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How ideas ofPT-symmetric quantum mechanics can be applied to quantum graphs is analyzed, in particular to the star graph. The class of rotationally symmetric vertex conditions is analyzed. It is shown that all such conditions can effectively be described by circulant matrices: real in the case of odd number of edges and complex having particular block structure in the even case. Spectral properties of the corresponding operators are discussed.

In this paper we generalize some Riemann-Liouville fractional integral inequalities of Ostrowski-type for h-convex functions via Katugampola fractional integrals, generalizations of the Riemann-Liouville and the Hadamard fractional integrals. Also we deduce some known results by using p-functions, convex functions and s-convex functions.

In this paper we obtain sharp Lieb-Thirring inequalities for a Schrödinger operator on semi-axis with a matrix potential and show how they can be used to other related problems. Among them are spectral inequalities on star graphs and spectral inequalities for Schrödinger operators on half-spaces with Robin boundary conditions.are known as Lieb-Thirring bounds. Here and in the following, V ± = (|V | ± V )/2 denote the positive and negative parts of the function V .It is known that the inequality (1.2) holds true with some finite constants if and only if γ ≥ 1/2, d = 1; γ > 0, d = 2 and γ ≥ 0, d ≥ 3.There are examples showing that (1.2) fails for 0 ≤ γ < 1/2, d = 1 and γ = 0, d = 2.Almost all the cases except for γ = 1/2, d = 1 and γ = 0, d ≥ 3 were justified in the original paper of E.H.Lieb and W.Thirring [LT]. The critical case γ = 0, d ≥ 3 is known as the Cwikel-Lieb-Rozenblum inequality, see [Cw, L, Roz]. It was also proved in [Fe, LY, Con] and very recently by R. Frank [Fr] using Rumin's approach. The remaining case γ = 1/2, d = 1 was verified by T.Weidl in [W1]. The sharp value of the constants R γ,d = 1 in (1.2) are known for the case γ ≥ 3/2 in all dimensions and it was first proved in [LT] and [AizL] for d = 1 and later in 1991 Mathematics Subject Classification. Primary: 35P15; Secondary: 81Q10.

In this paper, we investigate the existence of a fixed point for modified multivalued α *ψ−contractive type mapping in the context of complete metric space. We also construct some examples to illustrate the main result. Our results extend, improve and generalize the results on the topic in the literature.