In this paper we give more results about inner radius spectrum of operators on Hilbert spaces with several examples. Also, we established an inequality for inner radius spectrum of a Positive operator matrix and its minimum moduli block matrix.

In this work, we establish the existence of at least two Positive solutions for a coupled system of $p$-Laplacian fractional-order boundary value problems. Establishing the existence of Positive solutions to the problem is challenging for a variety of reasons, the most important of which is a lack of compatibility with the kernel. To address these issues, we have included the necessary conditions for overcoming certain methodological hurdles on the kernel as well as adapting to the problem's nature of positivity. The method is based on the AH functional fixed point theorem.

Some necessary and sufficient conditions are given for the existence of a G-Positive (G-rePositive) solution of adjointable operator equations AX=C, AXA (*) =C and AXB=C over Hilbert C*-modules, respectively. Moreover, the expressions of these general G-Positive (G-rePositive) solutions are also derived.Some of the findings of this paper extend some known results in the literature.

The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an operatorvalued inner product, we discuss operator versions for the above inequalities and give simple conditions that the equalities hold.

In this paper we derive some improvements of means inequalities for Hilbert space operators. More precisely, we obtain refinements and reverses of the arithmetic-geometric operator mean inequality. As an application, we also deduce an improved variant for the refined arithmetic-Heinz mean in- equality. We also present some eigenvalue inequalities for differences of certain operator means.