Best approximation and fixed-point theorems for discontinuous increasing maps in Banach lattices

In this paper, we extend and prove Ky Fan’s Theorem for discontinuous increasing maps f in a Banach lattice X when f has no compact conditions. The main tools of analysis are the variational characterization of the generalized projection operator and order-theoretic fixed-point theory. Moreover, we establish a sequence {xn} which converges strongly to the unique best approximation point. As an application of our best approximation theorems, a fixed-point theorem for non-self maps is established and proved under some conditions. Our results generalize and improve many recent results obtained by many authors

Positive solutions for Sturm-Liouville boundary value problems in a Banach space

We consider the existence of single and multiple positive solutions for a second-order Sturm-Liouville boundary value problem in a Banach space. The sufficient condition for the existence of positive solution is obtained by the fixed point theorem of strict set contraction operators in the frame of the ODE technique. Our results significantly extend and improve many known results including singular and nonsingular cases

Positive solutions for p-Laplacian fourth-order differential system with integral boundary conditions

This paper investigates the existence of positive solutions for a class of singular p-Laplacian fourth order differential equations with integral boundary conditions. By using the fixed point theory in cones, explicit range for λ and μ is derived such that for any λ and μ lie in their respective interval, the existence of at least one positive solution to the boundary value system is guaranteed

Positive solutions for nonlinear fractional differential equations with boundary conditions involving Riemann-Stieltjes integrals

We consider the existence of positive solutions for a class of nonlinear integral boundary value problems for fractional differential equations. By using some fixed point theorems, the existenceand multiplicity results of positive solutions are obtained. The results obtained in this paperimprove and generalize some well-known results

Multiple positive solutions of singular fractional differential system involving Stieltjes integral conditions

In this paper, the existence and multiplicity of positive solutions to singular fractional differential system is investigated. Sufficient conditions which guarantee the existence of positive solutions are obtained, by using a well known fixed point theorem. An example is added to illustrate the results

Multiple positive solutions of singular nonlinear Sturm-Liouville problems with caratheodory perturbed term

By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation Lu=p(t)f(t,u(t),u′′(t))-g(t,u(t),u′′(t)),0<t<1,α1u(0)-β1u'(0)=0,γ1u(1)+δ1u'(1)=0,α2u′′(0)-β2u′′′(0)=0,γ2u′′(1)+δ2u′′′(1)=0, with αi,βi,γi,δi≥0 and βiγi+αiγi+αiδi>0,    i=1,2, where L denotes the linear operator Lu:=(ru′′′)'-qu′′,r∈C1([0,1],(0,+∞)), and q∈C([0,1],[0,+∞)). This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed term g:(0,1)×[0,+∞)×(-∞,+∞)→(-∞,+∞) only satisfies the global Carathéodory conditions, which implies that the perturbed effect of g on f is quite large so that the nonlinearity can tend to negative infinity at some singular points

Positive Solutions for (n - 1,1) -Type Singular Fractional Differential System with Coupled Integral Boundary Conditions

We study the positive solutions of the (n - 1,1)-type fractional differential system with coupled integral boundary conditions. The conditions for the existence of positive solutions to the system are established. In addition, we derive explicit formulae for the estimation of the positive solutions and obtain the unique positive solution when certain additional conditions hold. An example is then given to demonstrate the validity of our main results

Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives

By establishing a maximal principle and constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of a class of fractional differential equations is discussed. Some sufficient conditions for the existence of positive solutions are established