To prove our first existence result for multivalued problem (1.1), we need the following known results.
Lemma 3.1
([30])
Let X be a Banach space. Let (F : [a, b] times mathbb{R}^{2} to {mathcal {P}}_{cp,c}(mathbb{R})) be an (L^{1})– Carathéodory multivalued map, and let φ be a linear continuous mapping from (L^{1}([a,b],mathbb{R})) to (C([a,b],mathbb{R})). Then the operator
$$ varphi circ S_{F,u} : C bigl([a,b],mathbb{R} bigr) to P_{cp,c} bigl(C bigl([a,b], mathbb{R} bigr) bigr),quad u mapsto ( varphi circ S_{F,u}) (u) = varphi ( S_{F,u})$$
is a closed graph operator in (C([a,b],mathbb{R}) times C([a,b],mathbb{R})).
Lemma 3.2
(Nonlinear alternative for Kakutani maps [31])
Let (mathcal{S}) be a Banach space, (mathcal{S}_{1}) be a closed convex subset of (mathcal{S}), U be an open subset of (mathcal{S}_{1}), and (0in U). Suppose that (F: overline{U}to {mathcal {P}}_{c,cv}(mathcal{S}_{1})) is an upper semicontinuous compact map; here ({mathcal {P}}_{c,cv}(mathcal{S}_{1})) denotes the family of nonempty, compact convex subsets of (mathcal{S}_{1}). Then either
-
(i)
F has a fixed point in U̅ or
-
(ii)
there are (uin partial U) and (lambda in (0,1)) with (uin lambda F(u)).
Now we are in a position to present our first main result.
Theorem 3.3
Assume that
- ((H_{1})):
(F,G:[a,b]times mathbb{R}^{2}longrightarrow mathcal{P}(mathbb{R})) are Carathéodory possessing compact and convex values;
- ((H_{2})):
There exist continuous nondecreasing functions (psi _{1},psi _{2},phi _{1},phi _{2}:[0,infty )longrightarrow (0, infty )) such that
$$ biglVert F(t,u,v) bigrVert _{mathcal{P}}:=sup bigl{ vert hat{f} vert :hat{f}in F(t,u,v) bigr} leqslant p_{1}(t) bigl[psi _{1} bigl( Vert u Vert bigr) +phi _{1} bigl( Vert v Vert bigr) bigr]$$
and
$$ biglVert G(t,u,v) bigrVert _{mathcal{P}}:=sup bigl{ vert hat{g} vert :hat{g}in G(t,u,v) bigr} leqslant p_{2}(t) bigl[psi _{2} bigl( Vert u Vert bigr) +phi _{2} bigl( Vert v Vert bigr) bigr]$$
for each ((t,u,v)in [a,b]times mathbb{R}^{2}), where (p_{1},p_{2}in C([a,b],mathbb{R}^{+}));
- ((H_{3})):
There exists a constant (N>0) such that
$$ frac{N}{mathcal{E}_{1} Vert p_{1} Vert [psi _{1}(N)+phi _{1}(N)]+mathcal{E}_{2} Vert p_{2} Vert [psi _{2}(N)+phi _{2}(N)]}>1,$$
where (mathcal{E}_{i}) ((i=1,2)) are given in (2.7).
Then problem (1.1) has at least one solution on ([a,b]).
Proof
Consider the operators (Theta _{1},Theta _{2}:mathcal{F} times mathcal{F} to { mathcal {P}}(mathcal{F} times mathcal{F})) defined by (2.5) and (2.6) respectively. It follows from assumption ((H_{1})) that the sets (S_{F,(u,v)}) and (S_{G,(u,v)} ) are nonempty for each ((u,v) in mathcal{F} times mathcal{F}). Then, for (hat{f} in S_{F,(u,v)}), (hat{g} in S_{G,(u,v)}) and (forall (u,v) in mathcal{F} times mathcal{F}), we have
$$begin{aligned} h_{1}(u,v) (t) =& int _{a}^{t} biggl( frac{mu _{1}}{p(u)} int _{a}^{u} hat{f}(z),dz biggr),du+ frac {1}{R} biggl[ -alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} biggl( frac{ mu _{1}}{p(u)} int _{a}^{u} hat{f}(z),dz biggr),du \ &{}+lambda _{1}(beta _{1}+beta _{2}) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}(z),dz biggr),du ,ds \ &{}-lambda _{1}beta _{2}(eta -a) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}(z),dz biggr),du ,ds biggr] \ &{}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2}( beta _{1}+ beta _{2}) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}(beta _{1}+ beta _{2}) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{4} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}(z),dz biggr)+ biggl(- E_{4}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds -E_{3}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta}frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz -E_{2} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}(z),dz biggr)+ biggl(- E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b}frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}(z),dz ,ds biggr) biggr] end{aligned}$$
and
$$begin{aligned} h_{2}(u,v) (t) =& int _{a}^{t} biggl( frac{mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du \ &{}+ frac {1}{R} biggl[ -alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} biggl( frac{ mu _{1}}{p(u)} int _{a}^{u}hat{ f}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3}(b-xi ) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du ,ds \ &{}-beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}(z),dz biggr),du \ &{}+lambda _{3}(alpha _{1}+alpha _{2}) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}(z),dz biggr),du ,ds biggr] \ &{}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2} lambda _{3}(b- xi ) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}lambda _{3}(b- xi ) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{4} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{3} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}(z),dz biggr)+ biggl(- E_{4}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{3}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{3} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta} frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{2} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{1} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}(z),dz biggr)+ biggl(- E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{1} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}(z),dz ,ds biggr) biggr], end{aligned}$$
where (h_{1}in Theta _{1}(u,v)), (h_{2}in Theta _{2}(u,v)), and hence ((h_{1},h_{2}) in Theta (u,v)).
