Takaŝi Kusano’s research while affiliated with Fukuoka University and other places

... It is also important to observe that equation of the form (1) arises naturally in the study of radially symmetric solutions (ground states) of semi-linear equations, and many works have been conducted in this area; see [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. ...

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Existence and Uniqueness of Positive Solutions for Semipositone Lane-Emden Equations on the Half-Axis

... = D° j^) > 0 for r ^ r^-. Thus we again obtain a contradiction to the oscillation of all nontrivial solutions of (11), because the function w(r) = z(r) + a x p 0 (r) is an eventually positive solution of (11). ...

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Oscillation criteria for forced nonlinear elliptic equations of arbitrary order

... 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

... Due first to the fact that q may change signs and secondly to the presence of the function ф our results difibr from those previously obtained for equations (El) and (E2) respectively. Moreover, our results will cover all solutions not just the bounded ones (for example, see some of the results in [3] and [11]). In addition, we do not require J°° q(s) ds = 00 as many authors do (see some of the results in [3], [11], or [17]), and in that respect even when i^(x) = 1, our results differ from some of those previously known for equation (E2). ...

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A correction on the oscillatory behavior of solutions of certain second order nonlinear differential equations

... 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

... where $r_{0}>0$ is a sufficiently large number. When $f(u)=|u|'-1u$, $y$ $>0,$ oscillation criteria for (1), which can be regarded as generalizations of earilier results in $[1, 5]$ , have been obtained in [4]. The case of $m$ $=2$ has been treated in [3]. ...

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Oscillation theorems of quasilinear elliptic equations with arbitrary nonlinearities

... , [15]) . The existence of positive solutions of the equation on exterior domains (including R n ) has been widely considered (for example, see [3], [4], [7], [10], [11], [14], and references therein). The main approach used to prove existence results is to construct super and sub solutions. ...

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Dirichlet problems for semilinear elliptic equations with a fast growth coefficient on unbounded domains

... Oscillation criteria obtained by Kusano andOnose 1973 andby Belohorec 1969 are extended to second-order sublinear impulsive differential equations of Emden-Fowler type: x t p t |x τ t | α−1 x τ t 0, t / θ k ; Δx t | t θk q k |x τ θ k | α−1 x τ θ k 0; Δx t | t θk 0, 0 < α < 1 by considering the cases τ t ≤ t and τ t t, respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions. ...

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Oscillation of Second-Order Sublinear Impulsive Differential Equations

... The study of oscillation of higher order nonlinear functional differential equations with deviating arguments was attempted for the first time by Onose [28,29]. A typical generalization of Onose's oscillation theorem can be found in [11]. Recently, wide attention of the researchers has been attracted to the investigation of oscillation (or nonoscillation) of differential equations whose principal differential operators involve nonlinear Sturm-Liouville type differential operators [12, 20, 22, 30 -34]. ...

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Oscillation and nonoscillation criteria for even order nonlinear functional differential equations

... were obtained in [6,11,13,14] and [25]. Here, λ, τ > 0 and ω ± (t + τ ) = ∓ω ± (t). ...

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Existence and asymptotic behavior of solutions of nonlinear neutral differential equations