Showing papers in "Nonlinear Analysis-theory Methods & Applications in 2012"
TL;DR: In this paper, the authors introduce a new concept of α - ψ -contractive type mappings and establish fixed point theorems for such mappings in complete metric spaces.
Abstract: In this paper, we introduce a new concept of α – ψ -contractive type mappings and establish fixed point theorems for such mappings in complete metric spaces. Starting from the Banach contraction principle, the presented theorems extend, generalize and improve many existing results in the literature. Moreover, some examples and applications to ordinary differential equations are given here to illustrate the usability of the obtained results.
749 citations
TL;DR: In this article, the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption, is established and a novel definition of Palais-Smale condition is presented.
Abstract: The aim of this paper is to establish the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption. Several special cases are also emphasized. Moreover, a novel definition of Palais–Smale condition, which is more general than the usual one, is presented and a mountain pass theorem is pointed out. As a consequence, multiple critical points theorems are then established. Finally, as an example of applications, an elliptic Dirichlet problem with critical exponent is investigated.
167 citations
TL;DR: In this article, the projection algorithm of Iiduka and Takahashi for finding a solution of the variational inequality problem for an inverse strongly monotone operator in a Banach space was studied.
Abstract: In this paper, we consider the projection algorithm studied by Iiduka and Takahashi (2008) [10] for finding a solution of the variational inequality problem for an inverse strongly monotone operator in a Banach space. We first remark that, under the assumptions imposed on the operator in their paper, the iterative sequence converges weakly to a zero of the operator, not just a solution of the variational inequality problem. In our proof, slightly modified from the original, we do not assume the uniform smoothness of a space as was the case there. Finally, using Halpern’s type method, we modify this algorithm to obtain the strong convergence to a zero of an inverse strongly monotone operator which is nearest to the initial element of the algorithm in the sense of the Bergman distance associated with the function 1 2 ‖ ⋅ ‖ 2 .
147 citations
TL;DR: In this paper, the authors apply the technique of measures of noncompactness to the theory of infinite system of differential equations in the Banach sequence spaces l p (1 ≤ p ∞ ).
Abstract: In this paper we apply the technique of measures of noncompactness to the theory of infinite system of differential equations in the Banach sequence spaces l p ( 1 ≤ p ∞ ) . Our aim is to present some existence results for infinite system of differential equations formulated with the help of measures of noncompactness.
137 citations
TL;DR: In this article, the existence of positive solutions for the following class of nonlocal problems was studied: λ f ( u ) + γ u τ in R N, where τ = 5 for N = 3 and τ ∈ ( 1, + ∞ ) for N= 1, 2, λ is a positive parameter and γ ∈ { 0, 1 }.
Abstract: This paper is concerned with the existence of positive solutions for the following class of nonlocal problem M ( ∫ R N | ∇ u | 2 d x + ∫ R N V ( x ) | u | 2 d x ) [ − Δ u + V ( x ) u ] = λ f ( u ) + γ u τ in R N , where τ = 5 for N = 3 and τ ∈ ( 1 , + ∞ ) for N = 1 , 2 , λ is a positive parameter and γ ∈ { 0 , 1 } . Moreover, M , V , and f are continuous functions satisfying some conditions.
131 citations
TL;DR: In this article, a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative is given.
Abstract: We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler–Lagrange equations to the basic and isoperimetric problems as well as transversality conditions are proved.
129 citations
TL;DR: In this paper, the notion of G -Reich type maps was defined and a fixed point theorem for such mappings was obtained for contractive type mappings on ordered metric spaces.
Abstract: Let ( X , d ) be a metric space endowed with a graph G such that the set V ( G ) of vertices of G coincides with X . We define the notion of G -Reich type maps and obtain a fixed point theorem for such mappings. This extends and subsumes many recent results which were obtained for other contractive type mappings on ordered metric spaces and for cyclic operators.
126 citations
TL;DR: In this paper, the authors considered the split feasibility problem in infinite-dimensional Hilbert spaces, and studied the relaxed extragradient methods for finding a common element of the solution set Γ of SFP and the set Fix (S ) of fixed points of a nonexpansive mapping S.
Abstract: In this paper, we consider the split feasibility problem (SFP) in infinite-dimensional Hilbert spaces, and study the relaxed extragradient methods for finding a common element of the solution set Γ of SFP and the set Fix ( S ) of fixed points of a nonexpansive mapping S . Combining Mann’s iterative method and Korpelevich’s extragradient method, we propose two iterative algorithms for finding an element of Fix ( S ) ∩ Γ . On one hand, for S = I , the identity mapping, we derive the strong convergence of one iterative algorithm to the minimum-norm solution of the SFP under appropriate conditions. On the other hand, we also derive the weak convergence of another iterative algorithm to an element of Fix ( S ) ∩ Γ under mild assumptions.
