THREE POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY VALUE PROBLEM FOR THE p-laplacian DYNAMIC EQUATION ON TIME SCALES 1.

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1 Commun. Oim. Theory 218 (218, Aricle ID 13 hs://doi.org/ /co THREE POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY VALUE PROBLEM FOR THE -LAPLACIAN DYNAMIC EQUATION ON TIME SCALES ABDULKADIR DOGAN Dearmen of Alied Mahemaics, Faculy of Comuer Sciences, Abdullah Gul Universiy, Kayseri 3839, Turkey Absrac. In his aer, we sudy a hree-oin boundary value roblem for -Lalacian dynamic equaions on ime scales. By using he Avery and Peerson fixed oin heorem, we rove he exisence a leas hree osiive soluions of he boundary value roblem. The ineresing oin is ha he non-linear erm f involves a firs-order derivaive exlicily. An examle is also given o illusrae our resuls. Keywords. Boundary value roblem; Time scales; Posiive soluion; Fixed oin heorem. 21 Mahemaics Subjec Classificaion. 34B15, 34B INTRODUCTION In his aer, we are concerned wih he exisence of osiive soluions of he -Lalacian dynamic equaion on ime scales or (φ (y ( = w( f (,y(,y (, [,T ] T, (1.1 y( B (y ( =, y (T =, (1.2 y ( =, y(t + B 1 (y ( =, (1.3 where φ (y is a -Lalacian oeraor, i.e., φ (y = y 2 y, for > 1, wih (φ = φ q and 1/+1/q = 1. For general basic ideas and background abou dynamic equaions on ime scales we refer he reader o he books [1, 2]. As we know, when he nonlinear erm f is involved in he firs-order derivaive, difficulies arise immediaely. In his work, we use a fixed oin heorem due o Avery and Peerson o overcome he difficulies. Throughou he aer, our invesigaion is under he following assumions: (H1 T is a ime scales wih,t T, (,T T ; (H2 Le ξ min T : T, and here exiss τ T such ha ξ < τ < T holds; 2 address: abdulkadir.dogan@agu.edu.r. Received November 29, 217; Acceed Aril 19, 218. c 218 Communicaions in Oimizaion Theory 1

2 2 ABDULKADIR DOGAN (H3 f : [,T ] T R + R R + is coninuous, and does no vanish idenically on any closed subinerval of [,T ] T ; (H4 w : T R + is lef dense coninuous (i.e. w C ld (T,R +, and does no vanish idenically on any closed subinerval of [,T ] T ; (H5 B (υ and B 1 (υ are boh coninuous odd funcions defined on R and saisfy ha here exis A,B > such ha Bυ B j (υ Aυ, υ, j =,1. In [3], Anderson esablished he exisence of mulile osiive soluions o he nonlinear second-order hree-oin boundary value roblem on ime scale T given by u ( + f (,u( =, u( =, au(η = u(t. (,T T He emloyed he Legge -Williams fixed oin heorem in an aroriae cone o guaranee he exisence of a leas hree osiive soluions o his nonlinear roblem. Anderson, Avery and Henderson [4] sudied he ime-scale, dela-nabla dynamic eqauion, wih boundary condiions, (g(u + c( f (u = for a < < b, u(a B (u ( = and u (b =. They esablished he exisence resul of a leas one osiive soluion by a fixed oin heorem of cone exansion and comression of funcional ye. In [5], He invesigaed he exisence of osiive soluions of he -Lalacian dynamic equaion on a ime scale saisfying he boundary condiions or [φ (u (] + a( f (u( =, [,T ] T, u( B (u (η =, u (T =, u ( = u(t + B 1 (u (η =, where φ (s is a -Lalacian oeraor, i.e., φ (s = s 2 s, > 1, (φ = φ q, 1/ + 1/q = 1, η (,ρ(t T. By using a new double fixed-oin heorem due o Avery, Chyan and Henderson [6] in a cone, he roved ha here exis a leas double osiive soluions of boundary value roblem. In [7], Sun and Li sudied he one-dimonsional -Lalacian boundary value roblem on ime scales (ϕ (u ( + h( f (u σ ( =, [a,b], u(a B (u (a =, u (σ(b =, where ϕ (u is a -Lalacian oeraor, i.e.ϕ (u = u 2 u, > 1.They found some new resuls for he exisence of a leas single, win or rile osiive soluions of he above roblem by using Krasnosel skii s fixed oin heorem, new fixed oin heorem due o Avery and Henderson and Legge-Williams fixed oin heorem.