Now, we will verify that the operator Θ satisfies the assumptions of the nonlinear alternative of Leray–Schauder type. In the first step, we show that (Theta (u,v)) is convex valued for each ((u,v) in mathcal{F} times mathcal{F}). Let ((h_{i}, tilde{h_{i}})in (Theta _{1},Theta _{2})), (i=1,2). Then there exist (hat{f}_{i}in S_{F,(u,v)}), (hat{g}_{i}in S_{G,(u,v)}), (i=1,2), such that, for each (t in [a,b]), we have
$$begin{aligned} h_{i}(t) =& int _{a}^{t} biggl(frac{mu _{1}}{p(u)} int _{a}^{u} hat{f}_{i}(z),dz biggr),du+frac {1}{R} biggl[ -alpha _{2}( beta _{1}+ beta _{2}) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u} hat{f}_{i}(z),dz biggr),du \ &{}+lambda _{1}(beta _{1}+beta _{2}) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}_{i}(z),dz biggr),du ,ds \ &{}-lambda _{1}beta _{2}(eta -a) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}_{i}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u} hat{f}_{i}(z),dz biggr),du ,ds biggr] \ &{}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2}( beta _{1}+ beta _{2}) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}(beta _{1}+ beta _{2}) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{4} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}_{i}(z),dz biggr)+ biggl(- E_{4}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds -E_{3}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta}frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}_{i}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz -E_{2} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}_{i}(z),dz biggr)+ biggl(- E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b}frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}_{i}(z),dz ,ds biggr) biggr] end{aligned}$$
and
$$begin{aligned} tilde{h_{i}}(t) =& int _{a}^{t} biggl( frac{mu _{2}}{q(u)} int _{a}^{u} hat{g}_{i}(z),dz biggr),du \ &{}+ frac {1}{R} biggl[ -alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{ f}_{i}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3}(b-xi ) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}_{i}(z),dz biggr),du ,ds \ &{}-beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}_{i}(z),dz biggr),du \ &{}+lambda _{3}(alpha _{1}+alpha _{2}) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}_{i}(z),dz biggr),du ,ds biggr] \ &{}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2} lambda _{3}(b- xi ) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}lambda _{3}(b- xi ) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{4} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{3} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}_{i}(z),dz biggr)+ biggl(- E_{4}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{3}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{3} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta} frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}_{i}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{2} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{1} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}_{i}(z),dz biggr)+ biggl(- E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{1} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}_{i}(z),dz ,ds biggr) biggr]. end{aligned}$$
Let (0 le omega le 1). Then, for each (t in [0,1]), we have
$$begin{aligned}& bigl[omega h_{1}+(1-omega )h_{2} bigr](t) \ & quad = int _{a}^{t} biggl(frac{mu _{1}}{p(u)} int _{a}^{u} bigl[omega hat{f}_{1}(z)+(1- omega )hat{f}_{2} (z) bigr],dz biggr),du \ & qquad {}+frac {1}{R} biggl[ -alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u} bigl[omega hat{f}_{1}(z)+(1- omega )hat{f}_{2} (z) bigr],dz biggr),du \ & qquad {}+lambda _{1}(beta _{1}+beta _{2}) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} bigl[omega hat{g}_{1}(z)+(1- omega )hat{g}_{2} (z) bigr],dz biggr),du ,ds \ & qquad {} -lambda _{1}beta _{2}(eta -a) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u} bigl[omega hat{g}_{1}(z)+(1- omega )hat{g}_{2} (z) bigr],dz biggr),du \ & qquad {}+lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u} bigl[omega hat{f}_{1}(z)+(1- omega )hat{f}_{2} (z) bigr],dz biggr),du ,ds biggr] \ & qquad {}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2}( beta _{1}+ beta _{2}) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}(beta _{1}+ beta _{2}) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ & qquad {}+E_{3}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{4} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \& qquad {}-RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} bigl[omega hat{f}_{1}(z)+(1- omega )hat{f}_{2} (z) bigr],dz biggr) \& qquad {}+ biggl(- E_{4}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz+E_{3} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}-E_{3}lambda _{1}beta _{2}(eta -a) int _{a}^{b} frac{1}{q(z)},dz+E_{4} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \& qquad {}+RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( int _{a}^{ eta}frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} bigl[omega hat{g}_{1}(z)+(1- omega )hat{g}_{2} (z) bigr],dz ,ds biggr) \& qquad {}+ biggl( E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{1}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz -E_{2} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \& qquad {}-RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} bigl[omega hat{g}_{1}(z)+(1- omega )hat{g}_{2} (z) bigr],dz biggr) \& qquad {}+ biggl(- E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz+E_{1} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}-E_{1}lambda _{1}beta _{2}(eta -a) int _{a}^{b} frac{1}{q(z)},dz+E_{2} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \& qquad {}+RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( int _{xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} bigl[omega hat{f}_{1}(z)+(1- omega )hat{f}_{2} (z) bigr],dz ,ds biggr) biggr] end{aligned}$$
and
$$begin{aligned}& bigl[omega tilde{h_{1}}+(1-omega )tilde{h_{2}} bigr](t) \& quad = int _{a}^{t} biggl(frac{mu _{2}}{q(u)} int _{a}^{u} bigl[omega hat{g}_{1}(z)+(1- omega )hat{g}_{2} (z) bigr],dz biggr),du \& qquad {}+frac {1}{R} biggl[ -alpha _{2}lambda _{3}(b- xi ) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u} bigl[omega hat{f}_{1}(z)+(1- omega )hat{f}_{2} (z) bigr],dz biggr),du \& qquad {}+lambda _{1}lambda _{3}(b-xi ) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} bigl[omega hat{g}_{1}(z)+(1- omega )hat{g}_{2} (z) bigr],dz biggr),du ,ds \& qquad {} -beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u} bigl[omega hat{g}_{1}(z)+(1- omega )hat{g}_{2} (z) bigr],dz biggr),du \& qquad {}+lambda _{3}(alpha _{1}+alpha _{2}) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u} bigl[omega hat{f}_{1}(z)+(1- omega )hat{f}_{2} (z) bigr],dz biggr),du ,ds biggr] \& qquad {}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2} lambda _{3}(b- xi ) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}lambda _{3}(b- xi ) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{3}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{4} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \& qquad {}-RE_{3} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} bigl[omega hat{f}_{1}(z)+(1- omega )hat{f}_{2} (z) bigr],dz biggr) \& qquad {}+ biggl(- E_{4}alpha _{2}lambda _{3}(b- xi ) int _{a}^{b} frac{1}{p(z)},dz+E_{3} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}-E_{3}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz+E_{4} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{3} int _{a}^{t} frac{1}{p(z)},dz biggr) \& qquad {} times biggl( int _{a}^{eta} frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} bigl[omega hat{g}_{1}(z)+(1- omega )hat{g}_{2} (z) bigr],dz ,ds biggr) \& qquad {}+ biggl( E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{1}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{2} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \& qquad {}-RE_{1} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} bigl[omega hat{g}_{1}(z)+(1- omega )hat{g}_{2} (z) bigr],dz biggr) \& qquad {}+ biggl(- E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz+E_{1} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}-E_{1}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz+E_{2} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \& qquad {}+RE_{1} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( int _{xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} bigl[omega hat{f}_{1}(z)+(1- omega )hat{f}_{2} (z) bigr],dz ,ds biggr) biggr]. end{aligned}$$
Since (S_{F,(u,v)}), (S_{G,(u,v)}) are convex valued as F and G are convex valued maps, therefore (omega h_{1}+(1-omega )h_{2} in Theta _{1}), (omega tilde{h_{1}}+(1- omega )tilde{h_{2}} in Theta _{2} ), and hence (omega ( h_{1},tilde{h_{1}})+(1-omega )(h_{2},tilde{h_{2}}) in Theta ).
Now, we show that Θ maps bounded sets into bounded sets in (mathcal{F} times mathcal{F}). For a positive number (nu ^{*}), let (B_{nu ^{*}} = {(u,v) in mathcal{F} times mathcal{F}: |(u,v)| le nu ^{*} }) be a bounded set in (mathcal{F} times mathcal{F}). Then, for each (h_{i} in Theta _{i}) ((i=1,2)), ((u,v)in B_{ nu ^{*}}), there exist (hat{f} in S_{F,(u,v)}), (hat{g} in S_{G,(u,v)}) such that
$$begin{aligned} h_{1}(u,v) (t) =& int _{a}^{t} biggl( frac{mu _{1}}{p(u)} int _{a}^{u} hat{f}(z),dz biggr),du+ frac {1}{R} biggl[ -alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} biggl( frac{ mu _{1}}{p(u)} int _{a}^{u} hat{f}(z),dz biggr),du \ &{}+lambda _{1}(beta _{1}+beta _{2}) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}(z),dz biggr),du ,ds \ &{}-lambda _{1}beta _{2}(eta -a) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}(z),dz biggr),du ,ds biggr] \ &{}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2}( beta _{1}+ beta _{2}) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}(beta _{1}+ beta _{2}) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{4} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}(z),dz biggr)+ biggl(- E_{4}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds -E_{3}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta}frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz -E_{2} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}(z),dz biggr)+ biggl(- E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b}frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}(z),dz ,ds biggr) biggr] end{aligned}$$
and
$$begin{aligned} h_{2}(u,v) (t) =& int _{a}^{t} biggl( frac{mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du+ frac {1}{R} biggl[ -alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} biggl( frac{ mu _{1}}{p(u)} int _{a}^{u}hat{ f}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3}(b-xi ) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du ,ds \ &{}-beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}(z),dz biggr),du \ &{}+lambda _{3}(alpha _{1}+alpha _{2}) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}(z),dz biggr),du ,ds biggr] \ &{}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2} lambda _{3}(b- xi ) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}lambda _{3}(b- xi ) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{4} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{3} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}(z),dz biggr)+ biggl(- E_{4}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{3}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{3} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta} frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{2} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{1} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}(z),dz biggr)+ biggl(- E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{1} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}(z),dz ,ds biggr) biggr]. end{aligned}$$
Then, for (tin [a,b]), we have
$$begin{aligned}& biglvert h_{1}(u,v) (t) bigrvert \& quad leq int _{a}^{t} biggl(frac{ vert mu _{1} vert }{ vert p(u) vert } int _{a}^{u} biglvert hat{f}(z) bigrvert ,dz biggr),du+frac {1}{ vert R vert } biggl[ biglvert alpha _{2}( beta _{1}+beta _{2}) bigrvert int _{a}^{b} biggl(frac{ vert mu _{1} vert }{ vert p(u) vert } int _{a}^{u} biglvert hat{f}(z) bigrvert ,dz biggr),du \& qquad {}+ biglvert lambda _{1}(beta _{1}+beta _{2}) bigrvert int _{a}^{ eta } int _{a}^{s} biggl(frac{ vert mu _{2} vert }{ vert q(u) vert } int _{a}^{u} biglvert hat{g}(z) bigrvert ,dz biggr),du ,ds \& qquad {}+ biglvert lambda _{1}beta _{2}(eta -a) bigrvert int _{a}^{b} biggl(frac{ vert mu _{2} vert }{ vert q(u) vert } int _{a}^{u} biglvert hat{g}(z) bigrvert ,dz biggr),du \& qquad {}+ biglvert lambda _{1}lambda _{3} (eta -a) bigrvert int _{xi}^{b} int _{a}^{s} biggl(frac{ vert mu _{1} vert }{ vert p(u) vert } int _{a}^{u} biglvert hat{f}(z) bigrvert ,dz biggr),du ,ds biggr] \& qquad {}+frac{1}{ vert ER vert } biggl[ biggl( biglvert E_{4}alpha _{2}(beta _{1}+ beta _{2}) bigrvert int _{a}^{b}frac{1}{ vert p(z) vert },dz+ biglvert E_{3}lambda _{1}(beta _{1}+ beta _{2}) bigrvert int _{a}^{eta } int _{a}^{s}frac{1}{ vert q(z) vert },dz ,ds \& qquad {}+ biglvert E_{3}lambda _{1}beta _{2}( eta -a) bigrvert int _{a}^{b} frac{1}{ vert q(z) vert },dz+ biglvert E_{4}lambda _{1}lambda _{3} (eta -a) bigrvert