Optimality and existence for Lipschitz equations

Solutions of certain boundary value problems are shown to exist for the nth order differential equation y(n)=f(t,y,y′,…,y(n−1)), where f is continuous on a slab (a,b)×Rn and f satisfies a Lipschitz condition on the slab. Optimal length subintervals of (a,b) are determined, in terms of the Lipschitz coefficients, on which there exist unique solutions

Smoothness of solutions with respect to multi-strip integral boundary conditions for nth order ordinary differential equations

Under certain conditions, solutions of the boundary value problem y(n) = f(x,y,y',...,y(n-1)), a < x < b, y(i-1)(x1) = yi, i=1,...,n-1, y(x2) ∑ i=1mγi ∫ ξiηiy(x)dx=yn, a<x1<ξ1<η1<ξ2<η2<...<ξm<ηm<x2<b, are differentiated with respect to the boundary conditions

Existence of positive solutions for a system of semipositone fractional boundary value problems

We investigate the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to coupled integral boundary conditions

Existence and Asymptotic Stability of Solutions of a Perturbed Quadratic Fractional Integral Equation

Mathematics Subject Classification: 45G10, 45M99, 47H09We study the solvability of a perturbed quadratic integral equation of fractional order with linear modification of the argument. This equation is considered in the Banach space of real functions which are defined, bounded and continuous on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions. Finally, we give an example to illustrate our abstract results

Eigenvalue characterization for a class of boundary value problems

We consider the $n$'th order ordinary differential equation $(-1)^{n-k} y^{(n)}=\lambda a(t) f(y)$, $t\in[0,1]$, $n\geq 3$ together with the boundary condition $y^{(i)}(0)=0$, $0\leq i\leq k-1$ and $y^{(l)}=0$, $j\leq l\leq j+n-k-1$, for $1\leq j\leq k-1$ fixed. Values of $\lambda$ are characterized so that the boundary value problem has a positive solution

Positive solutions for systems of second-order integral boundary value problems

We investigate the existence and nonexistence of positive solutions of a system of second-order nonlinear ordinary differential equations, subject to integral boundary conditions

Positive solutions of second order boundary value problems with changing signs Carathéodory nonlinearities

In this paper we investigate the existence of positive solutions of two-point boundary value problems for nonlinear second order differential equations of the form $(py^{\prime})^{\prime}(t)+q(t)y(t)=f(t,y(t),y^{\prime}(t))$, where $f$ is a Carathéodory function, which may change sign, with respect to its second argument, infinitely many times

Existence theory for nonlinear functional boundary value problems

In this paper the existence of a solution of a general nonlinear functional two point boundary value problem is proved under mixed generalized Lipschitz and Carath\'eodory conditions. An existence theorem for extremal solutions is also proved under certain monotonicity and weaker continuity conditions. Examples are provided to illustrate the theory developed in this paper

Existence results for nondensely defined semilinear functional differential inclusions in Fréchet spaces

In this paper, a recent Frigon nonlinear alternative for contractive multivalued maps in Fréchet spaces, combined with semigroup theory, is used to investigate the existence of integral solutions for first order semilinear functional differential inclusions. An application to a control problem is studied. We assume that the linear part of the differential inclusion is a nondensely defined operator and satisfies the Hille-Yosida condition

Existence of positive solutions for a singular fractional boundary value problem

We study the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a sign-changing nonlinearity, subject to multi-point fractional boundary conditions