Showing papers in "Nonlinear Analysis-theory Methods & Applications in 2010"
TL;DR: In this article, the authors consider a differential equation of fractional order with uncertainty and present the concept of solution, which extends, for example, the cases of first order ordinary differential equations and of differential equations with uncertainty.
Abstract: We consider a differential equation of fractional order with uncertainty and present the concept of solution. It extends, for example, the cases of first order ordinary differential equations and of differential equations with uncertainty. Some examples are presented.
556 citations
TL;DR: In this paper, the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation was investigated, and the uniqueness of the positive solution was obtained by the use of contraction map principle and some Lipschitz type conditions.
Abstract: In this paper, we investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation. Firstly, we give Green’s function and prove its positivity; secondly, the uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions; thirdly, by means of the fixed point index theory, we obtain some existence results of positive solution. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.
440 citations
TL;DR: An overview of the field of differential equations with non-standard growth, which considers both existence and regularity questions, as well as several of the most important results.
Abstract: Differential equations with non-standard growth have been a very active field of investigation in recent years. In this survey we present an overview of the field, as well as several of the most important results. We consider both existence and regularity questions. Finally, we provide a comprehensive list of papers published to date.
307 citations
TL;DR: In this article, the generalized Meir-Keeler type functions and coupled fixed point theorems for complete metric spaces with partial order were defined and proved under a generalized MEK contractive condition.
Abstract: Let X be a non-empty set and F : X × X → X be a given mapping. An element ( x , y ) ∈ X × X is said to be a coupled fixed point of the mapping F if F ( x , y ) = x and F ( y , x ) = y . In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir–Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir–Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393].
281 citations
TL;DR: In this article, the authors present fixed point theorems in a complete metric space endowed with a partial order by using altering distance functions and present some applications to first and second order ordinary differential equations.
Abstract: The purpose of this paper is to present some fixed point theorems in a complete metric space endowed with a partial order by using altering distance functions We also present some applications to first and second order ordinary differential equations
264 citations
TL;DR: In this paper, the notion of compatibility of mappings in a partially ordered metric space was introduced and used to establish a coupled coincidence point result, which was later extended to fixed point theorems.
Abstract: In this paper we introduce the notion of compatibility of mappings in a partially ordered metric space and use this notion to establish a coupled coincidence point result. Our work extends the work of Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393]. An example is also given.
255 citations
TL;DR: In this article, a fixed point theorem for generalized contraction in partially ordered complete metric spaces is presented, and an existence and uniqueness for the solution of a periodic boundary value problem is given.
Abstract: We present a fixed point theorem for generalized contraction in partially ordered complete metric spaces. As an application, we give an existence and uniqueness for the solution of a periodic boundary value problem.
230 citations
TL;DR: In this article, the equivalence of vectorial versions of fixed point theorems in generalized cone metric spaces and scalar versions of the Banach contraction principles in general metric spaces is investigated.
Abstract: The main aim of this paper is to investigate the equivalence of vectorial versions of fixed point theorems in generalized cone metric spaces and scalar versions of fixed point theorems in (general) metric spaces (in usual sense). We show that the Banach contraction principles in general metric spaces and in T V S -cone metric spaces are equivalent. Our theorems also extend some results in Huang and Zhang (2007) [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], Rezapour and Hamlbarani (2008) [Sh. Rezapour, R. Hamlbarani, Some notes on the paper Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345 (2008) 719–724] and others.
224 citations
TL;DR: In this paper, it was shown that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume-filling effect in the sense of Painter and Hillen.
Abstract: We consider the elliptic–parabolic PDE system { u t = ∇ ⋅ ( ϕ ( u ) ∇ u ) − ∇ ⋅ ( ψ ( u ) ∇ v ) , x ∈ Ω , t > 0 , 0 = Δ v − M + u , x ∈ Ω , t > 0 , with nonnegative initial data u 0 having mean value M , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n . The nonlinearities ϕ and ψ are supposed to generalize the prototypes ϕ ( u ) = ( u + 1 ) − p , ψ ( u ) = u ( u + 1 ) q − 1 with p ≥ 0 and q ∈ R . Problems of this type arise as simplified models in the theoretical description of chemotaxis phenomena under the influence of the volume-filling effect as introduced by Painter and Hillen [K.J. Painter, T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q. 10 (2002) 501–543]. It is proved that if p + q 2 n then all solutions are global in time and bounded, whereas if p + q > 2 n , q > 0 , and Ω is a ball then there exist solutions that become unbounded in finite time. The former result is consistent with the aggregation–inhibiting effect of the volume-filling mechanism; the latter, however, is shown to imply that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume-filling effect in the sense of Painter and Hillen.
