We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.
In (Bor in Int. J. Math. Math. Sci. 17:479-482, 1994), Bor has proved the main theorem dealing with |N, p n | k summability factors of an infinite series. In the present paper, we have generalized this theorem on the ϕ -|A, p n | k summability factors under weaker conditions by using an almost increasing sequence instead of a positive non-decreasing sequence. MSC: 40D15; 40F05; 40G99
Bor has proved a main theorem dealing with |N , pn| k summability factors of infinite series. In this paper, we have generalized this theorem to the ϕ − |A, pn| k summability factors, under weaker conditions by using an almost increasing sequence instead of a positive monotonic non-decreasing sequence.Mathematics Subject Classification 2010: 40D25, 40F05, 40G99.
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