@article{freedman_rodriguez_2021, title={EXISTENCE OF SOLUTIONS TO NONLINEAR STURM-LIOUVILLE PROBLEMS WITH LARGE NONLINEARITIES}, volume={13}, ISSN={["1848-9605"]}, DOI={10.7153/dea-2021-13-11}, abstractNote={In this paper, we present results which allow us to establish the existence of solutions to nonlinear Sturm-Liouville problems with unbounded nonlinearities. We consider both regular and singular problems. Our main results rely on a variant of the Lyapunov-Schmidt used in conjunction with topological degree theory. Mathematics subject classification (2010): 34A34, 34B15, 47H11.}, number={2}, journal={DIFFERENTIAL EQUATIONS & APPLICATIONS}, author={Freedman, Benjamin and Rodriguez, Jesus}, year={2021}, month={May}, pages={193–210} }
 @article{freedman_rodriguez_2021, title={On nonlinear discrete Sturm-Liouville boundary value problems with unbounded nonlinearities}, ISSN={["1563-5120"]}, DOI={10.1080/10236198.2021.2017426}, abstractNote={This paper is devoted to the study of nonlinear Sturm–Liouville problems in the discrete setting. We consider problems for which nonlinearities in both the dynamics and boundary conditions may be unbounded. For such problems, we provide a criteria for the existence of solutions.}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Freedman, Benjamin and Rodriguez, Jesus}, year={2021}, month={Dec} }
 @article{freedman_rodriguez_2020, title={ON WEAKLY NONLINEAR BOUNDARY VALUE PROBLEMS ON INFINITE INTERVALS}, volume={12}, ISBN={1848-9605}, DOI={10.7153/dea-202-12-12}, number={2}, journal={DIFFERENTIAL EQUATIONS & APPLICATIONS}, author={Freedman, Benjamin and Rodriguez, Jesus}, year={2020}, month={May}, pages={185–200} }
 @article{freedman_rodriguez_2019, title={EXISTENCE OF SOLUTIONS TO NONLINEAR LEGENDRE BOUNDARY VALUE PROBLEMS}, volume={11}, ISSN={["1848-9605"]}, DOI={10.7153/dea-2019-11-24}, abstractNote={In this paper, we consider nonlinearly perturbed Legendre differential equations subject to the usual boundary conditions. For such problems we establish sufficient conditions for the existence of solutions and in some cases we provide a qualitative description of solutions depending on a parameter. The results presented depend on the size and limiting behavior of the nonlinearities.}, number={4}, journal={DIFFERENTIAL EQUATIONS & APPLICATIONS}, author={Freedman, Benjamin and Rodriguez, Jesus}, year={2019}, month={Nov}, pages={495–508} }
 @article{freedman_rodriguez_2019, title={On nonlinear boundary value problems in the discrete setting}, volume={25}, ISSN={["1563-5120"]}, DOI={10.1080/10236198.2019.1641497}, abstractNote={ABSTRACT Results appearing in this paper can be used to establish the solvability of nonlinear discrete time systems subject to generalized nonlinear boundary conditions. Two separate sets of results are established, each of which can be used to establish existence of solutions to problems for which previous related work proves inconclusive. The first set of results imposes conditions on the sizes of nonlinearities, while the second framework requires geometric properties of the nonlinearity in the dynamics.}, number={7}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Freedman, Benjamin and Rodriguez, Jesus}, year={2019}, month={Jul}, pages={994–1006} }
 @article{maroncelli_rodriguez_2018, title={EXISTENCE THEORY FOR NONLINEAR STURM-LIOUVILLE PROBLEMS WITH NON-LOCAL BOUNDARY CONDITIONS}, volume={10}, ISSN={["1848-9605"]}, DOI={10.7153/dea-2018-10-09}, abstractNote={In this work we provide conditions for the existence of solutions to nonlinear SturmLiouville problems of the form (p(t)x′(t))′ +q(t)x(t)+λx(t) = f (x(t)) subject to non-local boundary conditions ax(0)+bx′(0) = η1(x) and cx(1)+dx′(1) = η2(x). Our approach will be topological, utilizing Schaefer’s fixed point theorem and the LyapunovSchmidt procedure.}, number={2}, journal={DIFFERENTIAL EQUATIONS & APPLICATIONS}, author={Maroncelli, Daniel and Rodriguez, Jesus}, year={2018}, month={May}, pages={147–161} }