3 A THREE-POINT BOUNDARY VALUE PROBLEM FOR THE -LAPLACIAN DYNAMIC EQUATION 3 Sun, Tang and Wang [8] considered he eigenvalue roblem for he following one-dimensional - Lalacian hree-oin boundary value roblem on ime scales (ϕ (u ( + λh( f (u( =, (,T T, u( βu ( = γu (η u (T =. They esablished some sufficien condiions for he nonexisence and exisence of a leas one or wo osiive soluions for he boundary value roblem. In [9], Wang sudied he exisence of hree osiive soluions o he following boundary value roblems for -Lalacian dynamic equaions on ime scales: [φ (u (] + a( f (u( =, [,T ] T, u ( = u(t + B 1 (u (η =, or u( B (u (η =, u (T =. He esablished he exisence resul for a leas hree osiive soluions by using he Legge-Williams fixed oin heorem. Dynamic equaion on ime scales is a fairly new oic. In recen years, here has been much curren aenion focused on sudy of osiive soluions of boundary value roblems (BVPs on ime scales. When he nonlinear erm f does no deend on he firs-order derivaive, nonlinear BVPs on ime scales have been sudied exensively in he lieraure; see [3, 4, 5, 7, 8, 9, 1, 11, 12, 13, 14, 15, 16, 17, 18] and he references. However, here are few aers dealing wih he exisence of osiive soluions for BVPs on ime scales when he nonlinear erm f is involved in he firs-order derivaive exlicily; see [19, 2]. In his aer, moivaed by he above resuls, we show ha BVP (1.1, (1.2 and (1.3 has a leas hree osiive soluions by alying he he fixed oin heorem due o Avery and Peerson. The highligh is ha he nonlinear erm f is involved wih he firs-order derivaive exlicily. Our resuls are new for he secial cases of difference equaions and differenial equaions as well as in he general ime scale seing. The res of aer is arranged as follows. In Secion 2, we sae some definiions, noaions, lemmas and rove several reliminary resuls. In Secion 3 and 4, by defining an aroriae Banach sace and cones, we imose he growh condiions on f which allow us o aly he fixed oin heorem in finding exisence of hree osiive soluions of (1.1, (1.2 and (1.1, (1.3. In he las secion, we give an alicaion o demonsrae our resuls. 2. PRELIMINARIES Definiion 2.1. Le E be a real Banach sace. A nonemy, closed, convex se P E is said o be a cone if i saisfies he following wo condiions: (i x P,λ = λx P; (ii x P, x P = x =. Every cone P E induces an ordering in E given by x y if and only if y x P.

4 4 ABDULKADIR DOGAN Le γ and θ be nonnegaive coninuous convex funcionals on P. Le α be a nonnegaive coninuous concave funcional on P, and le ψ be a nonnegaive coninuous funcional on P. For osiive real numbers a,b,c and d, we define he following ses: P(γ,d = x P : γ(x < d, P(γ,α,b,d = x P : b α(x,γ(x d, P(γ,θ,α,b,c,d = x P : b α(x,θ(x c,γ(x d, R(γ,ψ,a,d = x P : a ψ(x,γ(x d. Le E = C 1 ld ([,T ] T R be a Banach sace wih he norm y = max su y(, su y ( [,T ] T [,T ] T and define he cone P E by P = y E : y(, [,T ] T ; y (, y (, [,T ] T, y (T =. Funcion y( is a soluion o (1.1 and (1.2 if and only if i saisfies he inegral equaion y( = φ w(r f (r,y(r,y (r r s s ( +B φ w(r f (r,y(r,y (r r, [,T ] T. We define he inegral oeraor F : P E as follows (Fy( = φ w(r f (r,y(r,y (r r s s ( +B φ w(r f (r,y(r,y (r r, y P. (2.1 Lemma 2.2. If y P, hen (i y( T y(t for [,T ] T; (ii y(s sy( for,s [,T ] T, wih s. Proof. (i Since y (, we find ha y ( is nonincreasing. Thus, for < < T, y( y( = y (s s y ( and T y(t y( = y (s s (T y ( which yield ha y(t + (T y( y( y(t. T T (ii If = s, hen he conclusion is saisfied. Noe ha y( is concave, nonnegaive on [,T ] T and y (T =. If s <, hen we ge y( y( y(s y(. s,

5 A THREE-POINT BOUNDARY VALUE PROBLEM FOR THE -LAPLACIAN DYNAMIC EQUATION 5 I follows ha This comlees he roof. y(s sy( + ( sy( sy(. Lemma 2.3. If y P, here exiss a osiive real number M such ha where M = max 1,(A + T. su y( M su y (, [,T ] T [,T ] T Proof. In view of y( = y( + y ( and y ( y (T =, we ge From Lemma 2.2, we have y(t = su y( = [,T ] T su [,T ] T (A + T su [,T ] T y (. B (y ( + y ( This comlees he roof. su y( = y(t (A + T su y (. [,T ] T [,T ] T Lemma 2.4. F : P P is comleely coninuous. Proof. The roof is divided ino ses. Se 1. We verify ha F : P P. From (H3, i is obvious ha (Fy( for [,T ] T. I is easy o find ha (Fy ( = φ w(r f (r,y(r,y (r r and (Fy (T =. We can see ha (Fy ( is coninuous and nonincreasing on [,T ] T. w(r f (r,y(r,y (r r = w( f (,y(,y (, [,T ] T. In addiion, φ (y is a monoone increasing coninuously differeniable funcion for y >. If T w(r f (r,y(r,y (r r > for [,T ] T, hen (Fy ( for [,T ] T. If for [,T ] T, hen (Fy ( = for [,T ] T. T w(r f (r,y(r,y (r r = Se 2. We rove ha F mas a bounded se ino iself. Suose ha c > is a consan and y P c = u P : y = max su y(, su y ( c. [,T ] T [,T ] T Since f (,y,v is coninuous, here exiss a consan C > such ha f (,y,v φ (C for (,y,v [,T ] T [,c] [,c]. We can see ha φ w(r f (r,y(r,y (r r < +, (2.2

6 6 ABDULKADIR DOGAN and φ w(r f (r,y(r,y (r r s s ( + B φ w(r f (r,y(r,y (r r < +. (2.3 Se 3. Leing 1, 2 [,T ] T and 1 < 2, we have 2 (Fy( 1 (Fy( 2 = φ φ C 1 2 φ w(r f (r,y(r,y (r r s s w(r f (r,y(r,y (r r s w(r r. By alying he Arzela-Ascoli heorem on ime scales [21], we immediaely find ha FP c is relaively comac. Se 4. We claim ha F : P c P is coninuous. If y n n=1 P c and lim n y n y, hen lim y n y and lim y n n n y. Since (Fy n ( n=1 is uniformly bounded and equiconinuous on [,T ] T, here exiss a uniformly convergen subsequence in (Fy n ( n=1. Le (Fy n(m( m=1 be a subsequence which converges o g( uniformly on [,T ] T. We examine ha (Fy n ( = φ w(r f (r,y n (r,y n (r r s s ( +B φ w(r f (r,y n (r,y n (r r. From (2.2 and (2.3, uing y n(m ino he above and hen aking m, we find ha g( = φ w(r f (r,y (r,y (r r s s ( +B φ w(r f (r,y (r,y (r r. Using he definiion of F, we see ha g( = Fy ( on [,T ] T. This shows ha each subsequence of (Fy n ( n=1 uniformly converges o (Fy (. So he sequence (Fy n ( n=1 uniformly converges o (Fy (. This imlies ha F is coninuous a y P c. Since y is arbirary, we find ha F is coninuous on P c. The following fixed oin heorem due o Avery and Peerson lays an imoran role in he roof of our main resuls. Theorem 2.5. ([22]. Le P be a cone in a real Banach sace E. Le γ and θ be nonnegaive coninuous convex funcionals on P Le α be a nonnegaive coninuous concave funcional on P and le ψ be a

7 A THREE-POINT BOUNDARY VALUE PROBLEM FOR THE -LAPLACIAN DYNAMIC EQUATION 7 nonnegaive coninuous funcional on P saisfying ψ(λy λψ(y for λ 1 such ha, for some osiive numbers h and d, α(y ψ(y and y hγ(y for all y P(γ, d. Suose ha F : P(γ, d P(γ, d is comleely coninuous and here exis osiive real numbers a,b and c wih a < b such ha (S1 y P(γ,θ,α,b,c,d : α(y > b = /, and α(fy > b for y P(γ,θ,α,b,c,d; (S2 α(fy > b, for y P(γ,α,b,d wih θ(fy > c; (S3 / R(γ,ψ,a,d and ψ(fy < a for all y R(γ,ψ,a,d wih ψ(y = a. Then F has a leas hree fixed oins y 1,y 2,y 3 P(γ,d such ha γ(y i d for i = 1,2,3; b < α(y 1 ; a < ψ(y 2 wih α(y 2 < b; ψ(y 3 < a. 3. SOLUTIONS OF (1.1 AND (1.2 IN A CONE Le ξ T be such ha < < ξ < T, and define he nonnegaive coninuous convex funcionals γ and θ, nonnegaive coninuous concave funcional α, and nonnegaive coninuous funcional ψ, resecively, on P by In view of Lemma 2.3, we find α(y = inf [ξ,t ] T y( = y(ξ, ψ(y = inf [ξ,t ] T y( = y(ξ, γ(y = su [,T ] T y ( = y (, θ(y = su [τ,t ] T y ( = y (τ. su y( M su y ( = Mγ(y for all y P. [,T ] T [,T ] T We also see ha θ(λ y = λ θ(y, for λ [, 1]. For noaional convenience, we define β, and δ by β = φ w(r r, = (B + ξ φ w(r r, δ = (A + ξ φ ξ w(r r. Nex, we rove he following exisence resuls. Theorem 3.1. Assume ha here exis consans a,b,d such ha < a < ξ T b < ξ d, βξ >, and T β suose ha f saisfies he following condiions: ( d (A1 f (,y,v φ for (,y,v [,T ] T [,Md] [ d,d]; β

8 8 ABDULKADIR DOGAN ( b (A2 f (,y,v > φ for (,y,v [ξ,t ] T [b,md] [ d,d]; ( a (A3 f (,y,v < φ for (,y,v [,T ] T [, T a] [ d,d]. δ ξ Then, here exis a leas hree osiive soluions y 1,y 2 and y 3 of (1.1 and (1.2 such ha y i d for i = 1,2,3, b < y 1 (ξ, a < u 2 (ξ and y 2 (ξ < b wih y 3 (ξ < a. (3.1 Proof. BVP (1.1 and (1.2 has a soluion y = y( if and only if y solves he oeraor equaion y = Fy. Thus we se ou o verify ha oeraor F saisfies he Avery and Peerson s fixed oin heorem which will be used o rove he exisence of hree fixed oins of F. Firs, we asser ha If y P(γ,d, hen F : P(γ,d P(γ,d. (3.2 γ(y = su y ( d. [,T ] T From Lemma 2.3, we obain ha su [,T ]T y( Md. From (A1, we have f (,y,v φ ( d β. Hence, γ(fy = su (Fy ( [,T ] T = φ w(r f (r,y(r,y (r r φ w(r r d β = d. Second, we rove ha condiion (S1 in Theorem 2.5 holds. Le Then α(y = 2b > b, θ(y = βb and γ(y = βb y = βb βb ξ + 2b. < d, ha is, ( y P γ,θ,α,b, βb,d : α(y > b /. On he oher hand, for any ( y P γ,θ,α,b, βb,d : α(y > b,

