EIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON TIME SCALES

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1 Elecronic Journal of Differenial Equaions, Vol. 27 (27, No. 37, pp. 3. ISSN: URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu EIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON TIME SCALES ABDULKADIR DOGAN Communicaed by Mokhar Kirane Absrac. In his aricle, we sudy a singular muli-poin dynamic eigenvalue problem on ime scales. We find exisence of posiive soluions by consrucing he Green s funcion and sudying is posiiviy eigenvalue inervals. Two examples are given o illusrae our resuls.. Inroducion In his aricle, we consider he following singular m-poin dynamic eigenvalue problem on ime scales αu( βp(u ( (p(u ( f(, u(, (, ] T, (. a i u(ξ i, γu( δp(u ( b i u(ξ i. (.2 Some basic definiions on dynamical sysems on ime scales can be found in [5, 6]. Throughou his paper, i is assumed ha (H p : (, T (, and p(s s exiss; we le Q( : p(s s; (H2 ξ i (, T wih < ξ < ξ 2 < < ξ <, a i, b i [, wih < a i < α, < b i <, α, β, δ, γ and < b i Q(ξ i γq( ( α a γ i b i (β (H3 f : (, T (, [, is a coninuous funcion and < Q(sf(s, w s <. b i Q(ξ i < δ; Recenly, some auhors have proved he exisence of posiive soluions o boundary value problems on ime scales; see for example [, 2, 4, 9,,, 2, 3, 5, 7, 2, 22, 23, 24, 25, 3] and he references herein. However, very lile work has been 2 Mahemaics Subjec Classificaion. 34B5, 34B6, 34B8, 39A. Key words and phrases. Time scales; singular muli-poin dynamic equaions; eigenvalue problems; Green s funcion; posiive soluions; fixed poin index heory. c 27 Texas Sae Universiy. Submied March 8, 26. Published February 2, 27.

2 2 A. DOGAN EJDE-27/37 done on he exisence of posiive soluions of singular dynamic boundary value problem on ime scales [7, 8, 8, 9]. Oher relaed resuls on singular ordinary differenial equaions and singular difference equaions appear in [3, 6, 2, 26, 27, 28, 29]. We would like o menion he following resuls. DaCunha e al. [8] proved he exisence resuls for he singular hree poin boundary value problem on ime scales y f(x, y, x (, ] T, y(, y(p y(σ 2 (, where p (, T is fixed and f(x, y is singular a y and possibly a x, y. Liang e al. [8] considered he singular wo poin dynamic eigenvalue problem on ime scales [ρ(x (] m(f(x(σ(, [a, b] T, αx(a βx (a, γx(σ(b δx (σ(b, where ρ( > on [a, σ(b], such ha boh he dela derivaive of ρ( and he inegral ρ(b (/ρ(τ τ exis, m( and f(, are given funcions, α, β, γ, δ, a such ha d : γβ ρ(a αδ ρ(b ρ(ρ(b αγ τ >. a ρ(τ Zhang and Wang [29] considered he exisence and mulipliciy of posiive soluions o singular muli-poin boundary value problem (p(u ( F (, u(, < <, m m u( a j u(x j, w( b j w(x j, j where w( : p(u (, a j b j [, wih < m j a j < and m j b j <, x j (, wih < x < x 2 < < x m <, under cerain condiions on p and F. The argumens were based upon he posiiviy of he Green s funcion and Krasnosel skii fixed poin heorem. Moivaed by [8, 8, 29, 9], in his aricle, we sudy he exisence of posiive soluions for a singular muli-poin dynamic eigenvalue problem on ime scales. We allow f(, w o be singular a and w. We find eigenvalue inervals in which here exiss a leas one posiive soluion of problem (.-(.2 by making use of he fixed poin index heory. The consrucion of a new Green s funcion and is posiiviy are imporan o our discussion. This aricle is organized as follows. In Secion 2, we consruc he Green s funcion and give some lemmas based on he posiiviy of he Green s funcion. In Secion 3, we find eigenvalue inervals in which here exiss a leas one posiive soluion of problem (.-(.2. In Secion 4, we sudy he exisence of posiive soluions o boundary value problem (.-(.2 wih. Finally, in secion 5, we give wo examples o illusrae our exisence heorems. j 2. Green s funcion and some lemmas Lemma 2.. Le h : (, T [, be coninuous and saisfy < Q(sh(s s <.

