<abstract><p>In this paper, we analyze the behavior of the neutral integro-differential equations of fractional order including the Caputo-Hadamard fractional derivative. The results and solutions are obtained using the topological degree method. Furthermore, some specific numerical examples are given to ascertain the wide applicability and high efficiency of the suggested fixed point technique.</p></abstract>

We propose a modified version of the classical Cesáro means method endowed with the hybrid shrinking projection method to solve the split equilibrium and fixed point problems (SEFPP) in Hilbert spaces. One of the main reasons to equip the classical Cesáro means method with the shrinking projection method is to establish strong convergence results which are often required in infinite-dimensional functional spaces. As a consequence, the convergence analysis is carried out under mild conditions on the underlying shrinking Cesáro means method. We emphasize that the results accounted in this manuscript can be considered as an improvement and generalization of various existing exciting results in this field of study.

In this work, we establish an approach to constructing compact operators between arbitrary infinite-dimensional Banach spaces without a Schauder basis. For this purpose, we use a countable number of basic sequences for the sake of verifying the result of Morrell and Retherford. We also use a nuclear operator, represented as an infinite-dimensional matrix defined over the space 1 of all absolutely summable sequences. Examples of nuclear operators over the space 1 are given and used to construct operators over general Banach spaces with specific approximation numbers.