Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions

In this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions $$\begin{array}{rlll} y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1, 2, \ldots,m,\\ y(0)&=&y(b), \end{array}$$ where $J=[0,b]$ and $F: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, 2, \ldots,m$). The topological structure of solution sets as well as some of their geometric properties (contractibility and $R_\delta$-sets) are studied. A continuous version of Filippov's theorem is also proved

First order impulsive differential inclusions with periodic conditions

In this paper, we present an impulsive version of Filippov's Theorem for the first-order nonresonance impulsive differential inclusion $$\begin{array}{rlll} y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\ y(0)&=&y(b), \end{array}$$ where $J=[0,b]$ and $F: J \times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m.$). Then the relaxed problem is considered and a Filippov-Wasewski result is obtained. We also consider periodic solutions of the first order impulsive differential inclusion $$\begin{array}{rlll} y'(t) &\in& \varphi(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\ y(0)&=&y(b), \end{array}$$ where $\varphi: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a multi-valued map. The study of the above problems use an approach based on the topological degree combined with a Poincar\'e operator

Existence of positive solutions to multi-point third order problems with sign changing nonlinearities

In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results

Positive solutions of a fourth-order differential equation with integral boundary conditions

summary:We study the existence of positive solutions to the fourth-order two-point boundary value problem $$\begin {cases} u^{\prime \prime \prime \prime }(t) + f(t,u(t))=0, & 0 < t < 1,\\ u^{\prime }(0) = u^\prime (1) = u^{\prime \prime }(0) =0, & u(0) = \alpha [u], \end {cases}$$ where $\alpha [u]=\int ^{1}_{0}u(t){\rm d}A(t)$ is a Riemann-Stieltjes integral with $A \geq 0$ being a nondecreasing function of bounded variation and $f \in \mathcal {C}([0,1] \times \mathbb {R}_{+}, \mathbb {R}_{+})$. The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii's fixed point theorem and the Avery-Peterson fixed point theorem

Higher order boundary value problems with φ-Laplacian and functional boundary conditions

We study the existence of solutions of the boundary value problem φ(u^(n−1)(t))′ + f (t, u(t), u′(t), . . . , u^(n−1)(t))= 0, t ∈ (0, 1), g_i (u, u′, . . . , u^(n−1), u^(i)(0))= 0, i = 0, . . . , n − 2, g_n−1 (u, u′, . . . , u^(n−1), u^(n−2)(1))= 0, where n ≥ 2, φ and g_i, i = 0, . . . , n − 1, are continuous, and f is a Carathéodory function. We obtain an existence criterion based on the existence of a pair of coupled lower and upper solutions.Wealso apply our existence theorem to derive some explicit conditions for the existence of a solution of a special case of the above problem. In our problem, both the differential equation and the boundary conditions may have dependence on all lower order derivatives of the unknown function, and many boundary value problems with various boundary conditions, studied extensively in the literature, are special cases of our problem. Consequently, our results improve and cover a number of known results in the literature. Examples are given to illustrate the applicability of our theorems

Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces

The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness

Global existence and stability for second order functional evolution equations with infinite delay

In this article, the authors give sufficient conditions for existence and attractivity of mild solutions for second order semi-linear functional evolution equation in Banach spaces using Schauder's fixed point theorem. An example is provided to illustrate the result

Boundedness and convergence to zero of solutions of a forced second-order nonlinear differential equation

AbstractSufficient conditions for continuability, boundedness, and convergence to zero of solutions of (a(t)x′)′ + h(t, x, x′) + q(t) f(x) g(x′) = e(t, x, x′) are given

Higher order functional boundary value problems: existence and location results

This paper considers a nth-order phi-laplacian differential equation with functional boundary conditions satisfying certain monotonicity assumptions. We present sufficient conditions on the nonlinearity and the boundary conditions to ensure the existence of solutions. Moreover, from the lower and upper solutions method, some information is given about the location of the solution and its qualitative properties. Due to the functional dependence in the boundary conditions, this work generalizes several results for higher order problems with many types of boundary conditions. The main results are illustrated with examples