int _{ xi}^{b} int _{a}^{s}frac{1}{ vert p(z) vert },dz ,ds \& qquad {}+ vert RE_{4} vert int _{a}^{t}frac{1}{ vert p(z) vert },dz biggr) biggl( frac{ vert alpha _{4}mu _{1} vert }{ vert p(b) vert } int _{a}^{b} biglvert hat{f}(z) bigrvert ,dz biggr)+ biggl( biglvert E_{4}alpha _{2}(beta _{1}+beta _{2}) bigrvert int _{a}^{b} frac{1}{ vert p(z) vert },dz \& qquad {}+ biglvert E_{3}lambda _{1}(beta _{1}+ beta _{2}) bigrvert int _{a}^{ eta } int _{a}^{s}frac{1}{ vert q(z) vert },dz ,ds+ biglvert E_{3}lambda _{1}beta _{2}( eta -a) bigrvert int _{a}^{b}frac{1}{ vert q(z) vert },dz \& qquad {}+ biglvert E_{4}lambda _{1}lambda _{3} (eta -a) bigrvert int _{xi}^{b} int _{a}^{s}frac{1}{ vert p(z) vert },dz ,ds+ vert RE_{4} vert int _{a}^{t} frac{1}{ vert p(z) vert },dz biggr) \& qquad {}times biggl( int _{a}^{eta} frac{ vert lambda _{2}mu _{2} vert }{ vert q(s) vert } int _{a}^{s} biglvert hat{g}(z) bigrvert ,dz ,ds biggr) \& qquad {}+ biggl( biglvert E_{2}alpha _{2}(beta _{1}+beta _{2}) bigrvert int _{a}^{b} frac{1}{ vert p(z) vert },dz+ biglvert E_{1}lambda _{1}(beta _{1}+beta _{2}) bigrvert int _{a}^{ eta } int _{a}^{s}frac{1}{ vert q(z) vert },dz ,ds \& qquad {}+ biglvert E_{1}lambda _{1}beta _{2}( eta -a) bigrvert int _{a}^{b} frac{1}{ vert q(z) vert },dz + biglvert E_{2}lambda _{1}lambda _{3} (eta -a) bigrvert int _{ xi}^{b} int _{a}^{s}frac{1}{ vert p(z) vert },dz ,ds \& qquad {}+ vert RE_{2} vert int _{a}^{t}frac{1}{ vert p(z) vert },dz biggr) biggl( frac{ vert beta _{4}mu _{2} vert }{ vert q(b) vert } int _{a}^{b} biglvert hat{g}(z) bigrvert ,dz biggr)+ biggl( biglvert E_{2}alpha _{2}(beta _{1}+beta _{2}) bigrvert int _{a}^{b} frac{1}{ vert p(z) vert },dz \& qquad {}+ biglvert E_{1}lambda _{1}(beta _{1}+ beta _{2}) bigrvert int _{a}^{ eta } int _{a}^{s}frac{1}{ vert q(z) vert },dz ,ds+ biglvert E_{1}lambda _{1}beta _{2}( eta -a) bigrvert int _{a}^{b}frac{1}{ vert q(z) vert },dz \& qquad {}+ biglvert E_{2}lambda _{1}lambda _{3} (eta -a) bigrvert int _{xi}^{b} int _{a}^{s}frac{1}{ vert p(z) vert },dz ,ds+ vert RE_{2} vert int _{a}^{t} frac{1}{ vert p(z) vert },dz biggr) \& qquad {}times biggl( int _{xi}^{b} frac{ vert lambda _{4}mu _{1} vert }{ vert p(s) vert } int _{a}^{s} biglvert hat{f}(z) bigrvert ,dz ,ds biggr) biggr] \& quad le biggl{ frac{mu _{1}}{ vert Rbar{p} vert } biggl[frac{(b-a)^{2}}{2} bigl( vert R vert +alpha _{2}(beta _{1}+beta _{2}) bigr) + frac{lambda _{1}lambda _{2}(eta -a) [(b-a)^{3}-(xi -a)^{3} ]}{6} biggr] \& qquad {}+frac{1}{ vert RE vert } biggl[ biggl( frac{E_{4}alpha _{2}(beta _{1}+beta _{2})(b-a)}{bar{p}} + frac{E_{3}lambda _{1}(beta _{1}+beta _{2})(eta -a)^{2}}{bar{2q}}+ frac{E_{3}lambda _{1}beta _{2}(eta -a) (b-a)}{bar{q}} \& qquad {}+ frac{E_{4}lambda _{1}lambda _{3} (eta -a) [ (b-a)^{2}-(xi -a)^{2} ]}{2bar{p}}+ frac{RE_{4}(b-a)}{bar{p}} biggr) biggl( frac{alpha _{4}mu _{1} (b-a)}{ vert p(b) vert } biggr) \& qquad {}+ biggl( frac{E_{2}alpha _{2}(beta _{1}+beta _{2})(b-a)}{bar{p}}+ frac{E_{1}lambda _{1}(beta _{1}+beta _{2})(eta -a)^{2}}{2bar{q}}+ frac{E_{1}lambda _{1}beta _{2}(eta -a) (b-a)}{bar{q}} \& qquad {}+ frac{E_{2}lambda _{1}lambda _{3} (eta -a) [ (b-a)^{2}-(xi -a)^{2} ]}{2bar{p}}+ frac{RE_{2}(b-a)}{bar{p}} biggr) \& qquad {}times biggl( frac{lambda _{4}mu _{1} [(b-a)^{2}-(xi -a)^{2} ]}{2bar{p}} biggr) biggr] biggr} \& qquad {} times Vert p_{1} Vert bigl[psi _{1} bigl(nu ^{*} bigr) +phi _{1} bigl(nu ^{*} bigr) bigr] \& qquad {}+ biggl{ frac{mu _{2}}{ vert 2Rbar{q} vert } biggl[ frac{lambda _{1}(beta _{1}+beta _{2})(eta -a)^{3}}{3}+lambda _{1} beta _{2}(eta -a) (b-a)^{2} biggr] \& qquad {}+frac{1}{ vert RE vert } biggl[ biggl( frac{E_{4}alpha _{2} (beta _{1}+beta _{2})(b-a)}{bar{p}} + frac{E_{3}lambda _{1}(beta _{1}+beta _{2})(eta -a)^{2}}{2bar{q}} \& qquad {}+ frac{E_{3}lambda _{1}beta _{2}(eta -a) (b-a)}{bar{q}} \& qquad {}+ frac{E_{4}lambda _{1}lambda _{3} (eta -a) [ (b-a)^{2}-(xi -a)^{2} ]}{2bar{p}}+ frac{RE_{4}(b-a)}{bar{p}} biggr) biggl( frac{lambda _{2}mu _{2} (eta -a)^{2}}{2bar{q}} biggr) \& qquad {}+ biggl( frac{E_{2}alpha _{2}(beta _{1}+beta _{2})(b-a)}{bar{p}} + frac{E_{1}lambda _{1}(beta _{1}+beta _{2})(eta -a)^{2}}{bar{2q}}+ frac{E_{1}lambda _{1}beta _{2}(eta -a) (b-a)}{bar{q}} \& qquad {}+ frac{E_{2}lambda _{1}lambda _{3} (eta -a) [ (b-a)^{2}-(xi -a)^{2} ]}{2bar{p}}+ frac{RE_{2}(b-a)}{bar{p}} biggr) biggl( frac{beta _{4}mu _{2} (b-a)}{ vert q(b) vert } biggr) biggr] biggr} \& qquad {}times Vert p_{2} Vert bigl[psi _{2} bigl(nu ^{*} bigr) +phi _{2} bigl(nu ^{*} bigr) bigr] \& quad = mathcal{D}_{1} Vert p_{1} Vert bigl[psi _{1} bigl(nu ^{*} bigr) +phi _{1} bigl( nu ^{*} bigr) bigr]+ mathcal{D}_{2} Vert p_{2} Vert bigl[psi _{2} bigl(nu ^{*} bigr) +phi _{2} bigl( nu ^{*} bigr) bigr]. end{aligned}$$
Similarly, we can obtain that
$$ biglvert h_{2}(u,v) (t) bigrvert leq mathcal{D}_{3} Vert p_{1} Vert bigl[psi _{1} bigl( nu ^{*} bigr) +phi _{1} bigl(nu ^{*} bigr) bigr]+ mathcal{D}_{4} Vert p_{2} Vert bigl[psi _{2} bigl( nu ^{*} bigr) +phi _{2} bigl(nu ^{*} bigr) bigr].$$
Thus, we get
$$begin{aligned}& biglVert h_{1}(u,v) bigrVert leq mathcal{D}_{1} Vert p_{1} Vert bigl[psi _{1} bigl(nu ^{*} bigr) + phi _{1} bigl(nu ^{*} bigr) bigr]+ mathcal{D}_{2} Vert p_{2} Vert bigl[psi _{2} bigl(nu ^{*} bigr) + phi _{2} bigl(nu ^{*} bigr) bigr],\& biglVert h_{2}(u,v) bigrVert leq mathcal{D}_{3} Vert p_{1} Vert bigl[psi _{1} bigl(nu ^{*} bigr) + phi _{1} bigl(nu ^{*} bigr) bigr]+ mathcal{D}_{4} Vert p_{2} Vert bigl[psi _{2} bigl(nu ^{*} bigr) + phi _{2} bigl(nu ^{*} bigr) bigr], end{aligned}$$
where (mathcal{D}_{i}) ((i=1,ldots ,4)) are defined by (2.8). In consequence, we have
$$begin{aligned} biglVert (h_{1},h_{2}) bigrVert =& biglVert h_{1}(u,v) bigrVert + biglVert h_{2}(u,v) bigrVert \ leq& (mathcal{D}_{1}+ mathcal{D}_{3}) Vert p_{1} Vert bigl[ psi _{1} bigl(nu ^{*} bigr) +phi _{1} bigl(nu ^{*} bigr) bigr]+( mathcal{D}_{2}+ mathcal{D}_{4}) Vert p_{2} Vert bigl[psi _{2} bigl(nu ^{*} bigr) +phi _{2} bigl(nu ^{*} bigr) bigr] \ =& mathcal{E}_{1} Vert p_{1} Vert bigl[psi _{1} bigl(nu ^{*} bigr) +phi _{1} bigl( nu ^{*} bigr) bigr]+ mathcal{E}_{2} Vert p_{2} Vert bigl[psi _{2} bigl(nu ^{*} bigr) +phi _{2} bigl( nu ^{*} bigr) bigr] \ =& ellquad text{(constant),} end{aligned}$$
where (mathcal{E}_{i}), (i=1,2), are defined in (2.7).
Next, we verify that (Theta (u,v)) is equicontinuous. Let (t_{1}, t_{2} in [a,b]) with (t_{1}< t_{2}). Then, for (hat{f} in S_{F,(u,v)}), (hat{g} in S_{G,(u,v)}), we get
$$begin{aligned}& biglvert h_{1}(u,v) (t_{2})-h_{1}(u,v) (t_{1}) bigrvert \& quad = bigglvert int _{a}^{t_{2}} biggl(frac{mu _{1}}{p(u)} int _{a}^{u} hat{f}(tau ),dz biggr),du- int _{a}^{t_{1}} biggl( frac{mu _{1}}{p(u)} int _{a}^{u}hat{f}(tau ),dz biggr),du \& qquad {}+ biggl(frac{E_{4}}{E} biggl( int _{a}^{t_{2}} frac{1}{p(z)},dz- int _{a}^{t_{1}}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}(tau ),dz biggr) biggr) \& qquad {}+ biggl(frac{E_{4}}{E} biggl( int _{a}^{t_{2}} frac{1}{p(z)},dz- int _{a}^{t_{1}}frac{1}{p(z)},dz biggr) biggl( int _{a}^{ eta}frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}(tau ),dz ,ds biggr) biggr) \& qquad {}+ biggl(frac{E_{2}}{E} int _{a}^{t_{2}}frac{1}{p(z)},dz- int _{a}^{t_{1}}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b}hat{g}(tau ),dz biggr) ) \& qquad {}+ biggl(frac{E_{2}}{E} biggl( int _{a}^{t_{2}} frac{1}{p(z)},dz – int _{a}^{t_{1}}frac{1}{p(z)},dz biggr) biggl( int _{ xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s}hat{f}(tau ),dz ,ds biggr) biggr) biggrvert \& quad le biggl[ biggl(frac{mu _{1}}{ vert bar{p} vert } biggr) frac{(t_{2}-a)^{2}-(t_{1}-a)^{2}}{2}+ frac{E_{4}}{E vert bar{p} vert } biggl( frac{alpha _{4}mu _{1}}{ vert p(b) vert } biggr) (t_{2}-t_{1}) (b-a) \& qquad {}+frac{E_{2}}{E vert bar{p} vert } frac{ (lambda _{4}mu _{1} )(t_{2}-t_{1}) [ (b-a)^{2}-(xi -a)^{2} ]}{2} biggr] times Vert p_{1} Vert bigl[psi _{1} bigl(nu ^{*} bigr) +phi _{1} bigl(nu ^{*} bigr) bigr] \& qquad {} + biggl[frac{E_{4}}{E vert bar{p} vert } frac{ (lambda _{2}mu _{2} )(t_{2}-t_{1})(eta -a)^{2}}{2bar{q}}+ frac{E_{2}}{E vert bar{p} vert } biggl( frac{beta _{4}mu _{2}}{ vert q(b) vert } biggr) (t_{2}-t_{1}) (b-a) biggr] \& qquad {} times Vert p_{2} Vert bigl[psi _{2} bigl(nu ^{*} bigr) +phi _{2} bigl(nu ^{*} bigr) bigr] rightarrow 0 quad text{as } t_{2} rightarrow t_{1} text{ independent of } (u,v). end{aligned}$$
Analogously, it can be shown that
$$ biglvert h_{2}(u,v) (t_{2})-h_{2}(u,v) (t_{1}) bigrvert to 0quad text{as } t_{2} rightarrow t_{1} text{ independent of } (u,v).$$
Obviously, the right-hand sides of the above inequalities tend to zero independently of ((u,v)in B_{nu ^{*}}) as (t_{2}-t_{1}longrightarrow 0). Therefore, the operator (Theta (u,v)) is equicontinuous, and hence we deduce that (Theta (u,v):mathcal{F} times mathcal{F} to {mathcal {P}}( mathcal{F} times mathcal{F})) is completely continuous by the Arzelá–Ascoli theorem.
