Journal of Mathematical Analysis and Applications

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1 J. Mah. Anal. Appl ) Conens liss available a SciVerse ScienceDirec Journal of Mahemaical Analysis and Applicaions Posiive soluions o singular sysem wih four-poin coupled boundary condiions Naseer Ahmad Asif a,, Rahma Ali Khan b a Cenre for Advanced Mahemaics and Physics CAMP), College of Elecrical and Mechanical Engineering, Naional Universiy of Sciences and Technology NUST), Peshawar Road, Rawalpindi, Pakisan b Universiy of Malakand, Chakdara DirL), Khyber Pakhuoonkhwa, Pakisan aricle info absrac Aricle hisory: Received 7 April 2 Available online 23 Augus 2 Submied by J.J. Nieo Keywords: Posiive soluions Coupled singular sysem Coupled four-poin boundary condiions Exisence of posiive soluion o a nonlinear singular sysem wih four-poin coupled boundary condiions of he ype x ) f, x), y) ),, ), y ) g, x), y) ),, ), x), y), x) α yξ), y) βxη), is esablished. The nonlineariies f, g :, ) [, ) [, ) [, ) are coninuous and singular a,, while he parameers α, β, ξ, η saisfy ξ,η, ), < αβξη <. An example is included o show he applicabiliy of our resul. 2 Elsevier Inc. All righs reserved.. Inroducion Coupled boundary condiions BCs) arise in he sudy of reacion diffusion equaions and Surm Liouville problems, see [3,4,6] and [28, Chaper 3]. The sudy of coupled BCs of he following ype D f Ω D 2 f ν,.) where D and D 2 are differenial operaors from L 2 Ω; W ) o L 2 Ω; W ), Ω R n and W is a separable Hilber space, for he ellipic sysem has been iniiaed by Agmon and coauhors [2]. In he sudy of ineracion problems and ellipic operaors on polygonal domains, Mehmei [2], Mehmei and Nicaise [22] and Nicaise [23] have sudied coupled BCs. In [5, Secion 8.3], Krsic and coauhors presened he Timoshenko beam model wih free-end BCs εu, x) d )u xx θ x ),, x, ), μεθ, x) d ) εθ xx au x θ) ),, x, ), u x, ) θ, ), θ x, ),.2) * Corresponding auhor. addresses: naseerasif@yahoo.com N.A. Asif), rahma_alipk@yahoo.com R.A. Khan) X/$ see fron maer 2 Elsevier Inc. All righs reserved. doi:.6/j.jmaa

