EXISTENCE OF THREE SOLUTIONS FOR A KIRCHHOFF-TYPE BOUNDARY-VALUE PROBLEM

Similar documents
About Some Inequalities for Isotonic Linear Functionals and Applications

Three solutions to a p(x)-laplacian problem in weighted-variable-exponent Sobolev space

Integrals and Polygamma Representations for Binomial Sums

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013

Right-indefinite half-linear Sturm Liouville problems

Qualitative Analysis for Solutions of a Class of. Nonlinear Ordinary Differential Equations

KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION

9.4 The response of equilibrium to temperature (continued)

Michael Rotkowitz 1,2

On Natural Partial Orders of IC-Abundant Semigroups

A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM

Positive Solutions of Operator Equations on Half-Line

POSITIVE SOLUTIONS FOR SINGULAR THREE-POINT BOUNDARY-VALUE PROBLEMS

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by

LYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p 1, p 2,..., p n )-LAPLACIAN

A comparison principle for nonlinear heat Rockland operators on graded groups

Multiple Positive Solutions for the System of Higher Order Two-Point Boundary Value Problems on Time Scales

Research Article Hermite-Hadamard-Type Inequalities for r-preinvex Functions

Set Integral Equations in Metric Spaces

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x

Optimization. x = 22 corresponds to local maximum by second derivative test

Electronic Companion for Optimal Design of Co-Productive Services: Interaction and Work Allocation

A General Dynamic Inequality of Opial Type

Electronic Supplementary Material

Friedmannien equations

The Banach algebra of functions of bounded variation and the pointwise Helly selection theorem

1 The Lagrange interpolation formula

Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2.

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

EXISTENCE OF ENTIRE POSITIVE SOLUTIONS FOR A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system

SOME HARDY TYPE INEQUALITIES WITH WEIGHTED FUNCTIONS VIA OPIAL TYPE INEQUALITIES

FUNCTIONS OF α-slow INCREASE

SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS

Computing the first eigenpair of the p-laplacian in annuli

Fourier-Bessel Expansions with Arbitrary Radial Boundaries

Positive solutions for system of 2n-th order Sturm Liouville boundary value problems on time scales

The Clamped Plate Equation for the Limaçon

Review of Mathematical Concepts

Hadamard and Caputo-Hadamard FDE s with Three Point Integral Boundary Conditions

Solution of fuzzy multi-objective nonlinear programming problem using interval arithmetic based alpha-cut

FI 2201 Electromagnetism

Journal of Inequalities in Pure and Applied Mathematics

defined on a domain can be expanded into the Taylor series around a point a except a singular point. Also, f( z)

AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION

A basic logarithmic inequality, and the logarithmic mean

1 2-D Second Order Equations: Separation of Variables

ITI Introduction to Computing II

MEAN VALUE PROBLEMS OF FLETT TYPE FOR A VOLTERRA OPERATOR

Measure Estimates of Nodal Sets of Polyharmonic Functions

The Formulas of Vector Calculus John Cullinan

On Certain Classes of Analytic and Univalent Functions Based on Al-Oboudi Operator

Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )

Discrete Model Parametrization

A New Generalization of Lemma Gronwall-Bellman

Topics for Review for Final Exam in Calculus 16A

Fredholm Integral Equations of the First Kind Solved by Using the Homotopy Perturbation Method

S. S. Dragomir. 2, we have the inequality. b a

On the Continuous Dependence of Solutions of Boundary Value Problems for Delay Differential Equations

Entrance Exam, Real Analysis September 1, 2009 Solve exactly 6 out of the 8 problems. Compute the following and justify your computation: lim

Research Article On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation

Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS

MULTIPLICITY OF POSITIVE SOLUTIONS FOR SECOND-ORDER DIFFERENTIAL INCLUSION SYSTEMS DEPENDING ON TWO PARAMETERS

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3

Review of Calculus, cont d

f(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all

Czechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction

S. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:

Journal of Computational and Applied Mathematics. On positive solutions for fourth-order boundary value problem with impulse

Asymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function

Keywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.

