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Abstract: We consider an initial value problem for a fractional differential equation of Caputo type. The convergence of the Picard successive approximations is established by first showing that the Caputo derivatives of these approximations converge. The method utilized, originally introduced in [O. Kooi, The method of successive approximations and a uniqueness theorem of Krasnoselskii–Krein in the theory of differential equations, Nederi. Akad. Wetensch, Proc. Ser. A61; Indag. Math. 20 (1958) 322–327], is interesting in itself. [Copyright &y& Elsevier]

Abstract: The hybrid real number system consisting of terminating and nonterminating decimals, dark numbers, dual dark numbers, involving the notions of personal infinities and the impersonal infinity has been discussed. Some algebraic properties of dark numbers and dual dark numbers are discussed. [Copyright &y& Elsevier]

Abstract: We develop Heatlets, the fundamental solutions of heat equation using wavelets, for numerically solving inhomogeneous and homogeneous initial value problems of diffusion equation on unbounded domains. Unlike finite difference and finite element methods, diffusion into an infinite medium is satisfied analytically, avoiding the need for artificial boundary conditions on a finite computational domain. The approach is applied to a number of examples and the numerical results confirm the theoretical findings. [Copyright &y& Elsevier]

Abstract: We investigate, in this paper, the existence theory for functional differential equations with both retardation and anticipation as a first step by splitting the problem into two parts and using the existing theory of functional differential equations with delay and the contraction mapping theorem for operators with PPF (past, present, and future) dependence. [Copyright &y& Elsevier]

Abstract: The method of the monotone iterative technique together with coupled lower and upper solutions is employed to prove existence and uniqueness results for functional differential equations involving retardation and anticipation. [Copyright &y& Elsevier]

Abstract: We prove a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order, using a weak contractivity type of assumption. Besides including several recent developments, our theorem can be used to investigate a large class of problems. As an application, we discuss the existence and uniqueness of solution for a periodic boundary value problem. [Copyright &y& Elsevier]

Abstract: Nonlinear variation of parameters formula for set differential equations in a metric space is developed investigating necessary tools such as differentiability of solutions relative to initial values. Other needed results are also proven. [Copyright &y& Elsevier]

Abstract: We prove a comparison result in terms of vector Lyapunov-like functions relative to a set differential system. Using this, we provide sufficient conditions in terms of vector Lyapunov-like functions for the stability properties of the trivial solutions of set differential systems. [Copyright &y& Elsevier]

Abstract: Notions of stability for the solutions of set differential equations, using Lyapunov-like functions are considered. Criteria for the equistability, equiasymptotic stability, uniform and uniform asymptotic stability are presented. [Copyright &y& Elsevier]

Abstract: The set of compact convex subsets of a Fréchet space is studied in detail and is realized as a projective limit of metric spaces. The notion of Hausdorff metric on it is replaced by a family of corresponding “semi-metrics” which provide the necessary background for the support of continuity and Lipschitz continuity. Finally the notion of Hukuhara derivative suited to our study is developed. The proposed approach forms the appropriate environment within which the study of set differential equations for Fréchet spaces can be developed. A first example on is included. [Copyright &y& Elsevier]

We revisit the original formulation of fuzzy differential equations (FDEs) and provide examples to show that the theory of FDEs is richer than the corresponding theory of ordinary differential equations (ODEs) and therefore need not be rejected. The alternative formulation in terms of differential inclusions does not always have advantage over ODEs, as claimed. We also discuss a stability result via Lyapunov functions in this setup. [Copyright &y& Elsevier]

We prove a comparison theorem for the lower and upper solutions of a nonlinear two point boundary value problem on time scales. This theorem plays an important role in the development of the method of generalized quasilinearization on time scales. [Copyright &y& Elsevier]

Abstract: We shall develop, in this paper, the hybrid set theory extending some notions of the hybrid real number system considered in [V. Lakshmikantham, D. Kovach, Hybrid real number system, Nonlinear Analysis: Hybrid Systems (2006), doi:10.1016/nahs.2006.07.001], so as to shed some light on the existing set theory and strengthen it. We shall also list some ideas of set theory that existed long before Cantor’s discovery [G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (P.E.B., Trans.), The Open Court Publishing Co., la Salle, Illinois, 1952]. [Copyright &y& Elsevier]

Steam gasification of phenolic boards (PB) in the presence of ternary eutectic carbonates (LNK–carbonate mixture: Li 2 CO 3 , Na 2 CO 3 and K 2 CO 3 ) was carried out at 823–948 K under atmospheric pressure to produce clean hydrogen. Rates of hydrogen formation were accelerated in the presence of nickel metal powder in the low temperature range (823 K). The experimental results imply that unsupported nickel metal powder or small nickel pieces in molten carbonate could be used as catalyst to produce hydrogen on steam gasification of e-waste. [ABSTRACT FROM AUTHOR]