Impulsive first-order functional q k -integro-difference inclusions with boundary conditions

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1 Avalable onlne at J. Nonlnear Sc. Appl. 9 (2016, Research Artcle Impulsve frst-order functonal q k -ntegro-dfference nclusons wth boundary condtons Jessada Tarboon a,, Sotrs K. Ntouyas b, Weerawat Sudsutad a a Nonlnear Dynamc Analyss Research Center, Department of Mathematcs, Faculty of Appled Scence, Kng Mongkut s Unversty of Technology North Bangkok, Bangkok, Thaland. b Department of Mathematcs, Unversty of Ioannna, Ioannna, Greece; Nonlnear Analyss and Appled Mathematcs (NAAM-Research Group, Department of Mathematcs, Faculty of Scence, Kng AbdulazzUnversty, P.O. Box 80203, Jeddah 21589, Saud Araba. Communcated by P. Kumam Abstract In ths paper, we dscuss the exstence of solutons for a frst order boundary value problem for mpulsve functonal q k -ntegro-dfference nclusons. Some new exstence results are obtaned for convex as well as non-convex multvalued maps wth the ad of some classcal fxed pont theorems. Illustratve examples are also presented. c 2016 All rghts reserved. Keywords: q k -dervatve, q k -ntegral, mpulsve q k -dfference nclusons, exstence, fxed pont theorem MSC: 34A60, 26A33, 39A13, 34A Introducton In ths paper, we nvestgate the exstence of solutons for a boundary value problem of mpulsve functonal q k -ntegro-dfference nclusons of the form: D qk x(t F (t, x(t, x(θ(t, (K qk x(t, t J := [0, T ], t, x( = I k (x(, k = 1, 2,..., m, (1.1 αx(0 = βx(t γ x(sd q s, Correspondng author Emal addresses: jessada.t@sc.kmutnb.ac.th (Jessada Tarboon, sntouyas@uo.gr (Sotrs K. Ntouyas, wrw.sst@gmal.com (Weerawat Sudsutad Receved

2 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, where 0 = t 0 < t 1 < t 2 < < < < t m < t m1 = T, f : J R 3 R, θ : J J, (K qk x(t = φ(t, sx(sd qk s, k = 0, 1, 2,..., m, (1.2 φ : J 2 [0, s a contnuous functon, I k C(R, R, x( = x(t k x( for k = 1, 2,..., m, x(t k = lm x(t h 0 k h, α, β, γ, = 0, 1,..., m are real constants, 0 < q k < 1 for k = 0, 1, 2,..., m and φ 0 = sup φ(t, s. (t,s J 2 Fractonal dfferental equatons have been shown to be very useful n the study of models of many phenomena n varous felds of scence and engneerng, such as physcs, chemstry, bology, sgnal and mage processng, bophyscs, blood flow phenomena, control theory, economcs, aerodynamcs and fttng of expermental data. For examples and recent developments of the topc, see ([1, 4, 6, 8, 9, 11, 12, 23, 27, 28, 29, 30] and the references cted theren. The book of Kac and Cheung [22] covers many of the fundamental aspects of quantum calculus. In recent years, the topc of q-calculus has attracted the attenton of several researchers and a varety of new results can be found n the papers [2, 3, 5, 7, 10, 18, 34] and the references cted theren. On the other hand, for some monographs on mpulsve dfferental equatons we refer the reader to [13, 25, 31]. The notons of q k -dervatve and q k -ntegral on fnte ntervals were ntroduced recently by the authors n [32]. Ther basc propertes were studed and, as applcatons, exstence and unqueness results were proved for ntal value problems for frst and second order mpulsve q k -dfference equatons. Recently, n [33], the problem (1.1 was consdered n the case when F = {f}, thas, when F s sngle-valued. Exstence and unqueness results were obtaned by usng Banach s contracton prncple, Krasnoselsk s fxed pont theorem and Leray-Schauder degree theory. In ths paper we contnue the study on ths new subject to cover the mult-valued boundary value problem (1.1. We establsh some exstence results for the problem (1.1, when the rght hand sde s convex as well as non-convex valued. In the frst result we use the nonlnear alternatve of Leray-Schauder type, and n the second result we combne the nonlnear alternatve of Leray-Schauder type for sngle-valued maps wth a selecton theorem due to Bressan and Colombo for lower semcontnuous multvalued mappngs wth nonempty closed and decomposable values. The thrd result reles on the fxed pont theorem for contracton multvalued mappngs due to Covtz and Nadler. The methods used are well known, however ther exposton n the framework of problem (1.1 s new. 2. Prelmnares In ths secton we recall the notons of q k -dervatve and q k -ntegral on fnte ntervals. For a fxed k N {0} let J k := [, 1 ] R be an nterval and 0 < q k < 1 be a constant. We defne the q k -dervatve of a functon f : J k R at a pont t J k as follows: Defnton 2.1. Assume f : J k R s a contnuous functon and let t J k. Then the expresson D qk f(t = f(t f(q kt (1 q k, t, D qk f( = lm D qk f(t, (2.1 (1 q k (t t s called the q k -dervatve of functon f at t. We say that f s q k -dfferentable on J k f D qk f(t exsts for all t J k. Note thaf = 0 and q k = q n (2.1, then D qk f = D q f, where D q s the well-known q-dervatve of the functon f(t defned by D q f(t = f(t f(qt. (2.2 (1 qt In addton, we can defne the hgher order q k -dervatve of functons.

3 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, Defnton 2.2. Let f : J k R be a contnuous functon. If D qk f s q k -dfferentable on J k, we defne the second-order q k -dervatve of f by D 2 q k f : J k R, D 2 q k f = D qk (D qk f. Smlarly, we can consder hgher order q k -dervatves D n q k : J k R. The q k -ntegral s defned as follows: Defnton 2.3. Assume f : J k R s a contnuous functon. Then the q k -ntegral s gven by f(sd qk s = (1 q k (t for t J k. Moreover, f a (, t then the defnte q k -ntegral s gven by a a n=0 qk n f(qn k t (1 qn k, (2.3 f(sd qk s = f(sd qk s f(sd qk s = (1 q k (t qk n f(qn k t (1 qn k (1 q k (a qk n f(qn k a (1 qn k. Note thaf = 0 and q k = q, then (2.3 reduces to the q-ntegral of a functon f(t, defned by 0 f(sd q s = (1 qt n=0 n=0 q n f(q n t for t [0,. n=0 For the basc propertes of the q k -dervatve and q k -ntegral we refer the reader to [32]. Let J = [0, T ], J 0 = [t 0, t 1 ], J k = (, 1 ] for k = 1, 2,..., m. Let P C(J, R = {x : J R : x(t s contnuous everywhere except for some at whch x(t k and x(t k exst and x(t k = x(, k = 1, 2,..., m}. P C(J, R s a Banach space wth the norm x P C = sup{ x(t ; t J}. We now consder the followng lnear problem: D qk x(t = y(t, t J := [0, T ], t, x( = I k (x(, k = 1, 2,..., m, (2.4 αx(0 = βx(t γ x(sd q s, where y : J R. Lemma 2.4 ([33]. Let α β γ (1. The unque soluton of problem (2.4 s gven by x(t = k β m1 y(sd qk 1 s β γ I k (x( y(sd q sd q r 1 γ (1 γ (1 y(sd qk 1 s I k (x( 1 ( y(sd qk 1 s I k (x( y(sd qk s, 1 0< <t (2.5 wth b =a ( = 0 for a > b, where ( 1 = α β γ (1. (2.6