2 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) where u, x) denoes he displacemen and θ, x) denoes he angle of roaion due o he bending. The posiive consans a and μ are proporional o he nondimensional cross-secional area and he nondimensional momen of ineria of he beam, respecively. The parameer ε is inversely proporional o he nondimensional shear modulus of he beam. The coefficien d denoes he possible presence of Kelvin Voig damping. The meaning of he firs boundary condiion is ha zero force is being applied a he ip, while he meaning of he second boundary condiion is ha zero momen is being applied a he ip. The model is conrolled a x hrough he condiions on u, ) and θ, ). Coupled BCs have also some applicaions in mahemaical biology. For example, Leung [6] sudied he following reacion diffusion sysem for prey predaor ineracion: u, x) σ u u a f u, v) ),, x Ω R n, v, x) σ 2 v v r gu, v) ),, x Ω R n,.3) subjec o he coupled BCs u η, v pu) qv) η on Ω,.4) where n 2 i, a, r, σ x 2, σ 2 are posiive consans, f, g : R 2 R have Hölder coninuous parial derivaives up o i second order in compac ses, η is a uni ouward normal a Ω and p and q have Hölder coninuous firs derivaives in compac subses of [, ). The funcions u, x), v, x) respecively represen he densiy of prey and predaor a ime and a posiion x x,...,x n ). Similar coupled BCs are also sudied in [5] for biochemical sysem. Exisence heory for boundary value problems BVPs) of ordinary differenial equaions is well sudied; we refer he readers o [,6,2,25,26] and he reference herein for wo-poin BVPs and [4,7 9,29,3] for hree-poin BVPs, while for muli-poin BVPs, we refer o [2,3,24,3]. Inspired by he above menioned work and wide applicaions of coupled BCs in various fields of sciences and engineering, we sudy exisence resul o a coupled singular sysem subjec o four-poin coupled BCs of he ype x ) f, x), y) ),, ), y ) g, x), y) ),, ), x), y), x) α yξ), y) βxη), where he parameers α, β, ξ, η saisfy ξ,η, ), < αβξη <. We assume ha he nonlineariies f, g :, ) [, ) [, ) [, ) are coninuous and singular a,. By singulariy we mean ha he funcions f, x, y), g, x, y) are unbounded a and.here,wesudyexisenceofaleasoneposiivesoluionforhesysemofbvps.5).by a posiive soluion of he sysem.5), we mean ha x, y) C[, ] C 2, )) C[, ] C 2, )), x, y) saisfies.5), x > and y > on, ]. Furher we remark ha, o he bes of our knowledge in lieraure here is no resul for a sysem of ordinary differenial equaions wih four-poin coupled BCs. Throughou he paper, we assume ha he following condiions hold: H ) f,, ), g,, ) C, ),, )) and saisfy a : ) f,, ) d <, b : )g,, ) d <. H 2 ) There exis real consans α i, β i wih α i β i <, i, 2; β β 2 <, such ha for all, ), x, y [, ), c β f, x, y) f, cx, y) c α f, x, y), < c, c α f, x, y) f, cx, y) c β f, x, y), c, c β 2 f, x, y) f, x, cy) c α 2 f, x, y), < c, c α 2 f, x, y) f, x, cy) c β 2 f, x, y), c. H 3 ) There exis real consans γ i, ρ i wih γ i ρ i <, i, 2; ρ ρ 2 <, such ha for all, ), x, y [, ), c ρ g, x, y) g, cx, y) c γ g, x, y), < c, c γ g, x, y) g, cx, y) c ρ g, x, y), c, c ρ 2 g, x, y) g, x, cy) c γ 2 g, x, y), < c, c γ 2 g, x, y) g, x, cy) c ρ 2 g, x, y), c..5)

3 85 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) Here we remark ha, in [27] an exisence resul for a coupled sysem of second order and fourh order BVPs has been esablished under hypoheses H ) H 3 ) ogeher wih some addiional assumpions on he nonlineariies. Bu in his paper, we prove exisence resuls for he sysem.5) under he hypoheses H ) H 3 ) only. This paper is organized as follows. In Secion 2, we presen a posiive cone, a fixed poin resul which will be used o prove exisence of posiive soluion, Green s funcion for he sysem of BVPs.5) and some relaed lemmas. In Secion 3, we presen main resul of he paper and finally an example is provided o show he applicabiliy of our heory. 2. Preliminaries For each u E : C[, ] we wrie u max{ u) : [, ]}. Define { } P u E: min u) γ u, [max{ξ,η},] where < γ : min{,αξ,αβξ,βη,αβη} min{ξ,η, ξ, η} max{, α,β, αβξ, αβη} Clearly, E, ) is a Banach space and P is a cone of E. Similarly, for each x, y) E E we wrie x, y) x y. Clearly, E E, ) is a Banach space and P P is a cone of E E. For any real consan r >, define Ω r {x, y) E E: x, y) < r}. The proof of our main resul Theorem 3.) is based on he Guo Krasnosel skii fixed-poin heorem. <. Lemma 2. Guo Krasnosel skii fixed-poin heorem). See [].) Le P be a cone of a real Banach space E, and le Ω, Ω 2 be bounded open neighborhoods of E, and assume Ω Ω 2. Suppose ha T : P Ω 2 \ Ω ) P is compleely coninuous such ha one of he following condiions holds: i) Tx x for x Ω P ; Tx x for x Ω 2 P ; ii) Tx x for x Ω 2 P ; Tx x for x Ω P. Then, T has a fixed poin in P Ω 2 \ Ω ). We need he following resuls in he sequel. Lemma 2.2. Le u, v C[, ], hen he sysem of BVPs x ) u), y ) v), [, ], x) y), x) α yξ), y) βxη), 2.) has inegral represenaion x) y) F ξη, s)us) ds F ηξ, s)vs) ds G αβξη, s)vs) ds, G βαηξ, s)us) ds, 2.2) where s) αβξη s) s), s, s η, s) F ξη, s) αβξη s), s, s η, 2.3) s) s), s, s η, s), s, s η, { αξ s) G αβξη, s) ξ s), s,, s ξ, αξ s), s,, s ξ. 2.4)

4 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) Proof. Inegraing he sysem 2.), we have x) c c 3 s)us) ds, y) c 2 c 4 s)vs) ds, 2.5) where c i, i,...,4, are consans. Now, using he BCs, we obain c, c 2, c 3 c 4 αξ c 3 βη c 4 β Solving for c 3 and c 4,wege c 3 c 4 s)us) ds α αβξη αξ αβξη βη αβξη αβξη η s)us) ds ξ s)vs) ds, s)vs) ds. s)us) ds αβξ αβξη s)vs) ds s)us) ds Thus sysem 2.5) becomes x) y) αβξη αξ αβξη αβξη βη αβξη α αβξη β αβξη η s)us) ds s)vs) ds αβη αβξη s)us) ds αβξ αβξη s)vs) ds ξ s)vs) ds, η s)us) ds α αβξη s)vs) ds αβη αβξη s)us) ds which is equivalen o he sysem 2.2). β αβξη ξ s)vs) ds. η s)us) ds ξ s)vs) ds, ξ s)vs) ds η s)us) ds, s)us) ds s)vs) ds Lemma 2.3. The funcions F ξη and G αβξη saisfy i) F ξη, s) max{,αβξ} s s),, s [, ], ii) G αβξη, s) s s),, s [, ]. α

5 852 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) Proof. For, s) [, ] [, ], we discuss various cases. Case. s η, s; from 2.3), we obain s F ξη, s) s αβξ ) αβξη. If αβξ >, he maximum occurs a, hence s η) F ξη, s) F ξη, s) αβξ αβξη and if α, he maximum occurs a s, hence F ξη, s) F ξη s, s) Case 2. s η, s; using 2.3), we have F ξη, s) s s αβξs η)) αβξη αβξ s s) αβξη s s) αβξη s) αβξη s) s) s s) αβξη αβξη αβξη αβξη Case 3. s η, s; using 2.3), we ge F ξη, s) s αβξη s) αβξη. If αβξη > s, he maximum occurs a, hence η s) F ξη, s) F ξη, s) αβξ αβξη and if αβξη s, he maximum occurs a s, so F ξη, s) F ξη s, s) s s) αβξη Case 4. s η, s; using 2.3), we ge F ξη, s) s) s s) αβξη αβξη αβξ s s) αβξη max{,αβξ} s s). αβξη max{,αβξ} s s). αβξη Now we prove ii). For, s) [, ] [, ], wediscusswocases. Case. s ξ ; using 2.4), we obain G αβξη, s) Case 2. s ξ ; using 2.4), we have G αβξη, s) s ξ) αβξη α s s). αβξη αξ s) αβξη α s s). αβξη Remark 2.4. In view of Lemma 2.3, we have F ηξ, s) max{,αβη} s s), αβξη, s [, ], β G βαηξ, s) s s), αβξη, s [, ]. Lemma 2.5. The funcions F ξη and G αβξη saisfy i) F ξη, s) ii) G αβξη, s) min{,αβξ} min{η, η} s s),, s) [η, ] [, ], αξ min{ξ, ξ} s s),, s) [ξ,] [, ]. max{,αβξ} s s), αβξη max{,αβξ} s s). αβξη max{,αβξ} s s). αβξη max{,αβξ} s s), αβξη