120 citations
TL;DR: A strong compactness result in the spirit of the Lions-Aubin-Simon lemma for piecewise constant functions in time ( u τ ) with values in a Banach space was shown in this article.
Abstract: A strong compactness result in the spirit of the Lions–Aubin–Simon lemma is proven for piecewise constant functions in time ( u τ ) with values in a Banach space. The main feature of our result is that it is sufficient to verify one uniform estimate for the time shifts u τ − u τ ( ⋅ − τ ) instead of all time shifts u τ − u τ ( ⋅ − h ) for h > 0 , as required in Simon’s compactness theorem. This simplifies significantly the application of the Rothe method in the existence analysis of parabolic problems.
116 citations
TL;DR: In this article, the existence, nonexistence and regularity results for the boundary value problem Δ λ u + f ( u ) = 0 in Ω, u ∣ ∂ Ω = 0, where Ω is a bounded subset of R N, N ≥ 2, and Δ κ is a Δ γ -Laplacian, i.e. a "degenerate" elliptic operator of the kind λ : = ∑ i = 1 N ∂ x i ( λ i 2 ( x ) ∂ �
Abstract: We prove some existence, nonexistence and regularity results for the boundary value problem Δ λ u + f ( u ) = 0 in Ω , u ∣ ∂ Ω = 0 , where Ω is a bounded subset of R N , N ≥ 2 , and Δ λ is a Δ λ -Laplacian, i.e. a “degenerate” elliptic operator of the kind Δ λ : = ∑ i = 1 N ∂ x i ( λ i 2 ( x ) ∂ x i ) , λ = ( λ 1 , … , λ N ) . Together with some assumptions made in Franchi and Lanconelli (1984) [1] , the family λ is supposed to verify a condition making Δ λ homogeneous of degree two with respect to a group of dilations in R N .
104 citations
TL;DR: In this paper, the coupled fixed point theorems for mixed monotone operators F : X × X → X obtained by Bhaskar and Lakshmikantham are extended by weakening the involved contractive condition.
Abstract: In this paper, we extend the coupled fixed point theorems for mixed monotone operators F : X × X → X obtained by Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393] and Luong and Thuan [N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011) 983–992], by weakening the involved contractive condition. An example as well as an application to nonlinear Fredholm integral equations is also given in order to illustrate the effectiveness of our generalizations.
TL;DR: In this article, sufficient conditions for the controllability of nonlinear fractional dynamical systems are established by using the recently derived formula for solution representation of systems of fractional differential equations and the application of the Schauder fixed point theorem.
Abstract: In this paper we establish a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. The results are obtained by using the recently derived formula for solution representation of systems of fractional differential equations and the application of the Schauder fixed point theorem. Examples are provided to illustrate the results.
TL;DR: In this article, the analysis of a Lane-Emden-Fowler equation with Dirichlet boundary condition and variable potential functions is presented. And the analysis developed in this paper combines monotonicity methods with variational arguments.
Abstract: The paper deals with the study of a Lane-Emden-Fowler equation with Dirichlet boundary condition and variable potential functions. The analysis developed in this paper combines monotonicity methods with variational arguments.
TL;DR: In this article, the boundary value problem for fractional p -Laplacian is considered and a new result on the existence of solutions is obtained, which generalize and enrich some known results.
Abstract: In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional p -Laplacian equation { D 0 + β ϕ p ( D 0 + α x ( t ) ) = f ( t , x ( t ) , D 0 + α x ( t ) ) , t ∈ [ 0 , 1 ] , D 0 + α x ( 0 ) = D 0 + α x ( 1 ) = 0 , where 0 α , β ≤ 1 , 1 α + β ≤ 2 , D 0 + α is a Caputo fractional derivative, and p > 1 , ϕ p ( s ) = | s | p − 2 s is a p -Laplacian operator. A new result on the existence of solutions for the above fractional boundary value problem is obtained, which generalize and enrich some known results to some extent from the literature.
TL;DR: In this paper, a Schrodinger-Kirchhoff-type problem (P) is studied and four existence theorems of nontrivial solutions and a sequence of high energy solutions are obtained by the Mountain Pass Theorem and symmetric Mountain Pass theorem.