10 Statistical Distributions Solutions

Mean Curvature and Shape Operator of Slant Immersions in a Sasakian Space Form

1 Using Integration to Find Arc Lengths and Surface Areas

For a continuous function f : [a; b]! R we wish to define the Riemann integral

SUPERSTABILITY OF DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS

New problems in universal algebraic geometry illustrated by boolean equations

Variational Techniques for Sturm-Liouville Eigenvalue Problems

Problem Set 4: Solutions Math 201A: Fall 2016

1.9 C 2 inner variations

g i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f

Some Hardy Type Inequalities with Weighted Functions via Opial Type Inequalities

Lecture 10. Solution of Nonlinear Equations - II

New Integral Inequalities for n-time Differentiable Functions with Applications for pdfs

Comparative Studies of Law of Gravity and General Relativity. No.1 of Comparative Physics Series Papers

Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.

ON CLOSED CONVEX HULLS AND THEIR EXTREME POINTS. S. K. Lee and S. M. Khairnar

A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES. 1. Introduction

New Expansion and Infinite Series

SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set

NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a

Improper Integrals, and Differential Equations

Sturm-Liouville Eigenvalue problem: Let p(x) > 0, q(x) 0, r(x) 0 in I = (a, b). Here we assume b > a. Let X C 2 1

The Wave Equation I. MA 436 Kurt Bryan

Lecture 1. Functional series. Pointwise and uniform convergence.

p n m q m s m. (p q) n

D-STABLE ROBUST RELIABLE CONTROL FOR UNCERTAIN DELTA OPERATOR SYSTEMS

Deterministic simulation of a NFA with k symbol lookahead

Radial geodesics in Schwarzschild spacetime

Transcription:

Electonic Jounl of Diffeentil Eutions, Vol. 20 (20, No. 9, pp.. ISSN: 072-669. URL: http://ejde.mth.txstte.edu o http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu EXISTENCE OF THREE SOLUTIONS FOR A KIRCHHOFF-TYPE BOUNDARY-VALUE PROBLEM SHAPOUR HEIDARKHANI, GHASEM ALIZADEH AFROUZI, DONAL O REGAN Abstct. In this note, we estblish the existence of two intevls of positive el pmetes λ fo which the boundy-vlue poblem of Kichhoff-type K` Z b u (x 2 dx u = λf(x, u, u( = u(b = 0 dmits thee wek solutions whose noms e unifomly bounded with espect to λ belonging to one of the two intevls. Ou min tool is thee citicl point theoem by Bonnno.. Intoduction In the litetue mny esults focus on the existence of multiple solutions to boundy-vlue poblems. Fo exmple, cetin chemicl ections in tubul ectos cn be mthemticlly descibed by nonline two-point boundy-vlue poblem nd one is inteested if multiple stedy-sttes exist. Fo ecent tetment of chemicl ecto theoy nd multiple solutions see [, section 7] nd the efeences theein. Bonnno in [3] estblished the existence of two intevls of positive el pmetes λ fo which the functionl Φ λψ hs thee citicl points whose noms e unifomly bounded in espect to λ belonging to one of the two intevls nd he obtined multiplicity esults fo two point boundy-vlue poblem. In the pesent ppe s n ppliction, we shll illustte these esults fo Kichhoff-type poblem. Poblems of Kichhoff-type hve been widely investigted. We efe the ede to the ppes [2, 5, 7, 9, 0,, 5] nd the efeences theein. Riccei [3] estblished the existence of t lest thee wek solutions to clss of Kichhoff-type doubly eigenvlue boundy vlue poblem using [2, Theoem 2]. Conside the Kichhoff-type poblem K ( u (x 2 dx u = λf(x, u, u( = u(b = 0 (. 2000 Mthemtics Subject Clssifiction. 35J20, 35J25, 35J60. Key wods nd phses. Kichhoff-type poblem; multiple solutions; citicl point. c 20 Texs Stte Univesity - Sn Mcos. Submitted Apil 26, 20. Published July 6, 20.

2 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 whee K : [0, + [ R is continuous function, f : [, b] R R is Cthéodoy function nd λ > 0. In the pesent ppe, ou ppoch is bsed on thee citicl points theoem poved in [3], which is eclled in the next section fo the ede s convenience (Theoem 2.. Ou min esult is Theoem 2.2 which, unde suitble ssumptions, ensues the existence of two intevls Λ nd Λ 2 such tht, fo ech λ Λ Λ 2, the poblem (. dmits t lest thee clssicl solutions whose noms e unifomly bounded in espect to λ Λ 2. Let X the the Sobolev spce H0 ([, b] with the nom ( /2. u = ( u (x dx 2 We sy tht u is wek solution to (. if u X nd K ( u (x 2 dx u (xv (xdx λ f(x, u(xv(xdx = 0 fo evey v X. Fo othe bsic nottions nd definitions, we efe the ede to [4, 6, 8, 4]. 2. Results Fo the ede s convenience, dist we hee ecll [3, Theoem 2.]. Theoem 2.. Let X be sepble nd eflexive el Bnch spce, Φ : X R nonnegtive continuously Gâteux diffeentible nd seuentilly wekly lowe semicontinuous functionl whose Gâteux deivtive dmits continuous invese on X, J : X R continuously Gâteux diffeentible functionl whose Gâteux deivtive is compct. Assume tht thee exists x 0 X such tht Φ(x 0 = J(x 0 = 0 nd tht lim (Φ(x λj(x = + x + fo ll λ [0, + [. Futhe, ssume tht thee e > 0, x X such tht < Φ(x nd x Φ (],[ w J(x < + Φ(x J(x ; hee Φ (], [ w denotes the closue of Φ (], [ in the wek topology (in pticul note J(x 0 since x 0 Φ (], [ w (note J(x 0 = 0 so x Φ w J(x 0. Then, fo ech (],[ the eution Φ(x λ Λ =] J(x x Φ (],[ w J(x, x Φ (],[ w J(x[, Φ (u + λj (u = 0 (2. hs t lest thee solutions in X nd, moeove, fo ech η >, thee exist n open intevl η Λ 2 [0, J(x Φ(x x Φ (,[ w J(x] nd positive el numbe σ such tht, fo ech λ Λ 2, the eution (2.2 hs t lest thee solutions in X whose noms e less thn σ.

EJDE-20/9 EXISTENCE OF THREE SOLUTIONS 3 Let K : [0, + [ R be continuous function such tht thee exists positive numbe m with K(t m fo ll t 0, nd let f : [, b] R R be Cthéodoy function such tht ξ s f(., ξ L (, b fo ll s > 0. Coesponding to K nd f we intoduce the functions K : [0, + [ R nd F : [, b] R R, espectively s follows nd F (x, t = K(t = t Now, we stte ou min esult. 0 t 0 K(sds fo ll t 0 (2.2 f(x, sds fo ll (x, t [, b] R. (2.3 Theoem 2.2. Assume tht thee exist positive constnts nd θ, nd function w X such tht: (i K( w 2 > 2, (ii (b (b F (x, tdx < + K( w 2 2, (iii (b 2 2m lim t + F (x,t t 2 < θ unifomly with espect to x [, b]. Futhe, ssume tht thee exists continuous function h : [0, + [ R such tht h(tk(t 2 = t fo ll t 0. Then, fo ech λ in the intevl Λ =] (b (b (b F (x, tdx[, (b F (x, tdx, poblem (. dmits t lest thee wek solutions in X nd, moeove, fo ech η >, thee exist n open intevl η Λ 2 [0, R b 2 F (x,w(xdx ] b K( w 2 F (x, tdx (b (b nd positive el numbe σ such tht, fo ech λ Λ 2, poblem (. dmits t lest thee wek solutions in X whose noms e less thn σ. Let us fist give pticul conseuence of Theoem 2.2 fo fixed test function w. Coolly 2.3. Assume tht thee exist positive constnts c, d, α, β nd θ with β α < b such tht Assumption (ii in Theoem 2.2 holds, nd (i K(d 2 ( α+β 4mc2 αβ > b, (ii F (x, t 0 fo ech (x, t ([, + α] [b β, b] [0, d], (iii c,c] F (x, tdx < 2mc2 b R b β +α F (x,ddx b + 2 K(d 2 ( α+β.

4 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 Futhe, ssume tht thee exists continuous function h : [0, + [ R such tht h(tk(t 2 = t fo ll t 0. Then, fo ech λ Λ =] K(d 2 2 ( α+β β +α F (x, ddx c,c] F (x, tdx, b c,c] F (x, tdx [, poblem (. dmits t lest thee wek solutions in X nd, moeove, fo ech η >, thee exist n open intevl Λ 2 [0, ( 2ηmc 2 b ( +α 4mc 2 F (x, d α (x dx + β +α F (x, ddx + b β F (x, d β (b xdx b K(d 2 ( α+β c,c] F (x, tdx ] nd positive el numbe σ such tht, fo ech λ Λ 2, poblem (. dmits t lest thee wek solutions in X whose noms e less thn σ. Poof. We clim tht the ll the ssumptions of Theoem 2.2 e fulfilled with d α (x if x < + α, w(x = d if + α x b β, (2.4 d β (b x if b β < x b nd = /(b whee constnts c, d, α nd β e given in the sttement of the theoem. It is cle fom (2.4 tht w X nd, in pticul, one hs w 2 = d 2 ( α + β. (2.5 αβ Moeove with this choice of w nd tking into ccount (2.5, fom (i we get (i of Theoem 2.2. Since 0 w(x d fo ech x [, b], condition (ii ensues tht so fom (iii we hve +α c,c] + b β F (x, tdx < 2mc2 b 2mc2 b = 0, β +α b F (x, ddx + 2 K(d 2 ( α+β b + +, so (ii of Theoem 2.2 holds (note c 2 = (b 2m. Next notice tht (b (b F (x, tdx

EJDE-20/9 EXISTENCE OF THREE SOLUTIONS 5 nd In ddition note tht Finlly note tht K(d 2 2 ( α+β αβ b β +α F (x, ddx c,c] F (x, tdx (b = (b F (x, tdx K(d 2 2 ( α+β αβ b β +α F (x, ddx c,c] F (x, tdx < = b c,c] F (x, tdx. K(d 2 2 ( α+β ( b + K(d 2 2 ( α+β c,c] F (x, tdx b b c,c] F (x, tdx. h R b 2 F (x,w(xdx K( w 2 (b (b F (x, tdx = ( 2hmc 2 b ( +α 4mc 2 F (x, d α (x dx + β +α F (x, ddx + b β F (x, d β (b xdx b K(d 2 ( α+β c,c] F (x, tdx, nd tking into ccount tht Λ Λ we hve the desied conclusion diectly fom Theoem 2.2. It is of inteest to list some specil cses of Coolly 2.3. Coolly 2.4. Assume tht thee exist positive constnts c, d, p, p 2, α, β nd θ with β α < b such tht Assumption (ii of Coolly 2.3 holds, nd (i p d 2 ( α+β αβ + p2 2 d4 ( α+β αβ 2 > 4pc2 b, (ii c,c] F (x, tdx < 2p c 2 b (iii (b 2 F (x,t 2p lim t + t 2 Then, fo ech λ Λ =] < θ 2p c 2 b p 2 d 2 ( α+β αβ + p2 4 d4 ( α+β αβ 2 β +α F (x, ddx c,c] F (x, tdx, β F (x, ddx +α + p 2 d2 ( α+β αβ + p2 4 d4 (, α+β αβ 2 unifomly with espect to x [, b]. 2p c 2 b c,c] F (x, tdx [,

6 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 the poblem (p + p 2 u (x 2 dxu = λf(x, u, u( = u(b = 0 (2.6 dmits t lest thee wek solutions in X nd, moeove, fo ech η >, thee exist n open intevl Λ 2 [0, ( 2ηp c 2 b ( 4p c 2 b c,c] +α F (x, d α (x dx + β +α F (x, ddx + b β F (x, d β (b xdx F (x, tdx ] p d 2 ( α+β αβ + p2 2 d4 ( α+β αβ 2 nd positive el numbe σ such tht, fo ech λ Λ 2, poblem (2.6 dmits t lest thee wek solutions in X whose noms e less thn σ. Poof. Fo fixed p, p 2 > 0, set K(t = p + p 2 t fo ll t 0. Being in mind tht m = p, fom (i (iii, we see tht (i (iii of Coolly 2.4 hold espectively. Also we note tht thee exists continuous function h : [0, + [ R such tht h(tk(t 2 = t fo ll t 0 becuse the function K is nondecesing in [0, + [ with K(0 > 0 nd t tk(t 2 (t 0 is incesing nd onto [0, + [. Hence, Coolly 2.3 yields the conclusion. Coolly 2.5. Assume tht thee exist positive constnts c, d, α, β nd θ with β α < b such tht Assumption (ii in Coolly 2.3 holds, nd (i d 2 ( α+β αβ > 4c2 b, (ii c,c] F (x, tdx < (iii (b 2 F (x,t 2 lim t + t < 2 θ Then, fo ech λ Λ =] the poblem d 2 2 ( α+β αβ β 2c2 F (x, ddx +α b 2c 2 b + d2 2 ( α+β αβ, β +α F (x, ddx c,c] F (x, tdx, unifomly with espect to x [, b]. u = λf(x, u, u( = u(b = 0 2c 2 b c,c] F (x, tdx [, (2.7 dmits t lest thee wek solutions in X nd, moeove, fo ech η >, thee exist n open intevl Λ 2 [0, ( 2ηc 2 b ( +α 4c 2 F (x, d α (x dx + β +α F (x, ddx + b β F (x, d β (b xdx b d 2 ( α+β αβ

EJDE-20/9 EXISTENCE OF THREE SOLUTIONS 7 c,c] F (x, tdx ] nd positive el numbe σ such tht, fo ech λ Λ 2, the poblem (2.7 dmits t lest thee wek solutions in X whose noms e less thn σ. We conclude this section by giving n exmple to illustte ou esults pplying by Coolly 2.4. Exmple 2.6. Conside the poblem ( 28 + 64 0 u (x 2 dxu = λ(e u u (2 u, u(0 = u( = 0 (2.8 whee λ > 0. Set p = 28, p 2 = 64 nd f(x, t = e t t (2 t fo ll (x, t [0, ] R. A diect clcultion yields F (x, t = e t t 2 fo ll (x, t [0, ] R. Assumptions (i nd (ii of Coolly 2.4 e stisfied by choosing, fo exmple F (x,t d = 2, c =, [, b] = [0, ] nd α = β = /4. Also, since lim t + t = 0, 2 Assumption (iii of Coolly 2.4 is fulfilled. Now we cn pply Coolly 2.4. Then, fo ech λ Λ 33 =] 2 4 e 2 8e, 64e [ poblem (2.8 dmits t lest thee wek solutions in H0 ([0, ] nd, moeove, fo ech η >, thee exist n open intevl Λ 2 [0, ( 8 33 8 2 4 0 e 8t t 2 dt + 2 e 2 + 8 2 3 η 4 ] e 8( t ( t 2 dt 64e nd positive el numbe σ such tht, fo ech λ Λ 2, poblem (2.8 dmits t lest thee wek solutions in H 0 ([0, ] whose noms e less thn σ. We begin by setting 3. Poof of Theoem 2.2 Φ(u = 2 K( u 2, (3. J(u = F (x, u(xdx (3.2 fo ech u X, whee K nd F e given in (2.2 nd (2.3, espectively. It is well known tht J is Gâteux diffeentible functionl whose Gâteux deivtive t the point u X is the functionl J (u X, given by J (uv = f(x, u(xv(xdx fo evey v X, nd tht J : X X is continuous nd compct opeto. Moeove, Φ is continuously Gâteux diffeentible nd seuentilly wekly lowe semi continuous functionl whose Gâteux deivtive t the point u X is the functionl Φ (u X, given by Φ (uv = K( u (x 2 dx u (xv (xdx

