ON A GENERALIZATION OF KRASNOSELSKII S THEOREM

Size: px
Start display at page:

Download "ON A GENERALIZATION OF KRASNOSELSKII S THEOREM"

Transcription

1 J. Austral. Math. Soc. 72 (2002), ON A GENERALZATON OF KRASNOSELSK S THEOREM DARUSZ DCZAK and ANDRZEJ ROGOWSK (Received 25 August 1999 revised 24 April 2001) Communicated by K. Ecker Abstract n this paper we prove a generalization of the well known theorem of Krasnoselskii on the superposition operator in which the domain of Nemytskii s operator is a product space. We also give an application of this result Mathematics subject classification: primary 49J50, 47H30, 47N10. Keywords and phrases: Nemytskii s operator, variational methods, continuous differentiability. 1. ntroduction Let f : R l R m, R s be a Carathéodory function (that is, measurable with respect to the first variable and continuous with respect to the group of others). n nonlinear analysis, the operator of superposition (often referred to as Nemytskii s operator) of the form F : L p.; R l / x. / f. ; x. // L 1.; R m /, p 1(see [5, 6, 7, 8]), plays an essential role. The well-known theorem of Krasnoselskii says that an operator F is continuous if it is bounded, that is, f.t; x/ a x p + b.t/ for some a > 0andb L 1./. For nonlinear operators, this is quite remarkable. The original proof of this theorem and its known modifications are long and not elementary (see for example [5, 6, 9]). A very simple and short proof of the theorem of Krasnoselskii can be found in [10]. n this paper we prove a generalization of Krasnoselskii s theorem in which the domain of Nemytskii s operator is a product space. There exists an extensive list of different kinds of generalizations of the theorem of Krasnoselskii (see [2, 1] and the references therein) but the authors have not found the results proved in this paper anywhere. The result we obtained is used to prove some differentiability properties of integral functionals. c 2002 Australian Mathematical Society /2000 $A2:00 + 0:00 389

2 390 Dariusz dczak and Andrzej Rogowski [2] 2. Some generalization of the Krasnoselskii theorem Recall the classical theorem of Krasnoselskii [5]. Let R s be a (Lebesgue) measurable set and let f : R l R m, s; l; m N, be a Carathéodory function. t is well known that for any measurable function x : R l ; the function t f.t; x.t// R m is measurable. THEOREM 1 (Krasnoselskii). Let p 1 ; p 2 [1; + /. f there exist a constant a > 0 and a function b L p2./, such that f.t; x/ a x p1=p2 +b.t/ for t a.e., x R l, then the Nemytskii operator F : L p1.; R l / x. / f. ; x. // L p2.; R m / is continuous. Let S i, i = 1;:::;k, be given sets of measurable functions on with values in R li. Assume L i S i, i = 1;:::;k, are topological Hausdorff spaces such that any convergent sequence in L i contains a subsequence convergent (to the same limit) pointwise a.e. in. PutL L 1 L k with the topology induced from L 1 L k. Assume that f : R l1 R lk R m is a Carathéodory function. THEOREM 2. f, for any convergent sequence.x 1 n ;:::;x k n / n N in L there exist a subsequence.x 1 n i ;:::;x k n i / i N and a function h L p.; R + 0 /, 1 p <, such that (2.1) f.t; x 1 n i.t// h.t/ for all i N and t a.e., then the Nemytskii operator F : L.x 1 ;:::;x k / f. ; x 1. /;:::;x k. // L p.; R m / is well defined and sequentially continuous in L,thatis,if.x 1 n ;:::;x k n / n.x 1 0 ;:::; x k 0 / in L, then F.x1 n ;:::;x k n / n F.x 1 0 ;:::;x k 0 / in L p.; R m /. PROOF. The assumption of the theorem, applied to a constant sequence, implies that F is well defined. Suppose now that.x 1;:::;xk/.x 1;:::;xk n n n 0 0 / in L and there exist a number ">0and a subsequence still denoted by.x 1;:::;xk/ n n n N, such that (2.2) f.t; x 1.t/;:::;x k.t// f.t; n n x1.t/;:::;x k.t// p 0 0 dt >" for n N: On the other hand, we can choose a subsequence.x 1 n i ;:::;x k n i / of.x 1 n ;:::;x k n / satisfying (2.1). We calculate f.t; x 1 n i.t// f.t; x 1 0.t/;:::;x k 0.t// p 2 p. f.t; x 1 n i.t// p + f.t; x 1.t/;:::;x k 0 0.t// p / 2 p..h.t// p + f.t; x 1.t/;:::;x k 0 0.t// p /