213 citations
TL;DR: In this article, the fixed point existence results of multivalued mappings defined on cone metric spaces are discussed and the new results are merely copies of the classical ones and do not necessitate the underlying Banach space nor associated cone.
Abstract: In this work we discuss some recent results about KKM mappings in cone metric spaces. We also discuss the fixed point existence results of multivalued mappings defined on such metric spaces. In particular we show that most of the new results are merely copies of the classical ones and do not necessitate the underlying Banach space nor the associated cone.
212 citations
TL;DR: In this article, a fixed point theorem for cyclic φ -contractions is presented in connection with data dependence, well-posedness of the fixed point problem, limit shadowing property and sequences of operators and fixed points.
Abstract: Following [W.A. Kirk, P.S. Srinivasan, P. Veeramany, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (1) (2003) 79–89], we present a fixed point theorem for cyclic φ -contractions and following [I.A. Rus, The theory of a metrical fixed point theorem: Theoretical and applicative relevances, Fixed Point Theory 9 (2) (2008) 541–559] we construct a theory of this fixed point theorem. This theory is in connection with data dependence, well-posedness of the fixed point problem, limit shadowing property and sequences of operators and fixed points. A Maia type fixed point theorem for cyclic φ -contractions is also given.
TL;DR: Agarwal et al. as discussed by the authors considered a general class of abstract fractional differential equations and showed that the existence of a solution for these problems is not always known, since the considered variation of constant formulas is not appropriate.
Abstract: This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1] , Belmekki and Benchohra (2010) [2] , Darwish et al. (2009) [3] , Hu et al. (2009) [4] , Mophou and N’Guerekata (2009) [6] , [7] , Mophou (2010) [8] , [9] , Muslim (2009) [10] , Pandey et al. (2009) [11] , Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13] ] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.
TL;DR: In this paper, the existence and uniqueness of a mild solution to impulsive fractional semilinear differential equations is studied. The results are obtained by means of fixed point methods.
Abstract: This paper is concerned with the existence and uniqueness of a mild solution to impulsive fractional semilinear differential equations. The results are obtained by means of fixed point methods. We also give an example of such problems.
TL;DR: In this article, the existence of bounded solutions of a boundary value problem on an unbounded domain for differential equations involving the Caputo fractional derivative was studied. And the results were based on a fixed point theorem of Schauder combined with the diagonalization method.
Abstract: We are concerned with the existence of bounded solutions of a boundary value problem on an unbounded domain for differential equations involving the Caputo fractional derivative. Our results are based on a fixed point theorem of Schauder combined with the diagonalization method.
TL;DR: In this paper, the convergence of two iterative algorithms for finding common fixed points of finitely many Bregman strongly nonexpansive operators in reflexive Banach spaces is studied.
Abstract: We study the convergence of two iterative algorithms for finding common fixed points of finitely many Bregman strongly nonexpansive operators in reflexive Banach spaces Both algorithms take into account possible computational errors We establish two strong convergence theorems and then apply them to the solution of convex feasibility, variational inequality and equilibrium problems
TL;DR: Nakano et al. as mentioned in this paper introduced the notion of a modular space, which is defined as a metric space with three properties: x = y if and only if w ( λ, x, y, y ) = 0 for all λ > 0; w( λ, x, y, x ) = w (λ, y, y, x ) for all ≥ 0; and w(λ + μ, X, y) = w(λ, λ + μ, x, Y ) + w (λ, λ
Abstract: The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w : ( 0 , ∞ ) × X × X → [ 0 , ∞ ] satisfying, for all x , y , z ∈ X , the following three properties: x = y if and only if w ( λ , x , y ) = 0 for all λ > 0 ; w ( λ , x , y ) = w ( λ , y , x ) for all λ > 0 ; w ( λ + μ , x , y ) ≤ w ( λ , x , z ) + w ( μ , y , z ) for all λ , μ > 0 . We show that, given x 0 ∈ X , the set X w = { x ∈ X : lim λ → ∞ w ( λ , x , x 0 ) = 0 } is a metric space with metric d w ∘ ( x , y ) = inf { λ > 0 : w ( λ , x , y ) ≤ λ } , called a modular space. The modular w is said to be convex if ( λ , x , y ) ↦ λ w ( λ , x , y ) is also a modular on X . In this case X w coincides with the set of all x ∈ X such that w ( λ , x , x 0 ) ∞ for some λ = λ ( x ) > 0 and is metrizable by d w ∗ ( x , y ) = inf { λ > 0 : w ( λ , x , y ) ≤ 1 } . Moreover, if d w ∘ ( x , y ) 1 or d w ∗ ( x , y ) 1 , then ( d w ∘ ( x , y ) ) 2 ≤ d w ∗ ( x , y ) ≤ d w ∘ ( x , y ) ; otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces.