 @article{freedman_rodriguez_2018, title={ON THE SOLVABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS SUBJECT TO GENERALIZED BOUNDARY CONDITIONS}, volume={10}, ISSN={["1848-9605"]}, DOI={10.7153/dea-2018-10-22}, abstractNote={In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability of a large number of nonlinear scalar boundary value problems. We begin by giving our general strategy which involves the reformulation of our boundary value problem as an operator equation. We then proceed to establish our results and compare them to closely related previous work.}, number={3}, journal={DIFFERENTIAL EQUATIONS & APPLICATIONS}, author={Freedman, Benjamin and Rodriguez, Jesus}, year={2018}, month={Aug}, pages={317–327} }
 @article{maroncelli_rodriguezb_2016, title={Periodic behaviour of nonlinear, second-order discrete dynamical systems}, volume={22}, ISSN={["1563-5120"]}, DOI={10.1080/10236198.2015.1083016}, abstractNote={In this work, we provide conditions for the existence of periodic solutions to nonlinear, second-order difference equations of the form where , and is continuous and periodic in . Our analysis uses the Lyapunov–Schmidt reduction in combination with fixed point methods and topological degree theory.}, number={2}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Maroncelli, Daniel and Rodriguezb, Jesus}, year={2016}, pages={280–294} }
 @article{maroncelli_rodriguez_2014, title={On the solvability of multipoint boundary value problems for discrete systems at resonance}, volume={20}, ISSN={["1563-5120"]}, DOI={10.1080/10236198.2013.805216}, abstractNote={We study nonlinear discrete-time boundary value problems at resonance. Criteria for the existence of solutions are established via topological degree theory and the Lyapunov–Schmidt procedure.}, number={1}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Maroncelli, Daniel and Rodriguez, Jesus}, year={2014}, month={Jan}, pages={24–35} }
 @article{maroncelli_rodriguez_2014, title={On the solvability of nonlinear impulsive boundary value problems}, volume={44}, DOI={10.12775/tmna.2014.052}, abstractNote={In this paper we provide sufficient conditions for the existence of solutions to two-point boundary value problems for nonlinear ordinary differential equations subject to impulses. Our results depend on properties of the nonlinearities as well as on the solution space of the associated linear problem. Our approach is based on topological degree arguments in conjunction with the Lyapunov-Schmidt procedure.}, number={2}, journal={Topological Methods in Nonlinear Analysis}, author={Maroncelli, D. and Rodriguez, J.}, year={2014}, pages={381–398} }
 @article{rodriguez_abernathy_2012, title={Nonlinear discrete Sturm-Liouville problems with global boundary conditions}, volume={18}, ISSN={["1023-6198"]}, DOI={10.1080/10236198.2010.505237}, abstractNote={This paper is devoted to the study of nonlinear difference equations subject to global nonlinear boundary conditions. We provide sufficient conditions for the existence of solutions based on properties of the nonlinearities and the eigenvalues of an associated linear Sturm–Liouville problem.}, number={3}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Rodriguez, Jesus and Abernathy, Zachary}, year={2012}, pages={431–445} }
 @article{rodriguez_abernathy_2012, title={Nonlocal boundary value problems for discrete systems}, volume={385}, ISSN={["0022-247X"]}, DOI={10.1016/j.jmaa.2011.06.028}, abstractNote={Our goal in this paper is to provide sufficient conditions for the existence of solutions to discrete, nonlinear systems subject to multipoint boundary conditions. The criteria we present depends on the size of the nonlinearity and the set of solutions to the corresponding linear, homogeneous boundary value problems. Our analysis is based on the Lyapunov–Schmidt Procedure and Brouwerʼs Fixed Point Theorem. The results presented extend the previous work of D. Etheridge and J. Rodríguez (1996, 1998) [5], [6] and J. Rodríguez and P. Taylor (2007) [18], [19].}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Rodriguez, Jesus and Abernathy, Kristen Kobylus}, year={2012}, month={Jan}, pages={49–59} }