3 [3] On Krasnoselskii s theorem 391 for all i N and t a.e. Since the topologies in L j, j = 1;:::;k, imply a pointwise convergence a.e. of a subsequence, the Carathéodory condition on f implies that, up to a subsequence, f.t; x 1 n i.t// is pointwise convergent a.e. in to f.t; x 1.t/;:::;xk 0 0.t//. Using the Lebesgue dominated convergence theorem, we get lim i f.t; x 1 n i.t// f.t; x 1.t/;:::;xk 0 0.t// p dt = 0, which contradicts (2.2) and the proof is completed. REMARK. n the above theorem any subsets of L pi.; R li / or C.; R li / with compact can be considered as topological spaces L i (with the induced topologies). 3. Application (3.1) Let us consider a second order system d dt. u 1.t/ + Þ 1.t// = au 1.t/ + þ 1.t/ + A.t/u 2.t/; d dt. u 2.t/ 2 u 2.t/ + Þ 2.t// = b u 2.t/ 2 u 2.t/ + þ 2.t/ + A T.t/u 1.t/ for t =[0;³] a.e., with Dirichlet boundary conditions u.0/ = u.³/ = 0. t is easy to notice that the functional of action for system (3.1) (which becomes the Euler-Lagrange equation for this functional) is of the form ( 1 (3.2).u 1 ; u 2 / = 2 u 1.t/ u 2.t/ 4 + Þ 1.t/; u 1.t/ + Þ 2.t/; u 2.t/ a 2 u 1.t/ 2 b 4 u 2.t/ 4 + þ 1.t/; u 1.t/ ) + þ 2.t/; u 2.t/ + u 1.t/; A.t/u 2.t/ dt; where a; b R, Þ 1 L 2.; R l1 / \ L 4.; R l1 /, Þ 2 L 4=3.; R l2 /, þ 1 L 1.; R ( l1 /, þ 2 L 1.; R l2 /, A L 1.; R l1 l2 1;p /, l 1 ; l 2 N. Recall that W 0.; R l /,where p [1; [, l N is the standard Sobolev space, that is, the space of all absolutely continuous functions u : R l such that u L p.; R l / and u.0/ = u.³/ = 0, equipped with the norm u = ( u.t/ p dt + u.t/ p dt ) 1=p). At the first glance it seems that the space W 1;4 0.; R l1 R l2 / is a natural space for investigating system (3.1) as well as functional (3.2). However, from the fact that Þ 1 L 2.; R l1 / \ L 4.; R l1 / it follows that there is no solution of system (3.1) inw 1;4 0.; R l1 R l2 /. On the other hand, we easily check that is a Gâteaux-differentiable, weakly lower semicontinuous and coercive functional on W 1;2 0.; R l1 / W 1;4 0.; R l2 /. By applying a direct method of the calculus of variations it is easy to prove the existence of a solution of (3.1)

4 392 Dariusz dczak and Andrzej Rogowski [4] in W 1;2 0.; R l1 / W 1;4 0.; R l2 /. n many variational and optimal control problems, an essential role is played by the theorem on the existence and continuity of the Fréchet derivative of integral functionals (see for example [7, 4]). Using Theorem 2, we can prove R THEOREM 3. Let be a finite interval in R, and let g : R l1 R l2 R l1 R l2 be a Carathéodory function continuously differentiable with respect to the last four variables for t a.e. f there exist functions a 1 ; a 2 C.; R + 0 /, b L 1.; R + 0 /, c L q1.; R + 0 /, d L q2.; R + 0 /, wherep q 1 1 = 1, p q 1 2 = 1, such that g.t; u 1 ; u 2 ; u 1 ; u 2 /.a 1. u 1 / + a 2. u 2 //.b.t/ + u 1 p1 + u 2 p2 /; u1 g.t; u 1 ; u 2 ; u 1 ; u 2 /.a 1. u 1 / + a 2. u 2 //.b.t/ + u 1 p1 + u 2 p2 /; u2 g.t; u 1 ; u 2 ; u 1 ; u 2 /.a 1. u 1 / + a 2. u 2 //.b.t/ + u 1 p1 + u 2 p2 /; u1 g.t; u 1 ; u 2 ; u 1 ; u 2 /.a 1. u 1 / + a 2. u 2 //.c.t/ + u 1 p1 1 + u 2 p2=q1 /; u2 g.t; u 1 ; u 2 ; u 1 ; u 2 /.a 1. u 1 / + a 2. u 2 //.d.t/ + u 2 p2 1 + u 1 p1=q2 /; then the functional : W 1;p1 0.; R l2 / R defined by the formula.u 1 ; u 2 / = g.t; u 1.t/; u 2.t/; u 1.t/; u 2.t// dt is continuously Fréchet-differentiable in W 1;p1 0.; R l2 /, and (.u 1 ; u 2 /.h 1 ; h 2 / = u1 g.t; u 1.t/; u 2.t/; u 1.t/; u 2.t//h 1.t/ for.h 1 ; h 2 / W 1;p1 0.; R l2 /. + u2 g.t; u 1.t/; u 2.t/; u 1.t/; u 2.t//h 2.t/ + u1 g.t; u 1.t/; u 2.t/; u 1.t/; u 2.t//ḣ 1.t/ + u2 g.t; u 1.t/; u 2.t/; u 1.t/; u 2.t//ḣ 2.t/ ) dt REMARK. This theorem is a generalization of [7, Theorem 1.4] in the case of product of spaces W 1;p1 0.; R l2 /. n a natural way it may be extended to the case of product of any finite number of spaces W 1;pi 0.; R li /. n the proof of Theorem 3 we shall use this very interesting lemma of Brézis ([3, Theorem V.9]). LEMMA 1(Brézis). From any sequence.x n / n N of elements of L p.; R l /, p [1; + /, converging in L p.; R l / to some x 0 L p.; R l /, one can choose a subsequence.x nk / k N converging to x 0 a.e. in, such that x nk.t/ g.t/ for t a.e. and k N with some function g L p.; R l /.

5 [5] On Krasnoselskii s theorem 393 PROOF OF THEOREM 3. t suffices to prove that there exist partial Gâteaux derivatives u 1.u 1 ; u 2 /, u 2.u 1 ; u 2 / of the functional, continuous in W 1;p1 0.; R l1 / W 1;p2 0.; R l2 /. t is easy to show (see for example [7, proof of Theorem 1.4]) that these derivatives exist and are given by the formulae ( u 1.u 1 ; u 2 /.h 1 / = u1 g.t; u 1.t/; u 2.t/; u 1.t/; u 2.t//h 1.t/ + u1 g.t; u 1.t/; u 2.t/; u 1.t/; u 2.t//ḣ 1.t/ ) dt for h 1 W 1;p1 0.; R l1 / and ( u 2.u 1 ; u 2 /.h 2 / = u2 g.t; u 1.t/; u 2.t/; u 1.t/; u 2.t//h 2.t/ + u2 g.t; u 1.t/; u 2.t/; u 1.t/; u 2.t//ḣ 2.t/ ) dt for h 2 W 1;p2 0.; R l2 /. To end the proof, we must show that the operators and W 1;p1 0.; R l2 /.u 1 ; u 2 / ui g. ; u 1. /; u 2. /; u 1. /; u 2. // L 1.; R li / W 1;p1 0.; R l2 /.u 1 ; u 2 / ui g. ; u 1. /; u 2. /; u 1. /; u 2. // L 1.; R li / for i = 1; 2 are continuous. Since W 1;p1 0.; R l1 /, W 1;p2 0.; R l2 / are metric spaces, it is enough to show that the above operators are sequentially continuous. Since the convergence of a sequence of functions in W 1;pi 0.; R li / implies the convergence of this sequence in C.; R li / together with the convergence of the sequence of derivatives in L pi.; R li /, therefore, by Theorem 2, putting L 1 = C.; R l1 /, L 2 = C.; R l2 /, L 3 = L p1.; R l1 /, L 4 = L p2.; R l2 / and L = L 1 L 2 L 3 L 4,wegettheassertion. ndeed, let.z n; 1 zn; 2 zn; 3 zn/ 4 n.z0; 1 z0; 2 z0; 3 z0 4 / in L. Applying Lemma 1, one can choose subsequences.z ni 3 / i N and.z ni 4 / i N such that.z ni 3.t// g 3.t/ and.z ni 4.t// g 4.t/ (t a.e., i N) for some functions g 3 L p1.; R + 0 / and g 4 L p2.; R + 0 /. Moreover, since L 1 = C.; R l1 /, L 2 = C.; R l2 /, there exists M 0 such that.z ni 1.t// M and.z ni 2.t// M; t ; i N: Hence the assumptions of the theorem yield, for j = 1; 2, the estimates u j g.t; z ni 1 ; z ni 2 ; z ni 3 ; z ni 4 / h j.t/; t a.e., i N; with h j.t/ =.a 1.M/ + a 2.M//.b.t/ + g 3.t/ p1 + g 4.t/ p2 / belonging to L 1.; R + 0 /.

6 394 Dariusz dczak and Andrzej Rogowski [6] n the same way, for j = 1; 2; one can check that u j g.t; z ni 1 ; zni 2 ; zni 3 ; zni 4 / h j.t/; t a.e., i N; with h 1 L q1.; R + 0 /, h 2 L q2.; R + 0 /. Consequently, each of the functions u j g, u j g, j = 1; 2, satisfies the assumptions of Theorem 2. Therefore we conclude that if.u n 1 ; un 2 / n.u0 1 ; u0 1;p1 2 / in W0.; R l1 / W 1;p2 0.; R l2 /, then u j g. ; u n 1. /; un 2. /; un 1. /; un 2. // n u j g. ; u 0 1. /; u0 2. /; u0 1. /; u0 2. // in L 1.; R l j / for j = 1; 2, and u j g. ; u n 1. /; un 2. /; un 1. /; un 2. // n u j g. ; u 0 1. /; u0 2. /; u0 1. /; u0 2. // in L q j.; R l j / for j = 1; 2. This ends the proof. References [1] J. Appell and P. P. Zabrejko, Boundedness properties of the superposition operator, Bull. Polish Acad. Sci. Math. 37 (1989), [2], Continuity properties of the superposition operator, J. Austral. Math. Soc. Ser. A 47 (1989), [3] H. Brézis, Analyse fonctionelle. Théorie et applications (Masson, Paris, 1983). [4] A. offe and V. Tikhomirov, Theory of extremal problems (North-Holland, Amsterdam, 1979). [5] M. K. Krasnoselskii, Topological methods in the theory of nonlinear integral equations (Pergamon Press, New York, 1964). [6] M. K. Krasnoselskii, P. P. Zabreiko, J.. Pustylnik and P. J. Sobolevskii, ntegral operators in spaces of summable functions (Noordhoff, Leyden, 1976). [7] J. Mawhin and P. Willem, Critical point theory and Hamiltonian systems (Springer, New York, 1989). [8] M. Struwe, Variational methods (Springer, Berlin, 1990). [9] M. M. Vainberg, Variational method and method of monotone operators in the theory of nonlinear equations (Halsted Press, New York, 1973). [10] M. Willem, Minimax theorems (Birkhäuser, Basel, 1996). Faculty of Mathematics University of Łódź S. Banacha Łódź, Poland idczak@math.uni.lodz.pl, arogow@math.uni.lodz.pl

ON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM

ON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM Journal of Applied Analysis Vol. 9, No. 1 23, pp. 139 147 ON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM M. GALEWSKI Received July 3, 21 and, in revised form, March 26, 22 Abstract. We provide

More information

EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM

EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using

More information

INTEGRABLE SOLUTIONS OF A FUNCTIONAL-INTEGRAL EQUATION. In this paper we consider the following functional-integral equation. k(t, s)g(s, y(s)) ds

INTEGRABLE SOLUTIONS OF A FUNCTIONAL-INTEGRAL EQUATION. In this paper we consider the following functional-integral equation. k(t, s)g(s, y(s)) ds JOURNAL OF INTEGRAL EQUATIONS AN APPLICATIONS Volume 4, Number 1, Winter 1992 INTEGRABLE SOLUTIONS OF A FUNCTIONAL-INTEGRAL EQUATION G. EMMANUELE ABSTRACT. We consider a very general functional-integral

More information

CHARACTERIZATION OF CARATHÉODORY FUNCTIONS. Andrzej Nowak. 1. Preliminaries

CHARACTERIZATION OF CARATHÉODORY FUNCTIONS. Andrzej Nowak. 1. Preliminaries Annales Mathematicae Silesianae 27 (2013), 93 98 Prace Naukowe Uniwersytetu Śląskiego nr 3106, Katowice CHARACTERIZATION OF CARATHÉODORY FUNCTIONS Andrzej Nowak Abstract. We study Carathéodory functions

More information

THE BANG-BANG PRINCIPLE FOR THE GOURSAT-DARBOUX PROBLEM*

THE BANG-BANG PRINCIPLE FOR THE GOURSAT-DARBOUX PROBLEM* THE BANG-BANG PRINCIPLE FOR THE GOURSAT-DARBOUX PROBLEM* DARIUSZ IDCZAK Abstract. In the paper, the bang-bang principle for a control system connected with a system of linear nonautonomous partial differential

More information

NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction

NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou

More information

PERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian

PERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian Electronic Journal of Differential Euations, Vol. 24 (24), No. 64, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PERIODIC SOLUIONS FOR NON-AUONOMOUS

More information

EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM. Saeid Shokooh and Ghasem A. Afrouzi. 1. Introduction

EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM. Saeid Shokooh and Ghasem A. Afrouzi. 1. Introduction MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69 4 (217 271 28 December 217 research paper originalni nauqni rad EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM Saeid Shokooh and Ghasem A.

More information

On the fixed point theorem of Krasnoselskii and Sobolev

On the fixed point theorem of Krasnoselskii and Sobolev Electronic Journal of Qualitative Theory of Differential Equations 2011, No. 5, 1-6; http://www.math.u-szeged.hu/ejqtde/ On the fixed point theorem of Krasnoselskii and Sobolev Cristina G. Fuentes and

More information

NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS

NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University

More information

Multiple positive solutions for a class of quasilinear elliptic boundary-value problems

Multiple positive solutions for a class of quasilinear elliptic boundary-value problems Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive

More information

Periodic solutions of a class of nonautonomous second order differential systems with. Daniel Paşca

Periodic solutions of a class of nonautonomous second order differential systems with. Daniel Paşca Periodic solutions of a class of nonautonomous second order differential systems with (q, p) Laplacian Daniel Paşca Abstract Some existence theorems are obtained by the least action principle for periodic

More information

CHRISTENSEN ZERO SETS AND MEASURABLE CONVEX FUNCTIONS1 PAL FISCHER AND ZBIGNIEW SLODKOWSKI

CHRISTENSEN ZERO SETS AND MEASURABLE CONVEX FUNCTIONS1 PAL FISCHER AND ZBIGNIEW SLODKOWSKI PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 79, Number 3, July 1980 CHRISTENSEN ZERO SETS AND MEASURABLE CONVEX FUNCTIONS1 PAL FISCHER AND ZBIGNIEW SLODKOWSKI Abstract. A notion of measurability

More information

2 Statement of the problem and assumptions

2 Statement of the problem and assumptions Mathematical Notes, 25, vol. 78, no. 4, pp. 466 48. Existence Theorem for Optimal Control Problems on an Infinite Time Interval A.V. Dmitruk and N.V. Kuz kina We consider an optimal control problem on

More information

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER NONLINEAR HYPERBOLIC SYSTEM

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER NONLINEAR HYPERBOLIC SYSTEM Electronic Journal of Differential Equations, Vol. 211 (211), No. 78, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS

More information

AN EXTENSION OF THE LAX-MILGRAM THEOREM AND ITS APPLICATION TO FRACTIONAL DIFFERENTIAL EQUATIONS

AN EXTENSION OF THE LAX-MILGRAM THEOREM AND ITS APPLICATION TO FRACTIONAL DIFFERENTIAL EQUATIONS Electronic Journal of Differential Equations, Vol. 215 (215), No. 95, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu AN EXTENSION OF THE

More information

VARIATIONAL AND TOPOLOGICAL METHODS FOR DIRICHLET PROBLEMS. The aim of this paper is to obtain existence results for the Dirichlet problem

VARIATIONAL AND TOPOLOGICAL METHODS FOR DIRICHLET PROBLEMS. The aim of this paper is to obtain existence results for the Dirichlet problem PORTUGALIAE MATHEMATICA Vol. 58 Fasc. 3 2001 Nova Série VARIATIONAL AND TOPOLOGICAL METHODS FOR DIRICHLET PROBLEMS WITH p-laplacian G. Dinca, P. Jebelean and J. Mawhin Presented by L. Sanchez 0 Introduction

More information

A Concise Course on Stochastic Partial Differential Equations

A Concise Course on Stochastic Partial Differential Equations A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original

More information

FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi

FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi Opuscula Math. 37, no. 2 27), 265 28 http://dx.doi.org/.7494/opmath.27.37.2.265 Opuscula Mathematica FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi

More information

Nonlocal Cauchy problems for first-order multivalued differential equations

Nonlocal Cauchy problems for first-order multivalued differential equations Electronic Journal of Differential Equations, Vol. 22(22), No. 47, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Nonlocal Cauchy

More information

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze

More information

ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM. Paweł Goncerz

ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM. Paweł Goncerz Opuscula Mathematica Vol. 32 No. 3 2012 http://dx.doi.org/10.7494/opmath.2012.32.3.473 ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM Paweł Goncerz Abstract. We consider a quasilinear

More information

FEEDBACK DIFFERENTIAL SYSTEMS: APPROXIMATE AND LIMITING TRAJECTORIES

FEEDBACK DIFFERENTIAL SYSTEMS: APPROXIMATE AND LIMITING TRAJECTORIES STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLIX, Number 3, September 2004 FEEDBACK DIFFERENTIAL SYSTEMS: APPROXIMATE AND LIMITING TRAJECTORIES Abstract. A feedback differential system is defined as

More information

POTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem

POTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem Electronic Journal of Differential Equations, Vol. 25(25), No. 94, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) POTENTIAL

More information

On the Coercive Functions and Minimizers

On the Coercive Functions and Minimizers Advanced Studies in Theoretical Physics Vol. 11, 17, no. 1, 79-715 HIKARI Ltd, www.m-hikari.com https://doi.org/1.1988/astp.17.71154 On the Coercive Functions and Minimizers Carlos Alberto Abello Muñoz

More information

WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES

WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES Fixed Point Theory, 12(2011), No. 2, 309-320 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES S. DHOMPONGSA,

More information

Existence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1

Existence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1 Journal of Mathematical Analysis and Applications 257, 321 331 (2001) doi:10.1006/jmaa.2000.7347, available online at http://www.idealibrary.com on Existence and Multiplicity of Solutions for a Class of

More information

On the Euler Lagrange Equation in Calculus of Variations

On the Euler Lagrange Equation in Calculus of Variations On the Euler Lagrange Equation in Calculus of Variations Ivar Ekeland Vietnam Journal of Mathematics ISSN 235-221X Vietnam J. Math. DOI 1.17/s113-18-285-z 1 23 Your article is protected by copyright and

More information

Math 742: Geometric Analysis

Math 742: Geometric Analysis Math 742: Geometric Analysis Lecture 5 and 6 Notes Jacky Chong jwchong@math.umd.edu The following notes are based upon Professor Yanir ubenstein s lectures with reference to Variational Methods 4 th edition

More information

New York Journal of Mathematics. A Refinement of Ball s Theorem on Young Measures

New York Journal of Mathematics. A Refinement of Ball s Theorem on Young Measures New York Journal of Mathematics New York J. Math. 3 (1997) 48 53. A Refinement of Ball s Theorem on Young Measures Norbert Hungerbühler Abstract. For a sequence u j : R n R m generating the Young measure

More information

Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives

Existence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 1, pp. 3 12 (2013) http://campus.mst.edu/adsa Existence of Minimizers for Fractional Variational Problems Containing Caputo

More information

Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.

Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological

More information

PERIODIC PROBLEMS WITH φ-laplacian INVOLVING NON-ORDERED LOWER AND UPPER FUNCTIONS

PERIODIC PROBLEMS WITH φ-laplacian INVOLVING NON-ORDERED LOWER AND UPPER FUNCTIONS Fixed Point Theory, Volume 6, No. 1, 25, 99-112 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm PERIODIC PROBLEMS WITH φ-laplacian INVOLVING NON-ORDERED LOWER AND UPPER FUNCTIONS IRENA RACHŮNKOVÁ1 AND MILAN

More information

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n.

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n. Scientiae Mathematicae Japonicae Online, e-014, 145 15 145 A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS Masaru Nagisa Received May 19, 014 ; revised April 10, 014 Abstract. Let f be oeprator monotone

More information

COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS

COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS Dynamic Systems and Applications 22 (203) 37-384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute

More information

PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC

PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC LE MATEMATICHE Vol. LXV 21) Fasc. II, pp. 97 17 doi: 1.4418/21.65.2.11 PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC JEAN MAWHIN The paper surveys and compares some results on the

More information

AW -Convergence and Well-Posedness of Non Convex Functions

AW -Convergence and Well-Posedness of Non Convex Functions Journal of Convex Analysis Volume 10 (2003), No. 2, 351 364 AW -Convergence Well-Posedness of Non Convex Functions Silvia Villa DIMA, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy villa@dima.unige.it

More information

On Systems of Boundary Value Problems for Differential Inclusions

On Systems of Boundary Value Problems for Differential Inclusions On Systems of Boundary Value Problems for Differential Inclusions Lynn Erbe 1, Christopher C. Tisdell 2 and Patricia J. Y. Wong 3 1 Department of Mathematics The University of Nebraska - Lincoln Lincoln,

More information

CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction

CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction Korean J. Math. 16 (2008), No. 2, pp. 215 231 CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES Jong Soo Jung Abstract. Let E be a uniformly convex Banach space

More information

Weakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1

Weakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1 Int. Journal of Math. Analysis, Vol. 4, 2010, no. 37, 1851-1856 Weakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1 Hong Bin Bai School of Science Sichuan University of Science

More information

A NOTE ON A FOURTH ORDER DISCRETE BOUNDARY VALUE PROBLEM. Marek Galewski and Joanna Smejda

A NOTE ON A FOURTH ORDER DISCRETE BOUNDARY VALUE PROBLEM. Marek Galewski and Joanna Smejda Opuscula Mathematica Vol. 3 No. 1 01 http://dx.doi.org/10.7494/opmath.01.3.1.115 A NOE ON A FOURH ORDER DISCREE BOUNDARY VALUE PROBLEM Marek Galewski and Joanna Smejda Abstract. Using variational methods

More information

Convergence results for solutions of a first-order differential equation

Convergence results for solutions of a first-order differential equation vailable online at www.tjnsa.com J. Nonlinear ci. ppl. 6 (213), 18 28 Research rticle Convergence results for solutions of a first-order differential equation Liviu C. Florescu Faculty of Mathematics,

More information

UNIFORMLY CONTINUOUS COMPOSITION OPERATORS IN THE SPACE OF BOUNDED Φ-VARIATION FUNCTIONS IN THE SCHRAMM SENSE

UNIFORMLY CONTINUOUS COMPOSITION OPERATORS IN THE SPACE OF BOUNDED Φ-VARIATION FUNCTIONS IN THE SCHRAMM SENSE Opuscula Mathematica Vol. 3 No. 01 http://dx.doi.org/10.7494/opmath.01.3..39 UNIFORMLY CONTINUOUS COMPOSITION OPERATORS IN THE SPACE OF BOUNDED Φ-VARIATION FUNCTIONS IN THE SCHRAMM SENSE Tomás Ereú, Nelson

More information

EXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS

EXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journal of Inequalities in Pure and Applied athematics http://jipam.vu.edu.au/ Volume 4, Issue 5, Article 98, 2003 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES D. ESKINE AND A. ELAHI DÉPARTEENT

More information

Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces

Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua

More information

Functions of several variables of finite variation and their differentiability

Functions of several variables of finite variation and their differentiability ANNALES POLONICI MATHEMATICI LX.1 (1994) Functions of several variables of finite variation and their differentiability by Dariusz Idczak ( Lódź) Abstract. Some differentiability properties of functions

More information

In this paper we study periodic solutions of a second order differential equation

In this paper we study periodic solutions of a second order differential equation Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 5, 1995, 385 396 A GRANAS TYPE APPROACH TO SOME CONTINUATION THEOREMS AND PERIODIC BOUNDARY VALUE PROBLEMS WITH IMPULSES

More information

A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS

A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS Fixed Point Theory, (0), No., 4-46 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS A. ABKAR AND M. ESLAMIAN Department of Mathematics,

More information

DIFFERENTIABILITY VIA DIRECTIONAL DERIVATIVES

DIFFERENTIABILITY VIA DIRECTIONAL DERIVATIVES PROCEEDINGS OK THE AMERICAN MATHEMATICAL SOCIETY Volume 70, Number 1, lune 1978 DIFFERENTIABILITY VIA DIRECTIONAL DERIVATIVES KA-SING LAU AND CLIFFORD E. WEIL Abstract. Let F be a continuous function from

More information

HIGHER INTEGRABILITY WITH WEIGHTS

HIGHER INTEGRABILITY WITH WEIGHTS Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 355 366 HIGHER INTEGRABILITY WITH WEIGHTS Juha Kinnunen University of Jyväskylä, Department of Mathematics P.O. Box 35, SF-4351

More information

SOME PROPERTIES OF ESSENTIAL SPECTRA OF A POSITIVE OPERATOR

SOME PROPERTIES OF ESSENTIAL SPECTRA OF A POSITIVE OPERATOR SOME PROPERTIES OF ESSENTIAL SPECTRA OF A POSITIVE OPERATOR E. A. ALEKHNO (Belarus, Minsk) Abstract. Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ ew (T ) of the

More information

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian

More information

SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES

SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), 167 174 SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES ABDELHAKIM MAADEN AND ABDELKADER STOUTI Abstract. It is shown that under natural assumptions,

More information

ON ψ-direct SUMS OF BANACH SPACES AND CONVEXITY

ON ψ-direct SUMS OF BANACH SPACES AND CONVEXITY J. Aust. Math. Soc. 75 (003, 43 4 ON ψ-direct SUMS OF BANACH SPACES AND CONVEXITY MIKIO KATO, KICHI-SUKE SAITO and TAKAYUKI TAMURA Dedicated to Maestro Ivry Gitlis on his 80th birthday with deep respect

More information

CONTROL SYSTEM DEFINED BY SOME INTEGRAL OPERATOR. Marek Majewski

CONTROL SYSTEM DEFINED BY SOME INTEGRAL OPERATOR. Marek Majewski Opuscula Math. 37, no. (7), 33 35 http://dx.doi.org/.7494/opmath.7.37..33 Opuscula Mathematica CONTROL SYSTEM DEFINED BY SOME INTEGRAL OPERATOR Marek Majewski Communicated by Zdzisław Jackiewicz Abstract.

More information

MULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL

MULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL Electronic Journal of Differential Equations, Vol. 2015 (2015), o. 274, pp. 1 9. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIOS

More information

MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN

MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS

More information

THE FIXED POINT PROPERTY FOR CONTINUA APPROXIMATED FROM WITHIN BY PEANO CONTINUA WITH THIS PROPERTY

THE FIXED POINT PROPERTY FOR CONTINUA APPROXIMATED FROM WITHIN BY PEANO CONTINUA WITH THIS PROPERTY PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 91, Number 3, July 1984 THE FIXED POINT PROPERTY FOR CONTINUA APPROXIMATED FROM WITHIN BY PEANO CONTINUA WITH THIS PROPERTY AKIRA TOMINAGA ABSTRACT.

More information

PERIODIC SOLUTIONS OF NON-AUTONOMOUS SECOND ORDER SYSTEMS. 1. Introduction

PERIODIC SOLUTIONS OF NON-AUTONOMOUS SECOND ORDER SYSTEMS. 1. Introduction ao DOI:.2478/s275-4-248- Math. Slovaca 64 (24), No. 4, 93 93 PERIODIC SOLUIONS OF NON-AUONOMOUS SECOND ORDER SYSEMS WIH (,)-LAPLACIAN Daniel Paşca* Chun-Lei ang** (Communicated by Christian Pötzsche )

More information

EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem

EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS

More information

THE SEMI ORLICZ SPACE cs d 1

THE SEMI ORLICZ SPACE cs d 1 Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 269 276. THE SEMI ORLICZ SPACE cs d 1 N. SUBRAMANIAN 1, B. C. TRIPATHY 2, AND C. MURUGESAN 3 Abstract. Let Γ denote the space of all entire

More information

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.

More information

REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS. Julien Frontisi

REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS. Julien Frontisi Serdica Math. J. 22 (1996), 33-38 REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS Julien Frontisi Communicated by G. Godefroy Abstract. It is proved that a representable non-separable Banach

More information

Overview of normed linear spaces

Overview of normed linear spaces 20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural

More information

HOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS

HOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS Electronic Journal of Differential Equations, Vol. 010010, No. 9, pp. 1 10. ISSN: 107-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HOMOCLINIC SOLUTIONS FO

More information

A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction

A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction Comm. Korean Math. Soc. 16 (2001), No. 2, pp. 277 285 A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE Myung-Hyun Cho and Jun-Hui Kim Abstract. The purpose of this paper

More information

On the validity of the Euler Lagrange equation

On the validity of the Euler Lagrange equation J. Math. Anal. Appl. 304 (2005) 356 369 www.elsevier.com/locate/jmaa On the validity of the Euler Lagrange equation A. Ferriero, E.M. Marchini Dipartimento di Matematica e Applicazioni, Università degli

More information

ON THE DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS

ON THE DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume, Number, Pages S -9939(XX- ON THE DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS N. S. HOANG AND A. G. RAMM (Communicated

More information

ITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI. Received December 14, 1999

ITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI. Received December 14, 1999 Scientiae Mathematicae Vol. 3, No. 1(2000), 107 115 107 ITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI Received December 14, 1999

More information

PERIODIC SOLUTIONS FOR A KIND OF LIÉNARD-TYPE p(t)-laplacian EQUATION. R. Ayazoglu (Mashiyev), I. Ekincioglu, G. Alisoy

PERIODIC SOLUTIONS FOR A KIND OF LIÉNARD-TYPE p(t)-laplacian EQUATION. R. Ayazoglu (Mashiyev), I. Ekincioglu, G. Alisoy Acta Universitatis Apulensis ISSN: 1582-5329 http://www.uab.ro/auajournal/ No. 47/216 pp. 61-72 doi: 1.17114/j.aua.216.47.5 PERIODIC SOLUTIONS FOR A KIND OF LIÉNARD-TYPE pt)-laplacian EQUATION R. Ayazoglu

More information

AN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W 1,p FOR EVERY p > 2 BUT NOT ON H0

AN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W 1,p FOR EVERY p > 2 BUT NOT ON H0 AN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p FOR EVERY p > BUT NOT ON H FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT Abstract. In this note we give an example of functional

More information

A DUAL DIFFERENTIATION SPACE WITHOUT AN EQUIVALENT LOCALLY UNIFORMLY ROTUND NORM

A DUAL DIFFERENTIATION SPACE WITHOUT AN EQUIVALENT LOCALLY UNIFORMLY ROTUND NORM J. Aust. Math. Soc. 77 (2004), 357 364 A DUAL DIFFERENTIATION SPACE WITHOUT AN EQUIVALENT LOCALLY UNIFORMLY ROTUND NORM PETAR S. KENDEROV and WARREN B. MOORS (Received 23 August 2002; revised 27 August

More information

WEAK LOWER SEMI-CONTINUITY OF THE OPTIMAL VALUE FUNCTION AND APPLICATIONS TO WORST-CASE ROBUST OPTIMAL CONTROL PROBLEMS

WEAK LOWER SEMI-CONTINUITY OF THE OPTIMAL VALUE FUNCTION AND APPLICATIONS TO WORST-CASE ROBUST OPTIMAL CONTROL PROBLEMS WEAK LOWER SEMI-CONTINUITY OF THE OPTIMAL VALUE FUNCTION AND APPLICATIONS TO WORST-CASE ROBUST OPTIMAL CONTROL PROBLEMS ROLAND HERZOG AND FRANK SCHMIDT Abstract. Sufficient conditions ensuring weak lower

More information

ON THE REGULARITY OF ONE PARAMETER TRANSFORMATION GROUPS IN BARRELED LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES

ON THE REGULARITY OF ONE PARAMETER TRANSFORMATION GROUPS IN BARRELED LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES Communications on Stochastic Analysis Vol. 9, No. 3 (2015) 413-418 Serials Publications www.serialspublications.com ON THE REGULARITY OF ONE PARAMETER TRANSFORMATION GROUPS IN BARRELED LOCALLY CONVEX TOPOLOGICAL

More information

The small ball property in Banach spaces (quantitative results)

The small ball property in Banach spaces (quantitative results) The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence

More information

arxiv: v1 [math.ap] 28 Mar 2014

arxiv: v1 [math.ap] 28 Mar 2014 GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard

More information

STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS

STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS ARCHIVUM MATHEMATICUM (BRNO) Tomus 45 (2009), 147 158 STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS Xiaolong Qin 1, Shin Min Kang 1, Yongfu Su 2,

More information

ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS

ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS Chin. Ann. Math.??B(?), 200?, 1 20 DOI: 10.1007/s11401-007-0001-x ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS Michel CHIPOT Abstract We present a method allowing to obtain existence of a solution for

More information

Tiziana Cardinali Francesco Portigiani Paola Rubbioni. 1. Introduction

Tiziana Cardinali Francesco Portigiani Paola Rubbioni. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 32, 2008, 247 259 LOCAL MILD SOLUTIONS AND IMPULSIVE MILD SOLUTIONS FOR SEMILINEAR CAUCHY PROBLEMS INVOLVING LOWER

More information

Sobolev spaces. May 18

Sobolev spaces. May 18 Sobolev spaces May 18 2015 1 Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references

More information

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,

More information

LERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION

LERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION 2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

arxiv: v1 [math.fa] 18 Feb 2016

arxiv: v1 [math.fa] 18 Feb 2016 Arzelà-Ascoli theorem in uniform spaces Mateusz Krukowski arxiv:1602.05693v1 [math.fa] 18 Feb 2016 Technical University of Łódź, Institute of Mathematics, Wólczańska 215, 90-924 Łódź, Poland February 19,

More information

A fixed point theorem for multivalued mappings

A fixed point theorem for multivalued mappings Electronic Journal of Qualitative Theory of Differential Equations 24, No. 17, 1-1; http://www.math.u-szeged.hu/ejqtde/ A fixed point theorem for multivalued mappings Cezar AVRAMESCU Abstract A generalization

More information

On the minimax inequality and its application to existence of three solutions for elliptic equations with Dirichlet boundary condition

On the minimax inequality and its application to existence of three solutions for elliptic equations with Dirichlet boundary condition ISSN 1 746-7233 England UK World Journal of Modelling and Simulation Vol. 3 (2007) No. 2. 83-89 On the minimax inequality and its alication to existence of three solutions for ellitic equations with Dirichlet

More information

ON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR TERM

ON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR TERM Internat. J. Math. & Math. Sci. Vol. 22, No. 3 (999 587 595 S 6-72 9922587-2 Electronic Publishing House ON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR

More information

ON DENSITY TOPOLOGIES WITH RESPECT

ON DENSITY TOPOLOGIES WITH RESPECT Journal of Applied Analysis Vol. 8, No. 2 (2002), pp. 201 219 ON DENSITY TOPOLOGIES WITH RESPECT TO INVARIANT σ-ideals J. HEJDUK Received June 13, 2001 and, in revised form, December 17, 2001 Abstract.

More information

arxiv: v1 [math.ap] 16 Jan 2015

arxiv: v1 [math.ap] 16 Jan 2015 Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities Vladimir Lubyshev January 19, 2015 arxiv:1501.03870v1 [math.ap] 16 Jan 2015 Abstract This paper studies a nonlinear

More information

Continuous Functions on Metric Spaces

Continuous Functions on Metric Spaces Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0

More information

ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES

ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 3, 2018 ISSN 1223-7027 ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES Vahid Dadashi 1 In this paper, we introduce a hybrid projection algorithm for a countable

More information

THE NEARLY ADDITIVE MAPS

THE NEARLY ADDITIVE MAPS Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between

More information

Existence of homoclinic solutions for Duffing type differential equation with deviating argument

Existence of homoclinic solutions for Duffing type differential equation with deviating argument 2014 9 «28 «3 Sept. 2014 Communication on Applied Mathematics and Computation Vol.28 No.3 DOI 10.3969/j.issn.1006-6330.2014.03.007 Existence of homoclinic solutions for Duffing type differential equation

More information

EXISTENCE AND UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH LINEAR PROGRAMS EMBEDDED

EXISTENCE AND UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH LINEAR PROGRAMS EMBEDDED EXISTENCE AND UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH LINEAR PROGRAMS EMBEDDED STUART M. HARWOOD AND PAUL I. BARTON Key words. linear programs, ordinary differential equations, embedded

More information

YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS

YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 3, July 1988 YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS JOHN RAINWATER (Communicated by William J. Davis) ABSTRACT. Generic

More information

ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES

ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 167 176 ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES Piotr Haj lasz and Jani Onninen Warsaw University, Institute of Mathematics

More information

MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS OPERATORS AND DISCONTINUOUS NONLINEARITIES

MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS OPERATORS AND DISCONTINUOUS NONLINEARITIES MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS OPERATORS,... 1 RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo L (21), pp.??? MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS

More information

Abstract. The connectivity of the efficient point set and of some proper efficient point sets in locally convex spaces is investigated.

Abstract. The connectivity of the efficient point set and of some proper efficient point sets in locally convex spaces is investigated. APPLICATIONES MATHEMATICAE 25,1 (1998), pp. 121 127 W. SONG (Harbin and Warszawa) ON THE CONNECTIVITY OF EFFICIENT POINT SETS Abstract. The connectivity of the efficient point set and of some proper efficient

More information

THE CARATHÉODORY EXTENSION THEOREM FOR VECTOR VALUED MEASURES

THE CARATHÉODORY EXTENSION THEOREM FOR VECTOR VALUED MEASURES PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 72, Number 1, October 1978 THE CARATHÉODORY EXTENSION THEOREM FOR VECTOR VALUED MEASURES JOSEPH KUPKA Abstract. This paper comprises three advertisements

More information