Generalized quasilinearization method for nonlinear boundary value problems with integral boundary conditions.

Sun, Li; Zhou, Mingru; Wang, Guangwa

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In the first part, we investigate the singular BVP ${\frac{d}{d t}}^{c} D^{α} u + {(a / t)}^{c} D^{α} u = ℋ u$ , u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where $ℋ$ is a continuous operator, α ∈ (0, 1) and a < 0. Here, c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems ${\frac{d}{d t}}^{c} D^{α_{n}} u + {(a / t)}^{c} D^{α_{n}} u = f (t, u,^{c} D^{β_{n}} u)$ , u(0) = A, u(1) = B, ${^{c} D^{α_{n}} u (t)|}_{t = 0} = 0$ where a < 0, 0 < β n ≤ α n < 1, limn→∞ β n = 1, and solutions of u″+(a/t)u′ = f(t,...

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This paper studies the existence of solutions to the singular boundary value problem $\}\begin{matrix} - u^{''} = g (t, u) + h (t, u), t \in (0, 1), u (0) = 0 = u (1), \end{matrix}$ where $g (0, 1) \times (0, \infty) \to ℝ$ and $h (0, 1) \times [0, \infty) \to [0, \infty)$ are continuous. So our nonlinearity may be singular at $t = 0, 1$ and $u = 0$ and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and lower solutions.

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