Existence to singular boundary value problems with sign changing nonlinearities using an approximation method approach

Haishen Lü; Donal O'Regan; Ravi P. Agarwal

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The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem ${(φ (u^{'}))}^{'} = λ f (t, u, u^{'})$ , $u (0) = u (T) = A$ . Here $λ$ is the positive parameter, $A > 0$ , $f$ is singular at the value $0$ of its first phase variable and may be singular at the value $A$ of its first and at the value $0$ of its second phase variable.

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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,...