TL;DR: In this article, the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces is proved and Cauchy problems with nonlocal initial conditions are discussed.
Abstract: In this paper we prove the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces. Further, Cauchy problems with nonlocal initial conditions are discussed for the aforementioned fractional integrodifferential equations. At the end, an example is presented.
TL;DR: In this paper, the authors discuss some new positive properties of the Green function for boundary value problems of nonlinear Dirichlet-type fractional differential equations, and its applications are also given.
Abstract: In this paper, we discuss some new positive properties of the Green function for boundary value problems of nonlinear Dirichlet-type fractional differential equations. Its applications are also given.
TL;DR: In this paper, Miyagaki and Souto studied the existence of a nontrivial solution to the nonlinear elliptic boundary value problem of p -Laplacian type.
Abstract: In this paper, we study the existence of a nontrivial solution to the following nonlinear elliptic boundary value problem of p -Laplacian type: ( ( P ) λ ) { − Δ p u = λ f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω where p > 1 , λ ∈ R 1 , Ω ⊂ R N is a bounded domain and Δ p u = d i v ( | ∇ u | p − 2 ∇ u ) is the p -Laplacian of u . f ∈ C 0 ( Ω × R 1 , R 1 ) is p -superlinear at t = 0 and subcritical at t = ∞ . We prove that under suitable conditions for all λ > 0 , the problem ( ( P ) λ ) has at least one nontrivial solution without assuming the Ambrosetti–Rabinowitz condition. Our main result extends a result for ( ( P ) λ ) for when p = 2 given by Miyagaki and Souto (2008) in [8] to the general problem ( ( P ) λ ) where p > 1 . Meanwhile, our result is stronger than a similar result of Li and Zhou (2003) given in [15] .
TL;DR: In this article, the differentiability properties of the solutions for nonlinear fractional differential equations were obtained, and sufficient conditions for the local asymptotical stability of nonlinear FDEs were derived.
Abstract: This paper first obtains the differentiability properties of the solutions for nonlinear fractional differential equations, and then the sufficient conditions for the local asymptotical stability of nonlinear fractional differential equations are also derived.
TL;DR: A two-point boundary value problem for a second order fuzzy differential equation is interpreted by using a generalized differentiability concept and the problem of finding new solutions is investigated.
Abstract: In this paper, we interpret a two-point boundary value problem for a second order fuzzy differential equation by using a generalized differentiability concept. We present a new concept of solutions and, utilizing the generalized differentiability, we investigate the problem of finding new solutions. Some examples are provided for which the new solutions are found.
TL;DR: In this article, the existence of multiple solutions for a class of second-order impulsive Hamiltonian systems with a perturbed term has been studied and some new criteria for guaranteeing that these systems have at least three solutions.
Abstract: In this paper, we study the existence of multiple solutions for a class of second-order impulsive Hamiltonian systems. We give some new criteria for guaranteeing that the impulsive Hamiltonian systems with a perturbed term have at least three solutions by using a variational method and some critical points theorems of B. Ricceri. We extend and improve on some recent results. Finally, some examples are presented to illustrate our main results.
TL;DR: In this paper, the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative by using the monotone iterative method was discussed.
Abstract: In this paper, we shall discuss the properties of the well-known Mittag–Leffler function, and consider the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann–Liouville fractional derivative by using the monotone iterative method.