6 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) Proof. Here for, s) [η, ] [, ], we discuss various cases. Case. s η, s; using 2.3), we obain s F ξη, s) s αβξ ) αβξη. If αβξ <, he minimum occurs a, hence s η) F ξη, s) F ξη, s) αβξ αβξη and if αβξ, he minimum occurs a η, hen F ξη, s) F ξη η, s) s η) αβξη Case 2. s η, s; using 2.3), we have F ξη, s) s s αβξη) αβξη. If s > αβξη, he minimum occurs a, hence F ξη, s) F ξη, s) αβξη s) αβξη αβξs η) αβξη and if s αβξη, he minimum occurs a s, herefore F ξη, s) F ξη s, s) Case 3. s η, s; using 2.3), we have F ξη, s) min{,αβξ} min{η, η} s s). αβξη s s) η s) αβξη αβξη s) η s) αβξη αβξη min{,αβξ} min{η, η} s s), αβξη min{,αβξ} min{η, η} s s), αβξη min{,αβξ} min{η, η} s s). αβξη min{,αβξ} min{η, η} s s). αβξη Now we prove ii). For, s) [ξ,] [, ], wediscusswocases. Case. s ξ ; using 2.4), we have G αβξη, s) s ξ) αβξη Case 2. s ξ ; using 2.4), we ge G αβξη, s) αξ s) αβξη αξ s ξ) αβξη αξξ s) αβξη Remark 2.6. In view of Lemma 2.5, we have αξ min{ξ, ξ} s s). αβξη αξ min{ξ, ξ} s s). αβξη min{,αβη} min{ξ, ξ} F ηξ, s) s s),, s) [ξ,] [, ], αβξη βη min{η, η} G βαηξ, s) s s),, s) [η, ] [, ]. αβξη Remark 2.7. From Lemma 2.3 and Remark 2.4, for, s [, ] we have F ξη, s) μs s), F ηξ, s) μs s), G αβξη, s) μs s), G βαηξ, s) μs s). Also, from Lemma 2.5 and Remark 2.6, for, s) [max{ξ,η}, ] [, ] we have F ξη, s) νs s), F ηξ, s) νs s), G αβξη, s) νs s), G βαηξ, s) νs s), where μ max{,α,β,αβξ,αβη} and ν min{,αξ,αβξ,βη,αβη} min{ξ,η, ξ, η}.

7 854 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) Lemma 2.8. The Green s funcions F ξη and G αβξη can be expressed as: F ξη, s) H, s) G αβξη, s) αβξ Hη, s), αβξη Hξ, s), αβξη 2.6) where { s ), s, H, s) s), s. Proof. From 2.2), consider he inegral equaion x) F ξη, s)us) ds αβξη αξ αβξη G αβξη, s)vs) ds s)us) ds αβξ αβξη H, s)us) ds s )us) ds s)us) ds H, s)us) ds αβξ αβξη H, s)us) ds s)vs) ds αβξη η s)us) ds ξ s)vs) ds s)us) ds αβξ αβξη αβξη s)us) ds αβξη αβξη ξ s)vs) ds s)us) ds s)us) ds αβξη αβξ η s)us) ds αβξ αβξη αβξη η s)us) ds αβξ αβξη η s)us) ds ξ s)vs) ds η s)us) ds s ξ)vs) ds αβξη αβξη ξ η s)us) ds αβξ αβξη s ξ)vs) ds αβξη αβξη ξ H, s)us) ds αβξ αβξη ξ s)vs) ds s η)us) ds αβξ αβξη s ξ)vs) ds αβξη αβξη ξ η ξ s)vs) ds η s)us) ds η s)us) ds ξ s)vs) ds

8 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) This proves 2.6). H, s)us) ds H, s) αβξ αβξη αβξ Hη, s) αβξη Hη, s)us) ds ) us) ds Employing Lemma 2.2, he sysem.5) can be expressed as x) y) F ξη, s) f s, xs), ys) ) ds F ηξ, s)g s, xs), ys) ) ds Hξ, s)vs) ds αβξη Hξ, s)vs) ds. αβξη G αβξη, s)g s, xs), ys) ) ds, [, ], G βαηξ, s) f s, xs), ys) ) ds, [, ]. 2.7) By a soluion of he sysem.5), we mean a soluion of he corresponding sysem of inegral equaions 2.7). Define an operaor T : P P P P by T x, y) Ax, y), Bx, y) ), where operaors A, B : P P P are defined by Ax, y)) Bx, y)) F ξη, s) f s, xs), ys) ) ds F ηξ, s)g s, xs), ys) ) ds G αβξη, s)g s, xs), ys) ) ds, [, ], Clearly, if x, y) P P is a fixed poin of T,henx, y) is a soluion of sysem.5). G βαηξ, s) f s, xs), ys) ) ds, [, ]. 2.8) Lemma 2.9. Under he hypoheses H ) H 3 ),hemapt : Ω r P P) P P is compleely coninuous. Proof. Firs we show ha AP P) P.Forx, y) P P, [, ], using 2.8) and Remark 2.7, we have Ax, y)) μ F ξη, s) f s, xs), ys) ) ds s s) f s, xs), ys) ) ds μ G αβξη, s)g s, xs), ys) ) ds s s)g s, xs), ys) ) ds, 2.9) which implies ha Ax, y) μ s s) f s, xs), ys) ) ds μ Also, for x, y) P P, [max{ξ,η}, ], using 2.8), Remark 2.7 and 2.), we obain Ax, y)) ν F ξη, s) f s, xs), ys) ) ds s s) f s, xs), ys) ) ds ν s s)g s, xs), ys) ) ds. 2.) G αβξη, s)g s, xs), ys) ) ds s s)g s, xs), ys) ) ds

9 856 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) γμ s s) f s, xs), ys) ) ds γμ s s)g s, xs), ys) ) ds γ Ax, y). 2.) Consequenly, Ax, y) P. Similarly, we can show ha BP P) P. Hence, T P P) P P). Now, we show ha he operaor A : Ω r P P) P is uniformly bounded. Choose a real consan c, ] such ha cr. For x, y) Ω r P P), [, ], using 2.8), Remark 2.7 and H ) H 3 ),wehave Ax, y)) μ μ F ξη, s) f s, xs), ys) ) ds μc β s s) f s, xs), ys) ) ds μ G αβξη, s)g s, xs), ys) ) ds s s) f s, cxs), cys) ) ds μ μc β β 2 s s) f s s)g s, xs), ys) ) ds s, cxs), cys) ) ds μc ρ c s s)g s, cxs), cys) ) ds s s) f s, cxs), cys) ) ds μc ρ ρ 2 s s)g s, cxs), cys) ) ds c s s)g s, cxs), cys) ) ds μc α β β 2 s s) xs) ) α f s,, cys) ) ds μc γ ρ ρ 2 μc α α 2 β β 2 μc γ γ 2 ρ ρ 2 s s) xs) ) γ g s,, cys) ) ds s s) xs) ) α ys) ) α2 f s,, ) ds s s) xs) ) γ ys) ) γ2 gs,, ) ds μ ac α α 2 β β 2 r α α 2 μ bc γ γ 2 ρ ρ 2 r γ γ 2, which implies ha AΩ r P P)) is uniformly bounded. Similarly, using 2.8), Remark 2.7 and H ) H 3 ),wecanshow ha BΩ r P P)) is also uniformly bounded. Thus, T Ω r P P)) is uniformly bounded. Now we show ha AΩ r P P)) is equiconinuous. For x, y) Ω r P P), [, ], using 2.8) and Lemma 2.8, we have Ax, y)) F ξη, s) f s, xs), ys) ) ds H, s) f s, xs), ys) ) ds αβξη G αβξη, s)g s, xs), ys) ) ds αβξ αβξη Hξ, s)g s, xs), ys) ) ds Hη, s) f s, xs), ys) ) ds

10 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) s ) f s, xs), ys) ) ds αβξ αβξη Differeniaing wih respec o, we obain Ax, y) ) sf s, xs), ys) ) ds Hη, s) f s, xs), ys) ) ds s) f s, xs), ys) ) ds αβξη s) f s, xs), ys) ) ds Hξ, s)g s, xs), ys) ) ds. which implies ha αβξ αβξη Hη, s) f s, xs), ys) ) ds α αβξη Hξ, s)g s, xs), ys) ) ds, Ax, y) ) sf s, xs), ys) ) ds αβξ αβξη Now using H ) H 3 ),wehave Le Ax, y) ) sf s, cxs), cys) ) ds αβξ αβξη s) f s, xs), ys) ) ds s s) f s, xs), ys) ) ds s s) f c α α 2 β β 2 r α α 2 h) c α α 2 β β 2 r α α 2 Then using H ),wehave α αβξη s) f s, cxs), cys) ) ds s, cxs), cys) ) sfs,, ) ds ds α αβξη s) f s,, ) ds α βξac α α 2 β β 2 r α α 2 bc γ γ 2 ρ ρ 2 r γ ) γ 2. αβξη sfs,, ) ds s) f s,, ) ds α βξac α α 2 β β 2 r α α 2 bc γ γ 2 ρ ρ 2 r γ ) γ 2. αβξη h) d c α α 2 β β 2 r α α 2 α αβξη c α α 2 β β 2 r α α 2 sfs,, ) dsd βξac α α 2 β β 2 r α α 2 bc γ γ 2 ρ ρ 2 r γ ) γ 2 s s) f s,, ) ds ) s s)g s, xs), ys) ) ds. ) s) f s,, ) dsd s s) f s,, ) ds s s)g s, cxs), cys) ) ds ) )

11 858 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) α βξac α α 2 β β 2 r α α 2 bc γ γ 2 ρ ρ 2 r γ ) γ 2 αβξη 2 αβξ 2αβξη)acα α 2 β β 2 r α α 2 αbc γ γ 2 ρ ρ 2 r γ γ ) αβξη Thus for any given, 2 [, ] wih 2 and x, y) Ω r P P), wehave Ax, y) ) Ax, y) 2 ) 2 2 Ax, y) ) d h) d, 2.3) which ogeher wih 2.2) yields ha AΩ r P P)) is equiconinuous on [, ]. Similarly, we can also show ha BΩ r P P)) is also equiconinuous. Thus, T Ω r P P)) is equiconinuous. From his ogeher wih uniform boundedness of T Ω r P P)) and he Arzelà Ascoli heorem, i follows ha T Ω r P P)) is relaively compac. Hence, T is a compac operaor. Now we show ha T is coninuous. Le x m, y m ), x, y) Ω r P P) such ha x m, y m ) x, y) asm. Then by using 2.8) and Remark 2.7, we have Axm, y m )) Ax, y)) μ F ξη, s) f s, x m s), y m s) ) f s, xs), ys) )) ds G αβξη, s) g s, x m s), y m s) ) g s, xs), ys) )) ds F ξη, s) f s, xm s), y m s) ) f s, xs), ys) ) ds μ G αβξη, s) g s, xm s), y m s) ) g s, xs), ys) ) ds s s) f s, xm s), y m s) ) f s, xs), ys) ) ds s s) g s, xm s), y m s) ) g s, xs), ys) ) ds. Consequenly, Axm, y m ) Ax, y) μ μ s s) f s, xm s), y m s) ) f s, xs), ys) ) ds s s) g s, xm s), y m s) ) g s, xs), ys) ) ds. By he Lebesgue dominaed convergence heorem, i follows ha Axm, y m ) Ax, y) asm. 2.4) Similarly, by using 2.8) and Remark 2.7, we have Bxm, y m ) Bx, y) asm. 2.5) From 2.4) and 2.5), i follows ha T xm, y m ) T x, y) asm, ha is, T : Ω r P P) P P is coninuous. Hence, T : Ω r P P) P P is compleely coninuous.

