We study a mathematical model proposed in the literature with the aim of describing the interactions between tumor cells and the immune system, when a periodic treatment of immunotherapy is applied. Combining some techniques from non-linear analysis (degree theory, lower and upper solutions, and theory of free-homeomorphisms in the plane), we give a detailed global analysis of the model. We also observe that for certain therapies, the maximum level of aggressiveness of a cancer, for which the treatment works (or does not work), can be computed explicitly. We discuss some strategies for designing therapies. The mathematical analysis is completed with numerical results and conclusions.

We consider the boundary-value problem

where H: [0,+∞) → ℝ and f : [0, 1] × ℝ → ℝ are continuous and λ > 0 is a parameter. We show that if H satisfies a boundedness condition on a specified compact set, then this, together with an assumption that H is either affine or superlinear at +∞, implies existence of at least one positive solution to the problem, even in the case where we impose no growth conditions on f. Finally, since it can hold that f(t, y) < 0 for all (t, y) ∈ [0, 1]×ℝ, the semipositone problem is included as a special case of the existence result.

We study the periodic and Neumann boundary-value problems associated with the second-order nonlinear differential equation

where is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when

and λ > 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.

In this paper we consider the existence of a positive solution to boundary-value problems with non-local nonlinear boundary conditions, the archetypical example being −y″(t) = λf(t,y(t)), t ∈ (0, 1), y(0) = H(φ(y)), y(1) = 0. Here, H is a nonlinear function, λ > 0 is a parameter and φ is a linear functional that is realized as a Lebesgue—Stieltjes integral with signed measure. By requiring φ to decompose in a certain way, we show that this problem has at least one positive solution for each λ ∈ (0, λ0), for a number λ0 > 0 that is explicitly computable. We also give a separate result that holds for all λ > 0.

We give a unified approach for studying the existence of multiple positive solutions of nonlinear differential equations of the form

$$-u''(t)=g(t)f(t,u(t)),\quad \text{for almost every } t \in (0,1),$$

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

$$u(t)=\gamma(t){\alpha}[u]+\delta(t){\beta}[u]+\int_{0}^{1}k(t,s)g(s)f(s,u(s))\,ds$$

where $\alpha[u]$, $\beta[u]$ 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.

In this paper, we construct a generalized degree theory of Browder-Petryshyn or Petryshyn type for a class of semilinear operator equations involving a Fredholm type mapping with infinite dimensional kernel.

the degree defined by p. benevieri and m. furi (see ‘a simple notion of orientability for fredholm maps of index zero between banach manifolds and degree theory’, ann. sci. math. québec 22 (1998) 131–148) is used here to obtain some sharp lower bounds for the number of solutions of the $\lambda$-sections of the compact components of the set of non-trivial solutions of $\mf{f}(\lambda,x)=0$, where $\mf{f}$ is a $\cal{c}^1$ fredholm map of index 1 such that $\mf{f}(\lambda,0)=0$ for all $\lambda\in\mathbb{r}$. these bounds are given in terms of the parity of the linearized fredholm family $d_2\mf{f}(\cdot,0)$. the parity is a local invariant measuring the change of the orientation of $d_2\mf{f}(\lambda,0)$ as $\lambda$ crosses an interval. the set of eigenvalues of $d_2\mf{f}(\cdot,0)$ is not assumed to be discrete. therefore, even in the classical situation when $\mf{f}$ is a compact perturbation of the identity, the results presented here are completely new.

We study the existence of solutions of the semilinear equations (1) in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy are arbitrarily nonnegative constants, . The proofs are based upon degree theoretic arguments.

New existence conditions, under which an index at infinity can be calculated, are given for bifurcations at infinity of asymptotically linear equations in spaces of vector-valued functions. The case where a bounded nonlinearity has discontinuous principal homogeneous part is considered. The results are applied to 2π-periodic problems for two-dimensional systems of ordinary differential equations and to a vector two-point boundary value problem.