In the next step, we show that (Theta (u,v)) is upper semicontinuous. Instead it will be established that (Theta (u,v)) has a closed graph in view of the fact that a completely continuous operator is upper semicontinuous if it has a closed graph. Let (( u_{k},v_{k})longrightarrow ( u_{ast},v_{ast})) and ((h_{k} ,tilde{h_{k}})in Theta (u_{k},v_{k})) and ((h_{k} ,tilde{h_{k}})longrightarrow ( h_{ast},tilde{h_{ast}}) ). Then we have to show that (( h_{ast},tilde{h_{ast}}) in Theta (u_{ast},v_{ast})). Associated with ((h_{k} ,tilde{h_{k}}) in Theta (u_{k},v_{k})) and (hat{f}_{k} in S_{F,(u,v)}), (hat{g}_{k} in S_{G,(u,v)}), for each (t in [a,b]), we have
$$begin{aligned}& h_{k}(u_{k},v_{k}) (t) \& quad = int _{a}^{t} biggl(frac{mu _{1}}{p(u)} int _{a}^{u} hat{f}_{k}(z),dz biggr),du+frac {1}{R} biggl[ -alpha _{2}(beta _{1}+ beta _{2}) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u} hat{f}_{k}(z),dz biggr),du \& qquad {}+lambda _{1}(beta _{1}+beta _{2}) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}_{k}(z),dz biggr),du ,ds \& qquad {}-lambda _{1}beta _{2}(eta -a) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}_{k}(z),dz biggr),du \& qquad {}+lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}_{k}(z),dz biggr),du ,ds biggr] \& qquad {}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2}( beta _{1}+ beta _{2}) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}(beta _{1}+ beta _{2}) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{3}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{4} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \& qquad {}-RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}_{k}(z),dz biggr)+ biggl(-E_{4}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \& qquad {}+E_{3}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds -E_{3}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \& qquad {}+E_{4}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( int _{a}^{eta}frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}_{k}(z),dz ,ds biggr) \& qquad {}+ biggl( E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{1}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{2} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \& qquad {}-RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}_{k}(z),dz biggr)+ biggl(- E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \& qquad {}+E_{1}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \& qquad {}+E_{2}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) \& qquad {}times biggl( int _{xi}^{b}frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}_{k}(z),dz ,ds biggr) biggr] end{aligned}$$
and
$$begin{aligned}& tilde{h_{k}}(u_{k},v_{k}) (t) \& quad = int _{a}^{t} biggl(frac{mu _{2}}{q(u)} int _{a}^{u} hat{g}_{k}(z),dz biggr),du+frac {1}{R} biggl[ -alpha _{2}lambda _{3}(b- xi ) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{ f}_{k}(z),dz biggr),du \& qquad {}+lambda _{1}lambda _{3}(b-xi ) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}_{k}(z),dz biggr),du ,ds \& qquad {}-beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}_{k}(z),dz biggr),du \& qquad {}+lambda _{3}(alpha _{1}+alpha _{2}) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}_{k}(z),dz biggr),du ,ds biggr] \& qquad {}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2} lambda _{3}(b- xi ) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}lambda _{3}(b- xi ) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{3}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{4} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \& qquad {}-RE_{3} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}_{k}(z),dz biggr)+ biggl(- E_{4}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \& qquad {}+E_{3}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{3}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \& qquad {}+E_{4}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{3} int _{a}^{t} frac{1}{p(z)},dz biggr) biggl( int _{a}^{eta} frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}_{k}(z),dz ,ds biggr) \& qquad {}+ biggl( E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{1}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{2} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \& qquad {}-RE_{1} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}_{k}(z),dz biggr)+ biggl(- E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \& qquad {}+E_{1}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \& qquad {}+E_{2}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{1} int _{a}^{t} frac{1}{p(z)},dz biggr) \& qquad {}times biggl( int _{xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}_{k}(z),dz ,ds biggr) biggr]. end{aligned}$$
Consider the continuous linear operators (Psi _{1},Psi _{2}:L^{1}([a,b],mathcal{F}times mathcal{F}) longrightarrow C([a,b],mathcal{F}times mathcal{F}) ) given by
$$begin{aligned} Psi _{1}(u,v) (t) =& int _{a}^{t} biggl( frac{mu _{1}}{p(u)} int _{a}^{u} hat{f}(z),dz biggr),du \ &{}+ frac {1}{R} biggl[ -alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} biggl( frac{ mu _{1}}{p(u)} int _{a}^{u} hat{f}(z),dz biggr),du \ &{}+lambda _{1}(beta _{1}+beta _{2}) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}(z),dz biggr),du ,ds \ &{}-lambda _{1}beta _{2}(eta -a) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}(z),dz biggr),du ,ds biggr] \ &{}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2}( beta _{1}+ beta _{2}) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}(beta _{1}+ beta _{2}) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{4} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}(z),dz biggr)+ biggl(-E_{4}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds -E_{3}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta}frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{2} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}(z),dz biggr)+ biggl(- E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b}frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}(z),dz ,ds biggr) biggr] end{aligned}$$