 @article{rodriguez_abernathy_2012, title={On the solvability of nonlinear Sturm-Liouville problems}, volume={387}, ISSN={["1096-0813"]}, DOI={10.1016/j.jmaa.2011.08.079}, abstractNote={The focus of this paper is the study of nonlinear differential equations subject to general non-local boundary conditions. To establish sufficient conditions for the existence of solutions, we use properties of the nonlinearities and their relationship with the eigenvalues of an associated linear Sturm–Liouville problem.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Rodriguez, Jesus and Abernathy, Zachary}, year={2012}, month={Mar}, pages={310–319} }
 @article{kalhorn_rodriguez_2011, title={Global boundary value problems for nonlinear dynamic equations on time scales}, volume={17}, ISSN={["1563-5120"]}, DOI={10.1080/10236190903150161}, abstractNote={We consider nonlinear boundary value problems for dynamic equations on time scales. We study nonlinear dynamic equations subject to global boundary conditions. Criteria are provided for the solvability of such problems. In the case of weak nonlinearities, we also examine the dependence of the solution on parameters.}, number={4}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Kalhorn, Rebecca I. B. and Rodriguez, Jesus}, year={2011}, pages={541–553} }
 @article{rodriguez_taylor_2008, title={Multipoint boundary value problems for nonlinear ordinary differential equations}, volume={68}, ISSN={["0362-546X"]}, DOI={10.1016/j.na.2007.03.038}, abstractNote={In this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we establish existence results via the Schauder Fixed Point Theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.}, number={11}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Rodriguez, Jesus and Taylor, Padraic}, year={2008}, month={Jun}, pages={3465–3474} }
 @article{rodriguez_taylor_2007, title={Scalar discrete nonlinear multipoint boundary value problems}, volume={330}, ISSN={["0022-247X"]}, DOI={10.1016/j.jmaa.2006.08.008}, abstractNote={In this paper we provide sufficient conditions for the existence of solutions to scalar discrete nonlinear multipoint boundary value problems. By allowing more general boundary conditions and by imposing less restrictions on the nonlinearities, we obtain results that extend previous work in the area of discrete boundary value problems [Debra L. Etheridge, Jesús Rodriguez, Periodic solutions of nonlinear discrete-time systems, Appl. Anal. 62 (1996) 119–137; Debra L. Etheridge, Jesús Rodriguez, Scalar discrete nonlinear two-point boundary value problems, J. Difference Equ. Appl. 4 (1998) 127–144].}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Rodriguez, Jesus and Taylor, Padraic}, year={2007}, month={Jun}, pages={876–890} }
 @article{rodriguez_taylor_2007, title={Weakly nonlinear discrete multipoint boundary value problems}, volume={329}, ISSN={["0022-247X"]}, DOI={10.1016/j.jmaa.2006.06.024}, abstractNote={In this paper we study nonlinear, discrete, multipoint boundary value problems of the formx(t+1)=A(t)x(t)+ϵf(t,x(t)) subject toB0x(0)+B1x(1)+⋯+BNx(N)=0. We provide sufficient conditions for the existence of solutions and we present a qualitative analysis of the way the solutions depend on the parameter ϵ.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Rodriguez, Jesus and Taylor, Padraic}, year={2007}, month={May}, pages={77–91} }
 @article{rodriguez_2005, title={Nonlinear discrete Sturm-Liouville problems}, volume={308}, ISSN={["0022-247X"]}, DOI={10.1016/j.jmaa.2005.01.032}, abstractNote={In this paper we study nonlinear boundary value problems of the form Δ[p(t−1)Δy(t−1)]+q(t)y(t)+λy(t)=f(y(t));t=a+1,…,b+1, subject to a11y(a)+a12Δy(a)=0anda21y(b+1)+a22Δy(b+1)=0. The parameter λ is an eigenvalue of the associated linear problem; that is, there is a nontrivial function u that satisfies the boundary conditions and also Δ[p(t−1)Δu(t−1)]+q(t)u(t)+λu(t)=0 for t in {a+1,a+2,…,b+1}. We establish sufficient conditions for the solvability of such problems. In addition, in those cases where the nonlinearity is “small,” we provide a qualitative analysis of the relation between solutions of the nonlinear problem and eigenfunctions of the linear one.}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Rodriguez, J}, year={2005}, month={Aug}, pages={380–391} }
