa Arab, Muath Awadalla, Murugesan Manigandan, Kinda Abuasbeh, Nazim I. Mahmudov and Thangaraj Nandha Gopal

Fractal Fract. 2023, 7(3), 229; https://doi.org/10.3390/fractalfract7030229 - 4 Mar 2023

Cited by 3 | Viewed by 1479

Abstract

In this article, we study the existence of a solution to the mixed hybrid fractional differential equations of sequential type with nonlocal integral hybrid boundary conditions. The main results are established with the aid of Darbo’s fixed point theorem and Hausdorff’s measure of [...] Read more.

In this article, we study the existence of a solution to the mixed hybrid fractional differential equations of sequential type with nonlocal integral hybrid boundary conditions. The main results are established with the aid of Darbo’s fixed point theorem and Hausdorff’s measure of noncompactness method. The stability of the proposed fractional differential equation is also investigated using the Ulam–Hyer technique. In addition, an applied example that supports the theoretical results reached through this study is included. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Evolution Equations and Related Topics)

18 pages, 309 KiB

Open AccessArticle

Analytical Contribution to a Cubic Functional Integral Equation with Feedback Control on the Real Half Axis

by Ahmed M. A. El-Sayed, Hind H. G. Hashem and Shorouk M. Al-Issa

Mathematics 2023, 11(5), 1133; https://doi.org/10.3390/math11051133 - 24 Feb 2023

Cited by 4 | Viewed by 1621

Abstract

Synthetic biology involves trying to create new approaches using design-based approaches. A controller is a biological system intended to regulate the performance of other biological processes. The design of such controllers can be based on the results of control theory, including strategies. Integrated [...] Read more.

Synthetic biology involves trying to create new approaches using design-based approaches. A controller is a biological system intended to regulate the performance of other biological processes. The design of such controllers can be based on the results of control theory, including strategies. Integrated feedback control is central to regulation, sensory adaptation, and long-term effects. In this work, we present a study of a cubic functional integral equation with a general and new constraint that may help in investigating some real problems. We discuss the existence of solutions for an equation that involves a control variable in the class of bounded continuous functions

BC (R_{+}),

by applying the technique of measure of noncompactness on

R_{+}

. Furthermore, we establish sufficient conditions for the continuous dependence of the state function on the control variable. Finally, some remarks and discussion are presented to demonstrate our results. Full article

(This article belongs to the Special Issue Advances in Differential Analysis and Functional Analysis)

23 pages, 363 KiB

Open AccessEditor’s ChoiceArticle

Boundary-Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay via Kuratowski Measure of Noncompactness

by Benoumran Telli, Mohammed Said Souid and Ivanka Stamova

Axioms 2023, 12(1), 80; https://doi.org/10.3390/axioms12010080 - 12 Jan 2023

Cited by 9 | Viewed by 2198

Abstract

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. [...] Read more.

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

12 pages, 269 KiB

Open AccessArticle

Treatment of a Coupled System for Quadratic Functional Integral Equation on the Real Half-Line via Measure of Noncompactness

by Ahmed M. A. El-Sayed, Yasmin M. Y. Omar, Hind H. G. Hashem and Shorouk M. Al-Issa

Foundations 2023, 3(1), 37-48; https://doi.org/10.3390/foundations3010004 - 6 Jan 2023

Cited by 1 | Viewed by 1653

Abstract

This article is devoted to the solvability and the asymptotic stability of a coupled system of a functional integral equation on the real half-axis. Our consideration is located in the space of bounded continuous functions on

R_{+}

[...] Read more.

This article is devoted to the solvability and the asymptotic stability of a coupled system of a functional integral equation on the real half-axis. Our consideration is located in the space of bounded continuous functions on

R_{+}

(B C (R_{+})) .

The main tool applied in this work is the technique associated with measures of noncompactness in

B C (R_{+})

by a given modulus of continuity. Next, we formulate and prove a sufficient condition for the solvability of that coupled system. We, additionally, provide an example and some particular cases to demonstrate the effectiveness and value of our results. Full article

(This article belongs to the Section Mathematical Sciences)

18 pages, 432 KiB

Open AccessArticle

A New Result Concerning Nonlocal Controllability of Hilfer Fractional Stochastic Differential Equations via almost Sectorial Operators

by Sivajiganesan Sivasankar, Ramalingam Udhayakumar, Muchenedi Hari Kishor, Sharifah E. Alhazmi and Shrideh Al-Omari

Mathematics 2023, 11(1), 159; https://doi.org/10.3390/math11010159 - 28 Dec 2022

Cited by 12 | Viewed by 2016

Abstract

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors [...] Read more.

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications)

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