8 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 fo evey v X. We clim tht Φ dmits continuous invese on X (we identity X with X. To pove this fct, guing s in [3] we need to find continuous opeto T : X X such tht T (Φ (u = u fo ll u X. Let T : X X be the opeto defined by { h( v v T (v = v if v 0 0 if v = 0, whee h is defined in the sttement of Theoem 2.2. Since, h is continuous nd h(0 = 0, we hve tht the opeto T is continuous in X. Fo evey u X, tking into ccount tht inf t 0 K(t m > 0, we hve since h(t K(t 2 = t fo ll t 0 tht T (Φ (u = T (K( u 2 u = h(k( u 2 u K( u 2 u K( u 2 u = u K( u 2 u K( u 2 u = u, so ou clim is tue. Moeove, since m K(s fo ll s [0, + [, fom (3. we hve Φ(u m 2 u 2 fo ll u X. (3.3 Futhemoe fom (iii, thee exist two constnts γ, τ R with 0 < γ < /θ such tht (b 2 2m F (x, t γt2 + τ fo ll x (, b nd ll t R. Fix u X. Then F (x, u(x 2m (b 2 (γ u(x 2 + τ fo ll x (, b. (3.4 Fix λ ]0, + [. Then thee exists θ > 0 with λ ]0, θ]. Now since mx x [,b] fom (3.3, (3.4 nd (3.5, we hve Φ(u λj(u = 2 K( u 2 λ m 2 u 2 (b /2 u(x u, (3.5 2 2θm ( (b 2 γ F (x, u(xdx u(x 2 + τ(b m 2 u 2 2θm ( (b 2 (b 2 γ u 2 + τ(b 4 = m 2 ( γθ u 2 2θτm b, nd so lim (Φ(u λj(u = +. u + Also fom (3. nd (i we hve Φ(w >. Using (3.3 nd (3.5, we obtin Φ (], [ = { u X; Φ(u < } { u X; u < 2/m }

EJDE-20/9 EXISTENCE OF THREE SOLUTIONS 9 so, we hve { u X; u(x (b /(2m, fo ll x [, b] }, u Φ (],[ w J(u Theefoe, fom (ii, we hve u Φ (],[ w J(u (b (b F (x, tdx. (b (b F (x, tdx < + K( w 2 2 = + Φ(w J(w. Now, we cn pply Theoem 2.. Note fo ech x [, b], u Φ (],[ w J(u nd Φ(w J(w u Φ (],[ w J(u Note lso tht (ii immeditely implies Also < = J(w Note fom (ii tht ( + η (b (b (b (b (b (b F (x, tdx. (b F (x, tdx (b F (x, tdx. (b F (x, tdx (b F (x, tdx Φ(w u Φ (,[ w J(u η R b 2 F (x,w(xdx = ρ. b K( w 2 F (x, tdx (b (b 2 K( w 2 (b F (x, tdx (b

