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The notion of a generalized Riemann–Liouville fractional integral is introduced, and its domain, range, and properties are studied. The new notion and properties provide new insight and understanding into the classical Riemann–Liouville fractional integral and its properties. Based on the generalized Riemann–Liouville fractional integral, equivalences between linear first ‐order fractional differential equations (FDEs) and integral equations are established. These equivalence results are applied to obtain solutions of Abel type integral equations and to study the existence and uniqueness of generalized normal solutions of initial value problems for nonlinear first‐order FDEs. [ABSTRACT FROM AUTHOR]

New properties of the first‐order Riemann–Liouville fractional integrals are proved and first‐order fractional derivatives for the equivalence class of functions in L1(a,b)$$ {L}^1\left(a,b\right) $$ are analyzed. Solutions, normal solutions, and generalized normal solutions of the initial value problems (IVPs) for nonlinear first‐order fractional differential equations (FDEs) with fraction α∈(0,1)$$ \alpha \in \left(0,1\right) $$ are introduced. The equivalences in L1(a,b)$$ {L}^1\left(a,b\right) $$ between the first‐order FDEs and the corresponding (Volterra) integral equations are established based on normal solutions and generalized normal solutions. New results on the existence and uniqueness of generalized normal solutions or nonnegative generalized normal solutions in Lp(a,b)$$ {L}^p\left(a,b\right) $$ of the IVPs are obtained for p∈[1,1/α)$$ p\in \left[1,1/\alpha \right) $$. These results are applied to study the existence and uniqueness of generalized normal nonnegative solutions in L+p(a,b)$$ {L}_{+}^p\left(a,b\right) $$ of the IVP for the nonlinear first‐order FDE with nonlinearities related to the population models with growth rates of Ricker type. [ABSTRACT FROM AUTHOR]

The existence of nonzero nonnegative solutions of the population models of Ricker types governed by systems of second‐order elliptic boundary value problems is studied. New theory on limit inferiors and superiors of functions at points and ∞$$ \infty $$ in product Banach spaces is established. New results on the limit inferiors, limit superiors and limits for monotone and mixed monotone functions at points in ordered Banach spaces and at ∞$$ \infty $$ in the Euclidean spaces are obtained. These results on limit inferiors and superiors are used to obtain the existence of nonzero nonnegative solutions of the population models of Ricker types. [ABSTRACT FROM AUTHOR]

New coexistence fixed point theorems in product Banach spaces are established via the classical theory of fixed point index for compact maps defined on cones. Each component of a coexistence fixed point is nonzero. These coexistence fixed point theorems are applied to obtain results on coexistence solutions of systems of Hammerstein integral equations, second‐order ordinary differential equations with general separated boundary conditions, and one‐dimensional competition model of Ricker types. [ABSTRACT FROM AUTHOR]

The existence and nonexistence of semi-trivial or coexistence steady-state solutions of one-dimensional competition models in an unstirred chemostat are studied by establishing new results on systems of Hammerstein integral equations via the classical fixed point index theory. We provide three ranges for the two parameters involved in the competition models under which the models have no semi-trivial and coexistence steady-state solutions or have semi-trivial steady-state solutions but no coexistence steady-state solutions or have semi-trivial or coexistence steady-state solutions. It remains open to find the largest range for the two parameters under which the models have only coexistence steady-state solutions. We apply the new results on systems of Hammerstein integral equations to obtain results on steady-state solutions of systems of reaction-diffusion equations with general separated boundary conditions. Such type of results have not been studied in the literature. However, these results are very useful for studying the competition models in an unstirred chemostat. Our results on Hammerstein integral equations and differential equations generalize and improve some previous results. [ABSTRACT FROM AUTHOR]

Lyapunov type inequalities for (linear or nonlinear) Hammerstein integral equations are established and applied to second order differential equations (ODEs) with general separated boundary conditions. These new inequalities provide necessary conditions for the Hammerstein integral equations and these boundary value problems to have nonzero nonnegative solutions. As applications of these inequalities for nonlinear ODEs, we obtain extinction criteria and optimal locations of favorable habitats for populations inhabiting one dimensional heterogeneous environments governed by reaction-diffusion equations with spatially varying growth rates and external forcing. [ABSTRACT FROM AUTHOR]

One-dimensional logistic population models with quasi-constant-yield harvest rates are studied under the assumptions that a population inhabits a patch of dimensionless width and no members of the population can survive outside of the patch. The essential problem is to determine the size of the patch and the ranges of the harvesting rate functions under which the population survives or becomes extinct. This is the first paper which discusses such models with the Dirichlet boundary conditions and can tell the exact quantity of harvest rates of the species without having the population die out. The methodology is to establish new results on the existence of positive solutions of semi-positone Hammerstein integral equations using the fixed point index theory for compact maps defined on cones, and apply the new results to tackle the essential problem. It is expected that the established analytical results have broad applications in management of sustainable ecological systems.