4 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, Next, we recall some defntons and notatons about multfunctons ([17], [21]. For a normed space (X,, let P cl (X = {Y P(X : Y s closed }, P b (X = {Y P(X : Y s bounded}, P cp (X = {Y P(X : Y s compact}, and P cp,cv (X = {Y P(X : Y s compact and convex}. A mult-valued map G : X P(X s convex (closed valued f G(x s convex (closed for all x X. The map G s bounded on bounded sets f G(B = U x B G(x s bounded n X for all B P b (X (.e. sup x B {sup{ y : y G(x}} <. G s called upper sem-contnuous (u.s.c. on X f for each x 0 X, the set G(x 0 s a nonempty closed subset of X, and f for each open set N of X contanng G(x 0, there exsts an open neghborhood N 0 of x 0 such that G(N 0 N. G s sad to be completely contnuous f G(B s relatvely compact for every B P b (X. If the mult-valued map G s completely contnuous wth nonempty compact values, then G s u.s.c. f and only f G has a closed graph,.e., x n x, y n y, y n G(x n mply y G(x. G has a fxed ponf there exsts x X such that x G(x. The fxed pont set of the multvalued operator G wll be denoted by F xg. A multvalued map G : J P cl (R s sad to be measurable f for every y R, the functon t d(y, G(t = nf{ y z : z G(t} s measurable. 3. Man results Before studyng the boundary value problem (1.1, let us begn by defnng ts soluton. Defnton 3.1. A functon x P C(J, R s called a soluton of problem (1.1 f x( = I (x(, αx(0 = βx(t m γ x(sd q s, k = 1, 2,..., m, and there exsts a functon f L 1 (J, R such that f(t F (t, x(t, x(θ(t, (S qk x(t, a.e. t J and x(t = k β m1 f(sd qk 1 s β γ I k (x( f(sd q sd q r 1 γ (1 γ (1 f(sd qk 1 s I k (x( 1 ( f(sd qk 1 s I k (x( f(sd qk s. 1 0< <t 3.1. The Carathéodory case In ths subsecton, we consder the case when F has convex values and prove an exstence result based on the nonlnear alternatve of Leray-Schauder type, assumng that F s Carathéodory. Defnton 3.2. A mult-valued map F : J R 3 P(R s sad to be Carathéodory f ( t F (t, x 1, x 2, x 3 s measurable for each (x 1, x 2, x 3 R R R; ( (x 1, x 2, x 3 F (t, x 1, x 2, x 3 s upper sem-contnuous for almost all t J; Also, a Carathéodory multfuncton F : J R 3 P(R s called L 1 -Carathéodory f ( for each κ > 0, there exsts ϕ κ L 1 (J, R such that F (t, x 1, x 2, x 3 = sup{ v : v F (t, x 1, x 2, x 3 } ϕ κ (t t J for all x 1, x 2, x 3 κ and for almost all t J. For each x P C(J, R, defne the set of selectons of F by S F,x = {v L 1 (J, R : v(t F (t, x(t, x(θ(t, (K qk x(t for a.e. t J}. We defne the graph of a functon G to be the set Gr(G = {(x, y X Y, y G(x} and recall two results for closed graphs and upper sem-contnuty.

5 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, Lemma 3.3 ([17], Proposton 1.2. If G : X P cl (Y s u.s.c., then Gr(G s a closed subset of X Y,.e., for every sequence {x n } n N X and {y n } n N Y, f x n x, y n y when n and y n G(x n, then y G(x. Conversely, f G s completely contnuous and has a closed graph, then s upper semcontnuous. Next, we recall a well-known fxed pont theorem whch s needed n the followng. Lemma 3.4 (Nonlnear alternatve for Kakutan maps [20]. Let E be a Banach space, C a closed convex subset of E, U an open subset of C and 0 U. Suppose that F : U P cp,cv (C s a upper sem-contnuous compact map. Then ether ( F has a fxed ponn U, or ( there exst u U and λ (0, 1 wth u λf (u. Lemma 3.5 ([26]. Let X be a Banach space. Let F : J R 3 P cp,cv (X be an L 1 - Carathéodory functon and let Θ be a lnear contnuous mappng from L 1 (J, R to C(J, R. Then the operator Θ S F : C(J, R P cp,cv (C(J, R, x (Θ S F (x = Θ(S F,x s a closed graph operator n C(J, R C(J, R. Now we are n a poston to prove the exstence of the solutons for the boundary value problem (1.1 when the rght-hand sde s convex valued. Theorem 3.6. Suppose that: (H 1 F : J R 3 P cp,cv (R s an L 1 -Carathéodory multfuncton. (H 2 There exst contnuous nondecreasng functons ψ j : [0, (0, and functons p j, b C(J, R, 1 j 2, such that F (t, x 1, x 2, x 3 P := sup{ y : y F (t, x 1, x 2, x 3 } for each (t, x j J R 3, 1 j 3. (H 3 There exst constants c k such that I k (y c k, 1 k m for each y R. (H 4 There exsts a constant M > 0 such that 2 p j (tψ j ( x j b(t x 3, (1 Ψ b φ 0 M 2 Λ p j ψ j (M Φ > 1, Ψ b φ 0 < 1, where ( β Ψ = m1 Λ = β T ( q k 1 γ (1 2 (1 q ( β m Φ = c k γ (1 3 m (1 q q 2 γ (1 Then the boundary value problem (1.1 has at least one soluton on J := [0, T ]. T, (3.1 γ (1 ( 1 2, (3.2 (1 q k 1 γ (1 c k. (3.3

6 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, Proof. Defne the operator H : P C(J, R P(P C(J, R by h P C(J, R : m1 β k f(sd qk 1 s β 1 H(x = h(t = γ I k (x( f(sd q sd q r γ (1 f(sd qk 1 s 1 γ (1 I k (x( ( f(sd qk 1 s I k (x( 1 0< <t f(sd qk s, for f S F,x. We shall show that H satsfes the assumptons of the nonlnear alternatve of Leray-Schauder type. The proof conssts of several steps. As a frst step, we show that H s convex for each x P C(J, R. Ths step s obvous snce S F,x s convex (F has convex values, and therefore we omt the proof. In the second step, we show that H maps bounded sets (balls nto bounded sets n P C(J, R. For a postve number ρ, let B ρ = {x P C(J, R : x ρ} be a bounded ball n P C(J, R. Then, for each h H(x, x B ρ, there exsts f S F,x such that Then we have h(t β h(t = β m1 m1 k k 0< <t k β 0< <t m1 k b(s 1 s 1 f(sd qk 1 s β γ (1 ( k k I k (x( 1 f(sd qk 1 s 1 f(sd qk 1 s I k (x( 1 f(s d qk 1 s β γ (1 k f(s d qk 1 s 1 ( I k (x( 1 f(s d qk 1 s 0< <t I k (x( p 1 (sψ 1 ( x(s p 2 (sψ 2 ( x(s 1 φ(s, u x(u d qk 1 u d qk 1 s γ t1 f(sd q sd q r γ (1 I k (x( f(sd qk s, t J. γ t1 f(s d q sd q r γ (1 I k (x( f(s d qk s γ t1 ( r p 1 (sψ 1 ( x(s

7 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, p 2 (sψ 2 ( x(s b(s s φ(s, u x(u d q u ( p 1 (sψ 1 ( x(s p 2 (sψ 2 ( x(s b(s 1 ( m1 β β k 1 c k d q sd q r s p 1 (sψ 1 ( x(s p 2 (sψ 2 ( x(s b(s m1 k 2 p j ψ j (ρ 2 γ p j ψ j (ρ 2 p j ψ j (ρ b φ 0 ρ m1 k 2 p j ψ j (ρ γ (1 c k c k 1 d qk 1 s b φ 0 ρ β γ (1 γ (1 k m1 d q sd q r b φ 0 ρ 1 k s 1 d qk 1 s b φ 0 ρ 1 d qk 1 s γ (1 1 φ(s, u x(u d qk 1 u k s 1 1 d qk 1 ud qk 1 s m1 k β γ (1 c k c k c k ( β T γ (1 2 γ (1 = (1 q ( ( β m1 ( q k 1 γ (1 ( 1 2 b φ 0 ρ (1 q k 1 ( β m γ (1 c k c k =Λ for all t J. Thus we get 2 p j ψ j (ρ Ψ b φ 0 ρ Φ, h Λ 1 s γ (1 3 (1 q q 2 1 φ(s, u x(u d qk 1 u s 1 d qk 1 ud qk 1 s γ t1 1 d qk 1 ud qk 1 s 2 T p j ψ j (ρ 2 p j ψ j (ρ Ψ b φ 0 ρ Φ, s d qk 1 s d qk 1 s d q ud q sd q r whch mples that H maps bounded sets nto bounded sets n P C(J, R. Now we prove that H maps bounded sets nto equ-contnuous subsets of P C(J, R. Suppose that x B ρ

8 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, and τ 1, τ 2 J, τ 1 < τ 2 wth τ 1 J v, τ 2 J u, v u for some u, v {0, 1, 2,..., m}. Then we obtan k h(τ 2 h(τ 1 f(sd qk 1 s τ 1 < <τ 2 1 I k (x( τ2 τ1 f(sd qk s f(sd qk s τ 1 < <τ 2 t u t v k f(s d qk 1 s τ2 τ1 I k (x( f(sd qk s f(sd qk s. 1 τ 1 < <τ 2 t u t v τ 1 < <τ 2 Obvously the rght hand sde of the above nequalty tends to zero ndependently of x B ρ as τ 2 τ 1 0. As H satsfes the above three assumptons, t follows by the Arzelá-Ascol theorem that H : P C(J, R P(P C(J, R s completely contnuous. In the next step, we show that H s upper sem-contnuous. By Lemma 3.3, H wll be upper semcontnuous f we prove that has a closed graph, snce H was already shown to be completely contnuous. Thus we wll prove that H has a closed graph. Let x n x, h n H(x n and h n h. We need to show that h H(x. Assocated wth h n H(x n, there exsts f n S F,xn such that for each t J, k h n (t = β m1 f n (sd qk 1 s β I k (x n ( 1 γ (1 f n (sd qk 1 s 1 k f n (sd qk 1 s I k (x n ( f n (sd qk s. 1 0< <t 0< <t Thus t suffces to show that there exsts f S F,x such that for each t J, k h (t = β m1 f (sd qk 1 s β I k (x ( 1 γ (1 f (sd qk 1 s 1 k f (sd qk 1 s I k (x ( f (sd qk s. 1 0< <t 0< <t Let us consder the lnear operator Θ : L 1 (J, R P C(J, R gven by k γ t1 f n (sd q sd q r γ (1 I k (x n ( γ t1 f (sd q sd q r γ (1 I k (x ( f Θ(f(t = β m1 f(sd qk 1 s β I k (x( 1 γ (1 f(sd qk 1 s 1 k f(sd qk 1 s I k (x( f(sd qk s. 1 0< <t 0< <t γ t1 f(sd q sd q r γ (1 I k (x( Note that β h n (t h (t = m1 k 1 (f n (s f (sd qk 1 s β (I k (x n ( I k (x (

9 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, γ t1 (f n (s f (sd q sd q r γ (1 (f n (s f (sd qk 1 s 1 γ (1 (I k (x n ( I k (x ( k (f n (s f (sd qk 1 s (I k (x n ( I k (x ( 0< <t 1 0< <t (f n (s f (sd qk s 0, as n. Thus, t follows by Lemma 3.5 that Θ S F s a closed graph operator. Furthermore, we have h n (t Θ(S F,xn. Snce x n x, therefore, we have k h (t = β m1 f (sd qk 1 s β I k (x ( 1 γ (1 f (sd qk 1 s 1 k f (sd qk 1 s I k (x ( f (sd qk s 1 0< <t 0< <t γ t1 f (sd q sd q r γ (1 I k (x ( for some f S F,x. Fnally, we show there exsts an open set U P C(J, R wth x / H(x for any λ (0, 1 and all x U. Let λ (0, 1 and x λh(x. Then there exsts f L 1 (J, R wth f S F,x such that, for t J, we have k x(t = λβ m1 f(sd qk 1 s λβ λγ I k (x( f(sd q sd q r 1 λγ (1 λγ (1 f(sd qk 1 s I k (x( 1 λ k f(sd qk 1 s λ I k (x( λ f(sd qk s. 1 0< <t 0< <t Usng the computatons of the second step above we have whch mples that x Λ 2 p j ψ j ( x Ψ b φ 0 x Φ, (1 Ψ b φ 0 x Λ 2 1. p j ψ j ( x Φ In vew of (H 4, there exsts M such that x M. Let us set U = {x P C(J, R : x < M}.

