We give a unified approach for studying the existence of multiple positive solutions of nonlinear differential equations of the form −u (t) = g(t)f (t, u(t)), for almost every t ∈ (0, 1),where g, f are non-negative functions, subject to various nonlocal boundary conditions. We study these problems via new results for a perturbed integral equation, in the space C[0, 1], of the form β[u] are linear functionals given by Stieltjes integrals but are not assumed to be positive for all positive u. This means we actually cover many more differential equations than the simple equation written above. Previous results have studied positive functionals only, but even for positive functionals our methods give improvements on previous work. The well-known m-point boundary value problems are special cases and we obtain sharp conditions on the coefficients, which allows some of them to have opposite signs. We also use some optimal assumptions on the nonlinear term.
Abstract.We establish the existence of multiple positive solutions of nonlinear equations of the formwhere g, f are non-negative functions, subject to various nonlocal boundary conditions. The common feature is that each can be written as an integral equation, in the space C[0, 1], of the formwhere α[u] is a linear functional given by a Stieltjes integral but is not assumed to be positive for all positive u. Our new results cover many nonlocal boundary conditions previously studied on a case by case basis for particular positive functionals only, for example, many m-point BVPs are special cases. Even for positive functionals our methods give improvements on previous work. Also we allow weaker assumptions on the nonlinear term than were previously imposed.2000 Mathematics Subject Classification: Primary 34B18, secondary 34B10, 47H10, 47H30.
Motivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the formwhere G is a compact set in R n . Here α[u] is a positive functional and f is positive, while k and γ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.
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