Weak Condensing Frameworks and Best Proximity Methods for Nonlinear Operator Equations

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional (κ,ϕ)-Riemann–Liouville along with Erdélyi–Kober fractional operators on a Banach space C([1,T]) arising in biological population dynamics. The key findings of the article are based on theoretical concepts pertaining to the fractional calculus and the Hausdorff measure of non-compactness (MNC). To obtain this goal, we employ Darbo’s fixed-point theorem (DFPT) in the Banach space. In addition, we provide two numerical examples to demonstrate the applicability of our findings to the theory of fractional integral equations.

Substituting the usual growth condition by an assumption that a specific initial value problem has a maximal solution, we obtain existence results for functional nonlinear integral equations with variable delay. Appli- cation of the technique to initial value problems for differential equations as well as to integrodifferential equations are given. Nonlinear integral equations and nonlinear functional integral equations have been some topics of great interest in the field of nonlinear analysis for a long time. Since the pioneering work of Volterra up to our days, integral equations have attracted the interest of scientists not only because of their mathematical context but also because of their miscellaneous applications in various fields of science and technology. In particular, existence theory for nonlinear integral equations, strongly related with the evolution on fixed point theory, has been boosted ahead after the remarkable work of Krasnoselskii (6) which signaled a new era in the research of the subject. The present note is motivated by a recent paper by Dhage and Ntouyas (3) presenting some results on the existence of solutions to the nonlinear functional integral equation (E)

In this paper, we investigate the existence of solutions for a new class of nonlinear product-type fractional functional integral equations (FFIEs) involving the Riemann–Liouville fractional integral operator. To establish the existence of at least one solution, we employ Petryshyn's fixed-point theorem (PFPT) combined with the concept of the measure of noncompactness (MNC) in the Banach space C [ 0 , a ] of continuous functions. Unlike other approaches based on Darbo's or Schauder's fixed-point theorems in Banach algebras, our method does not require the operator to map a closed convex subset onto itself, nor does it rely on the commonly assumed “sublinear condition” for the functional involved in the equation. Therefore, our results generalize and unify several existing results in the literature under fewer conditions. Additionally, to support our theoretical findings, we provide an example of such nonlinear FFIEs, thereby illustrating the applicability of the proposed results.

In this article, we establish some results for the existence of solution of nonlinear functional integral equations by using Darbo’s fixed point theorem in Banach algebra. As an application, we discuss some examples of nonlinear functional integral equations and existence of solutions.

ABSTRACTWe prove a theorem on the existence of solutions of some nonlinear functional integral equations in the Banach algebra of continuous functions on the interval [0,a]. Then we consider a nonlinear integral equation of fractional order and give some sufficient conditions for existence of solutions of this equation. We use fixed point theorems associated with the measure of noncompactness as the main tool. Our existence results include several results obtained in previous studies. Finally we present some examples which show that our results are applicable.

<abstract><p>In this article, we consider a class of nonlinear functional integral equations, motivated by an equation that offers increasing evidence to the extant literature through replication studies. We investigate the existence of solution for nonlinear functional integral equations on Banach space $ C[0, 1] $. We use the technique of the generalized Darbo's fixed-point theorem associated with the measure of noncompactness (MNC) to prove our existence result. Also, we have given two examples of the applicability of established existence result in the theory of functional integral equations. Further, we construct an efficient iterative algorithm to compute the solution of the first example, by employing the modified homotopy perturbation (MHP) method associated with Adomian decomposition. Moreover, the condition of convergence and an upper bound of errors are presented.</p></abstract>

One parameter operator imbedding to modify Newton method for solution of nonlinear equations

This study focuses on the nonlinear fractional functional integral equation (FFIE) concerning the Riemann-Liouville operator. In certain weaker conditions, the authors demonstrate that the FFIE has a solution, which is defined within the Banach algebra 0 $ ]]> C [ 0 , a ] , a > 0 . Our analysis relies on the Petryshyn's fixed point theorem and the notion of measure of non-compactness (MNC). In addition, our results include numerous authors' work under less restrictive conditions. Furthermore, we provide an illustrative example of fractional functional integral equations to support our proven results.

Let E and F be nonempty disjoint subsets of a metric space (M,d). For a non-self-mapping φ:E→F, which is fixed-point free, a point ϰ∈E is said to be a best proximity point for the mapping φ whenever the distance of the point ϰ to its image under φ is equal to the distance between the sets, E and F. In this article, we establish new best proximity point theorems and obtain real extensions of Edelstein’s fixed point theorem in metric spaces, Krasnoselskii’s fixed point theorem in strictly convex Banach spaces, Dhage’s fixed point theorem in strictly convex Banach algebras, and Sadovskii’s fixed point problem in strictly convex Banach spaces. We then present applications of these best proximity point results to complex function theory, as well as the existence of a solution of a nonlinear functional integral equation and the existence of a mutually nearest solution for a system of integral equations.

In this paper, utilizing the technique of generalized Darbo's fixed‐point theorem associated with measure of noncompactness in Banach space, we analyze the existence of solution for a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator. The existing result was obtained to strengthen the ones mentioned previously in the literature. An example for a class of nonlinear functional integral equations is also presented to validate our main result. Finally, we propose the numerical method formed by the modified homotopy perturbation approach to resolving the problem with acceptable accuracy.

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.

This paper is concerned with existence results for a quite general nonlinear functional integral equation in $L^1$ spaces. For this purpose, making use of the De Blasi measure of weak noncompactness, we first establish a new fixed point theorem of the nonautonomous superposition operators. After that, our theorem is applied to prove the solvability of the mentioned nonlinear functional integral equation.

An algebraic fixed point theorem involving the three operators in a Banach algebra is proved using the properties of cones and they are further applied to a certain nonlinear integral equations of mixed type for proving the existence of maximal and minimal solutions. Our results include the earlier fixed point theorems of Dhage (1992 and 1999) as special cases with a different but simple method.

On a fixed point theorem in Banach algebras with applications