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We study the boundary differentiability of infinity harmonic functions with given differentiable boundary data on convex domains. At a flat point (the boundary point where the blow-up of the domain is the half-space), the infinity harmonic function u is differentiable due to a previous result of the first author in Hong (2013). At a corner point (the boundary point where the blow-up of the domain is not the half-space), an example shows that u is not necessarily differentiable. In this paper, we establish a slope estimate for u at corner points and provide a necessary and sufficient condition for the differentiability of u at corner points.

Let ( M , F ) be an n -dimensional compact Finsler manifold without boundary or with a convex boundary and λ 1 be the first (nonzero) closed or Neumann eigenvalue of the Finsler Laplacian on M with nonnegative weighted Ricci curvature. In this paper, we prove that λ 1 ≥ π 2 d 2 , where d is the diameter of M , and that the equality holds if and only if M is a 1 -dimensional circle or a 1 -dimensional segment, which generalize the well-known Zhong–Yang’s sharp estimate in Riemannian geometry (Zhong and Yang, 1984).

Problem (P) { − Δ p u + | u | p − 2 u = | u | r − 1 u x ∈ Ω | ∇ u | p − 2 ∂ u ∂ ν = λ | u | s − 1 u x ∈ ∂ Ω , where Ω ⊂ R N is a bounded smooth domain, ν is the unit outward normal at ∂ Ω , Δ p is the p -Laplacian operator and λ > 0 is a parameter, was studied in Sabina de Lis (2011) and Sabina de Lis and Segura de Leon (in press). Among other features, it was shown there that when exponents lie in the regime 1 s p r , a minimal positive solution exists if 0 λ ≤ Λ , for a certain finite Λ , while no positive solutions exist in the complementary range λ > Λ . Furthermore, in the radially symmetric case a second positive solution exists for λ varying in the same full range ( 0 , Λ ) provided r p ∗ . Our main achievement in this work just asserts that such global multiplicity feature holds true when Ω is a general domain. To show such result the well-known Brezis–Nirenberg variational result in Brezis and Nirenberg (1993) must be extended to the framework of (P) . This is the second main contribution in the present work.

The system of two inclusions ∂ u E ( t , u ( t ) , z ( t ) ) ∋ 0 and ∂ R ( z ) + ∂ z E ( t , u ( t ) , z ( t ) ) ∋ 0 with the dissipation potential R degree-1 homogeneous and with the stored energy E ( t , ⋅ , ⋅ ) separately convex is considered. The relation between conventional weak solutions and local solutions is shown, and a suitably integrated maximal-dissipation principle is devised to select force-driven local solutions and eliminate solutions with “too-early jumps” as it may occur in energy-driven ones. This is illustrated on scalar examples. An approximation by a simple and efficient semi-implicit time discretization of the fractional-step type is shown to converge to local solutions. On the scalar examples, the approximate solutions are shown to satisfy the integrated maximal-dissipation principle asymptotically, while in general it is devised only to serve as an a-posteriori tool to justify (or possibly adaptively adjust) thus obtained approximate solutions as, in fact, force driven. Applications of such solutions are illustrated on specific examples from continuum mechanics at small strains involving inelastic processes in a bulk or on a surface, namely damage and delamination.

A convex sweeping process with unbounded nonconvex perturbation term of a rather general form is investigated in a separable Hilbert space. We consider our sweeping process as an evolution inclusion with subdifferential operators. This allows to prove the theorem on existence of solutions under conditions generalizing the usual assumptions for convex sweeping processes. In particular, we show that multivalued mappings whose values are hyperplanes, half-spaces, etc. may be chosen as multivalued mappings generating the sweeping process. Such mappings do not satisfy the conditions under which convex sweeping processes are usually studied.