0 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 ( 2 > K( w 2 + K( w 2 2 ( 2 K( w 2 2 = 0 K( w 2 since 0 (note F (x, 0 = 0 so (b (b F (x, tdx 0 nd now pply (ii. Now with x 0 = 0, x = w fom Theoem 2. (note J(0 = 0 fom (2.3 it follows tht, fo ech λ Λ, the poblem (. dmits t lest thee wek solutions nd thee exist n open intevl Λ 2 [0, ρ] nd el positive numbe σ such tht, fo ech λ Λ 2, the poblem (. dmits t lest thee wek solutions tht whose noms in X e less thn σ. Refeences [] R. P. Agwl, H. B. Thompson nd C.C. Tisdell; On the existence of multiple solutions to boundy vlue poblems fo second ode, odiny diffeentil eutions. Dynm. Systems Appl. 6 (2007 595-609. [2] C. O. Alves, F. S. J. A. Coê nd T. F. M; Positive solutions fo usiline elliptic eutions of Kichhoff type, Comput. Mth. Appl., 49 (2005, 85-93. [3] G. Bonnno; A citicl points theoem nd nonline diffeentil poblems, J. Globl Optimiztion, 28 (2004 249-258. [4] G. Bonnno, G. Molic Bisci nd V. Rădulescu; Infinitely mny solutions fo clss of nonline eigenvlue poblem in Olicz-Sobolev spces, C. R. Acd. Sci. Pis, Se. I 349 (20 263-268. [5] M. Chipot nd B. Lovt; Some emks on non locl elliptic nd pbolic poblems, Nonline Anl., 30 (997, 469-4627. [6] M. Ghegu nd V. Rădulescu; Singul Elliptic Poblems. Bifuction nd Asymptotic Anlysis, Oxfod Lectue Seies in Mthemtics nd Its Applictions, vol. 37, Oxfod Univesity Pess, 2008. [7] X. He nd W. Zou; Infinitely mny positive solutions fo Kichhoff-type poblems, Nonline Anl. 70 (2009 407-44. [8] A. Kistály, V. Rădulescu nd C. Vg; Vitionl Pinciples in Mthemticl Physics, Geomety, nd Economics: Qulittive Anlysis of Nonline Eutions nd Uniltel Poblems, Encyclopedi of Mthemtics nd its Applictions, No. 36, Cmbidge Univesity Pess, Cmbidge, 200. [9] T. F. M; Remks on n elliptic eution of Kichhoff type, Nonline Anl., 63 (2005, e957-e977. [0] A. Mo nd Z. Zhng; Sign-chnging nd multiple solutions of Kichhoff type poblems without the P. S. condition, Nonline Anl. 70 (2009 275-287. [] K. Pee nd Z. T. Zhng; Nontivil solutions of Kichhoff-type poblems vi the Yng index, J. Diffeentil Eutions, 22 (2006, 246-255. [2] B. Riccei; A futhe thee citicl points theoem, Nonline Anl. 7 (2009 45-457. [3] B. Riccei; On n elliptic Kichhoff-type poblem depending on two pmetes, J. Globl Optimiztion 46 (200 543-549. [4] E. Zeidle; Nonline functionl nlysis nd its pplictions, Vol. II, III. Belin-Heidelbeg- New Yok 985. [5] Z. T. Zhng nd K. Pee; Sign chnging solutions of Kichhoff type poblems vi invint sets of descent flow, J. Mth. Anl. Appl., 37 (2006 456-463. Shpou Heidkhni Deptment of Mthemtics, Fculty of Sciences, Rzi Univesity, 6749 Kemnshh, In E-mil ddess: s.heidkhni@zi.c.i

EJDE-20/9 EXISTENCE OF THREE SOLUTIONS Ghsem Alizdeh Afouzi Deptment of Mthemtics, Fculty of Bsic Sciences, Univesity of Mzndn, 4746-467 Bbols, In E-mil ddess: fouzi@umz.c.i Donl O Regn Deptment of Mthemtics, Ntionl Univesity of Ielnd, Glwy, Ielnd E-mil ddess: donl.oegn@nuiglwy.ie

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback