Journal of Fixed Point Theory and Applications最新文献

Stochastic differential equations with Poisson jumps become very popular in modeling the phenomena arising in various fields, for instance in financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. The objective of this paper is to investigate the approximate controllability for a class of control systems represented by second-order stochastic differential equations with time delay and Poisson jumps. The main technique is the theory of fundamental solution constructed through Laplace transformation. By employing the so-called resolvent condition, theory of cosine operators and stochastic analysis, we formulate and prove some sufficient conditions for the approximate controllability of the considered system. In the end an example is given and discussed to illustrate the obtained results.

We introduce the novel concept of integral Read–Bajraktarević (iRB) operator and discuss some of its properties. We show that this iRB operator generalizes the known Read–Bajraktarević (RB) operator and we derive conditions for the fixed point of the iRB operator to belong to certain function spaces.

In this paper, we study a Schrödinger–Choquard equation involving the logarithmic Laplacian operator in (mathbb {R}^{n}):

where (0<s<1, p>1, r>0, nge 2, omega >0). Using the direct method of moving planes, we prove that if u satisfies some suitable asymptotic properties, then u must be radially symmetric and monotone decreasing about some point in the whole space. The key ingredients of the proofs are the narrow region principle and decay at infinity theorem; the ideas can be applied to problems involving more general nonlocal operators.

For any (lambda ge 0), (2le nle 4) and (mu _1in mathbb {R}), we will prove the existence of unique radially symmetric solution (hin C^2((0,infty ))cap C^1([0,infty ))) for the nonlinear singular elliptic equation (2r^{2}h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-lambda r-(n-1))), (h(r)>0), in ((0,infty )) satisfying (h(0)=1), (h_r(0)=mu _1). We also prove the existence of unique analytic solution of the about equation on ([0,infty )) for any (lambda ge 0), (nge 2) and (mu _1in mathbb {R}). Moreover we will prove the asymptotic behaviour of the solution h for any (nge 2), (lambda ge 0) and (mu _1in mathbb {R}setminus {0}).

In this paper, we consider the following nonhomogeneous quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities

where (4<p<+infty ,,varepsilon ,,lambda >0) are parameters, (mathrm{Delta _4 phi = div(|nabla phi |^2 nabla phi )}), (Omega subset {mathbb {R}}^2) is a bounded domain, and f has exponential critical growth. First, using reduction argument, truncation technique, Ekeland’s variational principle, and the Mountain Pass theorem, we obtain that the above system admits at least two solutions with different energy for (lambda ) large enough and (varepsilon ) fixed. Finally, we research the asymptotic behavior of solutions with respect to the parameters (varepsilon ).

We prove a generalization of Ekeland’s variational principal using the notion of (mathbb {P}) sets. Using this result, we give proofs for fixed point theorems on partially ordered sets. Furthermore, one can obtain theorems for coupled fixed points using this technique. We demonstrate the procedure for proving such theorems.

In this paper, the existence of solutions, ((lambda ,u)), of the problem

is explored for (0< p < 1). When (p>1), it is known that there is an unbounded component of such solutions bifurcating from ((sigma _1, 0)), where (sigma _1) is the smallest eigenvalue of (-Delta ) in (Omega ) under Dirichlet boundary conditions on (partial Omega ). These solutions have (u in P), the interior of the positive cone. The continuation argument used when (p>1) to keep (u in P) fails if (0< p < 1). Nevertheless when (0< p < 1), we are still able to show that there is a component of solutions bifurcating from ((sigma _1, infty )), unbounded outside of a neighborhood of ((sigma _1, infty )), and having (u gneq 0). This non-negativity for u cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described.

We present a version of O’Neill’s theorem (Theorem 5.2 in O’Neill in Am J Math 75(3):497–509, 1953) for piecewise linear approximations.

In this paper, we investigate the existence and asymptotic behavior of positive solutions for quasilinear Schrödinger equations involving p-Laplacian

where (2<p<N), (kappa ,) (lambda ) are parameters and A(x) is a potential. The problem is quite sensitive to the sign of (kappa ) and there have been many results for (kappa le 0.) By means of minimization on the Nehari manifold together with perturbation type techniques, we establish the existence of positive solutions for small (kappa >0) and large (lambda ). Moreover, we show that the solutions (u_{kappa ,lambda }) converge in (W^{1,p}) to a positive solution of p-Laplacian in a bounded domain as ((kappa ,lambda )rightarrow (0^+,+infty )). Our results extend some known results of (kappa le 0).

It is known that the global convergence of the method of successive approximations is obtained by means of the Banach contraction principle. In this paper, we study the global convergence of the method by means of a technique that uses auxiliary points and, as a consequence of this study, we obtain fixed-point type results on closed balls. We apply the study to nonlinear Fredholm integral equations of the second kind.