A unified existence theory for evolution equations and systems under nonlocal conditions

We investigate the effect of nonlocal conditions expressed by linear continuous mappings over the hypotheses which guarantee the existence of global mild solutions for functional-differential equations in a Banach space. A progressive transition from the Volterra integral operator associated to the Cauchy problem, to Fredholm type operators appears when the support of the nonlocal condition increases from zero to the entire interval of the problem. The results are extended to systems of equations in a such way that the system nonlinearities behave independently as much as possible and the support of the nonlocal condition may differ from one variable to another.Comment: 19 page

Compression–expansion fixed point theorem in two norms and applications

AbstractIn this paper we present a two-norms version of Krasnoselskii's fixed point theorem in cones. The abstract result is then applied to prove the existence of positive Lp solutions of Hammerstein integral equations with better integrability properties on the kernels

Implicit first order differential systems with nonlocal conditions

The present paper is devoted to the existence of solutions for implicit first order differential systems with nonlocal conditions expressed by continuous linear functionals. The lack of complete continuity of the associated integral operators, due to the implicit form of the equations, is overcome by using Krasnoselskii's fixed point theorem for the sum of two operators. Moreover, a vectorial version of Krasnoselskii's theorem and the technique based on vector-valued norms and matrices having the spectral radius less than one are likely to allow the system nonlinearities to behave independently as much as possible. In addition, the connection between the support of the nonlocal conditions and the constants from the growth conditions is highlighted

Implicit elliptic equations via Krasnoselskii–Schaefer type theorems

Existence of solutions to the Dirichlet problem for implicit elliptic equations is established by using Krasnoselskii–Schaefer type theorems owed to Burton–Kirk and Gao–Li–Zhang. The nonlinearity of the equations splits into two terms: one term depending on the state, its gradient and the elliptic principal part is Lipschitz continuous, and the other one only depending on the state and its gradient has a superlinear growth and satisfies a sign condition. Correspondingly, the associated operator is a sum of a contraction with a completely continuous mapping. The solutions are found in a ball of a Lebesgue space of a sufficiently large radius established by the method of a priori bounds

Positive radial solutions for Dirichlet problems via a Harnack-type inequality

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving -Laplacian operators in a ball. In particular, -Laplacian and Minkowski-curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack-type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutionsJorge Rodríguez-López was partially supported by the Institute of Advanced Studies in Science and Technology of Babeş-Bolyai University of Cluj-Napoca (Romania) under a Postdoctoral Advanced Fellowship, project CNFIS-FDI-2021-0061 and by Xunta de Galicia (Spain), project ED431C 2019/02 and AIE, Spain, and FEDER, grant PID2020-113275GB-I00S

Positive solutions for φ-Laplacian equations with discontinuous state-dependent forcing terms

This paper concerns the existence, localization and multiplicity of positive solutions for a -Laplacian problem with a perturbed term that may have discontinuities in the state variable. First, the initial discontinuous differential equation is replaced by a differential inclusion with an upper semicontinuous term. Next, the existence and localization of a positive solution of the inclusion is obtained via a compression-expansion fixed point theorem for a composition of two multivalued maps, and finally, a suitable control of discontinuities allows to prove that any solution of the inclusion is a solution in the sense of Carathéodory of the initial discontinuous equation. No monotonicity assumptions on the nonlinearity are requiredS

Positive periodic solutions for Lotka–Volterra systems with a general attack rate

The paper deals with a non-autonomous Lotka-Volterra type system, which in particular may include logistic growth of the prey population and hunting cooperation between predators. We focus on the existence of positive periodic solutions by using an operator approach based on the Krasnosel'skii homotopy expansion theorem. We give sufficient conditions in order that the localized periodic solution does not reduce to a steady state. Particularly, two typical expression for the functional response of predators are discussed.The research of Cristina Lois-Prados has been partially supported by grant MTM2016-75140-P (AEI/FEDER, UE) and grant ED481A-2018/080 from Xunta de Galicia.S