Simulation functions and Meir–Keeler condensing operators with application to integral equations

We propose a new concept of condensing operators by using a notion of measure of non-compactness in the setting of Banach spaces and establish a new generalization of Darbo’s fixed point theorem. We also show the applicability of our results to integral equations. A concrete example will be presented to support the application part.

On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria

In this paper, we consider a nonlinear sequential q-difference equation based on the Caputo fractional quantum derivatives with nonlocal boundary value conditions containing Riemann–Liouville fractional quantum integrals in four points. In this direction, we derive some criteria and conditions of the existence and uniqueness of solutions to a given Caputo fractional q-difference boundary value problem. Some pure techniques based on condensing operators and Sadovskii’s measure and the eigenvalue of an operator are employed to prove the main results. Also, the Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated. We examine our results by providing two illustrative examples.

Existence results for infinite systems of the Hilfer fractional boundary value problems in Banach sequence spaces

The main aim of this paper is to present some existence criteria for an infinite system of Hilfer fractional boundary value problems of the form $$ \mathcal{D}_{a^{+}}^{\alpha,\beta }u_{i}=-F_{i}(t,u),\quad u_{i}(a)=u_{i}(b)=0, a< t< b,i=1,2,\ldots, $$ D a + α , β u i = − F i ( t , u ) , u i ( a ) = u i ( b ) = 0 , a < t < b , i = 1 , 2 , … , in Banach sequence spaces of $c_{0}$ c 0 and $l_{p},p\geq 1$ l p , p ≥ 1 types. Our approach is based on the Darbo-type fixed point theorems acting on the condensing operators. The obtained existence results in each of the above sequence spaces are illustrated by presenting some numerical examples.

Meir-Keeler condensing operators and applications

Motivated by the open question posed by H. K. XU in [39] (Question 2:8), Belhadj, Ben Amar and Boumaiza introduced in [5] the concept of Meir-Keeler condensing operator for self-mappings in a Banach space via an arbitrary measure of weak noncompactness. In this paper, we introduce the concept of Meir- Keeler condensing operator for nonself-mappings in a Banach space via a measure of weak noncompactness and we establish fixed point results under the condition of Leray-Schauder type. Some basic hybrid fixed point theorems involving the sum as well as the product of two operators are also presented. These results generalize the results on the lines of Krasnoselskii and Dhage. An application is given to nonlinear hybrid linearly perturbed integral equations and an example is also presented.

Existence of Solutions of Infinite Systems of Nonlinear Functional Integral Equations of N-Variables in C(I × I ×⋯ × I,m(ϕ))

The aim of this paper is to investigate the solvability of infinite systems of nonlinear functional integral equations of [Formula: see text]-variables in [Formula: see text] by using the Hausdorff measure of noncompactness with the help of Meir–Keeler condensing operators. We also provide an illustrative example in support of our existence theorems.

Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving ψ -Caputo Derivative in Banach and Fréchet Spaces

Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involving ψ -Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger’s fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Grönwall’s inequality, the Mittag–Leffler–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo’s fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.

Family of mappings with an equicontractive-type condition

In a real Banach space X and a complete metric space M, we consider a compact mapping C defined on a closed and bounded subset A of X with values in M and the operator $$T:A\times C(A) \rightarrow X$$ T : A × C ( A ) → X . Using a new type of equicontractive condition for a certain family of mappings and $$\beta $$ β -condensing operators defined by the Hausdorff measure of noncompactness we prove that the operator $$x\mapsto T(x,C(x))$$ x ↦ T ( x , C ( x ) ) has a fixed point. The obtained results are applied to the initial value problem.

New classes of condensing operators and application to solvability of singular integral equations

In this paper, we introduce the notion of Krasnoselskii and Dugundji-Granas condensing operators in Banach spaces. In order to pave the way for a study the solvability of some classes of singular integral equations in the Banach algebra C[a,b], we provide some results for the existence of fixed points for such condensing operators. An example is presented to show the applicability of the results.