3 EJDE-27/37 EIGENVALUE PROBLEMS 3 Le Q( : p(s s, y( : Q( h(s s, [, ] T, where he funcion p( is a nonnegaive measurable on (, ] T. Then y( : lim y(. Proof. If h(s s <, hen he lemma is clearly rue. We now assume ha h(s s. In his case, he funcion y( can be wrien in he form y( Q(sh(s s H(s Q(sh(s s ( Q(s Q( h(s s H(s ( Q(s Q( h(s s for all (, ] T, where H( is he Heaviside funcion, i.e., H(s, for s > and H(s, for s. Le f n (s H(s n Q(sh(s, g n (s H(s n (Q(s Q( n h(s, s [, ] T, where n is an arbirary decreasing sequence ha approaches as n. Then lim f n(s lim g n(s Q(sh(s. n n Applying he Levi monoone convergence heorem or he Lebesgue dominaed convergence heorem, we have y( lim y( lim This complees he proof. n lim f n(s s n Q(sh(s s f n (s s lim n lim g n(s s n Q(sh(s s. g n (s s Lemma 2.2. Le he assumpions of Lemma 2. be saisfied. Then he boundaryvalue problem αu( βp(u ( (p(u ( h(, (, ] T, (2. has a unique soluion given by u( a i u(ξ i, γu( δp(u ( Here he Green s funcion is defined by G(, s : D(, s [ ( d α a b i γ i ][ β b i D(ξ i, s γq(s b i u(ξ i (2.2 G(, sh(s s, [, ] T. (2.3 ( a iq(ξ i α a i a i D(ξ i, s β ] Q(

4 4 A. DOGAN EJDE-27/37 where α a i D(, s : min{q(, Q(s}, d γq( δ ( b i Q(ξ i a i D(ξ i, s β, ( α a γ i Proof. We suppose ha C and C 2 are arbirary consans. Le u( : Q(sh(s s Then we have ( u ( Q(sh(s s b i (β Q(h(s s C Q( C 2 a i Q(ξ i. D(, sh(s s C Q( C 2, [, ] T. (2.4 ( Q(sh(s s C 2 (Q( Q(h(s s C Q( C 2 ( ( Q(sh(s s Q(σ( h(s s Q ( h(s s C p( Q(σ(h(σ( Q(σ(h(σ( p( p( h(s s C p(, (, ] T h(s s C Q ( h(s s C p(, (, ] T, (2.5 where we used he dela derivaive produc rule, and Hence p(u ( ( (p(u ( h(s s C ( h(s s C, (, ] T. (2.6 h(s s h(, (, ]T, (2.7 which shows ha he funcion u( defined by (2.4 is a general soluion of (2.7. We are going o find a soluion o problem (2. and (2.2. From (2.2, (2.4-(2.6 and (H2, we find he sysem of wo equaions αc 2 β h(s s βc

5 EJDE-27/37 EIGENVALUE PROBLEMS 5 C 2 a i C γc 2 γc Q( γ C 2 b i C Rearranging hese equaions, we have ( α a i C 2 a i Q(ξ i Q(sh(s s δc b i Q(ξ i C (β a i Q(ξ i a i ( γq( δ b i Q(ξ i b i Solving for C and C 2 yields and C 2 C { C D(ξ i, sh(s s C 2 { b i d α a i γ ( α a i α a i γ ( α a i α a i ( a i D(ξ i, sh(s s, b i D(ξ i, sh(s s. D(ξ i, sh(s s β b i γc 2 γ a i D(ξ i, s β a i D(ξ i, s β ( d{ b i α a i ( a i D(ξ i, s β ( }( γq(s } a i D(ξ i, s β h(s s. Subsiuing C and C 2 in (2.4, we find ha u( h(s s, Q(sh(s s. b i D(ξ i, s } γq(s h(s s a i D(ξ i, s β β G(, sh(s s, [, ] T, a i Q(ξ i b i D(ξ i, s where G(, s is defined as in Lemma 2.2. Clearly, he consans C and C 2 are uniquely deermined by he boundary condiions, he funcion u is a unique soluion o problem (2. and (2.2. We discuss he posiiviy of G(, s. I is clear ha G(, s >, (, s [, ] T (, ] T. (2.8

6 6 A. DOGAN EJDE-27/37 Lemma 2.3. The unique soluion u of problem (2. and (2.2 saisfies where ηu( u( u( u(, [, ] T, { G(, s η : inf G(, s : Proof. From Lemma 2.2 we know ha u( } s (, ] T >. G(, sh(s s, [, ] T, where G(, s is defined as in Lemma 2.2 and saisfies (2.8. From (2.5, we know ha u ( h(s s C p( p(, (, ] T, herefore u( u( u( on [, ] T. Noe ha and G(, s d ( ( α a i b i γ ( ( β b i D(ξ i, s γq(s ( α a i a i D(ξ i, s β G(, s Q(s ( ( d α a b i γ i ( β b i D(ξ i, s γq(s ( α a i a i D(ξ i, s β a i D(ξ i, s β a iq(ξ i α a i, ( a iq(ξ i α a i I is clear ha G(, s < G(, s, s (, ] T. We obain where As a resul, we have ηu( < η { G(, s η : inf G(, s : ηg(, sh(s s G(, s G(, s <,. } s (, ] T. a i D(ξ i, s β Q( G(, sh(s s u(.

7 EJDE-27/37 EIGENVALUE PROBLEMS 7 Le E C[, ] T be a Banach space equipped wih he supremum norm and P : {u E : η u u(, [, ] T }, where η > is given by Lemma 2.3. Then P is a cone in E. For u P, we define where (Au( G(, sf (s, u(s s, u P, <, (2.9 f (, w : f(, max{w, ητ} (2. Here τ is a small posiive number o be deermined. Noe ha f has effecively removed he singulariy in f(, w a w, herefore Au is well defined. By Lemma 2.3, we have for each fixed u P, η Au Au (Au( G(, sf (s, u(s s, (2. G(, sf (s, u(s s (Au( (Au(, [, ] T. (2.2 Therefore, A : P P. Addiionally, i is easy o check ha A is a compleely coninuous mapping. Theorem 2.4 ([4]. Le P be a cone in a Banach space E, Ω E a bounded open se, Ω, and A : P Ω P a compleely coninuous operaor. (A If Ax µx, x P Ω µ <, hen he fixed poin index i(a, P Ω, P. (A2 If here exiss y P, y, such ha x Ax vy, for all x P Ω, v, hen he fixed poin index i(a, P Ω, P. 3. Posiive soluions o eigenvalue problems (.-(.2 For an arbirary consan r, we le Ω r {u E : u < r}. Then M (x : M 2 (x : M 3 (x : M 4 (x : Ω r {u E : u r}, min w [ηx,x] min w [ηx,x] max w [ηx,x] max w [ηx,x] G(, sf(s, w s, x >, G(, sf(s, w s, x >, G(, sf(s, w s, x >, G(, sf(s, w s, x >. For eigenvalue problem (.-(.2, we have he following exisence heorems for posiive soluions. Theorem 3.. Suppose ha (H (H3 hold. If fm 4 : lim x M 4 (x/x and fm : lim x M (x/x exis and < fm 4 < fm, hen problem (.-(.2 has a leas one posiive soluion f < <. M f M 4

8 8 A. DOGAN EJDE-27/37 Proof. Since < /f M 4, we know ha here exiss < τ where, for τ x >, Therefore, G(, sf(s, w s < x, ηx w x. G(, sf(s, w s < τ, ητ w τ. We show ha Au µu, u P Ω τ µ <. From u P Ω τ, we obain u τ and ητ u( τ for all [, ] T. Thus, µ u Au (Au( G(, sf(s, u(s s < τ τ u. Therefore µ <. By Theorem 2.4 (A i follows ha G(, sf (s, u(s s i(a, P Ω τ, P. (3. Furher, since f M <, here exiss < τ < ρ where, for ρ x, Therefore, G(, sf(s, w s > x, ηx w x. G(, sf(s, w s > ρ, ηρ w ρ. (3.2 Taking y ρ, we show ha x Ax vy for all x P Ω ρ, v. Assume o he conrary ha here exis u P Ω ρ, v such ha u Au v ρ. From (2.9-(2.2, (3.2, u ρ and < ητ < ηρ u ( ρ for all [, ] T, we have u ( (Au ( v ρ G(, sf (s, u (s s v ρ G(, sf(s, u (s s v ρ > ρ v ρ ( v ρ ( v u u, which is a conradicion. By Theorem 2.4 (A2 we have i(a, P Ω ρ, P. (3.3 In view of (3., (3.3 wih he fac ha Ω τ Ω ρ, we obain i(a, P (Ω ρ \Ω τ, P i(a, P Ω ρ, P i(a, P Ω τ, P. (3.4 By (3.4 and he fixed poin index heory he operaor A has a fixed poin u P (Ω ρ \Ω τ wih ρ u τ >, herefore < ητ η u u( for all

