ψ−COUPLED FIXED POINT THEOREM VIA SIMULATION FUNCTIONS IN COMPLETE PARTIALLY ORDERED METRIC SPACE AND ITS APPLICATIONS

The Numerical Solution of Integral Equations on the Half-Line

Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations

Fractional differential equations and fractional integral equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines. Gronwall-Bellman inequalities are important tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of Fractional differential equations and fractional integral equations. In this paper, we discuss a class of integral inequalities with pth power, which includes a nonconstant term outside the integrals. Using the definitions and properties of modified Riemann-Liouville fractional derivative and Riemann-Liouville fractional integral, the upper bounds of the unknown function is estimated explicitly. The derived result can be applied in the study of qualitative properties of solutions of fractional integral equations. Introduction Fractional differential equations and fractional integral equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines. Gronwall-Bellman inequalities [1, 2] are important tools in the study of existence, uniqueness, boundedness, stability, invariant manifolds and other qualitative properties of solutions of fractional differential equations and fractional integral equations. In 2011, Abdeldaim et al. [3] studied a new iterated integral inequality with pth power ds d d g u h s u s u s f u t u s t ] ] ) ( ) ( [ ) ( ) ( )[ ( ) ( ) ( 0 0 0 0              . (1) In 2014, El-Owaidy, Abdeldaim, and El-Deeb [4] discussed a new nonlinear integral inequality with a nonconstant term outside the integrals ds s u s h ds s u s g t f t u t

By utilizing the technique of Petryshyn's fixed point theorem in Banach algebra, we examine the existence of solutions for fractional integral equations, which include as special cases of many fractional integral equations that arise in various branches of mathematical analysis and their applications. Also, the numerical iterative method is employed successfully to find the solutions to fractional integral equations. Lastly, we recall some different cases and examples to verify the applicability of our study.

In this paper, we present some results concerning the existence and the local asymptotic stability of solutions for a functional integral equation of fractional order, by using some fixed point theorems.

In this article, we consider a new fractional integral equation, namely, generalized proportional Hadamard fractional (GPHF) integral equations. Then as an application of Darbo’s fixed point theory (DFPT), we establish the existence of the solution of above-mentioned GPHF integral equations, using a measure of noncompactness (MNC). At the end, we have provided a suitable example to verify our obtained results.

In the present study, we work on the problem of the existence of positive solutions of fractional integral equations by means of measures of noncompactness in association with Darbo’s fixed point theorem. To achieve the goal, we first establish new fixed point theorems using a new contractive condition of the measure of noncompactness in Banach spaces. By doing this we generalize Darbo’s fixed point theorem along with some recent results of (Aghajani et al. (J. Comput. Appl. Math. 260:67-77, 2014)), (Aghajani et al. (Bull. Belg. Math. Soc. Simon Stevin 20(2):345-358, 2013)), (Arab (Mediterr. J. Math. 13(2):759-773, 2016)), (Banaś et al. (Dyn. Syst. Appl. 18:251-264, 2009)), and (Samadi et al. (Abstr. Appl. Anal. 2014:852324, 2014)). We also derive corresponding coupled fixed point results. Finally, we give an illustrative example to verify the effectiveness and applicability of our results.

This paper is concerned with the investigation of the existence of solutions to the nonlinear fractional Hadamard-type functional integral equations on C([1, a]). To achieve this goal, we employ the theory of measure of noncompactness, fractional calculus, and the fixed point theory in Banach algebra as the key tool to prove our result. Also, we verify the validity and applicability of our result by an appropriate example.

In this article, we propose some new fixed point theorem involving measure of noncompactness and control function. Further, we prove the existence of a solution of functional integral equations in two variables by using this fixed point theorem in Banach Algebra, and also illustrate the results with the help of an example.

Fractional equations and generalizations of Schaefer’s and Krasnoselskii’s fixed point theorems

<abstract><p>This article considers the existence and the uniqueness of monotonic solutions of a delay functional integral equation of fractional order in the weighted Lebesgue space $ L_1^N({\mathbb{R}}^+) $. Our analysis uses a suitable measure of noncompactness, a modified version of Darbo's fixed point theorem, and fractional calculus in the mentioned space. An illustrated example to show the applicability and significance of our outcomes is included.</p></abstract>

In this paper, Darbo’s fixed point theorem is generalized and it is applied to find the existence of solution of a fractional integral equation involving an operator with iterative relations in a Banach space. Moreover, an example is provided to illustrate the results.

In this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's and Schaefer's fixed point theorems are employed in the analysis. The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions. We employ Liapunov's direct method to obtain such an a priori bound. In the process, we compare these theorems in terms of assumptions and outcomes.

A solution of a nonlinear integral equation

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of β-Banach spaces. Moreover, we examine the β–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained.