Manabu Naito’s research while affiliated with Hiroshima University and other places

... In this regard, see [84] for an extensive complete study of a specific nonlinear equation and [85] for a bibliographical study of unforced equations of the form y ′′ (x) + F (x, y(x), y ′ (x)) = 0. For results which compare the non-oscillatory behavior of forced equations of the form (1) with those of the associated unforced equation, (6) below, and possible equations with delays, we refer the reader to [1], [33] and [73]. One should not forget that even though the literature is filled with sufficient criteria for oscillation/non-oscillation of unforced equations like ...

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Asymptotic solutions of forced nonlinear second order differential equations and their extensions

... The asymptotic integration problem for second-order ordinary differential equations is a classical research topic in mathematics. It has been widely investigated by many authors for the last several decades, see for instance [1][2][3][4][5][6][7][8][9][10] and the references cited therein. The problem is to find sufficient conditions to guarantee the existence of a solution with a prescribed behavior at infinity. ...

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Asymptotic representation of solutions for second-order impulsive differential equations

... Therefore, Theorem 3.1 can be used in combination with the theory of Noussair and Swanson to establish existence of oscillatory solutions vanishing at infinity in exterior domains in R n for the forced elliptic equations of the form ∆u + Φ(x, u) + Ψ(x) = 0 (cf. Kawano et al. [17], Kusano and Naito [19], and Noussair [23]). ...

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Oscillation of second‐order perturbed differential equations

... Perhaps one of the oldest studies in the literature, [25], that presented some oscillation criteria for equation ...

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Fourth Order Functional Differential Equations of Neutral Type: Enhanced Oscillation Theorems

... Existence of global solutions of the second-order nonlinear ordinary differential equation u + f (t, u) = 0, t ≥ t 0 ≥ 1 (1.1) that exhibit linear-like asymptotic behavior at infinity, that is, have asymptotic representations u(t) = At + o(t) or u(t) = At + B + o(1) as t → +∞, where A = 0 and B are real constants, has been discussed in a series of recent papers [1] - [11] . This problem is important not only for the oscillation theory of ordinary differential equations, but is also closely related with existence of solutions with certain asymptotic behavior for semi-linear partial differential equations of elliptic type, see Kusano and Trench [3], Yin [10], Constantin [12], and the references therein. ...

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On asymptotic integration of a nonlinear second-order differential equation

... Using some classical results (see [14,36] The focus and the original part of our study consists in analyzing the asymptotic behaviour of the solutions with large initial data d, which is determined by the parameter ℓ in (H ℓ ). In particular, we have the following bifurcation phenomenon, when (H ↑ ) is assumed ...

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A bifurcation phenomenon for the critical Laplace and $p$-Laplace equation in the ball

... The biharmonic equations originate from the study of traveling waves in suspension bridges [4] and static deflection of a bending beam [16], which have aroused the interest of many mathematicians. We refer the reader to [3,11,14,15,18,20,[24][25][26][27][28][29][31][32][33][34] and the references cited therein. In particular, when p > 1, Dalmasso [7] used maximum principle to analyze the existence and uniqueness of positive solutions for the following biharmonic problem: (1.2) where B R is the ball of radius R centered at the origin in R n (n ≥ 1). ...

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Triple positive radial solutions arising from biharmonic elliptic systems

... In regard to obtaining results about the asymptotic nature of the nonoscillatory solutions, the coefficient a(f) has been assumed to be of one sign by majority of authors. When α(ί)>0 for sufficiently large ί, then a decent account of the nonoscillatory solutions can be found in Kusano and Onose [3][4], Kitamura Kusano and Naito [2], Singh [6] and Kusano and Singh [7]. However when a(t) is oscillatory, nothing seems to be known about the asymptotic nature of the nonoscillatory solutions of equation ...

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Asymptotic nonoscillation under large amplitudes of oscillating coefficients in second order functional equations

... Since the beginning of last century, numerous contributions flourished within the topics of the existence and uniqueness of solution to problems related to (1.1). Here, we just mention, among many possible choices, the papers [3][4][5][6][7][8][9][10]. Moreover, we refer to [11][12][13] for the existence and uniqueness results of problems related to (1.1) set on R N and [14,15] set on an annuli. ...

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Uniqueness of nodal radial solutions to nonlinear elliptic equations in the unit ball

... A study of the oscillatory solutions to the Emden-Fowler equation (1.1) has a long history. See, for example, [1,2,6,7,8,10,17]. For instance, the following result is well-known. ...

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Box-counting dimension of oscillatory solutions to the Emden-Fowler equation