9 EJDE-27/37 EIGENVALUE PROBLEMS 9 [, ] T. This shows ha he fixed poin u is a posiive soluion of (.-(.2. The proof is complee. Corollary 3.2. Suppose ha (H (H3 hold. If fm 4 : lim x M 4 (x/x and fm 2 : lim x M 2 (x/x exis and < fm 4 < fm 2, hen problem (.-(.2 has a leas one posiive soluion f < <. M f 2 M 4 Theorem 3.3. Suppose ha (H (H3 hold. If fm 4 : lim x M 4 (x/x and fm : lim x M (x/x exis and < fm 4 < fm, hen problem (.-(.2 has a leas one posiive soluion < < fm f. M 4 Proof. Since > /f M, we know ha here exiss < τ where, for τ x >, Therefore, G(, sf(s, w s > x, ηx w x. G(, sf(s, w s > τ, ητ w τ. Since < f M 4, here exiss < τ < ρ where, for ρ x, Therefore, G(, sf(s, w s < x, ηx w x. G(, sf(s, w s < ρ, ηρ w ρ. In he same way as in he proof of Theorem 3., we have i(a, P (Ω ρ \ Ω τ, P i(a, P Ω ρ, P i(a, P Ω τ, P. (3.5 By (3.5 and he fixed poin index heory, he operaor A has a fixed poin u P (Ω ρ \Ω τ wih ρ u τ >, herefore < ητ η u u( for all [, ] T. This shows ha he fixed poin u is a posiive soluion of problem (.-(.2. The proof is complee. Corollary 3.4. Suppose ha (H (H3 hold. If fm 4 : lim x M 4 (x/x and fm 2 : lim x M 2 (x/x exis and fm 2 > fm 4 >, hen problem (.-(.2 has a leas one posiive soluion provided < < fm f. 2 M 4 4. Eigenvalue problem (.-(.2 for Theorem 4.. Suppose ha (H (H3 hold. If f(, w saisfies f M 4 < < f M, hen boundary value problem (.-(.2 has a leas one posiive soluion. The above heorem is a special case of Theorem 3.3 when. Corollary 4.2. Suppose ha (H (H3 hold. If f(, w saisfies f M 4 < < f M 2, hen he boundary value problem (.-(.2 has a leas one posiive soluion. Theorem 4.3. Suppose ha (H (H3 hold. Assume ha f(, w saisfies (H4 η R G(,s s < lim w f(,w w, uniformly for (, ] T. Then boundary value problem (.-(.2 has a leas one posiive soluion.

10 A. DOGAN EJDE-27/37 Proof. By a mehod similar o ha used o prove Theorem 3.3 wih some aleraions, we can complee his proof. Corollary 4.4. Suppose ha (H (H3 hold. Assume ha f(, w saisfies (H5 η R < lim G(,s s w f(,w w, uniformly for (, ] T. Then boundary value problem (.-(.2 has a leas one posiive soluion. 5. Examples In his secion, we illusrae our resuls wih some examples. Example 5.. Le T {} { 2 : n N n }, where N denoes he se of nonnegaive inegers. Take p(, α, γ /2, δ 3, β /4, a /2, a 2 /4, b /3, b 2 /6, ξ /4, ξ 2 /2, and choose f(, w ( w, w >. w We can see ha f(, w is singular a and w. Consider he boundary-value problem u ( I is easy o see by calculaion ha 2 a i 3 2 4, ( u(, u( T, (5. u( 4 u ( 2 u( 4 2 u( 3u ( 3 u( 4, ( 4 u 2 6 u( 2. (5.2 b i 2, Q(, β a iq(ξ i α a 2, d i 3, herefore condiions (H,(H2 and (H3 hold. By calculaions we obain [ G(, s 6 2 min { 4, s} min { 2, s} 3 min { 4, s} 6 min { 2, s} ] [ 2 s 4 2 min { 4, s} 4 min { 2, s} 4 27s 7, for s 4, 33 G(, s 2, for s 2, 8, for s, and [ G(, s s 9 2 min { 4, s} min { 2, s} 3 min { 4, s} 6 min { 2, s} ] [ 2 s 4 2 min { 4, s} 4 min { 2, s} 4 4s, for s 4, 97 G(, s 4, for s 2, 27, for s, { G(, s } η inf 2 s (,] G(, s 3. ], ],

11 EJDE-27/37 EIGENVALUE PROBLEMS Observe ha w w is increasing when w and decreasing when w. For x η, we obain For x, we obain and Therefore, f M 4 f M M 4 (x M (x M 2 (x f M 2 max w [ηx,x] G(, s max w [ηx,x] s ( x x min G(, sf(s, w s ( x x. w [ηx,x] G(, s min w [ηx,x] s ( x x min ( w w s G(, s s s G(, sf(s, w s ( x x w [ηx,x] G(, s min w [ηx,x] s ( x x ( w w s G(, s s s G(, sf(s, w s ( x 3 x. M 4 (x : lim 365 x x 8 ( w w s G(, s s s x 2 lim x x ; M (x : lim 365 x 2 lim x x 8 x x 2 ; M 2 (x x 2 : lim 3 lim x x x x 2. By Theorem 4. or Corollary 4.2, problem (5. and (5.2 have a leas one posiive soluion. Example 5.2. If we se ( w f(, w, w >, w

12 2 A. DOGAN EJDE-27/37 hen f(, w is singular a w. By example 5. and simple calculaions, we have G(, s s 2 4 n , Thus, η uniformly for (, ] T ; G(, s s 27 2 n2 n2 72 G(, s s 259 < lim f(, w w w 4 n lim w w 2 w 2, η G(, s s 6 7 < lim f(, w, w w uniformly for (, ] T. By Theorem 4.3 or Corollary 4.4, he singular boundary value problem ( u( u (, T, u( u( 4 u ( 2 u( ( 4 4 u, 2 2 u( 3u ( 3 u( 4 6 u( 2 has a leas one posiive soluion. Conclusion. In his aricle we have considered a singular muli-poin dynamic eigenvalue problem on ime scales. We have allowed f(, w o be singular a and w. We have found eigenvalue inervals in which here exiss a leas one posiive soluion of problem (.-(.2. We have consruced he Green s funcion and have given some lemmas based on he posiiviy of he Green s funcion. Our resuls generalize and improve he resuls in [9]. Moreover, we have given wo examples o indicae jus how our resuls differ from and generalize hose in oher recen papers. Acknowledgmens. The auhor would like o hank he anonymous referees and edior for heir helpful commens and suggesions. References [] D. Anderson, R. Ma; Second-order n-poin eigenvalue problems on ime scales, Adv. Difference Equ. 26 (26 Aricle ID 59572, 7 pages. [2] D. R. Anderson; Exisence of soluions for nonlinear muli-poin problems on ime scales, Dynam. Sys. Appl. 5 (26, [3] H. Asakawa; Nonresonan singular wo-poin boundary value problems, Nonlinear Anal. TMA 44 (2, [4] F. M. Aici, G. Sh. Guseinov; On Green s funcions and posiive soluions for boundary value problems on ime scales, J. Compu. Appl. Mah. 4 (22, [5] M. Bohner, A. Peerson; Dynamic Equaions on Time Scales: An Inroducion wih Applicaions, Birkhauser, Boson, Cambridge, MA 2. [6] M. Bohner, A. Peerson; Advances in Dynamic Equaions on Time Scales, Birkhauser, Boson, Cambridge, MA 23. [7] M. Bohner, H. Luo; Singular second-order mulipoin dynamic boundary value problems wih mixed derivaives, Adv. Differ. Equ. Volume (26 Aricle ID 54989, Pages -5. [8] J. J. DaCunha, J. M. Davis, P. K. Singh; Exisence resuls for singular hree-poin boundary value problems on ime scales, J. Mah. Anal. Appl. 295 (24,

13 EJDE-27/37 EIGENVALUE PROBLEMS 3 [9] A. Dogan, J. R. Graef, L. Kong; Higher order semiposione muli-poin boundary value problems on ime scales, Compu. Mah. Appl., 6 (2, [] A. Dogan, J. R. Graef, L. Kong; Higher order singular muli-poin boundary value problems on ime scales, Proc. Edinb. Mah. Soc. 54 (2, [] A. Dogan; Exisence of hree posiive soluions for an m-poin boundary-value problem on ime scales, Elecron. J. Differenial Equaions, 23 (49 (23, -. [2] A. Dogan; Exisence of muliple posiive soluions for p-laplacian mulipoin boundary value problems on ime scales, Adv. Differ. Equ., (23, -23. [3] A. Dogan; Eigenvalue problems for nonlinear hird-order m-poin p-laplacian dynamic equaions on ime scales, Mah. Meh. Appl. Sci., 39 (26, [4] D. Guo, V. Lakshmikanham; Nonlinear Problems in Absrac Cones, Academic Press, San Diego, 988. [5] W. Han, Z. Jin, S. Kang; Exisence of posiive soluions of nonlinear m-poin BVP for an increasing homeomorphism and posiive homomorphism on ime scales, J. Compu. Appl. Mah. 233 (29, [6] J. Henderson, E.R. Kaufmann; Focal boundary value problems for singular difference equaions, Compu. Mah. Appl. 36 (998, -. [7] S. Hilger; Analysis on measure chains-a unified approach o coninuous and discree calculus, Resuls Mah. 8 (99, [8] J. Liang, T. Xiao, Z. Hao; Posiive soluions of singular differenial equaions on measure chains, Compu. Mah. Appl. 49 (25, [9] H. Luo; Posiive soluions o singular muli-poin dynamic eigenvalue problems wih mixed derivaives, Nonlinear Anal. 7 (29, [2] R. Ma, D. O Regan; Solvabiliy of singular second order m-poin boundary value problems, J. Mah. Anal. Appl. 3 (25, [2] Y. Sang, H. Su; Several exisence heorems of nonlinear m-poin boundary value problem for p-laplacian dynamic equaions on ime scales, J. Mah. Anal. Appl., 34 ( [22] Y. Sang, H. Su, F. Xu; Posiive soluions of nonlinear m-poin BVP for an increasing homeomorphism and homomorphism wih sign changing nonlineariy on ime scales, Compu. Mah. Appl. 58 (29, [23] H. R. Sun, W. T. Li; Muliple posiive soluions for p-laplacian m-poin boundary value problems on ime scales, Appl. Mah. Compu., 82 (26, [24] H. R. Sun; Triple posiive soluions for p-laplacian m-poin boundary value problem on ime scales, Compu. Mah. Appl. 58 (29, [25] Y.H. Su; Arbirary posiive soluions o a muli-poin p-laplacian boundary value problem involving he derivaive on ime scales, Mah. Compu. Modelling, 53 (2, [26] P. J. Y. Wong, R. P. Agarwal; On he exisence of soluions of singular boundary value problems for higher order difference equaions, Nonlinear Anal. TMA, 28 (997, [27] Y. Zhang; Posiive soluions of singular sublinear Emden-Fowler boundary value problems, J. Mah. Anal. Appl., 85 (994, [28] Z. Zhang, J. Wang; The upper and lower soluion mehod for a class of singular nonlinear second order hree-poin boundary value problems, J. Compu. Appl. Mah., 47 (22, [29] Z. Zhang, J. Wang; On exisence and mulipliciy of posiive soluions o singular muli-poin boundary value problems, J. Mah. Anal. Appl., 295 (24, [3] Y. Zhu, J. Zhu; The muliple posiive soluions for p-laplacian mulipoin BVP wih sign changing nonlineariy on ime scales, J. Mah. Anal. Appl., 344 ( Abdulkadir Dogan Deparmen of Applied Mahemaics, Faculy of Compuer Sciences, Abdullah Gul Universiy, Kayseri 3839, Turkey. Tel: , Fax: address: abdulkadir.dogan@agu.edu.r

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