Positive solutions for mixed problems of singular fractional differential equations

We investigate the existence of positive solutions to the singular fractional boundary value problem: $^c\hspace{-1.0pt}D^{\alpha }u +f(t,u,u^{\prime },^c\hspace{-2.0pt}D^{\mu }u)=0$, u′(0) = 0, u(1) = 0, where 1 < α < 2, 0 < μ < 1, f is a L^q‐Carathéodory function, $q > \frac{1}{\alpha -1}$, and f(t, x, y, z) may be singular at the value 0 of its space variables x, y, z. Here $^c \hspace{-1.0pt}D$ stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones.     

An upper and lower solution approach for singular boundary value problems with sign changing non‐linearities

We obtain via Schauder's fixed point theorem new results for singular second‐order boundary value problems where our non‐linear term f(t,y,z) is allowed to change sign. In particular, our problem may be singular at y=0, t=0 and/or t=1. Copyright © 2002 John Wiley & Sons, Ltd.     

Positive and dead core solutions of singular BVPs for third-order differential equations

The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u^″))^′=λf(t,u,u^′,u^″) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y.     

Existence of Three Positive Pseudo-symmetric Solutions for a One Dimensional Discrete p-Laplacian

We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the discrete three point boundary value problem, ?(g(?u(t-1)))+a(t))f(u(t))=0, for t∈{a+1,…,b+1} and u(a)=0 with u(v)=u(b+2) where g(v)=|v| ^p-2 v, p>1, for some fixed v∈{a+1,…,b+1} and σ=(b+2+v)/2 is an integer.     

Nodal solutions of multi-point boundary value problems

We study the nonlinear boundary value problem consisting of the equation y^″+w(t)f(y)=0 on [a,b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes in the existence question for different types of nodal solutions as the problem changes.     

A Fixed Point Theorem of Krasnoselskii—Schaefer Type

In this paper we focus on three fixed point theorems and an integral equation. Schaefer's fixed point theorem will yield a T-periodic solution of (0.1) x(t)= a(t) + ^t_t-h D(t,s)g(s,x(s))ds if D and g satisfy certain sign conditions independent of their magnitude. A combination of the contraction mapping theorem and Schauder's theorem (known as Krasnoselskii's theorem) will yield a T-periodic solution of (0.2) x(t) = f(t,x(t)) + ^t_t-h D(t,s)g(s,x(s))ds if f defines a contraction and if D and g are small enough. We prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer's theorem which yields a T-periodic solution of (0.2) when / defines a contraction mapping, while D and g satisfy the aforementioned sign conditions.     

Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Dαu(t)+f(t,u(t),Dμu(t))=0, u(0)=u(1)=0, where 1<α<2, 0<μ?α−1, Dα is the standard Riemann-Liouville fractional derivative, f is a positive Carathéodory function and f(t,x,y) is singular at x=0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.     

Singular Nonlinear (n − 1,1) Conjugate Boundary Value Problems

Solutions are obtained for the boundary value problem, y ⁽ⁿ⁾ + f(x,y) = 0, y ⁽ⁱ⁾(0) = y(1) = 0, 0 i n – 2, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

Existence of three positive solutions for boundary value problems with p-Laplacian

By using fixed point theorem, we study the following equation g(u^′′(t))+a(t)f(u)=0 subject to boundary conditions, where g(v)=|v|^p−2v with p>1; the existence of at least three positive solutions is proved.     

Weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions

In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x₀+g(x) in Banach space X under some suitable hypotheses.     

Smooth center manifolds for nonuniformly partially hyperbolic trajectories

We establish the existence of unique smooth center manifolds for ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that v^′=A(t)v admits a nonuniform exponential trichotomy. This allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories. In addition, we prove that the center manifolds are as regular as the vector field. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space. To the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting.     

Positive solutions for systems of nonlinear eigenvalue problems for functional differential equations

Values of?λ?are determined for which there exist positive solutions of the system of functional differential equations, u″?+?λa(t)f(v _t )?=?0,v″?+?λb(t)g(u _t )?=?0, for 0?t?u(s)?=?v(s)?=?φ(s), ?r?≤?s?≤?0, and the boundary conditions u(0)?=?v(0)?=?φ(0)?=?u(1)?=?v(1)?=?0. A Guo–Krasnosel'skii fixed point theorem is applied.     

The method of lower and upper solutions for third order singular four point boundary value problems

We mainly study the existence of positive solutions for the following third order singular four point boundary value problem $$\begin{cases}x^{(3)}(t)+f(t,x,x',-x'')=0,\quad 0<t<1,\\x(0)-\alpha x(\xi)=0,\quad x'(1)-\beta x'(\eta)=0,\quad x''(0)=0.\end{cases}$$ where 0≤α<1, 0≤β<1, 0<ξ<1,0<η<1. And we obtain some necessary and sufficient conditions for the existence of C ²[0,1] positive solutions by means of the lower and upper solution method. Our nonlinearity f(t,x,y,z) may be singular at x,y,z,t=0 and/or t=1.     

Nonuniform nonresonance at the first eigenvalue for singular boundary value problems with sign changing nonlinearities

New nonresonant results are presented for the boundary value problem y″+f(t,y,y′)=0, 0<t<1 with Dirichlet boundary data. Our nonlinearity may be singular in its dependent variable and is allowed to change sign.     

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

By using a topological approach and the relation between rotation numbers and weighted eigenvalues, we give some multiplicity results for the boundary value problem u′′ + f(t, u) = 0, u(0) = u(T) = 0, under suitable assumptions on f(t, x)/x at zero and infinity. Solutions are characterized by their nodal properties. Supported by MIUR, GNAMPA and FCT.     

Boundary Value Problems for Singular Second-Order Functional Differential Equations

Abstract Positive solutions to the boundary value problem, y'=-f(x,y(w(x)) 0相似文献   

Scattering techniques for a one dimensional inverse problem in geophysics

A one dimensional problem for SH waves in an elastic medium is treated which can be written as v_tt = A^?1 (Av_y)_y, A = (?μ)^1/2, ? = density, and μ = shear modulus. Assume A ? C¹ and A′/A ? L¹; from an input v_y(t, 0) = ?(t) let the response v(t, 0) = g(t) be measured (v(t, y) = 0 for t < 0). Inverse scattering techniques are generalized to recover A(y) for y > 0 in terms of the solution K of a Gelfand-Levitan type equation, .     

Positive solutions to superlinear singular boundary value problems

New existence results are presented for the second-order equation y″ + f(t,y) = 0, 0<t<1 with Dirichlet or mixed boundary data. In our theory the nonlinearity f is allowed to change sign. Singularities at y = 0, t = 0 and t = 1 are discussed.