In this paper, we investigate the existence and uniqueness of solutions for a differential equation of fractional-order q ∈ (1, 2] subject to nonlocal boundary conditions involving Caputo derivative of the form0 < 1 < σ < β 1 < β 2 < 2 < 1, 0 < μ < 1, and δ, a, b, c are real constants. We make use of some standard tools of fixed point theory to obtain the desired results which are well illustrated with the aid of examples.

Abstract. In this paper, we study some new nonlinear boundary value problems of LiouvilleCaputo type fractional differential equations supplemented with nonlocal multi-point conditions involving lower order fractional derivative. We make use of some well known tools of the fixed point theory to establish the existence of solutions for problems at hand. For illustration of the obtained results, several examples are discussed.Mathematics subject classification (2010): 34A08, 34B15.

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In this study, an unknown input observer is proposed for a class of nonlinear GPFOSs. For this class of systems, both full-order and reduced-order observers have been established. The investigated system satisfies the one-sided Lipschitz nonlinear condition, which is an improvement of the classic Lipschitz condition. Sufficient conditions have been proposed to ensure the error dynamics’ Mittag–Leffler stability. The value of this work lies in the fact that, to the best of the authors’ knowledge, this is the first research work that investigates the issue of Observer Design (OD) for GPFOSs. To exemplify the usefulness of the suggested observers, an illustrative numerical example is suggested.