Leray–Schauder Fixed Point Theorems for Block Operator Matrix with an Application

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a 2 × 2 block operator matrix defined on nonempty, closed, and convex subsets Ω of Banach spaces. Note here that Ω need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.

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.

Measure of Weak Noncompactness and Fixed Point Theorems in Banach Algebras with Applications

In this paper, we prove some fixed point theorems for the nonlinear operator A · B + C in Banach algebra. Our fixed point results are obtained under a weak topology and measure of weak noncompactness; and we give an example of the application of our results to a nonlinear integral equation in Banach algebra.

A QUANTITATIVE EXTENSION OF SZLENK’S THEOREM

We show that for a bounded subset $A$ of the $L_{1}(\unicode[STIX]{x1D707})$ space with finite measure $\unicode[STIX]{x1D707}$, the measure of weak noncompactness of $A$ based on the convex separation of sequences coincides with the measure of deviation from the Banach–Saks property expressed by the arithmetic separation of sequences. A similar result holds for a related quantity with the alternating signs Banach–Saks property. The results provide a geometric and quantitative extension of Szlenk’s theorem saying that every weakly convergent sequence in the Lebesgue space $L_{1}$ has a subsequence whose arithmetic means are norm convergent.

Existence of Weak Solutions for Fractional Integrodifferential Equations with Multipoint Boundary Conditions

By combining the techniques of fractional calculus with measure of weak noncompactness and fixed point theorem, we establish the existence of weak solutions of multipoint boundary value problem for fractional integrodifferential equations.

Fractional differential inclusions of Hilfer type under weak topologies in Banach spaces

In this paper, we present some results concerning the existence of weak solutions for some functional Hilfer and Hadamard fractional differential inclusions. The Mönch’s fixed point theorem and the concept of measure of weak noncompactness are the main tools used to carry out our results.

Weak Solutions for Partial Random Hadamard Fractional Integral Equations with Multiple Delays

We present some results concerning the existence of weak solutions for some functional integral equations of Hadamard fractional order with random effects and multiple delays by applying Mönch’s and Engl’s fixed point theorems associated with the technique of measure of weak noncompactness.

Fractional Order Pettis Integral Equations with Multiple Time Delay in Banach Spaces

This paper is devoted to study the existence of solutions under the Pettis integrability assumption for an integral equation of fractional order with multiple time delay in Banach space by using the technique of measure of weak noncompactness. Mathematics Subject Classification 2010: 26A33, 34A08.

Integrable Solutions of a Nonlinear Integral Equation via Noncompactness Measure and Krasnoselskii's Fixed Point Theorem

We study the existence of solutions of a nonlinear Volterra integral equation in the space L1[0,+∞). With the help of Krasnoselskii’s fixed point theorem and the theory of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes on nonlinear integral equations. Our results extend and generalize some previous works. An example is given to support our results.