9 A THREE-POINT BOUNDARY VALUE PROBLEM FOR THE -LAPLACIAN DYNAMIC EQUATION 9 we find from Lemma 2.3 ha b y( Md, d y ( d, and [ξ,t ] T. Using (A2, we see ha α(fy = Fy(ξ = ξ φ +B ( φ > (B + ξ φ = (B + ξ φ s w(r f (r,y(r,y (r r s ξ ξ w(r f (r,y(r,y (r r ( b w(rφ w(r r b = b. ( Therefore, we have α(y > b, for all y P γ,θ,α,b, βb,d. Then r Third, we now asser ha (S2 of Theorem 2.5 is saisfied. Le u P(γ,α,b,d wih θ(fy > βb. I follows ha θ(fy = (Fy (τ = φ w(r f (r,y(r,y (r r > βb τ. α(fy = inf (Fy( [ξ,t ] T ξ φ s ( +B φ w(r f (r,y(r,y (r r s w(r f (r,y(r,y (r r > ξ φ w(r f (r,y(r,y (r r τ > ξ βb > b. Hence, condiion (S2 in Theorem 2.5 is saisfied. Finally, we rove ha (S3 in Theorem 2.5 is saisfied. Since ψ( = < a, we have / R(γ,ψ,a,d. Indeed, suose ha y R(γ, ψ, a, d wih In view of Lemma 2.2, we have ψ(y = inf (y( = y(ξ = a. [ξ,t ] T su y( = y(t T [,T ] T ξ y(ξ = T ξ a. Furhermore, we have y( T ξ a, [,T ] T.

10 1 ABDULKADIR DOGAN Moreover, one has γ(y = su y ( d. [,T ] T Hence, by he condiion (A3 of his heorem, we see ha α(fy = inf (Fy( [ξ,t ] T ξ φ ( +A φ ( a < (A + ξ φ w(rφ δ = (A + ξ φ = a. w(r f (r,y(r,y (r r s w(r f (r,y(r,y (r r w(r r a δ r As a resul, all he condiions of Theorem 2.5 are saisfied. This comlees he roof. 4. SOLUTIONS OF (1.1 AND (1.3 IN A CONE Le E = Cld 1 ([,T ] T R be a Banach sace wih he norm y = max su y(, [,T ] T su y ( [,T ] T and define he cone P 1 E by P 1 = y E : y(, [,T ] T ; y (, y (, [,T ] T, y ( =. Fix τ T such ha < τ <, and define he nonnegaive coninuous convex funcionals γ and θ, nonnegaive coninuous concave funcional α, and nonnegaive coninuous funcional ψ, resecively, on P 1 by, Se α(y = inf [τ,t ] T y( = y(t, ψ(y = inf [τ,t ] T y( = y(t, γ(y = su [,T ] T y ( = y (T, θ(y = su [τ,t ] T y ( = y (T. β 1 = φ ( τ 1 = Bφ δ 1 = Aφ w(r r, w(r r, w(r r.

11 A THREE-POINT BOUNDARY VALUE PROBLEM FOR THE -LAPLACIAN DYNAMIC EQUATION 11 Noe ha y( is a soluion of (1.1 and (1.3 if and only if T ( s y( = φ w(r f (r,y(r,y (r r s ( ( +B 1 φ w(r f (r,y(r,y (r r, [,T ] T. Theorem 4.1. Assume ha condiions (H1-(H5 are saisfied. Le < a < ξ T b < ξ 1 d, β 1 ξ > 1, and T β 1 suose ha f saisfies he following condiions: ( d (C1 f (,y,v φ for (,y,v [,T ] T [,Md] [ d,d]; β ( 1 b (C2 f (,y,v > φ for (,y,v [ξ,t ] T [b,md] [ d,d]; ( 1 a (C3 f (,y,v < φ for (,y,v [,T ] T [, T a] [ d,d]. δ 1 ξ Then BVP (1.1 and (1.3 has a leas hree osiive soluions y 1,y 2 and y 3, such ha y i d for i = 1,2,3, b < y 1 (ξ, a < y 2 (ξ and y 2 (ξ < b wih y 3 (ξ < a. (4.1 Proof. We define a comleely coninuous inegral oeraor F 1 : P 1 E by T ( s (F 1 y( = φ w(r f (r,y(r,y (r r s ( ( +B 1 φ w(r f (r,y(r,y (r r, y P 1, (4.2 for [,T ] T, and each fixed oin of F 1 in he cone P 1 is a osiive soluion of (1.1 and (1.3. Noe ha, from (4.2, if u P 1, hen (F 1 u( for [,T ] T, and ( (F 1 y ( = φ w(r f (r,y(r,y (r r, y P 1, [,T ] T. We can see ha (F 1 y ( is coninuous and nonincreasing on [,T ] T. Using Theorem 8.39 in [1], we find ha (F 1 y ( for [,T ] T. In addiion, (F 1 y ( =. This imlies F 1 y P 1, and so F : P 1 P 1. Wih an analogue o he roof of Theorem 3.1, we arrive a he conclusion immediaely. 5. AN APPLICATION ( 1 N Le T = 2, 18 3, 14, 16, 12,1, 54, 32, 74,2, where N denoes he se of all nonnegaive inegers. We consider he -Lalacian dynamic equaion saisfying he boundary condiions y( B ( where w( = + ρ( (φ (y ( = w( f (,y(,y (, [,2] T, (5.1 y ( 1 2 =, y (2 =, (5.2

