Abstract:
In this talk, we present a brief survey of theory and applications of measures of noncompactness. The classical measures of noncompactness are discussed and their properties are compared. The approaches for constructing measure of noncompactness in a general metric or linear space are described, along with the classical results for existence of fixed point for condensing operators. Also several generalization of classical results are mentioned and their applications in various problems of analysis such as linear equation, differential equations, integral equations and common solutions of equations are discussed. The most effective way in the characterization of compact operators between the Banach spaces is applying the Hausdorff measure of noncompactness. In this talk, we present some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary BK-space into the sequence spaces c₀, c, ℓ∞ and ℓ₁. Many linear compact operators may be represented as matrix operators in sequence spaces or integral operators in function spaces [J. Banas and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, 2014]. Recently the measures of noncompactness are applied in solving infinite system of integral equations [A. Das, B. Hazarika, R. Arab and M. Mursaleen, Solvability of the infinite system of integral equations in two variables in the sequence spaces c₀ and ℓ₁, Jour. Comput. Appl. Math., 326 (2017) 183-192] and differential equations [M. Mursaleen and S.M.H. Rizvi, Solvability of infinite system of second order differential equations in c₀ and ℓ₁ by Meir-Keeler condensing operator, Proc. Amer. Math. Soc., 144(10) (2016) 4279-4289] in sequence spaces.
This talk originates from the investigation of nonlinear functional-integral equation with Erdlyi-Kober fractional operator. Existence results of solutions in Banach algebra are obtained under some relevant results of fixed point theorems such as Darbo’s theorem concerning the mentioned goal in Banach algebra. Finally, some examples to illustrate the usefulness of our results.
Keywords: Sequence spaces; Erdlyi-Kober fractional integrals; functional-integral equation; compact operators, fixed point theorem; Banach algebra; measures of noncompactness, infinite system of differential equations.
References:
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