We consider one scenario of transformation of a cycle of a differential equation, which we call the blow up. In this scenario, a cycle born via the Hopf bifurcation grows to infinity as a parameter ranges over a finite segment. From another perspective, in the product of the phase space and the parameter axis there is a branch of cycles connecting the equilibrium and infinity. We first discuss ...

We study a boundary value problem for a fractional differential inclusion of order α ∈ (1, 2] with non-separated boundary conditions involving a nonconvex set-valued map. We establish a Filippov type existence theorem and we prove the arcwise connectedness of the solution set of the problem considered.

ABSTRACT. We present local existence theorem of the initial value problem for third order semilinear dispersive partial differential equations in two space dimensions. This type of equations arises in the study of gravity wave of deep water, and cannot be solved by the classical energy method. To solve the initial value problem, we make full use of pseudodifferential operators with nonsmooth co...

This paper deals with the existence of positive solutions for a boundary value problem involving a nonlinear functional differential equation of fractional order α given by Dαu(t) + f(t, ut) = 0, t ∈ (0, 1), 2 < α ≤ 3, u′(0) = 0, u′(1) = bu′(η), u0 = φ. Our results are based on the nonlinear alternative of Leray-Schauder type and Krasnosel’skii fixed point theorem.

In the present paper, we investigate the existence, uniqueness and continuous dependence on initial data of mild solutions of first order nonlocal semilinear functional impulsive integro-differential equations of more general type with finite delay in Banach spaces. Our analysis is based on semigroup theory and Banach contraction theorem.

By transforming the boundary value problem into the corresponding fixed-point problem of a completely continuous operator, the existence is obtained in the paper for two-point boundary value problem of fourth order superlinear singular semipositone differential system via the fixed point theorem concerning cone compression and expansion in norm type.

We prove a Filippov type existence theorem for solutions of a higher order differential inclusion in Banach spaces with nonconvex valued right hand side by applying the contraction principle in the space of the derivatives of solutions instead of the space of solutions.

In this paper, we extend a type of Strassen’s theorem for the existence of probability measures with given marginals to positive vector measures with values in the dual of a barreled locally convex space which has certain order conditions. In this process of the extension we also give some useful properties for vector measures with values in dual spaces.

In the present paper, we investigate the existence, uniqueness and continuous dependence on initial data of mild solutions of second order nonlocal semilinear functional integrodifferential equations of more general type with delay in Banach spaces. Our analysis is based on the theory of strongly continuous cosine family of operators and modified version of Banach contraction theorem.