Hyperbolic beta integrals are analogues of Euler’s beta integral in which the role of Euler’s gamma function is taken over by Ruijsenaars’ hyperbolic gamma function. They may be viewed as (q, q̃ )-bibasic analogues of the beta integral in which the two bases q and q̃ are interrelated by modular inversion, and they entail q-analogues of the beta integral for |q| = 1. The integrals under considerat...

In this paper the fixed point theorem of Schauder is used to prove the existence of a continuous solution of the nonlinear fuzzy Volterra integral equations. Then using some conditions the uniqueness of the solution is investigated.

Integral equations have been of great theoretical importance for analyzing boundary value problems. There is a large amount of literature devoted to the classical potential theory and its applications on solving the boundary value problems of elliptic partial differential equations (see, for example, [5, 20, 23, 25, 26, 31, 32, 36, 37, 40]). For elliptic problems, integral equations have been c...

We report a new type of Hamiltonian equations and maximum principle for the optimal control of a Goursat system when the underlying domain of 2-D "time" is not rectangular and has non-smooth boundary. The resulting Hamiltonian equations form a system of impulsive hyperbolic partial differential equations, with the impulses induced by the conditions on the costate at points of discontinuity on t...