Abstract: In the present paper, the following Schrodinger–Kirchhoff-type problem (P) { − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( | x | ) u = Q ( | x | ) f ( u ) , in R N , u ( x ) → 0 as | x | → ∞ , is studied. Four existence theorems of nontrivial solutions and a sequence of high energy solutions for problem (P) are obtained by the Mountain Pass Theorem and symmetric Mountain Pass Theorem.
TL;DR: In this article, it was shown that f ∈ BMO implies that A ( ∇ u ) inherits the Campanato and VMO regularity of f, which is the limiting case of the nonlinear Calderon-Zygmund theory.
Abstract: We prove BMO estimates of the inhomogeneous p -Laplace system given by − div ( | ∇ u | p − 2 ∇ u ) = div f . We show that f ∈ BMO implies | ∇ u | p − 2 ∇ u ∈ BMO , which is the limiting case of the nonlinear Calderon–Zygmund theory. This extends the work of DiBenedetto and Manfredi (1993) [2] , which was restricted to the super-quadratic case p ≥ 2 , to the full case 1 p ∞ and even more general growth. Moreover, we prove that A ( ∇ u ) inherits the Campanato and VMO regularity of f .
TL;DR: A new concept of weighted pseudo almost automorphic functions using the measure theory is established and new results on weighted ergodic functions like completeness and composition theorems are presented.
Abstract: In this work, we establish a new concept of weighted pseudo almost automorphic functions using the measure theory. We present new results on weighted ergodic functions like completeness and composition theorems. The theory of this work generalizes the classical results on weighted pseudo almost periodic and automorphic functions. For illustration, we provide some applications for evolution equations which include reaction–diffusion systems and partial functional differential equations.
TL;DR: In this article, the uniqueness result of positive solutions for a class of quasilinear elliptic equations arising from plasma physics was studied and the existence of a positive radial solution for original equation under the suitable conditions on the power of nonlinearity and quasiliinearity.
Abstract: We are concerned with the uniqueness result of positive solutions for a class of quasilinear elliptic equation arising from plasma physics. We convert a quasilinear elliptic equation into a semilinear one and show the unique existence of positive radial solution for original equation under the suitable conditions on the power of nonlinearity and quasilinearity. We also investigate the non-degeneracy of a positive radial solution for a converted semilinear elliptic equation.
TL;DR: In this article, the authors present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed mixtures.
Abstract: The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.
TL;DR: In this article, the authors studied limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin.
Abstract: In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in e . In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.
TL;DR: In this paper, necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral integral were obtained.
Abstract: We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler–Lagrange type equations and natural boundary conditions, which provide a generalization of the previous results found in the literature. Isoperimetric problems, problems with holonomic constraints and depending on higher-order Caputo derivatives, as well as fractional Lagrange problems, are considered.
TL;DR: In this paper, the sharp estimate on the first nontrivial eigenvalue of the p-Laplacian on a compact Riemannian manifold with nonnegative Ricci curvature was proved.
Abstract: The aim of this paper is to prove the sharp estimate on the first nontrivial eigenvalue of the p -Laplacian on a compact Riemannian manifold with nonnegative Ricci curvature and to characterize the equality case. The estimate applies to manifolds with empty or convex boundary, and in this latter case we also assume Neumann boundary conditions for the p -Laplacian. The main tool used for the proof is a gradient comparison based on a generalized p -Bochner formula.
TL;DR: In this paper, a generalization of the MFCQ to an optimization problem with operator constraint is applied to each of these reformulations, hence leading to new constraint qualifications (CQs) for the bilevel optimization problem.
Abstract: This paper is mainly concerned with the classical KKT reformulation and the primal KKT reformulation (also known as an optimization problem with generalized equation constraint (OPEC)) of the optimistic bilevel optimization problem. A generalization of the MFCQ to an optimization problem with operator constraint is applied to each of these reformulations, hence leading to new constraint qualifications (CQs) for the bilevel optimization problem. M- and S-type stationarity conditions tailored for the problem are derived as well. Considering the close link between the aforementioned reformulations, similarities and relationships between the corresponding CQs and optimality conditions are highlighted. In this paper, a concept of partial calmness known for the optimal value reformulation is also introduced for the primal KKT reformulation and used to recover the M-stationarity conditions.
TL;DR: In this paper, the boundary value problem is considered in the setting where ϕ may be strictly non-positive for some y > 0 and the existence of at least one positive solution is assumed.