10 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, Note that the operator H : U P(P C(J, R s upper sem-contnuous and completely contnuous. From the choce of U, there s no x U such that x λh(x for some λ (0, 1. Consequently, by the nonlnear alternatve of Leray-Schauder type (Lemma 3.4, we deduce that H has a fxed pont x U whch s a soluton of the problem (1.1. Ths completes the proof The lower sem-contnuous case The next result concerns the case when F s not necessarly convex valued. Our strategy to deal wth ths problem s based on the nonlnear alternatve of Leray Schauder type together wth the selecton theorem of Bressan and Colombo [14] for lower sem-contnuous maps wth decomposable values. Let X be a nonempty closed subset of a Banach space E and G : X P(E be a mult-valued operator wth nonempty closed values. G s lower sem-contnuous (l.s.c. f the set {y X : G(y B } s open for any open set B n E. Let A be a subset of [0, T ] R. A s L B measurable f A belongs to the σ algebra generated by all sets of the form J D, where J s Lebesgue measurable n [0, T ] and D s Borel measurable n R. A subset A of L 1 ([0, T ], R s decomposable f for all x, y A and measurable J [0, T ] = J, the functon xχ J yχ J J A, where χ J stands for the characterstc functon of J. Defnton 3.7. Let Y be a separable metrc space and let N : Y P(L 1 (J, R be a multvalued operator. We say N has a property (BC f N s lower sem-contnuous (l.s.c. and has nonempty closed and decomposable values. Let F : J R 3 P(R be a multvalued map wth nonempty compact values. Defne a mult-valued operator F : P C(J R P(L 1 (J, R assocated wth F as F(x = {w L 1 (J, R : w(t F (t, x(t for a.e. t J}, whch s called the Nemytsk operator assocated wth F. Defnton 3.8. Let F : J R 3 P(R be a mult-valued functon wth nonempty compact values. We say F s of lower sem-contnuous type (l.s.c. type f ts assocated Nemytsk operator F s lower semcontnuous and has nonempty closed and decomposable values. Lemma 3.9 ([19]. Let Y be a separable metrc space and let N : Y P(L 1 (J, R be a mult-valued operator satsfyng the property (BC. Then N has a contnuous selecton, thas, there exsts a contnuous (sngle-valued functon g : Y L 1 (J, R such that g(x N(x for every x Y. Theorem Assume that (H 2 -(H 4 and the followng condton hold: (H 5 F : J R 3 P(R s a nonempty compact-valued map such that (a (t, x 1, x 2, x 3 F (t, x 1, x 2, x 3 s L B measurable, (b (x 1, x 2, x 3 F (t, x 1, x 2, x 3 s lower sem-contnuous for each t [0, T ]; Then the boundary value problem (1.1 has at least one soluton on J := [0, T ]. Proof. It follows from (H 2 and (H 5 that F s of l.s.c. type. Then from Lemma 3.9, there exsts a contnuous functon f : P C(J, R L 1 (J, R such that f(x F (x for all x P C(J, R. Consder the problem D qk x(t = f(x(t, t J := [0, T ], t, x( = I k (x(, k = 1, 2,..., m, αx(0 = βx(t γ x(sd q s. (3.4

11 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, Note thaf x P C(J, R s a soluton of (3.4, then x s a soluton to the problem (1.1. In order to transform the problem (3.4 nto a fxed pont problem, we defne the operator H as k Hx(t = β m1 f(x(sd qk 1 s β I k (x( 1 γ (1 f(x(sd qk 1 s 1 ( f(x(sd qk 1 s I k (x( f(x(sd qk s. 1 0< <t γ t1 f(x(sd q sd q r γ (1 I k (x( It can easly be shown that H s contnuous and completely contnuous. The remanng part of the proof s smlar to that of Theorem 3.6, so we omt The Lpschtz case Now we prove the exstence of solutons for problem (1.1 wth a not necessarly nonconvex valued rght hand sde, by applyng a fxed pont theorem for multvalued mappngs due to Covtz and Nadler [16]. Let (X, d be a metrc space nduced from the normed space (X;. Consder the Pompeu-Hausdoff metrc H d : P(X P(X R { } gven by H d (A, B = max{sup d(a, B, sup d(a, b}, a A b B where d(a, b = nf a A d(a; b and d(a, B = nf b B d(a; b. (P cl (X, H d s a generalzed metrc space (see [24]. Then (P b,cl (X, H d s a metrc space and Defnton A mult-valued operator H : X P cl (X s called: (a κ Lpschtz f and only f there exsts κ > 0 such that H d (H(x, H(y κd(x, y for each x, y X; (b a contracton f and only f f κ Lpschtz wth κ < 1. Lemma 3.12 ([16]. Let (X, d be a complete metrc space. If N : X P cl (X s a contracton, then F xn. Theorem Assume that: (L 1 F : J R 3 P cp (R s such that F (, x 1, x 2, x 3 : J P cp (R s measurable for each x j R, 1 j 3. (L 2 For almost all t J and x j, y j R, 1 j 3 we have H d (F (t, x 1, x 2, x 3, F (t, y 1, y 2, y 3 p(t wth p C(J, R and d(0, F (t, 0, 0, 0 p(t for almost all t J. Then the boundary value problem (1.1 has at least one soluton on J f p Λ(2 φ 0 T < 1. 3 x j y j,