12 12 ABDULKADIR DOGAN and v, f (,y,v = + v + (y, v, for (,y,v [,2] T [,4] [ 6,6]; for (,y,v [,2] T [4,4.1] [ 6,6]; for (,y,v [,2] T [4.1,2] [ 6,6]. Take T = 2, = 7, = 1 2,τ = 3 2. Here (y saisfies (4 = 1, (4.1 = 1586, (y : R R+ is coninuous and (y >. If we choose ξ = 1,A = B = 2 β = 6 w(r r 1.26, = δ = ( w(r r 1.261, 1 1 ( w(r r ,a = 2, b = 4.1, d = 6, hen we find 1 I can be readily seen ha < a < ξ T b < ξ d, βξ >, and f (,u,v saisfies ha T β ( d ( 6 6 f (,y,v < φ = , for 2, y 1.1, v 6, β 1.26 ( b ( f (,y,v > φ = , for 1 2, 4.1 y 1.1, v 6, ( a ( 2 6 f (,y,v < φ = , for 2, y 4, v 6. δ Therefore by Theorem 3.1, BVP (5.1 and (5.2 has a leas hree osiive soluions y 1,y 2,y 3 saisfying y i 6 for i = 1,2,3, 4.1 < y 1 (1, 2 < y 2 (1 and y 2 (1 < 4.1 wih y 3 (1 < 2. REFERENCES [1] M. Bohner, A. Peerson, Dynamic Equaions on Time Scales: An Inroducion wih Alicaions, Birkhauser, Boson, Cambridge, MA, 21. [2] M. Bohner, A. Peerson, Advances in Dynamic Equaions on Time Scales, Birkhauser, Boson, Cambridge, MA, 23. [3] D. Anderson, Soluions o second-order hree-oin roblems on ime scales, J. Difference Equ. Al. 8 (22, [4] D.R. Anderson, R. Avery, J. Henderson, Exisence of soluions for a one-dimensional -Lalacian on ime scales, J. Difference Equ. Al. 1 (24, [5] Z. He, Double osiive soluions of hree-oin boundary value roblems for -Lalacian dynamic equaions on ime scales, J. Comu. Al. Mah. 182 (25, [6] R. I. Avery, C. J. Chyan, J. Henderson, Twin soluions of a boundary value roblems for ordinary differenial equaions and finie difference equaions, Comu. Mah. Al. 42 (21, [7] H.R. Sun, W.T. Li, Posiive soluions for nonlinear hree-oin boundary value roblems on ime scales, J. Mah. Anal. Al. 299 (24, [8] H.R. Sun, L.T. Tang, Y.H. Wang, Eigenvalue roblem for -Lalacian hree-oin boundary value roblems on ime scales, J. Mah. Anal. Al. 331 (27, [9] D. B. Wang, Three osiive soluions of hree-oin boundary value roblems for -Lalacian dynamic equaions on ime scales, Nonlinear Anal. 68 (28,

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