Abstract: In this paper, we consider the boundary value problem y Δ Δ ( t ) = − λ f ( t , y σ ( t ) ) subject to the boundary conditions y ( a ) = ϕ ( y ) and y ( σ 2 ( b ) ) = 0 . In this setting, ϕ : C rd ( [ a , σ 2 ( b ) ] T , R ) → R is a continuous functional, which represents a nonlinear nonlocal boundary condition. By imposing sufficient structure on ϕ and the nonlinearity f , we deduce the existence of at least one positive solution to this problem. The novelty in our setting lies in the fact that ϕ may be strictly nonpositive for some y > 0 . Our results are achieved by appealing to the Krasnosel’skiĭ fixed point theorem. We conclude with several examples illustrating our results and the generalizations that they afford.
TL;DR: In this article, the well-posedness property in the setting of set optimization problems was characterized through the Tykhonov sense, in the context of a family of scalar optimization problems.
Abstract: This paper deals with the well-posedness property in the setting of set optimization problems By using a notion of well-posed set optimization problem due to Zhang et al (2009) [18] and a scalarization process, we characterize this property through the well-posedness, in the Tykhonov sense, of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well-posed Our approach is based just on a weak boundedness assumption, called cone properness, that is unavoidable to obtain a meaningful set optimization problem
TL;DR: In this paper, the existence of infinitely many weak solutions for a non-homogeneous eigenvalue Dirichlet problem is studied. But the authors focus on the problem of finding a sequence of non-negative weak solutions strongly converging to zero.
Abstract: In this paper, we are interested in the existence of infinitely many weak solutions for a non-homogeneous eigenvalue Dirichlet problem. By using variational methods, in an appropriate Orlicz–Sobolev setting, we determine intervals of parameters such that our problem admits either a sequence of non-negative weak solutions strongly converging to zero provided that the non-linearity has a suitable behaviour at zero or an unbounded sequence of non-negative weak solutions if a similar behaviour occurs at infinity.
TL;DR: In this article, the existence of solutions for the Schrodinger-Poisson equation with a general nonlinearity in the critical growth was studied, where f has critical growth.
Abstract: In this paper, we consider the following Schrodinger–Poisson equation: { − Δ u + V u + μ ϕ u = f ( u ) , in R 3 , − Δ ϕ = μ u 2 , in R 3 , where f has critical growth. The purpose of this paper is to study the existence of solutions for the Schrodinger–Poisson equation with a general nonlinearity in the critical growth.
TL;DR: In this article, a generalization of the delayed exponential defined by Khusainov and Shuklin (2003) for autonomous linear delay systems with one delay defined by permutable matrices is given.
Abstract: In this paper a generalization of the delayed exponential defined by Khusainov and Shuklin (2003) [1] for autonomous linear delay systems with one delay defined by permutable matrices is given for delay systems with multiple delays and pairwise permutable matrices. Using this multidelay-exponential a solution of a Cauchy initial value problem is represented. By an application of this representation and using Pinto’s integral inequality an asymptotic stability results for some classes of nonlinear multidelay differential equations are proved.
TL;DR: In this article, the existence and uniqueness of square-mean pseudo almost automorphic mild solutions to a linear and semilinear case of fractional stochastic differential equations in a Hilbert space were established.
Abstract: Fractional stochastic differential equations have gained considerable importance due to their application in various fields of science and engineering. This paper is concerned with the square-mean pseudo almost automorphic solutions for a class of fractional stochastic differential equations in a Hilbert space. The main objective of this paper is to establish the existence and uniqueness of square-mean pseudo almost automorphic mild solutions to a linear and semilinear case of these equations. A new set of sufficient conditions is obtained to achieve the required result by using the stochastic analysis theory and fixed point strategy. Finally, an example is provided to illustrate the obtained theory.
TL;DR: In this article, the existence of (L 2, L p ) -random attractor is established for a stochastic reaction-diffusion equation on the whole space R N, which is a compact and invariant tempered set which attracts every tempered random subset of L 2 in the topology of L p.
Abstract: In this paper, the existence of ( L 2 , L p ) -random attractor is established for a stochastic reaction–diffusion equation on the whole space R N . This random attractor is a compact and invariant tempered set which attracts every tempered random subset of L 2 in the topology of L p . The nonlinearity f is supposed to satisfy some growth of arbitrary order p − 1 , where p ≥ 2 . The ( L 2 , L p ) -asymptotic compactness of the random dynamical system is proved by an asymptotic a priori estimate of the unbounded part of solutions.