We use the cosine family of linear operators to prove the existence, uniqueness, and stability of the integral solution of a nonlocal telegraph equation in frame of the conformable time-fractional derivative. Moreover, we give its implicit fundamental solution in terms of the classical trigonometric functions.
is paper deals with the existence of mild solutions for the following Cauchy problem:where d α (.)/dt α is the so-called conformable fractional derivative. e linear part A is the infinitesimal generator of a uniformly continuous semigroup (T(t)) t≥0 on a Banach space X, f and g are given functions. e main result is proved by using the Darbo-Sadovskii fixed point theorem without assuming the compactness of the family (T(t)) t>0 and the Lipshitz condition on the nonlocal part g.
In this paper, we investigate second order evolution differential equation in the frame of sequential conformable derivatives with nonlocal condition. First, we establish Duhamel's formula in terms of a standard cosine family of linear operators. Then, we prove some results concerning the existence, uniqueness, stability, and regularity of mild solution concept. Moreover, we present a concrete application of the main results.
MSC: 34A08; 47D09
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