TL;DR: In this paper, the authors proved that the sequence { x n } generated by the iterative method x n + 1 = α n γ f ( x n ) + ( I − μ α n F ) T x n converges strongly to a fixed point x ∈ F i x (T ), which solves the variational inequality 〈 ( γ F − μ F ) x, x − x 〉 ≤ 0, for x ∆ F i X (T).
Abstract: Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient 0 α 1 , and F : H → H is a k -Lipschitzian and η -strongly monotone operator with k > 0 , η > 0 . Let 0 μ 2 η / k 2 , 0 γ μ ( η − μ k 2 2 ) / α = τ / α . We proved that the sequence { x n } generated by the iterative method x n + 1 = α n γ f ( x n ) + ( I − μ α n F ) T x n converges strongly to a fixed point x ∈ F i x ( T ) , which solves the variational inequality 〈 ( γ f − μ F ) x , x − x 〉 ≤ 0 , for x ∈ F i x ( T ) .
TL;DR: In this article, Miyagaki and Souto obtained existence and multiplicity results for superlinear p -Laplacian equations without the Ambrosetti and Rabinowitz condition.
Abstract: Existence and multiplicity results are obtained for superlinear p -Laplacian equations without the Ambrosetti and Rabinowitz condition. To overcome the difficulty that the Palais–Smale sequences of the Euler–Lagrange functional may be unbounded, we consider the Cerami sequences. Our results extend the recent results of Miyagaki and Souto [O. Miyagaki, M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008) 3628–3638].
TL;DR: In this article, a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems is introduced, i.e., the sets of common fixed points of a family of infinite non-expansive mappings.
Abstract: In this paper, we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems, the set of common fixed points of a family of infinite non-expansive mappings. Strong convergence theorems are established in a real Hilbert space under suitable conditions. As some applications, we consider convex feasibility problems and equilibrium problems. The results presented improve and extend the corresponding results of many others.
TL;DR: The main purpose of as discussed by the authors is the study of the generalization of some results given in [M. Berinde, V. Kannan, and V. Chatterjea] and references therein.
Abstract: The main purpose of this paper is the study of the generalization of some results given in [M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl. 326 (2007) 772–782] and references therein. Some generalizations of the Mizoguchi–Takahashi fixed point theorem, Kannan’s fixed point theorems and Chatterjea’s fixed point theorems are established by using our new fixed point theorems.
TL;DR: In this paper, the identity or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary B K -space into the sequence spaces c 0, c, l ∞ and l 1, and into the matrix domains of triangles in these spaces.
Abstract: In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary B K -space into the sequence spaces c 0 , c , l ∞ and l 1 , and into the matrix domains of triangles in these spaces. Furthermore, by using the Hausdorff measure of noncompactness, we apply our results to characterize some classes of compact operators on the B K -spaces.
TL;DR: In this article, the existence and multiplicity of the non-local p ( x ) -Laplacian Dirichlet problems with non-variational form were studied.
Abstract: This paper deals with the nonlocal p ( x ) -Laplacian Dirichlet problems with non-variational form − A ( u ) Δ p ( x ) u ( x ) = B ( u ) f ( x , u ( x ) ) in Ω ; u ∣ ∂ Ω = 0 , and with variational form − a ( ∫ Ω | ∇ u | p ( x ) p ( x ) d x ) Δ p ( x ) u ( x ) = b ( ∫ Ω F ( x , u ) d x ) f ( x , u ( x ) ) in Ω ; u ∣ ∂ Ω = 0 , where F ( x , t ) = ∫ 0 t f ( x , s ) d s , and a is allowed to be singular at zero. Using ( S + ) mapping theory and the variational method, some results on existence and multiplicity for the problems are obtained under weaker hypotheses. Our results are also new even for the case when p ( x ) ≡ p is a constant.
TL;DR: In this paper, the authors studied S -asymptotically ω -periodic solutions of the abstract fractional equation u = ∂ − α + 1 A u + f ( t, u t ), 1 α 2, where A is a linear operator of sectorial type μ 0.
Abstract: We study S -asymptotically ω -periodic solutions of the abstract fractional equation u ′ = ∂ − α + 1 A u + f ( t , u t ) , 1 α 2 , where A is a linear operator of sectorial type μ 0 .