In this paper we deal with the four-point singular boundary value problem $$ \begin {cases} (\phi _p(u'(t)))'+q(t)f(t,u(t),u'(t))=0,& t\in (0,1),\\ u'(0)-\alpha u(\xi )=0, \quad u'(1)+\beta u(\eta )=0, \end {cases} $$ where $\phi _p(s)=|s|^{p-2}s$, $p>1$, $0<\xi <\eta <1$, $\alpha ,\beta >0$, $q\in C[0,1]$, $q(t)>0$, $t\in (0,1)$, and $f\in C([0,1]\times (0,+\infty )\times \mathbb R,(0,+\infty ))$ may be singular at $u = 0$. By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.
In this paper we examine nonlinear periodic systems driven by the vectorial $p$-Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the “sublinear” problem. For the semilinear problem (i.e. $p = 2$) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem for the “superlinear” problem. Our work generalizes some recent results of Tang (PAMS 126(1998)).
In this paper we study the existence of classical solutions for a class of abstract neutral integro-differential equation with unbounded delay. A concrete application to partial neutral integro-differential equations is considered.
In this paper, we discuss the existence of solutions for a boundary value problem of fractional differential inclusions with nonlocal Riemann-Liouville integral boundary conditions. Our results include the cases when the multivalued map involved in the problem is (i) convex valued, (ii) lower semicontinuous with nonempty closed and decomposable values and (iii) nonconvex valued. In case (i) we apply a nonlinear alternative of Leray-Schauder type, in the second case we combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo, while in the third case we use a fixed point theorem for multivalued contractions due to Covitz and Nadler.
In this paper we establish the existence of nontrivial solutions to \[\frac{\mathrm d}{{\mathrm d}t}L_{x^{\prime }}(t,x^{\prime }(t))+V_{x} (t,x(t))=0,\quad x(0)=0=x(T),\] with $V_x$ superlinear in $x$.
A class of impulsive boundary value problems of fractional differential systems is studied. Banach spaces are constructed and nonlinear operators defined on these Banach spaces. Sufficient conditions are given for the existence of solutions of this class of impulsive boundary value problems for singular fractional differential systems in which odd homeomorphism operators (Definition 2.6) are involved. Main results are Theorem 4.1 and Corollary 4.2. The analysis relies on a well known fixed point theorem: Leray-Schauder Nonlinear Alternative (Lemma 2.1). An example is given to illustrate the efficiency of the main theorems, see Example 5.1.
We prove existence results for the Dirichlet problem associated with an elliptic semilinear second-order equation of divergence form. Degeneracy in the ellipticity condition is allowed.
We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation which nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem for the full parabolic equation. We summarize our results on existence of solutions in these cases and propose an open problem related to the existence of self-similar solutions.
The purpose of the present paper is to study the existence of solutions to initial value problems for nonlinear first order differential systems subject to nonlinear nonlocal initial conditions of functional type. The approach uses vector-valued metrics and matrices convergent to zero. Two existence results are given by means of Schauder and Leray-Schauder fixed point principles and the existence and uniqueness of the solution is obtained via a fixed point theorem due to Perov. Two examples are given to illustrate the theory.
In this paper we consider the existence, multiplicity, and nonexistence of positive solutions to fractional differential equation with integral boundary conditions. Our analysis relies on the fixed point index.