and
$$begin{aligned} Psi _{2}(u,v) (t) =& int _{a}^{t} biggl( frac{mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du+ frac {1}{R} biggl[ -alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} biggl( frac{ mu _{1}}{p(u)} int _{a}^{u}hat{ f}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3}(b-xi ) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du ,ds \ &{}-beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}(z),dz biggr),du \ &{}+lambda _{3}(alpha _{1}+alpha _{2}) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}(z),dz biggr),du ,ds biggr] \ &{}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2} lambda _{3}(b- xi ) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}lambda _{3}(b- xi ) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{4} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{3} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}(z),dz biggr)+ biggl(- E_{4}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{3}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{3} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta} frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{2} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{1} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}(z),dz biggr)+ biggl(- E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{1} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}(z),dz ,ds biggr) biggr]. end{aligned}$$
From Lemma 3.1, we know that ((Psi _{1},Psi _{2}) circ (S_{F},S_{G})) is a closed graph operator. Moreover, we have ((h_{k},tilde{h_{k}}) in (Psi _{1},Psi _{2}) circ (S_{F,(u_{k},v_{k})},S_{G,(u_{k},v_{k})})) for all k. Since ((u_{k},v_{k}) longrightarrow (u_{ast},v_{ast})), ((h_{k}, tilde{h_{k}}) longrightarrow (h_{ast},tilde{h_{ast}})), it follows that (hat{f}_{ast }in S_{F,(u,v)}), (hat{g}_{ast }in S_{G,(u,v)}) such that
$$begin{aligned}& h_{ast}(u_{ast},v_{ast}) (t) \& quad = int _{a}^{t} biggl(frac{mu _{1}}{p(u)} int _{a}^{u} hat{f}_{ast}(z),dz biggr),du+frac {1}{R} biggl[ -alpha _{2}(beta _{1}+ beta _{2}) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u} hat{f}_{ast}(z),dz biggr),du \& qquad {}+lambda _{1}(beta _{1}+beta _{2}) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}_{ast}(z),dz biggr),du ,ds \& qquad {}-lambda _{1}beta _{2}(eta -a) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}_{ast}(z),dz biggr),du \& qquad {}+lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}_{ast}(z),dz biggr),du ,ds biggr] \& qquad {}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2}( beta _{1}+ beta _{2}) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}(beta _{1}+ beta _{2}) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{3}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{4} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \& qquad {}-RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}_{ast}(z),dz biggr)+ biggl(-E_{4}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \& qquad {}+E_{3}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds -E_{3}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \& qquad {}+E_{4}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( int _{a}^{eta}frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}_{ast}(z),dz ,ds biggr) \& qquad {}+ biggl( E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{1}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{2} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \& qquad {}-RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}_{ast}(z),dz biggr)+ biggl(- E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \& qquad {}+E_{1}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \& qquad {}+E_{2}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) \& qquad {}times biggl( int _{xi}^{b}frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}_{ast}(z),dz ,ds biggr) biggr] end{aligned}$$
and
$$begin{aligned}& tilde{h_{ast}}(u_{ast},v_{ast}) (t) \& quad = int _{a}^{t} biggl(frac{mu _{2}}{q(u)} int _{a}^{u} hat{g}_{ast}(z),dz biggr),du+frac {1}{R} biggl[ -alpha _{2}lambda _{3}(b- xi ) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{ f}_{ ast}(z),dz biggr),du \& qquad {}+lambda _{1}lambda _{3}(b-xi ) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}_{ast}(z),dz biggr),du ,ds \& qquad {}-beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}_{ast}(z),dz biggr),du \& qquad {}+lambda _{3}(alpha _{1}+alpha _{2}) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}_{ast}(z),dz biggr),du ,ds biggr] \& qquad {}+frac{1}{ER} biggl[ biggl( E_{4}alpha _{2} lambda _{3}(b- xi ) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}lambda _{3}(b- xi ) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{3}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{4} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \& qquad {}-RE_{3} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}_{ast}(z),dz biggr)+ biggl(- E_{4}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \& qquad {}+E_{3}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{3}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \& qquad {}+E_{4}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{3} int _{a}^{t} frac{1}{p(z)},dz biggr) biggl( int _{a}^{eta} frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}_{ast}(z),dz ,ds biggr) \& qquad {}+ biggl( E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \& qquad {}+E_{1}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{2} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \& qquad {}-RE_{1} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}_{ast}(z),dz biggr)+ biggl(- E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \& qquad {}+E_{1}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \& qquad {}+E_{2}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{1} int _{a}^{t} frac{1}{p(z)},dz biggr) \& qquad {}times biggl( int _{xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}_{ast}(z),dz ,ds biggr) biggr], end{aligned}$$
which leads to the conclusion that ((h_{k},tilde{h_{k}})in Theta (u_{ast},v_{ast})).