12 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, Proof. Consder the operator H : P C(J, R P(P C(J, R defned n the begnnng of the proof of Theorem 3.6. Note that the set S F,x s nonempty for each x P C(J, R by the assumpton (L 1, so F has a measurable selecton (see Theorem III.6 [15]. Now we show that the operator H satsfes the assumptons of Lemma To show that H(x P cl (P C(J, R for each x P C(J, R, let {x n } n 0 H(x be such that x n x (n n P C(J, R. Then x P C(J, R and there exsts f n S F,xn such that, for each t J, k x n (t = β m1 f n (sd qk 1 s β I k (x n ( 1 γ (1 f n (sd qk 1 s 1 ( f n (sd qk 1 s I k (x n ( f n (sd qk s. 1 0< <t γ t1 f n (sd q sd q r γ (1 I k (x n ( As F has compact values, we pass to a subsequence (f necessary to obtan that f n converges to f n L 1 (J, R. Thus, f S F,x and for each t J, we have k f n (t f(t = β m1 f(sd qk 1 s β I k (x( 1 γ (1 f(sd qk 1 s 1 ( f(sd qk 1 s I k (x( f(sd qk s. 1 0< <t γ t1 f(sd q sd q r γ (1 I k (x( Hence, x H(x. Next we show that H s a contractve multfuncton wth constant κ = p Λ(2 φ 0 T < 1. Let x, y P C(J, R and h 1 H(x. Then there exsts f 1 S F,x such that, for each t J, By (L 2 we get k h 1 (t = β m1 f 1 (sd qk 1 s β I k (x( 1 γ (1 f 1 (sd qk 1 s 1 ( f 1 (sd qk 1 s I k (x( f 1 (sd qk s. 1 0< <t γ t1 f 1 (sd q sd q r γ (1 I k (x( H d (F (t, x(t, x(θ(t, (K qk x(t, F (t, y(t, y(θ(t, (K qk y(t p(t ( x(t y(t x(θ(t y(θ(t so, there exsts z F (t, x(t, x(θ(t, K qk x(t such that (K qk x(t (K qk y(t, f 1 (t z p(t ( x(t y(t x(θ(t y(θ(t (K qk x(t (K qk y(t, for almost all t J. Defne the multfuncton T : J P(R by T (t ={z R : f 1 (t z p(t( x(t y(t x(θ(t y(θ(t

13 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, (K qk x(t (K qk y(t, for almost all t J}. Is easy to check that the multfuncton T ( F (, x(, x(θ(, (K qk x( s measurable. Hence, we choose f 2 S F,x such that f 1 (t f 2 (t p(t ( x(t y(t x(θ(t y(θ(t (K qk x(t (K qk y(t, for almost all t J. Consder h 2 H(x whch s defned by Thus, Hence, k h 2 (t = β m1 f 2 (sd qk 1 s β I k (x( 1 γ (1 f 2 (sd qk 1 s 1 ( f 2 (sd qk 1 s I k (x( f 2 (sd qk s. 1 h 1 (t h 2 (t β p 0< <t m1 k ( β T 1 f 1 (s f 2 (s d qk 1 s γ (1 = p Λ(2 φ 0 T x y. k γ (1 2 (1 q 1 f 1 (s f 2 (s d qk 1 s γ t1 f 2 (sd q sd q r γ (1 I k (x( γ t1 f 1 (s f 2 (s d q sd q r γ (1 h 1 h 2 p Λ(2 φ 0 T x y. Analogously, nterchangng the roles of x and y, we obtan m1 H d (H F (x, H F (y p Λ(2 φ 0 T x y. T k f 1 (s f 2 (s d qk 1 s 1 (2 φ 0 T x y Snce κ = p Λ(2 φ 0 T < 1, H s a contracton, and t follows by Lemma 3.12 that H has a fxed pont x whch s a soluton of (1.1. Ths completes the proof. 4. Examples In ths secton, we wll llustrate our man results wth the help of some examples. Let us consder the followng boundary value problem for mpulsve frst-order functonal q k -ntegro-dfference nclusons D k1 k3 x(t F (t, x(t, x(θ(t, (K k1 x(t, k3 t [0, 2], t (k 1 x( x( = (k 2( x( 1, = k, k = 1, 2, 3, x(0 = 2 3 ( 2 t1 3 x(2 x(sd 1 s. 3 3 (4.1

