Existence results for superlinear semipositone BVP’s

Abstract

Abstract. We consider the existence of positive solutions to the BVPwhere λ > 0. Our results extend some of the existing literature on superlinear semipositone problems and singular BVPs. Our proofs are quite simple and are based on fixed point theorems in a cone.

Population models with quasi-constant-yield harvest rates

“…The conditions (C 1 ), (C 2 ) and (C 5 ) are the standard conditions used in [12,13,14,28], and (C 4 ) with a constant η was used in some of these references. (C 3 ) allows f to take negative values and is more general than those in [2,12,13,14,21], where the lower bound function η is a constant. (C 3 ) was used in [28], where η is integrable and its main result can not be applied to treat the biological models in section 2.…”

“…Proof of Theorem 2.3. It is sufficient to show that for each case, Corollary 2.1 (1) and (2) where η ρ = 8 − ρ 64 . We define a function D λ : (0, 8 9 ) → R by…”

“…where the nonlinearity f satisfies the semi-positone condition: [2,12,13,14,21] and the references therein). By employing the well-known nonzero fixed point theorems for compact maps defined on cones obtained via the fixed point index [1], we prove a result on the existence of nonzero nonnegative solutions of (1.5) with λ = 1 and then apply the result to obtain a new result on the eigenvalue problem (1.5).…”

Existence of positive solutions for fourth-order semipositone multi-point boundary value problems with a sign-changing nonlinear term

“…For further background information of multi-point boundary value problems we refer the reader to [11,12,16]. However, in previous work, the positivity which imposed on nonlinear term plays an important role for boundary value problems.…”

“…Inspired and motivated greatly by the above mentioned works, the present work may be viewed as a direct attempt to extend the results of [3,13] to a broader class of nonlinear boundary value problems in a general Banach spaces. When the nonlinearity is negative, such kinds of the problems are called semipositone problems, which occur in chemical rector theory, combustion and management of natural resources, see [11,[13][14][15][16]. To our best knowledge, few results were obtained for the problem (1.1).…”