 @article{rodriguez_etheridge_2005, title={Periodic solutions of nonlinear second-order difference equations}, ISSN={["1687-1839"]}, DOI={10.1155/ade.2005.173}, abstractNote={We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form y(t + 2) + by (t + 1) + cy(t) = f (y(t)), where f: ℝ → ℝ and β > 0 is continuous. In our main result we assume that f exhibits sublinear growth and that there is a constant uf (u) > 0 such that |u| ≥ β whenever c = 1. For such an equation we prove that if N is an odd integer larger than one, then there exists at least one N-periodic solution unless all of the following conditions are simultaneously satisfied: |b| < 2, N across-1(-b/2), and π is an even multiple of c ≠ 0.}, number={2}, journal={ADVANCES IN DIFFERENCE EQUATIONS}, author={Rodriguez, Jesus and Etheridge, Debra Lynn}, year={2005}, pages={173–192} }
 @article{rodriguez_2003, title={Nonlinear discrete systems with global boundary conditions}, volume={286}, DOI={10.1016/S022-247X(03)00536-5}, number={2}, journal={Journal of Mathematical Analysis and Applications}, author={Rodriguez, J.}, year={2003}, pages={782–794} }
 @article{rodriguez_2003, title={On the solvability of nonlinear discrete boundary value problems}, volume={9}, ISSN={["1023-6198"]}, DOI={10.1080/1023619031000062934}, abstractNote={In this paper, we establish sufficient conditions for the existence of solutions to nonlinear, second-order, discrete boundary value problems of the form: subject to and The nonlinearity is assumed to be continuous, but not necessarily differentiable. We provide conditions for the solvability of the boundary value problem when f is of the “perturbation type” as well as when f is of “slow growth”.}, number={9}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Rodriguez, J}, year={2003}, month={Jul}, pages={863–867} }
 @article{rodriguez_sweet_2002, title={Nonlinear boundary value problems on sequence spaces}, volume={8}, DOI={10.1080/10236190290003608}, number={2}, journal={Journal of Difference Equations and Applications}, author={Rodriguez, J. and Sweet, D.}, year={2002}, pages={153–162} }
 @article{etheridge_rodriguez_2002, title={On perturbed discrete boundary value problems}, volume={8}, ISSN={["1023-6198"]}, DOI={10.1080/10236190290017432}, abstractNote={In this paper, we study nonlinear discrete boundary value problems of the form x ( t +1)= A ( t ) x ( t )+ h ( t )+ k f ( t , x ( t ), k ) subject to Bx (0)+ Dx ( J )= u + k g ( x (0), x ( J ), k ) where k is a "small" parameter. Our main concern is the case of resonance, that is, the situation where the associated linear homogeneous boundary value problem x ( t +1)= A ( t ) x ( t ), Bx (0)+ Dx ( J )=0 admits nontrivial solutions. We establish conditions for the solvability of the nonlinear boundary value problem when k is "small". We also establish qualitative properties of these solutions.}, number={5}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Etheridge, DL and Rodriguez, J}, year={2002}, month={May}, pages={447–466} }
 @article{rodriguez_sweet_2001, title={Discrete boundary value problems on infinite intervals}, volume={7}, ISSN={["1023-6198"]}, DOI={10.1080/10236190108808280}, number={3}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Rodriguez, J and Sweet, D}, year={2001}, pages={435–443} }
 @article{etheridge_rodriguez_1998, title={Scalar discrete nonlinear two-point boundary value problems}, volume={4}, ISSN={["1023-6198"]}, DOI={10.1080/10236199808808133}, abstractNote={In this paper we establish conditions for the existence of solutions to nonlinear boundary value problems of the forms subject to where mis a fixed integer in {1,2.....n}. The solvability of this boundary value problem depends on the nonlinearity fas well as on the structure of the solution space to the linear homogeneous problem subject to the same boundary conditions .The most delicate as well as the most interesting case is the one where the linear homogeneous problem has nontrivial solutions.This problem is analyzed through the Altenative Method (Lyapunov-Schmidit Procedure)and we establish conditions, based on the limition behavior of the nonlinear term, which establish the existence of solutions to the nonlinear problem.}, number={2}, journal={JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS}, author={Etheridge, DL and Rodriguez, J}, year={1998}, pages={127–144} }