12 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) Main resul: Exisence of a leas one posiive soluion Theorem 3.. Under he hypoheses H ) H 3 ),hesysem.5) has a leas one posiive soluion. Proof. Choose a consan R > such ha R { } max,4aμ) β β 2,4bμ) ρ ρ 2. 3.) Le cr, for some real consan c. Then, for any x, y) Ω R P P), [, ], using 2.8), Remark 2.7, H ) H 3 ),we have Ax, y)) F ξη, s) f s, xs), ys) ) ds G αβξη, s)g s, xs), ys) ) ds μ μ s s) f s, xs), ys) ) ds μ s s) f s, cxs), cys) ) ds μ μc α β R α s s) f μc α β α 2 β 2 R α α 2 aμr β β 2 bμr ρ ρ 2. Thus, in view of 3.), we have s s)g s, xs), ys) ) ds s,, cys) ) ds μc γ ρ R γ c s s)g s, cxs), cys) ) ds s s)g s s) f s,, ) ds μc γ ρ γ 2 ρ 2 R γ γ 2 s,, cys) ) ds c s s)gs,, ) ds Ax, y) x, y), for all x, y) Ω R P P). 3.2) 2 Similarly, using 2.8), Remark 2.7, H ) H 3 ),wehave Bx, y) x, y), for all x, y) Ω R P P). 3.3) 2 From 3.2) and 3.3), i follows ha T x, y) x, y), for all x, y) Ω R P P). 3.4) Choose a real consan r, R) such ha r min {, 4νγ β β 2 max{ξ,η} s s) f s,, ) ds ) β β 2, 4νγ ρ ρ 2 max{ξ,η} Then, for any x, y) Ω r P P), [, ], using 2.8), Remark 2.7, H 2 ) H 3 ),wehave Ax, y)) ν ν F ξη, s) f s, xs), ys) ) ds s s) f s, xs), ys) ) ds ν G αβξη, s)g s, xs), ys) ) ds s s) xs) ) β f s,, ys) ) ds ν s s)g s, xs), ys) ) ds s s)g s, xs), ys) ) ds s s)gs,, ) ds ) } ρ ρ )

13 86 N.A. Asif, R.A. Khan / J. Mah. Anal. Appl ) ν ν ν ν s s) xs) ) β ys) ) β2 f s,, ) ds ν s s) xs) ) β ys) ) β2 f s,, ) ds ν s s) xs) ) β ys) ) β2 f s,, ) ds ν max{ξ,η} ν max{ξ,η} νγ β β 2 r β β 2 Thus in view of 3.5), we have s s) xs) ) β ys) ) β2 f s,, ) ds s s) xs) ) ρ ys) ) ρ2 gs,, ) ds max{ξ,η} s s)g s, xs), ys) ) ds s s) f s,, ) ds νγ ρ ρ 2 r ρ ρ 2 s s) xs) ) ρ g s,, ys) ) ds s s) xs) ) ρ ys) ) ρ2 gs,, ) ds max{ξ,η} s s)gs,, ) ds. Ax, y) x, y), for all x, y) Ω r P P). 3.6) 2 Similarly, using 2.8), Remark 2.7, H 2 ) H 3 ), in view of 3.5), we have Bx, y) x, y), for all x, y) Ω r P P). 3.7) 2 From 3.6) and 3.7), i follows ha T x, y) x, y), for all x, y) Ω r P P). 3.8) Hence in view of 3.4) and 3.8), by Lemma 2., T has a fixed poin x, y) Ω R \ Ω r ) P P). Tha is, x Ax, y) and y Bx, y). Moreover,x, y) is posiive. In fac, by concaviy of x and by consrucion of he cone P,wehave x) min x) γ x >, [max{ξ,η},] which implies ha x) > for all, ]. Similarly, y) > for all, ]. Hence,x, y) is a posiive soluion of.5). Example 3.2. Le f, x, y) m i j n p i ) q j x r i y s j, g, x, y) p k ) q l x r k y s l, m n k l where he real consans p i, q j, r i, s j saisfy p i, q j > 2, r i, s j <, i, 2,...,m; j, 2,...,n, wihmax im r i max jn s j <, and he real consans p k, q l, r k, s l saisfy p k, q l > 2, r k, s l <, k, 2,...,m ; l, 2,...,n,wih max km r k max ln s l <. Clearly, f and g saisfy hypoheses H ) H 3 ). Hence, by Theorem 3., he sysem of BVPs.5) has a posiive soluion.

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