14 J. Tarboon, S. K. Ntouyas, W. Sudsutad, J. Nonlnear Sc. Appl. 9 (2016, Here we have q k = (k 1/(k 3, γ k = (k 2/(k 3, k = 0, 1, 2, 3, m = 3, T = 2, α = 1/2, β = 2/3. By usng the Maple program, we can fnd = , Λ = , Ψ = Snce I k (x = ((k 1 x /((k 2( x 1, we have I k (x (k 1/(k 2 = c k, whch mples Φ = (a Consder the mult-valued map F : [0, 2] P(R gven by x F (t, x(t, x(2t/3, K k1 x(t = k3 [ x 2 (t x(t 1 0, 6(t 8( x(t 1 e t (x(2t/3 2 t 2 4(2t 5 8 Then, we have sup{ x : x F (t, x 1, x 3, x 3 } x(s cos 2 st 10 ] d k1 s k3. ( (t 8 ( x (2t 5 ( x 2 1 t 2 8 x 3, and φ 0 = 1/10. Choosng p 1 (t = 1/(6(t 8, ψ 1 (x = x 1, p 2 (t = 1/(4(2t 5, ψ 2 (x = x 1, b(t = (t 2/8 we have p 1 = 1/48, p 2 = 1/20, b = 1/2. From all nformaton, there exsts a postve constant M > whch satsfes the condton (H 4. Thus all the condtons of Theorem 3.6 are satsfed. Therefore, by the concluson of Theorem 3.6, the problem (4.1 wth the F (t, x 1, x 2, x 3 gven by (4.2 has at least one soluton on [0, 2]. (b Let F : [0, 2] R 3 P(R be a mult-valued map gven by x F (t, x(t, x(t/2, K k1 k3 = [ 0, e t2 8(t 6 x(t ( x 2 (t 2 x(t x(t 1 cos πt x(t/2 2(t t 3(4t 8 e s t 2 x(sd k1 k3 s 1 48 ]. (4.3 Is easy to see that F (, x 1, x 2, x 3 s measurable for each x R, = 1, 2, 3. For x, y R, = 1, 2, 3, we have 1 3 H d (F (t, x 1, x 2, x 3, F (t, y 1, y 2, y 3 x y, 3(4t 8 wth p(t = 1/(3(4t 8. Note that d(0, F (t, 0, 0, 0 = 1/48 p(t for t [0, 2] and φ 0 = 1/2. We can fnd that p Λ(2 φ 0 T = < 1. Therefore, all the condtons of Theorem 3.13 are satsfed, so the problem (4.1 wth F (t, x 1, x 2, x 3 gven by (4.3 has at least one soluton on [0, 2]. Acknowledgement: Ths research was funded by Kng Mongkut s Unversty of Technology North Bangkok. Contract no. KMUTNB-GOV References [1] R. P. Agarwal, Y. Zhou, Y. He, Exstence of fractonal neutral functonal dfferental equatons, Comput. Math. Appl., 59 (2010, [2] B. Ahmad, Boundary-value problems for nonlnear thrd-order q-dfference equatons, Electron. J. Dfferental Equ., 2011 (2011, 7 pages.1 [3] B. Ahmad, A. Alsaed, S. K. Ntouyas, A study of second-order q-dfference equatons wth boundary condtons, Adv. Dfference Equ., 2012 (2012, 10 pages. 1 [4] B. Ahmad, J. J. Neto, Remann-Louvlle fractonal ntegro-dfferental equatons wth fractonal nonlocal ntegral boundary condtons, Bound. Value Probl., 2011 (2011, 9 pages. 1 [5] B. Ahmad, J. J. Neto, On nonlocal boundary value problems of nonlnear q-dfference equatons, Adv. Dfference Equ., 2012 (2012, 10 pages.1

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