In this paper, we consider the existence and some qualitative properties of solutions of variational inequalities of the form: Find u ∈ D ( J ) : 〈 A ( u ) , v − u 〉 + 〈 F ( u ) , v − u 〉 + J ( v ) − J ( u ) ≥ 0 , ∀ v ∈ X , and of quasi-variational inequalities of the form: Find u ∈ D ( J u ) : 〈 A ( u ) , v − u 〉 + 〈 F ( u ) , v − u 〉 + J u ( v ) − J u ( u ) ≥ 0 , ∀ v ∈ X , where A is a second-order elliptic operator of Leray–Lions type, F is a multivalued lower order term, J and J u are convex functionals, and J u also depends on u . We concentrate here in noncoercive cases and use sub-supersolution methods to study the existence and enclosure of solutions, and also the existence of extremal solutions between sub and supersolutions.

We establish the exact multiplicity of positive solutions, and the global solution structure for two classes of problems on circular domains. The first class involves non-autonomous concave–convex problems on a ball in R n , and the second one deals with concave–convex problems on a “thin” annulus in R n . We illustrate our results by numerical computations, using a novel algorithm, which involves continuation in a global parameter.

Let X be a uniformly convex and uniformly smooth Banach space. Let T : C → C be an asymptotic pointwise nonexpansive mapping, where C is a bounded, closed and convex subset of X . In this paper we investigate conditions sufficient for the weak convergence of the generalized Mann and Ishikawa processes to a fixed point of T .

We study here the deterministic homogenization for a particular family of convex integral functionals, namely problems of minimal hypersurfaces. By the sigma-convergence method relooked especially in appropriate representation of non-periodic functions and in the extension of sigma-convergence on the L 1 -space, we obtain a general homogenization result, and thanks to the particularity of minimal hypersurfaces, we derive a macroscopic integral functional. We then provide a number of physical applications (including the periodic case) of this result.

In this paper, our main purpose is to establish the existence of multiple solutions of a p – q -Laplacian equation with convex and Sobolev critical nonlinearity by some standard variational methods, whose key is to construct homotopies between Ω and levels of the functional I θ , and some analytical techniques.

In previous work [J. Warga, A proper relaxation of controls with variable shifts, J. Math Anal. Appl. 196 (1995) 783–793], it has been shown that controls with k variable and unrelated time shifts form a set ℜ whose closure ℜ ¯ in the weak star topology of L 1 ( T , C ( Ω k + 1 ) ) ∗ is a convex and compact space. This provides a basis for establishing the existence of an optimal solution of the shifted control problem by essentially copying the assumptions and arguments used for control problems without shifted controls. Similarly, the necessary optimality conditions, for original and relaxed minima, that define an extremal, previously studied for differential and functional equations, apply directly to such problems with shifted controls except that the maximum principle follows directly in integral form only. In the present note we first establish the form of dependence of the relaxed shifted controls on the shifts and then use the integral form of the maximum principle to derive a corresponding pointwise maximum principle for optimal (ordinary or relaxed) shifted controls acting through ordinary differential and certain functional–differential equations.

This paper is devoted to investigate multiple positive solutions to a singular elliptic system where the nonlinearity involves a combination of concave and convex terms. By exploiting the effect of the coefficient of the critical nonlinearity and a variational method, we establish the main result which is based on the argument of the compactness.

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let Ti:C→C,i=1,2,⋯,N, be a finite family of Lipschitz pseudocontractive mappings. It is our purpose, in this paper, to prove strong convergence of Ishikawa's method to a common fixed point of a finite family of Lipschitz pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on T or on C. Moreover, computation of the closed convex set Cn for each n

In this paper, we prove existence of radially symmetric minimizers u A ( x ) = U A ( | x | ) , having U A ( ⋅ ) AC monotone and l ∗ ∗ ( U A ( ⋅ ) , 0 ) increasing, for the convex scalar multiple integral (∗ ) ∫ B R l ∗ ∗ ( u ( x ) , | ∇ u ( x ) | ρ 1 ( | x | ) ) ⋅ ρ 2 ( | x | ) d x among those u ( ⋅ ) in the Sobolev space A + W 0 1 , 1 ( B R ) . Here, | ∇ u ( x ) | is the Euclidean norm of the gradient vector and B R is the b a l l { x ∈ R d : | x | R } ; while A is the boundary data. Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian l ∗ ∗ : R × R → [ 0 , ∞ ] is just convex lsc and ∃ min l ∗ ∗ ( R , 0 ) and l ∗ ∗ ( s , ⋅ ) is even ∀ s ; while ρ 1 ( ⋅ ) and ρ 2 ( ⋅ ) are Borel bounded away from 0 and ∞ . Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since l ∗ ∗ ( s , v ) = ∞ is freely allowed.

In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the S -iteration process recently introduced by Sahu in [14] converges strongly to a unique fixed point of a mapping T , where T is κ -strongly pseudocontractive mapping from a nonempty, closed and convex subset C of a smooth Banach space into itself. It is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of λ i -strictly pseudocontractive mappings from C into itself. Our results extend and improve some very recent theorems in fixed point theory and variational inequality problems. Particularly, the results presented here extend some theorems of Reich (1980) [1] and Yamada (2001) [15] to a general class of λ -strictly pseudocontractive mappings in uniformly smooth Banach spaces.

This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖ T x n − x n ‖ → 0 due to Chidume and Zegeye in 2004 [14] where ( x n ) is a certain perturbed Krasnoselski–Mann iteration schema for Lipschitz pseudocontractive self-mappings T of closed and convex subsets of a real Banach space. We also give a qualitative strengthening of the theorem by Chidume and Zegeye, by weakening the assumption of the existence of a fixed point. For the bounded case, our bound is polynomial in the data involved.

Let Ω ⊂ R N ( N ≥ 3 ) be a bounded smooth domain containing the origin. In this paper, by using variational methods, the multiplicity of positive solutions is obtained for a quasilinear elliptic problem − Δ p u − μ | u | p − 2 u | x | p = | u | p ∗ ( t ) − 2 | x | t u + λ | u | q − 2 | x | s u , u ∈ W 0 1 . p ( Ω ) , with Dirichlet boundary condition, where Δ p u = div ( | ∇ u | p − 2 ∇ u ) , 1 p N , 0 ≤ μ μ = ( N − p p ) p , λ > 0 , 0 ≤ s , t p , 1 ≤ q p and p ∗ ( t ) = p ( N − t ) N − p is the critical Sobolev–Hardy exponent.

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Lusternik–Schnirelmann category to prove that a semilinear elliptic equation in R N involving sign-changing weight function has at least three positive solutions.

In this paper we consider a convex-composite generalized constraint equation in Banach spaces. Using variational analysis technique, in terms of normal cones and coderivatives, we first establish sufficient conditions for such an equation to be metrically subregular. Under the Robinson qualification, we prove that these conditions are also necessary for the metric subregularity. In particular, some existing results on error bound and metric subregularity are extended to the composite-convexity case from the convexity case.

In this paper, let Σ be a C 3 compact convex in R 2 n satisfying the reversible condition N Σ = Σ with N = diag ( − I n , I n ) . We prove that if there are exactly n geometrically distinct closed characteristics on Σ and all of them are nondegenerate, then all of them must be brake orbits up to a suitable translation of time. Moreover, for n = 2 or 3, we prove that if there are exactly n geometrically distinct closed characteristics on Σ , then all of them must be brake orbits up to a suitable translation of time.

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.

In this paper, we deal with the existence and multiplicity of positive solutions for the multi-singular semilinear elliptic equations − Δ u − ∑ i = 1 k μ i | x − a i | 2 u = | u | 2 ∗ − 2 u + λ | u | q − 2 u , x ∈ Ω , where Ω ⊂ R N ( N ≥ 3 ) is a smooth bounded domain such that the points a i ∈ Ω , i = 1 , 2 , … , k , k ≥ 2 , are different, 0 ≤ μ i μ = ( N − 2 2 ) 2 , λ > 0 , 1 ≤ q 2 , and 2 ∗ = 2 N N − 2 . The results depend crucially on the parameters λ , q and μ i for i = 1 , 2 , … , k .

In the KKM theory, some authors adopt the concepts of the compact closure (ccl), compact interior (cint), transfer compactly closed-valued multimap, transfer compactly l.s.c. multimap, and transfer compactly local intersection property, respectively, instead of the closure, interior, closed-valued multimap, l.s.c. multimap, and possession of a finite open cover property. In this paper, we show that such adoption is inappropriate and artificial. In fact, any theorem with a term with “transfer” attached is equivalent to the corresponding one without “transfer”. Moreover, we can invalidate terms with “compactly” attached by giving a finer topology on the underlying space. In such ways, we obtain simpler formulations of KKM type theorems, Fan–Browder type fixed point theorems, and other results in the KKM theory on abstract convex spaces.

In this paper, we prove a scattering theorem for the critical wave equation outside convex obstacle. The proof relies on generalized Strichartz estimates.

The purpose of this paper is to prove the existence of a fixed point for a pointwise eventually nonexpansive mapping in a nearly uniformly convex Banach space. This provides an affirmative answer to a question given by Kirk and Xu [W.A. Kirk, Hong-Kun Xu, Asymptotic pointwise contraction, Nonlinear Anal. 69 (2008), 4706–4712].

In this paper, we prove the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over compact convex subsets in a reflexive Banach space with a Frechet differentiable norm. Moreover, we give some conditions that guarantee the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over unbounded closed convex subsets. The result obtained in this paper improves and extends the recent ones announced by Yu and Yang [J. Yu, H. Yang, Existence of solutions for generalized variational inequality problems, Nonlinear Anal., 71 (2009) e2327–e2330] and many others.

We study multiple solutions of an even-order nonlinear partial differential equation with a Dirichlet boundary condition. A related class of nonlinear systems is investigated.

This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solution maps entirely via their initial data. This is done on the basis of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis.

If K and L are mutually dual closed convex cones in a Hilbert space H with the metric projections onto them denoted by P K and P L respectively, then the following two assertions are equivalent: (i) P K is isotone with respect to the order induced by K (i.e. v − u ∈ K implies P K v − P K u ∈ K ); (ii) P L is subadditive with respect to the order induced by L (i.e. P L u + P L v − P L ( u + v ) ∈ L for any u , v ∈ H ). This extends the similar result of A.B. Nemeth and the author for Euclidean spaces. The extension is essential because the proof of the result for Euclidean spaces is essentially finite dimensional and seemingly cannot be extended for Hilbert spaces. The proof of the result for Hilbert spaces is based on a completely different idea which uses extended lattice operations.

Let C be a nonempty, closed and convex subset of a uniformly convex and smooth Banach space and let { T n } be a family of mappings of C into itself such that the set of all common fixed points of { T n } is nonempty. We consider a sequence { x n } generated by the hybrid method by generalized projection in mathematical programming. We give conditions on { T n } under which { x n } converges strongly to a common fixed point of { T n } and generalize the results given in [12] , [14] , [13] , [11] .

In this paper, we investigate the exponential time decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations (E) u t − u t x x − ν u x x + β u x + f ( u ) x = 0 , t > 0 , x ∈ R with prescribed initial data (I) u ( 0 , x ) = u 0 ( x ) → u ± , as x → ± ∞ . Here ν ( > 0 ) , β ∈ R are constants, u ± are two given constants satisfying u + ≠ u − and the nonlinear function f ( u ) ∈ C 2 ( R ) is assumed to be either convex or concave. Based on the existence of traveling waves, the local stability and the algebraic decay rate to traveling waves of solutions to the Cauchy problem (E) and (I) established in Yin et al. (2007) [13] , we show an exponential decay rate of the solutions to the Cauchy problem (E) and (I) toward the traveling waves mentioned above, by employing the space–time weighted energy method which was initiated by Kawashima and Matsumura in (1985) [14] and later elaborated by Matsumura and Nishihara (1994) [15] and Nishikawa (1998) [16] .

We propose an abstract variational inequality with time-dependent constraints in a Banach space with a uniformly convex dual and establish the existence, uniqueness, and regularity of the solution thereof. The abstract result is applied to concrete variational inequalities with time-dependent constraints, thereby obtaining the L p -regularity of solutions.

This paper deals with uniform convexity of Musielak–Orlicz–Sobolev spaces and its applications to variational problems. Some sufficient conditions and examples for uniform convexity of Musielak–Orlicz–Sobolev spaces are given. Some special properties relative to the uniformly convex modular for uniformly convex Musielak–Orlicz–Sobolev spaces are presented. As an application of these abstract results, the local minimizers and the mountain pass type critical point of an integral functional with more complicated growth than the p ( x ) -growth are studied.

In the framework of the stability analysis of real functions, the Lipschitz continuity of copulas w.r.t. L p -norms is investigated. Emphasis is put on 1-Lipschitz continuity, as it is the strongest type possible for copulas. After illustrating how to identify 1-Lipschitz copulas w.r.t. some L p -norm, the preservation of this 1-Lipschitz continuity by certain construction methods, such as ordinal sums and convex sums, is demonstrated. Special attention is given to three important classes of copulas, namely Archimedean copulas, extreme value copulas and Archimax copulas.

Given a monotone and homogeneous self-mapping f of the n -dimensional positive cone, a communication matrix M f is introduced and a family F of functions is obtained by multiplying each component of f by arbitrary positive numbers. Assuming that f satisfies a weak form of convexity, a necessary and sufficient criterion on the structure of M f is given so that each function in F has a (nonlinear) eigenvalue. An alternative necessary and sufficient condition in terms of the recession function of f is also provided.

In this paper, we prove the existence and uniqueness of Hermitian harmonic maps from complete Hermitian manifolds into convex balls.

Consider the variational inequality (VI) of finding a point x ∗ such that (∗ ) x ∗ ∈ F i x ( T ) and 〈 ( I − S ) x ∗ , x − x ∗ 〉 ≥ 0 , x ∈ F i x ( T ) where T , S are nonexpansive self-mappings of a closed convex subset C of a Hilbert space, and F i x ( T ) is the set of fixed points of T . Assume that the solution set Ω of this VI is nonempty. This paper introduces two schemes, one implicit and one explicit, that can be used to find the minimum-norm solution of VI (∗) ; namely, the unique solution x ∗ to the quadratic minimization problem: x ∗ = arg min x ∈ Ω ‖ x ‖ 2 .

In this paper, for commuting pair consisting of a single-valued mapping t and multi-valued mapping T , which are satisfying Suzuki’s condition C , in a C A T ( 0 ) space X , we look for a solution to the problem of existence of z ∈ K ⊂ X such that z = t ( z ) ∈ T ( z ) , where K is a bounded closed convex subset of X . Our results improve and extend some recent results of Shahzad and Markin (2008) [12] .

Let H be a Hilbert space and C be a nonempty closed convex subset of H , { T i } i ∈ N be a family of nonexpansive mappings from C into H , G i : C × C → R be a finite family of equilibrium functions ( i ∈ { 1 , 2 , … , K } ) , A be a strongly positive bounded linear operator with a coefficient γ and B λ -Lipschitzian, relaxed ( μ , ν ) -cocoercive map of C into H . Moreover, let { r k , n } k = 1 K , { α n } satisfy appropriate conditions and F ≔ ( ∩ k = 1 K E P ( G k ) ) ∩ V I ( C , B ) ∩ ( ∩ n ∈ N F i x ( T n ) ) ≠ 0 ; we introduce an explicit scheme which defines a suitable sequence as follows: x n + 1 = α n γ f ( x n ) + ( 1 − α n A ) W n P C ( I − s n B ) S r 1 , n 1 S r 2 , n 2 ⋯ S r K , n K x n ∀ n ∈ N and { x n } strongly converges to x ∗ ∈ F which satisfies the variational inequality 〈 ( A − γ f ) x ∗ , x − x ∗ 〉 ≥ 0 for all x ∈ F . The results presented in this paper mainly extend and improve a recent result of Colao [V. Colao, An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Analysis (2009), doi:10.1016/j.na.2009.01.115 ] and Qin [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis 69 (2008) 3897–3909].

The existence of cat Ω ( Ω ) positive solutions for the p -Laplacian system with convex and Sobolev critical nonlinearities is obtained by some standard variational methods, whose key is to construct homotopies between Ω and levels of the functional J λ , μ , and some analytical techniques.

The principal aim of this paper is to show that weakly cone-convex vector-valued functions, as well as weakly cone-quasiconvex vector-valued functions, can be characterized in terms of usual weakly convexity and weakly quasiconvexity of certain real-valued functions, defined by means of the extreme directions of the polar cone or by Gerstewitz’s scalarization functions.

In cone uniform spaces X , using the concept of the D -family of cone pseudodistances, the distance between two not necessarily convex or compact sets A and B in X is defined, the concepts of cyclic and noncyclic set-valued dynamic systems of D -relatively quasi-asymptotic contractions T : A ∪ B → 2 A ∪ B are introduced and the best approximation and best proximity point theorems for such contractions are proved. Also conditions are given which guarantee that for each starting point each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and the limit is a best proximity point. Moreover, D -families are constructed, characterized and compared. The results are new for set-valued and single-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces. Various examples illustrating ideas, methods, definitions and results are constructed.

In this work, we obtain some properties for the family of some convex domains. Based on these, we prove the existence of solutions of some shape optimization for stationary Navier–Stokes equations.

This paper is devoted to the study of the pseudo-Lipschitz property of the efficient (Pareto) solution map for the perturbed convex semi-infinite vector optimization problem (CSVO). We establish sufficient conditions for the pseudo-Lipschitz property of the efficient solution map of (CSVO) under continuous perturbations of the right-hand side of the constraints and functional perturbations of the objective function. Examples are given to illustrate the obtained results.

We consider an elliptic system of the form − Δ u = λ f ( v ) in Ω − Δ v = λ g ( u ) in Ω u = 0 = v on ∂ Ω , } where λ > 0 is a parameter, Ω is a bounded domain in R N with smooth boundary ∂ Ω . Here the nonlinearities f , g : [ 0 , ∞ ) → R are C loc 0 , σ , 0 σ 1 , functions that are superlinear at infinity and satisfy f ( 0 ) 0 and g ( 0 ) 0 . We prove that the system has a positive solution for λ small when Ω is convex with C 3 boundary and no positive solution for λ large when Ω is a general bounded domain with C 2 , β boundary. Moreover, we show that there exists a closed connected subset of positive solutions bifurcating from infinity at λ = 0 .

Let E be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, D a nonempty closed convex subset of E , and T : D → K ( E ) a nonself multimap such that F ( T ) ≠ 0 and P T is nonexpansive, where F ( T ) is the fixed point set of T , K ( E ) is the family of nonempty compact subsets of E and P T ( x ) = { u x ∈ T x : ‖ x − u x ‖ = d ( x , T x ) } . Suppose that D is a nonexpansive retract of E and that for each v ∈ D and t ∈ ( 0 , 1 ) , the contraction S t defined by S t x = t P T x + ( 1 − t ) v has a fixed point x t ∈ D . Let { α n } , { β n } and { γ n } be three real sequences in ( 0 , 1 ) satisfying approximate conditions. Then for fixed u ∈ D and arbitrary x 0 ∈ D , the sequence { x n } generated by x n ∈ α n u + β n x n − 1 + γ n P T ( x n ) , ∀ n ≥ 0 , converges strongly to a fixed point of T .

In this paper, we consider a quasilinear elliptic system with both concave–convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

In this paper, new contraction type non-self mappings in a metric space are introduced, and conditions guaranteeing the existence of a common fixed-point for such non-self contractions in a convex metric space are established. These results generalize and improve the recent results of Imdad and Khan [M. Imdad, L. Khan, Some common fixed point theorems for a family of mappings in metrically convex spaces, Nonlinear Anal. Theory 67 (9) (2007) 2717–2726] and several others.

Many problems encountered in applied mathematics can be recast as the problem of selecting a particular common fixed point of a family of nonexpansive mappings in a Banach space. In this paper, under the hypotheses that E is a reflexive and strictly convex Banach spaces with a uniformly Gâeaux differentiable norm, the strong convergence of a iteration algorithm is proved for finding a common fixed point of a finite family of nonexpansive self-mappings defined on a closed convex subset K of E .

Let X be a Banach space while ( Y , ⪯ ) is a Banach lattice. We consider the class of “upper separated” set-valued functions F : X → 2 Y and investigate the problem of the existence of convex and locally Lipschitz selections of F . We discuss some applications of the selection results obtained in the paper to the theory of deterministic and stochastic differential inclusions.