Mountain pass solutions to equations of p-Laplacian type

This work is devoted to study the existence of solutions to equations of p-Laplacian type. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. In order to apply mountain pass results, we introduce a notion of uniformly convex functional that generalizes the notion of uniformly convex norm.     

Nonlinear elliptic problems with superlinear reaction and parametric concave boundary condition

We study parametric nonlinear elliptic boundary value problems driven by the p-Laplacian with convex and concave terms. The convex term appears in the reaction and the concave in the boundary condition (source). We study the existence and nonexistence of positive solutions as the parameter λ > 0 varies. For the semilinear problem (p = 2), we prove a bifurcation type result. Finally, we show the existence of nodal (sign changing) solutions.     

State Dependent Sweeping Process in p-Uniformly Smooth and q-Uniformly Convex Banach Spaces

We prove the existence of solutions for the state depending convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces via new recent results on proximal analysis in reflexive smooth Banach spaces.     

Absolute optimal solution for a compact and convex game

This paper investigates the existence of absolute optimal solutions for a partition P in continuous and quasiconcave games. We show that the P-consistency property introduced in the paper, together with the quasiconcavity and continuity of payoffs, permits the existence of P-absolute optimal solutions in games with compact and convex strategy spaces. The P-consistency property is a general condition that cannot be dispensed with for the existence of P-absolute optimal solutions. We also characterize the existence of P-absolute optimal solutions by providing necessary and sufficient conditions. Moreover, we suggest an algorithm for efficiently computing P-absolute optimal solutions.     

On the existence of periodic solutions for differential inclusions

In this paper we study the existence of periodic solutions for differential inclusions. We prove existence theorems under various sets of hypotheses for both the nonconvex and convex problems. Also we show the existence of extreme solutions. Some feedback control systems are also considered.     

On stably quasimonotone hemivariational inequalities

We establish some existence results for variational-hemivariational inequalities of the Hartman-Stampacchia type involving stably quasimonotone set-valued mappings on bounded, closed and convex subsets in reflexive Banach spaces. We also derive a sufficient condition for the existence and boundedness of solutions.     

Weingarten hypersurfaces with prescribed gradient image

We prove the existence of globally smooth convex solutions u of a class of curvature equations subject to the boundary condition where and are smooth uniformly convex domains in . The results generalize some of our previous work on the two dimensional case, and on Hessian equations in all dimensions.     

Radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations

In this paper, we study the existence of radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations. By applying a well‐known fixed point theorem in cones, we shall establish several new criteria for the existence of nontrivial radially symmetric convex solutions for the systems of Monge‐Ampère equations with or without an eigenvalue parameter. Copyright © 2015 John Wiley & Sons, Ltd.     

一般化凸空间上截口定理及其对择一不等式的应用

我们根据一般化凸空间上的KKM型定理得到了截口定理,然后作为它的应用讨论了若干个择一不等式.最后,引进了一个具体的一般化凸空间并在该空间上讨论了择一不等式解的存在性问题.     

Existence of Viscosity Solutions of Second Order Fully Nonlinear Elliptic Equations

We consider the problem of existence for viscosity solutions of seoond order fully nonlinear elliptic partial differential equations F(D²u, Du, u, z) = 0. We prove existence results for viscosity solutions in W^{1,∞} under assumptions that function F satisfies the natural structure conditions. We do not assume F is convex.     

Solution viable d'une inclusion différentielle non convexe

We present a result on the existence of viable solutions for a class of multivalued differential equations with right-hand side not necessarily compact and convex.     

Example of a displacement convex functional of first order

We present a family of first-order functionals which are displacement convex, that is convex along the geodesics induced by the quadratic transportation distance on the circle. The displacement convexity implies the existence and uniqueness of gradient flows of the given functionals. More precisely, we show the existence and uniqueness of gradient-flow solutions of a class of fourth-order degenerate parabolic equations with periodic boundary data. Moreover, positivity of the absolutely continuous part of the solutions is preserved along the flow.     

Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications

In this paper, we propose a new notion of ‘exceptional family of elements’ for convex optimization problems. By employing the notion of ‘exceptional family of elements’, we establish some existence results for convex optimization problem in reflexive Banach spaces. We show that the nonexistence of an exceptional family of elements is a sufficient and necessary condition for the solvability of the optimization problem. Furthermore, we establish several equivalent conditions for the solvability of convex optimization problems. As applications, the notion of ‘exceptional family of elements’ for convex optimization problems is applied to the constrained optimization problem and convex quadratic programming problem and some existence results for solutions of these problems are obtained.     

Second Order Nonlinear Evolution Inclusions Ⅰ： Existence and Relaxation Results

This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.     

Evolving convex curves

We consider the behaviour of convex curves undergoing curvature-driven motion. In particular we describe the long-term behaviour of solutions and properties of limiting shapes, and prove existence of unique solutions from singular or non-strictly convex initial curves, with sharp regularity estimates. Received May 21, 1997 / Accepted December 5, 1997     

Minimax results for semicoercive functionals with application to differential equations

We give minimax theorems for some class of generalized convex and semicoercive functions. We define semicoercive saddle points and give sufficient conditions for functionals to have such critical points. Then we apply this method to show the existence of solutions for partial differential systems at resonance.      

Fixed points and quasi-equilibrium problems

A fixed-point theorem for compact acyclic maps defined on convex subsets of notnecessarily locally convex topological vector spaces is applied to the existence of solutions of quasi-equilibrium problems. Such existence theorems extend known ones which were used for unified approaches to quasi-variational inequalities in [1–5] and others.     

Convex solutions to polynomial-like iterative equations on open intervals

The paper deals with the polynomial-like iterative functional equation $\lambda_1 f(x)+\lambda_2 f^2(x)+\cdots+\lambda_n f^n(x)=F(x).$ By using Schauder’s fixed point theorem and a version of the uniform boundedness principle for families of convex (respectively higher order convex) functions as basic tools, the existence of nondecreasing convex (respectively higher order convex) solutions to this equation on open (possibly unbounded) intervals is investigated. The results of the paper complement similar ones established by other authors, concerning the existence of monotonic or convex solutions to the above equation on compact intervals. Some examples illustrating their applicability are provided.     

Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients

We deal with a Hamilton-Jacobi equation with a Hamiltonian that is discontinuous in the space variable. This is closely related to a conservation law with discontinuous flux. Recently, an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection. In this paper, we characterize these infinite classes of solutions in terms of explicit Hopf-Lax type formulas which are obtained from the viscosity solutions of the corresponding Hamilton-Jacobi equation with discontinuous Hamiltonian. This also allows us to extend the framework of infinitely many classes of solutions to the Hamilton-Jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law. We have considered the case where both the Hamiltonians are convex (concave). Furthermore, we also deal with the less explored case of sign changing coefficients in which one of the Hamiltonians is convex and the other concave. In fact in convex-concave case we cannot expect always an existence of a solution satisfying Rankine-Hugoniot condition across the interface. Therefore the concept of generalised Rankine-Hugoniot condition is introduced and prove existence and uniqueness.     

A new existence theory for periodic solutions to evolution equations

ABSTRACT

This paper deals with a new existence theory for periodic solutions to a broad class of evolution equations. We first establish new fixed point theorems for affine maps in locally convex spaces and ordered Banach spaces. Our new fixed point results extend, encompass and complement a number of well-known theorems in the literature, including the famous Chow and Hale fixed point theorem. With these obtained fixed point results, we investigate the existence of periodic solutions for some class of nonhomogeneous linear systems in Banach spaces with lack of compactness. Some illustrative examples are also given.