Finally, we show that there exists an open set (Usubseteq mathcal{F} times mathcal{F} to {mathcal {P}}( mathcal{F} times mathcal{F})) with ((u,v) notin epsilon Theta (u,v)) for any (epsilon in (0,1)) and all ((u,v) in partial U ). Let (epsilon in (0,1)) and ((u,v) in epsilon Theta (u,v)). Then there exist (hat{f}in S_{F},_{(u,v)}) and (hat{g}in S_{G},_{(u,v)}) such that, for (t in [a,b]), we have
$$begin{aligned} u(t) = &epsilon int _{a}^{t} biggl( frac{mu _{1}}{p(u)} int _{a}^{u} hat{f}(z),dz biggr),du+ frac {epsilon}{R} biggl[ -alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u} hat{f}(z),dz biggr),du \ &{}+lambda _{1}(beta _{1}+beta _{2}) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}(z),dz biggr),du ,ds \ &{}-lambda _{1}beta _{2}(eta -a) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}(z),dz biggr),du ,ds biggr] \ &{}+frac{epsilon}{ER} biggl[ biggl( E_{4}alpha _{2}( beta _{1}+beta _{2}) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1}( beta _{1}+beta _{2}) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{4} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}(z),dz biggr)+ biggl(-E_{4}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds -E_{3}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{4} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta}frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}lambda _{1}beta _{2}(eta -a) int _{a}^{b}frac{1}{q(z)},dz-E_{2} lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s} frac{1}{p(z)},dz ,ds \ & {}-RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}(z),dz biggr)+ biggl(- E_{2}alpha _{2}(beta _{1}+beta _{2}) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}(beta _{1}+beta _{2}) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}lambda _{1}beta _{2}( eta -a) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{1}lambda _{3} (eta -a) int _{xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{2} int _{a}^{t}frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b}frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}(z),dz ,ds biggr) biggr] end{aligned}$$
and
$$begin{aligned} v(t) = &epsilon int _{a}^{t} biggl( frac{mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du+ frac {epsilon}{R} biggl[ -alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{ f}(z),dz biggr),du \ &{}+lambda _{1}lambda _{3}(b-xi ) int _{a}^{eta } int _{a}^{s} biggl(frac{ mu _{2}}{q(u)} int _{a}^{u} hat{g}(z),dz biggr),du ,ds \ &{}-beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} biggl( frac{ mu _{2}}{q(u)} int _{a}^{u}hat{g}(z),dz biggr),du \ &{}+lambda _{3}(alpha _{1}+alpha _{2}) int _{xi}^{b} int _{a}^{s} biggl(frac{ mu _{1}}{p(u)} int _{a}^{u}hat{f}(z),dz biggr),du ,ds biggr] \ &{}+frac{epsilon}{ER} biggl[ biggl( E_{4}alpha _{2} lambda _{3}(b-xi ) int _{a}^{b}frac{1}{p(z)},dz-E_{3} lambda _{1} lambda _{3}(b-xi ) int _{a}^{eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{3}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{4} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{3} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{alpha _{4}mu _{1}}{p(b)} int _{a}^{b} hat{f}(z),dz biggr)+ biggl(- E_{4}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{3}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{3}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{4}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{3} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{a}^{eta} frac{lambda _{2}mu _{2}}{q(s)} int _{a}^{s} hat{g}(z),dz ,ds biggr) \ &{}+ biggl( E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz-E_{1} lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds \ &{}+E_{1}beta _{2}(alpha _{1}+alpha _{2}) int _{a}^{b} frac{1}{q(z)},dz-E_{2} lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds \ & {}-RE_{1} int _{a}^{t}frac{1}{p(z)},dz biggr) biggl( frac{beta _{4}mu _{2}}{q(b)} int _{a}^{b} hat{g}(z),dz biggr)+ biggl(- E_{2}alpha _{2}lambda _{3}(b-xi ) int _{a}^{b} frac{1}{p(z)},dz \ &{}+E_{1}lambda _{1}lambda _{3}(b-xi ) int _{a}^{ eta } int _{a}^{s}frac{1}{q(z)},dz ,ds-E_{1}beta _{2}(alpha _{1}+ alpha _{2}) int _{a}^{b}frac{1}{q(z)},dz \ &{}+E_{2}lambda _{3} (alpha _{1}+alpha _{2}) int _{ xi}^{b} int _{a}^{s}frac{1}{p(z)},dz ,ds+RE_{1} int _{a}^{t} frac{1}{p(z)},dz biggr) \ &{}times biggl( int _{xi}^{b} frac{lambda _{4}mu _{1}}{p(s)} int _{a}^{s} hat{f}(z),dz ,ds biggr) biggr]. end{aligned}$$
Using the arguments employed in the second step, we find that
$$ Vert u Vert leq mathcal{D}_{1} Vert p_{1} Vert bigl[psi _{1} bigl( Vert u Vert bigr)+ phi _{1} bigl( Vert v Vert bigr) bigr]+mathcal{D}_{2} Vert p_{2} Vert bigl[psi _{2} bigl( Vert u Vert bigr)+ phi _{2} bigl( Vert v Vert bigr) bigr] $$
and
$$ Vert v Vert leq mathcal{D}_{3} Vert p_{1} Vert bigl[psi _{1} bigl( Vert u Vert bigr)+ phi _{1} bigl( Vert v Vert bigr) bigr]+mathcal{D}_{4} Vert p_{2} Vert bigl[psi _{2} bigl( Vert u Vert bigr)+phi _{2} bigl( Vert v Vert bigr) bigr]. $$
Then we have
$$begin{aligned} biglVert (u,v) bigrVert =& Vert u Vert + Vert v Vert \ leq& (mathcal{D}_{1}+mathcal{D}_{3}) Vert p_{1} Vert bigl[psi _{1} bigl( Vert u Vert bigr)+ phi _{1} bigl( Vert v Vert bigr) bigr] \ &{}+( mathcal{D}_{2}+ mathcal{D}_{4}) Vert p_{2} Vert bigl[psi _{2} bigl( Vert u Vert bigr)+phi _{2} bigl( Vert v Vert bigr) bigr] \ leq& mathcal{E}_{1} Vert p_{1} Vert bigl[psi _{1} bigl( Vert u Vert bigr)+phi _{1} bigl( Vert v Vert bigr) bigr]+mathcal{E}_{2} Vert p_{2} Vert bigl[psi _{2} bigl( Vert u Vert bigr)+ phi _{2} bigl( Vert v Vert bigr) bigr], end{aligned}$$
where (mathcal{E}_{i}), (i=1,2 ), are given by (2.7). Consequently, we have
$$ frac{ Vert (u,v) Vert }{mathcal{E}_{1} Vert p_{1} Vert [psi _{1}( Vert u Vert )+phi _{1}( Vert v Vert )]+mathcal{E}_{2} Vert p_{2} Vert [psi _{2}( Vert u Vert )+phi _{2}( Vert v Vert )]} leq 1. $$
According to ((H_{3})), there exists N such that (|(u,v)|neq N ). Let us set
$$ U= bigl{ (u,v)in (mathcal{F} times mathcal{F}): biglVert (u,v) bigrVert < N bigr} . $$
Observe that the operator (Theta :bar{U}longrightarrow mathcal{P}_{cp,cv}(mathcal{F}) times mathcal{P}_{cp,cv}(mathcal{F}) ) is completely continuous and upper semicontinuous. From the choice of U, there is no ((u,v)in partial U ) such that ((u,v) in epsilon Theta (u,v)) for some (epsilon in (0,1)). Therefore, by the nonlinear alternative of Leray–Schauder type (Lemma 3.2), we deduce that Θ has a fixed point ((u,v)in bar{U}) which is a solution of problem (1.1). □
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Disclaimer:
This article is autogenerated using RSS feeds and has not been created or edited by OA JF.
Click here for Source link (https://www.springeropen.com/)
