In this article, we discuss a conformable fractional Sturm‐Liouville boundary‐value problem. We prove an existence and uniqueness theorem for this equation and formulate a self‐adjoint boundary value problem. We also construct the associated Green function of this problem, and we give the eigenfunction expansions. Finally, we will give some examples.

In this paper, we introduce a q-analog of 1-dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of L 2 q ((0, a) ; E). Finally, we give some examples. KEYWORDS eigenfunction expansions, eigenvalues and eigenfunctions, Green matrix, q−Dirac operator, self-adjoint operator Math Meth Appl Sci. 2017;40:7287-7306.wileyonlinelibrary.com/journal/mma

In this article, we investigate the resolvent operator of singular Sturm‐Liouville problem with transmission conditions. We obtain integral representations for the resolvent of this operator in terms of the spectral function. Later, we discuss some properties of the resolvent operator, such as Hilbert‐Schmidt kernel property, compactness. Finally, we give a formula in terms of the spectral function for the Weyl‐Titchmarsh function of this problem.

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In this article, we investigate the resolvent operator of Sturm-Liouville problem on unbounded time scales. We obtain integral representations for the resolvent of this operator. Later, we discuss some properties of the resolvent operator, such as Hilbert-Schmidt's kernel property and compactness. Finally, we give a formula for the Titchmarsh-Weyl function of the Sturm-Liouville problem on unbounded time scales. MSC 2010. 34N05, 34L05, 47A10.